NUMERICAL MODELS OF THE HEAT FLOW FIELD IN SOUTH-WEST ENGLAND
by
Mark S . Sams, B.Sc., M.Sc., D.I.C.
November, 1987
A thesis submitted for the degree of Doctor of Philosophy of the University of London. ABSTRACT
A suite of computer programs using the finite element technique was written to investigate the effect of three- dimensional geology on the flow of heat and fluid on models of the South-west of England. The results from models of conductive heat transport show that the surface heat flow pattern is dominated by the three-dimensional shape of the granite batholith which underlies the region. The shape of the batholith controls the relative importance of two opposing effects: heat refraction due to conductivity contrasts enhancing flow in the granite margins and heat egression due to heat production contrasts diminishing it.
The temperature field at any depth shows widely ranging lateral variations over relatively short distances, the highest values occurring in areas of granite which are covered by insulating country rock and which lie towards the centre of the batholith. Heat flows vary substantially across granite outcrops and do not necessarily indicate areas of high or low subsurface temperatures. A remarkable correlation can be seen between areas of exceptionally high heat flow and regions of mineralization.
The introduction of convective heat transport is able to throw light on two aspects of the ore-field associated with the batholith. Firstly, the regional distribution of the mineralization is a result of the fluid flow pattern which is determined by the heat flow field and thus the shape of the batholith. Secondly, the regional zonation of minerals, with high temperature minerals occurring closer to the outcrops than the lower temperature, later minerals, is due to the changing fluid flow patterns brought about by the erosion of the cover rock above the granite. The fluid flow patterns also show that the more recent stages of kaolinisation are caused by downward flowing fluid.
i ACKNOWLEDGEMENTS
I would like to thank the many people who have given me help and advice during the time spent on this research. Particular thanks go to my supervisor Anna Thomas-Betts who has not only given much of her time and effort to help me, but has also been a great support.
Other members of staff at Imperial have been very helpful: Peter Harris for help with the computing, Andy Rankin for advice on fluid inclusions and making available his data, Angus Moore and Jim Wheildon for valuable discussion, and Carol Kilby for typing corrections to the text.
Thanks must also go to Dr. Tony Batchelor and the Camborne School of Mines for their hospitality on many occasions and for providing data from their research at the Geothermal Station. I would also like to thank Jonathan Willis- Richards of CSM for the gravity models he produced.
This research was supported by a bursary from Rio Tinto Zinc. I gratefully acknowledge this support.
Finally, I would like to thank my parents for their encouragement and enthusiasm.
ii TABLE OF CONTENTS
Page
Chapter 1 Introduction 1
1.1 Heat Flows and Geothermal Energy 1
1.2 Heat Flows and Mineralization 3
1.3 Heat Flows and Finite Elements 4
Chapter 2 Finite Elements 5
2.1 Introduction 5
2.2 The Finite Element Method 6
2.3 Shape Functions 10 •
2.4 Mapping 11
2.5 Heat Conduction and the Finite Element Method 14
2.6 Convective and Conductive Heat Transport and the Finite Element Method 18
2.7 Model Building 23
2.8 Matrix Inversion 27
2.9 Data Output 30
iii Chapter 3 Geology and Geophysics of South-west England 34
3.1 Introduction 34
3.2 Geology 35
3.2.1 Granite 38
3.2.2 Mineralization 41
3.2.3 Kaolinisation 46
3.3 Geophysics 48
3.3.1 Seismics 48
3.3.2 Gravity 49
3.3.3 Heat Flows 56
3.3.4 Heat Production 60
3.3.5 Thermal Conductivities 62
3.3.6 Permeability 65
Chapter 4 Models of Conductive Heat Flow 67
4.1 Introduction 67
4.2 Heat Flow in General 68
4.2.1 The Effect of Conductivity Contrasts 70
iv 4.2.2 The Effect of Heat Production Contrasts 73
4.2.3 The Effect of Anisotropic Conductivities 74
4.2.4 The Effect of Fluid Flow 74
4.3 Model Testing and Accuracy 75
4.4 The Basic Model 78
4.5 Model Parameters 79
4.6 Model Results: Heat Flows 84
4.6.1 Regional Model 84
4.6.2 Carnmenellis Model 87
4.6.3 Quality of Results 89
4.7 Model Results: Temperatures 95
4.8 Conclusions 106
Chapter 5 Models of Fluid Flow and Mineralization 108
5.1 Introduction 108
5.2 The Effect of Fluid Flow on Modelled Heat Flows 109
5.3 Introduction to the Proposed Explanation of the Regional Aspects of the Ore Field 116
v 5.4 Two-Dimensional Models 119
5.4.1 Results from Synthetic Models 120
5.4.2 Results from a Model of St. Agnes 125
5.5 Results from Three-Dimensional Models of Convective Fluid Flow 129
5.5.1 Comparison of Fluid Flow and Fluid Inclusion Patterns 135
5.5.2 Comparison of Fluid Flow and Mineralization Patterns 143
5.5.3 Comparison of Fluid Flow and Kaolinisation Patterns 148
5.5.4 Comparison of Fluid Flows and Regional Geochemical Patterns 150
5.6 Conclusions 160
Chapter 6 Conclusions 162
References 168
Appendix I Temperature-Dependence of the Thermal Conductivity of Granite from the Rosemanowes Deep Borehole Data 180
vi LIST OF TABLES
Page
3.1 Summary of contract boreholes. 55
4.1 Rock properties used in the numerical models. 82
4.2 Summary of models. 84
4.3 Comparison of measured heat flow values with those predicted by the conductive models. 94
5.1 Comparison of measured heat flow values with those predicted by the convective model. 114
5.2 Maximum temperatures attained for different depths of cover in the St. Agnes model. 128
vii LIST OF FIGURES
Page 2.1 Structure of the computer programme to solve the combined conductive and convective heat flow equation. 31
3.1 Simplified geology of South-west England. 36
3.2 Location map of the exposures of the Cornubian granite batholith. 39
3.3 Distribution of mineralized lodes in South-west England. 45
3.4 Known regions of kaolinised granite in South west England. 47
3.5 Simplified Bouguer anomaly map of South-west England. 51
3.6 Depth to the top of the granite as determined by Willis-Richards. 52
3.7 Depth to the top of the granite as determined by Tombs. 53
3.8 Location of heat flow measurement sites. 54
4.1 Graph showing variation in surface heat flow with depth of heat production anomaly. 69
4.2 Diagram to show the effect of refraction for different shaped boundaries. 72
4.3 Location of the various models. 80
viii 4.4 Surface heat flow predicted by Model 1. 86
4.5 Surface heat flow predicted by Model 2. 88
4.6 Profile across the Carnmenellis outcrop comparing measured and modelled heat flows. 90
4.7 Graph of difference between modelled and measured heat flows versus the standard deviation in the thermal conductivity measured at that site. 93
4.8 Temperatures at 2000m predicted by Model 2. 96
4.9 Temperatures at 2000m predicted by Models 3a & 3b. 97
4.10 Paleoclimate model used to correct measured temperatures. 99
4.11 Corrections applied to temperatures to remove palaeoclimatic effects. 100
4.12 Temperatures at 6000m predicted by Models 3a & 3b. 102
4.13 Region assigned a higher thermal conductivity in Models 3a & 3b. 103
4.14 Temperatures at 6000m predicted by Models 3a & 3b including area of higher thermal conductivity. 104
5.1 Conductive heat flows predicted by Models 3a & 3b. 110
ix 5.2 Convective heat flows predicted by Models 3a & 3b. Ill
5.3 Vertical fluid flow at the surface predicted by Models 3a & 3b. 112
5.4 Surface Temperature Gadient and Emanative Centres. 118
5.5 Vertical fluid flow associated with a symmetric regular intrusion. 122
5.6 Vertical fluid flow associated with a symmetric intrusion with outwardlysloping sides. 123
5.7 Vertical fluid flow associated with an asymmetric intrusion. 124
5.8 Location of mineralized lodes in the St. Agnes • .... Head mining district. 126
5.9 Vertical fluid flow for a cross-section through St. Agnes Head. 127
5.10 Vertical fluid flow predicted by Models 3a & 3b with 3km of additional cover. 131
5.11 Vertical fluid flow predicted by Models 3a & 3b with 2km of additional cover. 132
5.12 Vertical fluid flow predicted by Models 3a & 3b with 1km of additional cover. 133
5.13 Vertical fluid flow predicted by Models 3a & 3b with zero aditional cover. 134
x Absolute abundances of type 2a fluid inclusions for Land's End, Carnmenellis, Bodmin and Dartmoor. 138
Absolute abundances of type 3 fluid inclusions for Land's End, Carnmenellis, Bodmin and Dartmoor. 139
Absolute abundances of type 1 fluid inclusions for Land's End, Carnmenellis, Bodmin and Dartmoor. 140
Absolute abundances of type 4 fluid inclusions for Land's End,Carnmenellis, Bodmin and Dartmoor. 141
Distribution of tin and lead-zinc deposits (after Dines, 1956). 145
Comparison of distribution of tin centres with the fluid flow patterns for 3km of additional cover. 146
Comparison of distribution of lead mines with the fluid flow patterns for no additional cover. 147
Comparison of distribution of known kaolinised granite with the fluid flow patterns for no additional cover. 149
Anomalous distribution of tin. 153
Anomalous distribution of copper. 154
Anomalous distribution of lead. 155
xi 5.25 Anomalous distribution of zinc. 156
5.26 Anomalous distribution of arsenic. 157
xii Chapter One
Introduction
The South-west of England is dominated geologically and in many other ways the the presence of a large granitic intrusion, the Cornubian batholith, which underlies the region from east of Dartmoor to west of the Scilly Isles. Associated with the batholith and formed as a direct result of its intrusion is a highly complex ore-field. The minerals from this area have been exploited since Phoenician times and for some periods during the last century the mines of the region were producing over half of the World's tin and copper. Even though only small quantities of tin are now being produced, the granites are still of economic importance to the area due to their large reserves of high quality china clay of which almost 2.5 million tonnes are extracted per annum. Recent interest in the exploitation of geothermal energy in the U.K. is concentrated in an experimental station situated on one of the granite outcrops.
1.1 Heat Flows and Geothermal Energy
The investigation of the heat flow field of the U.K. in general accelerated after the oil crisis of the early seventies financed by the European Communities as part of a European exploration programme and the South-west of England was soon identified as the prime target for detailed study. It was to determine the geothermal potential of the batholith that a series of heat flow boreholes was drilled in the exposed granites and
1 surrounding country rocks under a prqgramme managed by the geothermal group at Imperial College. Using commercial and specially drilled boreholes, thirty-six heat flow measurements were made in the region (Tammemagi & Wheildon, 1974 & 1977, and Wheildon et al., 1980). Many of the heat flow boreholes drilled under this programme were very short (of the order of a hundred metres) but this was considered to be acceptable provided they could be satisfactorily corrected for perturbing effects. This decision has been vindicated to a great extent by recent measurements (Sams and Thomas-Betts, 1986) in much deeper boreholes which confirm the accuracy of the original heat flow measurements on the granite outcrops. The high heat flow anomaly delineated by this research is caused by the high heat production of the batholith (Francis, 1980).
Partly as a result of this study an experimental geothermal station was developed at Rosemanowes Quarry on the Carnmenellis outcrop to examine the potential of extracting energy from the granite by the Hot Dry Rock technique. The Hot Dry Rock (HDR) technique involves creating a fracture system at depth in the rock between two boreholes to act as a heat exchanger, with water being circulated via the boreholes. The hot water or steam extracted can then be used to provide energy for a variety of purposes depending on its temperature. The first experiments at depths of about 300m have been superceded by a 2000m system which is still being tested. The commercial development of the HDR method to produce steam at high enough temperatures to generate electricity will require the development of a similar system but at much greater depths. The next phase of the experiment is expected to comprise a system at about 6000m, tapping temperatures in excess of 200°C.
The additional information obtained from the deep boreholes at the HDR site together with the heat flows, thermal conductivities and heat production measurements from the
2 regional heat flow survey can be used to constrain numerical models within reasonable limits. These models can then be used to predict the temperatures at depth within the granite and thus delineate those areas which might be suitable for future, commercial development of the HDR method. They can also help to explain the features of the observed surface heat flows which are not predicted by two- dimensional models and those factors which primarily determine the variations in surface heat flows.
1.2 Heat Flows and Mineralization
Brown et al. (1980) found a correlation between epigenetic mineralization and high heat flows in the U.K.. They proposed that this was due to intrusives with high heat production and thermal conductivity focusing the development of hydrothermal systems responsible for some types of mineralization. Indeed, it is believed that the majority of mineral lodes of the ore-field associated with the Cornubian batholith were formed by the action of several episodes of convective flow (e.g. Moore, 1982) stimulated by the radiothermal contrast between the granite and its host rocks. So it may be possible to predict the locations of the mineral deposits by numerical modelling of the flow of fluid around the batholith at different stages during its history.
The convective flow of (mineralizing) fluids in this region has been modelled in two dimensions by Fehn (1985) . However, his models are poor in their representation of the actual situation and can not explain some of the more fundamental aspects of the ore-field. Two features which are of relevance are the overall distribution of mineralized lodes and the apparent horizontal zoning of different minerals seen within this distribution (Dines, 1934 & 1956). Three-dimensional numerical models will lead to an improved understanding of the influence the shape of
3 the batholith has over the location of convection cells and thus mineralization.
1.3 Heat Flow and Numerical Modelling
The numerical technique employed to solve these models was the finite element method. The improved efficiency and capability of modern computers has meant that the finite element technique has become a widely used and tested method for solving physical problems. The first reported attempt to employ this technique was in the solution of elliptical partial differential equations using a piecewise approximation (Courant, 1943), but the name "finite element" was not introduced until much later by Clough (1960). Geertsma (1971) was the first to use this particular technique to solve for subsurface temperatures in the upper crust. Since then it has been used to solve for temperatures in many different geological environments. The accuracy of the method was investigated by Lee and Heyney (1974) when they compared the results of the numerical method with analytical solutions (to the conductive heat flow problem in a geological environment) and concluded that good agreement was obtained provided that the approximations due to the finite size of the elements were satisfactory. It is also now a standard technique for solving problems which include the convective transport of heat (Huyakorn and Pinder, 1983).
4 Chapter Two
Finite Elements
2.1 Introduction
This chapter serves to outline the essential aspects of the finite element method, its application to the transport of heat by conduction and convection, and its implementation on a computer.
Since the finite element method can nowadays be considered a standard tool for solving physical problems and many texts exist which detail the development of the concept and the mathematics involved, only a brief synopsis of the technique is given here. This aims at informing the reader of the fundamentals of the method and the approximations which must be made. Although there are many approaches which can be adopted in the general finite element method, it is only the Galerkin weighted residual scheme which is used here and, hence, discussed. The equation for the conduction of heat through a solid is developed using this approach to a stage where it can be coded for use on a computer. The final forms of the equations for the conductive and convective transport of heat through a porous medium are also given.
Even though it is a standard technique, the implementation of the finite element method to a new physical problem on a particular computer will always raise difficulties and the latter half of the chapter deals with these. For instance, a major problem in dealing with large models is the initial
5 input of the shape of the model and a program to allow easy, efficient construction of a finite element grid that could represent three-dimensional variations of geology within a regularly-shaped portion of the Earth's crust had to be developed. The techniques applied to maximize the efficiency of the program which solves the equations by matrix inversion are described. Finally, the means by which three-dimensional results are displayed as two-dimensional diagrams is explained.
2.2 The Finite Element Method
In applying the finite element method, it is assumed that the problem to be solved is well posed: that is, that the physical problem is defined by a partial differential equation with appropriate auxiliary, initial and boundary conditions. The region of interest, in which the partial differential equation is to be solved, is divided into sub- regions. The partial differential equation is assumed to hold in each of these sub-regions whose shape and size allow an approximation to the physical reality.
Within each sub-region, or element, the unknown, desired function, u, in the partial differential equation is replaced by a finite series such that
M 2 .2.1 m=l
The Nm ( m = 1 to M ) are functions of space and, if necessary, time. The Um ( m = 1 to M ) are the unknown, desired coefficients. The Nm are chosen to be polynomials
6 that satisfy some, if not all, of the boundary conditions imposed on the problem. Even if these functions - known as the shape functions - are chosen such that they satisfy all of the boundary conditions, they will not necessarily satisfy completely the partial differential equation as well.
Consider, for example, the linear differential equation
G(u) = Pu + s = 0 2.2.2
where P is a linear differential operator and s is independent of u. This equation is to be solved in the volume V with the associated boundary condition
H(u) = Qu + t = 0 2.2.3
on the boundary S. Again, Q is a linear differential operator and t is independent of u. Substitution of equation 2.2.1 into equation 2.2.2 will yeild a residual,
M G (u) 2.2.4 m=l
and if u does not satisfy all of the boundary conditions then
M 2 .2 .5 m=l
7 The approximation can be optimized by minimizing these residuals in some fashion. One scheme would be to set the integrals of the residuals of the volume and boundary surface equal to zero over each element, thus:
2 .2.6
However, this generates only one equation for the M unknowns. By the introduction of M arbitrary weighting functions, say wR and vn , for n = 1 to M, M equations can be made available to solve for the M unknown coefficients,
n = 1 to M, 2.2.7
where wn and vn may be chosen independently. This procedure is known as the the method of weighted residuals and yields a set of equations that in theory can be solved for the unknown coefficients. It is clear that the choice of weighting functions can lead to different schemes within the weighted residual approach. Many schemes are available and the choice will affect the accuracy and efficiency of the solution. In the Galerkin method, the only scheme considered here, the shape functions themselves are used as the weighting functions. Thus the shape functions play a vital role in the Galerkin finite element method.
Since the function which is being approximated over each element must be continuous over the entire region, not only must the approximating functions from adjoining elements which are defined at a mutual boundary match at that boundary, but also there should be a degree of continuity, which depends on the order of the differentiation involved,
8 across that boundary. Substituting equations 2.2.4 and 2.2.5 into equation 2.2.7 and combining the results for all elements yields
M I wn < Z 'mP m + s dV + V' m=l
M Z v Um Q 'm + t ) dS o , m=l
for n = 1 to M. 2.2.8
If P or Q contain derivatives which are of such an order that the integrand contains infinities then the integrals may diverge. To avoid such difficulties, it is necessary that if the partial differential equation or the boundary conditions contain derivatives of the order p, then the derivatives of order p - 1 must be continuous in the shape functions.
It is therefore useful if higher order differential operators can be expressed in terms of lower order differential operators. This is often possible using Green's lemma. If the operator P in equation 2.2.8, say, contains second order derivatives, it can be written as
wR Pu dV = l(Awn )(Bu) dV + /wR Cu dS 2.2.9 where A, B and C are linear differential operators containing only first order derivatives. This means that a lower order of continuity is sufficient for the shape functions. The expression of the equations including this transformation is known as the weak formulation of the
9 weighted residual statement.
Equation 2.2.8 represents a series of M simultaneous equations which can be solved to find the M unknown coefficients Um .
2.3 Shape Functions
As stated previously, the shape functions, Nm , are chosen to be polynomials of the space and , if necessary, time domains. If the actual function, u, is to be approximated by a linear function, u, over an element, then in the one dimensional case, two points are required to define the function. If the approximating function is to be quadratic in any one direction, then three points are required to define the function. Thus the degree of the approximating polynomial will determine the number of points required to define the approximating function. Conversely, the degree of the approximating polynomial in an element will be determined by the number of points associated with that element. These points are known as nodes.
Equation 2.2.1 can be written for the one-dimensional linear case as
u(x) = u(x) = U 1 NX(X) + u 2 N2 (X) 2.3.1
One-dimensional linear elements have two nodes, each with a shape function associated with it. The shape functions are of the same order as the approximating function, u. So if the shape functions are given a value of unity at their associated node and zero at the other node then the Um represent the value of the approximating function at the
10 nodes. If the nodes lie at the extremes of an element then some nodes can be situated on more than one element. The shape function associated with such a node takes non-zero values on both elements, but each shape function is defined only in that element which contains its associated node.
The shape functions can now be readily found by imposing the above conditions. Hence
N^x) = x2 - x / x2 - x^ 2.3.2
N2 (x ) = x - x -l / x2 - x-^, 2.3.3
where x-^ and x2 are the coordinates of nodes 1 and 2. The extension of this technique to higher order approximation functions and higher dimensions is discussed in standard texts (S.S.Rao, 1982; 0.C.Zienkiewicz and K.Morgan, 1983; 0.C.Zienkiewicz, 1977), but the basic concepts shown above remain unaltered. When considering two or three dimensions there are a number of standard shapes which the elements can take and for which the shape functions are known. In two dimensions the most common elements are triangles and quadrilaterals. The shape functions for these elements and for varying numbers of nodes are also to be found in the standard texts.
2.4 Mapping
The shape of the element will determine how well the physical reality is approximated. For instance a diagonal boundary cannot be represented accurately by rectangular elements, but may be by triangular ones. Hence it is up to
11 the modeller to choose those elements which best suit the problem. Indeed, there is no restriction to just one type of element per model, just as there is no restriction on having different degree approximation functions for different elements in the same model.
Even more flexibility is allowed if elements are not required to be of a regular shape; that is, if the nodes which define the outline of the element are allowed to be moved with respect to one another. Then it would be possible for the side of an element with more than two nodes to represent a non-linear boundary with a higher degree of accuracy.
However, this introduces a computational problem in that the integrals in equation 2.2.8 must now be calculated over irregular volumes and surfaces. This problem can be obviated if a function can be found which allows such distorted elements to be mapped from this 'global' coordinate system to a ' local' coordinate system where the elements regain their regular shape. The integrations can then be more easily performed in this local space.
Looking at the problem in reverse, the question can be asked: what functions can be utilised in order to map from the simple local space to a more complex global space? A convenient answer is the shape functions. This type of mapping (known as parametric mapping) allows a boundary of an element, on which lie three nodes, to represent a quadratically shaped boundary.
For a two-dimensional element with M nodes, the mapping relationship between the local coordinates (r,s) and the global coordinates (x,y) is given by
12 M X = Z xi Ni
1 = 1 N ^ N -^ i^ s) 2.4.1 M y = Z y± Ni ' i=i where (Xi/Yi) are the global coordinates and N^(r,s) the shape function of the i^*1 node of the element. Since the shape functions possess some inter-element continuity, the coordinate maps will be continuous, even if different local origins are used in each element.
The derivatives of the shape functions must also be transformed:
M • = 6n - bx + M..* by 2.4.2 cbr bx br by br
M i = M i M + M i M . 2.4.3 bs bx bs by b s
The required derivatives can be found in terms of local co-ordinates by
provided that J, the Jacobian matrix of the transformation, is non-singular. In addition, in the two-dimensional case, for example,
13 det(J)drds. 2.4.5
That is the integration takes place in the local space with all functions in the integrand being defined in local coordinates.
Even now the integration is not simple and has to be carried out numerically. The most common technique involves Gauss-Legendre quadrature. The method requires the integrand to be evaluated at a fixed number of points in the volume or on the surface. A weighted sum of these values is the approximation to the integration. The number of points at which the value of the integrand is to be found is determined by the shape of the element and the number of nodes on the element. These points are commonly called Gauss points. The location of the Gauss points and the weights associated with them for different types of elements can be found in the standard texts.
2.5 Heat Conduction and the Finite Element Method
As discussed before, in order to implement the finite element method the problem must be well posed; that is, the problem must be described by a partial differential equation and have appropriate boundary conditions. For the conduction of heat in three dimensions this may be expressed as follows (Carslaw and Jaeger, 1946);
3 3 + A = 0 in V 2.5.1
14 T = Tb on st 2.5.2
K' • 6t = -q on srr 2.5.3 . 3 2m q where T is the temperature field, K^j the thermal conductivity tensor, A the heat generation, x^ are the coordinates, and n is distance along the outward normal to Sg. Equation 2.5.2 is a Dirichlet boundary condition which fixes the temperature at the boundary nodes and equation 2.5.3 is a Neumann boundary condition which fixes the flow of heat perpendicular to the boundary.
Following the method described in section 2.2 the desired function is replaced by an approximating function
M T(x,y,z) ^ ¥ = X ! Nm Tm 2.5.4 m=l which, when substituted into equations 2.5.1 and 2.5.3, gives the residuals
3 3 Z Z Kij + A = 2.5.5 1=1 3=1_n OX; OX-;
Kj^j 6T + q = R, 2.5.6 dn
These equations may be substituted into the weighted residual equation
15 wn Kv dv + vn Rs dS = 0 2.5.7 Sqe to give
(«nZ Z ^ Kii $£ ) dV + wn A dV _n xa 1 = 1 1 = 1 bxA ±3 6 V'
+ vn K-J-; bT dS = 0 for n =1 to M. 2.5.8 r Sqe J 6 n
In order to reduce the order of the equation, Green's lemma can be applied to the second order term in equation 2.5.8
3 3 3 3 wn Z Z K±i hr ) + Kij |$ wn dS. 2.5.9 i = l j = l 3 b x on It can be seen that the Neumann boundary condition is, after a fashion, a natural boundary condition (compare the last term of equation 2.5.8 with that of 2.5.9). By making vn = -wn on Sg and wn = 0 on S-Sg and substituting equation 2.5.9 into equation 2.5.8 the weighted residual equation becomes 16 3 X > i j bwn M. dV w.n A dV bx^ 6 xj V* j=l wn q dS = 0 2.5.10 Following the Galerkin scheme the weighting functions wn are replaced by the shape functions Nn and writing the approximating function in full gives 3 3 M L I I E Kij Tm dv = Jv i-l j«i m=l *xi Nn A dV + Nn q dS , for n = 1 to M. 2.5.11 V* Sqe Equation 2.5.11 may be written in matrix form KT = f 2.5.12 with r 3 3 M K = 2 / y \ y > i i &n. dV 2.5.13 nm e Jve z-' *-• >^n i=l j=l m=l b x &x™ ( T-j , Tjr Tm,...,TM■M ) 2.5.14 and 17 Nn q dS. 2.5.15 Equation 2.5.12 can be solved to give the unknown coefficients, Tm , which are in fact the temperatures at the nodes. It can be seen from equation 2.5.13 that the matrix K is symmetric about the diagonal, that is Knm =Kmn. This means that only half the matrix need be calculated and stored. If the thermal conductivities of the materials which constitute the elements are temperature-dependent, then the system of equations becomes non-linear and an iterative process must be followed in order to determine the temperature field. 2.6 Conductive and Convective Heat Transport and the Finite Element Method The steady state flow of heat by convection, as well as conduction, is described by the following partial differential equation p Cv v VT - V.(KVT) - A = 0. 2.6.1 The first term represents the convective flow of heat where p is the density of the fluid, Cv the heat capacity of the fluid and v the velocity of the fluid. The second term describes the conductive flow of heat where K is the hydrodynamic thermal dispersion tensor for the combined fluid and solid medium. A is the heat generation. The equation as a whole is founded on the conservation of energy. The velocity, v, is given by Darcy’s law 18 V 2 .6.2 where k is the permeability tensor, u the viscosity of the fluid, P the hydrostatic pressure and g the gravitational acceleration. In order to find the velocity of the fluid in the region of interest, the pressure, P, must first be found. This is done by solving the equation for the conservation of mass V. (yov) = 0 2.6.3 which states that the divergence of the mass flux is zero. The two equations 2.6.1 and 2.6.3 can be solved sequentially: the pressures are solved for first which yield the velocities. These velocitites are then used to solve for the temperatures. Since many of the terms involved in the equations are temperature- and pressure- dependent, the solution procedure must be iterative. The problem is thus posed as 3 3 in V 2.6.4 with boundary conditions P = Pb on Sp 2.6.5 19 bp - /O g bz. = f o n S ^ 2.6.6 bn bn and T . T , ( p cv |t b K,*bT ) i=l j=l dxj - A = 0 in V 2.6.7 with boundary conditions T = Tb on St 2 . 6 .8 ^ bT = -q on S„. 2.6.9 3 bn q By substituting M P ( x,y,z ) = P = Y, pm Nm 2.6.10 m=l in equation 2.6.4 and M T ( x,y, z ) S t = Y. Tm Nm 2.6.11 m=l in equation 2.6.7, and following the same procedure as for the conductive heat flow equation as shown in section 2.5 (see also Huyakorn and Pinder, 1982, pp 198-200) the final 20 matrix form of the two sets of equations can be written S P = g 2 . 6 .1 2 where 3 3 nm ■ • ? / , . ^ % dv i= l 3 = 1 2.6.13 t ' ' * * ' ^ 2.6.14 Xj M ij( + )ni Nn ds 2.6.15 with Mij = p ki j / p 2.6.16 and 21 ° i j = /> 2 <3 ki j / P 2.6.17 and RT = h 2 . 6 . 1 8 where (Bi M n Nm + Kij |Mn ^ J d V * » . - z X e i £ e v i=l j=l 1 } 2 . 6 . 1 9 TT = ( Tlf T2, T3, . . . , Tm ) 2 .6.20 3 3 n = Z / e A Nn dv + Z [ Z ZKij |T ni Nn dS e q 1=1 j =1 2 .6.21 with B. P CV vi‘ 2.6.22 A third set of equations is needed which describe the variation of the properties of the fluid with temperature and pressure. These may be found in the Steam Tables 22 (ERA, 1967) . The matrix R is not symmetric and so both halves of the matrix must be calculated and stored. 2.7 Model Building The first step in solving the problem posed is the discretization of the region of interest. This may not seem a difficult task at first, but as the problem is a three-dimensional one it soon becomes evident that a great deal of data must be generated with a computer either interactively or by direct input, and stored. Further difficulties arise when the three-dimensional shapes of the elements need to be visualized and changes to the model need to be made. The method developed here is based on simplicity and efficiency. All models to be considered have an overall shape of a regular hexahedron as this allows for sensible boundary conditions to be applied with ease. The top surface will usually represent the surface of the earth. The interior of the model is divided into smaller hexahedra which are not necessarily regular in shape and are the elements of the finite element method. By assigning different properties to the elements and by changing their shapes they can be made to represent various geological structures. To begin with each model consists of a large hexahedron (which is the whole model) with one node at each corner: the model is initially built with linear elements, though this can be changed at a later stage. The coordinate system is so arranged that each of the sides of the hexahedron is parallel to one of the axes and the origin of the coordinate system coincides with one of the nodes. 23 The nodes are numbered and these numbers must be stored in such a fashion that they are easily retrieved when the finite element grid is changed in any way. The method used here is as follows: each node belongs to a column of nodes such that all nodes in the same column have the same x and y coordinates. So, there are two nodes to each column and four columns. The columns are also numbered. The node number is stored in a two-dimensional array, NNODE(a,b), one dimension, b, referring to the column on which it lies and the other, a, to the its position along the column. In exactly the same manner the columns are said to belong to different rows of columns. That is, at the start, all columns with the same x coordinate belong to the same row, there being two columns per row and two rows. Again the rows are numbered and the column numbers are stored in a two-dimensional array, NCOL(a,b), with one dimension, b, referring to the number of the row on which it lies and the other, a, its position along the row. The row numbers are stored in a one-dimensional array NROW(a). Thus, say, the number of the third node of the second column of the fourth row will be NNODE(3,(NC0L(2,NR0W(4)))). The region can then be progressively split up into smaller subregions by the introduction of new rows of columns, new columns of nodes in each row or new nodes in each column. There is one restriction, however, that there must be the same number of nodes in each column, and also the number of columns in each row must be the same. The addition of a new node in each column will, then, introduce a new level dividing each of a group of elements into two elements in the z direction. Care must be taken that all the new nodes fall between the same consecutive pair of nodes of each column to avoid different levels crossing and leaving different elements sharing the same space. Levels may also be removed likewise. The logistics of this method are quite simple. In the 24 beginning the array of row numbers (NROW) contains two numbers: (1,2). If a new row is introduced between these rows the array will contain three numbers: (1,3,2). The array of column numbers (NCOL) initially contains four numbers, one for each column: (1,2; 3,4). If a new column is added to each row this matrix will then read (1,5,2; 3,6,4), and if a new row is added (1,2; 3,4; 5,6), and after the addition of a new row, and a new column in each row, (1,7,2; 3,8,4; 5,9,6). The advantage of this method is that each time new rows, columns or nodes are added, it is not necessary to renumber the nodes and hence have to reorganize all arrays relating to the node numbers like boundary conditions, coordinates and, since the element number is taken to be the same as the number of the node in that element which would lie closest to the origin if the grid were regular, the properties of the elements. New nodes take the next available node number and the arrays which store the node, row and column data can be manipulated with ease due to a routine which allows blocks of data within arrays to be shifted without recourse to multiple nested *D0' loops. (This is not a standard routine which is generally available but was written by Mr. Peter Harris, computer manager of the Geophysics Department.) However, re numbering and reorganization after deletion of rows, columns or nodes is advisable to avoid arrays becoming too large and containing redundant data. The model can be displayed on a VDU by plotting the outline of faces of elements one level at a time; for example the top of the model can be shown by drawing the outlines of the top faces of every element which lies on the surface. This is achieved by taking all the nodes in a particular level, that is, the n^*1 node of each column or all the nodes in the ntri column of each row or every node in a row and plotting the edges of the elements between the various 25 nodes (each node represents the corner of one or more elements). Once a picture has been displayed it is easy to make alterations to the model: nodes can be moved to change the shapes of the elements or allocated boundary conditions and elements may be assigned a material type. The program allows the outline of the elements to be overlain by a map depicting some aspect of the geology ( for example the outline of a granite batholith at depth ) in order that an accurate representation is achieved. However, care must be taken since it is not necessarily the case that all the nodes in a particular level, say in the xy plane, have the same third co-ordinate. The addition of new nodes and elements requires new properties to be allocated to those nodes and elements. An element that is subdivided passes on its material type to both elements created; properties of new nodes are linearly interpolated from existing nodal values. Likewise, if the linear elements are transformed to quaratic elements, the properties of new nodes are linear interpolations of existing values. Before the data is written to file to be stored the nodes are re-numbered and the data re-organized in order to reduce the amount of work required to invert the matrix. As stated before, the matrix is sparse and in order to optimize the inversion it is required that the non-zero values lie as close to the diagonal as possible. This reduction of the bandwidth is made possible by numbering the nodes in a given order. For the type of models presented here the optimal numbering scheme is as follows: any of the corner nodes of the entire model is numbered '1'. Numbering proceeds in the same direction (e.g. increasing x) for all lines of nodes. Starting with any one of the corner nodes of the entire model, numbering is carried out along the axis which contains the smallest number of nodes. The next line of nodes to be numbered is 26 adjacent to the previous one and starts on the axis which contains the next lowest number of nodes. Once all the nodes of this level have been re-numbered line by line, numbering continues in a similar manner on the adjacent (parallel) level in the direction of the most number of nodes starting with the node corresponding to the initial corner node. 2.8 Matrix Inversion The most computationally-intensive aspect of solving a problem by the finite element method is the inversion of a large matrix. The fact that the matrix is band-limited and, in the case of heat transport by conduction, symmetric, helps to some extent. However, even this reduction of storage requirements will not usually be enough to enable the matrix to be stored in its entirety in the active memory of the computer. Data must therefore be transferred into and out of memory during the calculations. It is important to reduce the number of data transfers to a minimum as they are inefficient. The standard technique is a wavefront approach which involves the inversion of parts of the matrix in the course of the assembly of the matrix so that each row of the matrix need only be written out once and read in once. A further improvement in efficiency was achieved by asynchronous block input/output which permits data to be input or output whilst calculations are being performed. (The routines to perform the asynchronous work are part of Digicon's seismic processing package, DISCO.) For the heat conduction problem the equation to be solved is of the form Ka = f 2 . 8 . 1 27 where K is a symmetric matrix and a i s unknown and it may be solved by the Choleski decomposition. In this method K can be decomposed into an upper matrix U where K = UTU 2.8.2 and U = ull* . .Uln 2.8.3 0 unn The elements of U are given by i-1 uij = i ( Kij - 2 ukiukj > j>i* i>2 2.8.4 uii k=l 1 -1 uii = ( Kii - 2 uki2 >1/2 2.8.5 k=l Ulj “ Klj / Ull* 2 .8.6 Then a can be found thus a = [ UTU ] 1f 2.8.7 a = U 1( UT ) 1f 2 .8.8 2 8 a = U 1 ( U 1 )T f 2.8.9 where U 1 = V 2 . 8 .1 0 and vl i 1 / uii 2 . 8 . 1 1 j vij = - ( Z uikvkj > / uii- 2 . 8 .1 2 k=i+l Thus the matrix a can be evaluated in just three steps: 1) Decomposition of K into U. 2) First elimination j-1 f'j = C fj - Z uijfi ) / ujj- 2.8.13 i=l 3) Second elimination n aj = < f,j - Z ujif,i ) / ujj- 2.8.14 i=j+l Parts 1 and 2 require data from previous lines and part 3 from succeeding lines in the matrix. However, as the matrix is band-limited not all preceding or succeeding 29 lines are needed. For instance, once j is greater than the bandwidth the first line is no longer required for parts 1 and 2 and may be written out and need only be read back in when part 3 is to be performed. As soon as the data have been written out the remaining data are shifted up a line to make room for new data at the bottom of the space allocated for the matrix. Also, the data storage is such that each line of this matrix starts with the element Ujj. These factors reduce the space required in the memory from MxM, where M is the total number of nodes, to BxL, where B is the bandwidth and L = 2xB + 1. A similar process can be used to solve a problem containing a non-symmetric matrix. In this case the matrix is decomposed into an upper matrix, as in the symmetric case, and a lower matrix (for details see Rao, 1982). The structure of the computer program used to solve the combined convective and conductive heat transport problem is given in Figure 2.1. 2.9 Data Output Whatever the problem solved or the type of element used, the result of the inversion gives the value of the desired function or functions at all the nodes. In order to display the results a contour map of the function, its derivatives or some combination of the two, on a given plane, must be produced. Since it is likely that only a few nodes have the exact x, y or z coordinate required, these values alone will be insufficient to produce such a contour map and additional values of the function must be sought. In theory, it is possible to calculate the value of the function at any point on an arbitrary plane from the nodal 3 0 FIGURE 2.1. Structure of the Computer Program to Solve the Combined Conductive and Convective Heat Flow Equtions. read data file call FILBLK_OPN set parameters call DIRICHLET if solving for pressure — i______t (call WATER end if ------=]------calculate element number (nel) continued... 3 1 Figure 2.1. Function and Source of Routines used to Solve the Combined Conductive and Convective Heat Flow Equations. DISCO* routines: FILBLK_OPN opens a mass storage file for the excess of the matrix data. FILBLK_PUT writes line of matrix to file asynchronously. FILBLK_GET fetches line of matrix from file asynchronously. FILBLK_WAIT waits unitl FILBLK_PUT or FILBLK_GET are complete. FILBLK_CLS closes mass storage file. Routines written by ^others: BLKSHIFT shifts a block of lines of matrix up or down(dn). Routines written by the author: DIRICHLET calculates the Dirichlet boundary conditions and assembles the results in the forcing matrix. NEUMANN imposes the Neumann boundary condition. WATER calculates the properties of the water at all nodes. JACOB calculates the Jacobian of transformation between the local and global coordinates of an element. ASSEMBLE calculates the matrix elements and assembles them in the matrix. DECOMP decomposes lines of the matrix which have been fully assembled. ELIMA completes the first part of the elimination on decomposed lines. ELIMB completes the second part of the elimination. DIFFER calculates the difference between this and the previous iteration. DISCO is• DIGICON's seismic • • processing • package. ^ Namely Mr. P. Harris, computer manager, Geophysics Dept., Imperial College. 32 values of the elements intersected by the plane. However, the value of the function within an element can only be calculated in terms of the local coordinate system, and the conversion from the global (x, y, z) system to the local system involves the inverse of the Jacobian of the transformation (2.4.1); therefore, in practice, this procedure would become far too complicated and time- consuming to be used routinely. There are, however, several points whose coordinates in the local system are already known and these are the Gauss points whose locations in global coordinates can readily be found (equation 2.4.1). As the plane of interest passes through an element, it will pass between a node and Gauss point or between two Gauss points at least once. On each such occasion, by linearly interpolating between the two known values, the value of the function at the point on the plane may be obtained. Sufficient values can be generated in this manner to define the variation of the function on that plane and, although these points will be irregularly spaced, they will be adequate for contouring to be performed after suitable interpolation. 3 3 Chapter Three Geology and Geophysics of South-west England 3.1 Introduction Results from numerical modelling will be reliable only if the models can be adequately constrained. The a priori information required to construct the model and to compare results with, is usually in the form of surface geology or results from geophysical experiments. The main feature of the region modelled herein is the large, high heat production granite batholith, which underlies the whole of the S.W. England peninsula from east of Dartmoor to west of the Scilly Isles, and is the reason for so much past and present interest in the region. Associated with the granite is mineralization and alteration products of the granite. These have been economically important for the area and, although exploited for many years, the processes by which they were formed are not completely understood. The presence of the Cornubian batholith is indicated by outcrops, but it is only by geophysical means that its form at depth can be ascertained. This is also true of the vertical disposition of other rocks which outcrop. The thermal properties of the various rock types can be measured from surface samples or from borehole samples, but the values at great depth must either be extrapolated from near-surface measurements or be solved for in the modelling. The results of the modelling need to be constrained by 3 4 measured values. Here the temperature field is first expressed in terms of vertical heat flows at the surface and then compared to the surface heat flows measured at various locations in the region, but are subject to errors which must be assessed before comparison with modelled results. 3.2 Geology The region of interest forms the peninsula of south-west England (fig 3.1) and is bounded by the Bristol Channel, the Atlantic Ocean and the English Channel. The geology is discussed in detail in the regional guide (Edmonds et al., 1975) and recent advances in the understanding of the geology are discussed by Dineley (1986). The following is a synopsis of these publications. The surface geology consists mainly of Palaeozoic sediments (now metamorphosed) and igneous rocks with some Mesozoic and Cainozoic deposits,but the area is dominated in many ways by the presence of a large granite batholith. The great interest in this region has been due largely to the economic resources of minerals and alteration products and, more recently, geothermal energy associated with the batholith. The emplacement of the granite occurred towards the end of the Variscan orogeny. Even though the geology of the region has been studied for over two centuries there still remain many unanswered questions and uncertainties concerning both the granite and the orogeny. However, the details of the Variscan orogeny and its causes are not relevant in the context of the present work. What is important is the distribution and physical properties of the rocks in the region at surface and at depth. 3 5 Figure 3.1 South-West England:Simplified Geological Map The Palaeozoic sediments were deposited into the Cornubian basin from Lower Devonian (Siegenian) until late Carboniferous (Westphalian) times. This basin was bordered to the north by the semi-desert Old Red Sandstone continent (St. George's Land) and to the south by a rising land mass. From the former were deposited the sandstones, shales, conglomerates, and some calcareous beds of the Lower Devonian. These were followed by Middle and Upper Devonian slates and mudstones. Limestones formed as coral banks and pillow lavas remain as relics of submarine volcanic activity. Vulcanicity continued into the Carboniferous forming lavas, tuffs, ashes and agglomerates. The Carboniferous sediments, known locally as the Culm, are resultant from the deposition of silts and muds in earlier times, and then, as the sea became shallower, sands and muds. They now lie at the centre of a large synclinorium (fig 3.1) which trends east-west with older rocks outcropping to the north and south. Towards the end of the Carboniferous these sediments were subjected to a north-south compress ional force. This may have been due to a continental collision following the closure of an ocean to the south of the the region. Whatever the cause, the forces were strong enough to produce overfolding, faulting and thrusting, and the development of slaty cleavage. Thrusts tend to dip south in the south and north in the north. The regional metamorphism began in the south where it is most intense and progressed northwards and it was at about this time the granite was emplaced with thermal metamorphism distributed as aureoles about the granite bosses. Many post-magmatic processes took place and these will be discussed in the next section. During the Permian, the granite was rapidly denuded and unroofed (Dangerfield and Hawkes, 1969) in arid conditions when the region lay close to the equator. Deposits were derived locally and for much 37 of its remaining history it was an area of positive character. This may have been due to isostatic uplift after the emplacement of the relatively light granite (Bott and Scott, 1964) though Badham (1980) suggests that there might not have been much isostatic imbalance. However, Mesozoic and later rocks are now restricted to the far east of the region. During the Tertiary there was movement along wrench faults (Dearman, 1963) which have changed the orientation of several features. Reconstruction of pre-faulted geology shows a more consistent east-west trend to the Palaeozoic rocks. These movements were associated with the Alpine orogeny. Marine erosion and uplift in these times have left a multitude of wave cut platforms. 3.2.1 Granite The granite which forms the Cornubian batholith was emplaced in the upper crust towards the end of the Variscan orogeny. The granite is now exposed as six major outcrops and several minor masses (fig 3.2). It is widely believed that the granite was formed from the partial melting of the lower crust but there is some disagreement as to the direction from which the magma came. Shackleton et al. (1982) propose a gently southward dipping decollement along and up which rose the magma to form the batholith. They argue that because the thickness of the crust at its present location is only 27 km, with little lost to erosion, the temperatures were not high enough to cause partial melting in the crust. However, Floyd et al. (1983) suggest that subduction related heating would have caused the necessary geothermal gradients and that the magma rose directly from beneath its present location. 3 8 o Cl 3 3 cr H- p 3 O H) ft cr (D o (D W W c Ci x O O (0 w rt 3 O Ml p H* o o w CO s fD n *3 H* iQ Granite Batholith OJ OJ VO ft 3* The initially dry magma rose rapidly by stoping and individual cupolae rose out of the main body by further stoping of more differentiated material (Exley and Stone, 1964) and was almost solid when emplaced. The granite rose to within a few kilometres of the surface (Floyd, 1971; Allman-Ward, 1985; Badham, 1980). After taking wrench faulting into account the batholith is oriented parallel to many Caledonian features in an ENE-WSW direction and may have exploited a pre-existing fault or weakness in the crust as it rose. Ghosh (1934) subdivided the granite of the Carnmenellis outcrop into three types. Al-turki and Stone (1977) revised this to two types, inner and outer, which are petrographically and chemically different though not necessarily separate intrusions, but have similarities with other centred complexes. Variations within the granite are most marked in the texture. The main variations are in terms of the overall grain size and size of the K-feldspar megacrysts (Dangerfield and Hawkes, 1981). Geochemically they show only minor variations, though six types have been identified by Exley and Stone (1964, 1982) and Stone and Exley (1985). About 90% of the outcropping rock is medium- to coarse-grained biotite granite which is typically megacrystic (Hawkes and Dangerfield, 1977). The predominant chemical and mineralogical features of the exposed B Type granites are those of the 'S' type granites of Chappell and White (1974). Late-magmatic and post-magmatic phenomena are extensive. Exley et al.(1983) recognise two sets of processes: primary and secondary which are distinguishable as being pre- and post-joint formation respectively. Tourmalinisation resulting from an increase in boron and fluorine in the residual aqueous fluids and greisenisation in younger varieties of granite are the primary processes. The secondary processes are greisening (resulting in greisen- 4 0 flanked veins) tourmalinisation and mineralisation. Also common are metallized and non-metallized pegmatites, quartz -porphyry dikes (locally known as elvans) and kaolinisation. Ages for these events are uncertain due to the re-working by later thermal events. Exley et al (1983) prefer, for the age of the granite emplacement, the value given by a dolerite dike at Meldon of 295 Ma and by the Scilly Isles granite of 303 M a . Jackson et al (1982) suggest an emplacement age between 290 Ma and 300 Ma and Stone (1982) puts it a bit earlier, between 300 Ma and 310 Ma. Halliday (1980) proposes the following ages: 295 Ma emplacement 285 Ma+ metallized pegmatites 285-280 Ma greisenisation 280-275 Ma porphyry dikes intruded 270 Ma main polymetallic mineralisation with meteoric waters entering the system at about 285 Ma. Kaolinisation and minor mineralisation are processes which have occurred up to recent times, and may still be continuing. 3.2.2 Mineralization The detail of the ore-field associated with the batholith is complex and not fully understood. There are accounts giving details of the mineralization and paragenesis of some particular regions e.g. Mount Wellington Mine (Kettaneh and Badham, 1978), the St.Just district (Jackson et al., 1982) and Cligga Head (Jackson et al., 1977) but there has been little published recently on the regional aspects of the distribution of the mineralization. This may be attributable to the complexities of local systems 4 1 which apparently contradict the conclusions of general theories of regional ore genesis. There is no doubt that ore deposition in different areas occurred under very different conditions, not the least of these being the chemistry of the mineralizing fluids, a factor that contributes significantly to the complexities of the local systems. From the study of the St. Just mining district, Jackson et al (1982) recognise five types of mineralization phenomena: metasomatism; barren pre- and post-joint pegmatites; mineralized sheeted vein systems; mineralized fissure systems; and irregularly-shaped replacement bodies. The minerals include Sn, Cu, and As and minor Pb, Zn, W, Sb, U, Ag, Au, Co, and Bi. They occur in the roof of the intrusion, in the flank of the intrusion, at the granite contact and in the outer aureole. A full understanding of the genesis of the ore-field is obviously not possible without detailed studies of these local systems; however it is also important to relate these systems to the regional development of the ore-field. The systematic development of a regional theory began with Dines who collated an enormous quantity of data related to mining in the district (Dines, 1956). His observation that the production was concentrated in small regions spawned the idea of the emanative centre, which were the locations of the centres of tin production, and were thought to be those places where the mineralizing fluids preferentially passed through the solid outer blanket of the consolidating batholith from the molten core. As had been noted before (Dewey, 1925, Davison, 1927, and Dines 1935) there appeared to be a zonation of minerals around these tin centres, with progressively lower temperature mineralization being found at greater distances from these centres. Dines explained this zonation in terms of purely vertical 4 2 zonation. His argument was as follows: the mineralising fluids emanating from the magma contained all the minerals which were progressively deposited at lower temperatures. The lower temperature minerals stayed in solution longer and thus had a greater opportunity to disperse into the country rock. Since the granite is hotter than the surrounding rock the isotherms dip into the country rock and thus the lower temperature mineralization when seen in cross-section also dips away into the country rock. Subsequent erosion will thus create the observed apparent horizontal zonation. This theory could also explain why the full suite of mineralization is not found beneath the lower temperature deposits. Dines' theory, based on a single pass of magmatic fluid through the fracture system, has been superceded, because by the 1960s it was recognized that the mineralization had occurred in many stages. Though still maintaining Dines' idea of emanative centres Hosking (1966) explained the distribution of mineralization in terms of ridges and cusps which were assumed to have formed in the roof of the granite and were controlled by the host rock. The residuum of the melt would have collected in these high points and at sufficiently high pressures rupturing would have released these fluids. The change in composition of the residuum with time accounted for the change from hypothermal to mesothermal deposits. Later deposits were also formed by the action of a convective meteoric system which extracted minerals from the micas of the consolidated granite and redistributed some of the earlier mineralization. It is worth noting, however, that there was little evidence at the time to suggest how much of the mineralization was of magmatic origin and how much came from the biotite and muscovite of the granite or indeed other sources such as the country rock. Neither Hosking's theory nor any of the subsequent 4 3 published work deals satisfactorily with the apparent horizontal zoning. The idea of ridges and cusps determining the distribution of mineral lodes was rejected by Moore when he produced a mechanical interpretation of the vein and dyke system (1975) , according to which the direction of the lodes was determined by the fluid pressures exerted by the mobile cores of the granite cupolae within a regional stress field. The asymmetrical pattern of the belts of mineralization would have resulted from the mechanical failure of one side of the pluton preferentially. In a later paper he discussed the mineral zoning (Moore, 1982) declaring that there was no regional zonation, but rather that zoning was a local phenomenon within individual vein swarms. He states that 'early (convective) systems were succeeded in different positions by younger generations', but gives no reason as to why this might be. Nor do Stone & Exley (1985) who declare that the 'focus of mineralization...migrated outwards with time'. As for the source of the minerals there is still no consensus of opinion, though the current tendency is towards the realisation that there was little contribution from magmatic sources. Stone (1982) claims that no economically important ore-bodies have been derived from granite differentiation and suggests that the pelitic country rocks were the principal source of the minerals emplaced into the fracture system by a meteoric convective system. The distribution of lodes is not simple in space (fig 3.3) nor has it been constant in time. The pattern of lodes is obviously asymmetric about individual outcrops. Jackson et al (1982) recognised four episodes of mineralization from the St. Just district and three from the area of Cligga Head (Jackson et al 1977). The repeated mineralization, with more recent events spatially overlapping older deposits, and the remobilization of minerals has confused 4 4 Figure 3.3 Distribution of Mineralized Lodes (after Dines, 1956) the picture. Even so it is possible to distinguish mineral zonation (Dewey, 1925? Moore, 1982) with a general pattern of high temperature mineralization occurring within or close to the granites and lower temperature mineralization at a greater distance from the outcrops. Some patterns can be discerned in the orientation of the lodes. The majority of the lodes lie roughly parallel to the batholith: E-W to the east of St. Austell and ENE-WSW to the west (after correction for wrench faulting they all generally trend ENE-WSW). These are called the main lodes and are cut at small angles by a slightly younger set called the caunter lodes. Both these sets are cut, approximately at right angles, by a set of cross-courses which are not heavily mineralized. When mineralized, the cross-courses contain minerals of a younger age and lower temperature than the main hypothermal orebodies. The minerals of this region have been exploited for at least 2,500 years and have been of great economic importance to the area for over two centuries. For some period the mines of Devon and Cornwall were producing over 40% of the world's output of copper and arsenic. Dines (1956) gives a comprehensive account of all known mines and their production. Recent falls in the price of tin make the majority of mines uneconomic and, at the time of writing, only two are in operation. 3.2.4 Kaolinisation Kaolin is the alteration of granite to china clay and occurs to some extent on all of the major outcrops of the Cornubian batholith (Figure 3.4). It occurs in troughs and pipes to depths of up to 400m. It is associated spatially with greisen-bordered quartz-tourmaline veins and faults some of which contain Sn and W (Bristow, 1977) . It is of 4 6 I o o Figure 3.4 Known Areas of Kaolinised Granite (after Highley, 1984) 4 7 great economic importance to the South-West of England with some 2.5 milion tonnes per annum being extracted from deposits in the region (Howe et al, 1985) . The method of formation of the kaolin is uncertain. Most recently, Durrance and Bristow (1986) suggested a two phase process. First the Na and K feldspars were altered to smectite- illite and illite clay mineral assemblages respectively by high temperature waters (200-300°C) about 260 Ma ago. Only much later did the alteration of smectite to kaolin occur at temperatures in the region of 50-150°C. For both processes, they suggest that the flow of fluids involved was downwards. Some alteration may also be due to weathering. 3.3 Geophysics Some of the earlier geophysical surveys in and around this region have been reviewed recently by Brooks et al. (1983). Many of the results reported there have been updated by more recent surveys and interpretations, the more relevant of these are summarised below. 3.3.1 Seismics Both reflection and refraction surveys have been carried out in this region, but have produced little information on the structure of the Cornubian granite batholith due to its apparent homogeneity. Bott et al. (1970) concluded from their refraction survey that the granite was homogeneous down to its base at a depth of 10-12 km. There is no apparent velocity discontinuity at the base of the granite but velocities gradually increase with depth in the lower crust. This velocity increase, they suggested, was due to the residue of the partially melted lower crust and stoped material which sank through the magma as it ascended. 4 8 Brooks et al, (1984), however, claim to have found a low velocity zone beneath a reflector at about 8 km depth within the batholith. Brooks et al (loc. cit.) detect a reflector (R2) at about the depth of the base of the granite which rises towards the north. They interpret this as a possible post emplacement thrust. They also suggest, on the basis of gravity modelling, that the base of the granite rises towards the west: being at a depth of 9-16 km in the west and 12-22 km in the east. However, if their reflector R2 is coincident with the base of the granite at all locations, the conclusion from the seismic data is that the granite would be thicker under Land*s End contrary to the evidence from their gravity model. The BIRPS and ECORS reflection and refraction surveys show no boundaries within the granite. In fact the only characteristic associated with the presence of the batholith is an unusually shallow portion of the lower crustal reflectors (BIRPS and ECORS, 1986). However, the only line to traverse the Cornubian batholith does so to the west of the Scilly Isles, close to the South-west margin of the batholith, and hence may not be representative of the batholith in general. This survey places the Moho at a depth of 27-30 km which is in accord with all other surveys. 3.3.2 Gravity A gravity survey by Bott et al (1958) delineated a belt of negative anomalies associated with the granite outcrops (Figure 3.5), indicating that the outcrops are just cupolae of a continuous batholith. The anomalies are caused by the lower density of granites relative to killas. 4 9 Two-dimensional modelling (Bott and Scott, 1964; Brooks et al, 1983) gives a depth to the base of the granite between 9 and 20 km depending on the density contrast used. The models of Al-Rawi (Brooks et al, loc. cit.) and the three- dimensional model of Tombs (1977) show a thinning of the batholith towards the west, with no dip to the base of the granite in the north-south direction. Tombs uses a variable density contrast in his model, which predicts 20 km of granite beneath Dartmoor thinning to 9 km beneath the Scilly Isles. The latest three-dimensional model (Willis-Richards, 1986) requires no thinning to the west, the main difference between this and earlier models being the use of a more sophisticated method for the removal of the regional gradient. The base of this model, which at its deepest is 13 km, does rise towards the north by various amounts along the length of the batholith. This sloping base may have some relationship to the sloping reflector seen by Brooks et al (loc. cit.). However, only a single density contrast is used to generate this model though the Devonian and Carboniferous, for instance, are known to have different densities (Bott et al, 1958). The model of Willis-Richards is shown in figure 3.6 and can be compared with that of Tombs in figure 3.7. The former shows more detail, though care must be taken in interpretation because negative density contrasts other than those of granite will have been modelled as granite (for example the small basin at Bovey Tracy just to the east of the Dartmoor outcrop). 5 0 Figure 3.5 South-West England.* Simplified Bouguer Anomaly 5 1 o o Figure 3.6 Depth in kilometres to the Top of the Granite from a Gravity Model by Willis Richards (1986). 5 2 Figure 3.7 Depth in kilometres to the Top of the Granite from a Gravity Model by Tombs (1977) 53 |r| " Figure 3.8 Sites of the Heat Flow Measurements up to 1980 (after Francis, 1980). Contract sites were drilled as part of a European Communities exploration programme (Wheildon et al., 1980). Table 3. 1 Summary of Contract Boreholes. Station Station Name Depth (m) Heat Flow Code (nominal) (mWm-2) CM-A Grillis Farm 100 119 CM-B Polgear Beacon 100 128 CM-C Medlyn Farm 100 120 CM-D Trevease Farm 100 118 CM-E Trerghan Farm 100 119 BD-A Bray Down 100 120 BD-B Blackhill 100 126 BD-C Pinnockshill 100 127 BD-D Browngelly 100 115 BD-E Gt. Hammet Farm 100 125 LE-A Newmill 100 130 LE-B Bunker's Hill 100 130 SA-A Tregarden Farm 100 132 SA-B Colcerrow Farm 100 133 DM-A Winter Tor 100 114 DM-B Blackingstone 100 112 DM-C Soussons Wood 100 139 DM-D Laughter Tor 100 121 DM-E Foggin Tor 100 118 CDD-1 Merrose Farm 100 84 CDD-2 Kestle Wartha 150 102 CDD-3 Callywith Farm 150 106 GAV Gaverigan 325 105 5 5 3.3.3 Heat Flows The heat flow has been measured at 39 sites in the region (Figure 3.8). 26 of the measurements were made in the granite and 13 in the country rock. 23 of the measurements (19 in granite and 4 in country rock) were taken in specially commissioned boreholes (Table 3.1) as part of a European Communities exploration program (Wheildon et al., 1980). Heat flow is the product of the thermal conductivity and the temperature gradient (Carslaw and Jaeger, 1946) q = KVT 3.3.3.1 where q is the heat flow, K the thermal conductivity tensor and T the temperature. Terrestrial heat flow is usually concerned with the vertical flow of heat only, so q = K dT 3.3.3.2 dz and K is assumed isotropic. The vertical temperature gradient is found by measuring temperatures down a borehole. Thermal conductivities are measured from samples of the rock taken from the boreholes (see section 3.3.5). However, since temperatures are not logged continuously down the borehole and measurements of thermal conductivity are made on samples which are not necessarily at the same location as the temperature measurements it is not possible to use equation 3.3.3.2 as it stands and the usual procedure is therefore to use the integral form of equation 3.3.3.2, which gives z 1 dz 3 . 3 . 3 . 3 L[0 K(z) which can be discretized to give the temperature Tz at 5 6 depth z, n 3.3.3.4 Tz = T0 - < l Z Zi / K: i=l where is the conductivity of the th layer and n z = ^ z_l . 3.3.3.5 i=l A plot of temperatures against thermal resistance ( ^ Zi/Ki) yields a straight line with a slope equal to the surface heat flow and an intercept with the value of the surface temperature. This is known as the Bullard plot (Bullard, 1939) and takes into account, as far as possible, variations in thermal conductivity with depth. However, unless variations in conductivity are related to another parameter which is logged continuously down the borehole, the exact nature of the variations will remain unknown and errors will be made in the calculation of the heat flow. The higher the frequency and amplitude of these variations the greater the error. Because of the assumptions involved in the heat conduction equation 3.3.3.2, the measured borehole temperature profile has to be corrected for distortion caused by other effects, the more important ones of which are irregular topography, uplift and/or erosion and fluctuation in the surface temperature in the past, resulting from palaeoclimate. These corrections are discussed in detail by Francis (1980) . However, it is important to make a comment about the nature and application of the palaeoclimate correction. In general, heat flows in the U.K. are corrected for only the most recent palaeoclimatic events: 5 7 that is, those which occurred in the last 500 years. This is satisfactory as a standardising procedure if the heat flows are only to be compared with one another. Data to be used in temperature predictions and constraining models, however, need to be corrected for earlier palaeoclimatic events also, because their effects still perturb temperatures at depth to a significant extent. The palaeoclimate function used for this purpose encompasses changes over the last 300,000 years (Lamb, 1965) and the extended correction is hereafter referred to as the full palaeoclimate correction. If the rocks contain heat-producing elements and the borehole is deep, then the heat generated between the surface and the location of the temperature measurement must be taken into account since it is the surface heat flow which is to be calculated. Again, if there are large changes in temperature down the borehole, the thermal conductivities must be corrected since they are temperature-dependent. Both these corrections are discussed in Appendix I which describes the heat flow determinations in the 2 km boreholes drilled by the CSM Geothermal Energy Project in the Carnmenellis pluton. Temperatures may also be influenced by water movement, which violates the assumption of heat transport being exclusively by conduction. Deviations from the straight line expected on the Bullard plot, due to water movement on a local scale, are to a great degree recognisable as such and can be dealt with by discarding data from the sections of the borehole that are disturbed. However, water movement on a regional scale will cause temperature gradients to decrease in regions of fluid downflow and increase in regions of fluid upflow. If the boreholes are small compared with the scale of the convection cell, it is impossible to detect such disturbances from the measured temperatures (Donaldson, 1962) and confusion may arise when 5 8 comparing such measurements with results from purely conductive models. Thus, there are many sources of error in heat flow measurements, the greatest being the uncertainty in the thermal conductivities. Richardson and Oxburgh (1978) state that the error in calculating the heat flow cannot simply be the error in fitting a straight line to the Bullard plot. They compute a mean (K) and standard (crK) for the conductivity of a petrographically uniform stratum and combine this with a mean (g) and standard deviation (o^) for the gradient such that °q = °g*K + 9 • ° k 3.3.3.6 It is not possible to apply this formula if the variations of rock type are more rapid than the sampling of temperature and the first term of equation 3.3.3.6 is often dropped as it is normally much smaller than the second. One would expect heat flow measurements in boreholes sited in close proximity in a granite to show a low variance. At the site of the Hot Dry Rock project (Rosemanowes Quarry, Carnmenellis) , five heat flow measurements have been made in boreholes, ranging in depth from 150 m to over 2000 m, and the calculated values have a variance of only 4.4 mWm”2 about a mean of 116.2 mWm and a standard deviation of 1.7 mWm . It is therefore assumed that heat flows measured m granite are relatively accurate and that the observed variations in heat flows across plutons, if greater than a few mWm —9 , are likely . to be physically . significant. . . . The same cannot be said of measurements in sedimentary rocks. The inhomogeneous nature of such rocks casts doubt on the accuracy of country rock heat flow values, although it is difficult to assess the degree of uncertainty because of 5 9 the dearth of experimental data. One country rock borehole, Cannington Park, which passes through four distinct petrographical strata but lies just outside the region of interest, has had the heat flow calculated from data from each layer; the variance is 12.9 mWm”2 about a mean of 50.5 mWm”2 with a standard deviation of 5.3 mWm “2 and there remains a large uncertainty in the heat flow despite the conductivity measurements showing only small variations within each stratum (Francis, loc.cit.). 3.3.4. Heat Production The heat production within a rock is usually determined by gamma ray spectrometry which measures the energy of gamma rays emitted by the decay of radioactive elements. There are three main sources of heat produced by radioactive decay and these are 238U, 232Th, and 40K. A discussion of the apparatus and the measuring and interpretation techniques is to be found in Francis (loc.cit.) along with the results for the boreholes in SW England. The measured values range from 3.1 to 7.4 microWm”3 for twenty-one granite sites and from 0.5 to 2.6 microWm”3 for eight country rock sites. The granites have a mean of 4.49+1.25 microWm”3 and the country rock a mean of 2.0+0.3 microWm. _ ' i . Francis , observed a general increase . m . heat production towards the granite contact from the centre of the outcrops and interpreted it as the result of concentration of Uranium at the edges of the granite due to fractionation. However, spectral logging of two of the deep holes at the Hot Dry Rock site (RH11 and RH12) show that the surface values of heat production are relatively low for the first 60 m (Withers, 1984) . Beyond 60 m, the heat production is generally constant at a value of about 5 ^iWm”3: a value much higher than the 3.6 ^pWm”3 quoted by Francis for this site. Therefore, it may be more recent 6 0 processes, such as hydrothermal circulation, which have caused the concentration of uranium to the edge of the outcrops. The low heat production at the surface is probably due to leaching caused by weathering or shallow convection (Ball and Balsham, 1978). The consistency of the heat production with depth beneath the depleted layer is remarkable in view of the general acceptance of an exponential decay of heat production with depth (Lachenbruch, 1970). The popularity of this hypothesis derives from considerations of the heat flow- heat production relation, the linear nature of which (Birch et al., 1968 and Roy et al., 1968) will be upheld under conditions of differential erosion only for an exponentially varying vertical distribution of heat production. However, Sawka and Chappell (1985) argue on geochemical evidence that there is no unique distribution of heat-producing elements with depth which can be applied to all granitic crust. They suggest that within I-type granites heat production does decrease with depth, but for S-type granites (like those of Cornubia), no such decrease need exist and that an increase with depth is possible. A similar conclusion is drawn by Webb et al. (1985) who state that heat production in the Carnmenellis pluton is unlikely to decrease significantly with depth. This does not completely undermine the linear relationship, but would suggest that it does not hold under differential erosion in certain regions. The distribution of heat producing elements with depth in the country rock and lower crust is unknown. If heat production remains at the mean value of 2.0 j j . Wm~3 down to the Moho, then the measured heat flows are entirely accounted for by heat production, with no contribution to the heat flow from the mantle which would seem unlikely. Evidence from radon isotope work by Hilton et al. (1985) suggests a contribution from the mantle of up to 35 mWm”2, 6 1 in fact, implying that the heat production in the lower crust is very much smaller than surface values. The distribution of heat production in the final models will depend to a certain degree on the choice of other parameters which are also inaccurately known. 3.3.5 Thermal Conductivities The total conductivity of rocks within the temperature range under consideration is the sum of a lattice component (K^) and a radiative component (Kr) such that K = K-l + Kr. 3.3.5.1 Investigations by Schatz and Simmons (1972) suggest that the radiative contribution is 50% for olivines in the temperature range of 900-1300°C. However, according to Cull (1975) radiation transfer is suppressed by the increasing absorption with increasing temperatures and, for studies within the crust, the radiative term is usually ignored (e.g. Balling, 1976). The conductivity (all references to conductivity now assume no radiative transfer) increases with pressure. Some experimental work has been done on the rate of increase but only within a limited pressure range. Cull (1975) used a linear relationship to approximate the increase and gave a value for pyroxene granulite lower continental crust of 5.9xl0“4 Wm”1K“1MPa”^. Sartori (1983) gives a value for Cornish granite of 1.5x10"*3 Wm"1K-1MPa-1. The work of Fujisawa et al. (1968) suggests that at higher pressures the inrease in conductivity is much smaller. Indeed, lateral variations in conductivity due to lateral variations in pressure are not likely to make any detectable difference to the modelled heat flow patterns. The temperature at depth would be influenced by changes in 6 2 conductivity but, since the models are not required to predict temperatures at great depth accurately, the pressure term can be ignored. The temperature-dependence can be written as K = (a + bT)-1 3.3.5.2 where a and b are constants (Birch and Clark, 1940) . Here the dependence is approximated by the equation K = d + c/T 3.3.5.3 where c and d are constants and T is now the absolute temperature. The rate of change of conductivity with temperature (c) will depend on the material. Birch and Clark's figures for the Rockport, Barre and Westerly granites give a range of about 180 - 600 W m ^ K ”2. These may well represent the extremes of the range expected for granites. Results from the Cornish and other HDR granites (Sartori, 1983 and Sibbit et al., 1979) tend to fall towards the lower end of this range at about 200 W m ^ K ”2. A value deduced by forcing heat flows measured at depth to be consistent with surface measurements (see Appendix I) at the HDR site gives a range between 300 and 400 Wm_1K”2. Measurements on non-granitic material (Cornish slate by Sartori and pyroxene granulite by Cull) give results towards the lower end of the granite range. Changes in values of c can affect the surface heat flows predicted by the models significantly, so that c may be regarded as an unknown parameter in modelling if the models can be constrained by good quality measured data. The intrinsic value of the conductivity will depend on the mineralogical content of the rock, e.g. an increase in quartz content will raise the conductivity of a granite (Birch and Clark, 1940). Measurements on granite samples 6 3 from boreholes in S.W. England show little deviation about the mean. The mean of the mean conductivities for 21 granite boreholes is 3.30 Wm-1^ 1 with a standard deviation of 0.16 Wm-1^ 1 (Francis ,1980). Samples from the 2 km deep holes at the HDR site show that there is no systematic change in conductivity with depth (all samples being measured at the same temperature) (Sams and Thomas-Betts, 1986). It has been suggested that a relationship exists between seismic velocity and thermal conductivity (Horai and Simmons, 1969) ; if this is true, the conclusion from seismic surveys would also be that there is little change in conductivity to the base of the Cornubian granite (Section 3.3.1). Measurements on other rock types in the area show a greater variability. Take, for example, Kestle Wartha where the borehole is drilled into Gramscatho Beds which comprise a sequence of shales, slates and fine sandstones. Thermal conductivities, determined using cored discs from samples taken approximately every 3 metres, are seen to range from 1.02 to 4.57 Wm^^K”1 , the variations depending on the mineral content of the discs. The conductivity, predominantly controlled by the ratio of shale to sandstone in the Gramascatho Beds, will depend on the depth of the sample, the repetition rate of the sequence and the location of the site relative to the original source of the sedimentation. With such rapid spatial variations in conductivity present, only a gross approximation to an effective conductivity is feasible when modelling large areas to great depths. The values of conductivity at depth have to be estimates based on assumptions about the rock content at depth. Smithson and Decker (1974) and Smithson (1978) propose a three-layered crust, with a surface zone of intermediate metamorphic rocks containing granitic intrusions, a zone grading downwards into a more felsic migmatic composition, 6 4 and a lower crustal zone of andesitic nature. Naturally, lateral inhomogeneities are present in all zones and the layering is not well defined. In their model, they suggest average thermal conductivities of 2.7, 2.7 and 2.9 W m ^ K -1 for upper, middle and lower crust respectively. Balling (1976) in modelling the crust and mantle around Southern Norway divided the crust into two: an upper crust of high- metamorphic migmatites decreasing in acidity with depth and a lower crust of high-density high-metamorphic elements of banded gneiss. He uses a thermal conductivity of 2.8-3.2 Wm_1K_1 for the upper crust and 2.3-2.6 Wm""1!^”1 for the lower crust. If, within the modelling, the lower crust is assumed homogeneous and the temperatures at great depth are not required to be predicted accurately, then the intrinsic conductivity of the lower crust is not of too great an importance and a rough estimate is all that is necessary. 3.3.6 Permeability Given the high heat production contrast between the granite and the country rock, the presence of sufficient permeability determines whether convection will take place (Fehn, 1978). Active hydrothermal circulation has been detected by Durrance et al. (1982) through the mapping of radionuclide migration and past circulation implied by the occurrence of substantial mineralization and kaolinisation indicates that at times the region has been permeable. Durrance (1985) recognized two scales of permeable pathway: slightly open fractures with separations of the order of metres, and open fractures and fracture zones separated by hundreds of metres. Heath (1985) used multi-packer hydraulic tests and radiation tracer experiments to show that flow through granites was restricted to narrow zones separated by zones of very low permeability. Permeability is not a function of fracture density (Heath, loc.cit.) and 6 5 although fluid may flow through the closely spaced small fractures it is the 'main drains' which dominate the effective rock permeability. Without these main drains the permeability of the granite would be as low as 3 mD, but fluid flow being along channels in the main drains the effective permeability is estimated to have a minimum value of 20 mD (Durrance, loc.cit. ) . This is more than sufficient for convection to take place provided the permeability does not decrease too rapidly with depth. It is difficult to assess the variation of permeability with depth. Although the fracture density decreases within the first few hundred metres of the surface, flow rates tend to remain fairly constant due to channel flow along the highly permeable main drains (Durrance, loc.cit.). In modelling this region Fehn (1985) chooses to treat the variation with depth as an exponential decay such that k(z) = kQ exp[ -f(z/h)] 3.3.6.1 where k is the permeability, h is the depth of the domain and f is the decay parameter of decrease which Fehn treats as a parameter to be solved for in the modelling. This expression is used to represent the permeability variation in all the models presented in chapter 5. 6 6 Chapter Four Models of Conductive Heat Flow 4.1 Introduction This chapter describes the models used with the conductive heat flow equation and the results from the models which have to be assessed in terms of known processes. Four processes which can cause variations in the surface heat flow are considered. Two of these, heat refraction and radiogenic heat egression are accounted for in the models and their effects are shown and discussed, both in general terms and in the context of the area under investigation. The other two processes, fluid flow and anisotropic thermal conductivity, are not catered for within the modelling and the effects of these are presented only in terms of possible discrepancies between modelled and measured values. The accuracy of the models is discussed in terms of the intrinsic accuracy of the numerical method and the inaccuracies due to the approximations made in the models. A description of the various models is given along with the parameters used in the models and the approximations that have been made. The results are presented in two sections. The first deals with predicted heat flows where modelled values are compared with measured values and possible explanations offered for significant discrepancies between the two sets where they exist. The second section deals with 6 7 temperature predictions at depth. The reliability of these predictions is discussed in terms of the effects produced on the temperatures by changes in the various model parameters. 4.2 Heat Flow in General The heat conducted to the Earth's surface from its interior averages about 60 mWm~2 (Pollack and Chapman, 1977), but surface heat flows differ from this markedly in different places, values as high as 200 mWm-2 being commonly encountered in some geothermal fields. This section describes some of the general features of the lateral variations of surface heat flows, their origin and their implications to some aspects of numerical modelling. The outward heat flow at the surface comprises contributions from the mantle and from radioactive decay in the Earth's crust, neither of which is uniform globally. However, the effect of the intervening thickness of crust is to smooth out the variations in mantle heat flow such that their significance to the lateral inhomogeneity of the surface heat flow field is much reduced. Lateral variations of crustal heat production will also produce anomalies in heat flow, but the deeper these variations occur, the smaller and broader their anomalies are (figure 4.1). It is also true that only heat production contrasts which have large depth-extents will produce detectable changes in surface heat flows. For example, enrichment by 1 microW m of the heat production of a 1 km-thick region of the crust will produce a maximum heat flow anomaly of 1 mW m , well within the errors of heat flow measurement. Nevertheless, some rocks are sufficiently enriched in heat- producing elements and occur in large enough volumes to produce observable heat flow anomalies. Granite intrusions 68 Figure 4.1 Graph showing the heat flow anomaly produced by a 1 km thick, 3 km wide heat production anomaly of 5 microW m“3 at depths of a) Okm, b) 3km, c) 5km, d) 9km. 6 9 are a prime example of this. The heat which flows out towards the surface may be re distributed in several ways. In the presence of circulating fluids, transport of heat will be controlled to a large extent by the fluid flow, the fluids gaining and losing heat as they flow across isotherms (Donaldson, 1962). Even in purely conductive situations, there are several processes which cause lateral variations in a heat flow field. Variations in the thermal conductivity of the rocks which comprise the crust can cause heat to be refracted, or focused and defocused (Lee, 1975); horizontal temperature gradients established by lateral variations in heat production will cause heat egression (Jaupart, 1983), and anisotropic thermal conductivities, depending on their orientation, will cause non-vertical heat flow (England et al., 1980). The effect of these processes on heat flow studies in general, and modelling in particular, will be considered in more detail now, starting with the purely- conductive regime. 4.2.1. The Effect of Conductivity Contrasts A temperature field is continuous at all points and so are temperature gradients parallel to boundaries between media of contrasting thermal conductivities provided there is no heat generation present. In two media separated by a non horizontal boundary the temperatures at the same depth and at large distances from the boundary will be different at least when the vertical heat flows away from the boundary are the same in both media. Therefore, there will exist a horizontal temperature gradient across the boundary and heat will flow from the medium of low thermal conductivity into the medium of high conductivity. This effect is known as refraction and results in enhanced vertical heat flows in the medium of high thermal conductivity in the vicinity 7 0 of the boundary. The size of this effect will depend on the shape of the boundary and the size of the contrast between the media. Of course refraction will not occur if the ratio of the nominal heat flows in the two media is larger than the ratio of thermal conductivities. If the conductivity contrast is of a limited depth extent, the temperatures beneath the the zone of high thermal conductivity will be lower than elsewhere (all other conditions being the same) and heat, which flows perpendicularly to isotherms, will be drawn into the region of high thermal conductivity. Figure 4.2 shows the effect of refraction derived from two- dimensional models for three different boundaries with a conductivity contrast of 2:3. Obviously the heat gained by one medium is always equal to the heat lost by the other. However the maximum heat flow depends on the form of the boundary as does the width of the resultant anomaly in each medium. The process of refraction may therefore cause heat flows to be enhanced or diminished over large areas and at some considerable distance from the contact. In South-west England there is an obvious thermal conductivity contrast between the granite and the surrounding country rock and refraction effects can be expected to exist provided temperatures in the country rock are higher than in the granite. There are also thermal conductivity contrasts in the country rock, the various stratigraphic units possessing different average thermal conductivities. Unfortunately, the lateral and vertical variations of the thermal conductivity of the country rocks are not known well enough to be modelled with confidence. Therefore, the average conductivity is used to represent the country rocks which will clearly be inadequate and may result in a poor match between measured and modelled heat flows at the country rock sites. But the differences in 7 1 HEAT REFRACTION ANOMALIES Figure 4.2 Graph showing the effect of refraction at a boundary between media with two contrasting thermal conductivities. A uniform heat flow of 30 mW m""2 is entering the base of the model and the ratio of the thermal conductivities is 2:3. 7 2 country rock thermal conductivities are small and often highly localised and so the resultant refraction effects should also be small. 4.2.2. The Effect of Heat Production Contrasts A region in which heat production exists will have an enhanced heat flow, causing greater temperature gradients to be established within that region (all other conditions being the same). The effect of these higher temperatures at equal depths (relative to regions of lower heat production) will be to promote lateral heat flow out of that region, an effect which will be referred to as egression. Also, beneath a region of heat production, temperatures will be relatively high and isotherms will be bent upwards, causing the heat flow from below to be deflected out of this region. Therefore, above a region of heat production, heat flows will rarely attain the maximum possible value, i.e. the sum of all the heat production directly beneath that point and the heat flux from the base (Fig.4.2). The effect on heat flow of heat production contrasts then is similar to that of conductivity contrasts, both causing lateral heat flow, a region of high heat production behaving similarly to a low conduction region. If a region has high heat production and high thermal conductivity, the processes of refraction and egression will work in opposite directions. Which effect occurs depends on the magnitude of the thermal conductivity and heat production contrasts and the shape of the boundaries. 7 3 4.2.3 The Effect of Anisotropic Conductivities England et al. (1980) suggest that the variation in heat flow with depth in the Beckermonds borehole could be due to anisotropic thermal conductivity. The effect of anisotropy can indeed be very large, especially in sedimentary rocks. The South-west England heat flow determinations (Francis, 1980) also considered anisotropy where appropriate, but the evidence from measurements on core is inconclusive, and the inadequate (in quantity and quality) nature of data precludes its use in modelling. Indeed, as much of the sedimentary strata are heavily contorted and folded, it may be that the effects are averaged out on a regional scale and so anisotropy would not play a large role in determining the regional pattern of heat flows in South west England. 4.2.4 The Effect of Fluid Flow The effect of fluid flow, whether under buoyancy or pressure differential, is to reduce heat flow in regions of downward flowing water and enhance it in regions of upward flowing water. Local perturbations to temperatures due to circulating fluids can be seen on down-hole temperature logs but regional scale flows are often not detectable from such measurements, especially in shallow boreholes (Donaldson, 1962). It may be impossible to distinguish between large scale water movement and any other cause if the number of surface heat flow measurements is low. It is well known that there is water movement in certain parts of the region under discussion (e.g. Durrance et al., 1982), and some heat flow measurements (e.g. at Sousson's Wood) are reported to be disturbed by fluid flow (Francis, 1980), but whether fluid circulation at the present level is sufficient to affect heat flow measurements significantly 74 is questionable. There is no conclusive proof of any large scale circulation at present. It is therefore assumed that, unless otherwise indicated by the temperature logs, there is no contribution to the measured heat flows from fluid flow. The modelling of the heat flow field in the first instance is also done assuming that there is no fluid flow and that the heat flow is purely conductive. Large scale fluid flows, if they exist, would then be expected to produce systematic mis-matches between the calculated and measured fields. 4.3 Model Testing and Accuracy It is self-evident that, in any modelling exercise, approx imations and assumptions have to be made about the physical situation and that a set of solutions obtained will only be valid for a particular set of approximations. Given that the accuracy of the finite element method is only limited by the accuracy of the matrix inversion, which here means that the only errors involved are due to round-off by the computer, some of the assumptions and approximations made, and justifications for them, are discussed in this section. One assumption whose validity is by no means certain or obvious for three-dimensional models, is the non singularity of the Jacobian of transformation between global and local co-ordinates for each element. This condition can be tested for in two ways. Since the Jacobian matrix is calculated at each Gauss point, its elements can be checked to see if they change sign within one element. If the Jacobian changes sign, then it must be zero somewhere inside the element and the transformation, using the inverse of the Jacobian, will not succeed since the zero will produce a singularity in the inverse. In the second test, all elements of the finite element model of 75 the region are given the same properties. Then, after inversion, all temperatures at the same depth must be equal and are predictable. A problem arises when using linear elements which contain heat-producing material. In linear elements, the temperature can only vary linearly in any direction, and hence the temperature gradient in any direction is constant. This is a poor approximation if heat is being generated within the element since the heat flow at the top of the element should be larger than at the bottom of the element. In the model, the heat flow assigned to such an element is the average of the correct heat flows at the top and bottom of the element, while the heat flowing from this element into the element above is taken to be the correct amount. This is a valid procedure for most elements, but surface elements have to be treated as a special case. If heat flows assigned to the surface elements are assumed to be the surface heat flows, the latter will clearly be deficient by half the amount of heat production within the element. This needs to be taken into account when matching modelled and measured heat flows or added on when finally calculating the modelled values using linear elements. However, in attempts to correct for this, it imperative to consider the effect the correction has on the temperatures. It is not sensible just to double the heat production in the surface elements as this will alter the temperatures at depth erroneously. The temperatures within such an element will be incorrect anyway except at the extreme top and bottom. For instance, at the midpoint of an element 1 km in depth with a heat production of 5 microW m-3 and thermal conductivity of 3 W m ^ K ”1, the temperature at the centre of a linear element is too low by about 3 degC. This problem can be avoided by using quadratic elements or alleviated to some extent by taking temperature predictions from close to the top and bottom of the element where the 7 6 errors are smaller. In real situations the accuracy of the models is impossible to assess. The shape of the granite will never be matched exactly by the elements used in the model. Indeed, the shape of the granite itself is only known from gravity modelling which also relies on a system of simplifying assumptions and approximations. So it is impossible to say how close the shape in the model is to the physical reality. Again, the use of a finite number of elements will affect the modelled variation of temperature over the region. The larger the number of elements the closer the convergence to the correct solution, but the number of elements is limited by the computer capacity and cannot be increased indefinitely. It is often not possible to say how much the restricted element number affects the solution given the complex geometries involved. Given the shape of the granite (as defined by the finite element mesh) and the number of elements, the solution to the problem is still not unique, being dependent on the thermal properties of the assumed model. There are many combinations of the various model parameters which would yield answers that lie within the error bounds of the constraining data that are often sparse and of a poor quality. To begin with, the simplest possible model which incorporates the basic geological and geophysical data is constructed. The various parameters are then systematically varied within the constraints placed on them by a priori information and geological feasibility until the resultant temperature field yields heat flows which lie within the error bounds of the measured data. 7 7 4.4 The Basic Model The models used for solving the heat conduction equation are based on one general form. The top surface of the model is planar and represents the surface of the Earth with the implication that measured values of heat flow must be fully corrected for topography, erosion/uplift and full palaeoclimate before comparison with modelled values. The surface is assumed to have a fixed temperature of 10°C which is a Dirichlet type boundary condition (see section 2.6). This may be regarded as slightly low considering the data obtained from the Bullard plots of the heat flow measurements (Francis, 1980), which give an average mean surface temperature of 11.1°C (27 values). However this discrepancy can be corrected for at a later stage if necessary. The base of the model is parallel to the top and represents the crust mantle interface. It is at a depth of 30 km. There is a contribution from the mantle to the surface heat flux. It is not certain what form this contribution takes, but here it is assumed to be uniform. The linear realationship between heat flow and heat production gives an intercept value of about 27 mWm”2 for this region (Richardson and Oxburgh, 1979). This figure is regarded as the heat flux at the base of the granite known as the reduced heat flow. However, this can only be regarded as such if the distribution of the heat producing elements with depth is the same in the granite and country rock. If the heat production in the country rock decreases more rapidly with depth than in the granite the value calculated for the reduced heat flow will be smaller than the actual value. Then the value of 27 mWm * can be used as a lower limit to the mantle heat flow since there is only a small contribution expected from heat production in the lower 78 crust (Smithson and Decker, 1974). An upper bound of 35 mWm-2 can be obtained from radon isotope work (Hilton et al., 1985) and this range of values is in agreement with the work of Pollack and Chapman (1977). Here a value of 30 mWm — 9 is . used for the Neumann boundary condition . , at the base of the model. The sides of the models are all planar and perpendicular to the top and bottom. It is assumed that they are at such a great distance from any lateral variations in heat production and thermal conductivity that the heat flux normal to the sides is zero. In some cases, this condition is not strictly fulfilled and then care must be taken in interpreting the results. 4.5 Model Parameters The approach to modelling was to start with a regional model using a very rough approximation to the shape of the batholith, which would then be altered until the regional broadscale variations in measured heat flows could be explained. More detailed models of smaller areas would be made later to try to explain the local variations in heat flow and to predict more accurately temperatures at depth. At the time when modelling began, there was only one three- dimensional gravity interpretation of the batholith available (Tombs, 1977) and so this was used as a first approximation. The model was 475 km long, 350 km wide and 30 km deep (Figure 4.3) consisting of 20 x 10 x 5 quadratic elements respectively: Model 1. The granite was assumed to be homogeneous and the country rock laterally homogeneous. This meant that whatever properties were assigned to the rocks, because Tombs* model has a depth of granite beneath Dartmoor of several kilometres greater than beneath Land's End, the predicted heat flows at Land's End 7 9 SOUTH-WEST ENGLAND Model Locations -100 Figure 4.3 Location map of the models. were always smaller than at Dartmoor. The average of the measured heat flows, however, for Land's End is 132.3 mWm~2 (3 values) and for Dartmoor is 116.0 mWm”2 (5 values ignoring the water disturbed value of Sousson's Wood). Surface heat production measurements do not suggest that this can by explained by a general increase in heat generation towards Land's End. There could be a layer of higher heat production beneath the surface which increases in thickness towards the west, but there is no evidence to suggest this. The heat flux at Land's End could be enhanced by either a greater contribution from the mantle or lower crust, or through a deep convective system of some sort. Again, there is no evidence for these processes. Another possibilty is that the shape of the batholith could be wrong. Since Tombs' model does not take into account any regional gradient in the gravity field, which is apparent from a cursory study of the gravity data, it seemed legitimate to reassign the depth to the base of the batholith. A constant depth of 15 km was chosen, which is a reasonable compromise of all the existing data (see Section 2.2.2). Indeed, this modified model was capable of predicting more accurately the regional distribution of heat flows. It was also able to predict trends in heat flow across outcrops as observed in measured values. This modification was given greater credibility by the publication of a new three-dimensional model of the shape of the batholith from gravity (Willis-Richards, 1986) which, due to the removal of a regional gradient, shows a more uniform depth to the base of the batholith along its length. With the base of the granite fixed at 15 km a detailed model of the area around the Carnmenellis pluton was made: Model 2 (figure 4.3). This area was chosen since it possessed the highest density of heat flow measurements and 8 1 included the HDR site where temperatures at a depth of greater than 2 km had been measured in the granite. Hence the results could be better constrained. The model measured 120 km long, 90 km wide and 30 km in depth and was represented by 12 x 10 x 8 quadratic elements respectively. The heat productions and thermal conductivities used in the final model are given in table 4.1. Table 4.1 Rock properties used in the numerical models Heat Production (W m“3 x 10"6) Depth (km) Granite Country Rock 0 - 3 5.0 2.0 3 - 9 6.0 1.5 9 - 1 5 6.0 1.0 Lower Crust 15 -30 0.5 Parameters used in thermal conductivity equation: K(T) = d + c/T c (W nT1) 326 326 d (W m-1K-1) 2.1 1.2 8 2 The increase in heat production within the granite at 3 km is of no physical significance: there is no implication that this increase actually takes place though it is not considered unlikely that hydrothermal circulation has caused leaching of radioactive elements in the first few kilometres of granite. It is equally possible that the heat production. remains . at 5 microWm . _ -} to the base of the granite with the base lying at a depth greater than 15 km. It is also possible that the thermal conductivity contrast at depth is greater than allowed for, causing more heat to be refracted into the granite. However the distribution of heat flow sites and the accuracy of the measurements does not enable any distinction to be made. Two linear models (Model 3a and Model 3b) which allowed the use of a larger number of elements and together covered the whole of the batholith (figure 4.3) were constructed to allow more accurate heat flow predictions than those from the original regional model. The larger number of elements allowed a better representation of the shape of the granite and even though linear elements have been used, the presence of a larger number of elements vertically ensures that temperature predictions are comparable in accuracy to those from the coarser regional model using quadratic elements. These models were also used to compare results with laterally inhomogeneous conductivities and other variations in parameters, which will be discussed in section 4.7. Model 3a is 110 km long, 80 km wide and 30 km deep and consists of 23 x 17 x 15 linear elements respectively. Model 3b is 100 km long, 80 km wide and 30 km deep and consists of 33 x 18 x 15 linear elements respectively. These last two models have depths to the top surface of the batholith taken from the model of Willis- Richards (1986). 8 3 Table 4.2 Summary of models Model Width Length Depth Rock Gravity (km) (km) (km) Properties Model 1 350 475 30 i JCT 2 90 120 30 i JCT 3a 80 110 30 ii JWR 3b 80 100 30 ii JWR JCT - depth to the top of the granite from Tombs'1 gravity model. JWR - depth to the top of the granite from Willis--Richards gravity model. i - rock properties given in table 4.1. ii - as i except heat production in the granite is 5.8 throughout. 4.6 Model Results: Heat Flows 4.6.1 Regional Model The heat flows predicted by Model 1 are shown in figure 4.4. The heat flows vary from 60 mWm“2 at a great distance from the outcrops up to over 13 0 mWm-2 on some outcrops. 8 4 The lowest value of heat flow on an outcrop is just under 100 mWm"2 on the south-eastern edge of the Carnmenellis outcrop. The sum of the basal heat flow and the vertical integration of the heat production beneath any outcrop is 127 mWm”2 . This value will only be achieved when the two opposing effects of heat refraction and heat egression due to radioactivity contrasts are equal. The effect of refraction can be seen to dominate on those edges of the plutons which face in towards the centre of the batholith and adjacent to those areas of country rock which have high heat flows. This is as expected since in order for refraction into the granite to take place the temperatures in the country rock must be greater than in the granite. This will only occur when the temperature gradient is greater in the country rock than in the granite which requires the ratio of heat flows at the surface to be at least 2.2:3.2 by virtue of the conductivity contrast assumed. When there is a substantial depth of granite (and hence heat production) underlying a layer of country rock, that is along the axis of the batholith, surface heat flows at country rock sites can achieve these high values. On all other sides of the plutons, heat is lost to the surrounding country rock, the relatively high heat production within the granite causing temperatures to be higher than in the country rock at all depths. Despite this seepage of heat out of the batholith, it can be seen that all the heat flows on the granites are at least 20 mWm”2 greater than in the adjacent country rock. In this model, the number of elements which represent the outcrops of the plutons are not enough to define the variation of the heat flow observed. Indeed the outcrops at Kit Hill have been ignored and the northern extension of the Carnmenellis pluton which actually outcrops only at St. Agnes Head is modelled as outcropping along its entire length. The St. Austell outcrop is only one element and the Bodmin exposure only two. Despite these and other 85 SOUTH-WEST ENGLAND Surface Heat Conductive Flow: Model Figure 4.4 Surface heat flow predicted by Model H Contour interval• is • 10 mW m _ o . restrictions, the similarity between modelled and measured values can be seen quite clearly (Table 4.2). For instance, the heat flows increase to the south on Bodmin, to the west on Dartmoor (ignoring Soussons Wood), and to the north on Carnmenellis, which are also trends in the measured data. The magnitudes of the heat flows are very similar to the measured values. Country rock values are less well predicted by this model. Some values like Wilsey Down and Predannack are very good, others like Wheal Jane and Bovey Tracey are very poor, though this is not unexpected. So, even with a very simple model with few elements it is possible to draw several conclusions: the high heat flows observed are due to the presence of a large volume of high heat producing granite; the heat production must remain fairly constant with depth? on a large scale the batholith is highly homogeneous and the base of the batholith is approximately flat. 4.6.2 Carnmenellis Model Figure 4.5 shows the heat flows predicted by Model 2. The central part of the model is shown which includes the Carnmenellis outcrop (heat flows on the St. Austell and Land's End outcrops are slightly affected by the boundary condition at the sides of the model) . The heat flow pattern is different from that of Model 1 in detail only. The general feature of heat flow increasing to the north across the outcrop is still present. The larger number of elements in Model 2 representing a smaller volume allows the variations in the temperature field to be approximated more accurately and therefore gives better definition to the heat flows. High heat flows are predicted on all sides of the pluton except the south-eastern edge. The heat flow reaches a maximum of just over 150 mWm“2 and a minimum of about 115 mWm-2. In the country rock, heat flows generally decrease away from the granite outcrops except above the buried northern extension of the Carnmenellis outcrop where 87 CARNMENELLIS Surface Heat Flow: Conductive Model • HDR site Figure 4.5 Surface heat flow predicted by Model 2. Contour interval is 5 mW m“2 8 8 values reach a maximum• of just . over 115 mWm _p . The predicted heat flow at the HDR site is about 116 mWm”2 which falls within the range of measured values at that location (116.2 + 2.2 mWm-2). The values at all the other sites on the outcrop fall within 3% of the measured heat flows. Figure 4.6 shows a profile taken across the pluton (the location of this profile is shown in figure 4.5) of heat flows predicted by the model and the individual measurements. This highlights how well they agree for granite sites. Variations in the measured heat flows across the pluton can be seen to have some physical basis and are not just the result of poor quality measurement; nor are they due to convective heat transport as suggested by Fehn (1985) and Durrance et al. (1982) . The extremely high values on the outcrop to the north are due to refraction effects between the granite and country rock. The heat flow gradually decreases to the south as the effect of refraction diminishes.Then there is a slight increase as the profile crosses another refraction effect caused by the conductivity contrast between the country rock and the bend in the granite between the Carnmenellis outcrop and the Godolphin mass. The heat flow then decreases as the effect of the heat egression begins to dominate in the south. These variations can not be seen in the results predicted by the two-dimensinal model, constructed with the same parameters as the three- dimensional model, which are also shown in figure 4.6. 4.6.3 Quality of Results This profile clearly shows the poorer match between the modelled and measured values at country rock sites. The mismatch at Kestle Wartha and Old Merrose Farm are both greater than 18% though with different signs. These differences are equivalent to about 5 km extra or less of granite thickness beneath these sites. It seems unlikely 8 9 HEAT FLOW (m W irf2) aus r mre b crls Te hr bek i the in breaks sharp The circles. by marked are values rdce vle ae u t te udn hne i thermal in changes sudden the to due are values predicted y todmninl oe wih ss h sm parameters same the uses which model two-dimensional a by conductivity across granite-country rockinterfaces.granite-countryacrossconductivity s oe 2 r dntd y h boe ln. h measured The line. broken the by denoted are 2 Model as r dntd y h cniuu ln. et lw predicted flows Heat line. continuous the by denoted are 2 Model by flowspredicted heat The outcrop.Carnmenellis iue . Ha fos ln te rfl ars the across profile the along flows Heat 4.6 Figure Maue Ha Fo (lgty f profile) off (slightly Flow Heat Measured o LS PROFILE LLIS E N E M N R A C • Measured Heat Flow (on profile) profile) (on Flow Heat Measured • 0 9 that the gravity model and its representation in this model are inaccurate to that extent. There is no evidence from temperature logs in either borehole that there is local water movement. Naturally, the use of a laterally uniform thermal conductivity for the country rock will preclude the prediction of all the variations that would be found in measured values. However, variations in country rock conductivities will only cause heat to be redistributed by refraction and, since the effect of the large contrast between the granite and country rock thermal conductivities (3.2:2.2) only increases the heat flow due to refraction by a maximum• of just • over 20 mWm _ o , the smaller contrasts between the various types of country rock are unlikely to account for the whole of the mismatch. Furthermore, the boreholes would have to be very close to the contact for the refraction effect to be significant. What is more likely is that the measurements themselves are in error. The variation in thermal conductivity with depth in the Kestle Wartha borehole has been discussed already (see section 2.2.5). The variation of conductivities down the Old Merrose Farm borehole is equally substantial. The error in measurement of the heat flow can be assessed by multiplying the mean gradient down the borehole by the standard deviation in the measured conductivities (Richardson and Oxburgh, 1978). This gives a standard deviation in the measured heat flow of 30.94 mWm”2 for Old Merrose Farm and 28.43 mWm"2 for Kestle Wartha. The predicted values lie well within these limits. Table 4.2 compares the modelled heat flows with the measured values. Wherever possible the modelled values have been taken from Model 2. Figure 4.7 is a plot of the difference between measured and modelled heat flows for all sites against the standard deviation in thermal conductivity for those boreholes, the measured values and the standard deviations in thermal conductivity are taken 9 1 from Francis (1980) and Tammemagi and Wheildon (1974 & 1976). The two lines represent the error in measurement of heat flow produced by the error in conductivity for a theoretical hole with a temperature gradient of 30 deg C/km; the majority of differences lie within these limits. Two holes, however, do not fit this scheme: Soussons Wood and Wheal Jane. Both these holes have temperature logs which indicate at least some locally flowing water. Work by Heath (Durrance et al., 1982) indicates that there are ascending fluids in the region around Sousson's Wood. Whether the heat transferred by the fluids is sufficient to raise the heat flow to the very high value measured is doubtful and will be assessed in the next chapter. It is more probable that the temperature gradient was not reliable because of local water disturbances equilibrium when measured. It is well known that hot waters seep into the mines at Wheal Jane (e.g. Alderton and Sheppard, 1972). The movement of water may have been caused by the mining activity itself but also may be part of a convective system established by the radiothermal contrast of the granite. As will be seen later, the area around Wheal Jane is predicted to be a region of upward flowing fluids if convection is taking place today. It will also be shown that the heat flow observed at Wheal Jane can be produced by such a system without affecting the heat flows on the granite outcrop to any great extent. There is one other factor that contributes to the (lack of) accuracy of heat flow measurements in rocks with highly variable thermal conductivities and that is the depth of the hole. Those boreholes in country rock which were drilled to depths of over 300 metres all show a good match with modelled values: Wilsey Down (726m) has a percentage difference between measured and modelled of 2.5, Predannack (304m) has a 3.9% difference and Gaverigan (325m) 5.96%. All other boreholes (save Wheal Jane) were less than 160m deep and have an average percentage difference of 15.7. 9 2 9 3 DIFFERENCE BETWEEN MODELLED AND MEASURED HEAT FLOW vs. STANDARD DEVIATION OF MEASURED CONDUCTIVITY for for a heat flow measurement with a thermal gradient of 30 Figure Figure 4.7 Graph of the difference between measured and degC/km versus the standard deviation in the thermal conductivity conductivity measurements. (Note: Wheal Jane is off the The The straight lines represents the theoretical error bounds modelled heat flows versus the standard deviation in the measured thermal conductivity from the heat borehole. flow vertical vertical scale.) .uimuj j h V Table 4.3 Comparison of Measured and Modelled Heat Flows Station Name HEAT FLOW (mW m 2) Standard Deviation Measured Modelled of Thermal Conductivity (W m"^-K"1) Grillis Farm 119 123 0.16 Polgear Beacon 128 127 0.35 Medlyn Farm 120 122 0.23 Trevease Farm 118 121 0.12 Trerghan Farm 119 119 0.17 Bray Down 120 115 0.17 Blackhill 126 119 0.18 Pinnockshill 127 121 0.13 Browngelly 115 125 0.17 Great Hammet Farm 125 130 0.16 Newmill 130 133 0.15 Bunker's Hill 130 125 0.21 Tregarden Farm 132 131 0.14 Colcerrow Farm 133 131 0.25 Winter Tor . 114 110 0.12 Blackingstone 112 110 0.32 Soussons Wood 139 115 0.28 Laughter Tor 121 117 0.19 Foggin Tor 118 119 0.17 Geevor 134 131 0.22 Troon 129 128 0.16 South Crofty 137 140 0.18 Rosemanowes A 113 117 0.20 Rosemanowes B 114 117 0.21 Longdowns 118 117 0.34 Hemerdon 114 119 0.50 9 4 Table 4.2 continued Country Rock Sites Merrose Farm 84 103 0.91 Kestle Wartha 102 84 0.92 Callywith Farm 106 95 0.50 Gaverigan 105 112 0.53 Wheal Jane 132 97 0.29 Newlyn East 111 92 0.58 Belowda Beacon 91 107 1.02 Lanivet 99 105 no data Wilsey Down 74 77 0.62 Meldon 120 115 0.35 Bovey Tracey 100 72 1.15 Predannack 68 66 0.22 Kennack Sands 79 65 0.36 4.7 Results: Temperatures The temperatures at depth depend on the temperature gradient and the thermal conductivity distribution. Highest heat flows do not necessarily indicate the largest temperatures at shallow depths. A case in point is the effect of refraction where heat flows in the medium of higher conductivity are greater than in the other medium but temperatures in the medium of lower conductivity are higher. Figure 4.8 which shows the temperatures predicted for a depth of 2000m from Model 2 confirms that the highest temperatures at shallow depths are under regions of very high heat flow in the country rock. It also confirms that where refraction has taken place temperatures in the country rock are higher than temperatures in the adjacent granite and that elsewhere the reverse is true. This model 9 5 CARNMENELLIS Temperatures at 2000m • HDR site Figure 4.8 Predicted temperaures at a depth of 2000m from M o d e l 2. Contour interval: 2 degC 96 o o • HDR site Figure 4.9 Predicted temperatures at a depth of 2000m fom M o d e l s 3a & 3b. Contour interval: 5 degC 97 predicts maximum temperatures of just over 100°C at this depth in this region. This maximum occurs beneath the region of highest heat flow in the country rock. Figure 4.9 shows the temperatures predicted by Model 3a & 3b for the whole of the region at a depth of 2000m. The differences between the predictions from the two models (2 and 3) are very slight but it is not possible to say whether they are due to the fact that one model uses quadratic elements and the other linear elements or if they are due to differences between the gravity interpretations of Tombs and Willis-Richards. The regional model shows four separate regions where temperatures reach 100°C which is at least 40 degC above the background temperature at this depth. Measurements of temperature at this depth have been made at the HDR site in three boreholes. The measurements from two of the boreholes have been corrected for the full palaeoclimate. The palaeoclimate used was that which was employed by Francis (1980) to correct all the heat flow measurements in this region and is shown in figure 4.10. The correction naturally depends on the depth of the measurement and here is about 4 degC as shown in figure 4.11. The corrected temperatures are 80.85°C and 80.86°C for boreholes RH11 and RH12 respectively. The predicted temperature at this site from Model 2 is 80.8°C. It is not very surprising that the match is so good since many of the parameters used in the models were based on measurements made down these holes. Also, for a given surface heat flow, temperatures at shallow depths do not vary widely for different thermal parameters (within reasonable limits) or different distributions of heat production with depth. For example, the one-dimensional extrapolated value for the temperature at 2000m depth for a site with a heat flow of 116 mWm-2 and a rate of change of thermal conductivity with temperature of 200 W/mK2 is 79.3°C. If the rate of change 9 8 fOo CL CQ co <0i — © >- • < Q. Figure 4.10 The full palaeoclimate model used to correct the measured temperatures. (After Francis, 1980.) 9 9 z o H o LU cc oc o o LU < _J o o LU < UJX 'H ld3a < CL Figure 4-11 Graph of temperatures to be applied to correct for palaeoclimatic effects versus depth. A thermal diffusivity of k/2.26 x 10 mm s was assumed where k is the thermal conductivity. 1 0 0 of thermal conductivity with temperature is increased to 600 W/mK2 the prediction is 81.2°C (the value of K at 297K being 3.24 in both cases). Examples using various distributions of heat production with depth are given in Francis (1980). A linear extrapolation at the HDR site gives a temperature at 2000m of 79.8°C using the same parameters as in Model 2. This indicates that there is not much non-vertical heat flow at this point within this depth range. In fact using a heat flow of 117.5 mWm“2 instead of 116 mWm”2 will give a temperature of 80.8°C at a depth of 2000m. As the depth increases and temperatures become higher, the use of different parameters in the models and extrapolations will produce greater differences in predicted temperatures. It is thus quite important to be able to put a lower limit on predicted temperatures. The use in these models of a rate of change of conductivity with temperature which lies in the lower range of expected values implies that temperature predictions are also towards the lower end of the scale. However, the use of the same value for the rate of change of thermal conductivity with temperature in the country rocks is somewhat questionable. If a value of 200 W/mK2 is used instead of 320 W/mK2 for country rock, the maximum temperature at a depth of 6000m decreases by about 10 degC whilst surface heat flows are altered negligibly. Of course if the absolute conductivity of the granite or surrounding rock were to increase with depth or if the average conductivity of the country rock were higher, then predicted temperatures would be lower than those predicted by these models. Predicted temperatures would also be too high if heat production were to increase with depth. Figure 4.12 shows the temperatures predicted by Models 3a & 3b for a depth of 6000m. Temperatures reach over 240°C in 1 0 1 Figure 4.12 Predicted temperatures at a depth of 6000m from Models 3a & 3b. Contour interval: 10 degC 1 0 2 Figure 4.13 Map showing the location and thickness of the surface layer assigned a higher conductivity. This layer is meant to represent the Carboniferous rocks and are allocated a thermal conductivity• • of 3.15 W m —1 K —i . 103 Figure 4.14 Predicted temperatures at a depth of 6000m from Models 3a & 3b, but with higher conductivities assigned to the Carboniferous rocks. 104 several regions and most of the batholith is above 200°c. The areas of highest temperatures are those which at surface are covered by the greatest thickness of country rock and lie towards the centre of the batholith. At this depth anomalous temperatures can be as much as 80 degC above the background value. It is in temperature prediction that the use of the wrong thermal conductivities for country rocks will have its greatest effect, as shown in the following example. There are only three boreholes drilled in Carboniferous sediments, two of which show unusually high average thermal conductivities: Bovey Tracey (3.16 W m^K"”1) and Meldon quarry (3.14 W m K x). If these results are indicative of the Carboniferous rocks in general then this will have an effect on the temperature predictions. In order to assess this effect, a layer of high conductivity (3.15 W/mK instead of 2.2 W/mK) country rock was used in Models 3a & 3b to represent the Carboniferous rocks to the north of Dartmoor and Bodmin (fig 4.13). Obviously this alteration to the model caused changes in the predicted heat flows, but the differences are small at the borehole sites and so it is not possible to say which of the models is more realistic. The temperatures at 6000m were also changed (fig 4.14) with lower temperatures predicted for the zone beneath the high conductivity country rock, but maximum temperatures in the granite were reduced by less than 10 degC. Although there may be other uncertainties, it is probably true to say that the error in predicting maximum temperatures at a depth of 6000m is no more than 20 degC and probably less than 10 degC. More importantly, the locations of the regions of maximum temperature do not change by more than a few kilometres even when severe changes are made to the thermal conductivities (compare figures 4.12 and 4.14). Thus it is possible to locate the 105 optimum drilling sites for future HDR work, as far as temperatures are concerned, with great certainty. 4.8 Conclusions The observed heat flow field of South-west England can be explained for the most part, in terms of the conductive flow of heat. The majority of the variations in the surface heat flows are due to two effects; heat refraction due to thermal conductivity contrasts and heat egression due to heat production contrasts. Differences between modelled and measured heat flows for granite sites are generally small and indicate that the granite is best represented as a homogeneous body with heat production remaining constant with depth. The results also indicate that the depth to the base of the batholith remains fairly constant along its length. The larger differences between modelled and measured heat flows for country rock sites are not thought to be due to a poor or over-simplified representation of the hetrogeneous country rocks. It is believed that they are in the main due to inaccuracies in the measured values brought about by the large variations in the thermal conductivities of the sedimentary and metamorphic rocks. The differences are notably smaller for those boreholes that are at least 300m deep, whether the thermal conductivities are highly variable or not. Temperature predictions for shallow depths are accurate and show that the highest temperatures are not found beneath the highest surface heat flows. They suggest that the HDR project is not ideally located as far as temperatures are concerned with some regions attaining temperatures of up to 20 degC higher at the same depth (2000m) than at its present location. Temperature predictions become prone to 106 greater uncertainty with increasing depth. At 6000m the uncertainties are of the order of, but probably less than, 20 degC with maximum temperatures in excess of 240°C. The highest temperatures are located towards the centre of the batholith and beneath regions insulated by a thickness of country rock. 107 Chapter Five Models of Fluid Flow and Mineralization 5.1 Introduction This chapter deals with the results of models which include convection as a means of heat transport. The first part assesses how convective fluid flow might affect the heat flow pattern observed today by comparing results from convective models with those from models without convection and with the measured data. The remainder of the chapter describes the possible connection between fluid flow patterns and the mineralization and other related phenomena associated with the batholith. The factors which affect the fluid flow patterns are discussed in general terms before the models are presented. Two-dimensional models are used to show how the fluid flow patterns are dependent on the shape of the contrasts in thermal parameters and the depths at which these contrasts exists. A model of a cross-section through the St. Agnes mining district is produced to indicate how the regional zonation of the mineralization can be related to changes in fluid flow patterns brought about by erosion. Three-dimensional models show the regional fluid flow patterns for the entire area for different levels of erosion. These patterns are then compared with the regional distribution of fluid inclusions, minerals, kaolin and certain geochemical elements. 108 5.2 The Effect of Convection on Heat Flows and a Comparison with Observed Data As indicated in the previous chapter fluid circulation on a large scale is able to alter the heat flow pattern quite dramatically whilst not causing changes in temperatures in any way that could be detected by measurements made in shallow boreholes. It was also suggested that the discrepency between measured heat flows in the country rock and modelled values might be partly due to the failure to take into account regional fluid flow. The introduction of a convective term into the heat flow equation, as indicated in Chapter 2, allows this possibility to be tested. As only linear elements were to be used in the convective models it was considered that the results should be compared with those from conductive models also with linear elements. The heat flows for the region from conductive and convective models are shown in figures 5.1 and 5.2 respectively. The conductive model is very similar to those models described in the previous chapter except that the heat production in the granite is averaged out to 5.8 microW/m3 at all depths. The convective model has the same parameters as the conductive model but there is a major difference between them in that the convective model is only 5 km in depth. The heat flow at the base of the model is assumed to be the same as the vertical heat flux at a depth of 5 km in the conductive model. The surface permeability is 0.1 mD and is assumed to be laterally homogeneous but decreases exponentially with depth according to equation 3.3.6.1 with f equal to 2. The differences in the predicted heat flows between the two models are naturally related to the direction of flow of fluid and the intensity of flow. The pattern of vertical 109 Figure 5.1 Surface heat flows predicted by Models 3a without convection. _ o Contour interval: 10 mW m 1 1 0 Figure 5.2 Surface heat flows predicted by Models 3a & 3b including convection with a surface permeability of 0.1 mD and a decay factor of f = 2. Contour interval:. 10 mW m—2 1 1 1 Figure 5.3 Vertical fluid flow at the surface predicted by Models 3a & 3b with a surface permeability of 0.1 mD and a decay factor of f = 2. Contour interval:• 2 x 10 _ Q kg m _o s_*i 112 fluid flow at the surface is shown in figure 5.3. Over most of the outcrops the fluid flow is downwards and thus the heat flows will be reduced. Heat flows will be raised in many areas of country rock especially close to the outcrops. Table 5.1 makes a direct comparison between the two sets of results and the measured values. The values from the convective model for granite sites show an overall deterioration in match with the measured values. Apart from two sites (Trevease Farm and Hemerdon) the heat flows from the convective model are lower than for the conductive model. However, increasing the modelled heat production of the granite, which intuitively may be expected to raise the heat flows, is unlikely to produce an improved overall match for the convective model since it would also reduce the counter-balancing effect of the sites where the convective model has produced an improved match. For the country rock sites the results are remarkably consistent, all sites showing an improvement (or no change) over the conductive model. Though the overall agreement is better there are still some large mismatches. Obviously it would be possible to improve matches at the country rock sites by increasing the permeability of the rocks but this would result in a degradation for granite sites. However, by making the granite to all intents and purposes impermeable, it is possible to achieve the improvements at the country rock sites without affecting the match at the granite sites. Nevertheless, no matter how large the permeability, there would remain some sites that showed little or no improvement. It would thus seem impossible to account for the mismatch between measured and modelled values by fluid flow alone and that the errors in measurement are quite significant for some boreholes. 113 Table 5-1 Comparison of Modelled and Measured Heati Flows BOREHOLE HEAT FLOW mW m 2 Measured Conductive Convective Improvement Modelled Modelled Granite Sites Grillis Farm 119 129 120 + 9 Polgear Beacon 128 123 118 - 5 5 Medlyn Farm 120 120 120 0 Trevease Farm 118 118 120 - 2 Trerghan Farm 119 119 112 - 7 _ _ _ * Bray Down 120 127 122 + 5 Black Hill 126 127 119 - 6 Pinnockshill 127 127 116 - 9 Browngelly 115 128 116 + 12 Great Hammet Farm 125 129 120 - 1 Newmill 130 129 120 - 9 Bunker's Hill 130 128 119 - 9 Tregarden Farm 132 140 136 + 5 Colcerrow Farm 133 130 124 - 6 Winter Tor 114 118 112 + 2 Blackingstone 112 118 112 + 6 Soussons Wood 139 115 115 0 Laughter Tor 121 118 113 - 5 Foggin Tor 118 125 118 + 7 Geevor 134 132 122 -10 Troon 129 129 120 - 9 South Crofty 137 140 133 - 1 Rosemanowes A 113 119 115 + 4 Rosemanowes B 114 119 115 + 4 Longdowns 118 119 115 - 2 Hemerdon 114 125 130 - 5 TOTAL for granite sites -32 114 Table 5.1 continued Country Rock Sites Merrose Farm 84 111 100 +11 Kestle Wartha 102 79 82 + 3 Callywith Farm 106 98 79 + 1 Gaverigan 105 111 105 + 6 Wheal Jane 132 88 110 +22 Newlyn East 111 101 109 + 8 Belowda Beacon 91 102 101 + 1 Lanivet 99 100 100 0 Wisley Down 74 69 69 0 Meldon 120 80 82 + 2 Bovey Tracey 100 73 81 + 8 Predannack 68 67 69 0 Kennack Sands 79 67 69 + 2 TOTAL for country rock sites +64 TOTAL for all sites +32 ^ 'improvement' means the absolute difference between conductive and measured heat flows minus the absolute difference between the convective and measured heat flows. * modelled values are estimated since the granite sites do not fall within the model granite outcrops. 1 1 5 There are, however, one or two sites where fluid flow has had a remarkable effect on the match. At Wheal Jane the match has been improved by 22 mW m“2. It is known from the temperature profile and from the mine workings that there is water-movement in this area and it is more than likely that the measured heat flow has a large contribution from fluid flow, though whether this flow is regional or more localised than that described by the model is a matter of conjecture. The other sites which may have a convective component in the measured heat flows are Merrose Farm and Gaverigan which lie in a region of downflow, and Newlyn East and Bovey Tracey which lie in regions of upflow. At Soussons Wood, however, there is no indication that the regional flow patterns could account for the very high heat flow measured and it must therefore be assumed that the temperatures in the borehole were disturbed by a fairly local convective system. 5.3 Introduction to the Proposed Explanation of the Regional Aspects of the Ore Field. The features of the ore field associated with the Cornubian batholith were presentd in chapter 3. It is evident that current theories are unable to explain some of the regional aspects of the ore field. Though not explicitly stated, it is assumed that the accepted explanation for the mineral zoning relies on changes in the horizontal distribution of permeabilities to account for the changing location of upwelling fluids with time. This is possible and it has been shown by Fehn (1985) that regions of extreme permeability can alter the fluid flow patterns. However, no mechanism has been proposed to explain how and why the fracture systems changed to produce the asymmetric distribution of mineral lodes about the outcrops and the 116 mineral zoning. A comparison of the distribution of tin centres with the predicted surface thermal gradient (figure 5.4) shows that there may be a connection between the two, with many of these centres lying within or close to the highest gradients. Given a medium of laterally homogeneous permeability, it will be the distribution of rock types with different thermal parameters, and, hence, the temperature field which will determine the fluid flow pattern. Thus, since hotter fluids rise, it would seem reasonable to expect mineralization associated with upwelling fluid to be found where the temperature gradients are highest. If the fluid flow pattern and hence the distribution of mineralization is dependent primarily on the temperature field, it must be this which changes in order to cause horizontal mineral zoning. It will be shown in the next section that changing the depths at which contrasts in thermal parameters exist can produce very different fluid flow patterns. Therefore, as a high heat producing granite which was emplaced at some depth beneath the surface of the Earth is exposed by erosion of the cover, the fluid flows associated with it will alter. Furthermore, as the granite is uncovered it will cool as will the fluid passing through a fixed level in the granite and so one would expect high and low temperature assemblages of minerals found at the same depth to have been deposited at different levels of erosion. There is some evidence for erosion between the main mineralizing event and the deposition of the later, lower temperature minerals from the St. Agnes Head district (Moore, pers.comm.) from studies of dips of mineral lodes. The argument is that since fractures induced by stress must be vertical at the surface of the Earth when formed but may 117 Figure 5.4 Comparison of the predicted surface temperature gradient and the distribution of emanative centres. 1 1 8 dip less steeply at depth, it is possible to infer at what depth they may have been formed. Observations of the different dips of the mineral lodes in the St. Agnes district, suggest that the high and low temperature mineral assemblages were deposited at different depths below the surface, and since they now lie at the same depth, erosion must have occurred between the two events. If the arguments presented above are valid it should be possible to explain the horizontal mineral zoning in terms of changing erosion levels provided it can be shown that there is a correlation between the observed locations of different temperature mineralization with fluid flow patterns associated with the batholith for different levels of cover. To this end four three-dimensional models have been studied representing the batholith at different depths within the upper crust (3 km, 2 km, 1km, and at the surface) which use the same thermal parameters and shape of the granite as described at the beginning of this chapter (section 5.2). The results from these models are then compared with the distribution of fluid inclusions, mineral deposits, kaolinisation and the geochemical distribution of certain elements. These comparisons are necessarily of a qualitative nature and there is no implication that all the tin centres, say, were formed at the same time and beneath the same depth of cover. 5.4 Two-dimensional Models Modelling in two dimensions has several advantages: relative to three-dimensional models they are easy to establish in the first place, they are quicker to run and they can use a large number of elements. However, in interpreting the results it should be remembered that the third dimension often has a large influence on the 119 temperature field. Nevertheless, two-dimensional models are extremely useful for determining the effects the shape and depth of a granite and other parameters have on the fluid flow patterns. Three synthetic models representing granites of various shapes and one model representing an actual geological situation are now presented in order to highlight these effects. Each model was run four times with different depths of cover above the intrusion (3 km, 2 km, 1 km, 0 km) . The thermal parameters of the granites and country rock are the same as those assigned to the convective model at the beginning of this chapter and are thus representative of the situation in South-west England. The surface permeabilities for the different models are chosen so that the maximum flow of fluid at the surface for the various depths of cover are comparable. 5.4.1 Results from Synthetic Models Model A is of a regularly shaped granite 20 km in width. Figure 5.5 shows the vertical fluid flow at the surface of the models. Apart from the case when there is no cover there is only one region of upward flow associated with the granite and that appears above the centre of the granite. (It should be noted that any asymmetry in what one would expect to be symmetric diagrams is an artefact of the plotting routines used and not a fault of the inversion technique.) For no cover there are two regions where the fluid is flowing upwards and these lie close to the edges of the granite. Thus there is a remarkable change in fluid flow pattern between an exposed regularly shaped granite and a covered one. Model B is a symmetric granite whose width at the top is also 20 km but whose sides slope outwards as one might 1 2 0 expect for an intrusive. The results are shown in figure 5.6. As expected the fluid flow patterns are symmetrical but they are very different from those of Model A. For all depths of cover there are least two regions where the fluid is flowing upwards. The location of these convection cells can be seen to change with the depth of cover. As the depth of cover is reduced from 3 km to 0 km so the regions of upflow move away from the centre of the granite by about 4 km in total. Model C is a combination of the two previous models with one side being vertical and the other sloping outwards. The vertical fluid flow for the different depths of cover are shown in figure 5.7. Again there are two regions of upflow for each depth of cover. However the patterns differ from those of Model B in several ways. The pattern is no longer symmetric, with the fluid flow intensity being greater above the sloping side of the granite. The region of upflow associated with the sloping side moves outwards with decreasing depth of cover but at a different rate than in Model B. The other region tends to move in towards the centre of the granite. These synthetic models show how the temperature field determines the fluid flow patterns in a homogeneously permeable medium. The strong control of the granite shape on flow patterns suggest that it is only possible to make useful predictions concerning fluid flow and hence mineralization by modelling with realistic shapes for the granite intrusive. It is also apparent that the asymmetric shape of a granite pluton could explain the asymmetric distribution of mineralized lodes associated with it. The outward migration of the region of upflow as predicted by these models suggests a mechanism that might be responsible for the horizontal zoning of minerals. 1 2 1 SYNTHETIC MODEL A DISTANCE (km) Figure 5.5 Vertical fluid flow through the surface for a model with a regular symmetric intrusion for different depths of cover. a) 3 km of cover with a surface permeability of 1.4 mD b) 2 km of cover with a surface permeability of 1.71 mD c) 1 km of cover with a surface permeability of 2.29 mD d) zero cover with a surface permeability of 4.6 mD The rate of decay of the permeability with depth is f = 5. 1 2 2 Figure 5.6 Vertical fluid flow through the surface of a model with a symmetric irregular intrusion for different depths of cover. a) 3 km of cover with a surface permeability of 1.4 mD b) 2 km of cover with a surface permeability of 1.71 mD c) 1 km of cover with a surface permeability of 2.29 mD d) zero cover with a surface permeability of 4.6 mD The rate of decay of the permeability with depth is f = 5. 123 DISTANCE (km) Figure 5.7 Vertical fluid flow through the surface of a model with an asymmetric intrusion for different depths of cover. a) 3 km of cover with a surface permeability of 1.4 mD b) 2 km of cover with a surface permeability of 1.71 mD c) 1 km of cover with a surface permeability of 2.29 mD d) zero cover with a surface permeability of 4.6 mD The rate of decay of the permeability with depth is f = 5. 124 5.4.2 Results from a Model of St. Agnes These ideas were also tested on a two-dimensional model of a real geological situation representing a cross-section through the northern limb of the Carnmenellis granite which outcrops at St. Agnes Head. This region was chosen as it is a prime example of the mineral zonation discussed above. Figure 5.8 is a map of the region showing the outcrop of the granite near the coast with the tin mineralization located within a short distance of the outcrop. The copper mineralization occurs out to a greater distance and the lead, zinc and iron deposits are at even greater distances. This model, whose subsurface shape of the granite was taken from the gravity model of Willis-Richards (1985), was also run for four depths of cover and the results are shown in figure 5.9. For each depth of cover there is only one region of upflow associated with the granite. This region tends to move inland as the depth of cover is decreased. An important feature in the argument being developed is the temperature of the mineralizing fluids flowing through the level that is today's surface for the different depths of cover. It has been shown by Fehn (1985), using two- dimensional models of South-west England, that there is a maximum temperature that can be reached in a convective system irrespective of the magnitude of the fluid flow, but dependent on the thickness of the permeable layer. This maximum temperature occurs at the base of the model directly below the upwelling area. It is also the case that at any level in the model there is a maximum temperature that can be attained. As the depth of cover decreases the maximum temperature at any level will also decrease. 125 Figure 5.8 Map of the St. Agnes Head mining district of South-west England showing the distribution of mineral lodes. 126 ST. AGNES PROFILE C L T 3 Figure 5.9 Vertical fluid flow through the surface of the model of a cross-section through St. Agnes Head for different depths of cover. a) 3 km of cover with a surface permeability of 0.2 mD b) 2 km of cover with a surface permeability of 0.5 mD c) 1 km of cover with a surface permeability of 1.0 mD d) zero cover with a surface permeability of 1.5 mD The rate of decay of permeability with depth is f = 5. 127 Table 5.2 Highest Temperature at the Top of the Granite for Different Depths of Cover Depth of Cover (km) Temperature (°C) 3 280 2 210 1 140 0 10 Table 5.2 gives the maximum temperatures at the top of the granite for the different depths of cover for this model. It can be seen that there is a drop in temperature of the mineralizing fluids which pass through the top of the granite for decreasing amounts of cover. Of course it is possible to increase these temperatures by increasing the depth of penetration of the fluids. However, assuming the base of the permeable layer remains fixed, it is possible to conclude that there exists a minimum depth at which a particular mineral can be deposited. It should also be noted that a decrease in intensity of flow will sometimes cause a drop in temperaure of the fluids at a given point.So, if different temperature minerals are found at the same depth either the intensity of fluid flow or the depth of cover must have changed. Changes in the intensity of the fluid flow do not seem, however, to cause changes in location of the major convection cells (Fehn, 1985) and, therefore, cannot lead to horizontal zoning. So, the correlation expected to 128 exist would be between the distribution of high temperature mineralization and the fluid flow patterns associated with large depths of cover irrespective of the intensity of fluid flow. Likewise low temperature mineralization should correlate with the fluid flow patterns for small depths of cover. Figure 5.9 does indeed show such correlations: the fluid flow migrating inland with decreasing cover, and the further inland one goes the lower the temperature that characterises the mineralization observed. Although temperature and upward fluid flow are not the only factors which determine the location of mineralization, they would appear to be capable explaining the general distribution of the lodes and the horizontal zoning of minerals from the results of the two-dimensional modelling carried out. This, of course, must be confirmed by three- dimensional modelling. 5.5 Results from Three-dimensional Models Three-dimensional models similar to that described in section 5.1 were constructed with varying amounts of cover. The vertical fluid-flows through the level that represents today's surface are shown in figures 5.10, 5.11, 5.12, and 5.13 for 3, 2, 1, and 0 km of cover respectively. It must be remembered that in these models the shape of the granite can only be approximated by a coarse grid of finite elements and therefore the results will only reveal general features of the fluid flow patterns. For 3 km of cover the areas of upwelling fluid are concentrated on or close to the outcrops. In fact they tend to lie along the axis of the batholith as a whole. For Land's End this region runs along the axis of the outcrop and can be likened to Model A of section 5.3 as it has fairly symmetrical steep sides. For the Carnmenellis 129 outcrop the upwelling occurs along the northern edge of the pluton from the Godolphin mass and bends north following the extension to the pluton which outcrops towards the coast. There is a smaller region of upwelling fluid associated with the southern edge of the outcrop. This pattern is similar to that produced by Model C of section 5.3 though the asymmetry of the pluton shape is more exaggerated here. This asymmetry can also be observed over the Bodmin outcrop with the maximum upwelling in the south. The upward flow over the St. Austell outcrop is concentrated towards the western end and above the covered half of the pluton. A maximum in the local upward flows is seen directly above the small outcrops in the area of Kit Hill. The intensity of flow through the Dartmoor outcrop is much reduced and shows a region of high around the western edge of the outcrop. There is quite a marked contrast from these patterns to the ones produced with 2 km of cover. Now the peaks of the upward flowing regions are beginning to appear around some of the edges of the outcrops. This is certainly true for Dartmoor and Carnmenellis. For Bodmin the main peak is now in the south-east corner and for St. Austell in the western lobe. Kit Hill remains a region of intense upward flow as does the region just off the coast to the south-west of Land’s End. There is now a separate high in the St. Agnes/ Cligga Head district. Two other features which should be noticed are the kink in the first level contour in the Wheal Jane area and the upflowing area to the west of Bodmin and north of St. Austell: the Camel estuary region of the Wadebridge district (see figure 5.18 for the location of mining districts). The overall shape of the upwelling region does not change much with the removal of another kilometre of cover, however there now exist two regions of downwelling: one over the Dartmoor outcrop and one over the Carnmenellis 130 o o Figure 5.10 Vertical fluid flow through the plane which represents today's surface predicted by Models 3a & 3b with an additional 3 km of cover. Surface permeability, kQ = 0.08 mD, f = 2. Contour interval: 2 x 10 kg m”2s_1. 131 Figure 5-11 Vertical fluid flow through the plane which represents today's surface predicted by Models 3a & 3b with an additional 2 km of cover. Surface permeability, kQ = 0.1 mD, f = 2. Contour interval: 1 x 10 kg m~2s~"1. 132 Figure 5-12 Vertical fluid flow through the plane which represents today's surface predicted by Models 3a & 3b with an additional 1 km of cover. Surface permeability, kQ = 0.1 mD, f = 2. Contour interval: 2 x 10 kg m_2s 133 o o Figure 5.13 Vertical fluid flow through the plane which represents today's surface predicted by Models 3a & 3b with no additional cover. Surface permeability, kQ = 0.1 mD, f = 2. Contour interval: 2 x 10 ~*9 kg m_2s_1. 134 outcrop. There are also local minima in the upwelling over the other major outcrops. Local maxima in upflow remain over the Kit Hill area, the south-east of Bodmin, the western edge of St. Austell, the Camel estuary , Cligga Head and the area just south-west of Land's End. New peaks have appeared on the southern edge of Godolphin, the south eastern edge of Carnmenellis, in the Wheal Jane district and along the north-western edge of Bodmin. It should also be noticed that the peaks of the upward fluid flow surrounding the outcrops have moved out a little. For 0 km of cover there are regions of downwelling over each of the major outcrops and one in the St. Agnes district. An interesting feature of these zones is the difference in intensity of flow between them. The most intense downward flow is associated with the St. Austell outcrop and is centred on the exposed half of the pluton. The next in intensity is the flow over the Bodmin outcrop which is concentrated in the western half. Over Dartmoor the maximum downflow is in the south-west, over Carnmenellis in the north, and over Land's End is split into two: one in the west and one in the east. The peaks of the upwelling regions have again moved outwards even though the outer limits of the region have not altered much. 5.5.1 Comparison of Fluid Flow and Fluid Inclusion Patterns Fluid inclusion studies provide details on the type of fluids which have passed through the rock, their temperature and their salinity. It is also possible that the abundances of fluid inclusions can give estimates of the quantity of fluid which has passed through the rock. There are many types of fluid inclusion which for the South-west of England have been separated into 6 types, though it may not always be possible to categorize with total confidence some inclusions (Rankin and Alderton, 135 1983). Rankin & Alderton (1982) and Alderton & Rankin (1983) produced maps showing the variation in percentage of the various fluid inclusion types and the overall abundance of fluid inclusions for the main outcrops of granite in South-west England. There appeared to be some evidence to suggest that the variation in overall abundance of fluid inclusions was related to mineralization. It is pertinent then to try to find any similarity between those fluid flow patterns produced by the models and the distribution of fluid inclusions. In brief, and following Rankin and Alderton's description: Type 1 inclusions are monophase aqueous inclusions with homogenisation temperatures of up to about 200°C. Type 2 inclusions are aqueous containing a moderate to large vapour bubble, with an arbitrary subdivision based on the proportion of vapour (2a, 2b). Homogenisation temperatures range between 200°C and 600°C. Type 3 inclusions are aqueous and characterised by chloride daughter minerals. Vapour-liquid homogenisation temperatures are generally between 200°C and 350°C. Halite solution temperatures may be as high as 500°C. Type 4 inclusions are normally totally filled with vapour. Homogenisation temperatures of the vapour phase are rarely determinable. Rankin & Alderton (1982) considered that the presence of type 2a inclusions in a granite signified that the area had been swamped by later (post-magmatic) fluids which were dominantly meteoric, although, when they plotted the percentage of type 2 fluid inclusions (type 2a and 2b) the results were not entirely conclusive. However, it would seem sensible to consider the absolute abundances rather 136 than the percentage values in making a comparison with modelled fluid flow patterns and figure 5.14 shows the absolute abundances of type 2a fluid inclusions for those outcrops where data, re-calculated from the values presented by Rankin and Alderton (1982), exist. There are two points to note: the first is the variation in density of this type of fluid inclusion between outcrops and the second is the variation over the individual outcrops. The most densely populated area is in the north of the Carnmenellis outcrop which is also the region of greatest productivity for tin and copper (Dines, 1956). The Dartmoor outcrop conversely has the lowest population and is also the least mineralized outcrop. The abundance of these inclusions, which were formed at temperatures between 200°C and 600°C, should be compared with fluid flow patterns for large depths of cover. The fluid flow pattern for 3 km of cover has some features similar to the pattern of type 2a inclusion density. It is evident that the flow of fluid through the Dartmoor outcrop is lower than through other outcrops and the greatest fluid flow is in the north of the Carnmenellis outcrop. Thus the model seems to be able to predict the intensity of the initial flow as indicated by the density of type 2a inclusions. For Carnmenellis and Bodmin there is a good correlation between fluid flow intensity and type 2a inclusion density. However, there is no such correlation on the Land's End or Dartmoor granites. For Land's End the fluid flow pattern suggests a fluid flow regime like that predicted for very shallow depths of cover or for a pluton with more gently dipping sides, and for Dartmoor a more deeply buried pluton. The type 3 fluid inclusions are thought to be older than the type 2 inclusions and may represent the flow of magmatic fluids. Figure 5.15 shows the variation in type 3 fluid inclusion abundance. Data does exist for 137 FLUID INCLUSIONS TYPE 2a Figure 5.14 Absolute abundance of type 2a fluid inclusions in quartz samples for the Land's End, Dartmoor, Bodmin and Carnmenel1is outcrops. Isoabundance contours are in absolute number of type inclusions/ unit volume Data from Rankin and Alderton (1982). 138 Figure 5.15 Absolute abundance of type 3 fluid inclusions in quartz samples for the Land's End, Dartmoor, Bodmin and Carnmenellis outcrops. Isoabundance contours are in absolute number of type inclusions/ unit volume Data from Rankin and Alderton (1982). 139 FLUID INCLUSIONS TYPE 1 NOT TO SCALE Figure 5-16 Absolute abundance of type 1 fluid inclusions in quartz samples for the Land's End, Dartmoor, Bodmin and Carnmenellis outcrops. Isoabundance contours are in absolute number of type inclusions/ unit volume Data from Rankin and Alderton (1982). 14 0 Figure 5.17 Absolute abundance of type 4 fluid inclusions in quartz samples for the Land's End, Dartmoor, Bodmin and Carnmenellis outcrops. Isoabundance contours are in absolute number of type inclusions/ unit volume Data from Rankin and Alderton (1982). 141 Carnmenellis and Bodmin but the values are very small. The high abundance of this type of inclusion on Dartmoor and Land's End indicates that some of the tin and tungsten deposits found therein are of magmatic origin and thus can not be predicted by this present kind of fluid flow modelling, Birch Tor on Dartmoor being a prime example. It must be noted that the mineralization of the St. Just district can not be explained in these terms. Rankin & Alderton (1983) noted a correlation between high abundance of type 1 inclusions and kaolinisation on the St. Austell outcrop. This parallel has not been noticed on any other outcrop and it was suggested that this was due to ignoring salinity variations within the type 1 category. The absolute abundances of type 1 inclusions, figure 5.16, are much less than type 2 but in general show the same overall pattern on Carnmenellis and Bodmin. However, this is not the case for Dartmoor or Land's End which have no similarity between any of the fluid inclusion patterns. This and the fact that the abundance of type 3 inclusions are comparatively negligible on both Carnmenellis and Bodmin suggest that the fluid histories for the two extreme outcrops were very different from those of Bodmin and Carnmenellis. It is therefore difficult to propose a general correlation between any fluid inclusion type, except type 2a, and mineralization or kaolinization. Type 4 fluid inclusions (figure 5.17) show a similar tendency to type 1 inclusions in as much as it may be possible to correlate their abundances with some form of hydrothermal activity on some outcrops but not on others. This could well be due to the different fluid histories of the plutons, but also may be due to poor subdivision of inclusion type. 142 5.5.2 Comparison of Fluid Flow and Mineralization Patterns It was indicated in the last section that over the outcrops there is a close correlation between the abundance of type 2a fluid inclusions, non-magmatic mineralization and fluid flow. It might therefore be expected that there will be a good correlation between the fluid flow patterns and mineralization in all regions. Since the fluid histories of the various outcrops are probably quite different, it could be the case that the hydrothermal activity responsible for a given type of mineralization was present at different times and with different amounts of cover for different areas. Therefore it would seem unreasonable to make detailed comparisons between fluid flow patterns and mineralization patterns, and so only regional correlations between fluid flow patterns for large and small depths of cover, and high and low temperature mineralization are sought. Tin is chosen as the high temperature mineral though it must be remembered that some of it may be magmatic in origin, and lead is representative of the lower temperature mineralization though this too might have high- and low- temperature forms according to Alderton (1978). Figure 5.18 reproduces the sketch map from Dines showing the distribution of tin, and lead-zinc deposits. The map shows clear evidence of the mineral zonation which have analogues in the fluid flow maps. Tin is located close to or on the outcrops, whereas the lead is generally found at greater distances from the outcrops. This is corroborated by the fact that over 90% of recorded output of tin came from within 1500 m of a granite contact, while the area of peak production for lead was between 1000 and 4000 m away from the contact. 143 The tin deposits on and around the Carnmenellis outcrop show a remarkable correlation with the fluid flow pattern for 3 km of cover (figure 5.19). The concentration along the north-western edge between Godolphin and C a m Brae with the extension along to St. Agnes could easily be explained by the strong flow of hot fluids in this region caused by the asymmetric shape of the batholith, that is because the Carnmenellis pluton outcrops to the South-east of the axis of the batholith as a whole. This good correlation includes the Camborne, Redruth and St. Day district which output more than 50% of the tin for the entire region (Dines, 1956). The majority of St. Austell tin deposits tend to fall south of the central location of the upward fluid flow predicted for large depths of cover, though there are deposits on all sides of the outcrop. The tin deposits of Bodmin all fall within the region of upward flowing fluid predicted for 3 km of cover. They generally lie towards the south of the outcrop which is also along the axis of the batholith. Again there is a coincidence of tin deposits, upward flowing fluid for 3 km of cover and the axis of the batholith in the Kit Hill distict. The fluid flows through Dartmoor and Land' s End are less intense than through the other plutons despite an assumed homogeneous permeability and, just as with the type 2a fluid inclusions, there is no correlation with mineralization patterns. The model has therefore failed to predict the heavily mineralized region of St. Just on the Land's End outcrop which has produced over 12% of the the tin for the whole region. Several reasons can be suggested for this deficiency, however the location of the mineralization points to a misrepresentation of the shape of the batholith. The mineralization is found on the flank of the pluton which in cross-section appears from the gravity model to be roughly symmetrical. The results from two- 144 Figure 5.18 Distribution of tin and lead-zinc deposits, and mining districts. (After Dines 1956.) 145 Figure 5-19 Comparison of fluid flow pattern with 3 km of additional cover and the distribution of tin centres. Regions of upward moving fluid only are contoured. 146 Figure 5.20 Comparison of fluid flow patterns for no additional cover and the locations of lead mines. Regions of upward moving fluid only are contoured. 147 dimensional modelling suggest that fluids will only flow up the flanks if the cover is small, when temperatures will be too low, or for large depths of cover if the pluton is broader. However, if the shape of the pluton as defined by the gravity models is incorrect and the northern flank of the Land's End pluton dips less steeply, the asymmetry would give a better correlation. For smaller depths of cover the peaks in the upward flow of fluid migrate outwards. For clarity the fluid flow pattern with no cover has been overlaid by a map of the approximate locations of lead mines and is shown in figure 5.20. The match is quite good. Especially noteworthy is the region around the Camel estuary which is underlain at a (present) depth of 3km by a small cusp in the roof of the granite, which causes the fluid to rise in this region. There is again good agreement between intensity of flow and those regions which have produced the largest quantities of lead: i.e. Liskeard (25.1%), Callington and Tavistock (38.3%), and St. Agnes (25.4%). 5.5.3 Comparison of Fluid Flow and Kaolinisation Patterns Kaolin occurs to some extent on all the major outcrops of the Cornubian batholith, though some areas are more heavily kaolinised than others. The St. Austell granite is the most altered granite with much of its central and western parts kaolinised. As far as present production is concerned the next most important region lies in the south west of the Dartmoor outcrop, and there are two producing pits on the Bodmin granite and one on the Land's End granite. It has recently been suggested by Durrance and Bristow (1986) that kaolinisation is caused by downward percolating fluids. If this is so, then according to the predictions of the fluid flow models the kaolinisation must have 148 Figure 5.21 Comparison of regions of downward flowing fluid predicted by Models 3a & 3b with no additional cover and the locations of kaolinised granite. 149 occurred when there was little or no cover above the present day outcrops. Indeed there is some correlation between the intensity of downward flow for no cover and the areas of kaolinisation (figure 5.21), the most intense downward flows occuring over the St. Austell outcrop which is the most productive area. Intense flow also occurs on the Bodmin granite where large areas of kaolinisation are found, though present production is less than on the Dartmoor granite. The concentration of kaolinisation is on the west of the outcrop as is the downward flow. The largest down- flows on the Dartmoor outcrop are in the south-west as is the kaolinisation although the two areas do not overlap entirely. On the other outcrops the kaolinisation generally falls within regions of most intense downwelling. Durrance and Bristow (loc.cit.) also suggest that an earlier phase of the kaolinisation occurred at very early times involving high temperature fluids (about 260°). Indeed a spatial correlation between kaolin and greisen bordered veins has been noted (Bristow, 1977). However, their proposal that these hot fluids were also percolating downwards would seem highly unlikely in view of the models presented here. Downward flowing fluids tend to decrease temperature gradients and hence only in the presence of very substantial cover would any fluids flowing down have reached the high temperatures required. But for large depths of cover there are no regions of down flow in the areas of observed kaolinisation. However, it can be noted that the kaolinised areas are situated where initially hot fluids were rising and later cooler fluids descending. 5.5.4 Comparison of Fluid Flow and Geochemical Patterns A regional geochemical reconnaissance stream sediment survey of South-west England, which formed part of a larger investigation of multi-element stream sediment data 150 covering all of England and Wales, was carried out in the early seventies by the Applied Geochemistry Research Group of Imperial College. The results are presented in the regional geochemical atlas (Wolfson, 1978) and the sampling techniques and data analysis are described by Dunlop (1973) . The survey, backed up by more detailed soil and stream sediment sampling, has provided data that delineate the regional anomaly patterns for many elements. Interpretation of these patterns is often complicated. A stream sediment is composed of the mechanically derived products of rock weathering, which have been washed into the drainage channel, and soluble products of weathering which have been leached from the bedrock and precipitated in the drainage channel. Thus the sample may be influenced by bedrock geology, overburden, mineralization, secondary environment and contamination. However, it is often possible to determine the cause of the anomaly by means of elimination. It is natural to expect that anomalous values of certain elements will occur in the vicinity of mineralization and that mining and related processes such as dumping and smelting will contaminate the immediate environment. It is also possible that enhancement will occur where fluids have flowed but have not been of sufficient concentration or for long enough durations to have produced mineralization. If this is so then it may be reasonable to compare regional geochemical patterns with the regional fluid flow predictions. It must be added that any enhancement above background values would have occurred only if the fluids were transporting those elements and the conditions were right for deposition. The use of anomalous values for comparison eliminates the regional variations due to bedrock geology and the method for determining the threshold values for the various rock units is given by Dunlop (loc. cit.). The maps presented are based on the threshold values determined by him, save for arsenic, the 151 anomalous pattern for which is taken from data presented by Aguilar Ravello (1974). The regional anomalous patterns for five elements are now presented: tin, copper, lead, zinc and arsenic. These elements have been chosen since their enrichment in certain areas suggest that at some stage they were mobile and transported by Circulating' fluids. Also it was observed by Nicol et al. (1971) that the distribution of these elements was in broad accord with the mineral zoning. i) Tin The distribution of anomalous levels of tin is shown in figure 5.22. This predominantly reflects contamination from areas of tin mineralization. However, areas of enhanced tin levels extend far beyond the margins of the known tin districts. Some of these have been interpreted as being due to detrital dispersion patterns formed during the exposure of the granites during the Permian (Dunlop, loc. cit.). This is especially true of the area between the Carnmenellis and St. Austell tin disticts. The highest tin values are found in the region around the Carnmenellis outcrop. Naturally there is a similarity between this pattern and the distribution of emanative centres and thus the fluid flow pattern for 3 km depths of cover as noted in section 5.5.2. ii) Copper The anomalous copper distribution is also influenced by the contamination from mineralized districts (figure 5.23). Of interest is the area around the Liskeard mining district on the south-east of Bodmin where anomalous levels appear to stretch from this region down to the coast. This is due to drainage along the River Seaton from the former copper mining area. Again of interest is the location of the peak 152 Figure 5.22 The anomalous regional geochemical distribution of tin. (After Dunlop, 1973) 153 o o Figure 5.23 The anomalous regional geochemical distribution of copper. (After Dunlop, 1973) 154 Figure 5.24 The anomalous regional geochemical distribution of lead. (After Dunlop, 1973) 155 Figure 5.25 The anomalous regional geochemical distribution of zinc. (After Dunlop, 1973) 156 o Figure 5.26 The anomalous regional geochemical distribution of arsenic. (After Aguilar Ravello, 1974) 157 anomaly in this area which has moved slightly east and south of the peak anomaly for tin in the same district. Around the Carnmenellis outcrop the copper anomaly does not stretch so far south as compared with that of tin but has moved further up the north coast towards the St. Agnes district. Similar features can be seen in the fluid flow patterns as the depth of cover is decreased. iii) Lead The anomalous distribution of lead is shown in figure 5.24. In the Liskeard district the peak anomaly has moved further east and south and the anomalous area around Carnmenellis is smaller and further north relative to the tin and copper anomalies. However the overall shape of the anomalous region is very similar to that for copper, except in the Camel estuary district to the north-west of Bodmin where fluids are predicted to rise for 2 km or less of cover only. iv) Zinc The regional zinc anomaly, shown in figure 5.25, is not significantly different to that of lead. Compared with the lead anomaly, however, the peak in the Liskeard district has moved further south. v) Arsenic The anomalous region of arsenic, shown in figure 5.26, forms an aureole around the granite outcrops, Nicol et al. (1971). They observed that this geochemical aureole was broader than the metamorphic aureole and also suggested that zinc shows some signs of forming such an aureole. The anomalous arsenic pattern is of great interest since it shows many similarities to the fluid flow patterns predicted for small depths of cover. The fluid flow 158 pattern for no cover shows upward flowing fluid forming aureoles around the granite outcrops. Noteworthy is the lack of anomalous arsenic in those areas of country rock where the fluid is predicted to be flowing downwards. These areas are: the region to the west of the Camel estuary, the Lizard district, the area to the south-west of St. Austell, the area to the north of the Bodmin, Kit Hill and Dartmoor outcrops, the coastal areas to the east and south of Dartmoor and, apart from the Seaton river valley, the area between Plymouth and St. Austell. It is also interesting to note that these areas have remained areas of downflow for all depths of cover and are not anomalous regions for any of the elements discussed here. Since the anomalous patterns reflect the mineralization in the region to a large extent, it is not surprising that zonation of the anomalies is seen. In some locations it is possible to relate this anomaly zonation with changes in the fluid flow pattern. The Liskeard mining district is one such example where the anomalies for the elements associated with the lower temperature mineralization are situated further east and south. The peak fluid flow in this region has also moved east and south as the cover has been eroded. Another good example is in the St. Agnes mining district where peak anomalies and fluid flows have moved east along the coast with decreasing temperature and cover. The biggest anomalies consistently occur around the Carnmenellis outcrop and in the Liskeard and Callington- Tavistock mining disticts? that is, they appear to be restricted to those areas where the surface geology is either the earliest or latest Devonian. Since large parts of these areas have values below threshold, the anomalies can not be simply reflecting the geology, but they might indicate the susceptiblilty of the rocks to mineralization. Indeed, the youngest Devonian rocks to the North of the 1 5 9 Culm synclinorium are also mineralized though this mineralization has no association with the batholith. By considering the paths of the circulating fluids, which can be deduced from the vertical flow patterns, it could also be suggested that these Devonian rocks were the source of the minerals. In general, the Carboniferous rocks show very few geochemically anomalous areas. However, this is not surprising since the fluid flow patterns predict that they have always been in regions of downward flowing fluids. 5.6 Conclusions Models including convective transport of heat have shown how the fluid flow patterns are determined by the depth and shape of a granite in the absence of variations in lateral permeability. There is little evidence to suggest that there is a significant contribution to the regional heat flow by convective means, though there would appear to be some regions where localised fluid movement is influencing the subsurface temperatures. It is possible to explain the regional features of the ore field by considering that mineralization was emplaced by convective systems which existed at different times and with different depths of cover. High temperature mineralization was deposited at depths of several kilometres beneath the surface when fluids were rising along the axis of the batholith. Low temperature mineralization was emplaced at shallow depths when the fluids were rising in regions forming aureoles around the plutons. The numerical modelling is able to predict not only the regions where fluids were ascending for different depths of cover, but also trends in the intensity of mineralization (and kaolinisation) as corroborated by 1 6 0 geochemical element and fluid inclusion distributions. 1 6 1 Chapter Six Conclusions Modelling of the conductive heat flow regime in the South west of England has shown that the anomalously high heat flows which have been measured can be explained by assuming that the Cornubian batholith has a heat production which is not only high at the surface but remains constant with depth, a departure from earlier theories. However, there is increasing evidence, especially from geochemistry, to support this claim. In general, the pattern of heat flows can be explained in terms of a homogeneous batholith, which has a constant depth along its entire length, set in a laterally homogeneous host rock. Recent gravity modelling indicates that previous models, which predicted a rise in the base of the granite to the west, were probably wrong due to incorrect regional gradient removal and heat flow modelling supports this. The good match between modelled and measured heat flows for granite sites indicates the reliability of not only the model but also the measured values. Indeed, the measurement of the heat flow from two more boreholes at the HDR site during the course of this project (making five in total) shows that the granite values are consistent to within one or two percent and that variations in heat flows observed across plutons are mostly real and not just due to errors in measurement. The large mismatches between some of the modelled and measured values for country rock sites reflect the problems involved in measuring heat flows in shallow boreholes in sedimentary rocks. However, the 1 6 2 mismatches fall within the error limits of the measurements and it is noticeable that those boreholes deeper than about 300m show smaller mismatches. Improvements in the match can not be achieved by introducing realistic inhomogeneities into the model parameters for the country rock. An improvement is achieved by allowing convective heat transport, but only at a limited number of sites. The pattern of heat flows is determined by the space form of the thermal conductivity and heat production contrasts; that is, by the three-dimensional shape of the granite. The high heat production of the granite causes temperatures to be higher in the granite than outside causing egression of heat and a lowering of surface heat flow towards the sides of the granite. In some locations, the lower thermal conductivity of the country rock produces in it higher vertical temperature gradients relative to adjacent granite leading to refraction of heat into the edges of the granite. The maximum modelled temperatures at a particular depth are not necessarily to be found beneath the highest heat flows. The highest temperatures in South-west England at a depth of 2000m are predicted to be about 20 km to the north of the HDR site and at over 100°C they are about 20°C above the measured temperature for this depth at the HDR site. So the present HDR site is not ideally located as far as temperatures are concerned. At a depth of 6000m the temperature predictions are only accurate to within about 10°C. The highest temperatures are in excess of 240°C and are found towards the axis of the batholith, beneath areas covered by a thickness of insulating country rock. These temperatures at this depth are higher than the value regarded as economically viable for the HDR method. In retrospect, it would seem to have been more sensible to have carried out such a modelling exercise before the 1 6 3 majority of the heat flow boreholes had been drilled. It would then have been possible to identify sites which would have been useful for resolving ambiguities. As it is, many of the contract boreholes are not in locations which help to eliminate uncertainties. It is suggested, therefore, that before any future heat flow exploration of granites for geothermal energy is embarked upon, three- dimensional modelling is carried out and the models are continually updated as results become available. It is also suggested that all heat flow boreholes drilled in sedimentary rocks be at least 300m in depth. The convective flow of fluid caused by the radiothermal and thermal conductivity contrast between the granite and its surrounds is greatly affected by the depth of the contrast as well as its three-dimensional shape. So, if the amount of cover above a granite is changed by erosion or sedimentation the fluid flow patterns will also change. If the sides of the granite slope outwards the regions of upwelling fluid will tend to move outwards during erosion at a rate that depends on the slope of the contact. If erosion takes place the granite will cool and the maximum temperature of the fluid flowing through a fixed level within the granite will decrease with erosion. On a regional scale there exists a correlation between fluid flow patterns and mineralization, fluid inclusion abundances, geochemical anomalies and kaolinisation. The results suggest that high temperature mineralization such as tin, when not magmatic in origin, was emplaced by the action of convective systems which were established soon after the consolidation of the granite, whilst there were several kilometres of cover rock above today*s outcrops. These deposits lie close to the edges of the outcrops which face the axis of the batholith due to the fact that fluids will tend to rise above the hottest regions when such a depth of cover exists. The intensity of upward flow varies 1 6 4 with temperature at depth, so that the most intense flows are found on the northern side of Carnmenellis and the least intense on Dartmoor and Land's End. The intensity- patterns match well with the abundance patterns of type 2a fluid inclusions, though the fluid inclusion data are more detailed. The low temperature lead-zinc deposits were formed by the action of convective systems established when the cover rock above the granite had been almost completely eroded. These deposits are found within aureoles that surround the pluton outcrops, formed as a result of the regions of upwelling fluid moving away from the outcrops as the plutons became exposed. The variation in intensity of flow within this pattern seems to depend on the shape of the granite. In both cases there is good correlation between those areas of intense upward flow of fluids and the location of the most economically-important deposits. The main exception to this is the St. Just mining district on Land's End. Further work is necessary to determine the cause of the discrepency between the modelled fluid flows and those suggested by the mineralization and fluid inclusion patterns, but it is thought likely that the problem is caused by a poor approximation to the shape of the granite in this region in the gravity model. The predicted fluid flow patterns suggest that the most recent stages of kaolinisation were brought about by the downward percolation of fluids as also indicated by workers in other fields. The intensity of downward flow points to those areas where kaolinisation is most widespread or intense. The earlier stages of kaolinisation, if caused by fluids with high temperatures, could not be the result of downward flowing fluids. 1 6 5 Of course there are many problems which have been avoided by assuming that there existed at some time, or times, a homogeneously and laterally isotropic permeable medium. Laterally anisotropic permeabilities will only tend to highlight those parts of the fluid flow patterns caused by contrasts which lie perpendicular to the principal direction of anisotropy. Inhomogeneities may well alter the patterns more dramatically. The influence of these and other factors, such as the depth of the permeable layer, on the fluid flow patterns ought to be examined in greater detail. In terms of using this type of modelling as a regional mineral exploration tool for similar environments, the results are very encouraging. The intensity of fluid flows predicted by the models, despite their simplicity, is able to locate the regions with the most economic reserves. Even though the South-west of England has far better constraints for modelling purposes than most regions, much of the information is redundant and the predictions could have been made with less a priori information. Nevertheless, the models highlight the need for a good approximation to the shape of the batholith from gravity data which is not always available. Further work should be directed towards confirming that modelling is a reliable tool for mineral exploration. It would be appropriate to attempt to predict the mineralized regions of a similar province having access only to the shape and size of the thermal contrasts. It would also be sensible to establish the relationship between fluid flow and fluid inclusion type and abundances by carrying out a joint study of granite outcrops from various regions. The presentation of fluid inclusion abundances in absolute terms rather than as percentages of total abundances has shown that this technique is capable of locating 1 6 6 mineralization on outcrops in a more efficient manner than originally proposed from the study of percentage abundances. A combination of fluid inclusion studies, geochemical stream sediment analysis, and fluid flow modelling would seem to be ideal for regional exploration of minerals deposited by convective systems associated with granite batholiths. In conclusion it can be said that even though the models presented have been very simple and crude approximations to the physical reality, they are capable of improving our understanding of some of the physical processes at work in the upper crust. They highlight the importance of the third dimension which is so often ignored in modelling exercises and they help to explain the regional distribution of the mineralization associated with the granite batholith of South-west England. 1 6 7 REFERENCES AHMED,S.N. 1977. The Geochemical Distribution and Source of Copper in the Metalliferous Mining Region of South-west England. Min. Dep.,12, pp 1-21. ALDERTON,D.H.M. 1978. Fluid Inclusion Data for Lead-Zinc Ores from South-west England. Instn. 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Exploration and Interpretation of the SW England Geothermal Anomaly pp 456-465 in: Proc. 2nd International 1 7 8 Seminar on the results of the E.C. Geothermal Energy Research, ed. Strub,A.S. & Ungemach,P. Dordrecht, Holland, D- Reidel pub- co. WILLIS-RICHARDS,J. 1986. Three Dimensional Modelling from Gravity of the Carnmenellis Granite Pluton, West Cornwall. Geophys. J. R. astr. Soc.,85, 1 pp 251 (abstract only). WITHERS,T.M. 1984. Radioactive Logging Techniques and Interpretation of Two Gamma Ray Logs from the Cornwall Geothermal Energy Project. M.Sc. Thesis, Imperial College. ZIENKIEWICZ,O.C. 1977. The Finite Element Method. McGraw- Hill. ZIENKIEWICZ,0.C . and MORGAN,K. 1983. Finite Elements and Approximations. Wiley. 1 7 9 Appendix I Temperature-Dependence of the Thermal Conducitivity of Granite from the Rosemanowes Deep Borehole Data. AI.l Introduction If after correction for heat production, palaeoclimate, topography, erosion and/or uplift the temperature gradient is multiplied by the measured conductivity at various depths within a borehole to give the heat flow at the surface, the value obtained will only be constant when there is purely vertical conductive heat flow and if the thermal conductivities have been corrected for their temperature-dependence. So it is possible, at least in theory, to find a temperature-dependence that will give a consistent suface heat flow with depth assuming of course that all the other corrections have been applied correctly and that heat is being conducted only in the vertical direction. Although this applies to data from any borehole, it is only in deep boreholes that the conductivity variations with temperature will produce sufficiently large deviations from a constant heat flow which can be used to establish the nature of this dependence. AI.2 The Data Set The drilling of two deep boreholes (RH11 and RH12) at the HDR site in the Rosemanowes Quarry on the Carnmenellis outcrop provided the necessary information to test this method. The data consisted of two temperature logs, one for each borehole, which were run by the Camborne School of Mines Geothermal Energy Project staff approximately four months after the cessation of drilling (CSM, 1984). Log T11010100 (RH11) consists of 3970 temperature measurements down to a total vertical depth of 2036 m with a measured 1 8 0 bottom hole temperature of 78.0257°C. Log T12013100 (RH12) consists of 5438 temperature measurements to a total vertical depth of 2054 m with a measured bottom hole temperature of 78.9122°C. Chippings of rock flushed out with the drilling fluid were gathered as samples from different depth ranges down the holes. The thermal conductivities of these samples were measured on the divided bar apparatus (Francis, 1980) using the pill-box technique (Sass et al., 1971). No correction for porosity was applied since the granite is non-porous. In general only one measurement per depth range (~ 100m) was carried out. Table AI.l gives the average values of the conductivities measured along with data from the same site available from previous measurements (Francis, 1980). The measurements were carried out at 24°C. Gamma ray spectral analysis was carried out on chip and core samples available for the different depths, and data were available from spectral logs run in the borehole (Withers, 1984). From these measurements heat production values were calculated. A summary is presented in table AI.2. Even though direct comparison at specific depths can not be made, it can be seen from the table that heat production values from chip samples are consistently lower than from the other two methods which show close agreement. It is also evident that the reason for this is the lower concentration of uranium (and thorium) in the chip samples which would have suffered uranium loss while being flushed to the surface by the drilling fluids. The spectral logs and core samples show that the heat production remains remarkably constant over the depth of the boreholes (Withers, loc.cit.) with an average value of about 5 microW 1 8 1 Table AI.1 Conductivity Data from Boreholes at the Rosemanowes Site. Borehole No. of Arithmetic Mean Standard Samples of Conductivity Deviation (W nT1^ 1) (W m-1K-1 RH11 29 3.26 0.19 RH12 48 3.16 0.21 RH15 149 2.97 0.37 Hole D* 10 3.09 0.21 Hole E* 44 3.30 0.20 Data taken from Francis (1980) 1 8 2 Table AI.2 Comparison of Radio Element and Heat Production Data between Core Samples, Chip Samples and Spectral Logs for the two boreholes RH11 and RH12. Data Type Depth U Th K Heat Production (m) (ppm) (ppm) (%) (10“6 W m“3) RH12 CO Core 2111 15.2 8.4 3.7 • Logs 2110 16.5 8.5 6.25 5.3 Chips 2027- 6.5 7.9 5.1 2.7 2109 RH11 Core 1689 15.1 7.2 3.5 4.7 Logs 1690 15.5 8.0 4.8 5.0 Chips 1648- 11.3 5.8 4.3 3.7 1690 RH11 Core 2183 13.4 8.9 4.0 4.4 Logs 2160 12.5 10.5 5.3 4.4 Chips 1968- 8.2 6.3 4.4 2.9 2110 1 8 3 AI.3 Heat Flow Calculation In calculating the heat flow from borehole data it is assumed that the heat flow is purely conductive and in a vertical direction. So at any depth, z, down the borehole q(z) = K(z,T) dT AI.l dz where q is the vertical heat flow and K is the thermal conductivity. q is independent of depth (and equal to its value at the surface, qQ) unless heat is being generated in the rocks through which the borehole has been drilled, in which case the surface heat flow can be determined from any point in the borehole by qQ = k (z ,t ) dz AI.2 where A(z) is the heat production variation with depth. Hence a plot of qQ against z should yield a straight line parallel to the z axis. However a problem arises in practice in that conductivities and temperature have not been measured continuously down the borehole. This is normally solved, to a limited extent, by adopting the Bullard approach (Bullard, 1939) wherein equation AI.2 is integrated to give ds) dz AI.3 1 8 4 and then discretised to the form <*o = T + AI.4 K: L I v i > where Z = and average conductivities are assigned to each of the i intervals. Rearranging the terms gives T« T, AI.5 where T* AI.6 Then a plot of temperature corrected for heat production (and palaeoclimate, topography, erosion and/or uplift) against thermal resistance will, in the ideal case, give a straight line with a slope equal to the surface heat flow and an intercept equal to the mean surface temperature. This is known as the Bullard plot (Bullard, 1939). Taking the gradient of this slope will again yield a set of values which are estimates of the surface heat flow from all depths and is known as the modified Bullard plot. Such values can be seen in figure AI. 1 which compares the results for raw data and data which have been corrected for heat production and palaeoclimate for RH12. These graphs show large variations in the predicticted heat flow which are caused by fluid movements through fractures in the rocks. These are most prominent near the top and bottom of the holes where the casing was either absent or not cemented. All subsequent diagrams and calculations deal 1 8 5 DEPTH, Km DEPTH, Km O o tO ? O >40 OO i?0 t'O too DO rdcin ny () sn tmeaue cretd for corrected temperatures (d) Using temperatures only, perturbations Using palaeoclimatic (c) only, production raw Using (a) depths, (b) temperaturecorrectedfortemperaturesdata, Using all heat at calculations flow heat the correctedforandheatpalaeoclimate production. both et t ilsrt, h efcs f h different the of in used wasvalues, measured the of mean conductivity, the effects the illustrate, to depth orcin apid o H2 A ige au o thermal of value single A RH12. to applied corrections Figure AI.l The modified Bullard plot offlowsBullardplotheat The modified versus Figure AI.l L m m mW . W FLO T A E H T A E H flow, flow, n" HE FOW. W m' mW . W FLO T EA H " in w m 6 8 1 O O O ? 10 140 130 i?0 nO OO DO HE AT FLO W . mW m m mW . W FLO AT HE * only with those parts of the boreholes with cemented casing. Also in this figure an average thermal conductivity has been used for all depths because of the significant problem of an insufficient set of data: since there is no continuous record of conductivities there is no way of determining variations in conductivity on a scale smaller than the depth range nor is it sensible to say that the measured value represents the average conductivity for that range. Thus enormous errors will occur if the measured conductivity is not representative of the depth range to which it is allocated. This can quite clearly be seen if the measured conductivities are compared with the theoretical values: that is, those which will produce a constant surface heat flow with depth. Figure AI.2 shows this comparison using a surface heat flow of 115 mW m”2 to calculate the theoretical values. Even though similarities can be seen between the measured and theoretical values, the measured conductivities have a much larger range and would thus yield heat flows which vary greatly with depth as is evident in figure AI.3. It should be noted that the general decrease in theoretical values with depth is due to the temperature dependence of the thermal conductivities being ignored in the calculation. This problem is solved by assuming that the granite is homogeneous on a large scale and assigning the mean of the measured thermal conductivities to the complete depth range. Figure AI.3 also shows the predicted surface heat flows when this approach is made. It is clear that a more uniform result is obtained. Ignoring the general trend due to the omission of the temperature-dependence term, variations are fairly small and could easily be due to actual variations in conductivity or might be due to other 1 8 7 CONDUCTIVITY. W m*' K'' CONDUCTIVITY. W m" K' 2 8 3 0 3 2 3 4 3 6 3 8 Figure AI.2 Comparison of measured conductivities (dashed lines) with the inferred "theoretical" conductivities (solid lines), i.e. the conductivites required to produce a constant heat flow of 115 mWm”2 throughout the hole when combined with the corrected thermal gradient at each point. Since temperatures were smoothed using a 100-point moving average, the conductivities also have been similarly smoothed, (a) RH11 and (b) Rhl2. 1 8 8 HEAT FLOW. mW mf1 105 110 115 120 125 130 135 140 HEAT FLOW. mW m" 9 0 100 110 120 130 140 Figure AI.3 Heat flow variation with depth (a) in RH11; (b) in RH12, using fully corrected temperatures: (i) using the measured conductivity in each depth range (dahed line) (ii) using the mean of the measured conductivities to calculate the heat flow at all depths (solid line). 1 8 9 effects such as fluid movement. It is now possible to determine a variation of thermal conductivity with temperature. Here it is assumed that the variation follows the relation K(T) = d + c/T AI.7 where T is the absolute temperature and c and d are constants. A combination of c and d can be found which gives the smallest standard deviation for the calculated surface heat flows and which also gives the mean of the measured conductivities at 24°C. The results show that RH11 and RH12 with mean conductivities of 3.26 and 3.16 W m_1K_1 have c as 400 and 320 W m respectively with corresponding values for d of 1.92 and 2.09 W m_1K_1. Using these parameters the heat flows come out to be 117.7 and 116.3 mW m~2 and the final plots are shown in figure AI.4. These parameters for the temperature dependence fall towards the lower end of values determined by Birch and Clarke (1940) for granites, 180 < c < 600, and close to the values detemined from experimental work on Cornish granites, c = 200, by Sartori (1983) and Mexican granites by Sibbet et al. (1977). These values yield standard deviations for the surface heat flow of 2.5 and 2.0 mW m”2 for RH11 and RH12 respectively which are very low but which do not represent the total error in measurement of the heat flow (Sams and Thomas-Betts, 1987). 1 9 0 i i DEPTH, their temperature-dependence.their the mean of the measured conductivities taking intoaccountconductivitiestakingof the measured mean iueA. ia oiidBladpos (a) forplots, RH11,FinalBullard Figure modified AI.4 b fr H2 uig h fly orce tmeaue and temperatures corrected fully the using RH12, for (b) 1 10 3 n 10 130 120 no 130 120 110 ET LW m m7 ET LW m rrf7 mW FLOW, HEAT m'7 mW FLOW, HEAT 1 9 1