TERRESTRIAL. HEAT FLOW STUDIES IN KENYA

a thesis

by

Kenneth Harry Williamson

submitted for the degree

of

Doctor of Philosophy

to the

University of London

The Geology Department, Imperial College, LONDON SW7. December 1975 ABSTRACT

Temperature logs have been made at 26 sites in Kenya, in boreholes 100 to 300 metres deep, and thermal conductivities were determined from laboratory measurements on drill cuttings from 13 of the sites. Seven of the logs were seriously disturbed by moving water, and minor disturbances were observed in many others.

On the floor of the Gregory rift near the equator, the measured - heat flow is very erratic with values ranging from 40 to 230 mW.m 2 in a pattern characteristic of a thermal regime dominated by recent igneous activity and hydrothermal circulation in fault zones. On the flanks, approximately 80 km east and west of the rift axis, -2 heat flows of 40 to 50 mW.m have been recorded. Isolated high flank values are apparently associated with zones of weakness in the basement. At a distance of 400 km east of the rift in the Lamu Embayment local highs in the vicinity of sub-surface horst- like features are superimposed on a background heat flow of 50 to 75 mW.m 2.

Model studies have been used to investigate the effect on surface heat flows of high temperature partial melt zones in the upper mantle, inferred from seismic and geomagnetic variation studies. Steady-state two-dimensional models predict surface heat flows significantly higher than observed flank values, but one-dimensional time-dependent studies have shown that the rift flank heat flows could still be unaffected by a high temperature melt zone in the mantle.

TABLE OF CONTENTS

page 1 Chapter 1 INTRODUCTION 3 . 1.1 Terrestrial heat flow : an introduction 3 1.2 Sources of heat in the earth. 4 1.3 Transfer of heat in the earth 5 1.4 Heat flow through the surface of the earth 7 1.5 Heat flow patterns on continents 9 1:6'. Heat flow in rift valleys

12 Chapter 2 MEASUREMENT OF HEAT FLOW 12 2.1 Introduction 14 2.2 Measurement of the geothermal gradient 14 2.2.1 Thermistor sensor 16 2.2.2 Resistance bridge 19 2.2.3 Cables 20 2.2.4 Calibration of the thermistors 22 2.3 Measurement of thermal conductivity 23 2.3.1 The divided bar 27 2.3.2 Calibration of the divided bar 28 2.3.3 Conductivity measurements on rock discs 28 2.3.4 Conductivity measurements on rock fragments

33 2.3.5 The porosity problem

40 Chapter 3 HEAT FLOW CORRECTIONS AND CALCULATIONS 41 3.1 Surface disturbances and corrections 41 3.3.3 Surface topography 44 3.1.2 Past climate 48 3.1.3 Uplift and erosion 49 3.2 Disturbance by convection 49 3.2.1 Convection in boreholes 52 3.2.2 Groundwater circulation 54 3.3 Disturbances from other sources 54 3.3.1 Disturbance due to drilling 55 3.3.2 Lateral variations in thermal conductivity 55 3.4 Calculation of heat flow 55 3.4.1 Methods of calculation 58 3.4.2 The accuracy of heat flow data

ii page

60 Chapter 4 THE THERMAL EFFECTS OF IGNEOUS INTRUSIONS

61. 4.1 Analytical solution for a cooling rectangular intrusion 67 4.2 Latent heat and convection in intrusions. 67 4.2.1 Latent heat mathematical approximations 73. 4.2.2 Convection in the magma : effect on the surroundings 75 4.2.3 Melting of country rock 75 4.3 Finite difference methods . 78 4.3.1 Numerical solution of time - dependent problems. 83 4.3.2 The ADI method in practice 87 4.3.3 Numerical solution of problems involving latent heat and simulated convection 90 4.3.4 Numerical solution of steady - state problems 92 4.4 Study of the surface temperature gradient disturbance above a cooling igneous intrusion 95. 4.4.1 The effect of latent heat 98 4.4.2 Simulated convection in the intrusion 103 4.4.3 Summary

107 Chapter 5 GEOLOGY AND GEOPHYSICS OF KENYA 107 5.1 Background to rifting 108 5.2 Geology of Kenya 108 5.2.1 A geological map of Kenya 110 5.2.2 Tectonics 112 5.2.3 Volcanicity 113 5.3 Geophysical studies in Kenya 113 5.3.1 Seismology 117 5.3.2 Gravity . 119 5.3.3 Geomagnetic variations 121 5.3.4 Seismicity 121 5.3.5 s Magnetism 121 5.3.6 Summary 122 5.4 Other rift valleys 122 5.4.1 The Ethiopian rift 124 5.4.2 The Western rift 126 5.4.3 The Red Sea 127 5.4.4 The Rhinegraben 130. 5.4.5 The Baikal rift 132 5.4.6 Discussion

iii

page 135 Chapter 6 HEAT FLOW MEASUREMENTS. IN KENYA 135 6.1 Collection of the data 136 6.2 Presentation of the data 140 6.2.1 Region 1 : Rift flank (west) 144 6.2.2 Region 2 : Rift f14,nk (east) 145 6.2.3 Region 3 : Gregory rift floor 149 6.2.4 Region 4 : Kavirondo rift floor 150. 6.2.5 . Region 5 : Lamu Embayment 153- 6.2.6 Region 6 : Coastal belt 154 6.3 Classification of the results 155 6.4 Previous heat flow studies in Kenya-, 156 6.5 Discussion of the heat flow results 156 6.5.1 Rift and flanks 160 6.5.2 Lamu Embayment 167 6.5.3 Summary

168 Chapter 7 INTERPRETATION OF THE RESULTS 168 7.1 Heat flow patterns in continental rifts 168 7.1.1 Western rift 169 7.1.2 Red Sea 169 7.1.3 Rhinegraben 172 7.1.4 Baikal rift 172 7.1.5 Discussion 175 7.2 Discussion of Kenya heat flow data : applicability in model studies 177 7.3 A tentative thermal model of an igneous intrusion in the Gregory rift 183 7.4 The geothermal implications of partial' Melting in the upper mantle 183 7.4.1 Development of two possible (*geothermal models 189 7.4.2 Finite difference solutions for models 1 and 2 194 7.4.3 Comparison of observed and model heat flows 197 7.4.4 Dynamic models of the geothermal regime 203 7.4.5 Summary of model results

205 CONCLUSIONS

207 RECOMMENDATIONS FOR FUTURE WORK

209 .REFERENCES

iv page 227 . Appendix 1 SI units 229 Appendix. 2 Correction for the effects of topography 231 Appendix 3 Temperature disturbance near an intrusion 233 Appendix 4 Finite difference solutions for the diffusion equations 238 Appendix 5 Solution of a tridiagonal system of equations 239 Appendix 6 Tabulation of heat flow data 255 Appendix 7 Heat flow data : plots and calculations 282 Appendix 8 Theory of temperature distribution in a flowing well 284 Appendix 9 Heat flow anomaly due to a buried disc-shaped source 286 Appendix 10 Heat flow anomaly caused by vertical movement of groundwater - 289 Appendix 11 Finite difference mesh sizes.

LIST. OF FIGURES

Page 2 1 - 1 The East African Rift System

6 1 - 2 Histogram of equal area grid (5° x 5o) heat flow through the surface of the earth

10 1 - 3 Heat flow and heat production data from plutons in a Caledonian province (New England), Precambrian platform (Central Stable Region) and Precambrian shields

15 2 - 1 Borehole temperature logging equipment

17 2 - 2 Field operations: temperature logging at a borehole site in Northern Kenya

18 2 - 3 Circuit diagram of resistance bridge amplifier

24 2 - 4 Vertical cross-section of the divided bar

32 2 - 5 Vertical cross-section of a conductivity cell

36 2 - 6 Effect of porosity on thermal conductivity

43 3 - 1 Plot of the function A(R, Z) in the range (WO for 3. values of Z

45 3 - 2 Variations in surface temperature (10 - 3.105 years B.P.)

47 3 - 3 Temperature gradient corrections for past climate: variations with depth

62 4 1 Rectangular intrusion model: initial and boundary conditions 66 4 - 2 Fractional increase in temperature gradient-above intrusion plotted against dimensionless time for 3 values of dimensionless distance 69 4 - 3 Error in the calculated temperature gradient anomaly above the intrusion of figure 4 - 2 when the geothermal gradient is ignored

70 4 - 4 Maximum fractional increase in temperature gradient above a rectangular intrusion plotted against dimensionless distance for 3 values of A and B 71 4 - 5 Maximum fractional increase in temperature gradient above a rectangular intrusion plotted against dimensionless distance for 3 values of C and two values of excess temperature T x

vi

Page

85 4 - 6 Effect of time step on discretisation error in computed surface temperature gradients above a cooling rectangular intrusion

93 4 - 7 Number of iterations required for convergence of the finite difference solution to the steady- state intrusion problem, for a range of values of the SOR factor

96 4 - 8 Comparison of solutions for temperature gradient anomalies above the postulated rift intrusion for 4 values of time after emplacement (magma convection ignored)

99 4 - 9 Comparison of solutions for temperature gradient anomalies above the postulated rift intrusion for 4 values of time after emplacement (magma convection included: K = 5Ko) max 101 4 - 10 Comparison of solutions for temperature gradient anomalies above the postulated rift intrusion for 4 values of time after emplacement (magma convection included: Koran = 10Ko)

104 4 - 11 Comparison of solutions for maximum values of temperature gradient attained above the rift intrusion

•105 -4 - 12 Comparison of steady-state and maximum ETA and NOLH solutions for temperature gradient above rift intrusion 109 5 - 1 Geo?_ogical map of Kenya

111 5 - 2 Fault pattern in central Kenya

118 5 - 3 Three models of anomalous structures beneath the Gregory rift which fit gravity and geomagnetic variation data

123 5 - 4 Rift valleys in north-eastern Africa

125 5 - 5 Western and Eastern branches of the East African Rift system

128 5 - 6 The tectonic setting of the Rhinegraben

131 5 - 7 Fault.pattern of the Baikal Rift System

141 6 - 1 Location.of heat flow sites in Kenya

142 6 - 2 Heat flow sites on western flank of Gregory rift and Kavirondo rift

vii Page

146 6 - 3 Heat flow sites and groundwater temperatures' on the Gregory riTt floor

157 6 - 4 Histograms of corrected heat flow values from Kenya 159 6 - 5 Histograms of corrected heat flow values from the flanks of the Gregory rift and the South African Shield

162 6 - 6 Contours of the depth (in metres below sea level) of the top of Mesozoic sediments in the Lamu Embayment

163 6 - 7 Cross-section (west-east) through the Lamu Embayment

170 7 - 1 Histograms of heat flow data from five sections of the Afro-Arabian Rift System

171 7 - 2 Histograms of heat flow data from the floors and flanks of the Baikal Rift, Rhinegraben and Gregory. Rift

178 7 - 3 Models of a rift intrusion, based on gravity data

182 7 - 4 Results of iterative solution for time of emplacement of intrusion model B

184 7 - 5 Two postulated models of the extent of partial melting within the upper mantle in the vicinity of the Gregory rift

188 7 - 6 Variation of solidus temperature with pressure in anhydrous and water-saturated pyrolite

191 7 - 7 Number of iterations required for. convergence of the finite difference solutions of models 1 and 2, for a range of values of the SOR factor

192 7 - 8 An example of the steady-state solution to model 1 (wet pyrolite, isothermal lower boundary case). Isotherms in 50°C increments are shown and the variation in surface heat flow is plotted above

193. 7 - 9 An example of the stead-state solution to model 2 (dry pyrolite, constant flux lower boundary case). Isotherms in 50°C increments are shown and the variation in surface heat flow is plotted

195 7 - 10. Effect of two types of lower boundary condition on the ' • model 1 solution for surface heat flow

viii Page 199 7 - 11 Increase of surface heat flow with time after an isothermal layer of infinite horizontal extent, at the (wet or dry) pyrolite solidus temperature is instantaneously raised to a level 24 km (zone 1) or 48 km (zone 2) below- the earth's surface. -2 Initial surface heat flow = 47 mlq.m (see text)

202 7 - 12 Surface heat flows above an isothermal layer (at the wet or dry solidus temperature) which is rising from 150 km below the earth's surface at a constant rate. The heat flows shown were calculated for the instant at which the layer reached 24 km (zone 1) or 48 km (zone 2), for a range of rise rates. Initial surface heat flow flow = 47 mW.m-2 (see text)

234 A 4 - 1 Schematic representation of a finite difference mesh 287 A10 - 1 Simple thermal model for groundwater flow over an anticline (from Bullard and Niblett, 1951)

LIST OF TABLES

1-1 Relationship between terrestrial heat flow and age of tectonic province 29 2-1 Calibration results for divided bar 34 2-2 Comparison of thermal conductivity measurements on rock discs and fragments _ . 56 3-1 Data on drilling times for Kenya boteholes 114 5-1 Generalised sequence of major volcanic and tectonic events associated with the formation of the Kenya rift 137 6-1 Borehole sites : this survey 138 6-2 Heat flow results (Morgan, 1973) 139 6-3 Heat flow results (this study) 152 6-4 Effect of porosity correction on heat flow in the Lamu Embayment 179 7-1 Dimensions of rectangular intrusion models A and B 179 7-2 Calculated time of emplacement of intrusions for two levels of assumed background heat flow 187 7-3 Crustal parameters for models 1 and 2 196 7-4 Comparison of measured heat flows with predictions of steady state models 1 and 2

227 A1-1 SI base units 227 A1-2 SI derived units 228 A1-3 SI prefixes 228 A1-4 Additional units (non-coherent) 228 A1-5 Superceded geophysical units and SI equivalents ACKNOWLEDGEMENTS

The research for this thesis was made possible by the co-operation and assistance of a large number of people and regrettably only a few can be mentioned here. The author expresses his gratitude to the following persons: The many Kenyans who, by their warmth and hospitality, made the fieldwork for this project a pleasurable experience. Dr. A.I'Brock and the staff of the Physics Department of the University of Nairobi. The Director and staff of the Water Development Division of the Ministry of Agriculture in Kenya. The Director and staff of the Mines and Geological Department in Kenya. Mr. J. Wheildon, the project supervisor who introduced me to the subject and has remained helpful and friendly throughout. Dr. P. Morgan for his assistance during the initial stages of the project. Dr. T. R. Evans for many fruitful discussions and for drawing my attention to numerous publications. Dr. A. Thomas-Betts and Dr. J. Milsom for their comments on the thesis. Mis8 R. Spragg for assistance with the thermal conductivity measurements. The Exploration and Production Division of the British Petroleum Company Ltd. for assistance with the porosity measurements. Shell International Petrbleum Company Ltd. for financial support. Mr. and Mrs. T. L. Williamson for support and encouragement. Karren Devlin for patience and understanding during the typing of the thesis. The research was funded by the Natural Environment Research Council.

xi CHAPTER 1

INTRODUCTION. .

The. East African Rift (figure 1-1) has received considerable attention from geologists and geophysicists in recent years, and details of its deep structure and development are beginning to emerge. Occuriencesof hot springs and fumaroles on the floor of many sections of the rift have long been known, and recent geophysical investigations point to the existence of a major thermal disturbance in the upper mantle beneath much of the eastern branch of the rift system. Surface heat flow is a potential source of infor- mation on the extent of this disturbance which has not been fully exploited. The main objective.of -this study was to extend a preliminary heat flow survey by P. Morgan (Morgan, 1973) and interpret the results in the light of recent discoveries (Searle, 1970; Long et al , 1972; Banks and Ottey, 1974).

Terrestrial heat flow studies have received a stimulus in recent years following the discovery of relationships between heat flow and the age of a tectonic province, and between heat flow and heat generation in surface rocks under certain conditions. Unfortunately temperatures within a few hundred metres of the land surface are subject to potentially large disturbances from a variety of sources and accurate determinations of heat flow are difficult to achieve unless carefully sited deep boreholes are available. Prohibitive drilling costs excluded the possibility of sinking boreholes in specially selected sites for this study, and the available holes in which measurements could be made were often prone to thermal disturbances. However a relatively largenumber of boreholes have been logged (49 in total, including the results, of Morgan, 1973) and an interesting pattern has emerged. The results of other geophysical investigations in the vicinity of the Kenya rift valley can be explained if an extensive zone of partial melting exists in the upper mantle below the rift and a basic igneous intrusion has been injected into the crust beneath the graben floor. Mathematical models have been developed to examine the geothermal implications of these structures and determine whether the observed surface heat flow pattern is compatible with them. W60T13 1° sangWOJJ) w,i4=1/43 4J:d ua311.11f Wea 34,1, 1.1 Terrestrial heat flow: an introduction.

The rate of heat loss from the interior of the earth has recently been 13 estimated at 4.2.10 W by Williams and Von Herzen (1974). Internal heat sources are the only known forms of available energy capable of maintaining tectonic energy at its present level, and therefore determining the. temperature distribution in the outer layers of the earth should be a major step towards understanding the mechanics of large scale geological events. One method of obtaining estimates of temperature at depth is to measure the heat flow on the surface of the earth, and use the theory of heat transfer in solids (Carslaw and Jaeger, 1959) to extrapolate the surface temperature gradient into the crust and upper mantle. However this approach involves assumptions about the heat transfer and heat generation properties of the crust and upper mantle which are necessarily based on indirect information. A complication, particularly relevant to this study arises when crustal temperatures are changing in response to some thermal disturbance at depth, because the diffusivity of common rock types is so low that thermal effects of a major event in the upper mantle, which may have produced dramatic tectonic activity in the upper crust, will not be detectable as surface heat flow anomaly for several tens of millions of years. Rift formation in Kenya appears to have begun in the Miocene, and therefore major changes in the geothermal regime may still be taking place. •

Indirect estimates of temperature within the earth are possible because of the temperature dependence of certain mechanical and electrical properties of rocks. Thus zones of anomalously high electrical conductivity, low seismic velocity and low density can often be caused by local elevations of temperature in the crust or mantle. Geomagnetic deep sounding, , magnetotelluric, seismic and gravity studies are therefore complementary to surface heat flow studies in investigations of the geothermal regime and are essential in a dynamic situation where surface heat flow can be insensitive to deep thermal anomalies for several million years.

1.2 Sources of heat in the earth.

Lord Kelvin considered the thermal history of the earth in terms of a semi- infinite solid with zero surface temperature cooling from its melting point. A simple calculation based on measured geothermal gradients led him to postulate that the earth was created 10 Ma ago, a hypothesis unacceptable to geologists. The reason for such a gross under-estimate became clear when it was recognised that a considerable quantity of heat was being produced within the earth from the decay of natural radioactive isotopes. This source was sufficient to maintain high temperatures over a geologically 3 .

acceptable period of time.

The exact distribution of the long-lived radioactive isotopes of uranium, thorium and potassium within the earth, and the significance of contributions from other sources is largely unknown at present, although recently discovered relationships between heat flow and heat generation in granitic plutons (section 1.5) have provided valuable clues to the nature of the distribution within the upper crust. Most work has been concentrated on this aspect because quantitative estimates of the contributions from other sources such as the energy released during the formation of the earth, heat produced by the decay of short-lived radioactive isotopes, heat derived from tidal friction or released during differentiation of the core and mantle are necessarily speculative. The situation is futther confused by uncertainties in the heat transfer characteristics of the core and mantle.

1.3 Transfer of heat in the earth.

The temperature T at a point in an isotropic medium is governed by the following equation: = 1C7.( kV + ecpy..VT + A (1.1)

where density specific heat at constant pressure thermal conductivity V = velocity of medium A = heat'generation.

In the absence of sources ( A = 0) and convective terms (p;VN17, =0), equation (1.1) may be written:

= V. (1.2)

where f is the heat flux vector.

In an isotropic solid, f is defined as:

f = -k V T (1.3) In an anisotropic solid, the components of f can be written:

f.I _- k-u xjT where k1- is the conductivity tensor.

Anisotropic rocks are common but details of anisotropic structures are rarely well enough known to justify the inclusion of this additional complication in thermal models of geological situations. Only isotropic media will be considered in this text, and thermal conductivity is therefore treated as a scalar quantity.

The thermal conductivities of rocks exposed on the surface of the earth lie in the range 1 to 7 W. m-1. K-1. Below the surface, conductivities change in response to increasing temperature and pressure in quite a complex manner. According to Clark (1969), lattice conduction and radiative transfer are the dominant modes of heat transfer in the crust and upper mantle, .so that thermal conductivity can be considered to be composed of two terms:

k = k1L+ kR

where k • lattice conduction • radiative transfer

The nature of kL and kR is such that they respond differently to changes in temperature and pressure, with the result that the thermal conductivity of mo-st rocks first decreases with depth below the surface and then increases rapidly. Experiments by Schatz and Simmons (1972) on olivines confirmed this trend. This predicted change in conductivity with depth is important In studies of the thermal state of the mantle since widespread melting would occur unless the conductivity increased. Seismic studies have demonstrated that most of the mantle is below its solidus temperature.

The term (DCpV.VT in equation (1.1r is potentially the largest on the right hand side if the effective viscosity of the medium is sufficiently low to permit convection. Large scale convection in the mantle has been proposed as a driving force for continental drift (Holmes, 1965) but the present state of knowledge of the thermal and mechanical properties of the upper mantle is inadequate to establish whether this is possible. The convection term also dominates the heat transfer equation (1.1) in parts of the upper crust where heat is carried by fluids moving through permeable rock formations (section 3.2).

1.4 Heat flow through the surface of the earth.

Horai and Simmons (1969) derived a global mean heat flow of 69.1 mW.m-2 from 2822 measurements. Lee (1970) took account of the extremely uneven geographical distribution of measuring sites by computing a global mean of 61.5 mW.m-2 from 50 (latitude) x 50 (longitude) averages using 3127 data points, having first weighted the data according to their estimated accuracy. Williams and Von Herzen (1974) arrived at a higher value (83.8 mW.m-2) from .theoretical considerations. They considered that heat flow measured by

5 30

20 >-

LLI CS Crr 10 w

Q r 50 100 150 200 HEAT FLOW (mWm-2)

Figure 1-2 Histogram of equal area grid(5cx 5° ) heat flow through the surface of the earth(from Lee,1970) normal ocean floor techniques (section 2.1) in the vicinity of mid-ocean ridges was too low because in these regions heat was being transferred to the surface by hydrothermal circulation as well as conduction, and therefore attempted a theoretical estimate of heat loss from the vicinity of active spreading ridges.

A histogram of equal area grid (weighted) averages of heat flow through the surface of the earth compiled by Lee (1970) is shown in figure 1-2. Heat flows more than an order of magnitude larger than the maximum on this histogram have been recorded in anomalous regions of limited areal extent. In most cases these occurrenas can be related to conditions favouring rapid transfer of heat (the termeq,VNT in equation (1.1) being large), such as exist in volcanic regions where hot liquid magma can rise relatively quickly from the mantle into the upper crust, producing a thermal anomaly for several million years until it cools to the temperature of its surroundings (chapter 4). However heat loss from these anomalous areas is much less than the total lost by normal conduction through the crust.

The possibility of bias in the location of continental measurement sites deserves consideration, because the expense of drilling to suitable depths has limited most continental sites to boreholes originally drilled for mineral or groundwater investigations. These are commonly located in unusual geological environments. The data gathered for this study is no exception since all measurements of geothermal gradient were carried out in boreholes drilled for groundwater exploration. In addition to the possiblity of geological bias, the boreholes themselves are prone to special types of thermal disturbance which could produce a systematic error in the heat flow results (chapter 3).

1.5 Heat flow patterns on continents.

Two recent discoveries have greatly contributed to the understanding of variations in heat flow on continents. The first, by Polyak and Smirnov (1968), was that the mean heat flow within a tectonic province decreases with increasing age of the province. Their conclusions are summarised in table 1-1. Sclater (1972) argued that the exact value of the means in each tectonic province should not be considered too significant, and in particular that the Precambrian value may be biased on the low side because of possible systematic'measurement errors and climatic disturbances.

The second discovery, by Roy et al (1968) and other workers in North' America, was that a linear relationship between heat flow and surface heat Age of Tectonic Province Average heat flow (mW.m 2)

• Precambrian 38.9 Caledonian 46.5 Hercynian 51.9 Mesozoic 59.4

Table 1-1 Relationship between terrestrial heat flow and age of tectonic proN;ince (from Polyak and Smirnov, 1968)

* Sclater (1972) suggests this value is c. 3 mW.m 2 too low production existed within plutons in a particular tectonic province.

Sclater (197.2) noted that heat flow and surface heat production in plutons within the Central Stable Region (Precambrian province) and the New England area (Caledonian province) in the United States, and the Australian and Canadian Precambrian shields, plotted on the same straight line (figure 1-3). Lachenbruch (1970) showed that this linear relationship, combined with certain reasonable assumptions, leads to the conclusion that the concentration of heat producing elements in the plutons decreases exponentially with depth below the surface. However no simple law governs the distribution of heat sources in the typically heterogeneous crust outside such plutons (Smithson and Decker, 1974). The decrease in heat flow with age, at least in plutons, can be explained in terms of the model of Lachenbruch as due to the effect of erosion in removing much of the radioactive upper layers from ancient crust. Sclater .1972) suggested that higher heat flow in Mesozoic and younger provinces could be due to the thermal- energy of intrusion.

1.6 Heat flow in rift valleys.

The East African Rift acquired a special significance in global tectonic theories after the discovery of a worldwide mid-oceanic ridge-rift system, because it appeared to be one of the few continental branches of this system (Girdler, 1964). Oceanic rifts occur along plate boundaries where new crust is being formed, and recent geological and geophysical investigations in East Africa have largely been directed towards an attempt to establish whether this continental rift is a developing plate boundary, and the site of a future oceanic ridge-rift system.

One of the characteristics of a mid-oceanic ridge is a large, irregular, positive heat flow anomaly which reaches a maximum over the rift and falls to around 45 to 50 mW.m in adjacent ocean basins. This anomaly has been interpreted as due to the dissipation of heat flow from newly created oceanic crust, which forms by dyke injection and cools as it is displaced laterally by continuing intrusion along the rift axis (McKenzie, 1967). It should be noted that a model of this type can not be applied directly to explain geothermal anomalies known to be associated with continental rifts such as the Rhinegraben (Haenel, 1971) or Kenya Rift (Morgan, 1973), because the flanks and floors of these rifts are known to be composed of continental crust (although the Kenya Rift. floor may have been extensively intruded by basic material). Other continental- rifts, such as the Baikal Rift in the . USSR, have no obvious connection with the mid-oceanic ridge system but are also characterised by distinct heat flow anomalies (Moiseenko et al, 1973). Interpretation of heat flow in continental rifts must rely heavily on infor- mation about the geothermal regime obtained from other geophysical studies

9 0 5 '10 Heat Production ( pW.rn-3 )

o USA (New England) o USA (Centrdl Stable Region) e Australian and Canadian Shields

Figure 1-3 Heat flow and heat production data from plutons in a Caledonian province(New England),PrecaMbrian,platform (Central Stable Region) and Precambrian shields (from Sclater,1972).

10 (sections 1.1, 5.3). Attempts to relate surface heat flow variations to deep thermal anomalies inferred from seismic, geomagnetic variation or gravity studies are complicated by the effects of convection in rift faults and associated zones of weakness. Surface manifestations of convection such as volcanic activity and.hot springs have been identified in most Cenozoic continental rift valleys, but subsurface occurrences can often only be inferred from the presence of local'heat flow anomalies of limited areal extent. Very detailed measurements of heat flow are therefore required in an environment which has been extensively faulted and intruded, such as the floor of the Kenya Rift, if heat flow data are to be used in a quantitative study of the geothermal regime in the crust and upper mantle in that area.

Thermal models of the crust and upper mantle in Kenya, based on seismic and geomagnetic deep sounding studies, have been developed for this study and are presented in chapter 7. Only selected heat flow sites on the flanks of the rift, away from faults and recent volcanoes, can be usefully compared with the predictions of large scale models, because the heat flow pattern on the rift floor is not known in sufficient detail to delineate regions disturbed by convection.

-11 CHAPTER 2

MEASUREMENT OF HEAT FLOW

2.1 Introduction,

The vertical component of heat flow near the surface of.the earth can be determined by'measuring two parameters at some point below the surface: the temperature gradient in the vertical direction dT and the thermal conductivity. of the rock (K). A heat flow value (0 may then, be dT .calculated from the relationship: q = K . The simplifying assumptions used in the derivation of this equation are discussed in the next chapter.

Thermal perturbations on the surface of the earth (Chapter 3) induce variations in temperature gradient below it; and measurements of heat flow are usually made as far below the surface as possible, in an attepmt to minimise surface-derived disturbances. Normally a series of temperature gradient and thermal conductivity determinations are made over a finite depth range so that variations in gradient which may be indicative of surface and subsurface disturbances can' be observed.

In deep oceans, stable sea bottom temperatures permit a relatively undisturbed determination of heat flow within a few metres of the sediment surface. This can. be conveniently-achieved by dropping a narrow cylindrical probe into soft surface sediments on the ocean floor, recording the temperature at two or more points along it End simultaneously recovering a sediment sample for laboratory measurements•of thermal conductivity. On continents, however, the measuring interval should usually be at least 100 m below the surface to avoid serious disturbance from surface perturbations or moving groundwater. Ideally a temperature or temperature gradient log is made in carefully sited boreholes which have been cased and grouted (Roy et wal 1972) and from which a section of rock core has been preserved, but in practice boreholes are expensive to drill and most continental heat flow measurements have been made in holes originally drilled for mineral or groundwater investigations, in mines or in tunnels, often in conditions which are far from perfect. Most: measurements have been made:in borebbles: and the following discussion is restricted to borehole techniques.

Geothermal gradients are usually calculated from a series of discrete

12 temperature measurements at regular depth intervals, but continuous temperature and temperature gradient logging methods can also be used. Early logging equipment employed mercury-in-glass maximum thermometers, but electrical sensors are now most popular. Thermocouples or pairs of thermistors have been used for direct gradient logs (Simmons, 1965) and platinum resistance thermometers (Reiter and Costaini 1973) or thermistors (Roy et al, 1968) for discrete temperature logs. Simple single thermistor probes were constructed for discrete temperature logging throughout the present study.

The thermal conductivities of the rocks penetrated by a borehole can be determined either in situ or by laboratory measurements on rock core or cuttings recovered during drilling. Beck et al (1971) described an in situ method in which the thermal conductivity of the rock formations in the vicinty of a borehole was deduced from the rate of increase of temperature of a heated cylinder inside the hole. This dire&L approach has obvious advantages, since core recovery and careful preparation of rock discs are not required, but it is not practicable in boreholes with irregular diameters, such as were encountered in the present study.

Laboratory techniques can be broadly classified as transient or steady state. Von Herzen and Maxwell (1959) described a transient method in which a heated needle probe was inserted into sediment samples. Its thermal conductivity can be determined from the rate of increase of temperature in the probe, according to the theory of a continuous line source of heat in an infinite medium. The most convenient steady state method is some form of divided bar apparatus, and a type similar to that described by Birch (1950) was used in this study. The apparatus was originally designed for disc-shaped'rock samples, but can be adapted for measurements on rock fragments' (Sass et al, 1971a).

The data gathered for this study (Chapter 6) are of highly variable quality for two reasons. Firstly because the only available boreholes had been drilled for groundwater exploration and it was to be expected that they would be prone to disturbances by moving water (section 3.2). Roy et al (1972) demonstrated that water movement within boreholes, the most serious source of error, could be effectively removed by grouting, but this was not practicabe for the present study. The second potential source of error exists because the boreholes have been drilled by the percussion method (Todd, 1959) and only fragmented rock samples were recovered. The technique employed to determine the thermal conductivity of rock chips (Sass et al, 1971a) actually measures the rock matrix conductivity and a calculation involving the rock porosity, which is

13 often difficult to estimate, is required to obtain the whole-rock thermal conductivity.

2.2 Measurement of the geothermal gradient.

Temperature logging equipment constructed and used by Morgan (1973) was modified for this study. An additional temperature sensor and winch were also assembled. The system was developed for use in boreholes 100 to 500 metres deep, in remote and often inhospitable conditions. Simple, rugged and lightweight apparatus was therfore necessary, and discrete temperature logging equipment based on the design of Beck (1963) was constructed. Since much of the time and expense involved in field operations was taken up in locating and travelling to boreholes, it was essential to minimise the possibility of equipment failure. Two complete logging systems were therfore carried on field excursions. Since the equipment was lightweight and compact, this could be done without " inconvenience.

Geothermal gradients were calculated from differences in temperature over regular depth intervals and therefore precise but not necessarily accurate temperature measurements were required. In practice a precision of 0.01 K was sufficient since temperature fluctuations of this order were common in the boreholes logged (section 3.2). Fast sensor response was particularly useful for identifying unstable sections in a borehole, where fluid movements often caused fluctuations in temperature.

2.2.1 Thermistor sensor.

Two types of thermistor manufactured by Standard Telephones and Cables Ltd. (STC) were examined for their suitability, a bead type G and a mounted bead type F. In each case the temperature coefficient of resistance was approximately -4%/K and the nominal resistance 10k) at 20°C. A G-type thermistor was used by Morgan (1973) and had proved rugged and reliable when mounted in tufnol and encased in a protective metal probe (figure 2-1). The F-type was less robust, and required careful handling until mounted in a probe, but had a very short thermal time constant-(5 seconds in air, as compared with several minutes for the G-type) and could detect short period fluctuations in temperature due to air or water movement. It also permitted detailed logs to be made in- air-filled boreholes, since the necessary observation time at each point was reduced from 15 minutes (G-type) to 2-3 minutes'(F-type). Figure 2-1 Borehole temperature logging equipment.

15 The resistance of a thermistor could be conveniently measured in a Wheatstone bridge configuration, although it was necessary to amplify the out-of-balance current in the bridge, to minimise the thermistor current and keep self-heating below 0.01 K. (Morgan,1973). A possible disadvantage in using thermistors for accurate temperature measurements is that changes in their characteristics can occur in response to mechanical and thermal shocks. Since only temperature differences were required in geothermal gradient determinations, this was not a problem providing changes did not occur between successive readings in any particular log. The calibrations of eight thermistors were checked regularly against a reliable laboratory standard (described in the discussion on calibration) between several short field excursions in England,. and no shift in their calibrations was observed. In fact no discernible shift (i.e.less than 0.01 K) in the calibration of the two thermistors used throughout the present survey has been observed over a period of three years.

Each thermistor was mounted in a stainless steel probe to provide weight, protection and a watertight interchangeable connection to the cable which completed the electrical circuit between a resistance bridge on the surface and a thermistor inside the borehole. The thermistor was positioned inside four metal rods connecting the rear end of the probe, in which the cable connection was made, to a protective metal front end (figure 2-1).The metallic probe caused a small thermal perturbation in the vicinity of the thermistor, but the effect could be made negligible in practice by the use of careful field procedures (Jaeger, 1965). The cables in the two equipment sets had different diameters, so that a special probe rear end had to be constructed for each cable, but the front ends, containing the thermistors, were made to be interchangeable. In the event of damage or suspected faults, .a thermistor could be easily replaced in the field.

2.2.2 Resistance bridge.

•

Morgan (1973) described an electrical circuit for measuring the resistance of a thermistor which incorporated a Wheatstone bridge, DC amplifier and galvanometer. An undesirable characteristic of this apparatus was an inherent error of approximately 0.2% in the measured value of resistance due to a small but significant input offset current in the DC amplifier. To overcome this problem, that amplifier was replaced by one in which the input offset current was negligible, and which was less sensitive to changes in ambient temperature. The improved

16 Figure 2-2 Field operations at a borehole site in Kenya.

17 +12V

• Al

.1144-R7 D2 ED 3

1k11.

D1 R3 110k11 -12V

Al Operational amplifier:type E7OB(Computing Techniques Ltd.) AP Operational amplifier:type 741-0PA(Radiospares Ltd.)

Figure 2-3 Circuit diagram of resistance bridge amplifier. circuit, designed by G.A. Ricketts at Imperial College, is shown in . figure 2-3 and is described below.

Two resistance bridges, as described by Morgan (1973), but equipped with the improved amplifiers, were capable of an accuracy of 0.02% in the range 5 kn. to 15 kfl , and were capable of resolving temperatures to better than 0.01 K. Laboratory checks were made periodically against a Cropico Resistance Box, type RB6 (accuracy = .02%) and field checks were made before each borehole log against a 4.91

A circuit diagram of the amplifier is shown in figure 2-3. The first stage amplifier Al was chosen for its extremely low input bias current characteristics. A second stage was added to improve the overall thermal stability. This was achieved using a temperature sensitive diode D1 in the potential divider circuit used to zero the second stage amplifier A2. The amount of compensation required to offset the drift due to changes in ambient temperature was set by adjusting R2 and R6. The completed bridge was tested over a range df ambient temperatures from 0°C to 50°C.

Before a resistance measurement, the amplifier was zeroed by connecting the input to, earth and adjusting R5 until the galvanometer reached its null position. The components and controls of the resistance bridge and the waterproof case in which it was enclosed, were described by Morgan (1973).

2.2.3 Cables.

A traditional disadvantage in using electrical thermometers for precise measurements in boreholes, the necessity for a heavy and expensive cable to provide a low resistance connection between the sensor in the hole and the resistance bridge on the surface (Bullard, 1960), was removed when Beck (1963) designed a system to compensate for the series and shunt resistances of a cable, and their variation with temperature. It then became posdible to construct simple and reliableportable equipment using thermistors. The system of Beck (1963) required a four conductor cable, of which two were used to connect the thermistor to the resistance bridge, a third compensated for the series resistance of the conductors and a fourth compensated for the shunt resistance between the conductors. The compensation for shunt resistance made it possible to use high resistance thermistors.

19 The cables were wound on simple hand winches constructed for the project. One winch and cable set was assembled by P. Morgan for a previous survey (Morgan, 1973).. This cable was manufactured by Standard Telephone and cables Ltd. (STC) and consisted of four high density polythene insulated 7/.15mm stainess steel cores which served both as conductors and strain members. The cable was sheathed by high density polythene and had an outside diameter of 2.8mm. Its nominal lead 1 resistance was 0.4flm and a series resistance asymmetry of 6n, was - compensated for by a balancing resistor in the circuit. A 670 m length; was assembled on the winch shown in figure 2-1. A second winch and cable set was constructed for the present study. Its 450 m long cable, manufactured by STC, comprised five polythene insulated 6/.177 mm tinned copper conductors and a 7/.5 mm stainless steel member inside a polythene sheath 4.3 mm in diameter.

The depth measurement system was described in detail by Morgan (1973). The cable was fed through a sheave and pinch wheel arrangement, attached to the borehole casing, and revolutions of the sheave wheel as the sensor was lowered were recorded on a counter. The number of revolutions between points of measurement was proportional to the depth interval. Regular checks against cable markers at 15 m intervals were performed as a routine part of the logging procedure. The equipment is shown in operation in figure 2-2.

2.2.4 Calibration of the thermistors.

The thermistors used in this study were carefully calibrated in the laboratory against a platinum resistance thermometer. Morgan (1973) described a procedure in which each thermistor had to be assembled on the cable before calibration against a set of mercury-in-glass thermometers. This was necessary because of asymmetries in the series resistance of the cable conductors and the sensitivity of the amplifier in the resistance• bridge to changes in input impedance, and had the obvious disdvantage that a separate calibration was required for each combination of thermistor, cable and resistance bridge, if these parts were to be interchangeable. However in the present study the improved amplifier design and the use of external resistances to compensate for cable asymmetries removed this necessity so that thermistors could be calibrated together before assembly in the probes and no adjustment to the calibration was required ifsensors, cables or• bridges were interchanged in the field if a fault was suspected.

The standard used in the calibrations was a platinum'resistance therm- ometer

20 type WS104, manufactured by the Rosemount Engineering Company Limited, and calibrated to + 0.01 K by the National Physical Laboratory (NPL). Its resistance was measured by a Cropico Smith Number 3 Resistance Bridge (accuracy = 0.005%) and Cropico Null Detector. The thermistors were calibrated in a thermostatically controlled water bath, Cobra Ultra Cryostat type KT20S. The bath could not maintain a steady temperature, but oscillated by approximately 0.03 K about a preset mean value. Therefore to ensure that the temperatures df each thermistor and the platinum resistance thermometer were within 0.01 K at each calibration point, the thermistors and the standard were positioned in a cylindrical aluminium block, with sufficient thermal inertia to reduce the oscillations below 0.01 K.

The resistance, Rp, of a platinum resistance thermometer at a temperature Tp 6:e (above 273 K) may be represented by the following equation (manufacturers' specification)

Rp = c% + Tp + 6 Tp2 (2.1)

are constants determined from the calibration data.

The NPL calibration data for the platinum resistance thermometer consisted of 186 points between 273 K and 773 K. A computer program was written to fit a curve of the form of equation (2.1) to the NPL calibration data by a least squares method, and the following values of CX, (i and 6 were calculated:

Cg. = 25.495 (3 = 0.102 -0.154 10 4

The temperature Tp corresponding to a resistance Rp is then the following root of equation (2.1) :

Tp = (-P + ( -4 (0(- Rp))1 )/2S (2.2)

The temperature dependence of the resistance of a thermistor can be adequately representedi over ranges of 30 K by an equation of the form (Morgan, 1973) :

1 -2 . RT, = 6clo (a + bT + cT ), ' (2.3)

21 where a,b and c are constants and T is the absolute temperature. The thermistors used in this study were calibrated using the apparatus described above over the range 278 K to 333 K in 1 K steps. A computer program written to fit curves of the form (2.3) to the calibration data by a least squares method, calculate two sets of values of a, b and c (one for the range 278 K to 303 K, and the other for the range 303 K to 333 K) and produce a temperature against resistance table for field use, over the range 278 K to 333 K at 0.01 K increments, for each thermistor. The temperature T corresponding to a thermistor resistance RT is the following solution of equation (2.3) :

T = 2c (-b+ ( b2 - 4 ( a -ln RT)c ) 2 ) (2.4)

It was noted above that thermistor characteristics can change with time, and are sensitive to mechanical and thermal shocks. Regular calibration checks were therefore essential and, since it was not practicable to remove the platinum resistance thermometer and associated equipment from the laboratory, a set of three mercury-in-glass thermometers were prepared as secondary standards for field checks. However these proved unsatisfactory because a systematic change of 0.02 K to 0.03 K was . apparent between calibrations carried out in London and Nairobi, although subsequent remeasurement in London indicated no change in the thermistor calibrations. This apparent shift in the mercury-in-glass thermometer calibration may be due to a difference in atmospheric pressure between -2 the two locations (20 .kN.m ) corresponding to an altitude difference of approximately 1800m. Field calibration checks were conveniently carried out at regular intervals by comparing the responses of the probes with those of unmounted thermistors in a stirred water bath.

2.3 Measurement of thermal conductivity.

The thermal conductivities of common rock types lie in the range 1 - 1 -1 to 7 W.m .K and careful measurement of the conductivities of rock formations penetrated by a borehole is essential for accurate heat flow determinations.

Because the boreholes available for this study had irregular diameters, in-situ-type conductivity measurements (Beck et al, 1971) were not feasible, and rock samples recovered during drilling were collected for laboratory measurements. The samples consisted of rock fragments ranging in size from fine powder to particles several millimetres in diameter, generated by the percussion method of drilling (Todd, 1959). The Water

22- Development division of the Ministry of Agriculture in Kenya had collected and catalogued chip samples recovered from several thousand boreholes drilled during the past few decades, and those relevant to this study were kindly made available for measurements, so, long as enough material remained in the original collection. Unfortunately, in some cases, insufficient material remained or no samples could be found.

Sampling intervals in different boreholes ranged from 2m to 30m. Special problems associated with using rock chip samples gathered during percussion drilling to obtain whole-rock thermal conductivities include the likelihood of sample contamination from higher levels in a borehole and the possibility that the action'of the drill has destroyed some quality of the rock fabric which contributes to its thermal conductivity (e.g. anisotropy).

Porous rocks require special consideration because the technique employed in this study for determining whole-rock thermal conductivities from fragments required an accurate knowledge of porosity for the calculation. A method developed by Morgan (1973) to measure -rock porosity using fragmented samples proved to be inadequate for this study, and it was necessary to obtain estimates of porosity from published values for various rock types encountered. Many of these exhibit a wide range of porosities and a relatively large error can arise because of a poor estimate of porosity (figure 2-6). The effect of porosity is treated here as a correction applied to the measured thermal conductivities, and the magnitude of the correction is displayed with the results in appendix 7. In general, where a correction is large, the probable error is also large. •

The thermal conductivity measurements were performed on a *divided bar apparatus, using a procedure described by Sass et al (1971a) to replace a rock disc (which.would normally have been prepared for the divided bar), by a specially constructed cell containing a rock fragment and water mixture. The principle and construction of the divided bar, and the operating procedure with rock discs and rock fragments are described below.

2.3.1 The Divided Bar.

The apparatus shown schematically in figure 2-4 is similar in design to that described by Jessop (1970). The major difference is that a plastic, a commercially available polycarbonate with the trade nethe

23 [Hot Water Shower _ — 29.1°C T1. ------T2 ZONE1

• • • • . • • •• • • • • • • • • • • • • • • • • • • a• • • ZONE 2 .•• •• • SPECIMEN . . .• • - - f ° a. • •, - • • • • • • T3 T4 ZONE 3

18.9 °C

*■ ■■■ ■■• ■■ ■■ ** [Gold Water Shower

Brass

Lexan damper mzazzo Lexan sub-standards

Figure 2-4 Vertical cross section of the divided bar(scale x2 approx.)Horizontal cross- section circular.. .

211- Lexan, is used as a reference material instead of glass.

The principle of operation of the divided bar requires that a negligible;- quantity of heat is lost to the surroundings along the region of the bar spanned by thermocouples (T1 to T4). If this condition .is satisfied, and the apparatus has attained a steady state, the voltage developed between a pair'of thermocouples is proportional to the thermal resistance of the section of bar, or specimen, between them. The known thermal resistance of sub - standards in each half bar can then be compared directly with the thermal resistance of a specimen.

A component of the thermal resistance between the thermocouples which span the specimen is due to a finite contact resistance between the faces of thedivided bar and the specimen. This was significant during calibration of the bars and when high conductivity rock discs were being measured, but the thermal resistance of a cell containing rock fragments and water was usually sufficiently large to make the contact resistance negligible by comparison.

Water temperatures in the hot and cold showers, within the upper and 'lower half bars, respectively, were fixed at prescribed values by Colora Model NB thermostatically controlled baths. The shower temperatures were set at 29.1°C (upper) and 18.9 C (lower) and minor fluctuations in water temperature were damped by 1.3mm thick discs of a material with low diffusivity, the polycarbonate Lexan, between the shower and first thermocouple in each half bar.

To minimise radial heat loss to the surroundings, it was. important to use high thermal resistance reference discs, thereby minimising the necessary length of half bar and the surface area exposed for heat loss. -1 -1 With Lexan reference discs (conductivity 0.2W.m .K ) the bar length (the distance between T1 and T4 in figure 2-4, excluding specimen) was approximately 10mm. Additional precautions included surrounding the bar and specimen with thermally insulating material during measurement, and adjusting the ambient temperature of the surroundings to be roughly mid - way between the upper and lower shower temperatures. The effects of radial heat - loss' were discussed by Jessop (1970). Copper - constantan 40SWG twin thermocouples, which generated approximately 40pV/K, were inserted in thin brass discs in the bar assembly, and a Cropico Vernier Potentiometer and Cropico Null Detector were used to measure voltage difference between the thermocouple junctions to an accuracy

25 of - 0.211V. The apparatus reached,eqUlibrium approximately 10 minutes after insertion of a rock disc specimen, and 20 minutes after insertion of a cell containing rock fragments and water.

To reduce the thermal contact resistance between the bar faces and the specimen, a hydraulic pump'was used to transmit an axial pressure of 2 0.69 MNm along the bars, and a water glycerol mixture smeared on the contacts before insertion of the specimen. Values of contact resistance for the calibration discs are given in table 2-1.

The thermocouple voltages, V1, V2 and V3 are proportional to the thermal resistances R1, R2 and R3 of zones 1,2 and 3 of the divided bar (figure 2-3).

It follows that:

R2/Rb V2/(V1 + V3) (2.5) where Rb (= R1 + R3), the bar reference resistance, is a constant. R2, the thermal resistance of zone 2, contains both the specimen resistance Rs, and the contact resistance, Rc. Equation (2.5) can therefore be written:

Rs = Rb.V2/(V1 + V3) - Rc (2.6 ) so that:

Ks = ds/ fRb.V2/(V1 + V3) - Rc] (2.7) where Ks . thermal conductivity of specimen ds = thickness of specimen.

When several rock discs of the same material but of different thicknesses are available, a plot of disc thickness against RbY2/(V1 V3) should have slope Ks and intercept - Ks.Rc on the disc thickness axis, so that the thermal conductivity and contact resistance may be obtained directly. A calculation of this type is referred to as a whole set analysis. When a number of whole set analyses have been performed for a variety of rock types, it should be possible to define an average value of contact resistance which may then be used in equation (2.7) to calculate the thermal conductivity of a rock using only one disc. This approach is referred to as a single disc analysis.

Equation (2.6) is also a starting point for the calculation of rock fragment conductivities, where Rs is defined as the total thermal,

26 resistance of a cell (figure 2.5) containing rock fragments and water.

2.3.2 Calibration of the divided bar.

An accurate knowledge of Rb, the bar resistance, is obviously a prerequisite for accurate conductivity determinations. Rb is measured using discs of known thermal conductivity in whole set type analyses in which equation (2.7)is rearranged so that the variables for plotting become ds.e. and V2/(V1 + V3), and the slope of the line through a set of points should be Rb and the intercept on the clsIs axis should be - Rc.

Fused quartz and Y-cut crystalline quartz discs are useful calibration 1 1 standards since their thermal conductivities (fused quartz = 1.36Wm .K , crystalline quartz = 6.30 W.mK-1) span the range of most rock types.. Sets of 5 discs each of fused and crystalline quartz were used in this study. A minor problem arises in the calculation of Rb from the calibration data, because there is some uncertainty in the exact values of thermal conductivity for fused and crystalline quartz (Horai and Simmons, 1969). Also, the contact resistances of fused quartz and crystalline quartz discs are different and the calculation of Rb should take account of'this.

A simple method of combining both sets of calibration data to obtain an optimum value of Rb and distinct values of contact resistance for the two materials was devised. A computer was used to recalculate Rb in an iterative fashion, allowing small changes Sf and Sc in the assumed conductivities Kf and Kc of fused and cyrstalline quartz respectively until the values of Rb obtained in seperate calculations for the two materials agreed within a prescribed error. It was assumed in the calculation that the fractional errors in Kf and Kc were the same, so that 6f and Sc were related as follows:

Si c =

The values of Kf and Kc stated above (published by Ratcliffe, 1959) were chosen as a starting point in the calculation. For the first iteration, 5 pairs of values of Rb were calculated, corresponding to each of 5 pairs of assumed fused and crystalline quartz conductivities of the form:

(Kf 4. n6f, Kc - nSf) n = -2,71,0,1;2

27

and that pair chosen which gave the closest agreement for Rb. This pair (Kf', Kc') was then selected as the mid-point of the range for the next iteration, in which the deviations . Sf and Sc were reduced by half, and another set of pairs of values of Rb was computed. This procedure was repeated until adjusted values of fused and crystalline quartz conductivities were found which agreed on the value of Rb (within prescribed limits) and gave reasonable values of contact resistance for the two materials.

The values of bar resistance Rb and contact resistances obtained in three seperate calibration experiments performed during the period of use of the apparatus are shown in table 2-1. In these calculations Kf was increased by 1.5% and Kc was decreased by 1.5%.

2.3.3 Conductivity measurements on rock discs.

Where possible, rock samples were taken from surface outcrops in the vicinity of a borehole within which a temperature log had been made, cut into discs by the techniques described by Misener and Beck (1960), and their conductivities were measured on the divided bar. The values obtained were then used to supplement unreliable or poorly sampled conductivity data from rock fragments, where sufficient evidence existed to correlate the lithological formations penetrated by a borehole with nearby surface outcrops. The discs were cut in thicknesses of 5mm to 10mm, accurate to ± 0.025mm, with diameters matching the divided bar diameter of 34.90mm within ± 0.05mm. Before a conductivity measurement, the discs were saturated with water under vacuum, as described by Morgan, (1973). The necessity for this procedure was demonstrated by comparing the results of measurements on a sample of lava which had been soaked in -1 water overnight (1.37 W.m .K), with those in the same sample after - 1 saturation under vacuum (1.59 W.m 1.K ). In this case, incomplete saturation has caused a 14% decrease in thermal conductivity. Combs and Simmons (1973) found that the conductivity of a shelf - dried rock with a porosity of 5 to 10% could be as little as 60% of the saturated rock value. This severely limits the value of heat flow measurements above the water table, at least until a convenient in situ method for measuring thermal conductivity is available.

2.3.4 Conductivity measurements on rock fragments.

Woodside and Messmer (1961) demonstrated that the thermal conductivity

28 ,,,,...,... • •

, Run Bar resistance contact resistance (x 03) No. Rb(x 103) fused quartz crystalline quartz

7.45 0.17 0.12

•2 - 7.43 0.19 0.12 '

3 7.43 0.12 0.05

4 7.43 0.17 0.10

Table 2-1 : Calibration results for divided bar

Thermal resistance units are m2.K.w 1

Run is actually a combined calculation using the data from runs 1, 2 and 3.

29 of the solid phase in a fluid - solid mixture can be determined if the the fluid phase and mixture conductivities are known. Their equation can be rearranged to allow the thermal conductivity of rock fragments to be calculated if the conductivity of a rock fragment - water mixture can be measured. Sass et al (1971a) described a convenient way of achieving this using divided bar apparatus. The method was adopted by Morgan (1973), proved satisfactory, and has therefore been used throughout this work. An alternative technique for measuring the conductivity of a rock fragment - water mixture, is the needle probe method (Horai and Baldridge, 1972).

Robertson and Peck (1974) described a variety of mathematical expressions relating the thermal conductivity of a two - phase mixture to the conductivities of its components. Experiments by Woodside and Messmer (1961) and Sass et al (1971a) demonstrated the appliOability of the weighted geometric mean, to a mixture in which the conductivity contrast between its components was less than an order of magnitude. The geometric mean is.therefore adequate'for a rock - water mixture. It is expressed as follows:

Ka = Kri-° KV125 (2-8)

where Ka o mixture conductivity Kr = rock conductivity Kw = water conductivity 0 = fraction of water in the mixture.

In the application of equation (2-8) in this study, Kw is known -1 -1 (0.62W.m .K ), 0 and Ka are measured, and Kr is calculated from:

Kr = (K9111)17_•9S . .(2-9)

Kr is the thermal conddctivity of the solid phase in the mixture, and for a crushed porous rock it represents the rock matrix conductivity. The whole rock conductivity (Kpr) can be obtained by a further application of the geometric mean formula as follows:

p Kpr = Kr17 KWP (2-10) where Kr and Kpr are defined above and p is the fractional porosity. Kr can therefore be determined from measurements of Ka and k5 by substitution in (2-9). Following Sass et al (1971a) and Morgan (1973),

30

the rock chip and water aggregate conductivity Ka is measured inia disc shaped cell ilSitig the divided bar. The cell is constructed of copper ends and perspex walls (figure 2-5). Rock fragments are packed inside the cell with a spatula and saturated under vacuum according to the procedure established for rock discs. The volume fraction of water. is calculated from the cell weight empty (We), filled with dry rock.;.- fragments (Wd) and after saturation (Ws) as follows:

0 = Ws - Wd

Ws - We (2.11)

Ka is calculated from the measured thermal resistance of a saturated cell Rs from the relation:

Ka = Ady/1Rs - R ) + B (2.12) xz

where A and B are constants (determined below), R is the combined xz thermal resistance of zones x and z (figure 2-5) and d. is the thickness 7 of zone y. The thermal resistance of zone y, R is related to Rs and

R z as follows:

Rs = R y+ Rxz

The value of was calculated from the known dimensions and conductivities of the cell ends, and a mean contact resistance. Equation (2.12) can be written

Ka = AK+ B (2.13) where K = d /R , the appatent conductivity of zone y. Y'

Zone y includes both.the rock water aggregate and the perspex cell wall. The constants A and B in (2.13) can be calculated approximately by assuming that lines of heat flux within the cell are parallel to the bar axis, so that the thermal resistance of the aggregate and perspex wall can be considered to be combined in parallel. It follows that:

A . DJ4",d 2

- (D2 - d2). Kp B = d2 zone x Izone y zone... .. z.

copper rock fragments + water perspex

Figure 2-5 Vertical cross-section of a conductivity cell(scale x1.5 approx.)Horizontal cross-. section circular.

32 where D and. d are the outer and inner diameters respectively of the perspex cell wall, and.Kp. is the thermal conductivity of perspex.

To establish the validity of equation (2.12) all 20 cells were filled with distilled water and measured in the divided bars. Calculated cell Constants A,B and Rxz were then used to compute Ka, which in this case should have equalled Kw the conductivity of water, for each of the cells. 1 -1 1 The results ranged from 0.607 W.m 1.K to 0.624 W.m .K , i.e. within 2% of the required value, confirming that the assumptions_used in the derivation of equation (2.12) were reasonable.

To check the applicability of the weighted geometric mean model in the calculation of the thermal conductivity of a two - phase mixture, a cell was packed with powdered fused quartz and saturated with water. The value of thermal conductivity obtained using equations (2.9) and (2.12) were within 3% of the value quoted by Ratcliffe (1959) (1.36W.m-1.K 1) in each of three sets of measurements.

Next, the use of the weighted geometric mean model in equation (2.10) was investigated by comparing rock disc and rock fragment thermal ' conductivity measurements from 10 porous rock samples, whose measured porosities ranged from 4% to 27%. The results are summarised in table 2-2. There appears to be no systematic difference and errors of 10% can be considered tolerable in this study. The largest error, 14%, was measured on a sample of pillow lava which had proved to be relatively impermeable during experiments on the saturation process. Sass et al (1971a) and Norgan,(1973) considered that large differences in conductivity between measurements on discs and fragments (where porosities are known) may have been related to saturation problems.

It should be emphasised here that much larger errors may be associated with conductivity determinations on the rock fragments for the reasons . outlined at the beginning of this section and'because the porosity values required in equation (2.10) can only be estimated, and may be subject to considerable error. The likely extent of the error due to porosity inaccuracies, and problems encountered in an attempt to.measure porosity, are discussed below. •

2.3.5 • The Porosity Problem.

Because the thermal conductivity of water is less than that of the common rock forming minerals, the conductivity of a water - saturated

33 Porosity Kd Nd Kt Nr Kpr (Kpr-7,1).100 % Kpr

• 3.9 4.88 1 4.81 3 4.43 -10 5.1 2.12 3 2.35 3 2.19 3 5.2 2.37 1 2.54 3 2.36 0 7.7 2.29 1 2.53 3 2.27 - 1 8.9 3.98 3 4.77 3 3.97 0 10.4 4.11 1 4.58 4 3.71 -11 11.1 2.33 3 2.84 3 2.39 3 17.2 1.59 3 2.19 3 1.76 9 26.7 1.67 2 2.93 3 1.93 14

mean disc conductivity (Nd•runs) Kr mean chip conductivity (Nr runs)

Kpr = chip conductivities corrected for porosity

- 1 Thermal conductivity values 'in W.m1.K .

Table 2-2 Comparison of thermal conductivity measurements on rock discs and fragments.

34 porous rock is less than that of the rock matrix. It is this matrix conductivity (Kr) which is determined when fragments are measured by the techniques outlined above.

Specimens recovered from below the water table can in general be considered to have been fully water - saturated under normal conditions, and for these the water - saturated state can be recreated in the laboratory. Above the water table, the factors governing the degree of water - saturation are quite complex (Todd, 1959), and conductivities determined in the laboratory under fully saturated conditions may differ significantly from the in situ values.

The relationship between the rock matrix (Kr) and water - saturated rock conductivities (Kpr) can be adequately represented by the equation (2.10), and a series of graphs of Kpr against porosity are shown in figure 2-6.

Since the rock formations penetrated by the boreholes used in this study had variable and often high porosities, a method for determining whole rock porosities from measurements on the available rock fragments was sought. Morgan (1373) attempted to adapt a technique used by the British Petroleum Ccimpany for porosity measurements on rock cores. The basic equipment was a Kobe Porosimeter, a single cell Boyle's Law apparatus for measuring the bulk and grain volumes of 25.4mm diameter rock cores. Porosity (P) is related to bulk volume (Vb) and grain volume (Vg) as follows:

P(%) = (Vb - Vg) . 100 (2.13)

• Vb

In a Kobe porosimeter,'Vb is measured by displacement of mercury._. and Vg by displacement of helium gas. Although' Morgan (1973) had obtained satisfactory comparisons between porosities measured using 25.4mm diameter 25.4mm long rock cores and porosities measured using fragments from the same sample for 8 different rocks collected in . Cyprus, further work by the present author using samples collected in Kenya revealed inconsistencies which demonstrated that the technique was unsuitable for porosity measurements on a routine basis for this study. Two problems may be distinguished. The first concerns the nature of the available samples and the second a difficulty which arises when the Kobe porosimeter is used for fragments instead of solid core.

35 6 (a) 5 4

3

1

5.00 10.00 15.00 20.00 25.00 POROSITY

K r (b )

20.00 310.00 40.00 50.00 POROSITY

Kr=rock matrix conductivity KPR=whole rock conductivity Figure 2-6 Effect of porosity on thermal conductivity. (a)Error in conductivity due to an error in porosity (b)Effect of porosity on the conductivity of a rock of matrix conductivity Kr.

36 The factors which determine whether a rock sample retains its in situ porosity depend upon the nature of the interstices in the rock and the degree of destruction of the rock fabric which took place during sampling. A percussion drill can reduce a competent sedimentary formation to its constituent grains, thereby destroying all the intergranular porosity, but the porosity of a fine grained hard rock may remain preserved in fragments 1 mm in diameter. Each sample must therefore be considered on its own merits before a porosity measurement is attempted.

The second problem concerns a difficulty encountered in the use of the Kobe porosimeter for measuring the bulk volume of fragments. The operation of the Kobe porosimeter, and the modified procedure required to handle rock fragments, are described in detail by Morgan (1973). In a bulk volume measurement, rock fragments were required to displace mercury in a chamber of known volume. It was necessary to ensure that fragments were completely surrounded by mercury when the reading was taken, because voids between rock,chips had the effect of increasing the apparent bulk volume. Morgan (1973) attempted to force mercury into all the inter-chip spaces by evacuating the chamber before filling it with mercury, and then releasing it tq atmospheric pressure before measuring the volume of mercury displaced by the fragments. This had the effect of applying an external pressure equal to atmospheric pressure to the mercury, which forced it into gaps between the rock chips. The critical pore diameter (dc) which can be penetrated by mercury under external pressure (Pe) is as follows (Drake, 1949) :

dc _ _ 4 otos 9 Pe where a = surface tension of mercury e = contact angle (140°approximately)

According to this equation, pores greater than 10pm diameter will be 2 entered under a pressure of 100 kN.m . Therefore, to obtain a -2 satisfactory result, a pressure of 100 kN.m must be sufficient to force mercury between the chips without being large enough to cause it to enter the pores.

Experimental tests by Morgan (1973) in which porosity measurements on rock fragments and core were compared for eight rock types from Cyprus with porosities in the range 1% to 27%, showed agreement within 2.5%

37 porosity. Similar investigations by the present author, in which the vacuum applied to the sample chamber was carefully regulated, revealed serious differences in test comparisons on certain samples. Two sandstone samples from Kenya were cored and porosities of 20% for sample 1 and 15% for sample 2 were determined in the normal way (without vacuum). When the measurements were repeated with a vacuum of 0.2 -2 kN.m (absolute) during the bulk volume determination, the apparent porosities were reduced to 1% (sample 1) and 2% (sample 2). The weight of sample 1 increased from 24.376g before, to 37.434 g after, the bulk density measurement in which a vacuum was applied. Similarly, sample 2 increased in weight from 27.418 g to 40.215 g. Evidently mercury had entered the rock pores of both samples when external pressure (atmospheric pressure) was applied, demonstrating that the porosities of fragmented samples of these rock types could not be measured by the procedure proposed by Morgan (1973).

In another case it appeared that atmospheric pressure was insufficient to force mercury into all the inter - chip voids. Core and chips were prepared from a fine - grained sandstone sample and the core porosity (7%) determined with and without application of a vacuum, during bulk volume measurement, verifying that mercury did not enter the pores of this particular sample under atmospheric pressure.

Chip sample porosities were determined using two different vacuums, -2 2 0.2 kN.m and 30 kN.m (absolute) and in both cases the apparent porosities were high (11%) suggesting that gaps between the fragments had not been completely filled by mercury during the bulk volume measurements, and therefore the apparent bulk volumes were too high. A similar experiment on samples of a pillow lava gaye a core porosity of 10% with and without -2 2 vacuum, and chip porosities of 15% under 30 kN.m and 9% under 0.2 kN.m . These results would suggest incomplete penetration of the inter - chip -2 spaces at 30 kN.m and complete penetration, without pore entry, at -2 0.2 kN.m . Evidently this method of determining rock porosities from measurements on fragmented samples is not generally applicable, and unfortunately the samples for which porosities were required in this survey spanned a wide variety of rock types. Tuffs, vesicular lavas and unconsolidated coarse-grained sediments were common, encompassing a wide range of porosity and pore entry diameters. Since only fragmented samples were available, serious errors due to the problems outlined above may be anticipated. The technique was therefore rejected as unsuitable for porosity measurements on Kenyan chip samples.

38 Unfortunately it was not possible to develop an alternative to the Kobe porosimeter in the time available. In situ measurements of porosity should be possible using equipment developed for the petroleum industry (Schlumberger, 1972) and the possibility of utilising these techniques (Neutron, Sonic and Density logging tools) deserves consideration in future studies.

For this work, it has therefore been necessary to estimate a value of porosity (p) for equation (2.10), based on the available information. A useful estimate of the error introduced in Kpr (S Kpr ) from a porosity error 6 p can be obtained from equation (2.10) by differentiation b.s follows:

k„ 41:0094KW/Kr) (2.14) . kpr

Since Kr > Kw, an overestimate of porosity leads to an underestimate of whole rock conductivity. The magnitude of the error S Kpr increases rapidly with Kr as shown in figure 2.11.

Thus largest errors are likely in porous rocks which have a high matrix conductivity Kr. Fortunately most Kenyan volcanics are relatively poor conductors, but the problem is quite acute in the case of Quaternary sediments of the Lamu Embayment (section 6.2). However, perhaps the most interesting results to emerge from this study are low heat flow values on the flanks of the rift valley in Kenya which are based on measurements in non -.porous phonolites and basement gneisses.

The porosities of a large variety of sedimentary rocks have been catalogued by Manger (1963) and this has been used as a guide in estimating the porosities of sedimentary formations encountered in this survey. Davis (1969), Schoeller (1962) and Keller (1960) have been used as sources of information on the porosities of volcanic rocks.

39 CHAPTER 3

HEAT FLOW CORRECTIONS AND CALCULATIONS.

A wide variety of factors can disturb the geothermal regime within. 300 metres of the earth's surface, the region in which heat flow measurements for this study were made. These include perturbations associated with the shape and thermal history of the surface of the earth, and with the movement of. groundwater. Special types of disturbance can also occur inside the boreholes in which heat flow is measured. In this chapter, these and other problems involved in the measurement of heat flow on land are reviewed, and where relevant, methods of estimating corrections to compensate for the effects of disturbances are described.

The following assumptions are usually made when a section of a borehole temperature log is used to calculate the geothermal gradient:

1. The geothermal regime is steady state- 2. The thermal flux vector is vertical 3. Heat is transferred by conduction only 4. No significant heat sources exist in the depth interval in which measurements were made.

The heat flow equation:

V(kVT) + ecpy.VT S = P (3.1) then reduces to: =o ( 3.2) and a value of terrestrial heat flow, q can be defined as follows:

(3.3) k YT" 1■z where thermal conductivity mass velocity density

Cp = specific heat

S = heat production

The deviation of the surface of the earth from a fixed isothermal plane, introduces horizontal components of heat flow in the earth, so that assumption 2 is no longer valid, and (3.1) does not reduce to the simple form (3.2). The effect is obviously greater at shallow depths, and since all the boreholes available for measurement in this study were less than 300 metres deep, the recorded geothermal gradients often show evidence of disturbance related to thermal conditions on the surface. In particular, past climate, topography, upliftlerosion and sedimentation can cause a significant change in temperature gradient within this depth range. In section 3.1 simple methods of computing the magnitude of these disturbances are indicated, and these are used to apply corrections to the data presented in appendix 7.

In some cases, shallow groundwater circulation may have a much greater effect on temperatures in the depth range over which measurements were made in this study, but a quantitative treatment of the thermal effects of groundwater movement was not attempted because the important hydrogeologic parameters are virtually unknown. A qualitative discussion of the thermally disturbing effects of moving groundwater is given in section 3.2..

3.1 Surface disturbances and corrections

3.1.1 Surface topography

Several mathematical methods have been used to estimate the effect of topography on the geothermal gradient, and their relative merits are discussed by Jaeger (1965). These methods consider either exact solutions for simple topographic shapes (e.g. Lees, 1910) or approximate solutions for more realistic representations (e.g. Jeffreys, 1938). It is usually assumed in the calculation of topographic effects that surface temperature varies linearly with elevation whereas it is in fact a complex function of elevation, hill slope orientation and inclination, and surface vegetation type (Blackwell, 1973). However disturbance by surface topography is of fairly minor importance in the present study and only a simple linear relation is considered here. The method of Jeffreys (1938) is useful where the topography can not be represented as a simple shape for which an analytical solution is available. Details of the method are given in appendix 2. A simplifying assumption, which makes this method an approximate one, is that there is no thermal interaction between the various elements of topography. With this assumption, the disturbance to the temperature gradient at depth z can be written:

c° Sg(z) = 0(r z)V(r)(Jr- (3.4) i0 2Tr

where: V(r) = .1 h(rle)(g -gide (3.5 ) 2rr 0

0 (r,z) = r ( r2- 2 22)( rt+ 4-5/2 ( 3.6)

radial distance from the borehole.

h(rp)= topographic height at (r,e ) above a horizontal reference plane through the borehole collar

undisturbed geothermal gradient

adiabatic lapse rate in the atmosphere.

The function V(r) therefore represents the temperature disturbance on the reference plane due to topography (assuming no interaction between the various elements), integrated over a circle radius r, centred on the borehole. Surface temperatures are assumed to decrease linearly with height, according to the adiabatic lapse rate

In practice the integral in equation (3.4) is approximated by a finite sum of discrete terms, computed from topographic maps using a polar grid centred on the borehole (Kappelm eyer and Haenel, 1974). A problem arises when large topographic features exist near shallow boreholes because large variations in the function k)(1,Z) (equation 3.6) occur for small values of r and z, and nearby topography can make a relatively large contribution to the sum. This can be deduced from figure 3-1 where the function A(r,z) defined as :

0,z)(r A(r) z):::/ 0(1)z) ( 3.7 ) 42 00 M CNI ri--1e. 1 50m CC , • o

Z =100 M

CO ; 0.00 0.20 0.40 0.60 0.80 1.00 R (KM)

Figure 3-1 Plot of the function A,(R, Z) in the range (0‹:R‹.1) for .3 values of Z. A(R,Z) = 0 (R,Z) / 0 (1,Z) (see text) R(Km) = radial distance from borehole collar to an element of topography. g(m) = depth below surface (where 011)Z) is the value of1)(rZ) at r. 1 kilometre), is plotted against r for three values of z within the range of interest to this study. The topographic maps available in Kenya were of scale 1:50,000 and were therefore only adequate for this type of correction where major features did not occur within 0.5 kilometres of the borehole. Fortunately most boreholes had been sited in areas with gentle topography, and the associated disturbance could usually be neglected. Where it appeared significant, the magnitude of the disturbance was estimated by the method outlined above and a correction applied to the geothermal gradient in the heat flow calculation (appendix 7).

• 3.1.2 Past climate.

Temperature variations on the surface of the earth affect the geothermal gradient, but are attenuated with depth according to frequency, and only long period variations need be considered here. The effects of a diurnal variation are usually insignificant below 1 metre, and an annual variation below 20 metres (Jaeger, 1965). 5 Fluctuations with periods of 10'to 10 years are of interest to this - study.

. Mitchell (1961) noted a warming trend of about 0.5 K since 1882 AD in tropical zones, and a curve of annual mean temperatures between latitudes 30° N and 30° S from 1882 AD to 1955 AD calculated by him is shown in figure(3-2a). No reliable climatic data for the preceding ' millenniumin Africa or the tropics has been found, but Lamb (1965) compiled evidence from Europe for a warm period around 1200 AD and a cold period between 1500 AD and 1700 AD (figure 3-2b), which may reflect global trends.

Global temperaturevariationsduring the Pleistocene have been estim- ated by Emiliani (1961.) (figure 3-2c), and Van Zinderen Bakker Sr. and Coetzee (1972) list evidence for cold periods in East Africa during this epoch. Flint (1959) had used data on changing snowlines on Mount Kenya to deduce a temperature drop of 5 K during cold periods in the Pleistocene, and Van Zinderen Bakker Sr. and Coetzee confirmed that this figure was reasonable from studies of pollen analyses. They estimated a likely range of 4 K to 7 K.

Birch (1948) described a method of correcting heat flow data for the

44

(c ) AT (K) 0 -47

0 ioo 2100 300 x103 Yrs BP

0_ (b),

1111111111 1 100 300 500 700 900 1100 Yrs BP

(a)

Yrs BP

Figure 3-2 Variations in surface temperature(1073.105yeais 3.P.) . .(a) Pecent warminE. trcnd in trclical zonr'n(fron Yitch^11 ,1961) (b)Variations in England in tic past milicnnium(from Lamb,1965) (c)Attenuated form of the Pleistocene curve of Emiliani. (1961)(see text).

effects of past climate. The temperature disturbance at depth z, 'ST (z), due to past surface temperature variations which can be represented as a series of deviations from the present-day mean:.

A. ...A 1 n In time intervals: ( 0,a.0 , (cipa2), - (a.l-1' • (an -Il an) ,whereeach.al is measured backwards from the present, can be written (Birch, 1948) :

T(z) A. ( R - R. ) 1 I (3.8) i =1 where Ro = 1 FR. = erf ) [4 Kai] 2 diffusivity

z depth below surface

e rf error function

The temperature gradient disturbance, S G(z) , can be calculated from equation (3.8).as.follows:

)n G(z).= A.I(R. R ) (3.9) T i i=1 where Ro= 0

f = (rrkaifYlexpr z2 ) 1 i n I 4 ai

Equation (3.9) has been used to estimate temperature gradient corrections for the heat flow data in appendix 6. The three surface temperature curves in figure 3-2 were each approximated by a series of constant

deviations (Ai ; = 1, n) and the sum in equation (3.9) calculated using a computer. The middle ages curve (figure 3-2b) produced a small gradient correction ( <0.5 mK. m 1), and since these temperature variations have not been definitely established in Africa it has been ignored. The effect of the Pleistocene temperature variations (figure 3-2c) were calculated for depths up to 200 m, assuming an attenuated . form of the curve predicted by Emiliani in which the maximum temperature drop (Tp) was 5 K, as suggested by van Zinderen Bakker Sr. and Coetzee (solid curve in figure. 3 -3a). The dashed curves in figure.3-3a

46

______--_------T. p 7 K ______

(a)

o

NJOg • (=I Tp = 5K CD

40.00 80.00 120.00 160.00 200.00 DEPTH(M)M )

Total N2 (b) . O Recent

o ,\;--Pleistocene(Tp5K)______

• 0.00 40.00 80.00 120.00 160.00 200.00

DEPTH( M ) • Figure 3-3 Temperature gradient corrections for past climate: variation with depth (assumed diffusivity = (a)Pleistocene correction for 3 values of T p(see text). (b)Total climate correction showing recent and Pleistocene components.

47 correspond to maximum temperature drops of 4 K, and 7 K. A correction to compensate for the recent warming trend (figure 3-2a) is shown in figure (3-3b) along with the 5 K drop Pleistocene correction and the combined effect of the recent warming and the Pleistocene cooling, 2 -1 shown as a solid curve (a rock diffusivity of 1 mm s was assumed). This combined correction, and values of diffusivity(Kappelmeyer and Haenel, 1974) appropriate to the rock type in which individual boreholes had been drilled were used to compute the corrections in the heat flow data in appendix 7.

It should be noted that the effect of past climate is not always visible in borehole temperature logs, even when the contribution from other sources of disturbance should be relatively small. Sass et al (1971b) observed no systematic variation in heat flow with depth in a 2.9 km deep borehole in Canada, despite strong evidence that the surface temperature was at least 3.5°C below the present-day mean for long periods during the Pleistocene. In fact no consistent variation in heat flow with depth which could be related to past climate was evident in the Kenya results. 'However groundwater movement and surface vegetation changes can have a greater effect on the temperature gradients and disturbances from these and other sources would probably mask any small variation due to past climate.

3.1.3 Uplift and erosion.

Vertical movements of the surface of the earth are accompanied by 1 temperature changes because a temperature gradient (c.6 mlf.m ) exists in the atmosphere, so that uplift and downwarping disturb the geothermal gradient. Since this gradient in the atmosphere is much less than typical geothermal gradients, erosion and deposition also disturb subsurface temperatures.

Benfield (1949) devised a method of correcting for uplift and erosion. Steady uplift has the same effect as a linearly decreasing surface temperature. The temperature change at some depth z below a surface undergoing erosion is equivalent to, the change produced when material at z is moving upwards towards the surface (at which it is annihilated without evolution of heat) with a velocity equal in magnitude to the speed of erosion. Deposition and downwarp can be treated as negative erosion and uplift respectively.

tag The expression for the corrected temperature gradient.gc at a depth z below a'sUrface which has been eroded at a constant speed, e and whose temperature has decreased at a constant rate u'for a period of time t, can be written (Benfield, 1949)

gc = ( '41u/elx) / (1 -. x) (3.10) where equilibrium geothermal gradient [erfi: (2JA ) 2 e4- --t ,)V2exp( LIP4 fl ) 2.[Ht ]k Tr n 2B n4 - (1 — if (z-et)) exp ) erfcg,,z)

k = diffusivity

erf , = error function erfc = 1 erf

Several major phases of uplift have occured in western and central Kenya, several kilometres of sediments and volcanics have accumulated in the rift valley, and downwarping and sedimentation have been taking place in eastern Kenya (section 5.2). Corrections were computed using equation (3.10) and are displayed with the heat flow data in appendix 7. The largest:correction is that for sedimentation on the floor of the rift valley ( 30% increase in geothermal gradient in the inner graben).

3.2 Disturbance by convection.

If the term eC.pV.VT in equation (3.1) becomes large; heat flows calculated using equation (3.1) may be greatly in error. This situation can arise in two connections in the present study : when fluid movement occurs within the borehole used for a heat flow measurement, or when groundwater movement takes place on a larger scale within permeable formations in the vicinity of a measuring site. In each case two types of convective motion can be distinguished, natural and forced.

3.2.1 Convection in boreholes.

The theoretical conditions under which natural convection can occur in a borehole were examined by Krige (1939). He adapted formulae derived by Jeffreys (1930) and.Bales (1937) to the problem of instability in the column of air or water enclosed by a borehole. The. critical temperature gradient, (3 , for the onset of natural convection can be

49 calculated as follows:

g ccT BvK (3 (3:11) cr, (gad')

where thermal diffusivity acceleration due to gravity OC = . volume coefficient of thermal expansion of the fluid CO = specific heat v = kinematic viscosity Q = radius of fluid column B = constant (=216) T = absolute temperature

Equation (3.11) predicts instability in every borehole logged'in this survey! However. experiments by other workers have demonstrated that satisfactory geothermal gradient determinations can be made in boreholes in which the critical gradient is exceeded, because temperature anomalies produced by natural convection are typically very small.

Gretener (1967) detected temperature fluctuations of .01 K to .05 K at several depth points within two large diameter boreholes (130 mm to 200 mm) in which the critical gradient was exceeded. He attributed these fluctuations to the effects of natural convection. Diment (1967) noted similar oscillations in a 250 mm diameter hole filled with water ( = 0.2 mK.m-1) in which gradients between 5 mK.m-1 and -1 40 mK.m had been recorded. He observed that the amplitude of the oscillations were roughly proportional to the temperature gradient in the vicinity of the point of measurement, and suggested that they were due to convection cells or eddies in which water movement occurred over a distance of no more than a few times the borehole diameter.

The presence of such fluctuations set a practical limit on the minimum size of depth interval over which an accurate determination of heat flow could be made, and it was obviously important to be able to observe any fluctuations which occurred. The very fast response of the temperature sensor (section 2.2) proved invaluable in this respect. In fact oscillations observed in the boreholes in Kenya rarely exceeded a few hundredths of a degree, although boreholes 150 mm or more in diameter were typical ( = 1.4 mK.m 1) and recorded gradients were as high as 70 mK.m-1, In many cases steady temperatures were recorded

50 where large oscillations might have been expected e.g. at site 24 (appendix 7) steady temperatures were recorded in the water-filled section of a borehole 152 mm in diameter, in which the gradient was 65 mK.m. •The occurrence of forced convection could also be recognised in several boreholes from the characteristic shape of the temperature - depth profiles. This situation arises when a borehole connects two or more permeable water - bearing zones and water flows vertically up or down the hole, lowering the temperature gradients in the region of movement and.producing sharp. changes, and sometimes erratic temperatures, where water enters or leaves the hole. Jaeger (1963) described temperature depth profiles in boreholes disturbed by water movement, and noted that the major anomalies in the profile did not appear to change with time, indicating that the flow of water was steady. The temperature log at site 32 (appendix 7) indicates that forced convection is taking place in the water - filled section. If the effects of forced convection are not recognised, heat flow calculated from measurements in a disturbed borehole could be seriously in error.

The above discussion has concentrated on the effects of convection. in water - filled boreholes. When the water level in a hole was 50 to 100 m below the surface, the air - filled sections were also logged. Diment (1967) reported an experiment to investigate the thermal stability of air - filled holes in which a 300 m deep dry hole was logged, filled with water and relogged after 11 months. The maximum observed temperature difference between the two logs was 0.05 K, possibly due to oscillations associated with natural convection in the water, indicating that satisfactory geothermal gradient determinations are possible in dry boreholes. However the hole investigated by Diment had been grouted after drilling, so that no fluid interchange between the borehole and its surroundings could take place. None of the holes logged in this survey had been grouted and air movement was apparent at several sites. In these boreholes, a steady drift in temperature, which continued as long as observations were made (2 -5 hours), was observed over a considerable depth range, and in some instances air' flow from the casing at the surface could be detected, indicating that the borehole was disturbed by forced convection. A plausible explanation for these occurrences'isthat the borehole provided the only effiCient connection between a reservoir of air, probably contained in dry porous. rocks beneath an impermeable formation, and the atmosphere. Diurnal variations in atmospheric pressure could then provide the energy to drive a forced convective system, producing temperature •

51 oscillations with a 24 hour period.

A notable example of this phenomenon, at site 46,.. was logged in the Northern Desert in Kenya. Air was blowing strongly through the surface casing and unsteady temperatures were recorded to a depth of 200 metres. The coincidence of a very low water table, large diurnal variations in atmospheric pressure and favourable geology (intercalated lavas and sediments) made the air-filled sections of boreholes in the area particularly susceptible to forced convection.

Combs and Simmons (1973) noted the occurrence of erratic temperatures in the air-filled section of a borehole, but found that reliable heat flow values could be obtained above the water table if the measured gradients were averaged over a considerable depth range. Direct comparisons between computed heat flows from air and water - filled sections of individual boreholes in this study are often confused by a large degree of uncertainty in. the thermal conductivities of dry or partially saturated porous rocks.

Forced convection in boreholes was the most serious source of disturbance to the Kenya heat flow data. The temperature - depth profiles in appendix 7 have been plotted as dashed curves over the regions where oscillating or drifting temperatures were recorded. Fluid movement between a borehole and its surroundings can be eliminated by using a chemical grout or cement to fill the annulus between the casing and the hole walls. Roy et al (1971) described how this treatment completely stopped artesian flow from a borehole, and removed large anomalies in the temperature log. Grouting would obviously have improved the quality of the data obtained in the present survey but it was not practicable for technical reasons. It should be seriously considered for any future work.

3.2.2 Groundwater Circulation.

Forced convection occurs within permeable formations where a hydraulic gradient exists in response to external groundwater sources and sinks; natural convection can take place locally where permeable zones and high geothermal gradients occur together.

Bullard and Niblett (1951) postulated forced convection in a groundwater system to explain a local heat flow anomaly in England. They proposed that the groundwater had attained a temperature several

52 degrees above the surrounding rock after moving vertically through a distance of 0.6 km, producing a heat flow anomaly at the surface of about 70 mW.m 2. Schneider (1964) used water well discharge temperatures to obtain rough estimates of the geothermal gradient and demonstrated the influence of moving water in a large acquifer system in Israel.

Parsons (1970) studied the thermal effects of forced convection in two dimensions using a mathematical model of a groundwater basin. He considered a steady-state situation, so that the heat flow equation (equation 3.1), in the absences of heat sources, reduces to:

V.(kVT) + O&,q,NST = 0 (3.12)

where: FL = density of water C = specific heat of water V = groundwater velocity = porosity To obtain a solution of (3.12) the groundwater velocity V must first be determined at each point in the medium by.solving the following 'equation:

V.(HVh) = 0 (3.13)

, where h = hydraulic head H = hydraulic conductivity

The groundwater velocity V is then calculated as follows:

V = -H Vh (3.14) Parsons (1970) showed that underground temperatures could be seriously distorted in a permeable medium where the vertical component of V was significant. A mathematical treatment of this kind could not be usefully applied to those parts of Kenya where groundwater disturbance was suspected, because of a lack of knowledge of the hydraulic conductivity. Fortunately no major -.aquifer systems existed in central Kenya, although vertical groundwater movement in permeable fault zones in the vicinity of the rift valley probably produces geothermal anomalies.

Natural convection is potentially a more serious source of disturbance in the rift valley, where numerous faults and recent volcanic activity provide a combination of high permeability and high geothermal gradients.

53 The movement of a naturally convecting fluid is due to density variations caused by differences in temperature. Elder (1965) summarised the processes of heat transfer in regions of very high heat flow, and emphasised the role of convection in providing an efficient means of cooling local hot spots in the upper crust. Hot springs and other evidence of hydrothermal circulation have been discovered in many parts of the rift valley in Kenya (Glover, pers.comm.) The amount of heat carried by moving fluids often greatly exceeds that transferred by conduction, invalidating the use of equation (3.3) in such circumstances. Mathematical analyses of convection in permeable' media are difficult (blooding, 1957) and would not be useful in the case of the Kenya rift floor because of a paucity of information on hydrogeological conditions, which are extremely complex due to extensive faulting. The simple theory (conduction only) was therefore assumed throughout so that heat flows deduced from measured temperature gradients and thermal conductivities in localities within which hydrothermal circulation is known to occur, may be greatly in error.

3.3 Disturbances from other sources.

3.3.1 Disturbance due to drilling.

Heat dissipated during the drilling of a borehole can significantly disturb the thermal environment for some time after the drilling process has been completed. The nature of the disturbance depends on the method of drilling, and the contrasting effects of rotary and dry percussion drilling are discussed below. All the boreholes used in this study were drilled by the dry percussion method.

The disturbance caused by rotary drilling is probably better understood and it will be considered first. The heat generated at the drill bit creates less disturbance than the heat lost to the circulating mud or water, with the result that the surrounding rock is cooled. Borehole temperatures soon after drilling will therefore be less than . corresponding equilibrium rock temperatures. Mathematical approximations to this cooling effect are discussed by Jaeger (1965).

In contrast, the resultant effect of dry percussion drilling is to heat the surrounding rocks, and borehole temperatures will be above the equilibrium values for some time after drilling. Loddo and Mongelli

514-

(1968) derived a mathematical expression for the heating effect of the percussion process, and concluded that after a period ten times greater than the total drilling time, the disturbance in temperature. is reduced by 95%. This period is likely to be an overestimate since• it is common practice to circulate water or mud in the well after drilling, which has a cooling effect and therefore compensates for heat generated by the drill bit. The analysis of Loddo and Mongelli also demonstrated that, away from the surface and bottom hole, even if temperatures are disturbed, the effect on temperature gradients is likely to be negligibly small. Since the temperature disturbance due to drilling is a transient phenomenon, changes in temperature with time should be observable, and some measure of the magnitude of the disturbance may be possible if several temperature logs are made over a period of time. Unfortunately boreholes which may have shown some evidence of the thermal effects of percussion drilling (sites 36 and 37) were relatively inaccessible, and it was not possible to repeat temperature logs in them. The drilling times and time lapses between completion and logging for the '26 boreholes logged in Kenya are shown in table 3-1.

3.3.2 Lateral variations in thermal conductivity.

Lateral variations of thermal conductivity in the earth can induce horizontal.components in the geothermal flux, and cause surface heat flow anomalies. Mundry (1966) described a mathematical technique fo.e calculating the disturbance due to a two-dimensional conductivity structure, similar to the finite difference scheme for solving the steady state heat flow equation which is outlined in section 4.3.4.

Thermal refraction operates on a smaller scale within a borehole • where conductivity contrasts occur at rock/water, air/water etc.. interfaces. Minor inflexions in a temperature log can sometimes be correlated with changes in hole diameter, the water level, or the bottom of casing. However the disturbances never exceed a few hundredths of a degree, and all major inflexions must be attributed to convective. effects.

3.4 Calculation of heat flow.

3.4.1 Methods of calculation The simplified heat flow equation (3.2), ---Z r 1, 211- -- .0 ))

55 site to .t site to t

24 0.5 216 37 1.7 2.0 25 0.8 9:5 38 2.2 5.5 26 0.2 214 39 1.3 6.0 27 0.9 221 40 - 28 0.8 184 41 1.6 4.0 29 1.2 142 42 1.2 288 30 1.9 29 43 1.1 291 31 - - 44 1.7 19 32 1.3 132 45 5.3 18 33 1.4 204 46 5.0 8.0' 34 2.6 10 47 1.8 134 35 2.1 23 48 5.0 130 36 - 0.5 49 8.3 39

to = duration of drilling (months) t = time lapse between completion and logging (months)

Table 3-1 Data on drilling times for Kenya boreholes

55 is assumed valid over that section of a borehole used for a heat flow . determination. In the absence of disturbances of the type discussed above, q should be independent of depth z and variations in conductivity K should be accompanied by inverse variations in geothermal gradient dT . The data collected in this survey (appendix dz 6) consist of temperat ures and conductivities sampled at discrete points .along boreholes, and the heat flow calculation combines data in an approximation of (3.2) over a selected depth interval. Three types of calculation from discrete data points are outlined below.

Bullard (1939) introduced a method, later referred to as the thermal depth method, for calculating q over a depth interval in which there are significant variations in K. Equation (3.2) can be simplified by introducing a thermal resistance defined as follows:

= d (3.41) 0 so that equation (3.2) can be written:

dT (3.42) d

A plot of T against should be a straight line of slope q, the geothermal flux.

Gough (1963) considered an alternative method in which heat flow is calculated over the intervals defined by each consecutive pair of temperatures, so that the regression of heat flow on depth can be examined to facilitate analyses of disturbances. Another commonly used method is one in which heat flows are calculated over depth sections defined by major lithologic boundaries, by computing the product of mean conductivity and mean temperature gradient within each lithologic unit. Before considering which calculation is most relevant to the data available in this study, two problems peculiar to measurements from shallow, percussion drilled boreholes will be restated.

The first concerns temperature disturbance by flowing water inside the boreholes. The disturbance can be very large and it was considered prudent to exclude all sections suspected of convection from heat flow calculations. The remaining sections, in which a calculation is likely to be meaningful, were often reduced to 30 metres or less in length. Since, in many cases the presence of flowing water could only be inferred

57 from the temperature profile, disturbed sections may have passed undetected introducing potentially large errors in the calculated heat flows.

The second problem concerns the lack of porosity information and the consequent uncertainty in thermal conductivities calculated from measurements on rock fragments (section 2.3). Fine structure in a heat flow against depth analysis may therefore be due entirely to variations in porosity or convection.

Therefore, unless the formations penetrated by a borehole are known to be non-porous, there is no evidence of water flow, and the rock conductivity variations are well known, a simple product of weighted harmonic mean conductivity and temperature gradient computed by a least squares method was used to calculate heat flow over the chosen depth interval.

It was convenient to perform the heat flow calculation, apply appropriate corrections and plot the results automatically using a digital computer, and a FORTRAN program was written for this purpose. The results of the heat flow calculations are shown along with the original data in appendix 7.

3.4.2 The accuracy of heat flow data.

Beck and Judge (1969) examined the significance of a single measurement of heat flow on land, by making detailed observations in a 600 metre deep borehole drilled specifically for heat flow purposes in a stable tectonic environment. Temperatures were recorded every 3 metres and the thermal conductivity of solid core samples was measured at 4 metre intervals (no porosity correction was necessary since measurements were made on discs). A conclusion of their experiment relevant to this study was that heat flow calculated over a randomly selected 30 metre section of the borehole could have been in error by up to 20%, due to causes not clearly understood. On the other hand, Combs and Simmons (1973) presented data from a 1040 metre deep'borehole in which heat flows calculated over 3 to 20 metre sections (gradients were computed from continuous temperature logs) were within 3% of the mean value for total depth section studied. Thus it appears that, in a carefully sited deep borehole, relatively large errors (20%) can be present if the heat flow determination is based on measurements over a single 30

58 metre section. Much larger variations in heat flow with depth are commonly encountered in shallow (<200 metres deep) boreholes. These can sometimes be correlated with changing surface temperature or topography but more often they must be attributed to the effect of hypothetical groundwater movements, or remain unexplained. Roy et al (1972) demonstrated that the movement of groundwater inside a borehole could be effectively prevented by grouting, and disturbances associated with surface conditions could be minimised by careful site selection, avoiding areas of human activity, vegetation contrasts and severe topography. Unfortunately the Kenya sites could not be preselected and grouting was not practicable in the present survey. The reliability of an individual heat flow determination from a short section of a shallow borehole therefore depends very much on conditions peculiar to the borehole, beyond the control of the observer in this survey, and no general statement on the accuracy of the data•is likely to be useful. In section 6.3criteria for assessing - the reliability of individual measurements are presented.

59 CHAPTER 4

THE THERMAL EFFECTS OF IGNEOUS INTRUSIONS.

The existence of localised' zones on the earth's surface through which heat flow is an order of magnitude or more above the global average was mentioned briefly in chapter 1. In these areas, heat has been focussed by mechanical means. The process considered in this chapter is the cooling of bodies of liquid magma in the crust which, 3 by virtue of their relatively low density (2.65 Mg.m for a basaltic liquid), have separated from a region of partial melt within the -3 mantle (where the density is typically 3.30 Mg.m ) and moved upwards until the forces resisting motion have balanced the buoyancy forces. The magmatic intrusions then cool to the temperature of their surroundings, and produce a transient surface heat flow anomaly.

Very high heat flow (section 6.2) and recent volcanic activity (section 5.2) characterise parts of the floor of the Gregory Rift Valley in Kenya, and the presence of a major dyke-shaped intrusion beneath the graben floor has been inferred from other geophysical investigations (section 5.3). A quantitative study of the thermal effects of'such an intrusion was therefore undertaken to provide a basis for the interpretation of the measured heat flow pattern;

Jaeger (1964) reviewed the mathematical theory used in calculating the thermal effects of intrusions of simple shapes, by analytical methods. He alo drew attention to important aspects of the cooling process which can not be satisfactorily modelled by analytical techniques, and in particular to the importance of natural convection within a liquid magma. Where the depth of cover is much less than the intrusion thickness, convection within the magma, and the effect of a background geothermal gradient, become particularly important. Other complicating factors include the possibility of metamorphism in surrounding country rocks, the transport of heat by volatiles from the magma, and the effects of efficient cooling by convecting groundwater in permeable zones above the intrusion.'

60 In section 4.1 an analytical solution for a cooling rectangular intrusion is presented which incorporates the effect of the earth's surface and the background geothermal gradient:- A crude treatment of the effects of latent heat can also be included. It is demonstrated in section 4.4, where the effects of latent heat in the intrusion and the country rock, and convection within the intrusion are simulated by a more sophisticated numerical method, that this crude analytical treatment is a better approximation for the purposes of surface heat flow than had originally been anticipated. This came about because the effect of convection in the intrusion appears to compensate for the deficiency in the analytical approximation to the latent heat problem.

Natural convection of groundwater in permeable formations above hot magma can seriously affect the pattern of surface heat flow (Elder, 1965), but a mathematical analysis of its effect is beyond the scope of this study. Indeed the complex hydrogeological environment within the Gregory rift (Glover,pers. comm.)wouldmake such an analysis virtually impossible. Numerous faults on the graben floor act as potential conduits for circulating groundwater, and local enhancement of heat flow in their vicinity may be anticipated.

4.1 Analytical solution for a cooling rectangular intrusion.

A solution of the two dimensional heat conduction equation (equation 4.1) has been derived in the region shown in figure 4-1. The effect of the background geothermal gradient has been included and the error associated with a simpler approach in which it was ignored (Simmons, 1967) has also been calculated.

A buried rectangular intrusion which extends downwards to the top of an isothermal magmatic or partial melt layer (c = 0), upwards to the isothermal ground surface (b = 0), and which is not hotter than the lower isothermal layer (Tx = 0) was considered by Reilly (1958).

The theroretical model is based on the following assumptions:

(d) magma injection is instantaneous (2) heat transfer is by conduction only

z ground surface. T=TO (t >0 , z =0 )

intrusion country rock T=To+ alh+g,,F +Tx T=1:+gez h (at t=0) (at t =0)

isothermal layer T=Toi-gb(t>0,z=h)

T = 0 (x=0)

Figure 4-1 Rectangular intrusion model:initial and boundary conditions.

G2 (3) the intrusion is underlain by an isothermal surface (e.g. the top of a magmatic layer or partial melt zone) (4) the intrusion is overlain by an isothermal surface (the surface of the earth) (5) before injection, temperature between the two isothermal surfaces increased linearly with depth, with gradient g (6) at the time of injection (t = 0), temperature within the intrusion varied linearly with depth, with gradient gm (7) changes of phase did not involve exchange of heat or variation in thermal properties (8) the thermal properties of the intrusion and surrounding . rock are identical.

The heat flow equation (3.1) therefore reduces to

g•T gT1 a T (4.1) 7(2' a Z2. = K t where k e c

The dimensions of the intrusion are specified by the parameters a,b,c and h, and rectangular co-ordinates (x,z) were chosen so that the model is symmetrical about the z axis (figure 4-1). It is therefore sufficient to obtain a solution in the region x:?..o,05,:z using the boundary condition I T = 0 at x = 0. /0( • The temperature at depth z is specified as follows:

before intrusion T = To goz (4.2)

after intrusion T = To + g,,z + 9 • (4.3)

The initial temperature within the intrusion may be written

T =To+(go Wh gmz + Tx (t =0) (4.4)

Temperatures at a time t>0 may be calculated from:

(4.5) at = V2T

63 From (4.3) and (4.5) 6;) = K V 9(x,ztt) (4.6)-

It is convenient to introduce the following dimensionless parameters = , = t )'11 h (4.7)

From (4.6) and (4.7)

1 219 ieq,4,11 2T YC (4.8)

The boundary conditions may be expressed as follows:

At =0, 9=0 1, 9 0 =0, YJIt =0 -4 0c) and the initial conditions (at T.0) : 04....cA 1 9=7+(,_gjh 5) (1- 13...5..(1-C) :( (4.13)

> A 8=0 (4.14)

A B 1 9=0 (4.15) where A= a/h, B= b/h, C= c/h. (4.16)

Equations (4.8) to (4.15) can be conveniently solved by an integral transform method (Sneddon, 1951) as shown in appendix 3. The solution may be expressed:

9( ,4,Z) = (g.„-g,)h q,21 Z(S,21 (4.17) where

X( ,Z) = ierf Vc-) + erf(—±P2';1c ) (4.18)

64 (erf is the error function, defined in appendix 3)

vo Z e-(111-TT) sin (rirr) (4.1_9) n=1

where G(n) = (1 -13) cos(rin_13.) • - Ccos(nrr[1-Cl) sin(nrr[1-C])-sin(nnE3))/ nrr (4.20) +Ecos(nnB) cos(nnI1-C))] -r„ h(ggr),

The temperature gradient disturbance at z = 0, Oslh..:0 is of primary interest in this study. From (4.17) and (4.19)

= (g0 gol 4z= X ,t) Z10,11 (4.21)

where

Z1 0 ;C) = EG(n) (4.22) n =1

Therefore; from (4.3) and (4.21)

,9 = X ( ,T) (4.23) ( g- g,o

where I z'z=0

In practical applications, the infinite series in equation (4.22) is approximated by a finite sum, and for the range of parameters examined in this study, convergence of the series was sufficiently rapid to require a maximum of 100 terms to be retained in the sum.

A computer program' was written to'rperform the calculation of equation (4.23) for a range of values of and 1: , and plot a smooth curve through the results.

In figure 4-2, the variation of (g - go) / go with 77 is shown for three values of , over two intrusions in which:

A = .1 B = .1 (4.24) Tx =, 0 gm =

65 (a) 13— C—

=0

O ' 93.02 0.06 0.10 0.14 0.16 0 42 TRU

O (b) R= .1 O 8=•1 C= 0

O

9).02 0.06 0.10 0.14 0.18 1.22 TRU

Figure 4.- 2 Fractional increase in temperature gradient above two cooling intrusions plotted against dimensionless time (TAU) for those values of dimensionless distance ( ). TAU = (t-{ = (see text) h

66 'In the top graph, C = 0.8 and in the bottom, C = 0. It is interesting to examine the error which arises if an additional simplifying assumption, that the effect of the initial geothermal gradient can be ignored, is included with those stated earlier in this section. This assumption was incorporated in the analysis of Simmons (1967).

In the subsequent analysis, it is convenient to assume gm = 0 Equation (4.21) may be expressed in the form:

= x( , -r) Z(o,t) (4.25)

where ZW(0)Z1 = Ee(n) nrr-rt (4.26) n=1

and G (n) = goI (1-B)cos(nnB)- Ccos[nn(1-C)1 --isin(nn[1-C])- sin(nnB)k-n

+Icos (nnB) - cos (nn[1-C]) ) -rx h (4.27)

If the initial geothermal gradient is ignored (4.27) reduces to

Go(n) = [ cos ( nnB) - cos(nn[1-C])1T (4.28)

The parameter Tx may be considered as the excess initial temperature of the intrusion. To construct a model which ignores the effect of the geothermal gradient, Tx may be replaced by

Tx = Tx + gc;h - goh (B 1.- c)/2. (4.29)

S Tx represents the initial temperature difference between the country rock and the intrusion at the depth of its mid point.

Thus the error in ( E4)Z=0 , may be written

error = E(O,Z) (4.30)

where

C•40

E(0,T) [(60(n)_6(n)) (4.31) n=1

67 * G (n) is defined by (4.27) and Go (n) by (4.28) with Tx replaced by Tx as in (4.29)

A plot of g error/go against T for the two intrusions considered in figure 4-2 is shown in figure 4-3. The error is most serious for small values of B and C, when the intrusion takes the form of a dyke.

The analysis of Simmons (1967) gives identical results to the above for the case in which the geothermal gradient go is ignored. For larger values of B and for longer times, the solutions diverge as the lower isothermal boundary condition, which does not appear in the.... analysis of Simmons (1967), plays a more significant role. For the intrusion specified by equation (4.24) the error curves are identical to those which would arise if the problem were tackled by the method of Simmons (1967).

For terrestrial heat flow studies, it is useful to have an estimate of the maximum values of temperature gradient over an intrusion. The nature of equation (4.21) precludes the calculation of a maximum by differentiation with respect to 7:, but since the curves (figure 4-2) are known to have a simple form, maximum values of (11k for given 1Z =0 • . values of can be calculated on a digital computer by a straightforward iterative procedure, within'prescribed error limits.

This was done for several values of A, B and C and the results are shown in figures 4-4 and 4-5 where (g* - go)/go is plotted against (= x/H) and g* = maximum value of g,obtained by an iterative method.

The effect of variations in A on the wavelength, and B on the amplitude of the anomaly can be clearly seen in figure 4-4. The maximum anomaly is relatively insensitive tovariations in C for x<2a (figure 4-5).

4.2 Latent heat and convection in intrusions.

4.2.1 Latent heat : mathematical approximations.

The total quantity of heat per unit mass liberated by a cooling -1 basic igneous intrusion is typically 1680 kJ.kg . This comprises 1 roughly 1260 kJ.kg due to the elevation of its initial temperature 1 above that of the surrounding country rock, and 420 kJ.kg -latent

68 C= .8

03 0 0 1 '0.02 0%08 01.10 01.14 0.18 0'.22 TAU

C=0

01.10 0'.14 TAU Figure 4-3 Error in the calculated temperature gradient anomaly above the intrusion of figure 4-2 when the geothermal gradient is ignored.

69 A= 1 0 cn C= 0

B=CM

B=0.2

B=0.3

01.08 01.16 01.24 01.32 0.40 X/H

B= C=

9.00 01.08 01.16 01.24 X/H Figure 4-4 Maximum fractional increase in temperature gradient les above a rectangular intrusion plotted against dimensiont distance for 3 values of A and B (see text):

70 A= • 1 B= .1 TX =0

^NO

Cho

C=0 C=0.7 r7-0.8 0.08 0.16 0'.24 0.32 0.40 X/H

O A= .1 B= .1 CO TX=200

C=0 ./C=0.7 cb.00 01.08 0-16 0.24 0'.32 0.40 • X/H Figure 4-5 Maximum fractional increase in temperature gradient above a rectangular intrusion plotted against dimensionless distance for 3 values of C and two values of excess temperature TX (see'text)

71 heat associated with the liquid - solid phase transition. Thus latent heat is likely to make a significant. contribution to the heat flow anomaly associated with a cooling intrusion and it must be incorporated in any realistic mathematical model of the thermal regime.

Jaeger (1957, 1964) discussed the effects of the latent heat of solidification of magma, and demonstrated that it contributed to the increase in temperature of country rock surrounding an intrusion. The simplest mathematical treatment of latent heat involves assigning an excess initial heat to the magma, equivalent to the total amount of heat liberated during solidification. In the case of a rectangular intrusion, this approximation can be readily incorporated in the analysis of section 4.1, by replacing the parameter Tx in equation (4.20) by Tx* = Tx 4- L/C (4.32) -1 where L = latent heat of solidification (c. 420 kJ.kg )

1 specific heat (c. 1.05 kJ.kg 1)

This simple approach requires the initial temperature of a basic igneous intrusion to be increased by approximately 400 K.

In fact the latent heat is not released immediately, but is liberated over the period of time taken for the intrusion to cool from its initial temperature (the liquidus TT.) to the solidus temperature (Ts) of the magma. The simple mathematical treatment therefore predicts a geothermal anomaly which is too large in the early stages of cooling. However, in the discussion below on the role of convection in the magma, it will be demonstrated that a convecting magma releases heat more efficiently to the upper isothermal (ground) surface, and therefore enhances the geothermal anomaly in the earlier stages of cooling, so that the simple approach to the effects of latent heat is more realistic than might otherwise have been expected.

A more accurate representation of the effects of solidification is based on the assumption that latent heat is liberated uniformly over the melting range (TL to Ts) so that over a temperature interval A T, within the melting range, the amount of latent heat ( A Q) liberated per unit volume of magma can be written :

da e LAT (4.33) (T, -TS)

72 p = density IL = liquidus temperature TS = solidus temperature and, in the limit OT-->0 , the rate of heat release within the melting range. can be written :

Vtt e ?)-1- (T,-T5) (4.34)

Therefore the equation of heat flow in two dimensions (3.1), in the absence of convection, may be expressed :

Pc t a NgkV + H (4.35) where

I C = C 0 71. = c + L Ts

Mundry (1970) used this representation to compute temperature changes in the vicinity of cooling intrusions by a finite difference method, the implicit Crank - Nicholson method. A more efficient finite difference technique, (Mitchell, 1969) for solving (4.35) is described in section 4.3.

4.2.2 Convection in the magma : effect on the surroundings.

A rigorous treatment of natural convection within intrusions is beyond the scope of this study, but a crude representation which may prove adequate when only surface heat fluxes are required, is described below.

Jaeger (1964) demonstrated that convection has the effect of accelerating the cooling process and increasing the maximum temperatures achieved in surrounding country rock. Hodge (1974) confirmed this using a finite difference technique to calculate temperatures in the surrounding country rock for the one-dimensional

73 problem of a cooling slab. He showed that convection increases the extent of partial melting in the surrounding rocks. Bartlett (1969) ' demonstrated that the role of convection is greatest in large magma chambers, where it is the dominant mode of heat transfer. A special case relevant to this discussion, that of large intrusions in which the depth of cover is less than the intrusion thickness, was shown by Jaeger (1964) to be particularly sensitive to the effects of convection. In this situation, heat loss is mainly from the upper contact, and the surface heat flux anomaly directly above the intrusion can be significantly increased.

Providing the temperature distribution inside the intrusion is not required, the effects of convection in liquid magma can be usefully approximated by replacing the thermal conductivity of the liquid portion by a higher "effective thermal conductivity", therebY reducing the problem to one of thermal diffusion. Since the diffusion coefficients are temperature dependent in this case the problem is non- linear, and special techniques, outlined below, must be employed in its solution. Jaeger (1964) suggested an effective conductivity 5 times greater than the rock conductivity for a molten intrusion. Tikhonov et al (1970) used this concept of an effective conductivity to simulate the effect of convection within the mantle on the cooling history of the earth, and estimated that the ratio of effective to actual thermal conductivity varied from 2 to 10 according to the depth of the melted layer.

A realistic value of effective thermal conductivity is difficult to estimate, and for this study, results have been computed for both a fivefold and tenfold increase in conductivity with melting. The conductivity is considered to increase linearly from a minimum at the solidus equal to the normal solid rock value (Ko), to a maximum. (5 Ko or 10 Ko) at the liquidus, assumed to be the initial temperature of the magma.

The implications for surface heat flow, and for partial melting of country rock, are examined for these two cases by a mathematical technique, based on the ADI method (section 4.3) and containing modifications to improve the solution of the non-linear equation (4.49)

L. 4.2.3 Melting of country rock.

Experimental studies on metamorphic rocks have shown that fusion is likely to commence at about 970 K.-% Since the initial temperature of a basic intrusion can be 500 K higher. than this, it may be necessary to consider partial fusion in the country rock adjacent to it in an analysis of the associated surface heat flow disturbance. Following Fyfe (1973), it is convenient to consider a metamorphic rock composed of 50% granite and 50% mafic material, in which only the granitic fraction is involved in partial melting. Since the 1 latent heat of granite is about 210 kJ.kg and only half the material melts, it is reasonable to assign a latent heat of 105 kJ.kg- 1 to the country rock in the vicinity of the intrusion.

This further complication can not be satisfactorily treated by analytical methods, but can be readily incorporated in the numerical method for solving equation (4.35) outlined in section 4.3, in which the following values are assigned to the parameters L, TL, Ts in the region outside the intrusion :

105 kJ.kg 1 Ts = 973 K TL = 1173 K

The above value for TL is based on an estimated temperature of granitic liquids (Brown and Fyfe, 1970).

4.3 Finite Difference methods.

The application of"finite difference methods to problems involving partial differential equations was described by Mitchell (1969). A continuous domain ) representing a prescribed regions of (x,y,t) space is replaced by a mesh of discrete points within $). (see figure A4-1 in appendix 41 A partial differential equation governing a dependent variable u (x,y,t) within ,) may then be discretized by replacing derivatives by appropriate finite difference operators. An infinite number of such operators can be defined, but only a few are in common use. To illustrate the application of finite difference

75 operators, it is convenient to define a function y = f (x), which after discretization takes the value yn = (xn) at mesh point n. When the mesh spacing has a constant value :

h =x zin. =1)2 N the following operators can be defined :

11 Yn •• Yn47 Yn (forward difference) (4.37)

8 Yn = Yn4: Yn (central difference) (4.38)

i Yr) Yn4-1/2. Yn -4_1 /2 (averaging) - (4.39)

Common combinations of (4.38) and e(4.39) are :

'y y - 2 y y e n n+1 n n-i (4.40)

PSYn y (Yn+i-Yn-i)/2 (4.41)

Corresponding operators for irregular meshes are defined in appendix 4. Equations 4.37, 4.38 and 4.41 can be used in an approximation of the differential operator D, where

( 4.42) D Y = d Y/dx

By Taylor's theorem at the grid point n :

Y04.1 = Yn + h dy/dx 144/dx?. dl Y/cix , (4.43)

Yn -1 = Yn + • • • (4.44) h d Y/dx d2-Y/dxa 11;d3Y/ dx Then, using (4.40)(4.41) and (4.42),

D y = Fiji& yn 4/dx3 (4.45)

= ht2-yn - dY‘,4 • • (4.46) 12

Equations (4.45) and (4.46) may be expressed in the form

Dy = + (4.47)

76

ETy. = hrz152yin + 0(1-121 (4.48) 2 where 0(h ) is the local order of accuracy of the finite difference approximation, and the quantities : 3. 3 h _h2d4y/ 12 dx4 6 d. yed x3

, in (4.45) and (4.46) respectively, are defined as the principal ra parts of the truncation errors which appear when flp0 and n 0 are used to approximate the differential operators D and D2 respectively.

Two types of error arise when a continuous problem is solved by finite difference methods, discretization error and round - off error. A discretization error arises when a differential operator in a continuous domain is replaced by a finite differenc&'operator on a discrete mesh. It is a function of the local truncation error described above, and a truncation error associated with approximations to the boundary conditions of the problem. Round - off error appears during the solution of the system of linear equations generated by the application of finite difference operators. This type of error actually becomes more serious if the mesh is refined and more equations are generated, whereas the discretization error decreases when the - mesh is refined.

The presence of these errors can lead to the phenomenon of numerical instability if the finite difference operators are not carefully constructed .(Mitchell, 1969). In the case of a solution to a diffusion problem, in which the solution at some time to is built up step by step from specified initial conditions, the operator applied at each step must be designed in such a way that errors are not amplified by repeated application. Such an operator may then be described as unconditionally stable. An example of the errors Which arise because of discretization is given in figure 4-6. The magnitude of the errors in this example decrease after each application of the operator, which is unconditionally stable.

The problems to be considered in this chapter involve calculating temperatures in the vicinity of some localised thermal disturbance. The computation begins by approximating some vertical section through the earth by a rectangular array of points (1000 to 2000 are typical in this study). Each point of this array is assigned values of thermal

77 conductivity, heat production (and diffusivity in time-dependent problems), and an initial value of temperature. Boundary conditions on the sides of the rectangle are chosen to represent, as closely as possible, conditions in the earth. A common requirement in the problems considered is that temperatures remain undisturbed at large distances from the thermal anomaly. To satisfy this requirement, and to obtain maximum resolution in the region of interest, it was . usually necessary to construct a mesh with variable spacing, where the grid increment increased rapidly towards boundaries far removed from the anomaly. Two types of boundary condition have been used in this work : prescribed temperature on the boundary, and prescribed flux perpendicular to the boundary. A special case of the latter, zero flux perpendicular to a boundary, is the pertinent condition in a case where the boundary represents an axis of symmetry. Symmetry properties of a problem are utilised wherever possible, since resolution is usually limited by the dimensions of the variables which can be stored in a computer. In a case where an axis of symmetry exists, only half the domain need be covered by a mesh, resulting in a valuable economy.

Finite differences methods have been applied in this study to the calculation of the heat flow anomaly above a cooling igneous intrusion in the crust (section 4.4) and also to the geothermal anomalies associated with major disturbances in the upper mantle (section 7:1 ). The anomalies associated with intrusions usually change with time (Jaeger, 1964) although a quasi- steady-state situation may be reached in a few cases, if an intrusive body can receive a constant supply of hot magma from a deep reservoir (Horai, 1974). Methods for solving time-dependent and steady-state problems are outlined in the following sections.

4.3.1 Numerical solution of time-dependent problems.

Various computational schemes utilising the finite difference techniques outlined above can be used to obtain an approximate solution to the following set of equations :

c = 3-(0-11) 10-1-1-) S x,yeR) (4.49) . 21t y .1y + (3W + = (x,y (4.50)

u = uclx, y) ,(t =0) (4.51)

RI is a rectangular region of (x,y) space, ande is the boundary of6. Equation (4.49) is a two dimensional form of (3.1) without convective terms, where temperature is represented by u .0 (x,y,t) and C = eCp where p , cp , K and S are as defined in section 3.1. Equation (4.50) incorporates the boundary conditions on ta and 0 (x,y) (x,y€ ). specifies either the flux across the boundary (when W = 0) or the temperature on the boundary (when (x. (3 = 0).

Equation (4.51) expresses the initial conditions (at time t = 0) within a,. The heat production S = S (x,y) may vary with position and C = C (x,y,u) and K = K (x,y,u) may vary with position and temperature. The cases in which C and K are functions of temperature make (4.49) non-linear, and special difficulties are encountered in the solution of non-linear problems which are described later in this section.

In each of the computational schemes, the solution u(x,y,to) at time t = t.0 is achieved in a stepwise fashion, starting from the initial values specified by equation (4.51) and proceeding by recalculating u (x,y,t) over a discretized approximation to the continuous domain R., at each of a series of time levels up to t = to. The schemes can be broadly classified into two types, implicit and explicit. It is convenient to consider here the case of a rectangular grid as the discretized approximation to the space variables x and y, so that the operator s defined by (4.38) can be applied directly. In the two dimensional case difference operators in the x ( e.g. S x) and y (e.g. S y) direction are distinguished. Similar equations derived for an irregular rectangular grid are given in appendix 4.

Following the development of Varga (1962), consider the ordinary differential equations which result at each point on the grid when only the space variables in equation (4.49) are discretized. Let the grid spacing be A x in the x-direction, and Ay in the y-direction.

79

Then define :

u.(t) = u(iilx,jAy IJ l t) (4.52)

The following operators (defined in appendix 4) can then be applied to LA4t) as an approximation to the spatial differential operators in (4.49) :

(Ax12 Sx( kS,)1 uji( t ) ( k rx- (4.53)

( (4) 2 ( kV) uji (t ) k 1-)u (4.54)

It is convenient to writeU-u as a vector and the operators in (4.53) and (4.54) as matrices. Equations (4.49) and (4.50) can then be approximated as follows :

C dui i Auk}uu + S (4.55) d t where C is a diagonal square matrix A = H +N is a square matrix Uu and S are vectors.

The matrices H and V incorporate the operators in (4.53) and (4.54) respectively and also a discretized approximation to the boundary conditions of (4.50) as described in appendix 4.

A solution to (4.55) can be written (Varga, 1962) :

uo(t) = + exp(t CA[ uji(0) +AIS) (4.56)

It follows that : (t ) +A'S] U0 (t +At) = + exp(AtCA)[uu (4.57)

Thus some approximation of (4.57) can be used as a basis for a stepwise calculation, proceeding from initial specified valuesUu( (defined by time incrementsAt to the required solution - at some time to, where to = p (where p is an integer). The various

8o computational schemes in common use arise from different possible methods of solving (4.57).

An expliCit solution results when the exponential term in (4.57) 'is apprOximated as follows :

exp(AJC4A) I. + At CIA (4.58)

, where I is the unit matrix, and (4.57) becomes :

wij(t +At) = (I + At CIA) wij(t) + At CIS (4.59)

Thus a relatively simple calculation is involved at each step. Howeverl explicit methods are not unconditionally stable (Mitchell, 1969) and the time stepAt must be less than some maximum value related to the spatial grid increment. In the case of a uniform square mesh of spacing h , At< h!' . This constraint is quite 4K. serious for the type of problem under consideration in this study and fortunately unconditionally stable methods do exist. These are the implicit methods.

The backward difference implicit method results from the following approximation of the exponential term in (4.57) :

exp(AteA) (I -Lt eAY4 (4.60)

, so that (4.57) 'becomes :

( I-At eA)wii(t+m) = wii(t) + At es (4.61)

To obtain a solution, the matrix P defined by :

P = I - At C-4A

must be inverted at each step, which is often uneconomical in terms of the necessary computation time. The truncation errors with respect to the time variable, t, for the explicit or forward difference solution (equation 4.59) and the backward difference (equation 4.61) are CO Crank and Nicholson (1947) devised another approximation to exp (At CAA) which reduced the truncation error to 0W1 and led to an unconditionally stable method. In this, the exponential term is approximated as : exp (Pt CA)I ; ( I Ai C4A)-1 (1 +A1C4A) (4.62) 2

so that (4.57) becomes :

(I -1C-4A)w.(t+At) (I+41C4A)w.(t) +AtCt (4.63) 2 U 2 U

In two dimensions the matrix B, where

B = I - C4A 2 is not tridiagonal and must be inverted by direct or iterative techniques which are generally laborious. However in one dimension (e.g. when A = H) the matrix B is tridiagonal and a solution for can be obtained easily by a Gaussian elimination method WI Mt) described in appendix 5.

Peaceman and Rachford (1955) proposed an alternative formulation which is unconditionally stable, requires only a tridiagonal matrix to be inverted in two dimensional problems, and is comparable in accuracy to the method of Crank and Nicholson (1947). It is known as the alternating direction implicit (ADI) method because it utilises an approximation of the exponential term of equation (4.57) which allows the calculation of - from VVii (t ) to be split WU it+Ati into two parts, the first implicit in one direction only (the x direction, say) and the second implicit in the other (y direction). The exponential term can be approximated as follows (Varga, 1962) :

.(4.64) exp(AtCA) -VV+H)e- +v)

The calculation can then be per'formed in:two stages by introducing an intermediate vector i(t+6A) as follows :

Stage 1

2 w t 2 C ki L i( r A-1 s t t w it U ^

This can be rearranged as follows :

82 — H A A u t+At) = +V)wij(t)+S (4.65) Stage 2

y2C_ P. wo (tf6t) At H I will; (t+At ) + AIS

, which after rearrangement and simplification becomes :

2C 2C — V) w.(t+At) = +H) (t+At) + At U At IJ (4.66)

It is shown in appendix 4 that H and V ( and thereforeg and (-2C —V ) ) take a tridiagonal form, so that equations (4.65) At and (4.66) can be solved by the Gaussian elimination technique • outlined in appendix 5. Finite difference approximations to the boundary conditions of the type equation (4.50) are incorporated in the matrices H and V (appendix 4). The intermediate solutions do not necessarily correspond to real approximate solutions of equation'

(4.49) at some time between t and t+ At , and both stages must be applied at each time increment if the calculation is to be unconditionally stable.

4.3.2 The ADI method in practice.

The ADI method outlined above has been widely applied to the solution of two dimensional problems, but recent studies by Briggs and Dixon (1968) and Rushton (1974) have demonstrated the serious errors that can develop under certain conditions. The analysis of Rushton (1974), on a hydrogeologic application of equation (4.49), indicated that the following situations must be treated with some caution, if serious errors are to be avoided :

1. When sudden changes of temperature occur 2. In the vicinity of fixed temperature boundaries 3. In the vicinity of changes of mesh increment; thermal conductivity, specific heat or density. All three situations arise in the problems to be considered in this

83 chapter and the ADI method has therefore been applied to the solution of a simple geological problem of a cooling igneous intrusion which could be solved by the analytical techniques of section 4.1, so that errors associated with the ADI solution can be evaluated. Analytical and ADI solutions for the temperature gradients above a rectangular intrusion 21 km wide, 20.7 km high and 2.75 km below the surface (section 4.4) were computed for a range of times using different time steps A t . The area on one side of the vertical symmetry axis was approximated by a 961 point rectangular mesh, with variable spacing (appendix 11) constructed to provide greatest resolution, near the surface in the vicinity of the intrusion. The magma was assigned an initial temperature of 1473 K and the background geothermal gradient was set at 20 mK.m I.

Rushton (1974) demonstrated that oscillations appear in the ADI solution when the temperature at a node changes suddenly, if the time step At is such that : OC > 0.5 where cx = 1-4At Ax)

= diffusivity = 'mesh spacing

and that oscillations become significant when > 10 The analytical solution for the surface temperature gradient above the intrusion 1 Ma after emplacement, is shown in figure 4-6a for distances of 0 - 20 km from the intrusion axis. In figure 4-6b the error in the ADI solution is plotted for 3 different time steps, one of which (LOG) is made to vary as the solution progresses.

It should be pointed out that the observed error includes not only the oscillatory component observed by Rushton (1974) which owes its existence to terms in the discretization error, but also a component due to changes in the vertical temperature gradient with depth, in the vicinity of the surface. The temperature gradient calculated by the finite difference method is computed from the temperature difference over a discrete depth interval (equal to the vertical mesh spacing), whereas that computed by the analytical solution is the actual value ,at the surface. This second component can be easily checked using

84 KENYA RIFT INTRUSION TIME=1 MA

8.00.0o 112.0o X ( K M )

• +

• af TIME STEP Of UJ + .1 MA CD 4_ +74- x .05 MR •;t1 gi LOG

0

'0.00 4.00 8.00 12.00 16.00 20.00 X ( K M ) Figure 4-6 Effect of time step on discretisation error on computed surface temperature gradients above a cooling rectangular intrusion (see text) Top graph: Analytical solution. Bottom graph: Error in ADI solution

85 equation (4.17) to calculate temperatures along the second row of the rectangular mesh, and using these to calculate gradients over the same depth interval as the numerical solution. In fact this potential.- source of error was quite small for the problem considered.

Curve 1 on figure 4-6b was calculated using 10 steps, each 0.1 Ma in duration, so that the oscillation parameter CK has the value 12.6 for the assumed diffusivity value (1 mm2.s -1 ), and is therefore above the value (10) at which oscillations were considered to become significant by Rushton (1974). Curve 2 was calculated in 20 steps, each0.05 Ma in duration. The errors are much smaller, as expected for this value of CK(6.3). A further improvement in the solution was made by using a variable time step, designed to be smallest when sudden changes in temperature occur (in this case at time t = 0). Curve 3 is the error after 20 steps when the time step varies according to : n/ At = a10 M (4.67)

where number of steps and Q and Nei are constants chosen to suit the problem. The form of (4.69) is such that Lit increased by an order of magnitude in M steps. Once M has been chosen, Q is calculated from the following expression : rY 3-10 M Q = logio[ 7r .1 t

where t is the time at which a solution is required and N is the total. number of steps. For the problem under consideration, M 10 20 1 Ma.

The first time step is therefore 0.0026 Ma, and for this value, Ot 0.33 and therefore below the recommended maximum of 0.5 (equation 4.67). Thus the solution has been improved with virtually no increase in computational effort.

86

4.3.3 Numerical solution of problems involving latent heat and simulated convection.

In section 4.2, it was noted that the inclusion of terms involving latent heat and an effective conductivity to simulate convection, give rise to a non - linear diffusion equation because, in equation (4.49), c and K are both functions of temperature u, defined by : f e (c„,+ L/(T,,-TS ) ) 5`u `Ti

I co u < Ts U >TL.

f ko u < T (4.68) s k(u) k0 + p ( u—Ts) Ts.<. u

k, P (TL—Ts) u >T L

, where ko = thermal conductivity Co = specific heat p = .density L latent heat T. = liquidus temperature Ts= solidus temperature.

Two values of p are tried

= g I-0T,— T3 ) (4.69)

p = 4 (-r,- (4.70)

The most appropriate value of p for a given situation will depend upon a variety of factors including .the dimensions of the intrusion, its depth below the surface etc. The choice of a linear relationship is also conjectural, based upon the assumption that convection will be more effective when the magma is 100% liquid at temperature , and its role will become less important as crystallisation proceeds until at the solidus temperature Ts , heat transfer is by conduction only and K takes the value Ko.

When the ADI method is applied directly to the non-linear form of- equation (4.49) a system of non-linear equations is created because the

87 coefficients of the matrices H and V in equations (4.65) and (4.66) are functions of temperature. There is no convenient solution to this set of equations. A technique for retaining the linearity of. the equations, and therefore the advantages of the ADI method, which involves some loss of accuracy and stability, is the predictor-corrector method (Ames, 1965). The method depends upon an ability to guess temperatures at an advanced time level (the predictor step) and to use these estimated values to update the coefficients of matrices C, V and H and then obtain more accurate temperatures at the advanced time level using equations (4.65) and (4.66) (the corrector steps). Several forms of predictor can be applied to obtain estimates of temperature at the required time level (Remson et al, 1971), and the one chosen for thiS. study employs an explicit finite difference formula of the type outlined in section 4.3 (equation 4.58). The procedure for advancing the solution from time levels n to n+1 (where level n corresponds to time t = n At) by an ADI method incorporating an explicit predictor, can be represented as four stages.

Let Cn, Vn and Hn represent the matrices C,V and H of (4.65) and (4.66) at time level n. Then the four stages are as follows :

1. Predictor for first ADI stage

Temperatures at level n +14 vv1 j(l+1,4) , are calculated from the following expression (c.f. 4.58)

wij(n+14) II +41 Cl Hn+ Vn)) wc(n) + At C'S (4.71) 4 4 n The values wij ni14) are then used to compute the coefficients ofCn+x)yi+41Fin+x•

2. Corrector for first ADI stage

, 41 The intermediate temperatures w.• are

calculated from equation (4.65) using the matrices

and H The values of W. Cn+1/4_ Vn+ % n+ 11°14 •

88

are then used to compute the coefficients of

Cic V and Hn+1_

3. Predictor for second ADI stage

An estimate of temperatures at n+3/4 is obtained

using the explicit predictor formula as follows :

n+3/4 ) = I 41_ Cl(-; + (Hn+1-)lwri (n )16_ 1:14.-1 S (4.72)

The coefficients of Cri 4 and +4, Hn+%

are then calculated using n )

4. Corrector for second ADI stage

Temperatures at the advanced time level n +1

are calculated using Cn+3,4 , V11-0,4 and H n+ 314

in (4.66). The coefficients of Cnil

and F111+1 are computed using viii(n+1)

preparation for the advancment of the solution from r1+1 to n +2 -

This modified ADI method for non-linear equations is not unconditionally stable, and a mathematical analysis of stability conditions in such a system is extremely complicated. The validity of the solutions for the cooling intrusion problem was checked by examining the dependence of the solution on the value of time step No significant change in the solution was observed for Lt< 0.025 Ma and this value was used throughout. Future studies should examine the applicability of variable time steps, of the form of equation (4.67) to this problem.

89 4.3.4 Numerical solution of steady-state problems.

In this section, various methods are. developed for obtaining an approximate solution to the following equations :

(k12-'1) + -s ax )x (x,yeR) (4,73)

111 01( du = 0 (x,y ES) --1 + \ (4.50)

, where the domain g, boundary8 and variables K, 96 70 are those used in the time - dependent case (section 4.3.1). Discretization may proceed as before, the continuous temperature variable u (x,y) being approximated at grid points by a vector uij , and the differential operators in (4.73) being represented as a square matrix A = H V as in equation (4.55) and appendix 4. A finite difference approximation to (4.73) and (4.50) can then be written :

A u.. = fij 3_3 (4.74) , where fij is a vector incorporating the grid values of S in (4.73) and an approxiMation of the boundary conditions given in (4.50).

Two possible types of solution exist for the system of equations corresponding to (4.74), a direct type and an iterative type. The I direct method of solution is particularly efficient when the matrix A is tridiagonal (appendix 5) and may be useful when A has a block tridiagonal form (Remson et al, 1971). However, an iterative method is generally simpler when A is not tridiagonal and an efficient type of iterative scheme, the successive - over - relaxation (SOR) method has been adopted for this study.

Development of an iterative scheme begins by rewriting equation (4.74) in the form :

(L D U) Uij = fij (4.75)

where L is a strictly lower triangular matrix 'D is a diagonal matrix U is a strictly upper triangular matrix.

90

Equation (4.75) may then be rearranged in a number of ways to enable an initial guess VDij of the vectorial to be continuously improved by successive applications of the equation.

The Jacobi method proceeds by rearranging (4.75) as follows :

D Uij = (L + U) Uij fij (4.76)

so that an initial guess Vij 'on the left hand side of (4.76) may be systematically improved by a series of iterations of the form

m+1 1 1 Vij D (L + U) Vij + D- fij

, where m = 0,1,2 M-1 .

The value of M is chosen so that the absolute difference between the last-two terms in the series, Vij M-1. and Vij M is sufficiently small, i.e.

i I NAM. -vi -1 < E ij (4.77)

for some E_>0 .

1 The matrix J = D (L + U) is known as the Jacobi Iteration Matrix. Another useful rearrangement of (4.77) gives/rise to the Gauss - Seidel Iteration Matrix G, defined as follows :

-1 - (D + L) U (4.78)

, which produces a more rapid convergence of the series Vij m

A further improvement in efficiency can be achieved by introduCing an

SOR factor W, (1 < w.< 2) so that (4.75) may be written in the following iterative form :

11 m 1 I -1 in+1 - (W 1D + L)--1 1- 14 D Va..) + D + Vij t U + fij

The value of W which gives the most efficient solution (Wopt) is that value which minimises the spectral radius of the matrix S(W)

(Remson et al, 1971) defined by :

-i r S(W). -(W + L) L U + (1-1r1) D ( 4.80 )

A theoretical determination of Wopt for the problems dealt with below is beyond the scope of this study because of complications which arise when variable coefficients K(x,y) and an irregular mesh are required. A suitable value of W for a particular problem can be found by investigating the convergence of the solution for a sensible range of values of W, and choosing that value for which convergence is most rapid (Mitchell, 1969).

In section 4.4 the SOR method is applied to a two-dimensional steady

-state intrusion problem in which a rectangular zone within the crust is held at its liquidus temperature (1473K). In figure 4-7 the number of iterations required to achieve convergence for this problem is plotted against a range of values of W, the SOR factor. Convergence in this case was considered to be complete for E. = 0.1 in equation

(4.77). It can be seen that changes in W of 0.1 can greatly reduce the computational ,effort required to reach the solution. Similar plots for models of more complex shapes (figure 7-5) are shown in figure 7-7.

In this case E in (4.77) was fixed at 0.01.

4.4 Study of the surface temperature gradient disturbance above

a cooling igneous intrusion.

The mathematical techniques described in section 4.3 can be used to study the geothermal regime in the vicinity of a cooling igneous intrusion, taking into account the effects of latent heat of solidification and convection within the magma, and partial melting in

92 240

200_

160

120

0 1.4 15 1•G 1.7 18 1.9 2.0 w

Figure 4-7. Number of iterations re7iuired for convergence of the finite difference solution to the steady-state intrusion problem, for a range of values of the SOR factor ('d)

93 the surrounding country rock.

Geophysical evidence for the existence of a major igneous intrusion beneath the floor of the Gregory rift valley will be outlined in section 5.3. A model similar to that suggested by Searle (1970) has been chosen for this study to demonstrate the effects of latent -heat and convection within a large intrusion using the mathematical techniques described above. The model shape has been approximated as a simple rectangle, since although the shapes suggested by Searle (1970) to satisfy the gravity data are more complex (figure 7-3.), lateral density variations of the graben infill could account for much of the fine structure in the positive Bouguer anomaly, upon which Searle's interpretation was based. The analytical solution for a cooling rectangular intrusion in a geothermal gradient (section 4.1) can therefore be applied, and numerical and analytical solutions are compared below. The dimensions assigned to the intrusion are as follows :

width = 21.00km

height = 20.70km

depth below surface = 2.75km

Other possible models are considered in section 7.3. The initial temperature of the magma was set at 1473 K. For the numerical solution, a rectangular area, bounded on the top by the surface of the earth and on the right-hand side by the vertical axis of symmetry through the intrusion, was covered by a 961 Point rectangular mesh of variable spacing (appendix 11). Fixed temperature boundary conditions were set on the horizontal sides of the rectangle, and zero horizontal flux on the vertical sides. The zero horizontal flux condition was applicable because one of the vertical sides was far removed from the anomaly, and the other was an axis of symmetry.

914 4.4.1 The effect of latent heat.

Two methods of incorporating the effects of the latent heat of solidification into solutions of the heat equation in the vicinity of a cooling intrusion were described in section 4'.2. The simplest approach, in which all the heat of solidification is considered'to be released immediately after intrusion, employs the analytical solution for a cooling rectangular body in a geothermal gradient (section 4.1) with an excess initial temperature,

Tx = L /C 400 K where L = latent heat of fusion (419 kJ. Kg -1 _ _ C = specific heat (1.05 kJ. Kg 1. K 1)

The second, more exact representation, uses the modified ADI method described in section 4.3.4 to solve the non-linear equation (4.49).

This technique can also handle cooling intrusions of complex shape and structures with varying thermal conductivities without difficulty.

For the sake of brevity, the excess - temperature analytical method will be referred to below as ETA and the excess - specific heat numerical method as ESHN. The ETA and ESHN solutions are shown, along with the analytical solution in which latent heat is.ignored.(NOLH) figures 4.8 (a), (b), (c) and (d). for times 0.125 Ma, 0.250 Ma,1.0 Ma and 2.5 Ma after emplacement. In the figures, (G - GO) is the anomalous surface temperature gradient, and X is the horizontal distance from the intrusion axis in kilometres. The four values of time were selected to illustrate the following :

1. NOLH is always less than ETA and ESHN

2. ETA produces higher gradients than ESHN during the early stages

of cooling (figures 4.8 (a) and (b)).

3. ESHN and ETA are roughly equivalent, for this particular

problem, at 1 Ma (figure 4.8 (C) )

95

(b) T= .250 MA

4-

4.00 8.00 12.00 16.00 20.00 X( KM )

(a) T= .125 MR

+ ••

+\, ETA +%, + + + ESHN NOLH

4,

0.00 4.00 8.00 12.00 16.00 20.00 X( KM )

Figure 4-8 Comparison of solutions for temperature gradient anomalies above the postulated rift intrusion for 4 values of time (T) after emplacement (magma convection ignored). NOLH = analytical solution (latent heat ignored) ETA = analytical solution with 400K excess temperature ESHN.= numerical solution including latent heat(419 J/kg ) 96 kg

(d) T=2.500 MR + M_ ESHN

• ▪ • ETA 00. • _ C) cb

Cb NOLH

• _

0 0 clb.00 4.00 8.00 12.00 16.00 20 .00 X( KM )

0 0 (c) T=1.000 MA 0

- +- O o

CO co 0 +-. 0_ -P-. ÷-,

O 0 1 1 0.00 4.00 81.00 12.06 116 .00 20.00. X( KM )

Figure 4-8 (continued)

97 4. ESHN eventually exceeds ETA (figure 4.8 (d) ).

The effects of partial melting and latent heat exchange within the country rock surrounding the intrusion, were investigated by the ESHN technique using the parameters specified in section 4.2. The surface gradient anomaly was not changed significantly by including the effects of partial melting in the country rock, although temperatures in the vicinity of the intrusion were modified.

4.4.2 Simulated convection in the intrusion.

The concept of an enhanced effective thermal conductivity within liquid magma as a useful simulation of the effects of convection for studies in which the region of interest is outside a cooling intrusion, was introduced in section 4.2, and the proposed temperature dependence of conductivity was expressed in equation (4.68). Two alternative values for the parameter p in (4.68) were suggested in equations (4.69) and (4.70) and solutions for both cases (figures 4.9 and 4.10) incorporating also an increased specific heat within the melting range t.6 approximate latent heat, were computed by the modified ADI technique.

Curves showing the NOLH and ETA solutions are included in the figures.

When p was specified by equation (4.70), so that a maximtm effective thermal conductivity of 5 Kb was achieved at the liquidus temperature, a particularly interesting result emerged (figure 4-9) ; the complicated numerical treatment incorporating latent heat and simulated convection, yielded the same surface temperature gradients as the ETA solution over the time range investigated (0 to 3.5 Ma after intrusion). The solutions for 0.125 Ma, 0.5 Ma, 2.5 Ma and 3.5 Ma are shown in figures

4-9 (a), (b), (c) and (a) respectively.

98

(b) T= .500 MR +.

O O

- M

O 0.00 4.00 8.00 12.00 .00 20.00 X ( )

(a) T= .125 MR

-----ETA NOLH ++++Kmax =5K0 O cn

O O O

O

0.00 4.00 8.00 12.00 15.00 20.00 X ( ) Figure 4-9 Comparison of solutions for temperature gradient anomalies above the postulated rigt intrusion for 4 values of time after emplacement (magma convection included: K = 5K ) (see text) max o

99

O (d) T=3.500 MR

C\I

O O , CD-- CD .4% CO ` O % 2IN,

O O 0.00 4.00 8.00 12.00 16.00 20.00 X( KM)

0 (C) T=2.500 MR rn_

O 0NI _ CD cn O Imo_

O O

ab.00 4.00 8.00 12.00 16.00 20.00 X( KM )

'figure 4-9 (continued)

100 ( b) T= .250 MR

O O CD O CID Cb cn O O cp co

0.00 4.00 8.00 12.00 16.00 20.00 X( KM)

(a) CIO I .125 MR

+ ------ETA ------NOLH + +++ Kmax =10K0

.00 4.00 8.00 12.00 16.00 20.00 X ( KM ) .Figure 4-10 Comparison of solutions for temperature gr"adient anomalies above the postulated rift intrusion for A values of time after emplacement (magma convection included: Kmax = 10 K )(see text)

101 (d) T=2.500 MR + +

+

4.00 8.00 12.00 16 .00 210.00 X( KM)

(c) .500 MR

51;

1 0.00 4.00 8.00 12.00 16.00 210.00 X( KM)

Figure 4-10 (continued)

102 When equation (4.69) was used to specify p in (4.68),

(K = 10 Ko at T = TL) surface gradients were higher during the early stages of cooling (figure 4-10a and b) but were overtaken by the

ETA solution at around 0.5 Ma. Inclusion of the effects of melting within surrounding country rock did not significantly change the calculated surface gradient anomaly.

In figure 4-11 a and b, the maximum surface gradients produced by the various types of solution described above are displayed for comparison. In figure 4-12 the SOR finite difference solution

(section 4.3) for the surface gradient anomaly above an intrusion which is being maintained at its liquidus temperature, and around which the temperature field is steady state, is compared with the maximum values from a NOLH and an ETA solution.

4.4.3 Summary

The important points to arise from this study are the following.

1. The latent heat of solidification made a significant

contribution to the surface temperature gradient anomaly

above the cooling intrusion.

2. Convection within the magma caused the intrusion to cool more

quickly, and increased the maximum surface gradient anomaly.

3. A simple analytical solution for the surface gradient anomaly,

in which all the latent heat of solidification was released at

the time of intrusion, was a good approximation to the

combined effects of latent heat and convection within a

rectangular intrusion.

103 ETA

1 4.00 8.00 12.00 116.00 210•00 X ( KM )

(a)

4.00 8.00 12.00 16.00 20.00 X ( KM )

Figure 4-11 Comparison of solutions for maximum values of temperature gradient attained above rift intrusion (see text).

104 O O

O _

CD NE • o •Steady-state

z ETA 1.1_1

0 • NOLH CC Da ..... s...... N. c\J .... ■ . N ■ N C.b ■ ....

• CD0 . 0 0 4.00 8.00 12.00 16.00 20.00 X(KM )

Figure 4-12 Comparison of steady-state and maximum ETA and NOLH solutions for temperature gradient above rift intrusion (see text). 4. The steady state gradient anomaly above a rectangular

intrusion maintained at its liquidus temperature, was

considerably greater than the maximum values produced above

the same body cooling from its liquidus temperature even

when the increases produced by latent heat and convection

were, included.

5. Partial melting, accompanied by latent heat exchange

within the country rock surrounding the intrusion had an

insignificant effect on the surface temperature gradient

anomaly.

Several heat flow measurements were made in the floor of the

Gregory rift near the equator, where a positive Bouguer gravity anomaly, and above - normal crustal seismic velocities have been

attributed to the presence of a major dyke-shaped basic igneous intrusion. The usefulness of the analytical solution (point 3,

above) is exploited in section 7.3, where an attempt is made to derive the time of intrusion by comparison of model results with measured heat flows.

106 CHAPTER 5

GEOLOGY AND GEOPHYSICS OF KENYA.

The Republic of Kenya, occupying an area of approximately 360,000 square kilometres bisected by the equator, is a land of diverse physical features. A coastal plain faces the Indian Ocean in the south-east, desert or semi desert covers much of northern Kenya, and high plateaus and mountains constitute the 1000 kilometre wide dome of western and central Kenya. This dome is bisected by the East African Rift Valley, which can be traced fron Mozambique to Ethiopia where it appears to link with the worldwide oceanic rift system. The rift valley is well developed in Kenya and its origin and development are of particular interest to this study.

5.1 Background to rifting.

The elliptical dome of central Kenya is superimposed upon another major topographic feature, the East African Plateau.. Holmes (1965) drew attention to the persistence of epeirogenic movements in Africa since the Paleozoic era, and the resulting development of the basin and swell features which now form distinctive physiographic provinces in the continent. The East African Plateau is one such swell.

An association of swells, rift valleys and alkaline volcanism has been observed in many localities on different continents. Le Bas (1971) has suggested that crustal swells are the primary features of this association, and that alkaline volcanism and rift faulting are secondary, influenced in many cases by structural grain in basement rocks. Burke and Whiteman (1973) attempted to explain this association as an evolutionary sequence in which uplift is followed by alkaline volcanism and the formation of three rifts in a triple 'junction on the crest of a swell. In their scheme one or more of these rifts may subsequently develop into active spreading axes, generating oceanic - type crust and initiating breakup of a continent.

McConnell, (1972) emphasised the role of structures within ancient basement rocks in determining the strike and location of Cenozoic rift

107 valleys. He introduded the term "perennial lineaments" to describe those relatively narrow mobile zones in the crust of eastern Africa which have been reactivated by successive phases of tectonothermal activity since the early Precambrian. The perennial lineaments surround ancient cratons which have been virtually undisturbed by tectonic activity for over 2500 Ma ,Clifford (1970) distinguished eight such ancient nuclei in Africa, now incorporated in three larger cratons which have been unaffected by orogenesis for 1100 Ma. The perennial lineament which has influenced rifting in eastern Africa is known as the Mozambique Belt. It is orientated north - south, extending from Ethiopia to Rhodesia, and was last affected by a major orogeny 450 Ma to 900' Ma ago (Cahen and Snelling, 1966). Shackleton (1967) considered that the Mozambique Belt had undergone several previous tectonothermal episodes in the Precambrian.

Unfortunately ancient structures in the Precambrian basement of Kenya have been largely obscured by a large volume of.Cenozoic volcanics associated with the rift formation, and their. relationship to the major graben faults can not be examined in detail.

5.2 Geology of Kenya.

5.2.1 A geological map of Kenya.

An outline of the geology is given in figure 5-1. The oldest rocks. in Kenya are part of the Dodoma - Nyanza nucleus (Clifford, 1970), one of the ancient stable nuclei of Africa which covers parts of Kenya, Uganda and Tanzania. The rocks consist of metamorphosed lavas and sediments and are known locally in Kenya as the Nyanzian System (the oldest) and the Kavirondian System. Both are considered to be older than 2,550 Ma. They were extensively intruded by granites during the Precambrian.

East of the craton are the. metamorphic rocks of the Mozambique Belt, known locally as the Basement System; now obscured in the vicinity of the rift by Cenozoic volcanics and sediments. The. Basement System dips steeply beneath Cenozoic. and Mesozoic sediments which fill the Lamu Embayment east of longitude 39.5°E.

In south - east Kenya, late Paleozoic - early Mesozoic sediments

.108 Ethiopia

Uganda

• • • • Somalia "=""----~."'>-: LAM U · • • • EMBAYMENT . . . Lake . . . . . Victoria ......

KEY . . rvolca ni cs fenozolcl sedi ments

~7'"'7'1 Mesozoic ~~ Paleozoic p.--..""'---'1 . r Mozambique Belt t::;:;:~ Precambnan· 1 t-+-HH-HJ ·l Dodoma-Nyanza Nuc eus

Figure 5-1 Geological map of Kenya.

109 accumulated in a trough along the eastern margin of the continent. These are contemporaneous with the extensive Karroo Series of southern Africa. Sediments of similar age also outcrop in north - east Kenya.

Much of Kenya is covered by Cenozoic volcanics which are intimately associated in both location and time of eruption with the development of the Gregory rift valley in Kenya. Activity commenced in the Neogene and continued into the Quaternary. The total volume of volcanic products exceeds 100,000 cubic kilometres (Baker et al, 1971). Broad variations in the composition of lavas, and in the intensity of volcanic activity, are a potential source of information on the thermal history of the upper mantle since the Miocene (section 7.4).

5.2.2 Tectonics.

The tectonic history of Kenya since the Mesozoic has been dominated by vertical movements. These include several phases of uplift of the East African Plateau and the Kenya dome, downwarping and graben formation along the rift valley, and downwarping of the Lamu Embayment (figure 5-1). Saggerson and Baker (1965) identified three major uplift phases in central Kenya as upper-Cretaceous, Miocene and Pliocene from a study of erosion surfaces in eastern Kenya. The Pliocene uplift had the greatest amplitude, reaching 1.5 km in central Kenya.

The rift valley began to form•during the Miocene by downwarping in the north, and progressed southwards in the early Pliocene as an asymmetric graben, faulted on its western margin (Baker and Wohlenberg, 1971). A late Pliocene to mid-Pliocene phase of faulting produced a complex graben 60 to 70 km wide, with swarms of closely spaced minor faults on its floor.

• The Kavirondo Rift developed perpendicular to the Gregory rift during the early Pliocene (Baker et al, 1972), intersecting it near the equator. The graben is much narrower than the Gregory rift (15 to 25 km in width) and did not develop swarms of minor faults on its, floor. The complex pattern of faulting in central Kenya is illustrated in figure 5-2.

Uplift and rifting in western and central Kenya were accompanied by downwarping in eastern Kenya. The results of a petroleum exploration

110 Figure 5-2 (a)Fault pattern in central Kenya. (b)Cross-section through XY in (a). (from Baker et a1,1972).

111 programme in the Lamu Embayment (figure 5-1) have confirmed that it is a deep sedimentary basin containing up to 10 km of sediment. The western margin of the basin is probably faulted•, but it is now obscured by Quaternary sediments. Subsurface exploration in the Lamu Embayment revealed several horst and graben type features beneath a sedimentary cover which appear to follow NNW - SSE and NNE - SSW trends probably associated with the rift structures to the west (Walters and Linton, 1973). These"basement highs" appear to have formed towards the end of the Cretaceous period.

Subsidence began in the basin during the late Paleozoic when a trough formed along the eastern margin of the continent in which sediments contemporaneous with the Karroo series of southern Africa are now preserved (figure 5-1). Downwarping of the rest of the Lamu Embayment began in the Jurassic and a major depositional phase continued through- out the Cretaceous. The rise of the Kenya dome to the west in the Miocene was accompanied by further downwarping and sedimentation in the Embayment and a hinge line can be identified along its western margin. Downfaulting of the southern coastal strip in which Cenozoic sediments have accumulated was contemporaneous with the final mid-Pliocene uplift of the Kenya dome.

Details of the subsurface structure of the Lamu Embayment which guide the interpretation of zones of unexpectedly high heat flow are described in section 6.5.

5.2.3 Volcanicity.

Most volcanic eruptions during the Cenozoic took place within the rift valley or along its margins. Important exceptions include Quaternary multicentre basalt fields east of the rift and several major central volcanoes on the rift flanks which are typically sited in regions of crustal flexure or other tectonic lineaments. Baker et al (1971) and Logatchev et al (1972) described the results of potassium - argon isotope age determinations on samples from major volcanic provinces in the vicinity of the rift valley and have distinguished variations in lava types in both space and time.. The main volcanic centres appear to have migrated eastward and compositional changes in their products seem to be related to stages in the development of the tectonic regime. Phases of phonolitic and trachytic volcanism in the'late Miocene. and again in the late Pliocene to mid-Pleistocene'are associated with

112 enhanced tectonic activity in the form of updoming and rift faulting. - Basaltic products however, were common during periods of relative tectonic quiesience. Bailey (1973) attributed these compositional variations to magma generation at different levels in the crust and upper mantle in response to the rise and fall of geoisotherms which followed major tectonic movements. A generalised sequence of the major volcanic and tectonic events associated with the development of the Kenya rift valley is shown in Table 5-1.

5.3 Geophysical studies in Kenya.

Earthquake and explosion seismology, gravity and geomagnetic variation studies have revealed anomalous structures in the crust and upper mantle in the vicinity of the Kenya rift valley. Interpretation of the results is confused by the complexity of the upper crust in this region and more detailed and more extensive surveys are required to resolve some of the existing ambiguities. Previous heat flow studies in Kenya by Morgan (1973) and Evans (1975) are discussed in chapter 6.

5.3.1 Seismology.

Gumper and Pomeroy (1970) used. surface wave dispersion measurements ,- on seismograms from observatories throughout Africa to compute a model of the average crust and upper mantle in stable parts of the continent. They noted that surface and body waves which had crossed the rift region generally did not conform to the predictions of their model. More detailed studies by other workers have subsequently helped to delineate the rift anomaly.

Knopoff and Schlue (1972) and Long et al (1972) studied the dispersion of Rayleigh waves on the paths between Addis Ababa and Nairobi to determine the average vertical distribution of seismic shear wave velocity over that region. The actual seismic waves sampled in these studies had passed some distance east of the rift valley, but across the- uplifted domes of Ethiopia and Kenya. - The dispersion curve indicates anomalously low seismic shear wave velocities at depths of 50 to 200 km in the upper mantle. A lid of normal velocity mantle may 'exist beneath the crust but its presence cannot be confirmed by the available evidence. Gumper and Pomeroy (1970) had reported that the S °S. was N phase from waves crossing the rift north of latitude 10

113 Volcanic Events Tectonic Events

Extensive fissure eruptions of Major faulting on W. margin, local

phonolite over a wide area faulting on E. Monoclinal down-flexing of rift floor Upper

and Alkali basalts and tuffs Gentle down - warping of rift Middle erupted from many small centres zone

Miocene Eruption of nephelinites, Up-doming of W - central part

melilites, emplacement of of Kenya alkaline igneous rocks and carbonatites in Kavirondo

Deposition of Miocene sediments. Stability.

Table 5-1 Generalised sequence of major volcanic and,tectOnic events associated with the formation of the Kenya rift. (from Bailey, 1973)

(continued on next page) Volcanic Events Tectonic Events

Recent Minor basaltic cones Minor renewals of movement in rift' Comendite and rhyolite plugs Grid faults - collapse and and breaking of rift floor Pleistocene Phonolitic - trachytic C'alderas in Graben faulting - early grid rift. Mount Kenya eruptions faults in rift floor

Trachytic ignim.brites and flood Down flexing of rift floor, trachytes in rift floor especially in N sector

Pliocene Basalts and basanites on rift floor

Nephelinite - trachyte - Uplift of rift shoulders. pyroclastic volcanoes on rift' floor

Table 5-1 (continued) attenuated or completely absent.

Station anomalies (P wave delay times for teleseismic events) at both Addis Ababa and Nairobi were found to be large and positive relative to observatories in stable parts of the continent, (Lilwall and Douglas, 1970) and this has been attributed to the existence of zones of low compressional velocity material in the upper mantle beneath the two stations, presumably associated with the nearby rift structures (Fairhead and Girdler, 1971; Long et al, 1972). However analyses of the spectral transfer ratios of long period body waves recorded at Addis Ababa, Nairobi and Lwiro in the western rift (which has a small positive station anomaly) led Bonjer et al (1970) and Mueller and Bonjer (1973) to postulate a crustal structure which could account for the relative station anomaly between Nairobi and Lwiro. A • . • pOsitiveanomaly, preiutably withitS.-origins%inthenpiper mantle, still - remained at Addis Ababa.--

More detailed studies of the crust and upper mantle in Kenya have ' been attempted by Long et al (1972) using a temporary L - shaped seismic array station on the western flank of the Gregory rift to receive short period body waves from both regional and teleseismic events. The results suggest that normal crust and upper mantle (according to Gumper and Pomeroy, 1970) may exist only 50 km west of the rift axis, and that anomalously low velocity upper mantle lies beneath a thin crust in the . vicinity of the rift. An azimuthal variation in the apparent velocities of teleseismic P waves was interpreted by Long et al as evidence for the existence of a wedge of low velocity mantle beneath the rift, thinning to the west away from the rift axis.

Griffiths et al (1971) performed a seismic refraction experiment on the Gfegory rift floor, using explosives in Lake Rudolph and Lake Hannington. However the quality of the data was such that only a crude one-dimensional approximation to the structure could be justified. The model features a crust approximately 20 km thick, of relatively high seismic velocity, underlain by relatively low seismic velocity upper mantle. There was also some evidence for a basic intrusion beneath the rift floor, reaching within 10 km of the surface. Future experiments in explosion seismology should resolve some of the existing .ambiguities.

116 5.3.2 Gravity.

The gravity field in the rift region was firSt studied by Bullard (1936) and has recently been measured in considerable. detail (Khan and Mansfield, 1971) An east - west profile along latitude 3°S from the Congo Basin to the Indian Ocean showed a 1000 km wide negative Bouguer -2 anomaly c.- 1500 pms in amplitude, coincident with the East African Plateau (Sowerbutts, 1969). In an analysis of the regional gravity field in the vicinity of the Eastern rift in Kenya and Tanzania, Fairhead (1972) described an oval-shaped negative Bouguer anomaly centred on 36.0°E, 0.5°S, c.350 km wide with an amplitude of c.- 500 -2 puns relative to the broad East African Plateau anomaly.

On the floor of the central part of the Gregory rift, McCall (1967) and Searle (1970) discovered a smaller positive Bouguer anomaly, 40 to 2 80 km wide and 300 to 600 puns in amplitude relative to the oval-shaped minimum. The axis of the positive anomaly extended along the rift floor o o from latitude 1.25 S to 0.25 N. More recent investigations suggest that it may continue north into Ethiopia (Khan and Mansfield, 1971; Swain, pers comm., 1975) and south as far as Lake Magadi (Darracott et al, 1972). The Kavirondo rift is marked by a 50 km wide negative Bouguer anomaly 500 punsr? in amplitude, but no axial positive anomaly.

Sowerbutts (1969) interpreted the 1000 km wide negative Bouguer anomaly as being due to a slab of low density material in the upper mantle, underlying the East African Plateau. Girdler et al (1969) proposed a model in which the low density layer was actually part of the asthenosphere, filling a gap in the thinned lithospheric plate. Further local thinning of the lithosphere beneath the rift zone then accounted for the 350 km wide negative anomaly over the rift.

Searle (1970), Baker and Wohlenberg (1971) and Fairhead (1972) used this concept as a basis for structural models which are compatible with the detailed pattern revealed in subsequent surveys. They interpreted the positive Bouguer anomaly over the rift as due to a high density basic intrusion in the crust. Interpretation of this anomaly is confused because of the uncertain contribution of thick deposits of low density volcanics which cover the area. A model proposed by Baker and Wohlenberg (1971) is shown in figure 5-3a. Guided by the results of the seismic refraction experiment of Griffiths et al (1971), / - Khan and Mansfield (1971) postulated a lozenge-shaped zone of anomalous

•

117

(1.0°N)

2.90 Mg.m-3

50: 3.4o Mg.m3 (b)

E

zone 1 L1S.m- (C ) 50_ A .zone 0.05S.m no:

50 100 km

Figure 5-3 Three models of anomalous structures beneath the Gregor:: rift which fit gravity(a and b) and geomagnetic variation (c) data. (a) density model(Baker and Wohlenberg,1971) (b)density model(Khan and Mansfield,1971) (c)electrical conductivity model(Banks and Ottey,1974)

118... low density mantle beneath the rift, rising into the crust within 20 km of the surface, and dropping into the mantle to approximately 60 kilometres (figure 5-3b).

The models of figure (5-3a and b) were constructed to fit gravity profiles over a section of the Gregory rift near the equator. The gravity field also varies significantly along the strike of the rift structures. Fairhead (1972) pointed out an apparent offset of the regional anomaly north of the equator, and Khan and Mansfield (1971) noted that the northward extension of the positive anomaly broadens to approximately 100 km.

Thus no unique model can be derived from the seismic and gravity data, even in two dimensions. However.; the presence of a narrow zone of positive density contrast in the crust underlain by a broader zone of negative density contrast in the upper mantle, is a common feature of all the models suggested for the central section of the Gregory rift. A zone of partial melt in the upper mantle could explain the low density, low velocity zone.

5.3.3 Geomagnetic variation studies.

Banks and Ottey- (1974) described a geomagnetic deep sounding experiment, in which the variations of each of three components of the geomagnetic field were recorded continuously at six sites along a profile across the Gregory rift near the equator. During magnetic storms, differences in the vertical components were observed at the six sites, which were attributed to the effects of electric current travelling in a north - south conductive zone beneath the rift. The essential features of a model which will explain the geomagnetic variation anomalies are as follows (Banks and Ottey, 1974) : 1. The western edge of a conducting body lies beneath the western rift scarp. 2. A conductor extends eastwards beyond the eastern rift scarp. .3. The top of a conductor beneath the rift is at a depth of not more than a few tens of kilometres.

It was not possible to construct a unique model from the geomagnetic variation data, so Banks and Ottey chose to make the top of the conducting bodies correspond to the depth at which seismic low velocity zones had

119 been encountered in the upper mantle (Griffiths, 1972; Long et al 1973). A model comprising two conductive rectangular prisms, one at 20 km depth beneath the rift and the other SO km below the surface 100 km to the east, (figure 5-3c) could produce the observed geomagnetic variation anomalies. The electrical conductivity of olivines increases with temperature,.but Banks and Ottey considered -1 that the high conductivities required in the rift model, 0.1 S.m in 1 zone 1 and 0.05 S.m in zone 2 (figure 5-3c) could only be produced by temperatures exceeding 1473 K, and probably only within regions where melting had occured. Tolland and Strens (1972) had shown that the bulk electrical conductivity of a partial melt, in which the liquid constituent had a relatively high conductivity, would increase dramatically when the proportion of interstitial fluid exceeded a critical value, possibly as low as 5%. This situation could occur in the upper mantle where small amounts of melting would produce a highly conducting (c. 1 S.m-1 at 1473 K) basaltic fluid. Banks and Ottey therefore interpreted zones 1 and 2 as parts of the mantle where the concentration of melt exceeded the critical value.

To illustrate the nonuniqueness of models based on the geomagnetic variation data, they considered a second model in which zone 1 of figure 5-3c was replaced by a 5 km thick conducting slab (conductivity 1 0.2 S.m ) immediately beneath the rift floor. This model has some geological foundation since the graben infill contains volcanics and sediments which might be expected to have a relatively high electrical conductivity by virtue of their water content.

The depth to the top of the conducting body can therefore not be well defined by the geomagnetic variation data. The decision to . place the top of the conducting zones at the depth of the low seismic velocity, low density zone resulted in a model about which Banks and Ottey (1974) commented: U... the lateral and vertical extent of the high conductivity zone beneath the rift valley appears to be much less than that of the low velocity low density zone detected by seismic and gravity observations...."

They went on to suggest that the greater width of the low density, low seismic velocity zone could simply reflect the sensitivity of density and seismic velocity to small degrees of melting, below the critical value at which a large increase in electrical conductivity

120

would occur. If this is indeed the case, the tops of the conducting zones may be expected to occur several kilometres below the top of the seismic low velocity zone. However, in view of the uncertainties involved in the modelling process,this refinement can probably not be justified. A geothermal model compatible with the model of figure 5-3c is discussed in section 7.4.

5.3.4 Seismicity.

The East African Rift System is known to be seismically active, characterised by shallow focus earthquakes. Epicentral determinations in East Africa have been carried out by Sykes and Landisman (1964), Wohlenberg (1969) and Fairhead and Girdler (1971). The main centres of activity are the Afar in Ethiopia, the northern part of the western rift, the southern end of the eastern rift in Tanzania, the Lake Malawi rift and its extension to the south and a zone south-west of Lake. Tanganyika. Fault plane solutions by Fairhead and Girdler (1971) and Maasha and Molnar (1972) show that the least compressive stress is oriented WNW - ESE in southern Africa and NE - SW in the Red Sea. and Gulf of Aden. Maasha and Molnar (1972) considered that an apparent paucity of earthquakes in the Gregory rift may have arisen because the sampling period was too short, or because deformation was occurring aseismically. Microearthquake surveys by Tobin et al (1969) and Molnar and Aggarwal (1971) showed an apparent increase in activity from north to south in the Gregory rift. No microearthquakes were recorded outside the rift except in a localised region near the southern boundary of the Kavirondo rift.

5.3.5 Magnetism.

The results of aeromagnetic surveys over two sections of the Kenya rift were described by Wohlenberg and Bhatt (1972). NW - SE trends were evident in both areas, but these did not correlate with surface features or other geophysical parameter variations. A strong magnetic low south of latitude 0.25oS appeared to be associated with the volcano Menengai.

5.3.6 Summary.

The available geophysical evidence indicates a local elevation in

121 the crust - mantle interface approximately 20 km below the rift floor underlain by upper mantle of anomalously low seismic velocity and density. Normal mantle underlies the crust 50 km west of the rift axis and the anomalous zone may take the form of a prism, thinning away from the rift. The crust beneath the rift axis appears to have been intruded by basic material, and there is some evidence for the existence of a major dyke - shaped basic intrusion, 10 to 20 km wide, rising to within 10 km (and perhaps even 3 km) of the surface beneath the rift floor. In section 7.3 heat flow data are used in a tentative calculation of the time of emplacement of such an intrusion. Geomag - netic variation anomalies can be accounted for by the presence of two electrically conductiire zones, one beneath the rift and the other to the east of it. An extensive region of partial melting could explain the gravity, seismic velocity and geomagnetic variation anomalies. A local elevation in the geoisotherms beneath the rift would be required to produce such a region, and the consequences of high temperatures in the upper mantle for surface heat flows are examined in section 7.4.

5.4 Other rift valleys.

The East African Rift System, of which the Gregory rift irs part, shows significant changes in character along its strike, in both surface expression and deep structure. The results of geological and geophysical studies in the Ethiopian rift, Western rift and Red Sea are discussed in this section together with the results of similar studies on two major rift valleys in Eurasia, the Rhinegraben and the Baikal rift. Heat flow studies from these regions are compared with the results of this study in section 7.1.

5.4.1 The Ethiopian rift.

North of the uplifted dome in Kenya, the rift valley is less distinct and continues beyond latitude 3°N as a zone of minor faulting until after en echelon.offsetstothe north-east, it joins the Ethiopian rift (figure 5-4) which is well developed on the Ethiopian plateau. The development of the Ethiopian rift was described by Baker et al (1972). During the late Eocene, parts of Ethiopia and south-western Arabia were uplifted into a dome structure, and the site of the present Ethiopian rift was occupied by a shallow trough. Details of the structural' development of the rift in Ethiopia have been largely obscured by

122

Key /major faults oceans and lakes

ARABIAN PLATE

NUBIAN PLATE

• .-- oe,\ •.... \ - • k‘ °‘ • - • Afar,' ::. I/ 1 '■, i — ••••••••• .

••r d ( lz): SOMALI 'I PLATE ••• 42`. .:.•

et

.• c.? -••

Figure 5-4 Rift valleys in north-eastern Africa (from Baker et a1,1972)

123 volcanics, but there is evidence that this early trough contained a considerable depth of lava by the Pliocene, and that a major phase of uplift and graben - faulting occured in, the early Pleistocene (Baker et al, 1972). Uplift and rift formation in Kenya began later, during the Miocene (section 5.2). Comparative studies of the composi- tions of the basic volcanics of Kenya.and Ethiopia indicate a deeper origin for most Kenyan lavas (Baker et al, 1972).

Gravity measurements in the vicinity of the Ethiopian rift valley by Searle and Gouin (1972) revealed a negative Bouguer anomaly over the Ethiopian dome, similar to that in Kenya (section 5.3), and a 100 km -2 wide, 500 to 600 pm.s positive Bouguer anomaly over the floor of the rift. A narrower, 40 - 80 km wide, positive Bouguer anomaly had been observed on the floor of the Kenya rift (Searle, 1970), and interpreted as the result of high density basic intrusions in the crust beneath the rift (section 5.3).

Searle and Gouin (1972) also adopted this interpretation to explain the Ethiopian anomaly, where the postulated intrusion was required to be wider than its Kenyan counterpart.

. Searle and Gouin (1971) reported that seismograms recorded at Addis Ababa of earthquakes in the Red Sea, Gulf of Aden, Afar, and Western Rift indicated a broad region of low seismic velocity in the upper mantle beneath the Ethio-Arabian Swell, coinciding with the apparent low density upper mantle zone which had been postulated as the source of the broad negative Bouguer anomaly.

5.4.2 The western rift.

The development of the western branch of the East African Rift (figure 5-5) was summarised by Logatchev et al (1972). An early phase of rifting in the southern part has been dated as Cretaceous, but most of the modern rift structures are considered to have been formed during the Neogene and Quaternary. Logatchev et al distinguished two phases of Cenozoic activity; during the first, in the Miocene and lower Pliocene, preliminary subsidence occurred along the future rift axis and in the second, Pleistocene to Recent phase, faulting and graben formation took place. McConnell (1974) described the present structure - as a 1500 km long arc, comprising four straight segments which follow

124 Figure 5-5 Western and eastern branches of the East African Rift System(from King,1970). ,

125 ancient lineaments around the western border of the Tanzania Craton. He emphasised the close relationship between rift faulting and ancient lineaments in the Precambrian basement.

Volcanic activity which accompanied rifting, was minor in comparison with the vast outpourings of lava in Kenya and Ethiopia, and the isolated volcanic centres appear to be associated with changes in level of the graben floor. Logatcheet al (1972) also noted a northward migration of activity since the process began. Recent eruptions have produced unusual strongly undersaturated or. carbonatitic lavas.

The Western rift has been less extensively surveyed by geophysical methods than the eastern branch. Bouguer gravity profiles crossing the graben show a pronounced minimum over the rift (Sowerbutts, 1969; Wong and Von Herzen, 1974) which is at least partly due to low density sediments beneath the rift floor, but no axial positive anomaly has been observed. The seismicity of the area has been investigated 'recently by Wohlenberg (1969) and Maasha (1975) and fault plane solutions for some large earthquakes were computed by Fairhead and Girdler (1971) and Maasha and Molnar (1972). High seismicity is closely associated with rift faulting and the least compressive stress appears to be aligned ESE 7 WNW indicating that the rift is under tension.

The intersection of the Eastern and Western rifts in northern Tanzania has been the site of intense volcanism, where the Rungwe volcanic complex has been active since the Pliocene(King, 1970). South of the junction, the rift is well developed in Lake Malawi and older faults, active during the Mesozoic, continue southward to the Limpopo river (Sowerbutts, 1972).

5.4.3 The Red Sea.

In the Afar region of Ethiopia, the Eastern rift intersects fault systems associated with the mid-oceanic ridge extension into the Gulf of Aden and with an axial rift in the Red Sea (figure 5-4).

The geology of the Red Sea area has been recently summarised by Coleman (1974). The Red Sea occupies a 2000 km long depression, 180 km wide in the north and 360 km in the south. It has a deep axial trough underlain by volcanics of a type commonly associated with mid-oceanic

126 ridges. Associated with the axial trough are linear magnetic anomalies which can be correlated with geomagnetic polarity history, numerous shallow earthquakes and very high heat flow. The trough therefore has the characteristics of an active sea floor spreading axis, underlain by oceanic crust. Away from the axial trough, the central Structure is uncertain although recent evidence (Girdler and Styles, 1974) suggests that much of the main depression in the southern Red Sea is also underlain by oceanic crust.

McKenzie et al (1970) postulated that the Afar region of Ethiopia is a triple junction, the point of intersection of 3 plate boundaries separating Arabian, Mubian and Somalian plates (figure- 5-4). However the role of the Eastern rift as a plate boundary in this context is not well understood. Further studies of the geology and geophysics of the rift, including heat flow studies, are essential if its role within the theory of plate tectonics is, to be properly defined.

Structures similar to the rift valley of East Africa have been identified in other continents. The Rhinegraben in Europe and the Baikal Rift in the USSR are well known examples and a brief review of their tectonic histories and deep structures may provide some insight into the problems of rift formation and development.

5.4.4 The Rhinegraben.

The Rhinegraben is a well-developed segment of a rift system which crosses Western Europe from the North Sea to the Mediterranean (figure 5-6). It is a 300 km long graben contained between normal faults c 36 km apart and filled with sediment to a maximum depth of 3.3 km (Illies, 1972).

Sedimentation began in the middle Eocene, rift faulting was initiated in the upper Eocene, and the whole trough was downfaulted in the lower Oligocene (Illies, 1974). Tertiary faults often followed ancient lineaments in the Hercynian basement (Illies, 1972). Subsidence of the rift floor was accompanied by uplift of the shoulders, and the - graben now appears to bisect an uplifted dame, c.-190 km wide. Volcanic activity preceded and accompanied rifting, and the volcanic products were chemically similar to many Kenya lava types although much smaller volumes were involved.

127 KEY coast

Alpine orogenic zone

Figure 5-6 The tectonic setting of the. Rhinegraben(from Illies,197h)

128 Seismic refraction and reflection experiments in the Rhinegraben area have revealed two important structural features in the crust; an elevation of the crust - mantle boundary beneath the rift(Edel et al, 1975) and a seismic low velocity layer in the crust (Mueller et al, 1973). The crust - mantle boundary takes the form of an arch, 150 to 180 km across, rising from 30 km to within 25 km of the surface beneath the southern Rhinegraben. Outside the rift, the moho is clearly defined and underlain by normal mantle (seismic P - wave velocity 1 8.0 - 8.1 km.S ) but within the graben the moho is replaced by a 4 janthick. transition zone below 21 km depth, in which P - wave velocities -1 change continuously from 7.1 to 8.1 km.S after an abrupt increase from 6.3 to 7.1 km.S-1 at 21 km. Edel et al (1975) interpreted this transition zone as a region of crust - mantle interaction.

Evidence for a seismic low velocity layer in the upper crust was reviewed by Mueller et al (1973). A P - wave velocity reversal, from -1 1 6.0 km.S to 5.5 km.S , appears to occur at a depth of c.10 km. The low velocity layer is 5 to 10 km thick and underlain by an intermediate -1_ layer with P - velocities of 6.7 km.S to 6.9 km.S 1.

2 A negative Bouguer gravity anomaly of about -300)um.s is associated with the Rhinegraben (Mueller and Rybach, 1974). Low density sediments filling the graben.can account for most of the anomaly, but an offset. of the minimum relative to the axis of maximum sedimentation may be due to low density-intrusions in the basement derived from the crustal low velocity layer. The uplifted crust - mantle boundary should have prodUceda pronounced gravity high which has not been observed. The absence of a broad positive Bouguer anomaly has been used as evidence for a compensating mass deficiency at greater depths in the upper mantle (Fuchs, 1974).

A zone of anomalously high electrical conductivity (0.03 to 0.04 - 1 S.m ) at a depth of 25 km beneath the rift floor has been detected by geomagnetic variation (Winter, 1974) and magnetotelluric (Scheelke, 1974) studies. Scheelke estimated its thickness to be 20 - 30 km, and Winter,50 - 65 km.

A complicating factor which must be taken into account in a study ' of taphrogenesis in this region is the influence of the Alpine orogenic belt.(figure 5-4) which was active when the rift was being formed..

129 Left-lateral strike-slip movements have occured along the Rhinegraben faults, and fault plane solutions of earthquakes have demonstrated that the maximum horizontal compressive stress is orientated roughly NW-SE, oblique to the main rift trend and radial to the Alpine fold belt (Illies, 1974). However Illies considered that a stress field in which the maximum horizontal compressive stress was orientated NNE- SSW may have prevailed during Mesozoic and early Tertiary times so that the least compressive stress could have been perpendicular to the strike of the Rhinegraben during the initial stages of its formation.

5.4.5 The Baikal Rift.

The Baikal Rift (figure 5-7) in the USSR was described by Florensov (1969). Subsidence during the Oligocene and Miocene preceded rift faulting, and sediments accumulated in broad troughs before the grabens were formed in the Pliocene and Pleistocene. Rifting was accompanied by uplift of the surrounding area into a complex dome, 150 - 270 km ' across. The grabens which comprise the Baikal Rift System are typically asymmetric, with major faults on the north side, and contain up to 4 km of sedimentary infill. Volcanic eruptions preceded and accompanied rifting but activity faded in the Pleistocene.

Deep seismic sounding experiments (Puzyrev et al, 1973) detected a relatively minor decrease in crustal thickness beneath the rift zone, 1 and a broad region of anomalously low seismic velocity (7.6 - 7.8 km.S ) below the moho. A negative Bouguer gravity anomaly over the rift could not be fully accounted for by the infill of low density sediments, and Artemjev and Artyushkov,(1971) considered it might have its origin in a low density zone within the upper mantle. Geomagnetic variation observations indicated a sharp rise in electrical conductivity 80 km beneath Lake Baikal (Lubimova, 1969) and further investigations using the magnetotelluric.rnethod delineated two anomalously high conductivity layers, 20 - 40 km thick; one located 80 - 120 km beneath the Siberian Platform rising to within 10 to 20 km of the rift floor, and the other 200 - 250 km beneath the ancient platform rising to between 100 - 140 km depth beneath the rift (Lubimova et al, 1972).

'A quantitative interpretation of magnetic anomalies suggested a thinnirig Of the magnetised layer in the crust which could be interpreted as a rise in the Curie point isotherm beneath "the rift,-(LUbiMova et al, 1972). Figure 5-7 Fault pattern of the Baikal Rift System(from Florensov ,1969).

131 5.4.6 Discussion.

Perhaps the most striking feature of continental rift systems is the parallelism of their bounding escarpments. Major changes in direction of one escarpment are usually closely followed by the other and en echelon features are common. This parallelism implies a characteristic width for each graben, be it 10 km in the Levantine Rift (Picard, 1966) or 65 km in the Gregory Rift. If variations in this characteristic width can be related to changes in some other measurable structural parameter, the mechanics of rift formation may become clear.

Almost all rift escarpments were created by normal faults, which commonly dip at 55° to 70° (Freund, 1966). Shallow focus type earthquakes are associated with the faulting, and fault plane solutions indicate in many cases that normal faults are still active, and that the least compressive stress is typically oriented perpendicular to the rift trend. A notable exception is the Rhinegraben where active transcurrent faults have been recognised, and the least compressive stress is oblique to the rift trend.

Rift faults often follow ancient dislocations in the basement and close relationships between Cenozoic and Precambrian structures have been reported in many parts of the East African Rift (McConnell, 1974). It appears, therefore, that the location of rifts may be strongly influenced by anisotropy in the mechanical properties of the upper crust.

Uplift of the rift flanks usually accompanied sinking of the graben floor and in some cases uplifting of a broad dome, which may be considered a more outstanding structure than the rift itself, preceded graben formation (Le Bas, 1971).

Seismic studies have established that the crust is typically thin in the vicinity of a rift valley, and in the case of the Kenya and Baikal rifts appears to be underlain by anomalously low velocity upper mantle. It should be noted that early interpretations of seismic refraction and reflection experiments in the vicinity of the Rhinegraben (Ansorge et al, 1970) also featured the existence of .a low velocity "mantle cushion" beneath the moho but subsequent more detailed studies established the presence of normal mantle below the rift flanks and a complex transition zone replacing the moho beneath the rift floor.

132 The current structural model of the Kenya rift (Griffiths, 1972) will probably be modified when more detailed experiments have been performed and interpreted. High density igneous intrusions in the Kenyan and Ethiopian rifts have be22n inferred from gravity observations. A low density crustal zone from which low density intrusions have risen into the basement has been proposed for the Rhinegraben area to explain seismic and gravity observations. Zones of high electrical conductivity -1 (0.01 - 0.1 S.m ) have been detected in the vicinity of the Kenya and Baikal rifts and the Rhinegraben.

Theories concerning the origin and development of continental rift systems are necessarily speculative, in view of the lack of detailed knowledge of their deep structure evident from the above discussion. Darracott et al (1973) considered the East African Rift to be a constructive plate boundary at an early stage in the evolution of a new ocean. In their proposed scheme of development, the Red Sea and Gulf of Aden represent respectively more advanced stages.

Le Bas (1971) and Burke and Whiteman (1973) emphasised the association of uplifted domes and rift valleys in Africa (section 5.1) viewing the rifts of Kenya and Ethiopia as distinct features, developing within their respective crustal domes. Burke and Whiteman (1973)'invoked the "mantle plume" concept of Morgan (1971) as the energy source behind each uplifted dome, and postulated a scheme of evolution incorporating doming, volcanicity, triple jjunction formation and continuing separation of one or more rifts as an active spreading centre. In this scheme, the intersection of the Kavirondo and Gregory Rifts in Kenya (figure 5-2) is seen as a triple junction, at which only two arms have remained active (the north and south sections of the Gregory Rift).

Oxburgh and Turcotte (1974) devised a fundamentally different theory for the origin of the East African Rift System. They proposed that the rift represents a propagatihg fracture system initiated by membrane stresses which can be developed in a rigid lithospheric plate moving over the surface of a non-spherical earth. Paleomagnetic results suggest that the African plate has been moving northwards. Oxburgh and Turcotte (1974) demonstrated that the locus of the onset of volcanic activity has migrated southwards as-the plate moved northwards, and interpreted this-ei,evidehce. to support their theory that the East African rift a propagating fracture zone moving south through the African continent.

133 The presence of a mantle plume beneath the Kenya dome obviously has important consequences for the geothermal regime within the crust and upper mantle, very different from those implied by the propagating crack theory. However the time scale of rift development in Kenya (c.25 Ma) is such that major thermal events within the upper mantle may be as yet undetectable at the surface (section 7.4). In this case surface heat flow observations can not be expected to resolve a geothermal disturbance within the upper mantle.

A feature of all three rift systems which has not been discussed above is the occurence of high heat flow on the rift floors. Heat flow data from the Rhinegraben, the Baikal and the Western rift and the Red Sea are compared with data acquired in Kenya for this study in section 7.1. The geothermal implications of the various interpretations of seismic velocity, gravity and geomagnetic variation anomalies associated with continental rifts are discussed in section 7.1 and models pertinent to the Gregory rift are examined in sections 7.3 and 7.4.

ti

134 CHAPTER 6

HEAT FLOW MEASUREMENTS IN KENYA.*

Heat flow results from 26 borehole sites in Kenya are presented in this chapter. The data comprise temperature logs from boreholes 71 to 253 metres deep, and the results of thermal conductivity measurements on 276 drill cuttings collected from 13 of the holes. Data of similar quality were collected by Morgan (1973) and his results are frequently referred to in this chapter.

6.1 Collection of the data.

The fieldwork for, this project was carried out in two phases; the first during July - November 1972 and the second during December 1973 - March 1974. The second phase'included an excursion to Ethiopia, the results of which will be presented in a future publication. Field operations in Kenya were based in Nairobi. The boreholes, most of which had originally been drilled for groundwater investigations, were visited by Landrover. While working in inhospitable regions in eastern and northern Kenya, daily radio contact was maintained with staff of the Physics Department at the University of Nairobi using a portable transmitter/receiver kindly loaned by them. A spare winch, probe and resistance bridge were carried on each excursion to minimise the possibility of delay due to equipment failure.

Every potentially suitable borehole (more than 100 metres deep) in Kenya which could be traced was investigated, but it turned out that only holes drilled for groundwater purposes were found to be in a usable condition; all the mineral exploration boreholes visited had collapsed. The assistance of the Water Development Division of the Ministry of Agriculture in tracing' suitable water boreholes is gratefully acknowledged.. Although drillers' reports could be found •for many of the holes, no up-to-date information on their condition was

135 readily available (except for that kindly provided by the United Nations - Kenya Government Geothermal Exploration Project, hereafter denoted GEP, for boreholes in the rift valley),' and each prospective site had to be visited. It transpired that many boreholes had been destroyed or blocked, some had been equipped with pumps which could not be removed, and a few were never found. The 26 boreholes logged represent a small fraction of the total number visited.

The condition of these 26 was also very variable. In arid parts of north and east Kenya, holes penetrating sediments were normally cased throughout, but on the rift floor and on the flanks only the upper sections were normally cased, and a few abandoned boreholes had been left without any casing. Borehole diameters ranged-from 152 mm to 356 mm but the deeper sections in which heat flow was determined were typically 152 mm to 203 mm. Occasionally minor temperature disturbances were observed at changes in hole diameter. The possibility of natural convection within large diameter wells was discussed in section 3.2. In several respects these holes were not ideal for geothermal gradient measurements and the results must be treated with caution. Problems associated with the determination of thermal conductivity were discussed in section 2.2.

6.2 Presentation of the data.

The 26 sites at which heat flow measurements were attempted have been numbered 24 to 49 in chronological order of logging, and their approximate locations are shown in figure 6-1, along with the 23 sites occupied by P. Morgan in 1971 (which were labelled 1 to 23 according to his nomenclature). The exact locations and depths are given in table 6-1, temperature and conductivity logs and details of the heat flow calculation, including corrections, are displayed in appendix 7, and the original data is listed for reference in appendix 6.

The results have been grouped into six regions for this discussion, defined essentially by major vertical movements which occured in Kenya during the Cenozoic era (section 5.2).

1. Rift Flank (west) 2. Rift Flank (east) 3. Gregory Rift Floor

136

.- , collar site latitude longitude depth (m) WDD no. elevation . (m) • 24 0.40°S 36.36°E. 183 C1361 2290 25 0.27°S 36.38°E 152 C3784 2350 o 26 0.31 S 36.06°E 105 C2278 1870 27 0.34°S 35.94°E 187 C1934 2160 o 28 0.30°S 35.91 E. 195 C2670 1980 o 29 0.28 S 35.43°E 163 C3096 2320 30 0.48°S 34.72°E . 92 C3671 1450 31 0.05°S 34.93°E 98 C2342 1220 32 0.14°S 35.040E 175 C3148 1240 o 33 0.14°S 35.26 E 103 C2444 1440 34 0.04°S 39.94°E 167 C3770 230 35 0.11°N 40.31°E 131 C3695 150 36 0.02°N 40.37°E 148 C3852 150 37 0.06°S 40.49°E • 136 ' C3831 150 38 0.70°N 39.47°E 160 C3804 300 39 0.76°N 39.61°E 113. C3792 300 40 0.43°N 40.02°.E 121 C3788 150 41 0.36°N 40.13°E 129 C3820 150 42 4.20°S 39.40°E • 135 ? 200 o 43 4.19 S 39.41°E 71 ? ' 180 . 44 0.66o N . 37.05°E 154 C3728 1280 45 2.75°N 38.10°E 183 C3724 590 46 3.12°N 38.28°E 253 C3793 610 47 0.18°N 35.46°E 243 C3142 2500 48 0.40°N 35.33°E 208 C3154 2160 49 0.45°N 35.30°E 184 C3619 2160

Table 6 - 1 Borehole sites : this survey

137 site: interval (m) K pr G G c Q Q c class

1 63 - 101 1.08 20 - 22 - C 2 79. - 167 1.04 65 74 67 77 ' A 3 32 - 240 1.17 47 65 55 76 B 4 79 - 155 1.26 82 94 103 119 B 5 166 - 239 1.29 150 143 194 185 B 6 79 - 151 .1.31 66 87 87 f 115 A 7 ,63 - 95 1.69 66 76 111 129 B 8 47' - -95 1.56 27 - 43 - C •9 -6 - 55 1.25 37. 43 47 53 B 10 61 - 183 0.96 199 244 190 234- B 11 41 - 114 1.48 15 -. 22 _. C 12 44 - 111 1.46 29 32 43 47 B 13 31 - 79 1.48 55 80 81 ' 118 B 14 30 - 250 1.08 36 - 39 - C 15 31 - .79 1.77 34 - 59 - C 16 31 - 221 5.25 17 . 19 87 102 B 17 31 - 47 1.76 30 - 53 - C 18 117 - 136 1.90 20 23 37 , 43 B 19 8 - 121. 1.34 48 55 64 74 A 20 43 - 60 1.49 51 56 76 83 B 21 61 - 123 1.48 28 ' 30 42' 45 A 22 41 - 151 1.13 5 - 6 - C 23 31 - 80 1.46 23 31 33 45 ' B

. .

—..

Table 6-2 Heat flow results (Morgan, 1973)

K pr - mean thermal conductivity over depth interval 1 • (W.m-1.K ) -1 G - uncorrected temperature gradient (mK.m ) G c corrected temperature gradient (mK.m-1). Q - uncorrected heat flow (mW.m 2). -2 Q:c - corrected heat flow (mW.m ) class - reliability classification (see section 6.3) 138 site interval Kr Kpr G . Q Gc Qc class

24 192 - 223 1.45 1.08 65 94. 74 80 A 25 52 - 89 • :-.:-:.(1.10) 58 64 -- -- C 26 73 - 104 - (1.30) 53 69 73 95 B 27 121 - 158 1.54 (1.16) 25 39 36 41 B 28 60 - 110 - (1.00) 56 56 -- -- C 29 115 - 151 1.70 1.19 70 119 72 85 B 30 15 - 91 - (1.50) 2 C 31 15 - 85 3.75 1.99' 45 169 47 93 A 32 18 - 54 - (1.00) 56 56 -- -- . C 33 60 - 85 1.90 1.28 21 40 25 32 B 34 144 - 160 3.86 2.03 28 108 30 62 A 35 115 - 130 - 7._ (2.00) 24 48 27 54 B 36 118 - 139 - (2.00) 51 102 55 111 B 37 110 - 133 - (2.00) 36 72 40 81 B 38 137 - 156 3.79 2.00 28 106 32 64 A 39 97 - 113 -. (2.00) 14 28 -- -- C 40 106 - 118 4.37 2.20 28 122 33 72 -B 41 94 - 127 3.76 1.99 26 98 30 61 A 42 97 - 133 5.92 3.36 21 124 24 79 B 43 42 - 66 - (3.40) 16 . 54 20 69 B 44 98 - 151 2:69 2.69 13 35 - 15 40 A. 45 160 - 175 :y-.--4(1.50) 16 24 -- -- C 46 224 - 249 - (1.50) 25 : 38 28 41 . B 47 165 - 213 - (1.50) 15, 22 -- -- C 48 80 - 171 1.60 1.60 30 48 31 -51 B 49 _ 68 - 144 1.35. 1.35 28 38 30 40 A .

Table 6-3 Heat-flow-results (this study)

: mean thermal conductivity (uncorrected for porosity)

Kpr, G, Q, Gc Qc, class : as in figure 6-2.

139 4. Kavirondo Rift Floor 5. Lamu Embayment 6. Coastal Belt

Sites 45 and 46 (figure 6-1) have been included in region 2 although they are c.150 km east of the main rift structures because the area has been the locus of extensive Quaternary volcanism, apparently associated with the development of the rift valley (Baker et a1,1971). The Kavirondo Rift is considered separately from the Gregory Rift because its strike is perpendicular to the main rift trend and factors which appear to dominate the geothermal regime in the Gregory rift (hydrothermal and lithothermal systems) are probably of secondary importance in the Kavirondo Rift. The sediments of the Coastal Belt, into which boreholes 42 and 43 were drilled, were deposited in a late Paleozoic - early Mesozoic trough whose formation preceded the main phase of downwarping in the Lamu Embayment (Walters and Linton, 1973).

The data are discussed region by region, and where relevant, comments are made on the results of Morgan (1973). A summary of the results •As presented in table 6-3, and an interpretation is attempted in chapter 7.

6.2.1 Region 1 : Rift Flanks (west).

Four holes on the west:flank. of::tha:Gregoy rift were logged. Three of these (47, 48 and 49) are on the Uasin Gishu Plateau c. 30 km west of the Elgeyo Escarpment, and the fourth (29) is south of the junction of the Gregory and Kavirondo Rifts, c. 50 km west of the Mau Escarpment (figure 6-2).

The Uasin Gishu Plateau consists of phonolite lavas overlaying the Basement System. The• lavas were erupted during the Miocene in the vicinity of the rift and flowed westwards (Lippard, 1973). The plateau has been uplifted c. 1.8 km since the Miocene and tilted upwards towards the rift (Baker ft al, 1972). Phonolite is virtually non - porous (Schoeller, 1962), but water movement can occur along fissures and weathered layers between successive lava flows (Jennings, 1964).

Borehole 47 also penetrates tuffs derived from the volcano Tinderet, and water movement within the borehole between a tuff horizon and

140

16 0.46 45 15' RI t / Rift fla k flank 017 wes i east 18 7 Gregory rift r - — 0 , 6 .44 c, ,I. Kavi r ndo E Lamu : oo rift 13Q 23042 --, Embaymnt 0Go • : r ' 011 , o 00 :0 • , , • : .. rea B , + • _, Area C Area A 1 L Eburr 0'14 1 22 01 ari I heat flow sites o this study o Morgan (1973) + Evans (1975) oaJ Belt, 2 3/

Figure 6-1 Location of heat flow sites in Kenya. Areas A,B,C are shown in detail in figures 6-2, 6-3 and 6-6 respectively.

35°E

I Mt Eon x \)( . 1 /// 1°N ..• g /ie. / \‘_. .,:. ucl / u) / 19 ID 1 \ o ›, 1 \ -'20 09 / o ma' 2148 Uasin Gishu Plateau o 1 +7 N 31wp 033 ,„ 32r/ e 030

KEY 470 heat .flow site (this study) 210 (Morgan 73) ,fault

hot spring line of weakness

Figure 6-2 Heat flow sites on the western flank o: the Gregory rift and Kavirondo rift(area A of figure 6-1).

142 fissures in the phonolite is probably the cause of the-serious. disturbances evident in the temperature - depth profile.

The temperature log of borehole 48 is interesting because, although there appears to be disturbance by flowing water in the upper and lower sections, the 90 m long middle section has a uniform temperature -1 gradient (30 mK.m ) which compares well with the value 28 mK.m-1 recorded in its undisturbed neighbour, borehole 49. The temperature log of 48 has similarities with one recorded by Birch (1947) in which moving water within the borehole had achieved a state of near thermal equilibrium with the surrounding rocks. A simple analysis of this situation suggested by Birch, in which the moving water is considered as a line source of heat, predicts a not unreasonable flow rate of -1 0.5 mm.s in borehole 48 (appendix 8).

Morgan (1973) made measurements at fcSur sites (9, 19, 20, 21) on the Uasin Gishu Plateau. Site 21, 5.5 km NW of site 49, had a - 1 i uniform gradient of 28 mK.m in phonolite, and is considered the most reliable of the four. The mean corrected heat flow of sites 21, 48 and -2 49-is 45 mW.m .

- 2 A significantly higher value (74 mW.m after correction) was calculated from the results at site 19, 28 km north of site 49, but the temperature gradient is very variable within this 125 m deep hole. The temperatures within boreholes 9 and 20 were disturbed by flowing water, and the corrected heat flows (53 MW.m-2; and 83 mW.m-2 respectively) are less reliable. It is worthy of note that sites 19 and 20 lie on an extension of a broad zone of NW - SE striking foliation in the Basement System now obscured by phonolite (Sanders; 1963). Cenozoic normal faults north of site 19 are also aligned NW SE and it is possible that the higher heat flows at sites 19 and 20 are associated with linear zones of weakness in the crust in_which lithothermal or hydrothermal systems are active.

The fourth new site on the west flanks of the rift, borehole 29, is in a complex tectonic environment near the junction of the Kavirondo Rift with the Gregory Rift. It is 22 km SE of-the Sondu flexure which defines the southern limit of the Kavirondo Rift in the area, and 1.5 km east of a volcanic vent, Tatwakapsigesa, which was "active during the Miocene and appears to lie on one of several N55°E

143 trending lines of tectonic weakness in this region indicated in figure 6-2 (Binge, 1962). The temperature log of borehole 29 indicates forced convection within the upper section of the hole. Indeed a groundwater source at 18 m depth had been noted by the driller, and unsteady temperatures (indicated on the temperature - depth profile as a dashed line) and a very low temperature gradient suggest that water was moving downwards within the borehole from 18 to 82 m depth. The rock porosity is unknown, and an assumed value of 35% was used in the 2 heat flow calculation. The high heat flow at this site (85 mW.m ) may have been anticipated in such an environment, but more measurements are required in this area before the results can be treated with confidence.

. 6.2.2 Region 2 ; rift flank (east).

Three of the twelve new sites east of the rift (44, 45, 46) are considered in this section. Although two of these (45, 46) are c. 150 km east of 'the rift zone, they are considered to be in a region physically associated with rifting since a major phase of volcanism connected with the rift took place 150 to 250 km east of the main graben at this latitude (Baker et al, 1971).

The third (44) was drilled in Precambrian metamorphic rocks c. 80 km east of the section of the Gregory rift occupied by Lake Baringo.

North of latitude 1°N, the zone of faulting which defines the rift funnels out, although a. narrow (20 km wide) inner graben can be _distinguished along the east side of this triangular region (Baker et al, 1972). Sites 45 and 46 are c. 150 km east of this inner graben.

The temperature log of borehole 45 is not useful because the water table was low (185 m below the surface) and the air column was unstable. In fact the instability was quite spectacular on the surface where air was blowing strongly through the borehole casing. Anexplanation of this behaviour was offered in section 3.2.

-1 A temperature gradient of 25 mK.m was recorded in the relatively short water section at site, 46, but no conductivity samples were available. The surface rocks were basaltic lava and a conductivity - 1 of 1.5 W.m1.K was assumed in the heat flow calculation. Morgan

144 (1973) recorded a gradient of 17 mic:m- 1 in the air section of borehole 16, 12 km NW of site 46 and used a surface sample from a nearby sandstone outcrop to obtain an estimate of thermal conductivity. - 1 1 In fact its conductivity was very high (5.25 W.m .K ) and the anomalously high value of heat flow calculated by Morgan (87mW.m-2 may have arisen because of an overestimate of conductivity, if the lower sections of the hole do not penetrate this highly conductive sandstone.

Borehole 44 was drilled in non porous Precambrian metamorphic rocks and the result.is considered reliable because below 100 m change in temperature gradient are accompanied by inverse variations in thermal conductivity. The thermal depth method (section 3.5) was therefore used to calculate a corrected heat flow of 40 mW.m- 2.

Two sites occupied by Morgan (1973) 60 km east of the Lake _Hannington section of the rift (12, 23) produced corrected heat flows -2 -2 of 47 mW.m and 45 mW.m respectively.

6.2.3 Region 3 : Gregory rift floor.

Measurements were made at five sites (24, 25, 26, 27, 28) on the floor of the Gregory Rift Valley between latitudes 0.25°S and 0.50°S (figure 6-3). A cross section through the rift at latitude 0.50°S in figure 5-2 b showed step platforms on the east side and a deep inner graben 30 km wide, filled with 3 km or more of volcanics and sediments. Three of the results in this section (26, 27, 28) were from the inner graben and two (24, 25) from the step platform.

The air section of borehole 24 was unstable but the water section was apparently undisturbed by convection. .A porosity correction reduced the conductivity by 25% but the heat flow remained significantly -2 high ( 80 mW.m ). The temperature - depth profile of borehole 25 is characteristic of a hole disturbed by forced convection, indicating water movement between water bearing zones at 90, 96 and 150 m. A tentative calculation above the disturbed sections gave a heat flow -2 of 75 mW.m .

Borehole 26 was air filled and the section above 66 m was unstable but below 70 m temperatures were steady; giving a gradient of 53 mK.M .

145

1 ,/ / 46. 0/E 1 / /

'\ Lake 0 \' Bolossat IL

km 25 \ totg E, ,. ...„.....7 el ,NIF,

0020's , .u. n ❑ pll'i ® ' 'F

KEY: groundwateri r113 >22037°C • 25 heat flow +■2 xis of positive \fault temperature( 0<25°C 010 sites Bouguer anomaly

Figure Heat flow site ;:r1 groundwater temperatures on the floor of the Gregory rift. (area B of figure 6.-1) The stability of. this lower section was established by a second temperature log one year later. The site is only 2.75 km south of the rim. of Menengai caldera, and heat from cooling magma beneath this recently active volcano probably contributes to the heat flow of -2 95 mW.m . It should be noted that a large ( > 30%) gradient correction for sedimentation has been applied to results from the inner graben, in which c. 3 km of volcanics and sediments accumulated in c. 10 Ma.

Site 27 is near the western margin of the rift. The air section was unstable and a sudden drop in temperature at 52 m coincides with the bottom of the casing. The driller had reported striking water -1 -1 at 122 and 195 m and a change in gradient from 25 mK.m to 45 mK.m at 168 m may be due to the effect of water movement. The calculated -2 heat flow (41 mW.m ) may therefore be low because of forced convection inside the borehole.

The temperature log of borehole 28 indicates water movement between 130 and 175 m and the calculated heat flow from the upper sections of the hole, in which quite large changes in gradient were evident, can not be considered reliable.

Morgan (1973) calculated uncorrected heat flows at eight sites (3, 4,5,6,7,8,10,13) on the rift floor, and the southernmost four (3,4,5,10) are included in figure 6-3. Sites 5 and 10 gave the highest values 2 2 recorded in Kenya (194 mW.m and 190 m W.m respectively), obviously associated with recent volcanic activity in Menengai and probably with narrow zones in which convection is the main mode of heat transfer -2 - 2 (see below). 103 mW.m was recorded at site 4 and 55 mW.m at site 3. The stability of the temperatures in boreholes 3 and 4 was confirmed by the present author from logs made one year later. Only one of the three sites (6,7,8) north of Lake Baringo yielded a useful 2 result (87 mW.m before correction). Corrections for the values calculated by Morgan (1973) were computed by the methods outlined in chapter 3, and the corrected heat flow values are listed in table 6-2.

Menengai, one of several central volcanoes which erupted on the rift floor during the Quaternary is a prominent feature in the vicinity . ' of sites 3,4,5,10,26 and 28 (figure 6-3)., The Quaternary period also heralded a phase of intense minor faulting on the rift floor, striking

147 roughly parallel to the graben boundary faults. Up to 2 or 3 sub- parallel minor faults per kilometre have been mapped in places (Baker et al, 1972). Hot springs and fumaroles are common in this region (McCall, 1967) and a United Nations - Kenya Government Geothermal Exploration Project (GEP) is currently investigating the economic potential of the heat sources which drive these geothermal systems. Some preliminary results from the GEP were kindly made available to this study.

A GEP survey of spring and groundwater temperatures in the rift between 0.50°N and 1.17°S revealed a complex pattern of local geothermal anomalies, typically associated with major fracture lines or Quaternary volcanicity. At two localities, Eburru and Olkaria, • 2 Glover (pers. comm.) estimated heat flows of 130 MW from 30 km and 2 377 MW from 20 km respectively (these values are approximately 70 and 300 times the global mean). pScess he6t appears to be derived from cooling igneous bodies in the upper crust, and'efficiently transferred to the surface by convecting fluids.

Steam and gas fumaroles have been observed within the Menengai caldera, and steam was encountered in two boreholes north of the caldera which, as McCall (1967) had indicated, lay on an extension of one of several linear thermal zones within the crater. McCann (1972) postulated a northerly movement of groundwater benea.th'114nengai, so the environment of sites 3 and 4 north of the caldera may be influenced by hot groundwater. Glover (pers. comm.) identified a warm water zone extending south from the western edge of the caldera, which he considered due to steam seepage along faults in this area (figure 6-3). This may disturb the environment of sites 20 and 28. Glover also noted a north - south linear warm water zone which intersects site 24. Its origin may be due to groundwater circulation along deep, permeable fractures. These local heat sources can seriously affect the surface heat flow pattern on the rift floor.

Thus the above results from the graben floor confirm that the thermal regime in this part of the rift is quite complex. Anomalously high heat flows over restricted areas are indicative of hydrothermal circulation or gas seepage above cooling igneous bodies.

Obviously a large number of heat flow observations will be required

1k8 to construct a geothermal anomaly pattern in any detail,-and to permit a useful average value of heat flow through the rift floor to be deduced.

6.2.4 Region 4 : Kavirondo rift floor.

Three sites were occupied on the floor of the Kavirondo Rift (31, 32,33) and one near the southern margin (30). This narrow graben (15 to 25 km wide) intersects the Gregory rift near the equator (figure 6-2), but the junction is now completely obscured by volcanics. The main phase of deformation which created the Kavirondo Rift took place in the late Pliocene or early Pleistocene, and observable faults now show a maximum vertical displacement of 700 m (Baker et al, 1972). The Kavirondo Rift floor does not appear to be extensively faulted and, away from its junction with the main rift valley, there are no major Quaternary central volcanoes on the graben floor. The only reported geothermal activity (Walsh, 1968) occurs near the southern margin in an area of intense microearthquake activity (Molnar and Aggarwa1,1971) related to movements on the Lambwe fault (figure 6-2).

Temperatures in borehole 32 are disturbed by water flow and only the portion above the water table appears useful. The rock conductivity is not known and, as was noted in chapter 2, a considerable range is possible in porous rocks above the water table. In addition the calculation interval is shallow (18 to 54 m) and the calculated heat -2 flow (56 mW.m ) may be greatly in error.

A uniformly high temperature gradient (45 mK.m- 1) was recorded at site 31, and there is no evidence of convection in the borehole The site is 1.5 km south of the. Nyando Escarpment (figure 6-2) in sediments. The rock matrix thermal conductivity was high (3.7W.m .K- 1 and the sediments quite porous so that a large conductivity error will result from a poor estimate of porosity (figure 2-5).' An assumed porosity of 35% made the whole rock conductivity 47% lower than the -2 - matrix conductivity, but the calculated heat flow (93mW.m ) was still 2 high. . If the porosity were as large as 50%, the heat flow (71mW.m ) would still be considerably above the average of sites 21, 48 and 49 - 2 (45 mW.m ), 55 km north on the Uasin Gishu Plateau. Unfortunately, other boreholes logged in'this- part of the Kavirondo Rift Floor were blocked at shallow depths and it was not possible to determine the

149 lateral extent of this heat flow anomaly.

One explanation for the high heat flow at this site, is that deep circulation of groundwater occurs in faults bounding the graben, producing local anomalies in the vicinity of permeable zones. However no surface manifestations of hydrothermal circulation (e.g. .hot springs) along the base of the Nyando Escarpment are known to the author. Another possibility is that the granites which outcrop on the escarpment and which underly the sediments on this part of the rift floor (granite was penetrated by a borehole 2 kilometres east of site 31), have relatively high concentrations of the heat producing elements uranium, thorium and potassium. Future studies should investigate this possibility by making heat production measurements on granite samples from the Nyando escarpment.

Site 33 lies on a line of tectonic weakness along an extension of the Sondu flexure (Binge, 1962) but the calculated heat flow is low -2 (31 mW.m ), in contrast to the high value at site 29 (85 mW.m 2) on a parallel line of weakness. More data are obviously required before any meaningful correlation with the complex tectonics of this area is possible.

At site 30 on the south flank of the graben, a temperature log was made inside a borehole which had recently been equipped with a pump designed to allow access to the hole. Unfortunately no information on the history of the borehole was available, but the extremely low recorded gradient may be a consequence of recent pumping. More measurements are required in the Kavirondo Rift to demonstrate the extent of the observed anomalies.

6.2.5 Region 5 : Lamu Embayment.

Eight boreholes were logged in the Lamu Embayment, 350 to 450 km o o . east of the rift valley, between latitudes 1.0 N and 0.1 S. Most of the Embayment is a flat, sandy plain, but subsurface exploration by the BP-Shell Petroleum Development Company of Kenya has revealed faulted structures in the basement beneath a thick layer of sediments. The location of the boreholes with respect to two buried horst-like structures, the Garissa and Wal Merer "basement highs", is shown in figure 6-6. The boreholes (34,35,36,37,38,39,40,41) penetrate Quaternary sediments consisting of sands, clay and sandstones. The recovered samples are typically very heterogeneous, and because the rocks had been effectively pulverised by the drilling process, it was extremely difficult to make an accurate estimate of their porosities. Manger (1963) catalogued the porosities of a variety of consolidated and unconsolidated sedimentary rocks, and most values for sand, clay, gravel and alluvium of Quaternary age lie in the range 25% to 45%. A value of 35% was used in the heat flow calculations shown in appendix 7, and the probable maximum values corresponding to 25% and 45% are listed in table 6-4.

Variations of porosity within the sediments penetrated by a particular borehole may give rise to apparent changes in heat flow with depth.

Porosity is obviously a serious unknown in this area where the rock matrix conductivity is relatively high (typically 3 to 4 W.m- 1.K 1 ), and the friable and heterogeneous nature of the sediments probably means that only in-situ measurements of thermal conductivity (section 2.3) will give satisfactory results.

Both air and water sections of each borehole were logged since the water table was typically 100 - 140 m below the surface in holes 130 7 170 m deep. Although the porosity effect is greatly enhanced in unsaturated rocks and the consequent uncertainty in thermal conductivity is too great to allow results from air sections alone to be useful, a reasonable comparison between the calculated heat flows over air and water sections was obtained for sites 34 and 38 when the fully saturated condition was assumed throughout. A zone of partial saturation, in which "pellicular" water is held in place by hygroscopic and capillary forces,'may extend c. 100 m above a water table (Todd, 1959), providing good thermal contact between the grains in the rock matrix, so that the large reduction in thermal. conductivity otherwise expected for porous rocks above a water table is not seen in practice. Alternatively it may be accounted for by variations in porosity.

The heat flow calculAions presented in appendix 7 have been restricted to water filled sections of the boreholes. Corrections for sedimentation were based on deposition rates calculated from the lithologic logs of nearby deep boreholes drilled by the BP-Shell

151 site Gc Kr Q25 . Q35 Q45

34 30.4 3.86 74 62 52 35* 26.9 - 65 54 45 36* 54.9 - 133 111 92 37* 40.0 - 97 81 67 38 .31.7 3.79. 76 64 53 39* 13.8 - 33 28 23 40 32.5 4.37 87 72 59 41 30.4 3.76 73 61 51

Gc corrected temperature gradient (mK.m-1) Kr rock matrix thermal conductivity (W.m-1.K-1) Qn corrected heat flow assuming n% porosity (mW.m-2)

Table 6-4 Effect of porosity correction on heat flow in the Lamu Embayment. (Results in column 5 are those given in table 6-5)

. - 1 * For these values, Kr = 3.8 WM7 K was assumed

.152 Petroleum Development Company. of Kenya.

Steady temperatures were recorded below 30 metres at sites 34,38 - and 41 and the recorded temperature gradients (28,28 and 26 mK.m 1 respectively) showed little variation between the three boreholes, although sites 34 and 38 are c. 100 km apart. A.porosity of 35% predicts a whole rock conductivity nearly 50% less than the measured matrix values. All three temperature - depth profiles show flexures at the air/water interface, and minor disturbances at the end of steel casing.

-1 The temperature gradient (28 mK.m ) recorded over a short section below the water table in borehole 40 agreed closely with the above three values, but a higher average matrix conductivity gave rise to a -2 heat flow c. 10 mW.m higher than sites 34, 38 and 41, when the same value of porosity (35%) was used in the calculation.

No rock samples were available from the four remaining boreholes (35,361 37,39) in the area. The temperatures in borehole 39 are considered too disturbed to be useful.

-1 The temperature gradient at site 35 (24 mK.m ) was comparable to the values at sited 34,38,40 and 41 but those at sites 36 and 37 were -1 significantly higher (51 and 36 mK.m respectively). Heat flows of -2 -2 111 mW.m at site 36 and 81 mW.m at 37 were calculated using an 1 estimated thermal conductivity of 2.0 W.m-1.K (corresponding to the water saturated average value at site 38 for 35% porosity). These two sites are near one of the buried horst-like structures in the Lamu Embayment, the Wal Merer basement high (figure 6-6). Evans (1975) calculated a high heat flow (93 mW.m 2) using bottom hole temperatures and rock sample conductivities from a 3.9 km deep petroleum exploration hole at Wal Merer, centred on the basement high, 13 km from site 37 and 30 km from site 36. He also calculated a high value (79 mW.m) from measurements in a well 1.2 km deep, 18 km NE of Garissa drilled above another basement high (figure 6-6).

6.2.6 Region 6: Coastal belt.

- 4 'Sites 42 and 43 were drilled in Mesozoic sediments in SE Kenya (figure 5-1) which are roughly contemporaneous with the Karroo series

153 of southern Africa. Faults separate these sediments, known locally as the Duruma Sandstone Series (Caswell, 1953), from Precambrian basement on the west and Mesozoic and Cenozoic .sediments on the east.

In borehole 42, the temperature-depth profile suggests disturbance by convection above 90 m, but a steady gradient below 100 m appears undisturbed. No conductivity samples were available for sites 42 and 43 and the conductivity values shown in appendix 6 were based on measured samples from a nearby borehole. Large variations in rock matrix conductivity with depth were evident because of highly conductive layers, and therefore if the assumed lithology is wrong, large errors in the calculated heat flows are possible. The -2 apparently high heat flow (79 and 69 mW.m at sites 42 and 43 respectively) may be entirely due to overestimated thermal conductivities and no interpretation of the results has been attempted.

6.3 Classification of the results.

The quality of the data presented in this chapter, and that presented by Morgan (1973), is extremely variable and it is considered necessary to attempt a classification of results according to their reliability, prior to the interpretation of the observed heat flow patterns in Kenya. Since the most important criterion for the reliability of a result in this study, the absence of serious convection in the borehole fluid, must be based on a subjective interpretation of the temperature-depth profile, only a simple threefold classification was considered. The basis for a cladsification into groups A, B and C is outlined below :

CLASS A :

1. There is no evidence of water movement within the borehole, or only minor disturbances outside the depth interval used for a heat flow calculation (i.e. the calculation interval).

2. Thermal conductivities are based on samples from the borehole.

3. The calculation interval is below the water level.

154 CLASS B :

1. Parts of the temperature-depth profile.may be disturbed by convection, but the calculation interval appears to be relatively undisturbed.

2. Drill cuttings from the borehole may not be available, and the thermal conductivity Can be based on measurements of surface samples, or samples from a nearby borehole.

3. The calculation interval may be .above the water table, if there is reasonable evidence that the air column is stable (e.g. site 26).

CLASS C :

1. The entire temperature-depth profile may be disturbed.

2. The thermal conductivity may be unknown.

3. The calculation interval may be very shallow (e.g. below 50 metres).

There is a considerable, variation in quality possible within any particular class. .

In threefold classifications of heat flow data from parts of the United States, Sass et al (1971c) and Reiter et al (1975) assigned a typical uncertainty of 10% for values in their most reliable classes. However errors of at least 20% must be considered possible for the'data assigned to class A in this study, and of 30% or more for data assigned to class B. Very large possible errors in class C data make it necessary to exclude them from a quantitativeitudy.

6.4 Previous heat flow studies in Kenya.

Two colleagues at Imperial College, P. Morgan and T.R. Evans have also been engaged in heat flow studies in Kenya. The results from Morgan (1973) are of similar quality to those presented above, and to make the two studies compatible, similar corrections for climate,

155 topography, uplift and erosion have been applied and the results classified according to the criteria of section 6.3 (table 6-2). The topographic corrections are those computed by Morgan (1973).

The geothermal gradients calculated by Evans(1975) from six sites in the Lamu Embayment were based on bottom hole temperatures recorded during interruptions in the drilling of oil exploration boreholes. Maximum mercury-in-glass thermometers had been used in each case, several hours after fluid circulation was stopped, during an interruption in drilling. Where the measurements had been made very close to the hole bottom, Evans and'Tammemagi (1974) considered that several such measurements taken at different stages during the drilling of a hole, could yield temperature gradients accurate to 10%. Thermal conductivities were estimated by Evans (1975) from measurements on drill samples (between 16 and 66 samples from each borehole) and lithologic information provided by the oil companies. Corrected heat flow values from the six sites (figure 6-1) are included in a histogram of heat flow in the Lamu Embayment (figure 6-4 c).

6.5 Discussion of the Heat Flow Results.

A total of 55 heat flow results, comprising 26 from the present study 23 from Morgan (1973), and 6 from Evans (1975), are presented in . histogram form in figure 6-4. The reliability classes A B and C, as defined in the last chapter, and the deep oil exploration borehole results are distinguished in the histograms. The global mean heat 2 flow, 61.5 mW.m , computed by Lee (1970) and the mean for the South -2 African Shield, 47 mW.m (Carte and van Rooyen, 1971) are indicated below each histogram.

6.5.1 Rift and flanks.

It is evident from figure 6-4 that the distribution of heat flow on the floor of the Gregory rift is markedly different from that on •the rift flanks and the Lamu Embayment. Very high values estimated by Glover (pers. comm.) for two sites on the rift floor, 18.8 W.m 2 -2 at Olkaria and 4 .3 W.m at EburrU (figure 6-1)., have not been included in the higtogram. The wide range of heat flow on the rift floor is particularly significant, reflecting the dominant role of localised

156 8- ( d) RIFT FLANKS 6- 4-

2-

0- 20 40 GO 80 100 1 0

(c) LAMU EMBAYMENT 4

2

0 20 40 GO 80 100 120

(b) GREGORY RIFT FLOOR 4 r-, 2]

0 r I I 0 20 401 GO 80 100 120 180 240

(a) KAVIRONDO RIFT FLOOR 20] r7 • 1 ` I 0 20 401 GO 110 100

class A B • C Oilwells(Evans,1974) South African Shield mean

Figure 6-4 Histograms of corrected heat flow values in Kenya (in mW.m-2). •

157 convection (in the form of hydrothermal circulation and volcanic activity) on the heat flow pattern. Obviously insufficient data are available to compute a meaningful avarage value for the rift floor region. Even less is known about the geothermal regime in the Kavirondo rift (figure 6-4 a).

In contrast, the results from the rift flanks are distributed over a much narrower range and the histogram of combined results has a -2 2 pronounced maximum in the range 40 mW.m to 50 mW.m (figure 6-4 d). It is considered particularly significant that the mode of this histogram is the same as that of a histogram of the South African Shield values taken from Carte and van Rooyen (1971) (figure 6-5 a). (Following these authors, 3 values near the southern boundary of the shield, which is not accurately defined, have been excluded from the histogram).

Two other important points are evident in the histograms of figure 6-5. The C class values (indicated by dashed lines) which are considered to be seriously disturbed by; moving water, tend to be distributed on the low side of the mode. This is to be expected where water movement occurs within the borehole along the depth • interval chosen for the heat flow calculation because flowing water has the effect of equalising temperatures and reducing temperature gradients.

The second point is that several isolated high values occur on the rift flanks. Several major central volcanoes have erupted on the flanks of the rift in the past 3 Ma, indicating that penetrative convection has been active in the ancient crust. It is proposed here that the three high values on the west flanks are associated with zones of weakness in which penetrative convection may have occured. One of the sites (Kenya 29) lies along a linear zone of weakness recognised by Binge (1962), and the other two (Kenya 19 and 20) lie along an extension of a broad zone of NW - SE striking foliation and normal faulting in the basement recognised by Sanders (1963) (figure 6-2). These last two boreholes are sited in Tertiary lavaZ.-which have obscured structures in the basement.

The single high value on the.east flanks (site 16) is based on a • -2 low measured gradient (19 mW.m ) but a very high estimated conductivity

158

(d) EAST FLANK 6_

4 - r --; 2_ I- I _-, L 0_ 1- -1 i - -= I ,,, n I 0 20 40 GO 80 100 120

(c) WEST FLANK

0] 0 20 40 GO 80 100 • (b) EAST AND WEST FLANKS 8- 6_

4- 2_

I 0_ I I 4 0 20 40 GO 80 100 120

(a) SOUTH AFRICAN SHIELD 8_ 6_

4- 2_ 0 I E 0 20 40I 60 80

class A B C

Figure 6-5 Histograms of corrected heat flow values(in mW.M-2) from the flanks of the.Gregory rift and South African Shield. 159 1 1 (5.25 W.m .K ) from a surface sample of a nearby outcrop (Morgan, 1973). The upper part of this hole penetrates basaltic lava, which has a much lower conductivity, and therefore the value may be greatly In.error due to a poor estimate of conductivity, or badly disturbed by refraction effects since large lateral variations in conductivity must occur in the vicinity of the borehole.

These 4 anomalous values and the disturbed C class values have therefore been excluded from analyses in which surface heat flows are to be used to investigate upper mantle conditions in section 7.4. A simple mean computed from the 9 remaining A and B class values is -2 45 mW.m . A weighted mean, in which B values are assigned half the weight of A values in the calculation, is insignificantly different -2 (44 mW.m ). This is very close to the mean value (47 mW.m-2) calculated by Carte and van Rooyen (1971) for the Precambrian shield 'of South Africa. It must be concluded that the effects of a major thermal anomaly postulated to exist in the upper mantle below this region (section 5.3) are not reflected in the heat flow results, except where penetrative convection has been possible through zones of weakness. The implications of this are examined further in chapter 7.

6.5.2 Lamu Embayment.

Eight results from the Lamu Embayment were presented in section 6.2.5, and these along with six values baSed on bottom hole temperatures from deep oil exploration wells (section 6.4) are summarised in histogram form in figure 6-4 c. Serious uncertainties in the calculation of heat flow from the shallow boreholes in this area were discussed in section 6.2, and because of these a detailed interpretation of the heat flow pattern can not be justified on the basis of the data from this study alone. However the presence of at least one anomalously high heat flow zone was also detected by the deep well results of Evans (1975) and therefore merits further investigation.

Details of the tectonic history of the Lamu Embayment are derived from the results of a subsurface exploration programme by the BP-Shell Petroleum Development Company of Kenya, summarised by Walters and Linton (1973) and outlined in section 5.2. The results of similar exploration programmes by other petroleum companies in neighbouring Somalia are described by Beltrandi and Pyre (1973). Subsurface

160 structures were discovered beneath the sedimentary cover which provide a basis for the interpretation of observed variations in heat flow. However the oil prospecting techniques could only resolve structures within a few kilometres of the surface and very little is known about the lower crust or upper mantle in the area.

The most interesting subsurface structures discoVered so far are dome - like features described by Walters and Linton (1973) as basement highs. Their presence was inferred from gravity measurements where they appeared as positive Bouguer anomalies, and from explosion seismic data. Exploratory drilling through the overlying sediments indicated that these horst and graben type features were initiated during the upper Cretaceous or Paleocene epochs, apparently as a result of normal faulting. The basement highs are roughly aligned NNW - SSE and NNE - SSW, showing a similar trend to the Gregory rift structures and also to the pattern of rifts and faults in coastal Tanzania (Walters and Linton). Beltrandi and Pyre (1973) postulated that a pre - Jurassic graben extended from coastal Kenya, beneath the Embayment, through north - east Kenya to the Danakil Alps in Ethiopia, but this theory must be considered highly speculative until the subsurface structure in the Embayment has been reslved in more detail.

In figure 6-6, the relationship of two basement highs to the heat flow sites can be inferred, from the contours marking the top of the Mesozoic sediments.which were deformed by the upthrust blocks (Walters and Linton, 1973). A cross section through A-B on figure 6-6 is shown in figure 6-7. The occurence of such a horst and graben pattern over a wide area may be indicative of large scale changes in the thermal state of the upper mantle, and indeed, the results of a geomagnetic variation study discussed in section 5.3 suggest that d region of partial melt may exist in the upper mantle 50 km beneath this area, so that high surface heat flows would not be unexpected. This.is discussed further in section 7.4.

Positive heat flow anomalies associated with the Garissa and Wal Merer basement highs are of too short a wavelength to be due to mantle sources, and an explanation must be sought in terms of structures in the upper crust. An upfaulted block of basement rock may contribute to a positive surface heat flow anomaly in the following ways :

161 o 407E 500 r ^- ■-• - o Habasrin - / -1000/ / ••••••.• / /* / 3 / ° I / " /*/ -3000 •. 0 /

--35;0 7/ 7.‘1 ( 35 2000 36 -1500 34 - 0-150.0 /0°- 1000- 37 g 3 , k // WAL-M.ERER H.,/// .\ ARISSA HIGH

Gari ss ( N a [ K EY 34 • heat flow site(this study) MI oilwell site(Evans,1975) approximate contours - 500L-- conjectured contours * magnetic anomalies(volcanics

Figure 6-6 Contours of the depth (::metres below sea level)of the top of Mesozoic sediments in the Lamu Embayment. (from Waiters and Linton,1973). -

162

319°E 410°E 411°E Km

'West East _2

B .1 it sea level base joce _0

Garissa high al-Meyer esozoic high

Figure 6-7 Crosssection (west=east)through the Lamu Embayment (see A-13 on figure 6-6) (from Walters and Linton,1973). 1. Its thermal conductivity may be higher than the surrounding sediments. Temperatures are increased above such a conductive channel (Mundry, 1966).

2. The horsts may be of granite composition, and contain sufficient radioactive material to create a local geothermal anomaly.

3. Cooling igneous intrusions may exist beneath the sedimentary cover.

4. Forced convection of groundwater.may occur in deep aquifers which were upwarped by the rising basement highs.

5. If the basement highs were formed by normal faulting, groundwater may convect naturally within permeable fracture zones.

Evans (1975) measured the thermal conductivities of 71 sediment samples in this area (to a maximum depth of 3.79 km) from two oil -1 - exploration boreholes in this area. The high mean values, 2.36 W.m .K 1 -1 1 and 2.25 W.m .K for the two holes exclude the first hypothesis, because the thermal conductivities of typical basement gneisses and granites lie in the range 2 to 3 W.m-1 .K-1 (Kappelmeyer and Haenel, 1974) and therefore the conductivity contrast between basement and sediments would be insufficient to produce the observed anomaly.

The magnitude of the heat flow anomaly due to an upthrust cylindrical granitic block which has a heat production above that of the surrounding sediments, can be calculated using the formula derived in appendix 9. The heat flow anomaly j q at the surface, on a projection of the axis of the buried disc, can be written :

A -(2.:+(h + 1) 2 (h2 + a2)

(6.1) -3 -where A heat production contrast ( p W.m ) 1 = thickness of disc a = radius of disc h = depth of 'top of disc below surface.

164. In the case of the Wal Merer basement high, 2 Lig 20 to 40 mW.m , and the upthrust block can 'be roughly represented as a disc with dimensions 1 = 2 km, a = 10 km, h = 4 km. The values of A required to cause anomalies of magnitude -2 -3 20.and 40 mW.m are 18.and 36 W.m respectively. Since the heat 3 production of typical granites are around 3 JJW.m (Kappelmeyer and Haenel, 1974), hypothesis 2 must be considered untenable. Even if the heat producing block extended to the base of the crust (1 . 30 km) -3 a heat production contrast of 3.75 14.m would be required to 2 produce a heat flow anomaly of 20 mW.m .

Some evidence exists to support the third hypothesis. Intense magnetic anomalies of shallow origin were detected in the north east of this region (Harrison and Haw, 1964) and these were interpreted as shallow volcanic plugs within the Cenozoic sediments. The locations of these postulated igneous bodies are shown in figure 6-6. Igneous intrusions may also be present beneath the sedimentary cover in other parts of the Lamu Embayment, and may locally enhance surface heat flows. If the basement highs are indeed horst blocks created by normal faulting, and if partial melting has taken place in the mantle at a depth of 50 km,(Banks and Ottey, 1974), then conditions favouring penetrative convection do exist and hypothesis 3 must be considered as a plausible explanation for the anomaly.

Forced or natural convection of groundwater in permeable formations could also produce the measured heat flow anomaly. Bullard and Niblett (1951) proposed a simple analytical model to calculate the anomaly produced by forced convection where groundwater flowing through a deep aquifer moves vertically upwards to pass over an anticlinal structure. This situation is likely in the Lamu Embayment where groundwater moving east or south-east from the Kenya dome into the Embayment may be forced to rise over the basement highs (figure 6-6). The disturbance to heat flow can be estimated from the following equation (appendix 10) : kx Q = AZ e mcz

(6.2)

165 where = disturbance to heat flow Q = background heat flow AZ = vertical distance moved by groundwater over step Z = vertical distance to top of step m = groundwater mass flow rate specific heat of water k • thermal conductivity of rocks (2.3 W.m.17 1) • horizontal distance from step in direction of- flow.

The depth to the top of the basement high (Z) is roughly 1 km and the vertical distance (AZ) through which the groundwater must move to cross it also approximately 1 km, as can be seen in figures 6-4 and -2 6-6. The background heat flow (Q) is chosen as 62 mW.m , the mean of three A class values in this region not located above the basement highs. Thus from equation (6.2) heat flow of 124 mW.m-2 might be expected above the step, and therefore this type Of disturbance could 2. account for the high value (110 mW.m ) at site 36. It can be shown 2 that moving water could also produce the high values (80 mW.m and 2 95 mW.m ) recorded 20.30 km south - east of site 36. The magnitude of the anomaly predicted by equation (6.1) falls off exponentially with distance x from the step. For the anomaly to have a value 1/e times its maximum at a distance of 50 km from the step, the mass flow of water, m, must be given'by :

.KX/(cZ)

1 = 0.274 kg.s (using the values given above)

In an acquifer 100 m deep, with a porosity of 1%, this would -1 require a groundwater velocity of 0.027 mm.s , a not unreasonable figure.

If the edge of the basement high is faulted, as suggested by Walters and Linton (1973), deep hydrothermal circulation in permeable faults may contribute extra heat to the forced convective system proposed above, increasing the magnitude of the surface anomaly.

Thus two possible explanations for local hot spots in, the-Lamu Embayment are that cooling igneous bodies exist locally beneath the

166 sedimentary cover, or that regional groundwater flow carries heat efficiently to the top of the anticlinal structures overlying the basement highs. Serious uncertainties in the thermal conductivities of the rocks in this area make it difficult to justify a more detailed analysis of the measured surface heat flow pattern in the Lamu Embayment.

6.5.3 Summary.

Several important facts have therefore emerged from this study which merit further investigation. It should be emphasised however, that the data is of very variable quality, and large errors are possible. In the case of the C class values, it has been demonstrated that water movement within boreholes may give rise to systematic errors. The subjective criteria for classification, and selection of calculation intervals, and the necessity of estimating rock porosities mean that much of the data is of questionable quality, and the possibility that some observed anomalies are due to systematic errors can not be ruled out.

In addition, the-area of investigation is large and structurally, complex and there is abundant evidence of both lithothermal and hydrothermal activity in many localities. A large number of heat flow measurements in carefully selected sites are required to construct a definitive pattern of surface heat flow.

The author considers that the most useful results from this survey and that of Morgan (1973) are those on the flanks of the rift. When values suspected of disturbance by convection are removed from the analysis, a close similarity between the Gregory. Rift flank and South African Shield can be seen. The implications of this fact are examined quantitatively in chapter 7.

Although a complex geothermal regime forbids a detailed analysis of the rift floor heat flow results, it is important to consider the thermal effects of the postulated axial intrusion (section 5.3), and these are examined in section 7.3 by techniques developed in chapter 4.

67 CHAPTER 7

INTERPRETATION OF THE RESULTS

Heat flow patterns in the Western Rift, the Rhinegraben and the Baikal Rift show similarities with the results of the present survey in the Gregory Rift. Seismic, gravity and geomagnetic variation studies have detected unusual structures in the crust and upper mantle beneath the Rhinegraben, Baikal Rift and Gregory Rift (section 5.4) and interpretations of rift valley heat flow anomalies (Haenel, 1971; Lubimova et al, 1972) have been largely based on theories connecting the deep structures with thermal disturbances.:. In section 7.4, the Kenya heat flow data are compared with the predictions of geothermal models based on the assumption that zones of high electrical conductivity, low density and low seismic velocity within the upper mantle are the result of partial melting.

7.1 Heat flow patterns in continental rifts.

Heat flow data from the Gregory Rift, the Western Rift and the Red. Sea are presented in histogram form in figure 7- 1 and data from the graben floors and flanks of the Gregory Rift, Rhinegraben and Baikal Rift are shown as histograms in figure 7-2. The Red Sea should strictly be classed as an intercontinental rift (Milanovsky, 1972) since the floor and shoulders of its axial graben are of oceanic composition. The Red Sea is included in this discussion because it appears to be a part of the Afro-Arabian Rift System (`which includes the Gregory Rift) which has reached an advanced stage of development (Baker et al, 1972.)

7.1.1 Western Rift

Several sections of the Western Rift are occupied by deep lakes in which heat flow can be conveniently measured by oceanic techniques (section 2.1). Results from Lake Malawi (Von Herzen and • Vacquier,'

168 1967), Lake Tanganyika (Degens et al, 1971) and Lake Kivu (Degens et al, 1973) are summarised in histogram form in figures 7-1 (a), (b) and (a respectively. The heat flows have not been corrected for the effects of topography, climate or sedimentation, and rapid sediment- ation within parts of Lake Malawi and Lake Tanganyika may be responsible for some of the low recorded values: High values in these lakes occur near offsets in the main graben faults and may be the result of hydrothermal circulation or igneous intrusion along transverse fracture zones (Von Herzen and Vacquier, 1967; Degens et al, 1971).

7.1.2 Red Sea.

Heat flow measurements have been made in the Red Sea by oceanic techniques (Sclater, 1966; Birch and Halunen, 1966; Langseth and Taylor 1967; Erickson and Simmons, 1969; Haenel, 1972; Scheuch, 1973) and by using bottom - hole temperatures obtained during interruptions in ' drilling of boreholes (Girdler, 1970; Girdler et al, 1974; Evans and Tammemagi, 1974). The published heat flow values, excluding those from brine deep zones (Erickson and Simmons, 1969; Scheuch, 1973), which are considered thermally unstable (Kappelmeyer and Haenel, 1974), are presented in a histogram in figure 7-1 (e). High heat flow was recorded throughout the downwarped depression which contains the Red Sea. Girdler et al (1974) noted that the highest values occurred near the oceanic rifted section along the axis of the depression.

7.1.3 Rhinegraben.

Heat flow data from the vicinity of the Rhinegraben have been summarised by Haenel (1970, 1971). Eight values from the graben floor and six from the flanks are included in the histograms of figure 7 -2(b). Supplementary geothermal gradient data, based on bottom - hole temperatures obtained during interruptions in the drilling of deep boreholes on the graben floor, have been published by Delattre et al (1970) and Werner and Doebl (1974). Their results have revealed many localised high heat flow zones, which have been attributed to the effects of deep groundwater circulation in permeable fracture zones.

With one exception, beat flows from the rift flanks were within -2 -2 - .11 mW.m of 70 mW.m , the mean value for Germany (excluding the ' - 2 • Rhinegraben). The exception, c..30 mW.m above the mean, was

169

e)RED SEA

6 -

4 -

2_

0 , n h r,. 1 E1 0 40 80 120 160' . 200 240

(d) GREGORY RIFT

n n 17 40 120 160 200 240

(c) LAKE KIVU

40 80 120 160

(b) LAKE TANGANYIKA

F-1 0 40 8010 12 0 160

(a) LAKE MALAWI 4_

2- 110•111■■

40 80 120

Figure 7-1 Histograms of heat flow data from five sections of the Afro-Arabian Rift System.(in- mW.M-2)

170 10_

S- G_ (c) GREGORY RIFT 4-

2- o_ ri 1 0 40 80 120 160 200 240 40 80 120

6-

4- RHINEGRABEN 2- o_ [-I T L:0 (30 120 160 200 u 40 80 120

12_ 10_

8_ (a) BAIKAL RIFT

4-

2-

o_ I 0 40 do 120 160

FLOOR FLANKS

-2 Figure 7-2 Histograms of heat flow data(in mW.m ) from the floors and flanks of the Baikal rift,Rhinegraben and Gregory rift.

171 recorded in the Swabian volcanic area 80 km SE of the graben boundary in a region which was extensively intruded by volcanits during the Upper Miocene (Carle, 1974) and faulted by the Bebenhausen Fracture Zone.

• The last major orogeny affecting the basement on the flanks of the Rhinegraben was completed during the Permian. The average heat flow on the rift flanks (excluding the Swabian volcanic area) is 67 mW.m 2, the same as the mean value determined by Lubimova and Polyak (1969) for Hercynian folded regions in Eurasia.

7.1.4. Baikal Rift.

Heat flow in the Baikal region Of the USSR has been studied by Lubimova (1969), Lubimova et al (1972) and Moiseenko et al (1973), and their results are summarised in the histograms of figure 7-2 (a). The rift system is flanked to the NW by the Precambrian Siberian -2 Platform where a mean heat flow of 42 mW.m has been measured, and to the SE by a Caledonian fold belt where the mean recorded heat flow is -2 59 mW.m (Moiseenko et al 1973).

A number of less reliable results which confirm the general pattern of elevated heat flow within the rift zone were also reported by Lubimova et al (1972) but these have not been included in figure 7-2(a).

7.1.5 Discussion.

The most striking common feature of the histograms in figures 7-1 and 7-2 is the wide range of heat flows measured on each of the graben floors. In the Gregory Rift (section 6.2), the Rhinegraben, Lake. Tanganyika and Lake Malawi the higher heat flows are associated with fracture zones or recent volcanic activity. Within the Rhinegraben, a relatively large number of rough temperature gradient determinations (Delattre et al, 1970; Werner and Doebl, 1974) have revealed many geothermal anomalies of limited areal extent, demonstrating that average heat flows deduced from measurements at a few scattered sites on the floor of the rifts, such as were used to construct the histograms in figure 7-2, may be quite misleading. The available evidence suggests that this also applies to results from the other rift valley floors. The histograms in figure 7-1 show a steady increase in both mean and maximum values of heat flow in-the East African Rift from •

172 south to north (figure 7-1 a to 7-1 e),' but in view of the previous statement, little significance should be attached to this fact until more detailed observations permit the areal extent of isolated anomalies to be estimated.

The heat flow pattern on the flanks of the Rhinegraben, Baikal and Gregory rifts is much less disturbed, and the heat flows appear to be characteristic of the tectonic age of the basement. Isolated high heat flows on the flanks of the Gregory rift and the Rhinegraben are associated with recent volcanic activity or zones of weakness in the basement.

A variety of explanations have been proposed for the anomalous heat flow in rift valleys. Since many different factors can contribute to the measured values (chapters 3 and 4), and their relative contributions are usually difficult to assess directly, most interpretations rely heavily on indirect evidence of the thermal state of the crust and upper mantle based on other geophysical and geological studies.

The Rhinegraben has probably received more attention from geologists and geophysicidts than any other continental rift, and recent findings have provided a basis for models of the geothermal regime. The discovery of a seismic low velocity layer in the crust, locally thickened beneath. the Rhinegraben (Mueller et al, 1973) was particularly significant. Haenel (1970) computed a number of crustal temperature models for the Rhinegraben area, assuming values of heat production and thermal conductivity within a layered crust, for surface heat flows -2 2 2 2 of 71 mW.m , 84 mW.m , 126 mW.m and 167 mW.m (approximately covering the range of the measured values). He deduced from these 2 models that in regions where the surface heat flow was above 126 mW.m , the seismic low velocity layer in the crust must be in a state of fusion or partial fusion. Haenel therefore proposed that cooling igneous intrusions existed within the crust and estimated the age of emplacement at 1 to 2 Ma (where the surface heat flow was 126 to 167 mW.m-2).

However Werner (1970) suggested that the.highest heat flows in the Rhinegraben were the result-of hydrothermal circulation within permeable fracture zones. Mueller and Rybach (1974) estimated an -2 average heat flow of 84 mW.m through the rift floor, based on a large number of geothermal gradient determinations (Werner and Doebl, 1974). They also calculated temperature profiles in the crust, estimating that temperatures 25 km beneath the rift were approximately 200 K above those at corresponding depths beneath the flanks, and that the contribution to surface heat flow from below 25 km was 13 mW.m 2 greater beneath the Rhinegraben. Their models did not predict large scale melting within the crust.

In an alternative model, Haenel (1971) considered the effects of a 16 km thick granite slab in the crust, 200 km wide with its top 4km below the surface, centred below the rift axis. When such a slab was -3 assigned a heat production 2 IAW.m higher than its surroundings the 2 model predicted a broad heat flow anomaly, 107 mW.m above the back- -2 ground (assumed 75 mW.m ) which showed similarities to the fifth-order trend surface of heat flow in this region. However one effect of this trend analysis was to reduce the amplitude of the maximum over'the rift and to increase its wavelength. In fact the measured values showed no evidence of high heat flow outside the boundary faults of the graben (except in the Swabian volcanic zone) and the predicted values - 2 of the granite slab model are over 100 mW.m higher than the recorded flank values nearest the rift. In the opinion of the present author, the use of such a trend surface analysis in situations of this kind can only be misleading.

Puzyrev et al (1973) detected a slight elevation in the moho beneath the Baikal Rift, underlain by anomalously low seismic velocity upper mantle, but no evidence exists for a low velocity layer within the crust. The discovery which has guided interpretations of the heat flow pattern was that of zones of high electrical conductivity (section 5.4) at depths of 10 to 20 km. Lubimova et al (1972) considered these conductive zones to be regions of elevated temperature and perhaps partial melting, but they observed that the heat flow anomaly was much narrower than the conducting zones and concluded that high heat flow was restricted to a zone of enhanced heat transfer due to hydrothermal circulation within the faults of the rift valley. Their attempt to calculate a limiting depth to the heat source from the shape of the anomaly can not yield a useful result in this case because this model assumes heat transfer by conduction in a homogeneous medium, ignoring the obviously important role of convection within the rift. An earlier model by Lubimova (1969), in which a postulated

174 zone of fusion in the crust reached the surface at the axis of the •rift, was not presented in sufficient detail to benefit this discussion.

Thus previous interpretations of a geothermal anomaly within the Rhinegraben have associated it with a locally thickened crustal low velocity layer, and the source of the Ba2ical Rift anomaly has been related to zones of high electrical conductivity in the lower crust or upper mantle. Quantitative estimates of the depth and lateral extent of the heat sources beneath the rifts using the available heat flow data have been unsuccessful because of local enhancement of the heat transfer properties of the crust beneath the rift floor where convection occurs within deep faults.

7.2 Discussion of Kenya heat flow data : applicability in model studies.

In this section, models of the geothermal regime in the vicinity of the Gregory rift, based on the theory of heat transfer through solids by lattice conduction (Carlslaw and Jaegar, 1959), are proposed. The important role played by convective processes in transferring heat within fault zones on the rift floor, and in zones of weakness on the rift flanks, was discussed in chapter 6. Heat flow studies in other continental rift valleys (section 7.1) have shown that the highest recorded values within rift zones are usually associated with recent volcanic activity, or with hydrothermal activity in nearby faults. Mathematical analyses of heat transfer in faulted zones are beyond the scope of this thesis, and indeed could not be usefully applied to the study of heat flow through the Gregory rift floor, because hydro - geological conditions are complex (Glover, pers. comm.) and the relevant parameters (e.g. permeability ) are not well enough defined. Therefore, in this preliminary study, only models involving lattice conduction are considered.

Criteria for a threefold reliability classification of the Kenya heat flow data were presented in section 6.3, and the classified values listed in tables 6-2 and 6-3. The purpose of the classification was to establish, at least in a qualitative sense, the extent to which temperature gradient and thermal conductivity determinations from a particular borehole were representative of the temperature gradients and thermal conductivities within the surrounding rock. Where

175 measured heat flows are to be compared with the predictions of large scale geothermal models, it is also important to establish whether the heat flow through the rock in the vicinity of a borehole is likely to

be representative of the flux through a larger area (10 to 100 km?).

On the west flank of the rift, sites 21, 48 and 49 lie within , a circle radius 6.5 km centred c. 85 km west of the rift axis in an environment apparently undisturbed by hydrothermal or volcanic activity. -2 -2 The measured values lie within - 6 mW.m of their mean (45 mW.m ), which was therefore judged to be a reliable and representative value for this location. On the east flank of the rift, measurements at two sites (12 and 23) 4.5 km apart, c. 80 km east of the rift axis, -2 produced heat flows of 45 and 47 mW.m . Both values were given a 2 B classification (section 6.3) and the mean (46 mW.m- ) is therefore 2 less reliable than the west flank value. A heat flow of 40 mW.m was recorded at site 44 (class A) 60 km NNE of sites 12 and 23, and 110 km east of the rift axis.

Three consistent results were recorded in eastern Kenya, within a circle radius 50 km centred 400 km east of the rift axis. The three -2 values (at sites 34, 38. and 41) were within - 2 mW.m of their mean 2 (62 mW.m ). However the accuracy in this area is poor because a large porosity correction was considered necessary and large errors are possible (table 6-4). If the porosities used to correct the thermal conductivities were reduced to 25%, or increased to 45%, the -2 -2 heat flow would be increased to 74 mW.m or reduced to 52 mW.m . Therefore the heat flow in this region, outside isolated hot spots -2 (section 6.5) probably lies in the range 52 to 74 mW.m .

The heat flows discussed above, and listed with model predictions in table 7-k, are considered reliable and representative. Sites in northern Kenya (16, 18 and 46) are not used for comparison with models because the rift is less well defined at that latitude and the

•models may not be applicable. Other values have been excluded where , they are unreliable (C category) or unrepresentative. because of local disturbances, described in the discussion of results in chapter 6. None of the rift floor sites were selected for comparison with the predictions of large scale thermal models because the evidence presented in section 6.2 suggests that all the sites are within range of potential disturbances by recent volcanic or hydrothermal activity. Selected rift floor values are, however, used in sectiDn 2.3 in a

176 tentative thermal model proposed to investigate the heat flow pattern around a cooling intrusion beneath the axis of the rift.

7.3 A tentative thermal model of an igneous intrusion in the Gregory rift.

The existence of a positive Bouguer anomaly along the axis of the Gregory rift was noted in section 5.3. Most authors (e.g. Girdler et al, 1969; Searle, 1970; Baker and Wohlenberg, 1971) have interpreted this anomaly as being due to a major igneous intrusion of positive density contrast in the crust beneath the rift floor, but estimates of its shape and depth below the surface differ considerably. A major problem in the interpretation is due to the strong influence on the gravity profile of a layer, several kilometres thick, of low density volcanics which covers much of the rift floor and flanksz; Data from a seismic refraction experiment on the rift floor could not definitely confirm the presence of an intrusion, but are consistent with a model in which a high velocity refractor is present at a depth of about 10 kilometres.

Two models which can satisfy the gravity data are shown in figure 7-3. Model A (figure 7-3 a) was proposed by Searle (1970) to fit the observed Bouguer gravity profile across the rift on a line through the Menengai caldera (figure '6-3) and model B (figure 7-3 b) by Baker and _ Wohlenberg (1971) to fit a similar profile just south of Menengai. . A number of heat flow measurements were made in this part of the rift (figure 6-3), and a tentative model to consider the thermal effects of the postulated intrusion is proposed below. It is convenient, for the purposes of developing a thermal model, to approximate the shapes in figure 7-3 by simple rectangular prisms, since the analytical method described in chapter 4 can then be applied to the solution. In view •of the uncertain contributions from low density volcanics this simplification is considered reasonable. The dimensions of the rectangles used to approximate the two models are given in table 7-1.

The importance of incorporating the effects of latent heat and convection in intrusion models, and the usefulness of an analytical solution which could include a crude simulation of latent heat and convection was demonstrated in section 4.4. This analytical solution has been used in an attempt to determine the time lapse since

177 20 40 km

km 0 surface

20_ ( b)

40,

0 20 40 km

4/ location of main rift faults

intrusion boundary

. Figure 7-3 Models of a rift intrusion based on gravity data. (a) Searle(1970) . (b) Baker and Wohlenberg(1971)

178 width height T latent model depth L heat (km) (km) (km) (K) (kJ.kg 1)

A 5.0 16 18 1473 419 B 1.5 10 35 1473 419

Table 7-1 Dimensions of rectangular intrusion models A and B

Background Model calculated time of • 2 heat flow (mW.m ) emplacement (Ma)

A 5.16 60 B 5.01

A 3.24 . 47 B 2:80

Table 7-2 Calculated time of emplacement of . intrusions for...two levels of assumed background heat flow.

179 intrusion by comparing observed and calculated heat flows for several values of time, in an iterative fashion. The value chosen to represent the time lapse since intrusion is that which minimises the sum of the squares of the differences between the observed heat flow and those predicted by the intrusion model. This value was arrived at automatically by using a computer to calculate the sum of the squares of the residuals over a sensible time range, selecting the minimum within that range and repeating the calculation for a new set of times, centred on the minimum and spanning a reduced range. This procedure was repeated until a best fit was obtained within prescribed error limits.

A total of nine heat flow measurements were made in the rift floor o between the equator and latitude 0.50 S, comprising 5 from this survey and 4 from that of Morgan (1973). The relationship of the heat flow sites to the axis of the positive Bouguer gravity anomaly was shown in figure 6-3. Of these nine values, one was classed as A (site 24), six as B (sites 3,4,5,10,26 and 27) and two as C (sites 25 and 28). Thermal disturbances associated with the rift valley sites were described in section 6.2. They include the effects of the rapid accumulation of volcanics on the rift floor, convection within fault zones and disturbances within individual boreholes. Four of the nine values were considered unsuitable for comparison with a simple intrusion model based on the assumption that heat is transferred by conduction only. They were the two C class sites, (25 and 28) and two sites near Menengai (5 and 10), apparently disturbed by convection within nearby fracture zones.

An important parameter required in the models is the background level of heat flow (Q) which would prevail in the absence of an Intrusion. Selection of a suitable value is particularly difficult. in the present situation because the results of seismic, gravity and geomagnetic variation studies (section 5.3) suggest that a major zone of thermal disturbance exists in the upper mantle below the intrusion . which could have raised the background:leirel of heat flow in the vicinity of the rift by a factor of three. .However the effects of such a deep-seated disturbance are investigated in section 7.4, where it is shown that the geothermal regime in the overlying crust has probably not yet attained a state of thermal. equilibrium and surface heat flow may still be relatively unaffected by the mantle disturbance.

i8o 2 Two levels for the background heat flow are considered, 60 mW.m and - 2 47 mW.m , (the South African shield mean value).

The theoretical heat flow profiles for model B which best fit the results are shown in figure 7-4. Both intrusion models were assigned initial temperatures of 1873 K, which included an excess of 400 K over the assumed.liqUidus (1473 K) to simulate the effects of latent heat (419 kJ.kg 1) and convection (where melt conductivity is enhanced by a factor of five), as described in detail in chapter 4.

The results of iterative solutions for the time of intrusion of models - 2 A and B, for assumed background heat flow levels of 47 mW.m and 60 2 mW.m are summarised in table 7-2. The times range from 2.80 M 2 -2 (model B, 47 mW.m ) to 5.16 Ma (model A, 60 mW.m ).

It is likely that the injection of a large basic igneous intrusion into the upper crust would have been accompanied by basaltic volcanism. The most significant phase of basaltic eruptions in the central part of the Gregory rift occurred during the Pliocene, approximately 5 Ma ago (Williams, 1972). Thus the time of intrusion calculated from gravity and heat flow data is consistent with available geologic evidence, if a background heat flow level'of 60 mW.m-2 is assumed.

In view of the importance of convection in the rift floor, as evidenced by the widespread occurrence of hot springs and warm ground- water (figure 6-3), and therefore the inadequacy of a thermal model based on the theory of lattice conduction, coupled with the uncertainty in the dimensions of the intrusion and its mode of emplacement ( e.g it may have been formed by several phases of dyke-injection over an extended period of time), the above analysis can'only be treated as highly speculative. It does however illustrate an efficient mathematical model for calculating the time of injection of an intrusion whose dimensions have been determined by geophysical techniques (e.g. gravity and seismic methods), which can be represented in two dimensions as a simple rectangle, and above which a discernible surface heat flow anomaly has been recorded. Intrusive bodies of more complex shape, which can be represented in a two-dimensional model, can be analysed by the numerical method outlined in section 4.3. In this case the sum of the squares of the differences between observed

181

MODEL B T_72.80 MA : t)==47.3 CD CD

LL cE Lug

co

O O 0 I_ M.00 8.00 16.00 ' 24.00 32.00 40.00 DISTANCE (km)

MODEL B T =-- 5.01,MR 0=60.0

30 CO 0

F- CC LLI • m_

CD

o 1 + 1 8.00 16.00 24.00 32.00 40.00 DISTANCE(krn) best fitting profile + heat flow data point

Figure 7-4 Results of the iterative solution for time of emplacement of intrusion model B.

182 and calculated heat flows would be computed after each time step (or specified number of steps, depending on the required accuracy), and the minimum value of the residual chosen as before. A much greater computational effort is required for such a numerical solution.

7.4 The geothermal implications of partial melting in the upper mantle.

The results of gravity, seismic and geomagnetic variation studies in the vicinity of the Gregory rift near the equator can be explained by the presence of an extensive region of partial melting near the top of the upper mantle. The temperatures at which partial melting of typical mantle material can occur are considerably higher than the values expected at these depths beneath a stable continent (Smithson and Decker, 1974). If elevated temperatures do indeed prevail within the crust and upper mantle beneath the Kenya rift, a significant increase in surface heat flow might be anticipated above the anomalous zones. The purpose of the models presented in this section was to obtain an estimate of the magnitude of the surface heat flow anomaly which could be expected, and to compare this with the observed heat flow values.

7.4.1. Development of two possible geothermal models.

Different, estimates of the extent of partial melting in the upper mantle have been derived from geomagnetic variation studies on the one hand, and seismic and gravity studies on the other (section 5.3). Banks and Ottey (1974) suggested that this could be due to the fact that a greater degree of melting is required to produce a discernible change in electrical conductivity than is required to produce an observable reduction in seismic velocity. Two models of the partial melt zone are considered in this section : the first (model 1) is based upon geomagnetic variation studies (Banks and Ottepi, 1974) and the second (model 2) upon seismic studies (Long et al, 1972). In model 1 (figure 7-5) the boundaries of the zones of high electrical 'conductivity proposed by Banks and Ottey (1974) (figure 5-3c) are taken to represent the limits of partial melting, and are therefore ' assumed to be at the appropriate solidus temperatures. Heat flow measurement sites on both flanks of the rift and in the Lamu Embayment•

183 rift axis West <— I —>East

80 40 40 80 460 500 (km) , i _ 'I

• "NO

40 zone1.1

120

model 2 boundary model 1 boundary

Figure 7-5 Two posLulated models of the extent of partial melting within the upper mantle in the vicinity of the Gregory rift. Model 1 is based on Banks and Ottey(1974) Model 2 is based on Lonc; et al(1972) are within the region considered by model 1. In model 2, the boundary of a wedge-shaped seismic low velocity zone of the type proposed by Long et al (1972) was taken as the limit of a partial melt zone west of the rift valley (figure 7-5). The seismic low velocity zone is not well defined east of the rift and model 2.does not consider the region east of the rift axis. If the reason suggested by Banks and Ottey for the difference in extent of the two tyes of anomaly is correct, the proposed partial melt zones . in7.model 1 may be too narrow, and therefore the surface heat flow anomaly should be wider than that predicted by model 1, providing the other assumptions used in its construction are valid.

Perhaps the most drastic assumption in this model study is that the geothermal regime can be treated as steady-state, so that the relevant differential equations governing the distribution of temperature are of the form (4.73) and (4.50), and a solution can be conveniently obtained using the finite difference SOR technique described in section 4.3.4. Although the observation by Baker.et al (1971) that the location of the main volcanic centres appears to be moving eastwards would suggest that the geothermal regime may still be changing, there is insufficient information on the thermal history of the crust and upper mantle to adequately define:the additional parameters required in a time - varying analysis. Such information might be expected to come from detailed studies of.the relationship of various magma types, . from which lava in Kenya have been derived, to the thermal state of the crust and upper mantle. tHarris (1969) outlined some of the considerations involved in this type of study. The possible consequences for surface heat flow of a time - varying geothermal regime are discussed in section 7.4.4.

An important parameter which must be specified for both models is the background level of surface heat flow which would be observed if the postulated thermal disturbances in the upper mantle did not exist. Polyak and Smirnov (1968) described a relationship between heat flow in a tectonic province and the time which had elapsed since the last major orogeny in that province (table 1-1). The basement exposed in western and central Kenya was last affected by a major orogeny during the late Precambrian (section 5.1) and the results of Polyak and. Smirnov (table 1-1) suggest that heat flow in this region should be close -2 to 40 mW.m , if rifting had never occurred. The mean value recorded 2 in a Precambrian shield environment in South -Africa, 47 mW.m was chosen as a realistic background level for models 1 and 2.

185 2 It should be noted however, that much lower heat flows (18 to 22 mW.m ) were measured in the Precambrian shield in West Africa (Chapman and -2 Pollack, 1974) and significantly higher values (55 to 76 mW.m ) have recently been reported from Precambrian sites in Zambia (Chapman and Pollack, 1975), so that the chosen background level may be inaccurate. Further heat flow measurements in stable parts of East Africa are required to establish its validity.

Heat transfer and heat production parameters of the crust and upper mantle also must be specified in models 1 and 2. Smithson and Decker (1974) proposed that the distribution of heat production and thermal conductivity in ancient stable continental crust could be roughly represented in a three - layer model. Their hypothesis was based on considerations of the physical and chemical properties of the metamorphic rocks of which most of the continental crust is composed. The heat production and thermal conductivity values in model A of Smithson and Decker (1974) were adopted for models 1 and 2 of this study, and are shown in table 7-3. In fact the effective layer thicknesses were slightly modified for models 1 and 2 to simplify construction of the finite difference grid, and associated changes in the temperature-depth profile are indicated in table 7-3.

The temperature at which melting will occur within the upper mantle depends upon composition, pressure and water content. Ringwood and Green (1969) proposed the substance pyrolite, composed of three parts peridotite and one part basalt, to explain the observed characteristics of the upper mantle. Green (1973) described the melting properties of pyrolite, under different degrees of pressure and water saturation. The experimentally determined solidus temperatures under anhydrous and water-saturated conditions are plotted against depth in figure 7-6.

Models 1 and 2 represent the geothermal regime in two dimensions only, because the available seismic and geomagnetic variation data were inadequate to resolve major changes in deep structure along the strike of the Gregory rift. However the models represent a relatively complex part of the rift, there it is joined by the Kavirondo branch and where immense volumes of lava have been erupted. A change in the gravity anomaly along the strike of the rift which may be due to variation in deep structure along the axis of the rift, was noted in . section 5.3. The restriction to two dimensions may therefore layer thermal heat layer base temperatures (°C)

thickness conductivity production . model A* model 1 model 2 (km) (W.m 1.K 1) (pW.-3)

8 2.72 1.26 144 154 149 18 2.72 2.09 229 26o 240 18 2.09 0.21 '390 418 Lloo

Table 7-3 Crustal parameters for models 1 and 2

* from Smithson and Decker (1974)

187 800 1000 1200 1400 1600 TEMPERATURE( °C )

Figure 7-6 Variation of solidus temperature with pressure in anhydrous and water-saturated pyrolite. (from Green,1973)

188 contribute to any discrepancy between calculated and observed surface heat flows.

7.4.2 Finite difference solutions for models 1 and 2.

Model 1.

The region covered by model 1 consists of an east - west rectangular vertical cross - section through the crust and upper mantle which interseets. the Gregory rift near the equator, and extends to a depth of 144 km below the surface of the earth. The region was represented •on a 2000 point finite difference mesh. Thermal conductivity and heat production were specified at each point in a discretised approximation of the vertically layered model proposed by Smithson and Decker (1974) (table 7-3). The finite difference mesh was constructed with a constant vertical spacing of 6 km, and a horizontal spacing of 12 km in the region of interest, expanding by five increments to a maximum of 768 km towards each vertical boundary of the rectangle (appendix 11). The vertical boundaries on both (east and west) sides of the rectangle were thus far removed from the anomalous zones.

Boundary conditions on the sides of the rectangle were chosen, as far as possible, to simulate the real case. Temperatures on the. upper boundary, which represents the surface of the earth, were fixed at 293K. Since both vertical boundaries were far removed from anomalies, zero horizontal flux conditions were appropriate. Temperatures did not change from their undisturbed values on the boundaries when this condition was imposed, confirming its validity. The most appropriate condition for the lower boundary'is less obvious and two possibilities were considered : a prescribed vertical flux condition, and a fixed temperature condition. The latter would be pertinent if the lower boundary was underlain by a partial melt zone whose upper surface could be considered isothermal. Since the state of the upper mantle is not well known at this depth, both types of condition were investigated for each of the various models described below. The values of vertical flux or temperature on the bottom boundary were.chosen. to produce a surface heat flow of 47 mW.m 2 in the absence of anomalies.

To minimise the computational effort required to obtain a satisfactory solution, it was necessary to determine an optimum relaxation factor

189 (section 4.3) and assign reasonable initial values of temperature at the grid points before commencing the iterative procedure. An optimum relaxation factor of 1.882 was obtained by experiment (figure 7-7 a), and initial temperatures outside the anomalies were set at •their undisturbed values, computed in a one-dimensional SOR calculation. Inside postulated partial melt zones (denoted zones 1 and 2 in figure 7-5) temperatures were fixed according to the pyrolite solidi curves of figure 7-6, and the iteration matrix was constructed in such a way that these temperatures did not change as the solution progressed. The tops of zones 1 and 2 were set at depths of 24 and 48 km respectively instead of 20 and 50 km as suggested by Banks and Ottey (1974), to facilitate the construction of the finite difference grid.

Model 2.

Model 2 covers an east - west rectangular cross section through the crust and upper mantle west of the Gregory rift axis, in the vicinity of the equator, reaching a maximum depth of 150 km. The region was represented on a 2040 point mesh, and thermal conductivity and heat production values were assigned to each point-as in model 1. The vertical mesh increment in this case was set at 3 km, and the horizontal at'6 km expanding to a maximum , of 384 km at the western vertical boundary in five steps (appendix 11)r

Boundary conditions on the sides of the rectangle were set as in model 1, but in model 2 the zero horizontal flux condition on the eastern vertical boundary had a different'significance. In'this ease thd boundary passed through the axis of the rift:which is treated here as a plane-of symmetry. The optimum relaxation: factor, 1.943, was obtained by experiment (figure 7-7b) and initial temperatures were specified as before. Solutions for wet and dry pyrolite partial melt zones, and both types of lower boundary condition were computed, and contours of isotherms in the crust and upper mantle, and surface heat flow over the anomalies were plotted automatically for each case. Figure 7-8 is an example of a solution to model 1 in which the partial melt temperatures were fixed at the wet pyrolite solidus. Theisotherm contour values are shown (in deg C) on the, left hand side of the plot. The numbers along the top and right hand boundaries refer to nodes on the finite difference mesh. Figure 7-9 is an example of a solution to model 2 in which the vertical heat flux across the lower- boundary was fixed, and

190 220 (a) Model 1 20

180 0 • .216 0 "2-,140

6120 z 100 4 1.86 8 1.90

600

(b)Model 2 500

• 40 C U) 0 Id 300

200. 1 9 1-91 1.93 1 5 Cu

Figure 7-7 Number of iterations required for convergence of the finite difference solutions of models 1 and 2,for a range of values of the SOR factor.

191 rift axis

west east 1 .2c._HERT FLOW

coco

• 1 3 S 7 9 11 13 16 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 1111111111111111111j1III1111111111111111 a

20 70

7 221 27

32

37

42

47

52

57

62

67

72

77

82

87

92

Figure 7-8 An example of the steady-state solution for model 1(wet. pyrolite ,isothermal lower boundary case),IsOtherms in 50 deg c increments are shown and the variation in surface heat flow plotted above. 192 west

rift axis HEAT FLOW

3 7 9 .11 13 15 17 119 21 23 25 27 29 31 33 35 37 39 O 20 70 120 170 220

270

320

370 0

420

470 U.

520

rs3 570

620

670

OCal 720

770

820

O 870

920

970

— CP

Figure 7-9 An example of the steady-state solution for model 2(dry pyrolite, constant flux lower boundary case),Isotherms in 50 deg C increments are shown and the variation in surface heat flow plotted above.

193 the partial melt zone temperatures set according to the solidus of dry pyrolite.

The change produced in the surface heat flow anomaly when the lower boundary condition is changed from .a prescribed vertical flux type to a fixed temperature type for the dry pyrolite version of modell is illustrated in figure 7-10. The prescribed vertical flux condition produces higher surface heat flows near the edges of the anomalous zones e.g. 85 km west of the rift axis near heat flow sites 21, 48 and 49, -2 the computed surface heat flows differ by 4.3 mW.m between the two boundary condition types. The choice of water-saturated or anhydrous pyrolite solidi also has a significant effect, amounting to 5.1 mW.m-2 between two versions of model 1 at this location (table 7-4).

7.4.3 Comparison of observed and model heat flows.

Four locations at which a meaningful comparison of observed heat flows with the predictions of large-scale models is possible were described at the beginning of this section, and averaged heat flow values from the four locations respectively 85 lilometreg west and 80 110 and 400 kilometres east of the rift axis are given in table 7-2, (denoted 85 W, 80 E and 400 E respectively). Computed values for each of four versions of models 1 and 2 (corresponding to partial melt zone temperatures consistent with anhydrous and water-saturated pyrolite mantle, and both lower boundary condition types) are shown in table 7-4, at 85 W, 80 E, 110 E and 400 E for model 1, and 85 W only for model 2.

The computed model heat flows are significantly higher than the observed values on both flanks of the rift but the lowest model 1 values (for a water-saturated pyrolite melt) are within the possible range of heat flow within the Lamu Embayment. On the west flank. of 2 the rift (85 W), model 1 predictions are 10 - 20 mW.m and model 2 -2 2 20 - 30 mW.m above the mean of three observed values (45 mW.m ). - 2 On the east flank the model 1 predictions are 30 - 40 mW.m higher than measured values.

The disOrepancy between observed and calculated values on the.xift- flanks is too large to be accounted for by errors in the data, particularly. in the case of model 2 at 85 W and model 1 at 80 E and .110"E. Lowering the background level of heat flow to 40 mW.m-2 did not

194

rift axis constant flux type O o _J isothermal type

• co

O O O -20.00 0.00 210.00 410.00 60.00 6.00 X (KM) E101

Figure 7-10 Effect of two types of lower boundary condition on thr model 1 solution for surface heat flow. Pyrolite Bottom Computed heat flow (mW m-2) . condition boundary model 1 model 2

, type 8 w 80E 110E 400E 85W . .

B F 65 86 85 85 78 anhydrous B T 60 86 85 85 77

water-saturated BF 60 74 73 73 67 B T 57 74 73 73 66

measured values 45 46 40 52-74 45

B = prescribed flux perpendicular to boundary F presctibed temperature on boundary

Table 7-4 Comparison of measured heat flows With predictions of steady-state models 1 and 2.

196 significantly change the model results at the four sites (maximum 2 change was - 2 mW.m at 85 W for model 1). The most likely reason for the discrepancy is that the geothermal regime has not reached a steady-state, and that the thermal disturbances within the upper mantle have not yet produced discernible changes in surface heat flow. The time lags involved are examined quantitatively in the. next section.

7.4.4 Dynamic models of the geothermal regime.

Studies of the volcanic history of the area are a potential source of information on the formation and development of partial melt zones in the upper mantle. Baker et al (1971) and Logatchev et al (1972) presented the results of age determinations on a large variety of Kenyan volcanics, and their evidence suggests that activity began 23 Ma ago within the rift and 5 Ma ago'east of the'rift zone.' It can be assumed that the magmas from which these lavas were derived must have separated from a partial melt zone in the upper mantle before the onset of surface volcanicity, so that the dates given above can be considered lower limits to the times of formation of melt zones in the upper mantle.

An estimate of the timeLaken-for'heat to be transferred by conduction from a disturbance in the upper mantle to the surface of the earth through a layered crust can be readily obtained for a one dimensional case by techniques outlined in section 4.3. A finite difference scheme proposed by Crank and Nicholson (1947) and outlined in section 4.3.1 (equation 4.63) is accurate and efficient for one dimensional problems since the system of equations generated at each time step can be solved by the Gaussian elimination method described in appendix 5.

The effects of the two partial melt zones in model 1 are considered separately in this case. Temperatures on an isothermal plane, corresponding to the top of a partial melt zone, are increased instantaneously (at time t = 0) from their equilibrium values (fixed by the parameters in table 7-3) to the appropriate solidus temperature, at which they remain fixed. The diffusivity of the crust and upper. 1 mantle was set at 1 mm.2s , and solutions were computed up to a maximum time of 25 Ma, in steps of 1 Ma on a grid with 3 km spacing. The change in surface heat flow with time above isothermal planes 24 km and 48 km below the surface (corresponding to the tops of melt zones'

197 1 and 2 in model 1) has been plotted in figure 7-11. The curves show that a measurable increase in surface heat flow (taken here to -2 be 5 mW.m ) should occur 1 to 1.5 Ma after the formation of zone 1 and 5 to 6 Ma after the formation of zone 2. The heat flows should -2 be within 5 mW.m of their final values 6.5 to 7.0 Ma after the formation of zone 1 and 15 to 17 Ma after the formation of zone 2. It must be noted that these times apply to zones of infinite horizontal extent, and that they can only be considered lower limits to the corresponding times applicable to changes in surface heat flow above the two dimensional zones in model 1. If the times given above for the onset of volcanism (14 - 23 Ma above zone 1, 5 Ma above zone 2) can be taken to be lower limits to the times of formation of zones 1 and 2, it would appear that a measurable increase in heat flow should have been recorded in the vicinity of zone 1 (at 85 W, 80 E 110 E) but not necessarily above zone 2 (at 400 E). In fact no increase at 85 W, 80 E and 110 E was aPpanalit and higher values were recorded at 400 E. However the situation represented by the one- dimensional dynamic model, in which a partial melt zone is instantane - ously created near the top of the upper mantle is not physically reasonable.

Harris (1969) discussed three possible'mechanisms which(could produce melting in the upper mantle'beneath the rift:

1. By raising the local temperature above that necessary for initial melting at the appropriate depth and pressure

2. Lowering the solidus temperature below the local temperature by release of pressure.

3. Mechanical uplift of hotter solid material into a zone of lower pressure where its temperature exceeds the pyrolite solidus temperature.

Although local elevations in temperature could be expected beneath an active rift zone where the mechanical energy associated with tectonism is converted into heat, the quantity involved ,is insufficient to produce widespread melting. Bailey (1964) favoured pressure release, following upwarping in response to horizontal compression of. the African continent, as a more likely cause of melting, invoking.

.198 c)

co_

co dry 3 ZONE 1 wet -

CC 123 dry . • co _ ------wet ZONE 2

0.00 5.00 10.00 115.00 210.00 215.00 TIME(MA)

Figure 7-11 Variation of surface heat flow with time after an isothermal layer of infinite horizontal extent,at the(wet or dry) pyrolite solidus temperature is instantaneously raised to a level 24 km(ZONE 1) or 48 km(ZONE 2) below the earth's surface.Initial heat flow=47 mId.m-2(see text). mechanism 2. However the amount of pressure release is unlikely to be sufficient to cause melting unless excess heating were also involved. (Bailey later discussed the possibility of an .influx of volatiles from the mantle into the lower-crust beneath the rift, inducing melting by carrying excess heat and causing chemical change). It should be noted that simple pressure release could not cause melting unless the mantle pyrolite were anhydrous, since the water - saturated pyrolite solidus is relatively insensitive pressure changes (figure 7-6). Harris pointed out that the rift faulting causes local pressure release and the faults provide convenient channels for ascending magma. However volcanic activity in Kenya has not been exclusively fault controlled and this can not be a major factor. Harris (1969) therefore postulated that uplift, volcanism and rifting were induced by a deep - seated mantle disturbance, such as a convective uprise, local heating or both.

In this case partial melting of the mantle could occur in response to heating from below or convective uprise, and a partial melt zone might be expected to extend upwards after formation. This type of process was postulated by Gass (1970) to explain the tectono - magmatic evolution of the junction area of the Red Sea , Gulf of Aden and Ethiopian rift. The early volcanic phases in Kenya may therefore have been derived from zones of melting much deeper than those postulated by Banks and Ottey (1974), so that the times of formation of zones 1 and 2 deduced above from the times of onset of volcanism may be incorrect, and the one dimensional model results in figure 7-11 inappropriate.

In an attempt to represent the geothermaleffects of a rising partial melt zone, a more sophisticated one-dimensional model is proposed: The model examines the change in surface heat flow above a rising partial melt zone in the mantle, which starts from a depth of 150 kilometres (taken to be the base of the lithosphere) at time t = 0, and moves by a series of 3 km steps (in specified time intervals) until its present day position is reached. A one-dimensional finite difference grid of 51 points with 3 km spacing was assigned the heat production and thermal conductivity parameters given in table 7-3 and the lower boundary prescribed heat flow condition was adjusted to give an 2 undisturbed heat flow of 47 mW.m (the mean value for the South African Shield). The equilibrium temperatures compatible with these parameters

200 were calculated by SOR, as in the two dimensional model 1 (section 7.4.2). The temperatures at the depth of a postulated partial melt zone were set at the appropriate pyrolitesolidus temperature (both water-saturated and anhydrous cases were considered) and a solution for the change in temperatures with time between this depth and the surface were computed over a series of discrete time increments. A rising melt zone was simulated by progressively resetting the position of the lower boundary (after a specified number of time steps), until this isothermal layer had reached the present depth of the partial melt zone (24 km depth for zone 1 and 48 km for zone 2).

In figure 7-12, the surface heat flow at the instant the melt zone reached its present day position is plotted against the total rise times from 10 - 100 Ma. In this case, no measurable increase in heat flow 5 mW.m 2) would have.occurred directly above zone 1, if the melt zone had risen from 150 km (approximately the base of the lithosphere) to 24 km below the surface in less than 18 Ma. If the melt zone rose from 150 - 24 km in 23 Ma an increase in heat flow of c. 10 mW.m 2 is. predicted by this one-dimensional model. In the case of zone 1,.which has a half-width of 30 km, this figure is applicable only to points within several kilometres of the rift axis. The effect is sharply attenuated at points beyond 30 km east or west of the axis, as was the two-dimensional steady-state solution (figure 7 -7).. Therefore the increase in heat flowat site 85 W would be considerably less than -2 10 mW.m . Similar comments apply to model 2, although the attenuation with distance from the rift axis is reduced because the apex of the prism extends to c. 90 km from the axis.

Surfaceheat flows above rising isothermal layers which reach the depth of zone 2 (48 km) are also plotted in figure 7-12. In this case the melt zone must not rise from 150 - 48 km in less than 40 Ma if the -2 surface heat flow is to be increased by more than 5 mW.m and not -2 less than 50 Ma if it is to rise by more than 10 mW.m .

Since errors of --!20% may be possible in the class A heat flow 2 results (section 6.3), a change of 10 mW.m from a background level of 40 - 50 mW.m- 2can not be distinguished with confidence from a single measurement. Attempts to improve the accuracy by averaging over a number of measurements made within a particular tectonic environment may be foiled by systematic errors due to the effect of porosity on thermal conductivity or groundwater movement on temperature

201

0 dry C _J ZONE 1 ------wet

■■■•

dry ZONE 2 - - —

0 • 0 0 20 .00 40.00 6.00 eib.00 ibo.00 RISE TIME(MR)MR ) Figure 7-12 Surface heat flpws above an isothermal layer(at the wet or dry oYrblite solidus temperature) which is rising from 150 km below the earth's surface at a constant rate.The heat flows shown were calculated for the instant at which the layer reached 24 km(ZONE 1) or 48 km(ZONE 2),for a range of rise rates.Initial heat flow = 47 mW.m-2. gradient. To guarantee an accuracy of better than 10 % it will be necessary to use boreholes at least 100 m in deep_",in .selected sites, which have been cased and grouted and from which solid core samples have been 'recovered.

7.4.5 Summary of model results.

It was shown in section 7.4 that the presence of extensive regions of partial melting in the upper mantle, such as have been proposed by Banks and Ottey (1974), would cause surfaceleat flows on the flanks of the Gregory rift to be higher than measured values, providing the peothermal regime had reached a steady state. A steady-state regime must be considered unlikely, however, since the development of the Kenya rift system appears to have begun around 23 Ma ago and heat may take several tens of millions of years to diffuse a distance of 100 km within the earth.

Perhaps the most reasonable process by which an extensive region of partial melting could be created at depths of 20 - 50 km, is one in which melting is initiated at the base of the lithosphere and spreads upwards through the upper mantle. To investigate the implications for surface heat flow of such a process, one-dimensional thermal models were devised to simulate the effects of a horizontal isothermal layer of infinite extent moving upwards from a depth of 150 km to 24 km (zone 1) or 48 km (zone 2) below the surface of the earth. These one- dimensional time-dependent models demonstrate that the partial melt zones postulated by Banks and Ottey (1974) could exist within the upper mantle, without causing any significant surface heat flow anomaly -2 ( 10 mW.m ) across the rift valley, away from regions of active volcanism and deep faulting where local convection has enabled heat to be efficiently transferred to the surface.

The apparent increase in heat flow from the rift flanks, where 2 recorded values in stable crust lie in the range 40 - 50 mW.m , to the Garissa area 400 km E of the rift, where heat flows In:the range 52.- 74 mW.m-2 were recorded (outside local anomalous zones) has not been explained. However, little is known about the structure of the deep crust and upper mantle in the area (the eastward extent of the ' high electrical conductivity region postulated by Banks and Ottey (1974) zone.2 in figure 7-5, is not well defined), and an interpretation of the apparently higher heat flow in the Lamu Embayment must await the

203 results of future geophysical studies of the deep structure in that region.

204 CONCLUSIONS

The main objective of this study was to extend a preliminary heat flow study by Morgan (1973),. and to interpret the results in terms .• of structures within the crust and upper mantle which had been delineated by other workers using a variety of.geophysical techniques. To achieve this, it was necessary to develop field equipment, investigate laboratory techniques for measuring thermal conductivity and porosity, carry out field excursions to Kenya, reduce and correct heat flow data, and develop and apply computer modelling techniques to interpret the results. The conclusions can be summarised as follows:

1. Modifications to borehole temperature logging equipment led to greater versatility, improved calibration procedures and reduced the thermal response time.

2. A laboratory technique for determining the thermalconductivity of a rock matrix from measurements on fragmented, samples was tested for a range of rock types and proved satisfactory. However, experiments on a method proposed by Morgan (1973) for determining whole - rock porosities from measurements on rock fragments demonstrated that it was unsuitable for application to many of the samples collected in Kenya. Whole - rock conductivities were therefore deduced from the matrix values using estimated porosity values.

3. Field operations ran smoothly, thanks to the cooperation of the Kenyans, and the equipment proved robust and reliable. A large proportion of the time was spent tracing boreholes and visiting potential measuring sites. Only a small fraction of the bore:-.4 holes visited could actually be logged, because many had been destroyed, could not be found, or were being pumped.

4. A large variation in quality was evident in the heat flow results obtained from 26 sites, and the data were therefore sorted into three classes according to reliability, the most reliable of which was judged to have an accuracy of better than - 20%. Corrections for the effects of past climate, topography, uplift, . erosion and sedimentation were applied. No corrections were applied to compensate for the potentially greater disturbance by

205 moving groundwater because of a lack of hydrogeologic information. Data from Morgan (1973) were corrected and class- ified according to the same criteria.

5. A highly erratic pattern of heat flow on the rift floor was characteristic of a geothermal regime dominated by recent igneous activity and hydrothermal circulation within faults. On the flanks of the rift, the pattern was less disturbed, with most values typical of a Precambrian shield environment. Isolated maxima were apparently associated with zones of weakness in the basement. In the Lamu Embayment of eastern Kenya, local highs in the vicinity of subsurface horst - like features were superimposed on a uniform background level, apparently above that recorded on the rift flanks.

6. Mathematical models were constructed to examine the geothermal implications of an igneous intrusion within the crust beneath the floor of the Gregory rift (Searle, 1970), and of broad regions of elevated temperature within the mantle in the vicinity of the rift zone (Long et al, 1972; Banks and Ottey, 1974). The rift intrusion was modelled in two dimensions by analytical and numerical techniques, and the validity of an analytical solution which incorporated a crude approximation of the effects of latent heat and magma convection was demonstrated. The time of emplacement of the intrusion was calculated using selected rift floor heat flow values, and the result shown to be compatible with geologic evidence. Two - dimensional steady ,.. state thermal models of partial melt zones within the upper mantle predicted heat flows on the rift flanks higher than measured values. However, time - varying one - dimensional models, featuring a l rising melt zone, showed that the postulated partial melt zones (Long et al, 1972; Banks and Ottey, 1974) could exist without causing a significant heat flow anomaly at measurement sites on the rift flanks which had not been affected by hydrothermal circulation or igneous activity.

206 RECOMMENDATIONS FOR FUTURE WORK

Future heat flow surveys utilising boreholes which have not been sealed from percolating groundwater could usefully employ , a sensitive downhole flowmeter to detect fluid movement within a borehole connecting permeable zones. Where only drill cuttings are available for thermal conductivity measurements, downhole porosity logging (Schlumberger, 1972) should be considered.

Model studies have shown that the effects of a thermal disturbance in the upper mantle, which could cause observed seismic and geomagnetic anomalies, may still be apparent at the surface 2 as a heat flow anomaly of only 10 mW.m . Extensive faulting and volcanic activity, and the associated convection, make the rift floor a totally unsuitable environment in which to attempt to discriminate an anomaly of this magnitude. Further heat flow studies aimed at investigating the thermal state of the upper mantle should therefore concentrate in making accurate (error<10%) measurements in tectonically stable areas on the flanks of the Gregory rift. To achieve this accuracy it-will be necessary to use carefully sited boreholes (i.e. sites selected to minimise the disturbances discussed in chapter 3), which have been cased and grouted, and from which solid core has been obtained. Depths of not more than 200 metres should be adequate. Several granite plutons exist in western Kenya where a combination of heat flow and surface heat production measurements would be a potential source of information on the mantle contribution to the surface flux.

The accuracy of the. present survey can be considered adequate for future rift floor measurements, where the only meaningful values are likely to be averages taken over a large number of measurements within areas of several hundred square kilometres.

Future measurements in the Lamu Embayment should investigate further the positive heat flow anomalies associated with the basement highs. Problems in obtaining accurate thermal conductivity measurements should be anticipated because of the friable nature of the sediments in that region. In-situ methods in specially drilled boreholes may be necessary.

207 The model intrusion studies could be usefully adapted to investigate the effects of metamorphism in country rock in the vicinity of cooling magma. Previdus studies have often ignored the effects of latent heat and convection Oithin magma in producing higher maximum temperatures in the surrounding rock.

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226

Appendix

SI Units

SI units have been used throughout this thesis. In this section, base, derived and associated units in the SI system relevant to the thesis are listed and equivalents of superceded units are given for reference.

Table A1-1 SI base units

Quantity Name Symbol length metre m' mass kilogram kg time second S electric current ampere A thermodynamic temperature kelvin K amount of subtance mol mol luminous intensity candela cd

Table Al-2 SI derived units

Quantity Name SI Symbol exponents of base units m kgr-- s ,A force newton N 1 1 -2 energy joule J.N.m 2 1 -2 - power watt W.J.s 1 2 1 -3 -2 pressure pascal. Pa-N.m -1 1 -2 density kilogram per - cubic metre kg.m 3 1 kinematic square metre viscosity per second m 2 .s -1 2 -1 dynamic viscosity pascal second Pa.s -1 1 -1_ e.m.f. volt V.W.A 1 2 1_ -3 -1 resistance ohm .V.A 1 2 .1 -3 -2 magnetic flux weber Wb.V.s 2 1 -2 -1 magnetic flux 2 density tesla T=Wb.m 1 -2 - conductance siemens S=A/V -2 -1 3 2

227

Quantity Name SI Symbol exponents of base units m kg G • specific heat capacity 2 -2 -1 thermal conductivity W.m•-1 .K - 1 1 1 -3 -1 -1 frequency hertz Hz=s -1 .

Table A1-3 SI prefixes factor .refix bol factor .refix bol

12 -1 10 tera T 10 deci d 9 10 giga G 10-2 centi 6 -3 10 mega M 10 milli m -6 103 kilo k 10 micro 9 102 hecto h 10 nano n 1 -12 10 deka da 10 pico 10-15 femto f 10- 18 atto

Table A1-4 Additional units (non-coherent) time : minute (min), hour (h), day (d), year (yr), century (cy), million years (Ma) temperature (scale) : °celsius = (Kelvin:-273.15 K) temperature (interval) : 1°C = 1K

Table A1-5 Superceded geophysical units and SI equivalents

Quantity SI symbol superceded units (equivalents) force N 105 dynes 5 pressure Pa 10 bar 7 energy J 10 ergs 0.24 calorie (cal) -2 acceleration tlITI.S 10-1 milligal dynamic viscosity Pa.s 10 poise 4 kinematic viscosity m 2.s -1 10 stokes magnetic flux density nT 1 gamma - - - thermal conductivity W.m-1 .K -1 2.389 mcal.cm .s1 1.o c 1 2 -2 -1 heat flux mW.m-2 2.389.10 ucal.cm .s

228 Appendix

Correction for the effects of surface topography

. A method of calculating the geothermal effeCts of topography devised by Jeffreys (1938) is described here.

Irregularities on the surface of the earth disturb the 'temperatures below it, and produce local changes in the geothermal gradient.

If the therMal interaction between various elements Of topography can be ignored, the problem can be reduced to one of a semi infinite solid with prescribed surface temperature, defined by:

F(x,y) (g - gI ) h (x,y) (A2.1)

where the xy plane is a reference plane from which h (x,y) of the topography is calculated, g is the undisturbed geothermal gradient and g' is the adiabatic lapse rate in the atmosphere.

The temperature disturbance.at depth z below the reference plane may be written (Carslaw and Jaeger, 1959) :

_ 1 c z F(x,y)dxdy T (x,y,t) f7r R3 -00 -00 (A2.2)

where R = ( zqs- (A2.3)

r = (xt+ (A2.4)

The temperature-gradient in the z direction can therefore be written 00 a, (R2- 3 z2)F(xy)dxdy _- 2 1 rt Rs -00 -00 (A2.5)

it is convenient to transform (x,y,z) into polar coordinates (r,e,z) and write : 2.1-r rsinO)de 1/(0= — F (rcose) (A2.6) 2 -rr 6

229 (A2.3) can therefore be expressed 00 2 ?)1- _ c'r(R -2)3z V(r)dr R 5 o and using (A2.4)

00 r ( -2 t) V(r)dr (A2.7) ( r2+ z2)4'2- 0

230

Appendix 3

Temperature disturbance near an intrusion

Following the discussion in section 4.1, a solution is developed for the following set of equations

1 9 2 2rr YL-= V G(C),Z) (4.8)

= 0 , 9 =0 (4.9) . 1 , 0. 0 = (4.10) o Y4/ =o ; (4.11) -4 00 e_? 0 (4.12).

=0,0.+A 9 .1-x-f- (g.--g„1 h — (4.13) 7=0,13 4c41-C =0, A 9=0 (4.14) 4/6, 1- =10 Os 4B 9 =0 (4.15) 1-C ss

Consider the following integral transform 00

0 ( pin,rt) C1(. , c, t) cos (p sin(nTrI)dc d (A3.1) 0 o The corresponding inverse transform may be written (Sneddon, 1951) 00

G (1),n;t)sin(m-r)cos(.p)dp (A3.2) n=3. 0 Transforming both sides of (4.8.) according to (A3.1)

1 = -(pz+ nzTr2) 9 ( p,n7) (A3.3) so that • 2 2+ p2r-C e = Q11) n ) a ( n'ir (A3.4)

231 where Pin) ( p,

A 1-c.

0.(p n) [T,t(9;gr) h (1-9 lcos(P)sin(n n- c)dcd 0 g (by 4.13,4.14f4.15)

G(n) (go_gri) h sin(pA) nn p where

6(n)= R1 - Ncos(nrrB ) - Ccos(nn-(1-C)) +(sinhn(1-C) - sin(h ak, Tr

+12( (cos(nrrB) - cos(nri{l-C)) h (go-g,r) (A3.5)

Using (A3.2) to transform (A3.4) 00 00(Rn) E ( n2T.T113rr sin (n n q)cos(tp)dp c0 = h ( go-g )r j(nrrtrsin(nTrIgerf AL:1 erf =1. n n 22* 22" using the relation (A3.6) 00 e-X2-Sif1(2XY)dX = :LT• erfy 2 0 Y 2 where erfy = eu d u (Sneddon, 1951)

0

232 Appendix 4

Finite difference solutions for the diffusion equation

A solution to the following equations :

x,y E (4.49)

u =. Xy e (4.50)

x,y ER u (4.51) where the parameters are as defined in section 4.3.1, R. . is a rectangular region of (x,y) space andqb is the boundary ofP.

The first step in the development of a finite difference solution for the above equations involves replacing the continuous domain representing R by a mesh of discrete points at which values of c,K S and u (at time t = o) are specified. Values of Ck lp, 6 and pare specified on the boundaries.

A corner of mesh used to approximate the rectangle is shown schematically in figure A4-1. The following variables are specified at each point on the mesh :

Uij = temperature Cij = product of density and specific heat Kij• = thermal conductivity Sij = heat production and the mesh spacing is specified by :

hj = horizontal increment (i,j) to (i,j + 1) Vi = Vertical increment (i,j) to (i + 1,j)

The grid consists of N rows and M columns of points in a

233 boundary A (0c=0,n=0,x=1,0=140) (1,i)

x x .(i-1,j)

vi \--hi--- kx(i+1,j) o >0 ( b)

x(N-1,j)

1) (N,j) (N,j+1) boundary C

-Figure A4-1 Schematic representation of a finite difference mesh(see text).

234 :rectangular array. In equation (4.55), C is a square diagonal, matrix of order P containing the variable Cij where

N.M,

and S (containing Sij) and Uij are vectors. V and H (and A) . are also square matrixes of order P containing Kij, hj, and Vi and incorporating finite difference approximations to the boundary conditions expressed in equation (4.50). Two types of boundary types are illustrated in figure A4-la : boundary A is a fixed temperature type (in 4.50AK = 0, be = 1, ef =g50 ) and boundary B is a zero perpendicular flux type, used in this study. to simulate the conditions at a symmetry axis (in 4.50, CX = 1, = 0, 6 = 0, 0 . 0).

In a regular grid, spacing h, the continuous differential operator 5i( ...k 1_7() at an internal point (i,j) (figure A4-1) can be

-2 . I replaced by the finite difference operator hNk.bx.i , defined by: ti( k 6,4) u6 = k- (u. - u< )] 2

where k .)/ (A4.1) .4i *)i 2 , . /2 4.-4F . ( L) k 4- )

In an irregular grid (figure A4-1b), v--( k can be replaced by Rx, defined as f011ows :

Rx u j Nc+ (A4-2)

Similarly can be replaced by F264 k 6,3) ic7)7)(1<

for a regular mesh, and Ry for an irregular mesh.

Equation (A4.2) can be written :

• =- bi C; (A4.3):

235

where

(A4.4) ve_d

The corresponding equation for Ry can be written :

.•= d.u.. + .e- u R u1-,3 L (A4.5)

where kw h,a

et: = k;441; (A4.6) h3 h.3.4) fi = 34T hi( h.i+ ,)

Two types of boundary condition will be considered, prescribed temperature type and prescribed (perpendicular) flux type.

Prescribed temperature boundary

Boundary conditions of this type are characterised by the following values of the parameters in equation (4.50) :

= o p =o =1 0= Olx,y)

Then temperature U(x,y) has a prescribed value 0 (x,y) on the boundary of the rectangle. In a finite difference formulation, (x,y) is approximated at points on the boundary by discrete values q5i,j .

Prescribed flux perpendicular to boundary

In the present study boundaries are parallel to the x and y axes so that two sets of parameters apply to this boundary condition :

0( = K(x,y), (3 X= 0, 0 .0(x,y) (boundary 1.1 y axis)

0( 0, = K(x,y), = 0, . (x,y) (boundary II. x axis).

236 . Consider the application of this condition at a. boundary point N,j) on the mesh (figure A4-1c). The flux across boundary C,

k (x y) aU , can be approximated as follows •

k • - kJ.1t, )16A g u ,- 114.11.4 ) 2

(A4.7) where (N 1,j) is an imaginary point outside the mesh, as shown in figure A4-1c. Equation (A4.7) can be used to substitute for the imaginary value U N4t,j when the operator Rx of equation (A4.3) is applied at a point on boundary C.

Tridiagonal Matrices

Equations of the type (A4.3) (with appropriate substitutions from equations of the type (A4.7) at boundary points) can be • written in matrix form, and it is clear from equation (A4.3) that the matrix operator will take a tridiagonal form. Thus the matrices H and V in equations (4.65) and (4.66) are tridiagonal.

237

Appendix 5

Solution of a tridiagonal system of equations

The procedure outlined below is discussed by Mitchell (1969). It is efficient and readily adaptable for machine computation and has therefore been used wherever applicable in the computer programs written to perform the finite difference solutions of equations (4.49) to (4.51).

Consider a matrix equation

Az where A is a given nxn tridiagonal matrix and k is a column vector.

Then if A and k have the following forms

A k

the matrix problem can be solved directly:by the following simple algorithm the components zi of the solution vector z are given recursively by

z„ =g,,

zi = g(.7"ivi 1 .11...; n-1 cl where

k, ki- 7-' 3" gi bt.

238 Appendix 6

Tabulation of heat flow data

A full listing of the borehole temperatures and thermal conductivity data obtained in this study are given below. The heat flow sites are numbered 24 to 49, and the latitude and . longitude of each are listed with the data. Temperatures (TEMP) are given in deg. C, and depths in metres below the surface. Depth sections in which unstable temperatures were recorded are indicated (UNSTABLE SECTIONS). The thermal conductivity values - 1 -1 shown (in W.m .K ) are uncorrected for porosity. The depth range in metres over which a conductivity value applies is shown (REGION).

239

36.-

TEMFF-1A _DAT Th th- P71- Tf t:F' - 0-r '04 f rrp 'DP114 :":7VmP , 15.10 2.r: .-E1 4.5.73 21.39 60.4: 75.50 23.%S4 25.22:. 105.70 PE. 63 1•23.:.27.72. • 151.•t'0 156,.14) '31.55 1 2f:'• 32.2 7 ' 32.-54 1st :32.71.- 60 32.72 1 a7.3 32. 79. 188.5.1 I+ 3 3 1(34./:,F. 19 E 33.97 197.0 33.37 19'4 .14 C. 7:F.4E 200.°,g 3:3. 55 2D 9,40 -9 03.(_-,0 33.77 215:.40., 33.91H 21.6.91 33r 96 %;'7, 1.0 34p t'a4 20 r cr 74.17 211-40 '34.1.8.-•• 214.5i 74,, 40 2 -1 r L 7 4f 51. 217.5: ::::=1.E5 219. 34475 _ 2243.5u 34. ("4 223. 5,1) 135. C7 225 .30..35..16-.; 22"6.5i: {5.24 UNSTAT'Ll:: 15.1 1b1 .2

I HE F A C pun vri: sip G I Of/ __ CONn GIG!, • F. r-- G1 CONn 224.E 2`9...7 _ . . 1.1tb. 2L9.7 _ 1.52 220.4 245.7 1.2 945.7 27 1 .0

KENYP "25 P.27 tV 1.;'• S.. 7E.77F -

TEV, PE_kPliP,,E

VIP7; 30.7E 19.65 . 3f- .60 1C.f u . 43.L0•• 20.16 /49.10 21.29 55.3;1. 2C. 53. 63.140 20.9:3 04 .!., 0 -21.17-E7.5::.; 21. 2$3H 7r:.F-7 21 . 4?t 73.73 21 .57 . 76.70 21.70 77,6': 21..96 ; F2ELG 22r 11-•,•-• -436x ;30 22, ?7-- Lii-22e- 56 -221. 59 95,20 22 e 58 90 29,;07 101.3r3 1L-. 4.4E 23.17 ' 1117. 4 J 27,20 110.51 23.2. 113.-60 23.25 119.70- 2.3.3 4 125.c..tu 23.35 • 1.37 2. Cif 27.36 . 130,16 23,56 -141.2G .23.59 104.30 23. 59 1/4 7.7r: 23.5° - 1 5r_41.4n 2 44,1:4 153r 50 - 214, • •;

• 36.7

TONt9UC=T'IV?T.Y -1JtT tt --1

REt:-.1ut■ 101 153r

:5E -,6 G

TEMPERi;.TURI-_7 1):7'.,-1- 11 DEPT Fi;.7• TEHP- • - "1-171'4P - • 15.13 21.E4 22, 22..1P. 31.7: 22.27 22.32: • -3.61j "?7.47 I- 77c. nn 22E49 . 22z51:: - ??,- 45L L5,:7-0 46.:10 22.71 51.4,;j 22.7. 52.9:1 22. b2 54.41j .5.90 57.4a 22.84: 5r -(7,1 92. P.7 22.C1' . n.12: 23. 2"r 26 • . LI) •23.t.2 • 72. rq; 23. R6 •75,5'0 - 214.29 .F1 u rn _: 24.42 E 4. EJU 24.63 7. r,6 •24 .7, 9L.Flf1 211. 9 0 1';24.09i3 25.715 .9r)..2rj- .25..1L 96.71 2E .22 G2c:t .2J C.70 25.79 1-01.22- 25.147•:: - 1'32,72 25.-57 1 C45% ls: 25,61 tit'Sik:ECTIOf\S• 66-;:4

PET: 1 - CC 01.:(.77TV7 171,.1 EST 1 ;-..P1 Eg .V.AL UFF: --r-ntmtit!TT-rr:T7r 71).1 105.7

K E VnY

TEV.1✓EI'./2Tlic.r DP 1 A-," CEF111 4t1',P H 15.70 .17,70 27.t../J 1.R.11. 70.70 _I c..44. 32.20 15.56 1",-. 5: -35.71 1.:::1 r 2 36.1_0 :1':.73 79,, q d 18.79 47.01! IS. F6- 4r3, .C1-19.1.:! 49.11 19.20 52.21.1 19.20 • 53.7C 13. E.-2. 51.7,70.,1-•%:‘ 5,-, 3..J 1::,,• ci:: 6:.. 4 r. lc ,15 24 00.1' 1.17 64.5 19i:4 .. , i1.,,bis- 4 57,-..r.:,:', lc 19,49' 70.5,1 1 : • _.7 r) i • / ,_ • 1 J l''.`:-., • , - 73.7J 1'43.74 75.25 2 76..71; •1'is. 87 _. .. 7 .:-.3ii .- 20.V1 • 79.t;0 20.12- ft1.14%.;. 243.22 82.11 23 27 -:.4.4'i. 7f1 .:..;'," P.,;71. t,11 2 -..i .37 r7.5". _650■A 2r.;. 46 • 9 r,r 6 11 2 0c. 5 2 92,. 19 29. 5.4 2 0,57 95e 2ij 2,.:.61 :'. " 9 r-, . 70 2.0 . F....4 . 951.2o 20.69 2U .72 101.h 21.73- - 102.30 7`,"2.73 '104.41 2'].7? 1 1'5.2 ?!:`.71 107.4ii 21.1. 61 :1-.1c.1.),71 2: . E1-.. -11.1.51 7 6.1...3, 112. ti? 2C . s4 113.6.. - 2:i.E.6 . 11.13:.12 i 2:3, 1::::-- , - .116,60 '.,.... 3,- Ecc -I 116r 6u 2Cr Fs, 11E.n 2i'?,- 119,70 - 20,,9,5 121,2D 21, Cr: 122, 21..07 124.30 21.I'.t.-- 121-...01-21-.:12- ': 127.40 21.16 123 ,c:L: 21.2.11 130.5b 21.24 132," 21 .:-.'F, • 137.50-21.72 .7i5. 1 J 21.35 135.61; 21.3.--. :.13.,..19. 21.41_ • 139..7ii 21.45 1L1.20 21.49 .142.70 71,' 57 •." ..• 144 r_ 3:1 21 i-c;.;.' ' 145c 1-_- :.. 21 f ri_ 1L7r 21,66 14B.c.; 21.71 • 15.71. tC3 21.7:7. - i 51.'--LI 21.77 153.9P 21.1_1 155. Jt 21.84 • - 15■..-,.1.7.fi 21. i._:, 158.10 21.92 161.1:-■ 21.99 .:-..162.79 27. E:-.7...- -164.20 - 22.09 165. 22.14 167.60 22. :t 16;__. t-. -' 27.23 i 7), 143 22, 7,1 171, .173.41j 22i. 44 I :17q, Or-1 22,5:: • 176., 50 22,57 172• IT.r; 'e•- 65 179466 22.72 - •1!--.1 ..10 27.1.0 1j:2.63 22.f. 7 r. 14 • 2t 72.92 1E5.7L 22.98 .,_.1.`:..?7.23 .23.114 _ 1.8..:, .6 ti 22.0E .193. .3u 23.26 191.E0 .23.32 - 193.40 23.32.._.._ UN Ti SECT 15.3 98.2

IF.E.R■;iL.L- .L:111\ r1N r, oNn 51.2 61.0 : .1..65 -1.60. 1.4S- 6503 ;7'71,4 53 01,4-101., 7 1. 13E) 106.7 :115, S 1,57 1113r 121c 9 1..5E- 121c q..17.-1 .1.62-. - 131./ 137.2 1..52 137.2 1 I 1. 1c2. 4 .170.7 1.5P • 17/1.7 1E;9.0..-....1.61.- 241

DEPTH- .".:.13E-F114 :TEMP. CEP1 Y_ TEMP 15.1' 2J,C6 221 60 21. 32. 'tit 21 21..77. 2 22t.07 .36c20. 22: 34 _ . 42..-31322r 7E_ 45.3u 22.96 430. 23.21 23. 52. c.;.T., 54.40 23.59 57 .4. 13 71.) /40 2H. 79 63.40 E5.46 24.01 65.50 24.17 7.2. 5.9f. 24..21 75.5: E1.6Lf 24,E. 0 '14 24..1 7, 6;3 2 6(1 25.. 47 25. 6D 96:7 6 25.5Z 102.70 26.01 2f-.11 - 11:5.71 26.22 1::7.20 26.34 2E .46 112.2D .76:57 .• 111. 20 2E.O5 11-3. 3':! 26.64 11/1c5 26. 69 116i 371 26E 7.5. - 117, EC 26,.7. 26&F.3 26.80 122.70 C7-‘ 1 ''77. 1.2 5 . • r: 26. q9 126.17.1, 27.04 12 ;_i 27.12- 131.40 •27. 23.h 132.53 77:44 1-311.40 - 1 27.F7 177.40 27..59 13(.;27. 6 • L7c.-J 27.fi f L 27.63. 1-6 3. 57 27.. E..4 145,00 27r E6 17-4bc 50 2717 03 '27r65 t40. _ 151 .03 27.73 157. 6P - 27. - 175.6.5 27..60•■ 157.10 27..1;1 15:- 27. 27.65 27.d7 163.1 6 27.1-1 5 1 6/1.an „ 1‘:,5.111 27..50 157.6 27.91 • 1623. 2U (.-.2 7.554 27 27. cb 17'7. l 27. S'? 175.26 2A, {14 176r 7'73 .2jir 14 .179..711 28 .-41 2'• . 56. 1s4 .3" 2c . 135.FU 29.13 187 29. 42 2Y.F-(5 1 9 Z. 31 29.67 1.(.)1.ED- .193.3t1 311 .11*- _ . . . . nr,"1 ES1 IMP1E0 .1lALUES . . F CI N CONC:Irl 3 VI TY 15.2 '1.3

KENYA 29 0.28 DEG S 35.43 DEG E

T8HPERATURE DATA . 'DEPTH TEMP DEPTH TEMP DEPTH TEtIP DEPTH TEMP 15.10 17.40 22.60 17.36 30.20 17.36 37,180 17.42 45.33 17.53 52.90 17.70 60..40 17.91 6340 17.199 66.40 18.11 69.50 19.22 72.50 19..33 75.50 16.49 77.00 18.55 78.50 16.64 80.00 18.74 81.60 18.80 83.10 18.88 84.60 18.94 65.10 19.04 87.60 19.22 89.10 19.30 90.60 19,41 92-10 19.57 93. 60 i5.71 95.20 19.79 96..70 19,90 98.20 20.02 99,70 20.12 101.20 20.24 102.70 20.38 104.20 20.49 115.70 20.64 107.20 20.83 108.80 20.91 110.20 20.97 111.60 21.06 113.30 21.21 114.90 21.25 116.30 21.34 117.80 21.44 119.30 21.53 120.80 21.62 122,30 21.71 123.80 21582. 125.40 21.96 126.90 22.03 128.40 22.15 129.90 22.26. 131.40 22.34 .132.90 22.45 134.40 22.60 135.90 22.67 127.40 22.79 179.80 22.87 140.50 27.02 142.470 23.12 143.50 23.24 145.00 23,76 1461.50 23.43 14600 23.53 149,50 23.63 151.00 23.75 152 , 60 27.77 154.00 23., 81 155.60 23.66 157.10 23.88 156.60 23.50 1E0.10 23.93 161.60 23.96 UNSTABLE SECTIONS 15.1 52.9 78.5 80.0

THERHAL CONDUCTIVITY DATA(NON-POPOOS) REGION COND REGION COND REGION COND • 51.8 54.9 1.76 54.9 57.9 1.65 57.9 61.1 1.75 61.0 64.0 1.70 64.0 67.1 1.75 67.1 70.1 1.69 70.1 73.2 1.77 77r2 76.2 1.65 76.2 79.2 1.73 79.2 82.3 1.82 92,3 85.7 1.65 85.3 ' 89.4 1.74 66..4 91.4 1.:52 91.4 94. 5 1.67 94.5 97.5 1.64 97.5 100.6 1.71 190.6 107.6 1.90 103.6 103.6 1.81 103.6 106.7 1.61 106.7 109.7 1.77 109.7 112.8 1.81 112.9 115cP 1.76 115.8 116.9 1.80 118. 9 121,9 1,55 121.9 125.0 1.72 125.0 129.E 1.77 12E.0 131.1 1.62 131.1 134.1 1.64 134.1 137.2 1.80 137.2 140.2 1.77 140.2 143.3 1.72 143.3 146.3 1.67 146.3 149.4 1.63 149.4 152.4 1.64 152 W /55.4 1.63 155.4 158.5 1.63

158.5 161c5 1.73 161:5 164.5 1.65•

KENYA 30 0.46 CEG S 34.72 DEG E

TEMFERATUPE DATA CEPTH TEMP DEPTH TEMP - DEPTH TEMP CEPTH TEMP 15.10 23.92. 18.10 23.94 21.20 23.95 24.2023.95 27.20 23.96 30.20 23.97 33.20 22.99 26.20 24.00 39 .30 24.01 42.30 24.01 45.30 24.02 48,.30 24.03 51.40 24.03 54.40 24.04 57.40 24.04 60.40 24.'04 63.40 24.04 66.40 24.05 65.59 24.06 72.50 24.06 75.50 24.06 78.50 24.07 81.60 24.07 84.60 24.07 87.60 24.08 90.60 24.09

THERMAL CONDUCTIVITY DATA ESTIMATED VALUES REGION CONDUCTIVITY' 15.2 90.6 1.50

243

KENYA 31 0005 DEG S 34.93 DEG E

TEMPERATURE DATA.. DEPTH TEMP DEPTH TEMP DEPTH TENT DEPTH •TEMP 15.10 23.38 16.60 23.44 18.10 2.50 19.60 23.59 11.20 23.64 22.60 23.69 24.20 23.76 25.70 22081 27,20 234 68 22*70 230.9.5 30,20 24,02 31,70 24.09 33.20 24.17 34.70 24.23 36.20 24.32 37.80 24.38 29.30 24.45 40.80 24.53 42.30 24.60 43,80 24.66 45.30 24.75 46.80 24.60 .48.30 24.84 49.80 24.91 51.40 24.97 52,90 25,01 54.40 25.09 55,90 25.16 57.40 25,23 58,90 25.31 6040 25.38 61;90 25444 63.40 25.51 65.01 25.57 66.40 25.64 68.60 25.76 69.50 25.81 71.00 25.89 72.50 25.97 74.00 26.01 . 75.50 26.06 77.00 26.13 7&.50 26.21 80.00 26.27 81066 26433 82019 26.141 84,60 26.49 86.10 26.56 87460 26.65 89,10 26.74 90.60 26.80 92.10 26.87 93.60 26.94 95.20 26.58 96.70 27.01

THERMAL CONDUCTIVITY OATA(NONFOROUS) REGION COND REGION COND REGION - COND 15.2 18.3 3.50 18.3 24.4 3.87 '24.4 30.- 5 3.75

.KENYA 32 - 0414 DEG S 35.04 DEG E

TEMPERATURE DATA • DEPTH TEMP DEPTH TEMP DEPTH TEMP DEPTH TEMP 15.10 25.69 18.10 25.81 21.20 28.96 24.20 26.05 27.20 2-6.28 30.20 26.57 33.20 26.75 36.20 26.88 39.30 27.03 42.30 - 27.16 45.30 27.31 49470 27.44 51.40 27.65 54,40 27.82 57,40 28,50 60.40 . 28,72 61.90 28.92 63.40 29.06 65.50 25.21 66.40 25.37 68.00 29.52 69.50 29.64 71.60 25.67 72.50 29.68 74.00 29.69 75.50 29.71 77.00 29.72 78.50 29.73 60.00 29.75 81.69 29.75 83,10 29.75 64,60 29.74 86.10 29.77 87,60 29.78 89.10 25:78 •90.60 29.77 92.10 29.79 93.60 29.86 55.20 25.61 96.70 29.81 58.26 29.83 99.70 29.61 101.20 25.82 1C2.70 29.84 104.20 29.84 105.70 25.65 107.20 - 25.64 108.80 29486 1100 20 29.06 111.80 25.87 113.30 25.87 114,'60 29.89 116.30 29.90 117.80 29.68 119.30 25.91 120.50 29.91 122.30 29.92 123.80 29.93 125.40 25.94 126.90 29.95 128.40 29.96 129.90 29.58 131.49 25.59 132.90 30.00 134.40 304 05 135,90 30425 137,40 -3E079 139,00 31.05 146.50 31.07 1426 00 31r10 143,50 31,12 145.00 31.13 146.50 31.14 148.00 31.15 145.50 31.16 151.00 31.17 152.E0 31.18 154.00 31.19 155.60 31.21 157.10 31.24 158.60 31.31 160.10 31.43 161.60 21.45 163.10 31.47 164,60 11448 166,.10 31.49 167. 60 31.51 165420 31.52 170.70 31.53 UNSTAELE SECTIONS - 134.4 139.0 - 154.0 158.6 4 THERMAL CONDUCTIVITY DATA •ES1IMATED VALUES . REGION - CONDUCTIVITY . 18.0 170.7 1.00

244

KENYA 33 0.14 'DEG S 35.26 DEG E •

'TEMPERATURE DATA

DEPTH TEMP DEPTH TEMP DEPTH TEMP DEPTH TEMP '60.40 24.07 61.90 24.10 63.40 24.14 65.00 24.18 66.40 24.20 68.00 24.24 • 69.50 24.27 71.00 24.30 72.50 24.34 74.00 24.37 75.50 24.39 77.00.24.43. 76.50 24.47 80e00 24.49 81.60 24.52 83.10 24.54 84.60 24.56 86.10 24.60 87.'60 24.67 89.10 24.74 90.60 24.76 92.10 24.86 93.60 24.94 95.20 24.95 96.70 24.97 98.20 24.99 99.70 25.01 101.20 25.02

THERMAL CONDUCTIVITY.CATA(NONPOROUS) REGION COND REGION COND REGION CONO 48.8 61.0 1.80 61.0 73.2 1.91 73.2 79.2 1.97 79.2 91.4 1.82 91.4 103.6 1.82

KENYA 34 0.04 DEG S 39.94 DEG E. ,

TEMPERATURE DATA

CEPTH TEMP DEPTH: TEMP DEPTH TEMP DEPTH TEMP - 30.40 33.59 35.20 33.79 39.30 33.98 42.30 34.11 45.30 34.22 . 46.30. 34.31 51.40 34.44 54.40 34.51 57.40 34.58 60.40 34.72 63.40 34.60 65.40 34.89 65.50 35,00 72.50 35.13 75.50 35.25 78-50 35.37 81.60 35.45 84.60 35.55 87.60 35.61 90.60 35.71 93.60 35.78 9E.70 35.85- 99.70 75.96 162.70 36.07 105.70 36.14 100.80 36.22 111.60 36.30 114.60 36.42 117.90 36.48 120:80 36.56 123.90 36.63 126.90 36.72 129.50 36.82 132 , 90 76.89 13590 36.95 139.00 37.17 140.50 376'18 142.00 37.22 143.50 37.24 145.00 37.28 146.50 37.32 14R.00 37.36 149.50 37.40 151.00 37.43. 152.60 37.48 154.00 37.52 155.60 37.56 157.10 37.61 156.60 37.66 160.10 37.70 161.60 37.78 163.10 37.85 164.60 37.90 166.10 37.95 UNSTAELE SECTIONS 30.4 36.2.

THERMAL CONDUCTIVITY - DATA(NONFOROUS) REGION COND PEGIOIS COND REGION ' CONO • 39.6 42.7 5.73 42.7 45.7 3.48 45.7 48.8 2.93 48.6 51.8 2.91 51.8 54.5 2.56 54.9 57.5 2.86 57.9 61.0 2.85 61.0 64.0 2.93 • 64. 0 70.1 4.32 70.1 71.9 3.00 71.9 73.2 2.87 73.2 76.2 3.36 76.2 79.2 3.60 79,2 82.3 3.11 82.3 65.3 3..59 85.3 88.4 3.73 88.4 91.4 2.70 °1.4 94.5 3.42 94.5 97.5 3.98 97.5 100.6 3.53 100.6 103.6 3.66 103..6 106.7 3.79 106.7 109.7 3.63 109.7 112.8 3.64 112.8 115.6 4.56 1154 6 118.° 4.02 116.5 121.5 3.46 121(9 125.0 4.37 125.0 128, 0- 4.22 128.0 1314,1 4.82 131.1 134.1 3.65 134.1 137.2 3.63 137.2 140.2 3.77 140.2 143.3 3.97 143.3 146.3 3.86 146.3 149.4 4.07 149.4 152.4 3.78 152.4 155.4 3.49 155.4 158.5 4.18 159.5 160.0 3.5.8 160',0 161.5 4.13 161.5 163,1 4.19 163.1 164.6 4.53 164.6 166.1 3.74

245

KENYA 35 0.11 DEG N 40.31 DEG E

TEMPERATURE DATA

DEPTH TEMP DEPTH TEMP DEPTH TEMP DEPTH TEMP 67.50 35.61 •73.70 25.71. 75.80 35.80 86.00 35.92 85.00 35.98 92.10 36.04 95.20 36.12 58,,20 36.22 101.30 36.29 104.40 36.39 107.40 36.52 110.50 36.65 112.00 36.69 113.60 36.73 115.10 3E.76 116.60 36.81 11.E.20 36.84 119.70 26.89 121.20 36.93 122.80 36.96 124.20 36.99 125.90 37.02 127.40 37.05 128.90 37.10 120.50 37.13 132.00 37.17 UNSTABLE SECTIONS 67.5 86.0

THERMAL CONDUCTIVITY DATA ESTIMATED VALUES REGION CONDUCTIVITY 30.5 1320 2.00

KENYA 0.02 DEG N 40.37 DEG E

TEMPERATURE DATA DEPTH 'TEMP DEPTH TEMP DEPTH TEMP DEPTH TEMP 3tj 20 33.32 33; 20 13.41' 36.20 33.51 39.30 33.63 42;,30 33.71 45.30 33.82 48.30 32.93 51.40 34.06 54.40 34.17 57.40 34129 60.40 34.42 63.40 34.54 66.40 34.61 69.50 34.69 72.50 34.77 75.50 34.84 78.50 34.57 81.60 35.11 54.60 35.23 87.65 35,35 50.60 35..50 53.6U 35.60 96.70 35.79 99,70 35.91 102.70 36.03 105.70 10.E0 36.15 110.20 36.21 111.80 36.24 113.30 36.31 114.80 36.40 116.30 36.50 117.80 36.56 119.30 36.64 120.60 36.71 122.30 36.79 123.60 36.87 125.40 36.53 126:90 37.02 123.40 37,08 129.50 37.17 131.40 37.24 132.90 77.32 134.40 37.42 135.90 37.50 13 .30 37.58 139.00 37.65 140.50 37.68 142.00 37.71 143.50 37.71 145.00 37.72 146.50 37.74

TEERMAL CONDUCTIVITY DATA ESTIMATED VALUES REGION CONDUCTIVITY 3045 146.5 • 2.00

246

KENYA 37 0.06 DEC S 40.49 DEG E -.

TEMPERATURE DATA

DEPTH-. TEMP DEPTH TEMP DEPTH TEMP DEPTH. TEMP , 15.10 33.81 22,.60 34,12 24.20 34.14. 27.20 34.29 .30.20 34.36 33.20 34.41 36.20 34.52 39.30 34.63 42.30 34.73 45.30 34.82 48.30 34.91 51.40 35.02 54.40 35.16 57.40 35.27 60.40 35.45 63,40 35,57 66,40 35.74 69,50 35,94 72,50 3E.12 7550 36.28 78.50 36.42 61.60 36.61 84.60 36.58 87.60 .37.10 50.60 37.48 93.60 37.68 96.70 37.84 99.70. 37.99 102.70 38.14 105.70 38.29 108.80 36.50 110.20 35.52 111.80 38,,57 113,30 38.64 114..80 38.70 116,30 38.78 117.80 38,83 119i, 30 28,87 120(80 3E692 123.20 38.97 123.80 39.03 125.40 39.09 126.90 39.14 128.40 39.20 129.90 39.24 131.40 35.29 132.90 39.33 134.40 39.35 UNSTABLE SECTIONS 15.1 22.6

THERMAL CONDUCTIVITY DATA ESTIMATED VALUES REGION CONCUCTIVITY 30.5 134.4 2.00

KENYA 38 0.70•1?EG N •79.47 DEC E:

.TEMPERATURE DATA DEPTH TEMP DEPTH TF-PP DEPTH TEMP 6EPTH TEMP 21.20 33.76 24.20 33.96 27.20 34.05 30.20 34.07 33.20 34.11 36.20 34.14 39.30 34.19 42.30 34.25 45.30 34.29 48.30 34.37 51.40 34.46 54.40 34.55 57.40 34.58 60,40 34,63 63,40 34.69 66,40 74,78 69.50 34,86 72-50 74.95 75.50 35.02 76.50 35.10. 61.60 35.16 84.60 35.23 87.60 35.31 90.60 35.38 53.60 35.44 96.70 35.50 99.70 35.70 101.20 35.73 102,70 35.76 104.20 35.80 105.70 35.0E6 107,20 .35.90 108.80 35,96 110,20 36.02 111.80 3E0'06 113.,30 36,11 114.80 36.16 116.30 36.20 117.80 36.25 119.30 36.29 120.80 36.33 122.30 36.35- 123.80 36.43 125.40 36.47 126.90 36.53 128.40 76.57 129.90 :36.63 131.40 36.65 132.20 36,71 134,40 36.73 135.90 36.64 137“+0 36.88 139.00 36.93 140.50 36,97 142.00 37.02 143.50 37.06 145.00 37.10 146.50 37.14 148.10 37.19 149.50 37.23' 151.00 37.28 152.60 37.21 154.00 37.35 155.60 37.40' 157.10 37.43 156.60 37.47 UNSTABLE SECTIONS 21.2 27.2

1HERMAL CONCUCTIVITY DA7A(NONPOROUS) REGION COND REGION COND PFGION COND 3.4 6.4 4.,85 6 4 61.0 4.92 61.0 67.1 4.51 67.1 70.1 5.12 70,. 1 73.2 4.94 73.2 76.2 4,94 76.2 79.2 4.96 79,2 82,3 4.46 82,3 85.3 4,22 6.5.3 88.4 4.14 88.4 91.4 4.51 91.4 94.5 4.41 94.5 97.5 4.03 97.5 100.6 4.02 100.6 103.6 4.30 103.6 106.7 3.58 106.7 109.7 3.18 109.7 112.8 2.93 112.8 114.3 2.93 114r 3 118.9 2.86 118.9 121.9 2.94 121,9 125.0 3.34 125.0 128.0 .3.11 129.0 131.1 3.44 131.1 134.1 3.60 134.1 137.2 3.80 137.2 140.2 3.13 140.2 143.3 3.68 143.3 144.2 3.68 144.2 147.8 3.83 147.6 149.4 4.38 149,4 150.3 4. 5 6 150.3 150.9 3.60 150.9 152,4 3,79 1524 153.9 3.97 153.9 155,4 3.79 155.4 158.5 3.94 156.5 161.6 3.52

247

•KENYA 39 0.76 DEG N 39.61 DEC E

. TEMPERATURE DATA

CEPTH TEMP DEPTH TEMP DEPTH TEMP CEPTH TEMP 60.140 35.12 63.40 35.15 66:40 35.18 69.50 35.21 72.50 35.26 75.50 35.29 78.50 35.29 81,60 35.29 84.60 35.33 87.60 35.36 90.60 35.40 93.60.35.45 96.70 35.31 98.20 35.23 99.70 75.35 101.20 35.36 102.70 35.39 10420 25.41 105c70 35.44 107.20 35,45 10t5.80 35.47 110.20 35.50 111.80 35.51 113.30 35.53

THEftAL CONDUCTIVITY DATA ESTIMATED VALUFS REGION CONDUCTIVITY ' 30.5 113.3 2.00

KENYA . 40 0.43 DEG N 40.02 DEG E.

TEFPF- RATUFE DATA DEPTH TEMP DEPTH TEMP DEPTH TEMP - DEPTH TEMP 60.40 35.60 66.4n 35.65 69.50 35.71 72.50 35.73 75.50 35.78 7F.50 35,68 51,60 35.93 F4.60 36.02 87.60 36.08 • 90.60 36.19 - 9 3.60 36.29 96.70 99.70 36.57 101.20 36.59 102.70 76.62 104.20 36.64 11-35.70 36.65 107.20 76.69 106.13 36.73. 110.20 36.77 111.80 36.F1 113.33 36.84 114,:,n 36.67 116.30 36.95 117.80 37.00 119.30 37.01 . 120.80 77.05

THERMAL CONDUCTIVITY OATAfNONPOROU.S,

REGION COND PEGION romn FEGTON COMO • 97.5 100.6 2.90 100.6 103.6 2.99 103.6 106.7 3.57 106.7 109.7 4.06 109.7 112.8 4.49 112.8 115.8 4.46 115.6 118.9 4.47 116.9 122.0 4.84

•248

KENYA 41 0.36 DEC N 40.13 DEC E.

TEMPERtTURE DATA

DEPTH TEMP DEPTH TEMP DEPTH TEMP DEPTH. TEMP 30.20 34.34 33.20 24.40 36.20 34.46 39.30 34.55 42,30 34:66 45.30 34.75 48.30 34082 51.40 34.90 54.40 35.01 57.40 75.07 60.40 35.14 63.40 35.24 66.46 35.36 62.50 25.44 72.50 35.52 75.50 35.59 78.50 35.66 78.50 35.66 81.60 35.75 84.60 35.85 67.60 35.94 90,60 26.07 93460 364 20 (:)670 26430 29470 36.39 10270 36.46 104i.20 26449 105470 36452 107.20 36.56 •108.80 36.61 110.20 36.64 111.E0 36.68 • 113.30 36.73 114.80.26.77 116.30 38.82 117.80 36.86' 119.30 36.90 120.80 36.94 122.30 36.96 123.80 37.01 125640 37,04 126.90 37,07 128,40 37.11

THERMAL CONDUCTIVITY:DATA(NON-FOROUS) REGION COND REGION COND REGION COND 27s4 30.5 4.10 305 33.5. 4.32 33.5 39.6 4.05 39.6 42.7 4.45 42.7 45.7 4.28 45.7 48.8 4.40 48.8 51.8 3.69 51.8 57.9 4.17 57.9 67.1' 4.34 67.1 70.1 4.53 70.1 73.2 4.45 73.2 79.2 4009 7942 854 3 4.29 85.3 9 445 4.31 94.5 10647 3488 106.7 112.8 3.68 112.8 121,9 3.36 121.9 125.0 4.55 128.0 131.1 4.40 131.1 134.2 4.22

KENYA 42 4.20 DEC S 32.40 DEG E

TEMPERATURE DATA DEPTH TEMP DEPTH TEMP DEPTH TEMP CEPTH TEMP 15.10 27.82 16.10 27.83 21.20 27.84 24.20 27.85. 27.20 27.86 30.20 27.87 33, 20 27:92 74.70 27.94 36.20 27.95 374 80 27.96 39.30 27.97 42.70 27.98 45.30 27..29 46.30 27.99 51.40 28.01 54.40 28.01 57.40 28.03 60.40 26.05 63.40 28.08 66.40 28.09 69.50 26.13 72.50 28.16 75.50 2 6 .29 79.50 28.23 61.60 23.25 844-60 2E429 57.63 28.25 X0.60 28.40 92.10 28.42 97.60 28.44 95.20 28453 X6.70 28.54 98.20 28.56 99.70 26.59 101.20 28.60 11-J2.70 28.63 104.20 28.67 105.70 26.70 107.20 26.73 108.80 28.76. 110.20 26.80 111.80 26.82 113.30 28.85 114460 26.87 116,30 20.91 117t.80 28.25. 119.30 28.99 120.80 29.01 122.30 29.05 123.6 0 29 .06 125.40 22.02 126.90 29.11 128.40 29.18 129.90 29.21 131.40 22.25 132.90 29.28

THERMAL CONDUCTIVITY DA17(NON•FORDUS) REGION CONO REGION COND REGION, COND 6.7 12.5 6.11 12.5 19.2 3.69 19.2 25.9 2.62 254 2 52.4 5.58 52:4 58.2 2489 5642 88.4 4.94 88,4 90.8 6.61 90.8 93.2 5.92

NOTE..4.DEPTH RANGE OF LAST CONE EXTENDED TO 131.4..

249

KENYA 43 ' 4.19 DEG S. 39.41 DEG E

TEMPERATURE DATA .DEPTH- TEMP DEPTH TEMP DEPTH TEMP CEPTH TEMP 36,20 27.24 42.70 27.30 45.30 27.75 46.60 27.30 48.30 27.39 49v50 27.42 51.40 - 27.45 52.90 27.47 54.40 27.49 55.90 27.51 57.40 27.53 58.90 27.56 60.40 27.58 61.90 27.60 63.40 27.64 65.00 27.66- 66.40 27.70 68.0G 27.71 69.50. 27.71 • 71000.27074

THERMAL CONDUCTIVITY [Wit ESTIMATED VALUES REGION CONDUCTIVITY 36.6 71.0 3.40

KENYA 44 0.66 DEG N 37.05 DEG E

TEMPERATURE DATA DEPTH TEMP DEPTH TEMP DEPTH TEMP DEPTH TEMP 36.20 27.69 39 .30 27.71 42.30 27.73 45.30 27.75 48.30 27.77 51.40 27.81 52.90 27.81 54.40 27.82 57.40 27.64 60.40 27.86 63.40 27.85 66.40 27.91 69.511 27, 93 72,50 27.96 7550 27.59 78.50 28.01 81.60 28.04 84.60 28.07 87.60 2-.10 90.60 28.14 93.60 23.16 56.70 28.20 98.20.28.21 99.70 28.23 '101.20 28.24 102.70 28.27 105.70 28.70 1t8.80 28.33 . 111.80 28.37 114.80 28.40 117.60 28.44 120,80 26.48 .123080 23.52 126.90 28.57 128 40 28.58 129-90 28.61 131.40 28.63 132.90 28.65 134.40 2F.67 135.9 0 28.69 137.40 28.70 139.00 28.73 140.50 28.74 142.00 26.76 143.50 28.79 145.00 26.81 146.50 28.62 148.00 26.84 145.50 25.86 151.00 28.87 152,.60 25.90

THERMAL CONDUCTIVITY DATA(NONPOROUS) REGION COND REGION . COND. REGION COND 97.5 103..6 2.87 103.6 106.7 2.03 106.7 109.7 3.08 109.7 115.8 2.59 115.e 118.9 3.12 118.9 125.0 2.92 125.0 128.0 2.76 128.0 131.1 2.56 131.1 134.1 2.78 134.1 137.2 2.76. 137.2 140.2 2.45 140,2 143.3 2.24 143.3 146.3 2.87 146.3 149.4 2.54 149.4 152.5 2.79

250

'KENYA 45 2.75 .DEG N 38.10 DEG E

TEMPERATURE DATA. .

CEFTH TEMP DEPTH TEMP DEPTH TEMP DEPTH TEMP 92.10 35.54 12 2:80 3672 125,90 36,83 128.90 36.98 132.00 371-10 13 5=10 37.19 138.10. 37.29 141.20 37.37 144.30 37.47 14 7.30 37.55 150.40 37.61 153.50 37.68 156.50 37.75 15 9.60 37.82 162.70 37.87 165.80 37.91 166.80 37.57 17 1.90 36.02 :175,00 36,06 178.00 38.13 181.10 36.33 16 4,20 25.28 185.70 3E.28 UNSUELE SECTIONS 92.1 184.2

THERMAL CONDUCTIVITY DATA - - - ESTIMATED VALUES REGION CONDUCTIVITY . 91.4 185.7 1.50

KENYA 46 3.12 0“'N 36.28 DEG E

TEMPERATURE DATA

CEPTH TEMP DEPTH TEMP DEPTH TEMP CEPTH TEMP 161.20 37.59. 187=30 38.00 19330 16.31 196.30 38.54 199.40 30.68 202.40 38.E0 '205.40 38.9? 206.40 39.06 211.40 39.19 214.50 39.31 217.50 79.47 220.50 39.63 223.50 "39.66 225.00 29.69 • 226.50 39.71 220.10 39.72 229,60.39.78 231.'10 39,83 232=60 39.85 23410 391,86 235.50 39..91 237.10 79.95 . ?38.60 39.99 240.10 40.02 241.60 40.08 .243.20 40.10 244.70 40.1.4 246.20 40.19 247.70 40.24 249.20 40.30 250.70•4C.33 tNSTAELE SECIIONS 181.2 193.3

THERMAL CONDUCTIVITY CATA ESTIMATED VALUES. REGION CONDUCTIVITY 182.9 250.7 1.50

251

KENYA 47 0.18 DEG N 35.46 DEG E

TEMPERATURE DATA DEPTH .TEMP DEPTH 'TEMP DEPTH TEMP DEPTH TEMP 60.40 18.61 66.40 18.54 72.50 18.46 78.50 18.44 84.60 18.20 90.60 18.22 93.60 1F.26 96.70 18.29 98.20 16.31 99.70 18.32 101.20 18.36 102.70 16.35 105.70 18.43 108,60 1F, 47 111,60 18.54 114u80 18.56 117.80 18.61 120.80 1.7.64 122.30 1P.67 123.60 16.68 125.40 18.71 126.90 18.73 128.40 16.75 129.90 18.77 131.40 18.80 132.90 18.82 134.413 18.83 135.90 18.86 137.40 18.67 139,-00 16.91 140.50 16r93 142,90 16.96 143e50 18.99 145: 00 19e01 146,50 15.02 146.00 19.05 149.50 1.9.07 151.00 19.11 152.69 19.14 154.00 19.16. 155.60 19.17 157.10 19.22 158.60 19.24 160.10 19.35 161.60 19.69 163.10 19.75 164.60 19,77 166c.10 19o80 167,60 19.83 169,20 19.66 170,70 19.68 172.20.19,90 173.70 19.93 .175.20 19.54 176.70 15.96 178.211 19.98 179.70 19.99 181.20 20.02 182.80 20.04 184.30 20.07 185.80 20.09 137.30 20.10 188.80 20.12 190.30 20.16 191.00 20.17 193,30 20.19 194,80 20021 196,30-20.23 197.80 20.26 ' 199,40 20.'28 200,90 20.30 202.40 20.33 203.90 20.36 205.40 20.37 207.t1 20.41 208.40 20.43 209.90 20.45 211.40 20.47 213.00 20.48 214.50 20.50 216.00 20.50 217.50 20.50 219.00 20.51 2201.50 20.52 222,00 20.54 223,50 21,55 225-00 20.56 226,50 20.58. 228.10 20.59 229.60 20.62 231,10 20.69 232.60 20.91- 234.10 21.23 235.60 21.50 237.10 21.73 238.60 21.88 240.10 22.09 UNSTABLE SECTIONS 60.4 84.6

THERMAL CONDUCTIVITY DATA ESTIMATED VALUES REGION ' CONDUCTIVITY 164.6 240.1 1.50

252 KENYA 48 0.40 DEp N 35.33 DEG E

TEMPERATURE DATA

DEPTH TEMP DEPTH .TEMP- .DEPTH TEMP CEPTH TEMP 15.10 19.34 18.10 19.35 21.20 19.36 24.20 19.38 27.20 19.39 30.20 19.40 33.20 15.42 36.20 19.43 35.30 19.46 42.30 19.47 45,3019.49 48,30 1.9050 51.40'19.52 54.40 19,53 57.40 15.56 60.40 19.60 63,40 19.66 66.40 15.73 68.00 19.75 69.50 19.80 71.00 19.83 72.50 19.87 74.00 19.91 75.50 19.95 77.00 19.99 76.50 20.02 80.00 20.06 81.60 20.10 83,10 20.14 84t60 20.20 86,10 20.23 87,60 20027 89.10 20.32 90.60 20.35 92,10 20.40 93.60 20.45 55.20 20.50 96.70 20.53 98.20 20.58 59.70 20.62 101.20 20.66 102.70 20.71 104.20 2(.75 105.70 20.80 107.20 20.85 108.80.20.88 110.20 20.94 111.80 21.00 113.30 21.03 114,:80 21.07 116.30 21.12 117.80 21.17 119.30 21.21 120.80 21.25 122.30 21.50 123.P.0 21.34 125.40 21.38 126.90 21.44 128.40 21.48 129.90 21.53 131.40 21.57 132.90 21.63 134.40 21.66 135.90 21.71 137,40 21.75 135,00 21.79 140,50 21,85 142.00 210 89 143.50 21.93 145.00 21.98 146.50 22.03 148.00 22.08 149.50 22.1? 151.00 22.17 152.60 22.19 154.00 22.25 155.50 22.29 157.10 22.34 158.60 22.37 1E0.10 22.42 161.60 22.47 163.10 22.50 164.60 22.54 166.10 22.59 167.60 22.64 169.20 22,67 170.70 22.74 172= 20 22.76 173.70 22.81 175.20 22.85 176.70 22.88 178.20 22.92 179.70 22.96 181.20 23.00 182.80 23.03 184.30 23.05 185.80'23.09 187.30 23.12 188.80 23.14 190.30 23.19 191.80 23.29 193.30 23..40 194.80 23.53 19 6.i.30 23.73 197.60 23,84 199.40 23,97 200,90 24.05 202,40 203.90 24.25 205.40 24.32 206.90 24.63

THERMAL CONDUCTIVITY CATA (NON-POPOUS) REGION COND REGION COND REGION. COND 15.2 18.3 1.59 18.3 • 21.3 1.59 21.3 24.4 1.62 24.4 27.4 1.56 27.4 30.5 1.56 30.5 36.6 1.60 36.6 39e6 1.65 39,6 42.7 1.63 42,7 45,7 1.72 45.7 48.8 1.71 48.8 51.8 1.63 51.8 54.9 1.62 54.9 57.5 1.54 57.9 61.0 1.58 61.0 64.0 1.59 64.0 67.1 1.56 67.1 70.1 1.31 70.1 73.2 1.69 73.2 76.2 1.47 76.2 79.2 1.36 79.2 82.3 1.59 €2.3 85e3 1060 85,3 58.4 1.58 88.4 91.4 1.48 91,4 94.5 1.69 94.5 °7.5 1.70 97.5 100.6 1.61 100.6 103.6 1.63 103.6 106.7 1.64 106.7 109.7 1.59 . . 109.7 112.8 1.66 112.8 115.8 1.67 115.6 116.9 1.62 116.9 121.° 1.47 121, 9 125,0 1.63 125,0 128,0 1.65 128.0 131.1 1.62 .171,1 134.1 1.55 134.1 137.2 1.59. 137.2 140.2 1.64 140.2 143.3 1.66 143.3 146.3 1.65 146.3 149.4 1.62 145.4. 152.4- 1.70 152.4 155.4 '1.57 155.4 158.5 1.55 158.5 158.5 1.59 158.5 161.5 1.49 161.5 164.6 1.61. 164.6 167.6 1.51 167.6 170.7 1.65 170.7 176.8 1.65 176.6 179.8 1.58 179.5 132.9 1.58 182.9 185.9 1.57 185.9 189.0 1.77 189.0 192.0. 1.77. 152.0 195.1 1.50 195.1 158.1 1.50 196.1 201.2 1.56 201.2 204.2 1.57 204 , 2 207.2 1.59

253 KENYA 49. 0.45 DEG: N 35.30 DEG E

TEMPERATURE DATA DEPTH TEMP DEPTH TEMP DEPTH TEMP DEPTH TEMP. 15.10 19.75 18.10 19.84 21.20 19.95 24.20 20.04 27.20 20+11,- 3020 20.26 33:20 20.36 34.70 20.41 36.2D 20.48 37.60 20,52 39.30 20.58 40.80 20.63 42.30 20.70 47.80 20.75 45.30 20.80 46.80 20.86 46.30 20.92 49.80 20.98 51.40 21.02 52.90 21.08 54.40 21.13 55.90 21.16 57.41 21.23 58.90 21.28 60.40 21+33 61.90 21.37 63,-40 21.42 65.00 21.46 66.40 21.51 68.00 21.55 69.50 21.59 71.00 21.64 72.50 21.69 74.00 21.73 75.50 21.78 77.00 21.81 78.50 21.85 80.00 21.90 81.60 21.93 53.10 21.98 64,60 22.03 86,.10 22.07 87.60 22.11 F9,10 22.15 90t60. 22,20 92,10 22.24 9360 22.28 95.20 22+32 96.70 22.36 9.20 22.41 99.70 22.44 101.20 22.49'. 102.70 22.53 104.20 22.57 105.70 22.61 107.20 22.65 106.80 22.70 110.20 22.73 111.60 22.77 113.30 22.82 114.80 22,66 116,30 22+90 117,50 22.95 119.30 22.99. 120.80 23.03 122.30 23.07 123.60 23.11 125.40 23.16 126.90 23.20 128.40 23.23 129.90 23.29 131.40 23.32 122.90 23.36 134.49 23.41 135.91 23.44 137.40 23.48 139.00 23..53 140-50 23,57 142,00 23.61 143,50 23.66 145.00 23+71 146+50 23.76 148.00 22+80 149..50 23.86 151.00 23.91 152.60 23.97 154.00 24.03 155.60 24.07 157.10 24.12 156.60 24.17 160.10 24.23 161.60 24.28 163.10 24.32 164.60 24.38 166.10 24.43 167.60 24.48' 169020 24.52 170, 70 24.56 172.20 24.62 173,70 24+67 175.20 24.72 176.70 24.77 178.20 24.80 179.70 24.84 181.20 24.66 181.50 24.88 181.80 24.92 182.10 24.96 182.50 25.01

THERMAL CONDUCTIVITY DATA(NON-POROUS) REGION CONO REGION CONO REGION CONO 15.2 18.3 1.57 16.3 30.5 1.47 30.5 45.6 -1.28 46+8 61.0 1.46 61.13 67.1 1+46 67,1 73,2 1.48 73.2 85.3 1.36 55.3 P1.4 1.40 91.4 110.0 1.23 110.0 115.8 1.52 115.8 126.0 1.30 128.0 170.7 1.43 170.7 178.3 1.67 178e.: 185.9 1.51

254 Appendix 7

Heat flow data : plots and calculations

The data given in appendix 6 are shown, along with lithologic information where available, in graph form. The water level in each borehole is indicated, and sections in which unstable temperatures were recorded are shown as dashed lines. In the right — hand box, details of the heat flow calculation are given, comprising the depth section upon 'Which the heat flow calculation was based (INTERVAL), uncorrected values for geothermal gradient, thermal conductivity, and heat flow over this section, the per cent change in gradient or conductivity due to individual corrections, and the final corrected heat flow value.

255 LOCATION..GILGIL \YR 24 GREGORY RIFT VALLEY L THOLOGY TEMPERATURE CONDUCTIVITY 0.40 DEG S 36.36 DEG E 211 23 25 27 29 31 33 35 1 1 ..4 1 15 0

INTERVAL 23- ► ► (192.0 TO 222.51 METRES

46- GEOTHERMAL GRADIENT =65 THERMAL CONDUCTIVITY=1.4 69- HEAT FLOW-54

92- CORRECTIONS • .4 GRADIENT TOPOGRAPHY CLIMATE 5 138- UPLIFT.SEDIMENT 9

CONDUCTIVITY 161- 34% POROSITY -25

WELDED TUFF 184 HATER LEVEL

GRIT 207- HEAT FLOW=80 (CORRECTED) LOCRTION..OL KRLOU /CE\ YR 2 GREGORY RIFT VALLEY LITHOLOGY TEMPERATURE CONDUCTIVITY 0.27 DEG S 38.38 DEG E 20 2,1 22 23 24 25 _WRIER LEVEL

16- INTERVAL (51.8 TO 89.0 ) METRES

32- GEOTHERMAL GRADIENT =58 THERMRL CONDUCTIVITY-1.1 x 48- HEAT FLOW-64.

64- CORRECTIONS

TUFF GRADIENT a. 80- w TOPOGRAPHY 0 CLIMATE a 96- UPLIFT,SEDIMENT 9

112-

128-

144- HEAT FLOW-75 (CORRECTED) •

* ESTIMATED VALUE 160- LOCATION..NAKURU /E\YR 26 GREGORY RIFT VALLEY LITHOLOGY TEMPERATURE CONDUCTIVITY 0.31 DEG S 36.06 DEG E 22 23 24 25

11 - INTERVAL (73.2 TO 101.2) METRES

22- GEOTHERMAL GRADIENT =53 'S THERMAL CONDUCTIVITY=1.3w 337- PUMICE HEAT FLOW=69

4 4 - CORRECTIONS GRADIENT TOPOGRAPHY -2 CLIMATE 7

66- UPLIFT.SEDIMENT 32

77-

PHONOL ITIC TRACHYTE 68- 1.3 )1(

99- HEAT FLOW=95 -WITERLEVEL (CORRECTED) )1( ESTIMATED VALUE 110- LOCATION..NJORO -\ YR 27 GREGORY RIFT FLOOR LITHOLOGY TEMPERATURE CONDUCTIVITY 0.34 DEG S 35.94 DEG E 18 19 20 21 22 23 1.5 1.6 1.7 0

20- INTERVAL (121.2 TO 158.1) METRES

40- TUFF GEOTHERMAL GRADIENT =25 THERMAL CONDUCTIVITY=1.5 60- HEAT FLOW=38

• 80- CORRECTIONS GRADIENT WRIER LEVEL Q.. 100- w TOPOGRAPHY 0 CLIMATE 12 UPLIFT, SEDIMENT 31 TUFF 120- CONDUCTIVITY % 140- 30% POROSITY -25

160-

CLAYEY TUFF 180- HEAT FLOW=41 (CORRECTED) TUFF

200- LOCATION..NJORO /E\YR 28 GREGORY RIFT . FLOOR L I THOLOGY TEMPERATURE • CONDUCTIVITY 0.30 DEG S 35.91 DEG E 21 23 25 27 29 0

20- INTERVAL (60.4 TO 110.2) METRES

40- -GEOTHERMAL GRADIENT =56 THERMAL CONDUCTIVITY=1.0 60- CLAY HEAT FLOW=56

80- CORRECTIONS GRADIENT - WATER LEVEL a_ 100- w 1 0 NI TOPOGRAPHY 0 TUFF WITH CLAY CLIMATE 7 120- UPLIFT.SEOIMENT 33

140-

160- TUFF

180- HEAT FLOW7_78 •TRACHYTE (CORRECTED) )1( ESTIMATED VALUE 200- LOCRTION....1ATOWSIGESR EI\ YR :KERICHO PLATEAU _ LITHOLOGY TEMPERATURE CONDUCTIVITY 0.28 DEG S 35.43 DEG E 18 19 20 21 22 23 1 15 1 16 1 17 0 ,IVITER LEVEL

17- INTERVAL (114.8 TO 151.0) METRES

34- GEOTHERMAL GRADIENT =70 THERMAL CONDUCTIVITY=1.7 51- HEAT FLOW=119

68- 1 CORRECTIONS GRADIENT a_ 85- TUFFS AND LAVAS w TOPOGRAPHY 0 CLIMATE 4

102- UPLIFT.EROSION -2

CONDUCTIVITY % 119- 35% POROSITY 1-30

136-

153- HEAT FLOW-785 (CORRECTED)

170- LOCRTION..0YUGIS /cE\YR 3n SOUTH NYRNZA LITHOLOGY TEMPERATURE CONDUCTIVITY 0.48 DEG S 34.72 DEG E 023 24 25

WATER LEVEL 10- INTERVAL (15.2 TO 90.5 l METRES

20- GEOTHERMAL GRADIENT =2 THERMAL CONDUCTIVITY=1.5m 30- HEAT FLOW=3

40- CORRECTIONS VOLCAN ICS NONE APPLIED a_ 50 - LIJ 1.5 If DISTURBED BY 60- RECENT PUMPING . 70-

80-

90-

100- IN ESTIMATED VRLUE LOCATION..MIWANI /C El \ y PI 3 1 KAVIRONDO RIFT FLOOD LITHOLOGY TEMPERATURE CONDUCTIVITY 0.05 DEG S 34.93 DEG E 24 25 26 27 3..5 3.16 3.17 3.8

INTERVAL (15.1 TO 84.6 ) METRES

GEOTHERMAL GRADIENT =45 THERMAL CONDUCTIVITY:3.7 HEAT FLOW=168

CORRECTIONS GRADIENT TOPOGRAPHY -6 DETRITUSS CLIMATE 11 UPLIFT.EROSION -0

CONDUCTIVITY % 35% POROSITY 1-47

HERT FLOk1:-- 93 (CORRECTED)

LOCRTION..KANO PLAINS . /E\YR 32 KAVIRONDO RIFT FLOOF LITHOLOGY TEMPERATURE CONDUCTIVITY 0.14 DEG S 35.04 DEG E 26 27 28 293 30 ,1 0 '

18- INTERVAL (18.0 TO 54.0 ) METRES

SEDIMENT AND TUFF 36- GEOTHERMAL GRADIENT =56 THERMAL CONDUCTIVITY=1.0m 54 WATER LEVEL HEAT FLOW=66

/2- CORRECTIONS PHONOLITE NONE APPLIED 1 .0 UNRELIABLE VALUE

ABOVE WATER TABLE 126- PEBBLE BED 144 -

• 162-

st ESTIMATED VALUE 180- LOCATION..MUHORON1 /E\YR 33 KAVIRONDO RIFT. FLOOP LITHOLOGY TEMPERATURE CONDUCTIVITY 0.14 DEG S 35.26 DEG E 23 * 24 25 26 1.po 1.ps 1.po 1.ps 0

11- INTERVAL (60.4 TO 84.6 ) METRES

. 22- GEOTHERMAL GRADIENT =21 THERMAL CONDUCTIVITY=1.9 337 HEAT FLOW=39

44--.1017ERLEYE1. CORRECTIONS

TUFF GRADIENT (1_55 - TOPOGRAPHY 0 CLIMATE 23

66 UPLIFT,EROSION ,1

CONDUCTIVITY % 77- 35% POROSITY 1-33

88-

99- HEAT FLOW= 32 (CORRECTED) /-• LOCATION..ALRNGOR: ARBR E Y 34 GRRISSA • LITHOLOGY TEMPERATURE CONDUCTIVITY 0.04 DEG 5 39.94 DEG E 34 35 36' 37 38 3 0

17- INTERVAL (143.5 TO 160.1) METRES

34- GEOTHERMAL GRADIENT =28 THERMAL CONDUCTIVIlY=3.9 • 51- HEAT FLOW=107

68- CORRECTIONS GRADIENT SAND FIND CLAY a_ 85- TOPOGRAPHY 0 M CLIMATE 8 102- SEDIMENT 2

CONDUCTIVITY 119- 35% POROSITY -48

136- WATER LEVEL

153- HEAT FLOW-762 (CORRECTED)

170- LOCRTION..ORDRR8 /E\YR GARISSA L I THOLOGY . TEMPERATURE CONDUCTIVITY 0.11 DEG N 40.31 DEG E 35 36 37 36 0

14- INTERVAL (114.8 TO 129.9). METRES

28- GEOTHERMAL GRADIENT :24 THERMAL CONDUCTIVITY=2.0' 42- HEAT FLOW=.47

56- CLAY AND SAND CORRECTIONS , GRADIENT a_ 70- TOPOGRAPHY 0 CLIMATE 12 2.0 84- xi SEDIMENT 3

98-

_HATER LEVEL 112-

SANDSTONE 126- HEAT FLOIx17-54 (CORRECTED) • m ESTIMATED VALUE 110 LOCATION..HAGADERA 'K EN Y R GARISSA LITHOLOGY TEMPERATURE CONDUCTIVITY 0.02 DEG N 40.37 DEG E 34 35 35 37 0

15- INTERVAL (117.8 TO 139.01 METRES

30- GEOTHERMAL GRADIENT =51 THERMAL CONDUCTIVITY-2.0'" 45- HEAT FLOW-102

60- CORRECTIONS GRADIENT SAND FIND CLAY a_ 75- w TOPOGRAPHY 0 0 CLIMATE_ 5 90- 2.0 ix SEDIMENT 3

105 - _WATER LEVEL

120-

135- HERT FLOW-1 1:0' (CORRECTED)

150- ESTIMATED VALUE LOCRTION..RLINJUGU c F YR 37 GRRISSR LITHOLOGY TEMPERATURE CONDUCTIVITY 0.06. DEG S 40.49 DEG E. 34 35 36 3? 39 39 o

14— INTERVAL (110.2 TO 132.9) METRES

28— GEOTHERMAL GRADIENT .:36 THERMAL CONDUCTIVITY:2.0m 42— HERT FLOW-773

56— CORRECTIONS

SAND AND CLAY GRADIENT 70- 7 TOPOGRAPHY .0 CLIMATE 8 84— 2.0 SEDIMENT 3

98—

WRIER LEVEL 112—

126— HEAT FLOW=80 • (CORRECTED) IN ESTIMATED VALUE 140—

LOCATION..ENDELA SWAMP / E \ YR 8 GRRISSR LITHOLOGY TEMPERATURE CONDUCTIVITY 0.70 DEG N 39..47 DEG E 3:1 ZS 36 3? 2.8 3 16 4.5 0 L

16- INTERVAL (137.4 TO 155.6) METRES

32- GEOTHERMAL GRADIENT =26 THERMAL CONDUCTIVITY=3'.8 48- HERT.FLOW.-107

64- CORRECTIONS

GRADIENT z SANDY CLAY 0_ 80- 0 TOPOGRAPHY 0 CLIMATE 8 SEDIMENT 4 96- _HATER LEVEL

CONDUCTIVITY % 112- 35% POROSITY -47

128-

144- HEAT FLOW=63 (CORRECTED)

160- LOCATION..KRLRLUT / E YR 39 GARISSA LITHOLOGY TEMPERATURE CONDUCTIVITY 0.76 DEG N 39.61 DEG E 035 36

12- INTERVAL (96.7 TO 113.31 METRES

24- GEOTHERMAL GRADIENT =14 THERMAL CONDUCTIVITY=2.0 36- HEHT FLOW=28

48- CORRECTIONS

SANDY CLAY • NONE APPLIED . 0_ 60- w DISTURBED NEAR 72- 2.0 )1( AIR/WATER INTERFACE 84-

9.6 WATER LEVFL

108-

111 ESTIMATED VALUE 120-

LOCRTION..MERI /ENYR GARISSA.. LITHOLOGY . TEMPERATURE CONDUCTIVITY .0.43 DEG N 40.02 DEGE

35 36 37 382.9 3.7 4.5 0

13- INTERVAL. (105.7 TO 117.8) METRES

26- GEOTHERMAL GRADIENT =28 THERMAL CONDUCTIyITY=4.4. 39- HEAT FLOW=124

52- CORRECTIONS SAND AND CLAY GRADIENT a_ 65 - TOPOGRAPHY CLIMATE 11 78- SEDIMENT 4

CONDUCTIVITY V.. 91- 357. POROSITY -50 _WATER LEVEL . 104-

117- HEAT FLOW= 1: (CORRECTED)

130- LOCATION..SRBULE \YR 41 GRRISSR LITHOLOGY TEMPERATURE CONDUCTIVITY 0.36 DEG N 40.13 DEG E 35 36 37 3.4 3 18 4 .I2 4.46 0

13- INTERVAL (93.6 TO 126.9) METRES

26- GEOTHERMAL GRADIENT =26 THERMAL CONDUCTIVITY-3.8 39- HEAT FLOW-99 SAND AND CLAY 52- CORRECTIONS

GRADIENT p_ 65 - TOPOGRAPHY 0 CLIMATE 12 78- SEDIMENT 4

CONDUCTIVITY 91- _HATER LEVEL 35% POROSITY I-47.

104-

SAND 117- HEAT FLOW L:61- (CORRECTED)

130- LOCATION..MRERE /E\ YR 4 MOMBASA LITHOLOGY TEMPERATURE CONDUCTIVITY 4.20 DEG S 39.40 DEG E 27 28 29 30 0

14 WATER LEY L INTERVAL (96.7 TO 132.9) METRES

WEATHERED SANDSTONE 28- GEOTHERMAL GRADIENT =21 THERMAL CONDUCTIVITY-5.9 42- HEAT FLOW-122

CLAY 56- CORRECTIONS GRADIENT TOPOGRAPHY 0 MEDIUM SANDSTONE CLIMATE 14

84- UPLIFT.EROSION 0

CONDUCTIVITY % 98- COARSE SANOSTONE 25% POROSITY -43

112- FINE SANDSTONE

126- HEAT FLOW=79 (CORRECTED)

140- LOCRTION..MRERE c E\ 43 MOMBASA. LITHOLOGY TEMPERATURE CONDUCTIVITY 4.19 DEG S 39.41 DEG E 28

WEATHERED SANDSTONE 8- INTERVAL (42.3 TO 65.5 ) METRES SANDSTONE

16- CLAY GEOTHERMAL GRADIENT =15 THERMAL CONDUCTIVITY=3.4' 24- HEAT FLOW-53 SANDY SHALE WRYER LEVEL

32- CORRECTIONS SANDSTONE GRADIENT 40- TOPOGRAPHY w SANDY SHALE CLIMATE . 31 UPLIFT.EROSION 0 48-

3.4 56- SANDSTONE 64-

72- HEAT FLOW L:69- (CORRECTED) - ESTIMATED VALUE 80 LOCATION..LONGOPITO /E\YA 44 EASTERN PLATEAU .LITHOLOGY . TEMPERATURE CONDUCTIVITY 0.66 DEG N 37.05 DEG E 027 28 29 30 2 10 2 %5 2 %9

INTERVAL 16- (97.5 TO 150.91 METRES

_WATER LEVEL 32- GEOTHERMAL GRADIENT =13 THERMAL CONDUCTIVITY=2.7 48 HEAT FLOW=35

64- CORRECTIONS GRADIENT z BASEMENT 80- w "TOPOGRAPHY 0 (GNEISS 1 m CLIMATE 21 UPLIpT.EROSION _5 96-

112-

128-

144-4 NEAT. FLOW-740 (CORRECTED) NI THERMAL DEPTH ANALYSIS 160- LOCATION..BUBISSA /cE\ YR N.F.D. LITHOLOGY TEMPERATURE CONDUCTIVITY 2.75 DEG N 38.10 DEG E 36 • 37 39 0

19- INTERVAL (159.6 TO 175.0) METRES.:

38- GEOTHERMAL GRADIENT =16 THERMAL CONDUCTIVITr=1.5m, 57- HEAT FLOW-2.4

76- . CORRECTIONS NONE APPLIED VOLCAN I CS 957

S. • INSTABILITY IN

1.14- • •

• . AIR SECTION 133- • 1.5

152- S. S.

171-

WRIER LEVEL ESTIMATED VALUE 190- LOCATION..HRWALA Y R 46 N.F.D. LITHOLOGY TEMPERATURE •t..! CONDUCTIVITY 3.12 DEG N 38.28 DEG E 38 39 40 0

26- INTERVAL (223.5 TO 249.2) METRES

52- • GEOTHERMAL GRADIENT =25 THERMAL CONDUCTIVITY=1.5° 78- HEAT FLOW=37

104- CORRECTIONS GRADIENT VOLCRN I CS a_ 130- w TOPOGRAPHY 0 CLIMATE 13

156- UPLIFT.EROSION -2

182-

208-

234- HEAT FLOW=41 (CORRECTED) ESTIMATED VALUE 260- LOCATION..NABKOI /E\YR 47 TINDERET HIGHLANDS . LITHOLOGY . TEMPERATURE •CONDUCTIVITY 0.18 DEG N 35.46 DEG E •p 20 2,1 22 0

25- INTERVAL (164.6 TO 213.41. METRES

• PHONOL I Tt 50- GEOTHERMAL GRADIENT '.=14 THERMAL CONDUCTIVIT1=1.5° 75- HEAT FLOW-22 WATER LEVEL TUFF 100- CORRECTIONS, NONE APPLIED a_ 125-

GRADIENT DISTURBED 150- BY WATER FLOW 175 - PHONOL TE

200.- 1 .S IN

225 -

PI ESTIMATED VALUE 250- LOCATION—PLATEU /E\YR 48 URSIN GISHU PLATEAU LITH.OLOGY TEMPERATURE CONDUCTIVITY 0.40 DEG N 36.33: DEG E 20 21 22 24 a 23 l•3 1.5 1 17 _HATER LEVEL

21- INTERVAL (80.0 TO 170.7) METRES

42- GEOTHERMAL GRADIENT =30 THERMAL CONDUCTIVITY=1.6 63- HEAT FLOW=48 r-- 84- CORRECTIONS GRADIENT z PHONOL 1 TE a_ 105- w TOPOGRAPHY 0 m CLIMATE 10

126- UPLIFT,EROSION —4

PP-

168-

189- HERT FLOIA1=50 (CORRECTED)

. 210- LOCATION..ELDORET /E\YR UPSIN GISHU PLATEAU LITHOLOGY TEMPERATURE CONDUCTIVITY 0.45 DEG N 35.30 DEG E 20 21 22 - 23 ' 24 25 1.,2 15 1 17 0 1 MURRUM SOIL ...E__ WRIER LEVEL

19 INTERVAL (66.0 TO 143.5) METRES

GEOTHERMAL GRADIENT =28 THERMAL CONDUCTIVITY=I:3 HEAT FLOW=37

CORRECTIONS GRADIENT PHONOL I TE TOPOGRAPHY 0 CLIMATE 12. UPLIFT,EROSION -4.

'HEAT FLOW-.40 (CORRECTED)

Appendix 8

Theory of temperature distribution in a flowing well.

The use of a line source (Carslaw and Jaeger, 1959) approximation to calculate the temperature disturbance created by a steady flow of water between two levels (widely spaced) in a borehole was suggested by. Birch (1947). The approximation can only be considered valid away from the levels of influx and efflux, and outside the region of flow.

The temperature increase at a distance r from a line source, which liberates H Joules of heat per unit length per unit time, can be written (Carslaw and Jaeger, 1959) : H (E; ( T( r,t )= 4K t (A8.1) where C = thermal conductivity of medium thermal diffusivity of medium oo6 x d x Ei(u) = LL (A8 . 2)

Values of the function Ei(u) can be obtained from tables (Carslaw and Jaeger, 1959), or for u<0.01 ,it can be closely approximated by the following expression:

(u) = 1°9e* ) (A8.3) .

where, = 1.7810

The disturbance at the casing(radius a) in 'a borehole through r/ .which water is flowing with velocity is(for— < 0.01 ) 4K t

2 a pc t..1vf". AT = tog%e 4Rt) (A8.4) 4 k

where p = density of water C = specific heat of water = geothermal gradient V = water velocity Substituting the following parameters for borehole site 48:

282 101.6 mm G =30 mic.m 1 F( =0.5 mm2.'71 k =1.5 mw.m-1.K-1 • LT =1 K

and using (A8.4) to calculate V (assuming t=time since drilling was completed = 39 months).

V =0.5 mm.s-1

283 Appendix 9

Heat flow anomaly due to a buried disc - shaped source.

The temperature rise T at a point P(x,y,z) due 'to a point

source of strength A at point P(x,y,zJ in an infinite medium

of uniform conductivity K, can be written :

(A9.1.)

where R =[(x-xri-(y-y1+ (z-z)2J/2

The heat flow at P is therefore :

= -k Ai 4nR2 • (A9.2)

The effect of a boundary condition T = 0 at the earth's

surface (z = 0) can be, conveniently calculated by the method of

images (Carlslaw and Jaeger, 1959), in which each point source

at P (x,y,z) is accompanied by an image source of strength -A

at S(x;y.,--). The vertical component of heat flow, qz, at

P(x,y,o) due to a source at P and its image at S can be written

_ zf(e4 Z9-3/z CIZ. 2 n (A9.3)

where • r = ( (x.- X)2 + •(y,- y)2

The vertical component of heat flow at point Po(0,0,0) on the isothermal surface z = 0, due to a horizontal disc source of strength A per unit volume per unit time, whose axis coincides with the z axis, at a depth h, of thickness 1 (h2 - h1), and of radilis a, can be found by integrating (A9.3) as follows :

2Tr a AL —A (1- '4. z'1512 r z d kir dz 2 n 56 :70 r'r. e wheke ,z) are cylindrical coordinates.

284 Integration gives

y 2 9_ 3. A [ 1 — hz+ a• 2 h+a )2- 1 (A9.4)

285 .Heat flow anomaly caused by vertical movement of groundwater

If moving groundwater is forced to rise over an anteclinal structure (e.g. a step, as shown in figure A10.1) it transfers heat efficiently by convection to higher levels in the crust and can produce a significant surface heat flow anomaly. A simple mathematical model to obtain rough indication of the anomaly produced under given conditions was considered by Bullard and Niblett (1951), and is outlined below..

Consider a sheet of water moving through ABCE in figure A10-1. If the background temperature gradient is (3 in a homogeneous medium of thermal conductivity k, the water temperature along AB is

TAB = ( Zi+AZI ) (A10.1)

where Z1 and Q z, are as shown and Ts is the ground surface - temperature.

Let the temperature at some point P on CD be T, then the heat flow above P is roughly

kI ( Let li=()) (k10.2) z,

Since the heat supplied from below is k (3 , the surface anomaly q can be written : k ( z, -(3 ) (A10.3)

This can be equated to the heat supplied by flowing water as

-mcST =, kr-1.43%x • L (A10.4)

where mass flow rate of groundwater specific heat In the limit E X -4 (21 becomes

286

Q=k.13

B m

Figure A10-1 Simple model for calculating heat flow anomaly caused - by groundwater flow over an antecline(from Bullard and Niblett,1951)

287

d T -k (T (3) d x MC Z

Integrating and using the condition , T-4 0 X • 00 gives :

_ kx Tp - (.3 (z, 4- A z,emcz, ) (A10.6).

The fractional change in heat flow at x is therefore

kx q A e-mcz, (A10.7) q

288• Appendix 11

Finite difference mesh sizes.

Rectangular intrusion (section 4.4)

961 point mesh (31 x 31)

.horizontal grid increments vertical grid increments index increment (Km) 1 index increment (Km) \ 1 57.67, 1-8 0.5 2 38.44 9-20 1.0 3 25.63 21 1.35 4 17.08 22 2.03 5 11.39 23 2.71 6 7.6 24 3.39 7 5.06 25 4.06 8 3.38 16 4.74 9 2.25 27 5.41 10 1.5 28 6.09 11-30 1 29 6.77 30 7.45

b) Partial melt zone model 1- (section 7.4)

2000 point mesh (25 x 80)

horizontal arid increments vertical grid increments index increment (Km) index increment (Km) 1 768 1-24 6 2 384 3 192 4 96 5 48 6 24 7-73 12 74 24 •75 48 76 96 77 192 78 384 79 768 289 Partial melt zone : model 2 (section 7.4)

2040 point mesh (51 x 40)

horizontal grid increments vertical grid increments index increment (km) index increment (km) 1 384 1-50 3 2 192 3 96 4 48 5 24 '6 . 12 7-39 12

290