129

Geologische Rundschau 78/1 I 129-154 I Stuttgart 1989

Modeling subsurface flow in sedimentary basins By CRAIG M. BETHKE, Urbana*)

With 16 figures

Zusammenfassung raines qui se manifestent dans les bassins sedimentaires, et qui resultent du relief topographique, de la convection, de la Grundwasserbewegungen in sedimentaren Becken, die compaction des sediments, de la decharge due l'erosio? et von dem topographischen Relief, konvektionsbedingtem a de la combinaison de ces divers facteurs. Dans ces modeles, Auftrieb, Sedimentkompaktion, isostatischen Ausgleichsbe­ on peut prendre en con~ideration les effets d~s tr:'nsferts d,e wegungen in Folge von Erosion und v~n ~mbinati~n~n chaleur et de matieres dlssoutes lors de la migration du pe­ dieser Kriifte gesteuert werden, konnen mit Hilfe quanutattv trole et ceux de I'interaction chimique de I'eau avec les modellierender Techniken beschrieben werden. In diesen roches. Toutefois la precision des previsions que I'on peut en Modellen kann man die Auswirkungen des Transports von deduire est limitee par la difficulte d'estimer I'echelle regia­ Warme und gel osten Stoffen, Petroleum-Migration und die a nale les proprietes hydrologiques des sediments, de reconsti­ chemische Interaktion zwischen Wasser und dem grund­ tuer les conditions anciennes, et de connaitre de quelle wasserleitenden Gestein beriicksichtigen. maniere les processus physiques et chimiques interferent Die Genauigkeit der Modell-Voraussagen ist allerdings a I'echelle des temps geologiques. La modelisation des bassins begrenzt wegen der Schwierigkeit, hydrologische Ei­ progressera dans la mesure ou la recherche hydrogeologique genschaften von Sedimenten in einem regionalen Rahmen ser mieux integree acelles d'autres disciplines telles que la se­ vorauszusagen, dem Schatzen vergangener Bedingungen und dimentologie, la mecanique des roches et la geochimie. dem Problem der Abschatzung von Wechselwirkungen physikalischer und chemischer Prozesse in geologischen Zeitriiumen. Fortschritte fUr das Modellieren von Becken KpaTKoe COAeplKaHHe werden mit der Integration hydrologischer Forschungsan­ C nOMO~blO MOAenH peKOHCTpYHpOBanH MHrpa~HIO strengungen in benachbarte Fachebiete wie Sedimentologie, rpYHTOBblX BOA B OCaj\O'lHbIX 6acceHHax, npHHHMaSi BO Gesteinsmechanik und Geochemie zunehmen. BHHMaHHe TonorpacpHIO pen be cpa , B03MOlKHOCTH KOHBeK­ ~HH, nnOTHoCTb ceAHMeHTHblX OTJlO>KeHHH, H30CTaTH'IeC­ Abstract KHe ABHlKeHHSI B pe3ynbTaTe 3P03HH, a TaKlKe KOM6HHa­ ~HIO Bcex 3THX cpaKTopOB. TIPH pa3paGoTKe TaKHX MOAe­ Groundwater flows that arise in sedimentary basins from neH cneAyeT Y'lHTbIBaTb BnHSlHHe BblCOKOH TeMnepayY­ the effects of topographic relief, buoyant convection, sedi­ Pbl, nepeHoca paCTBopeHHblX Be~eCTB, MHrpa~HIO HecpyH ment compaction, erosional unloading, and combinations H XHMH'IeCKOe B3aHMOAeHCTBHe MelKAY rpYHTOBblMH of these driving forces can be described using quantitative BOAaMH H oKpy>KalO~HMH HX nopOAaMH. modeling techniques. Models can be constructed to consider CaMO COGOH pa3YMeeTCSI, TO'lHOCTb nporH030B no the effects of heat and solute transport, petroleum migra­ TaKHM MOAenSiM orpaHH'IeHa H3-3a TPYAHOCTeH npeACKa- tion, and the chemical interaction of water and rocks. The 3aHHSI perHOHanbHbIX rHApOJlOrH'IeCKHX CBOHCTB ceAH­ accuracy of model predictions, however, is limited by the MeHTOB H H3-3a TonbKO npH6nH3HTenbHOH o~eHKH, KaK difficulty of predicting hydrologic properties of sediments npeAweBCTBYIO~HX B3aHMOOTHoweHHH, TaK H B3aHMO­ on regional dimensions, estimating past conditions such as AeHCTBHSI cpH3H'IeCKHX H XHMH'IeCKHX npo~eCCOB B Te'le­ topographic relief, and knowledge of how physical and che­ HHe reonoru'leCKOrO BpeMeHH. MOAenHpOBaHHe 6acceH­ mical processes interact over gelogic time scales. Progress in HOB oGe~aeT 3Ha'lHTenbHblH ycnex TonbKO npH KOM­ basin modeling will accelerate as hydrologic research efforts nneKCHblX HccneAoBaHHSlX no rHAponorHH, ceAHMeHTono­ are better integrated with those of other specialities such as rHH, MexaHHKe nopOA H reOXHMHH. sedimentology, rock mechanics, and geochemistry.

Resume 1. Introduction n est possible, par l'utilisation de techniques quantitatives In recent years the interest in describing quantita­ de modelisation, de decrire les mouvements des eaux souter- tively the subsurface movement of fluids in sedime~lt­ *) Author's address: Dr. C. M. BETHKE, Department of ary basins and the effects of such movements has In­ Geology, 1301 West Green Street, University of Illinois, creased rapidly. Fluids in basins, whether groundwa­ Urbana, Illinois 61801, USA. ters, hydrocarbons, or gases, are mobile over geologic no CRAIG M. BETHKE time periods. Moving fluids create and localize econo­ along the water table. Given adequate rainfall, the mic resources in sedimentary basins, including petro­ water table forms a subdued replica of the land sur­ leum and gas reservoirs and metallic ores. The increas­ face so that to a first approximation topography de­ ing interest in quantifying subsurface flow stems scribes the drive for flow. DARTON (1909), in his stu­ from escalating costs of locating new resources as well dy of the Dakota aquifer system in North America, as the need to isolate radioactive and chemical wastes was among the first to recognize the role of topogra­ from the biosphere for long periods of time. phy in causing groundwater flow on regional scales. There is a strong need in petroleum geology to pre­ The isotopic compositions of sedimentary brines dict the effects of hydrocarbon migration. Oil reser­ (e. g., CLAYTON et al. 1966) provide chemical eviden­ voirs sometimes are found more than 150 kilometres ce that meteoric water circulates deeply in many from source rocks; others are separated vertically basins. from their sources by kilometers of overpressured shale. The distribution of mature source beds is rou­ 2.1 Mathematical model tinely determined during basin exploration, but the The subsurface flow field can be predicted accord­ distance and even direction that oil migrates after be­ ing to a continuum model for any subsurface perme­ ing released from source rocks are commonly un­ ability distribution and water table configuration. known. Efficient mineral exploration also requires Ignoring the effects of varying fluid density, ground­ knowledge of present and past hydrologic conditions. water flows according to Darcy's law Sedimentary brines migrate for hundreds of kilo-' metres from deep strata onto basin margins where Ie, a~ they precipitate the metallic ores of Mississippi Val­ q, :;;: --;;:81 (1) ley-type deposits. Oxidizing surface waters infiltrate where q{ is specific discharge (the volumetric flow basins and leach uranium and other elements, and rate per unit area) in an arbitrary direction I, p, is fluid then precipitate ores in roll-front deposits as they viscosity, and k{ is directional permeability. Herein encounter reducing conditions at depth. Promising the directions of the coordinate axes are assumed to exploration targets could be identified more accurate­ corespond to the principal permeability values; in ly in each of these cases with knowledge of the pre­ this case permeability is described as a' vector (or, sent or past groundwater hydrology of the basin be­ more accurately, a diagonal tensor) rather than a full ing explored. tensor quantity. 4> is hydraulic potential Understanding present-day hydrologic conditions is also of economic and societal concern. Predicting ~ :;;: P - pgz (2) subsurface fluid pressures can be critically important when oil wells are drilled, especially in provinces the mechanical energy per unit volume of ground­ where overpressures can blowout wells. Safely dispos­ water (HUBBERT 1940). P and p are fluid pressure and ing of persistent toxins and radioactive elements with density, g is the acceleration of gravity, and z is depth half-lives of geologic duration requires knowledge of relative to a fixed datum. Hydraulic potential ac­ the rates and directions of transport by subsurface counts for the work of compression and elevation fluids. performed in moving a groundwater of constant den­ Quantitative modeling techniques combined with sity along a flow path; the vector - \7~ is the driving the results of observational and experimental studies force per volume of groundwater. have proved successful in analyzing flow and trans­ Given sufficient time, flow will adjust to a steady port in sedimentary basins on natural time and dist­ state for a given water table configuration. Combin­ ing Darcy's law with conservation of mass gives the ance scales (BETHKE et al. 1988). The purpose of this paper is to examine the variety of models that have equation for steady groundwater flow been applied to analyze basin processes and consider some of the principal uncertainties in applying these models to study groundwater hydrology in the pre­ sent and geologic past. in three dimensions, where x and yare horizontal di­ rections. If the medium can be assumed to be homo­ 2. Flow driven by topographic relief genous and viscosity constant, this equation can be simplified to give Topographic relief along basin surfaces can drive groundwater through deep strata. The flow is driven 2 2 Ie a2~ + Ie a c1> + Ie a c1> c: 0 (3 b) by variation in the potential energy of groundwater s az2 u ay2 • az2 Modeling subsurface flow in sedimentary basins 131 and to V2c1> =0 (3c) r~ if permeability everywhere is isotropic. Equation (3c) is Laplace's equation which has many known solu­ z tions from the theories of electrostatics, magnetics, diffusion, and heat conduction. Equations (3a-c) can be solved conveniently as boundary-value problems by writing appropriate boundary conditions (Fig. 1). Most commonly in = basin studies, the upper and lower boundaries are the P-pgz water table which is at known potential and a strati­ Fig. 1. Continuum model of groundwater flow in a sedi­ graphic contact above an aquitard, such as an evapo­ mentary basin. Potential distribution is given mathematic­ rite bed or crystalline basement rocks, which is taken ally as the solution to a boundary value problem that ac­ as a barrier to flow. Side boundaries are generally counts for relief on the water table and sediment permeabi­ taken as groundwater divides. lity. The flow field is then determined from Darcy's law.

2.2 Subsurface flow fields even in the presence of highly irregular relief, given a Solutions for the subsurface flow field can be ob­ permeable aquifer at depth. In Fig. 2(£), the flow tained analytically by symbolic manipulation for pattern reflects the effects of a discontinuity in the basins with simple boundaries and permeability distribution of an aquifer. Here, an aquifer that spans structures, or by numerical methods when the flow just part of the domain causes recharge and discharge system is more complex (FREEZE & WITHERSPOON in areas that would not be expected from the water 1966). The assumption of constant density can be table configuration. relaxed in obtaining numerical solutions (see Fig. 3 shows the evolution of groundwater flow sys­ Section 3). Fig. 2 shows calculated flow systems for a tems in two North American basins calculated using variety of hypothetical water table configurations and continuum models. Flow in the Western Canada Sedi­ permeability distributions. Equipotentials (dashed mentary Basin probably occurred as a basin-wide re­ lines) connect points of equal hydraulic potential. In a gional system (a) immediately aher the western basin well open to flow at a point intersecting an equipo­ margin was uplihed as the Canadian Rocky Moun­ tential, water will rise to the elevation at which the tains developed in the late Tertiary (GARVEN 1988). equipotential meets the top surface. As the basin surface was eroded, flow evolved into the Flow fields (a-d) in Fig. 2 are analytical solutions more localized regimes (b) present today. Plots (c-d) (T6TH 1962, 1963, 1978) that illustrate the salient fea­ show how Cenozoic erosion has produced under­ tures of flow systems arising from topographic relief. pressured conditions at depth in the Palo Duro basin In each case, groundwater recharges the flow system (SENGER & FOGG 1987, SENGER et al. 1987). Under­ at high elevation, and thus high potential. Ground­ pressures in the present day (d) have arisen from water everywhere moves toward regions of lower fluid change in the water table configuration due to dissec­ potential, eventually discharging at low elevation. As tion of the Pecos River Valley and the High Plains. a consequence, subsurface fluids in recharge areas are The underpressures have reversed the direction of underpressured with respect to a column of water ex­ flow across Permian evaporites that make up a region­ tending to the surface, whereas fluids in discharge al aquitard system. The reversal is significant because areas are overpressured. In Fig. 2(b), a layer at depth present-day flow across the evaporites, which have of relatively low permeability enhances the effect of been considered as sites for disposing of radioactive topography in creating overpressured and underpress­ waste, moves away from the biosphere. ured regions. In a system with irregular relief (Fig. These examples illustrate the importance of recon­ 2c), flow systems develop over a range of scales from structing topographic relief in simulating past local to regional. When the irregularity of the relief is groundwater flows. Past relief can sometimes be esti­ large compared to overall relief and to the thickness mated sedimentologically from the grain sizes of sedi­ of the flow system (Fig. 2d), the local flow systems ments shed from uplifted areas, and from the distri­ can overwhelm regional flow. bution of facies in the strata formed as these sedi­ Flow fields 2(e) and 2(£), which assume irregular ments are deposited (e. g., MARCHER & STEARNS water tables and permeability distributions, were ob­ 1962). Erosion rates can be inferred from the sedi­ tained numerically (FREEZE & WITHERSPOON 1967). ment loads of rivers draining exposed areas, or from Fig. 2(e) shows that regional scale flows can occur the volumes of these sediments deposited in strata of 132 CRAIG M. BETHKE

no v.e. v.e.=15 (c) nov.e.

(d) v.e.=2.5

(e) nov.e.

Fig. 2. Calculated groundwater flow patterns in hypothetical basins showing effects of various water table configurations and subsurface permeability distributions {v. e. = vertical exaggeration}. Solutions {a-d} were found analytically (T6TH 1962, 1963, 1978); plots (e, f) were determined numerically (FREEZE & WITHERSPOON 1967). Dashed lines are equipotentials and solid lines and arrows show flow directions. known stratigraphic ages. Stratigraphic projection can a flow path are considered separately rather than in help reconstruct erosional history. In addition, mea­ combined form within a hydraulic potential func­ surements of thermal maturity or shale compaction tion. In fact, there is no scalar hydraulic potential can be used to estimate past burial depths of preserv­ function applicable to flows of varying density (HUB­ ed sediments. The accuracy of models of past basin BEKf 1940). By equation (4), flow will occur in a verti­ flows will depend in large part on the extent to which cal cross-section given any pressure distribution in such geological techniques are integrated into paleo­ response to a lateral density gradient. In other words, hydrologic stu'dies. there is no pressure distribution that can balance a la­ teral density variation so that neither horizontal or vertical flow will occur. 3. Flow driven by buoyancy Free convection is the continual overturn of fluid Variations in fluid density create buoyant drives for by buoyant forces. Slow fluid circulation by convec­ fluid flow in the subsurface. Groundwater density tion provides an attractive explanation for the degree varies mostly because of thermal expansion and chan­ to which basin sediments are altered diagenetically ges in salinity. Pressure exerts only a small effect on (WOOD & HEWETT 1982, 1984). Considering the groundwater, but provides a major control on the quantity of cement and amount of secondary porosi­ densities of hydrocarbon and gas phases. When ty observed in the subsurface, pore fluids in some groundwater density varies, specific discharge is given rocks appear to have been replaced many thousands of times (e. g., SIBLEY & BLATT 1976, LAND & DUT­ by k, (ap az) (4) q,:: --; at - pg ai TON 1979). These quantities could be accounted for either by regional flows or fluids recirculating locally. This relationship differs from equation (1) in that the There is some field evidence indicating that fluids work of compressing and elevating groundwater along in reservoir rocks convect. RABINOWICZ et al. (1985) Modeling subsurface flow in sedimentary basins 133

WEST EAST

(a) Ikm~ 50 km

WEST I EAST Texas iOklahoma

10

400m~ 40km

Fig. 3. Past and present groundwater flows along cross sections of the Western Canada (GARvEN 1988) and Palo Duro (SENGER & FOGG 1987, SENGER et al. 1987) sedimentary basins as determined by numerical simulation. Western Canada example shows flow (a) after basin uplift in late Tertiary, and (b) after erosion to present-day topography. Palo Duro example shows flow predicted (c) before and (d) after Cenozoic erosion. Patterned area in (c-d) shows extent of Permian evaporite aquitard. 134 CRAIG M. BETHKE noted that the patterns of diagenesis in a reservoir three orders of magnitude smaller than the horizon­ from the North Sea seem to reflect the effects of con­ tal value. vective circulation. Convection also may have driven There is no stability criterion when a lateral oil migration within the field. AZIZ et al. (1973) temperature gradient is superimposed on the vertical found thermal gradients within a French oil field geothermal gradient (WOOD & HEWETT 1982); suggesting that reservoir fluids convect actively and systems where lateral thermal variations are maintain­ redistribute heat. ed will convect at any value of R". Flow rates when R" is small, however, may be too slow to be signifi­ 3.1 Thermal convection cant even on geologic time scales. Lateral temperature gradients occur due to variations in basement heat Free convection can arise spontaneously from flux, structures with large thermal conductivities thermal gradients in a basin. When cool fluids overlie such as salt domes, or thermal conductivity contrasts warmer fluids, the system will convect freely given between a sloping aquifer and surrounding sediments sufficient permeability and aquifer thickness. The cri­ (DAVIS et al. 1985). In the case of a sloping aquifer terion for the instability of a thermal stratification is (Fig. 4a), elongate convection cells can develop. These the filtration Rayleigh number cells may redistribute hydrocarbons and localize kga.p 2C,HI1T diagenetic alteration in areas of upward and down­ Ra = ...... :...... :....~-- ward flow where thermal gradients along flow paths .... K are steepest. The flow rates predicted for such cells, where ex and QCr are the coefficient of thermal however, are commonly quite slow, depending on per­ expansion and volumetric heat capacity for the pore meability and the lateral temperature gradient. fluid, flT is the temperature difference across an There has been little work to determine the extent aquifer of thickness H, and x is thermal conductivity to which convection persists in the presence of regio­ (LAPWOOD 1948). nal flow systems. PRATS (1966) pointed out that a la­ Convection is expected for R" > - 40; stratifica­ teral flow across the domain does not effect the Ray­ tions are stable at smaller values because ascending leigh criteria for convective stability. His analysis, fluids cool by conduction too rapidly to maintain however, does not consider the effects of thermal dis­ their buoyancy. Convective flow fields can be found persion. Dispersion works for hydrodynamic stabili­ analytically as 'solutions to boundary-value problems ty and against formation of convective cells (RUBIN, (PHILIP 1982) for many configurations and boundary 1974) because the process increases the apparent ther­ conditions, or numerically (MERCER et al. 1975) in mal conductivity of the medium. Since the rate of the general case. thermal dispersion varies with flow velocity, superim­ The critical Rayleigh number (about 40) is rather posing regional flow might eliminate the possibility large given the ranges of sediment permeability and of convection, or overwhelm the relatively slow con­ aquifer thicknesses likely to occur in basins. By equa­ vective flow rates so that coexisting convection is less tion (5), a 100-m thick aquifer under a normal geo­ significant. thermal gradient requires a permeability of about 2 darcys (2x10-12 m2) to satisfy the Rayleigh criterion; 3.2 Thermohaline convection a kilometre-thick aquifer requires about 20 milli­ darcys (2xl0-14 m2). Aquifers with large Rayleigh Thermohaline convection results when variations numbers may occur, for example, in sandstone fair­ in both solute concentration and temperature affect ways in basins at continental margins (BLANCHARD density. Solutes can retard convective circulation & SHARP 1985). when concentrations are highest in warmer fluids, or Determining Rayleigh numbers representative of enhance it when cooler fluids are most saline. Fig. the subsurface is complicated by heterogeneity in the 4(b) shows convective flow near a salt dome. Flow permeability structures of aquifers. A few percent may rise along the dome due to heat conducted from shale or silt partings in an otherwise homogeneous depth by the salt (KEEN 1983), or descend in response sandstone can cause strong anisotropy in permeabili­ to salt dissolving into the groundwater. ty (see section 8.2; Fig. 11). Anisotropy works to de­ HANOR (1987a) found evidence of complex convec­ crease R" significantly relative to an isotropic medium tive patterns surrounding piercement salt domes in of the same horizontal permeability (COMBARNOUS the Gulf of Mexico Basin in Louisiana. Flow in this & BORIES 1975). BEGG & CARTER (1987), for system is driven in part by salt dissolution at shallow example, found that the vertical permeability of a re­ depths, which causes warm flows that have ascended servoir in a fluvial sandstone formation was about along the dome to descend back into deeper strata. Modeling subsurface flow in sedimentary basins 135

(0)

(b)

Fig. 4. Basin environments in which variations in fluid density might control groundwater flow patterns. (a) convection in a sloping aquifer, (b) thermohaline convection near a salt dome, (c) seawater concentrated by evaporation infiltrating deep strata, and (d) brines pooled by density stratification.

Flow rates in the system may be as great as a few me­ In the latter case, the stratification maintains a dyna­ ters per year. mic stability if upwelling fluids cool too quickly to Thermohaline effects might also be important to maintain their buoyancy. The sum of the thermal and the formation and migration of sedimentary brines. solute Rayleigh numbers describes dynamic stability Fig. 4(c) shows how seawater concentrated into bit­ in thermohaline systems (NIELD 1968). Given a terns during deposition of evaporite beds might infil­ sufficient driving force such as a regional flow system trate into deep strata due to its greater density relative set up by topographic relief, the hydraulic gradient in to normal seawater. CARPENTER et al. (1974), on the deep aquifers can overwhelm buoyant forces and dis­ basis of the panitioning of chlorine and bromine dur­ place deep brines into shallow strata. Such a process is ing the formation of evaporite minerals, envisioned apparently responsible for forming the lead-zinc ores such a process as a possible origin of saline formation of the Mississippi Valley-type deposits (GARvEN & waters. FREEZE 1984a & b, BETHKE 1986a). Fig. 4{d) shows brines that have pooled by density stratification into deep strata. Thermal expansion off­ 4. Transient flow systems in evolving basins sets the increase in density due to salinity sufficiently that warm saline brines pooled at depth may be ei­ As basins evolve, groundwater flow systems can ther more or less dense that shallower, fresher fluids. develop that are significantly out of equilibrium with 136 CRAIG M. BETHKE the top?graphy of the .l~nd surface. Transient flow sys­ In .this equation, t is time, cf> is sediment porosity, and tems arIse from deposltlon of sediments that increases ~ IS flu.id compressibility. The top boundary in the t?e load on deep strata, and from rapid surficial ero­ sl~ulatl?n mov~ to accept sedimentation. Porosity SIO~ that decreases confining stress at depth. Once a varIes With effective stress, which is the difference be­ basin ceases to evolve at a geologically rapid rate, the tween the stress exerted by the weight of overlying disequilibrium hydraulic potentials dissipate toward sediments and the fluid pressure. the distribution that reflects the basin's topography Using this technique, DOLIGEZ et al. (1986) and and permeability structure. BETHKE et al. (1988) simulated development of over­ pressures in the Viking graben of the North Sea basin 4.1 Effects of rapid sedimentation and the Gulf of Mexico basin. Fig. 5 shows the evolu­ Large excess pressures develop in shaly basins un­ tion of fluid pressures calculated for a north-south dergoing rapid sedimentation (DICKINSON 1953). The cross-section through the Gulf of Mexico basin. Fluid overpressures pose a substantial risk to drillers and pressures are represented as average gradients, the ratio may play roles in localizing petroleum reservoirs and of pressure to burial depth. A hydrostatic gradient is about 10 MPa/km; lithostatic is about 23 MPa/km. ?re deposits. Overp.ressures are commonplace today The calculation shows that the expanse of overpress­ In the Gulf of MeXICO, North Sea, and Niger Delta basins. ured sediments increased over the past 30 m. y. to the A deep flow regime results in basins during sedi­ present maximum. Many sediments became over­ mentation from the fluid displaced as strata compact pressured within the past 2 m. y. under the weight of the accumulating sediments, and The phenomenon of young overpressures in older to a lesser extent from dehydration of clay minerals sediments poses special difficulty in modeling be­ (GALLOWAY 1984, BREDEHOEFT et al. 1988). Over­ cause the hydrologic properties of sediments must be pressures, fluid pressures greatly in excess of hydrosta­ ~stimat.ed when effective stress is decreasing as well as tic, are likely to develop in shaly basins where burial I?cre~slng. Fo~ exam~le, because sediment compac­ proceeds at rates of at least 0.1-1 mm/yr (BETHKE ~Ion IS largely Irreversible, microfractures can develop In shales that develop significant overpressures, de­ 1986b). In this case, burial proceeds so rapidly that enough fluids cannot escape to allow the sediments to pending on the confining stress field (DOMENICO & PALCIAUSKAS 1979). UNGERER et al. (1987), in their compact fully. The pore fluid, then, becomes over­ model of compaction-driven flow, assumed that the pressured as it bears part of the overburden weight permeabilities of sediments that develop overpress­ that would normally be supported by the sediment. ures increase due to microfracturing, assuming that Although the flow rates resulting from compaction lateral stress constitutes a fixed fraction of vertical are rather slow, generally centimetres or tens of centi­ load. metres per year, the large potential gradients in over­ pressured basins seem certain to block meteoric water from circulating into deep sediments. Only small ex­ cess potentials, on the other hand, are expected in ba­ 4.2 Effects of erosion sins that subside slowly or that contain deep aquifer Flow pattern~ that. are in disequilibrium with pre­ systems (BETHKE 1985, 1986a). s~nt topographic. rehef can develop in basins being The effects of sedimentation on pressure evolution dissected by erosIOn (T6TH & CORBET 1986). Two in basin sediments can be calculated as the solution to effects are significant in strata with sufficiently small a moving boundary problem. The problem can be permeabilities. First, subsurface flow patterns can solved analytically in the vertical dimension (GIBSON remain partly adjusted to a previous higher position 1958, BREDEHOEFT & HANSHAW 1968). In two di­ of the water table (T6TH & MIllAR 1983). This effect mensions, the fluid pressure distribution ~ z, P(x, t) produces pressures in excess of an equilibrium gra­ can be calculated by numerical solution (BETHKE dient from the current land surface. On the other 1985, BETHKE & CORBET 1988). In the reference hand, un?e~p~e~ures ca? develop in deep strata due frame of the sediments, flow through a deforming to the diminishing weight of overlying sediments medium is described by and, to a lesser extent, the thermal contraction of pore fluids (NEUZIL & POLLOCK 1983). On the basis of (6) ~p ~ = :. [~ [~: 1I + :. [~ [~~ Il available estimates. of material properties, underpres­ sur~s seem more hkely than overpressures in eroding basinS (NEUZIL 1986). The origin and persistence of + :. - (1 [~ [~: PIII- ~ ~) ~~ underpressured environments is of interest in waste Modeling subsurface flow in sedimentary basins 137

Oligocene (31 m.y.) Miocene (5.2 m.y.)

Pliocene (1.8 m.y.) Present Fig. 5. Calculated development of regional overpressures in the Gulf of Mexico basin due to rapid sedimentation (HARRISON & BETHKE 1986, BETHKE et a1. 1988). Cross section extends north-south in eastern Texas and extends about 300 kilometres offshore. Fine lines are contacts among time-stratigraphic units. Contours show subsurface pressure gradients (MPa/km). disposal efforts because potential gradients can be also representative of basins with more permeable expected to drive flow into deep strata and away from aquitards undergoing faster erosion. the biosphere (BRADLEY 1985), and because of the Hydraulic potential, contoured in MPa relative to possible effects of such environments on petroleum surface potential at the right boundary, is at a local migration. minimum within the aquitard due to its expansion Flow patterns in eroding basins can be calculated as and to a lesser extent the thermal contraction of pore solutions to an initial-boundary value problem. The fluids. The transient flow regime (a) can be compared scheme is similar to that applied in basins accepting to the pattern in equilibrium with the land surface sedimentation (equation 6), except that the upper (Fig. 6b). The equilibrium pattern was calculated boundary subsides during erosion. Fig. 6a shows the from the topographic relief and permeability distri­ transient flow pattern in a basin containing a 540 m­ bution (see section 2.1), ignoring the effects of ero­ thick aquitard of small permeability (10- 7 to 10-8 dar­ sion. Given sufficient time, the transient regime cys; 10- 19 to 10-20 m2) and an anisotropy of 10- 1 would approach this steady state. From the calcula­ (THOMAS CORBET, unpublished data). The basin has tions, erosion has generated hydraulic potentials as undergone erosion for 5 m. y. at rates between zero (at much as 0.6 MPa (60 m of hydraulic head) less than left boundary) and 0.1 mm/yr (right). Strata expand those expected at steady state. This class of models is somewhat as they are unloaded, so that the apparent limited, however, by poor understanding of the phy­ erosion rate is slightly less than these values. In the sical properties of sediments as they are unloaded simulation, sediment porosity increases at about 10% over geologic time. In particular, the degree to which of typical compaction rates during burial in basins to sediment porosity rebounds and to which fracture account for the fact that sediment compaction is only permeability develops during unloading is poorly partly reversible (NEUZIL 1986). The calculation is known. 138 CRAIG M. BETHKE

5 m.y. surface bacterial attack. Further, the presence of unusually ------p;;;;~t-l dilute pore water alters the electrical conductivity of deep sediments, complicating interpretation of the re­ sistivity logs used to locate reservoirs during petro­ leum exploration (DICKEY et al. 1987). Flows resulting from the combined effects of com­ paction and topographic relief can be modeled by solving the equation of flow in a deforming medium (6) subject to a boundary condition reflecting the to­ pography of the coastal plain (Fig. 1). The overall so­ lution can be found as the sum of two solutions. 4>, is the hydraulic potential function derived by solving the boundary value problem describing steady-state flow arising from topographic relief

.!.. [~ acz" ) + .!.. [~ acz" ] + .!.. [~ acz" ) = 0 (7a) az ,... az all,... a" a:,... a: where the upper boundary condition 4>ltly reflects the relief on the coastal plain according to equation (2). ell" the hydraulic potential resulting from compac­ (b) tion, is given as the solution to an initial-boundary Fig. 6. Transient flow system (a) resulting from erosional dis­ value governed by section of basin surface (THOMAS CORBET, unpublished data). Equipotentials (MPa) are contoured relative to surface .!.. (~ acz, e ) + .!.. (~ acz, e 1+ .L (~ acz, c ) potential at right of figure. Comparison to the steady-state az ,... az ay J1 a1l az,... a: (7b) flow system (b) shows that the transient system has not equi­ librated with the land surface. acz,c 1 1 ~ = ~ (at + pgv"" + (1 - <1» at

subject to the upper boundary condition 4>Wy = 5. Incursion of fresh water into compacting basins o. Equation (7b) is (6) recast in terms of hydraulic poten­ Each of the processes discussed to this point is like­ tial. ly to drive subsurface flow under certain conditions, The solution to the combined problem is but relatively little work has been aimed at determin­ (7c) ing how these processes interact. Consider the hydro­ logic regime near the margin of a basin accepting ra­ which can be seen to satisfy the equation of flow in a pid sedimentation. Sediment compaction within the deforming medium (equation 7b) as well as the boun­ basin drives fluids landward. Topographic relief on dary conditions reflecting topographic relief. By Dar­ the coastal plain, however, drives an opposing flow cy's law (equation 1), discharges predicted by solving system of fresh water basinward (GALLOWAY 1984). (7a) and (7b) are also additive to give the overall flow The basinward flow converges with the compaction­ rates. The decoupled solution (7a-c) is strictly valid flow system and discharges vertically. Flow in the when hydraulic potential does not affect permeabili­ freshwater regime can be relatively rapid and extend ty. In practice, the procedure is a good approximation for considerable distances subsea. because large potentials generally arise from sediment As fresh waters infiltrate basin strata, they cause sig­ compaction in strata too impermeable to host signi­ nificant chemical changes in the subsurface. Some ficant topographic flow. carbonate cements are associated with meteoric flows, Fig. 7 shows interaction of the flow regimes set up and the infiltration of dilute pore fluids can cause se­ by sediment compaction and relief on the coastal condary porosity by dissolving silicate grains (BJ0R. plain of the Texas Gulf Coast (HARRISON & BETHKE LYKKE 1984). Oxidizing flow systems form ore bodies 1986, BETHKE et al. 1988). The extent of the meteoric by precipitating uranium minerals as flow encounters flow is shown before and after a drop in sea level oc­ reducing conditions at depth (SANFORD 1982). Infil­ curring 31 m. y. ago during the Oligocene (HAQ et al. trating fresh water can degrade petroleum by leaching 1987). The drop increased the hydraulic potential rela­ the more soluble hydrocarbon compounds and by tive to sea level of groundwater along the coastal Modeling subsurface flow in sedimentary basins 139

plain, and exposed more of the coastal plain to mete­ 6.1 Solute transport oric precipitation. Fresh water invaded more basal Moving groundwaters redistribute their dissolved strata in the Oligocene than in the present day be­ load in the subsurface by advection, diffusion, and cause the strata were less deeply buried and wide­ hydrodynamic dispersion. The components carried spread geopressures had yet to develop. At the low in solution are the raw materials for precipitating dia­ stand of sea level, fresh water infiltrates into deeper genetic minerals and metallic ores. For this reason, strata and farther basinward along coastal aquifers. understanding transport processes in basins is basic to There is field evidence that changes in sea level in constructing quantitative models of sediment diage­ the geologic past have affected present-day flow sys­ nesis (WOOD & SURDAM 1979) and effective explora­ tems in coastal aquifers. MEISLER et al. (1985) found tion strategies for ore deposits (OHLE 1951, 1980). that pore fluids in the Atlantic Coastal Plain are di­ The distribution and transport of a solute through luted by fresh water 100 kilometres offshore of New a groundwater flow system is described by the equa­ Jersey. The authors interpreted the distal freshwater tion occurrences as remnants of low stands of sea level from the Pleistocene. ESSAID (1988) observed that :t (cf>C) = V'(cf>D VC) - v·(qC) + cf>A, (8) flow in the coastal aquifers of Monterey Bay, Califor­ nia, is still adjusting to Pleistocene fluctuations in the where C is solute concentration; Ar is the rate of reac­ sea level. tion with the medium per volume of fluid, where positive values denote dissolution reactions. The 6. Hydrologic transport tensor Many problems in basin hydrology are ultimately concerned with the ability of groundwater flow sys­ tems to transport dissolved mineral mass, thermal energy, or hydrocarbons within basins. Like ground­ water flow, the effects of hydrologic transport pheno­ mena can be studied as boundary value problems for­ describes the combined processes of molecular diffu­ mulated using continuum theory. sion and hydrodynamic dispersion.

(b)

2kmL 100 km

Fig. 7. Oligocene freshwater incursion driven basinward by relief on the coastal plain in Gulf of Mexico basin before (a) and after (b) a drop in sea level about 30 m. y. ago (HARRISON & BETHKE 1986, BETHKE et al. 1988). Shaded area shows regions of basinward flow. Contours show excess pressures due to sedimentation, expressed as pressure gradients (MPalkm). 140 CRAIG M. BETHKE

Hydrodynamic dispersion is a physical mixing pro­ be carried by advection into shallow strata to form cess. The process results primarily from the branch­ ore deposits. ing and joining of flow paths and variations in mi­ croscopic flow rates as the fluid moves around rock 6.2 Heat transfer grains and through heterogeneities in the medium. Basin thermal budgets are dominated by heat con­ Dispersion occurs both along and transverse to the ducted into the sedimentary pile from the underlying direction of bulk flow, but at different rates. Hence crystalline crust. Basement heat flow ultimately the process is described by a tensor that contains co­ moves to the surface by conduction, which is perhaps efficients for transport in each direction resulting most common, or through the advection of ground­ from the component of flow along each axis (BEAR water. When advective transfer dominates, discharge 1979). areas are warmed and recharge areas cooled relative to To date there has been little work on determining a conductive gradient (e. g., DOMENICO & PALCIAUS­ values for the coefficients of the dispersion tensor for KAS 1973, HITCHON 1984). heterogeneous media (e. g., SILLIMAN et al. 1987), and it is especially difficult to estimate these values for flow in natural systems. Dispersivity further varies with the scale of observation (e. g., WHEATCRAFT & TYLER 1988; see section 7). In addition, equation (8) ~ __-- 2500 represents dispersion as a Fickian process, that is in the form of Fick's law of diffusion, although detailed ~ ------2~----~ studies do not always support such an asumption ____ (MATHERON & DE MARSILY 1980; ANDERSON 1984). Transport theory has been used to study the origin of the salinity distributions observed in sedimentary basins, one of the longstanding problems in basin hydrology (HANOR 1987b). Many deep groundwaters ~~['----~ are brines many times more concentrated than sea­ 50km water. The brines are believed to be the residual (0) bitterns from precipitating evaporite beds (CARPEN. TER et al. 1974), groundwaters that have dissolved eva­ porite minerals in the subsurface (LAND & PREZBIN­ DOWSKI 1981), or waters concentrated by membrane filtration (BREDEHOEFT et al. 1963, GRAF 1982). Groundwaters in marine sediments that are more di­ lute than seawater are sometimes explained as con­ taining the water released as clay minerals dehydrate, but more commonly are attributed to infilrating me­ teoric water. Fig. 8(a) shows a salinity distribution at steady state calculated using equation (8). In the calculation, an aquifer filled initially with a brine is open to meteo­ ric recharge along a portion of the boundary (DOME­ NICO & ROBBINS 1985). At steady state the infiltrat­ 100km ing fresh water increases in concentration along its Fig. 8. Calculated effects of mass and heat transport by flow paths by dispersive mixing, forming a broad groundwater flows in sedimentary basins. Plot (a) is a map range of salinities. view showing the steady-state salinity distribution in an RANGANATHAN & HANOR (1987) studied the ef­ aquifer containing a connate brine and being recharged by meteoric water (DOMENICO & ROBBINS 1985). Contours fects of a basal salt layer on the salinity distribution in show solute concentration in mg/l, and the hatched region a basin undergoing sedimentation. Their results show is the recharge area. Plot (b) shows the calculated effects of a that diffusion over geologic time can significantly af­ Mesozoic groundwater flow system on the subsurface fect groundwater salinity in overlying strata. GARVEN temperature distribution along a cross-section through the & FREEZE 1984a, b) used transport theory to show Illinois basin (BETHKE 1986a). Contours give temperature in that metals leached from sediments deep in basins can °C. Modeling subsurface flow in sedimentary basins 141

Assuming that groundwater and sediment maintain 6.3 Petroleum migration thermal equilibrium locally, heat transfer by conduc­ It is clear from the distribution of source rocks and tion and advection is described by petroleum reservoirs that oil can migrate for remark­ able distances. Petroleum in some interior basins of North America migrated laterally for more than 150 kilometres (Dow 1974, CLAYTON & SWETLAND 1980). Gulf Coast oils have migrated from deep Creta­ Here, C and C are fluid and rock heat capacities, Qr f r ceous and Early Tertiary source rocks through thick is density of the rock grains, x is thermal conducti­ overpressured shale sections to present reservoirs, a vity (modified to account for thermal dispersion), and hw is fluid enthalpy. Variants of this equation can process that continues today (NUNN & SASSEN 1986). Discharge of an oil phase along I through a carrier also describe cases in which fluid and rock do not equilibrate thermally (COMBARNOUS & BORIES bed is described by 1975). Equation (9) can be solved for the temperature __ k,e k, (ap. _ az) field T(x, y, z, t) in a basin where groundwater q,. - IL. al P.g al (10) discharge q is known from solution of the flow equations (e. g., DOMENICO & PALCIAUSKAS 1973). where Po and Jlo are the oil-phase density and viscosi­ ty (PEACEMAN 1977). The buoyant drive on the oil Heat transfer theory has been applied to study how phase arises from the lesser value of Qo relative to the ores form in sedimentary basins. Mississippi Valley­ density of water. Equation (10) differs from the flow type lead-zinc deposits occur in shallow sediments on law for a single phase (equation 4) by accounting for basin margins. On the basis of fluid inclusion studies the pressure on the oil phase Po and by the factor kro, and isotopic analyses, the ores are believed to have the relative permeability of the medium to oil. precipitated from sedimentary brines at temperatures The oq-phase pressure is the sum of the water and ( - 50 to 200°C) characteristic of deep strata (SVER­ capillary pressures JENSKY 1986). GARVEN & FREEZE (1984a, b) and GARVEN (1985) showed that groundwater flow driven (11) by topographic relief in some basins is capable of transporting brines from deep strata to the sites of de­ Capillary pressure increases with oil saturation So in position. In the resulting hydrothermal system, the a given rock, and with decreasing apertures of the brines move rapidly enough to avoid complete cool­ pore throats for rocks in general. By equations ing by conduction to the surface. (10-11), oil will seek areas where capillary pressures are small; hence there is a strong drive for oil to Fig. 8(b) shows the temperature distribution in the migrate through rocks with broad pore openings. Illinois basin resulting from such a hydrothermal sys­ Relative permeability increases from zero at small tem. Flow is driven by uplift of the Pascola arch in saturation toward one as as oil saturates most of the the southern basin (BETHKE 1986a). In the calcula­ pore volume. tion, groundwater flowing through basal Paleozoic Like groundwater flow, petroleum migration in a aquifers entrains heat conducted into the basin from sedimentary basin can be modeled as a boundary va­ below. The groundwater advects the heat along flow lue problem. The governing equation may be repre­ paths, creating a significant thermal anomaly in the sented discharge area. A regional hydrothermal system of 1...( s) = ..!..[pokrokz (ap. lj + ..!..[pok,.kl/ (ap. lj this type probably formed the ores of the lead-zinc at cjlPo. az IL. az all IL. all district on the northern margin of the basin. Equation (9) can sometimes be used to infer ap r 6 (12) +..!..[P.k .k ( O + lj+A groundwater flow in basins with known temperature az IL. az P.g • distributions. In this case, the flow field q is the unknown variable. For example, WILLET & where Ao is the local rate of petroleum generation. CHAPMAN (1987) used temperature measurements Because capillary pressure and relative permeability from oil wells to constrain groundwater flow rates vary sharply with saturation, equation (12) is non­ and the permeability structure of the Uinta basin. linear in its coefficients and must be solved numeric­ WOODBURY & SMITH (1987) considered mathe­ ally. A variety of solution techniques have been de­ matical techniques to automatically find the most veloped in the petroleum industry to facilitate simu­ satisfactory flow regime by inverting field data for lation of multiphase flow in oil reservoirs (e. g., temperature and hydraulic head. PEACEMAN 1977). 142 CRAIG M. BETHKE

Much effort is expended during petroleum explora­ they are observed (BEAR 1972). Thus, measurements tion to delineate the extent of mature source rocks in made on small samples may describe poorly the beha­ basins; attempts at quantifying distances or even di­ vior of the rock unit sampled in the vicinity of a well. rections of secondary migration are less common. The local behavior, in turn, may not represent pro­ The theory of the kinetics of petroleum generation is perties of the same rock on the regional scale of inter­ relatively well established and calibrated (e. g.; TISSOT est in basin hydrology. Because there is no direct tech­ & WELTE 1984, LEWAN 1985), and the properties of nique for measuring the hydrologic properties needed evolving oils can be estimated (UNGERER et a1. 1981, to model flow on large scales, developing effective ENGLAND et al. 1987). These theories can be combin­ methods for inferring regional properties is among ed with the techniques already presented for calculat­ the most significant challenges in basin hydrology. ing flow and transport to describe in an integrated fashion the oil generation, migration, and evolution 7.1 Regional permeability of groundwater flow systems in basins. DOLIGEZ et a1. (1986) modeled pressure evolution, Permeability in sedimentary basins tends to in­ generation, and migration along a cross section crease (but can decrease) with breadth of the scale of through the Viking graben of the North Sea basin; observation. This phenomenon is attributed to the UNGERER et a1. (1986) applied similar techniques to effects of heterogeneities in the medium. Fracture study migration in the Suez Rift. Fig. 9 shows an ex­ sets, karst networks, and lenses of coarse grained sedi­ ample calculation of petroleum migration during ba­ ments contribute to the added permeability found on sin development (LEHNER et a1. 1987). The calcula­ macroscopic scales. Fig. 10 shows the effect of scale tion accounts for secondary migration and reservoir on hydraulic conductivity in carbonate aquifers of development due to buoyancy in a hypothetical car­ central Europe. Conductivities determined by labora­ rier bed. The thickness of the oil column at each tory measurement represent the effects of the prim­ point in the domain is tracked through time. As the ary porosity and microfractures of small samples. bed undergoes burial, oil is generated at rates deter­ Values determined from well tests, which encompass mined by the temperature and thermal history of the response of the area near the well-bore, are consi­ underlying source rocks. The calculation shown does derably larger because flow also moves through ma­ not account for groundwater flow through the bed or croscopic fracture sets. Conductivities inferred on re­ variation in capillary pressure, but the technique gional scales are about four orders of magnitude lar­ might be readily generalized. ger than the laboratory measurements because of the Considerable uncertainty remains, however, about effect of regional karst networks in the aquifers. the nature of migration. From equations (10-11), Scale effects can also be significant in aquitards. petroleum moves in response to buoyancy, the hydro­ Large volumes of low-permeability rocks can be more dynamic drive of groundwater flow, and gradients in conductive than small samples because of the pre­ capillary pressure resulting from heterogeneous distri­ sence of joints, fractures, and faults (NEUZIL 1986). butions of porosity or grain sizes (HUBBERT 1953). BREDEHOEFT et a1. (1983) found that the vertical con­ Although the relative magnitudes of these factors can ductivity of the Cretaceous Pierre Shale in South be calculated (e. g., DAVIS 1987, JENNINGS 1987), little Dakota is as much as a thousand times greater on a is known about how they interact. Buoyancy, for ex­ regional scale than values suggested by laboratory or ample, might provide the dominant driving force for in situ tests. Many aquitards, however, fail to develop migration, but hydrodynamic flow may be required significant permeability. to sweep oil from the structural irregularities of car­ Regional permeabilities sometimes can be inferred rier beds to traps of commercial size. In addition, he­ by matching results of numerical simulations to terogeneity in the capillary properties of a bed may historical observations of groundwater systems control whether oil can migrate for long distances during their exploitation (BREDEHOEFT et a1. 1983), through carrier beds without being dissipated as irre­ or indirectly by estimating flow rates geochemically ducible saturation (see Section 7.4). These questions in aquifers with known head gradients (PEARSON & might be answered by combining statistical descrip­ WHITE 1967). When these methods cannot be ap­ tions of irregularities and heterogeneities within car­ plied, as would be the case in attempts to study flows rier beds with detailed modeling studies. occurring in the geologic past, techniques are needed to determine regional permeabilities from observable 7. Hydrologic properties on regional scales properties of the medium. It is broadly recognized that the hydrologic proper­ Recently, efforts have been made to improve under­ ties of porous media vary with the scale on which standing of the nature of regional permeability by Modeling subsurface flow in sedimentary basins 143

Oil Column (m) iii >10 t------~1-10 ~ .1-1 (0) (b) gP-r;-J.01-.1 tw~Vm1.).003- .01 CJ <.003

Time before present (106 yr) (0) 73.5 (b) 40.5 (e) 15.0 (d) 0.0 (d) 10km

Fig. 9. Calculated buoyancy-driven migration and development of petroleum reservoirs during burial of a hypothetical carrier bed (LEHNER et al. 1987). Structure contours in kilometers. Petroleum is generated in underlying source beds pri­ marily along left side of figure. I Karst Network

Fracturet Sets Laboratory

.~ Porosity and ~ 1~8~~~~~~~~~~~~~~~~~~~~Microkadures* 1 1 2 3 ~ 10- 10° 10 10 10 ~ :t: Scale of Measurement (m) Fig. 10. Schematic representation of the effect of scale on typical hydraulic conductivities of carbonate rocks to water in cen­ tral Europe (GARvEN 1986, after KIRALY1975). 144 CRAIG M. BETHKE quantifying the heterogeneities in basin strata. For For example, anisotropies assumed in modeling topo­ example, probability distributions of fracture apertu­ graphy-driven flow can control whether regional or re, density, and lengths can be estimated by fracture local flow systems develop in a basin with irregular mapping techniques applied at well-bore or outcrop. topography (see Fig. 2). Most experimental studies of The distributions can then be used to define Monte groundwater flow in natural media have employed Carlo simulations (SMITH & SCHWARTZ 1984) of flow sandstones chosen for their homogeneity, such as the through media with statistically distributed fractures Berea in North America and the Bentheim in Eu­ sets. Such simulations give estimates of regional pro­ rope. Choice of these nearly isotropic sands has serv­ perties directly. Alternatively, heterogeneities arising ed to de-emphasize the amount to which permeabi­ from facies distributions might be defined using de­ lity varies with flow direction in sedimentary rocks. terministic models of sediment deposition (e. g.; Anisotropy in permeability results from microsco­ DOYLE et al. 1988, TETZLAFF 1988). pic factors such as the orientation of mica flakes and variations in grain size among individual laminae in the sediment, as well as from macroscopic heteroge­ 7.2 Anisotropies of basin strata neities. Heterogeneities include bedding shale part­ Permeability anisotropy of basin strata exerts a ings, interlayered formations, and fracture sets. Ani­ strong influence on the direction of subsurface flow. sotropy, like permeability, varies with the scale of ob-

140

-- =- -:- - 0.15 ~ 100 = =- -=--- ~

:t::q, (b) ~ ~ 60

(0) 20 5% shale kss = 700 md 0.01 O~------~------~------~------~ o 50 100 150 200 Most Likely Shale Length (m) Fig. 11. Calculated effect of most likely shale length on effective vertical permeability of a formation containing 5 percent shale interbeds (BEGG & KING 1985). Calculation treats three-dimensional flow through medium with randomly positioned shale interbeds with statistically distributed lengths and thicknesses. Formation is 125 m thick and sandstone permeability is 700 millidarcys (7xl0-11 m2). Modeling subsurface flow in sedimentary basins 145 servation (HALDORSEN 1986), tending to become these data suggest that formation anisotropy might be more pronounced as system dimensions increase. estimated even in the absence of detailed statistical For this reason, although the permeability of a la­ data on the basis of sedimentological study. boratory sample most commonly varies with the di­ rection of measurement by less than an order of mag­ nitude, sediments typically show greater anisotropies 7.3 Roles of faults and fractures when studied on larger scales. BEGG & CARTER Although faults can dominate the structures of se­ (1987) found good results by simulating production dimentary basins and may be clearly visible in seis­ in a reservoir composed of fluvial sandstones with mic profiles, their role in affecting subsurface flow is short shale interbeds by assigning an anisotropy poorly understood. Faults can provide a barrier to la­ (k/kj of 10-). FOGG (1986) estimated anisotropy in teral flow or channel flow across stratigraphy. Changes the Wilcox aquifer system in east Texas to be about in fluid pressure across faults and discontinuities in 10-" on a regional scale. the levels of oil-water contacts provide evidence that The influence of heterogeneities can be investigated faults can serve as subsurface seals (WEBER 1986). Be­ quantitatively by stochastic methods. Fig. 11 shows cause petroleum is retained in faulted reservoirs, seals the calculated effect of the most likely shale length on seem to be able to remain intact over geologic time the anisotrophy of a reservoir sandstone containing periods. just 5 percent shale interbeds (BEGG & KING 1985). On the other hand, faults clearly are capable of In the three-dimensional calculation, the thicknesses, channelling flow. Many hydrothermal ores in sedi­ lengths, and breadth were determined stochastically mentary basins are deposited in or near fault systems. from randomly distributed interbeds. The results Diagenetic cements precipitated from groundwaters show that for likely shale lengths, vertical permeabili­ are found localized along fault planes. In the Gulf ty is small relative to horizontal values even when Coast basin, faults comprise the most likely migra­ shale interbeds make up a small fraction of the forma­ tion pathways across the thick shale sections separat­ tion. ing deep source beds from reservoirs (NUNN & SA5- The marked effect of shale length on directional SEN 1986); many of the most productive fields here lie permeability in these calculations suggests that aniso­ near fault systems. BODNER et al. (1985) attributed tropy in basin strata might be better estimated by local thermal anomalies in this basin to the effects of considering the depositional environment of the for­ fluids upwelling along growth faults. Of course, faults mation in question. Fig. 12 shows the probability dis­ may be intermittently transmissive, with fractures tribution of the lengths of shale interbeds within opening during movement and later becoming sealed sandstones deposited in various fluvial and marine diagenetically. environments (WEBER 1982). Because shale length The varying hydrologic properties of faults may be varies so strongly with depositional environment, the result of differing stratigraphic settings. Fault pro­ perties depend on the plasticity of enclosing rocks, the compressive stress, the amount of gouge derived primarily from shales, and the extent to which aqui­ 100 ----..:::;;:::::~------....., fers become juxtaposed with aquitards. Thus fault hydraulics can be expected to be related to facies dis­ tributions and the sealing capacity determined in part by shale abundance. Better predictive techniques might arise from study of the relationship of hydro­ logic conditions, diagenetic alteration, and thermal patterns near faults to stratigraphic and tectonic en­ vironment.

7.4 Relative permeability to hydrocarbons 200 400 600 Darcy's law for flow of an immiscible phase (equa­ Length of Shale Interbed (m) tion 10) relates the mobility of a hydrocarbon phase Fig. 12. Cumulative frequency diagram of the lengths of to its relative permeability kro. The concept of rela­ shale interbeds in sandstone formations deposited in various tive permeability follows from the assumption that depositional environments (WEBER 1982). oil and water phases make up three-dimensional net- 146 CRAIG M. BETHKE works through the pore space of the rock (BEAR the most significant of which is probably differences 1972). Each phase moves within its portion of the in the capillary properties of laminae in the carrier pore network according to its own effective perme­ bed. These differences arise, for example, from varia­ ability. tions in grain size and cement distribution among the The value of kro varies with saturation So, approach­ laminae. Oil moves preferentially into laminae with ing zero when oil saturation is too small to maintain large pore openings because of their small capillary an interconnected network. This point is the irreduc­ pressures. ible oil saturation. Permeability to oil approaches the Even in a perfectly homogeneous carrier bed, how­ intrinsic permeability k as the oil phase nears ever, heterogeneities in saturation can develop from complete saturation. The relationship between kro viscous fingering and buoyant segregation. Fronts of and So in small samples is routinely measured in the one phase displacing another of contrasting viscosity laboratory to provide data for reservoir simulation. of relative permeability can be unstable, depending Measured values can be applied to study migration on the velocity of displacement (HILL 1952, CHUOKE in basins only to the extent that the distribution of et al. 1959). An unstable front seeks to lengthen itself oil saturation on the regional scale is analogous to the through viscous fingering, a common phenomenon roughly homogeneous distribution within the labo­ in petroleum reservoirs (BLACKWELL et al. 1959, ratory sample. The nature of two-phase flow, how­ PEACEMAN & RACHFORD 1962). Buoyant segregation ever, conspires to produce heterogeneous saturations works to redistribute less dense hydrocarbon phases within carrier beds (Fig. 13). There are several effects, toward the tops of carrier beds. The thin bitumen stains observed in deep aquifers are evidence of the extent to which petroleum can develop fine saturation structures as it migrates through carrier beds. Such structures act as conduits for migration. Migration along narrow conduits helps explain how hydrocarbons can move for long distan­ ces without being dissipated as the irreducible satura­ tion in the carrier beds. For example, ENGLAND et al. (1987) calculate that even in preferred cases oil must migrate through less than 10% of the volume of Channeling into heterogeneities carrier beds to avoid being dissipated completely. Migration conduits are significant to flow modeling because they allow petroleum to be mobile even when the average saturation of the carrier bed is very small. Petroleum engineers have long been aware that heterogeneities in saturation lead to anomalously large relative permeabilities on the reservoir scale compared to laboratory measurements. To compen­ sate, engineers use »pseudo-functions« that give rela­ tive permeability from saturation averaged over large sections of the reservoir (COATS et al. 1967, HALDOR­ Buoyant segregation SEN 1986). Pseudo-functions are determined by histo­ ry matching or simulating flow through heteroge­ neous domains. The need for pseudo-functions to describe relative permeability is even greater in modeling migration on a regional scale, where saturated conduits provide for flow even when average oil saturations are insigni­ ficant. For example, DOLIGEZ et al. (1986) assumed a pseudo-function that allows at least some oil move­ ment at any saturation in modeling migration in the Viking graben of the North Sea basin (Fig. 14). Fur­ Viscous fingering ther work is needed to establish a quantitative basis Fig. 13. Processes that can cause heterogeneities in oil satura­ for determining pseudo-functions for carrier beds, tion within a carrier bed. because these functions determine migration distan- Modeling subsurface flow in sedimentary basins 147

strongly influences sediment permeability and there­ fore production from reservoirs (GALWWAY 1979). 1.0 OHLE (1951) noted a relationship between permeabi­ \ lity structures and the localization of metallic ores in ...... ~ \ basins. Subsurface reactions further control the ex­ ~ 0.8 \ tent to which contaminants remain mobile as they ~ \ Cb \ migrate through aquifers. ~ \ Models of the distribution of diagenetic reactions ~ 0.6 \ Oil in the subsurface can be formulated by combining the transport equations already presented with descrip­ ~ \~~ \ tion of the equilibrium state or kinetics of the reac­ ~ 0.4 tions considered. The governing equations can be ...... :: \ , ~ , solved analytically for simple cases involving a single ,, reacting component. Examples of such systems in­ 0.2 , clude precipitation or dissolution of quartz or calcite in the absence of significant shifts in solution compo­ " ' .... sition. For example, P ALCIAUSKAS & DOMENICO 20 40 60 80 100 (1976) solved equations describing concurrent trans­ port and reaction of a groundwater flowing through a Water Saturation (%) carbonate aquifer under isothermal conditions. Their Fig. 14. Relative permeability curves for oil and water as analysis assumes that carbonate precipitation and dis­ functions of fractional water saturation of a sedimentary solution is described by rock. Solid line (left) shows the relative permeability to oil assumed by DOLlGEZ et al. (1986) to model migration on a regional scale. Dashed line is a typical laboratory determina­ (13) tion from a small rock sample (COATS et al. 1967). The left side of the equation describes the rate at which the reaction would change solute concentra­ tion in the absence of transport. krxn is the rate con­ ces and the amount of oil that reaches reservoirs in stant that accounts for the chemical kinetics of reac­ migration simulations. tion and the mineral surface area per unit mass of groundwater, and C<'q is the equilibrium solute con­ centration. 8. Unraveling the diagenetic record Subsurface alteration in more complicated geoche­ Diagenetic minerals within basin sediments record mical systems can be modeled by tracing the irrevers­ past thermal, chemical, and hydrologic conditions. ible reactions that accompany mass transfer, assuming Although groundwater flow has long been known to local or partial chemical equilibrium (HELGESON et affect sediment diagenesis (e. g., HAY 1966), the past al. 1970). For example, Fig. 15 traces the diagenetic dozen years has seen increasing appreciation for the effects of meteoric water infiltrating a feldspathic cumulative impact of slow fluid migration over geo­ sandstone originally saturated with a sedimentary logic time periods (DAVIS et al. 1985). For example, brine (BETHKE et al. 1988). More sophisticated calcu­ SIBLEY & BLATI (1976) used cathodoluminescent lations can be applied to determining the distribution microscopy to show that the Tuscarora orthoquartzite of reactions along flow paths (e. g., LICHTNER 1985, in the Appalachian basin, which had been viewed pre­ 1988). viously as an example of pressure welding, was tightly WOOD & HEWETI (1986) considered the patterns cemented by as much as 40% silica cement introduced of mineral precipitation and dissolution that would by advecting groundwater. occur in a polythermal system. They assume a single Predicting the distribution of subsurface reactions reacting component in a groundwater that remains in and the volumes of reactants consumed and products local equilibrium with an aquifer. Fig. 16 shows re­ created within basin strata is of considerable import­ sulting patterns in an aquifer with domal upwarps ance. The type and degree of diagenetic alteration in and structural depressions, considering groundwater aquifers can control where petroleum and natural gas flows in different directions. The geothermal gradient accumulate into reservoirs (WALDSCHMIDT 1941, is assumed to be constant so that the upwarps are LEVANDOWSKI et al. 1973). Diagenetic alteration cooler than the depressions. Minerals such as quartz, 148 CRAIG M. BETHKE

~ 01 I~~ 10-2S ~ -Ih~1 8 ~ IO-eo ~ t7~ ~ 10-75 ------,- , -6

~

o quartz albite -I k-feldspar calcite

nontronite

-4~------~----L---~--~--~~ __ U 0.65 0.45 0.25 0.05 Chlorinity (molal)

Fig. 15. Calculated diagenetic effects of the infiltration of meteoric water containing dissolved oxygen and carbon dioxide into a feldspathic sandstone (BETHKE et al. 1988). Temperature is 60°C and the sandstone is in equilibrium with a sedimentary brine before infiltration.

for which solubility varies in a prograde fashion with 9. Concluding remarks temperature, dissolve where fluids descend and preci­ pitate where they ascend to cooler conditions. Mine­ It is clear that quantitative models of flow and rals that can show retrograde solubilities such as cal­ transport in the subsurface can provide insights to the cite would be likely to react in antithetical patterns processes that shape sedimentary basins over geologic (WOOD 1986). time periods, but which may occur too slowly to be Increasingly, diagenetic patterns are evident in time observed in the field or laboratory. In many cases, as well as space. HEARN et al. (1987) used radiometric such models are the only available tool for studying age determinations of authigenic feldspar over­ basin processes on natural time and distance scales. growths to show that deep sedimentary brines migrat­ Important uncertainties remain, however, in for­ ed through the Appalachian basin during the Alleg­ mulating and applying hydrologic and paleohydro­ henian orogeny in the Permian. Absolute ages of dia­ logic techniques to basin studies. The permeabilities genetic alteration have also been estimated by of basin strata on regional scales are affected by hete­ paleomagnetic techniques (McCABE et al. 1983), and rogeneities such as the distribution of facies and inter­ temperatures at which alteration occurred can some­ beds and faults and fractures. Few quantitative tech­ times be determined from the stable isotopic compo­ niques are available, however, for estimating hydro­ sitions of diagenetic minerals (ESLINGER et al. 1979). logic properties over large scales of observation. In The timing of thermal events can be inferred by fis­ paleohydrologic modeling, records of critical vari­ sion-track dating of apatite and other mineral grains ables such as past topographic relief may have been (NAESER 1979). Such data provide important con­ destroyed by erosion. In addition, there are only straints on the nature of past hydrologic regimes. empirical methods of assessing changes in the Modeling subsurface flow in sedimentary basins 149

(a) ~

. ... ~...... lit::::::··········::::·-- ~(c) (d)

~ Fluid Cooling in;gnnJ Fluid Heating

Fig. 16. Effect of direction of groundwater flow on patterns of diagenetic alteration in an irregularly buried aquifer in which the groundwater remains in local equilibrium with the aquifer (WOOD & HEWETT 1986). Structure contours of the aquifer are shown in map view (a). Relative highs (+) and lows (-) are labeled. In (b-d), areas where fluid warms as it moves structurally upward and cools where it descends are shown for several directions of flow. Minerals with prograde solubilities (e. g., quartz) dissolve where the fluid warms and precipitates where it cools; minerals with retrograde solubilities (e. g., calcite, under many conditions) follow the opposite trend. hydrologic properties of sediments under conditions predictive models of the chemical interactions among of increasing and decreasing effective stress during groundwaters and rocks so that past flows can be in­ basin evolution. There have been few studies of how ferred from the diagenetic record. the various driving forces for fluid flow interact. These uncertainties underscore the importance of Much works remains, furthermore, to refine integrating techniques from a variety of specialties 150 CRAIG M. BETHKE into hydrologic analysis. For example, sedimentologic H Thickness of an aquifer (m) study can help in estimating past topographic relief krxn Rate constant for a chemical reaction (S-I) and describing the morphologies of heterogeneities kl Intrinsic permeability of a sediment along I (m2) k Relative permeability of a sediment to oil within strata; techniques of rock mechanics might be m P Pressure on a groundwater (Pa) used to predict the development of fracture perme­ P, Capillary pressure on an oil phase (Pa) ability; and progress in geochemical research will Po Pressure on an oil phase (Pa) improve the base of thermodynamic and kinetic data ql Specific discharge in an arbitrary direction I (m3 of and theoretical tools for understanding regional pro­ waterlm2 s) cesses of sediment diagenesis. ql. Specific discharge of an oil phase along I (m3 of water/m2 s) q Specific discharge vector (q", q)f' qJ (m3 of water/m2 s) Acknowledgements R.. Rayleigh number for a thermally stratified ground­ water of constant composition thank Thomas Corbet, University of Illinois, Grant So Oil saturation of a sediment, as a fraction of the pore Garven, Johns Hopkins University and Wendy Harrison, volume Colorado School of Mines, for sharing their recent calcula­ t Time (s) tions. Ken Belitz, John Bredehoeft, John Harbaugh, Charles T Temperature (0C) Kreider, Pat Leahy, Florian Lehner, John Sharp, and Phillipe V,m Velocity at which a sediment subsides relative to abso­ Ungerer provided interesting discussions and unpublished lute elevation (m/s) manuscripts. Joan Apperson drafted the figures. Much of x, y Lateral distance (m) the work described herein was supported by National Scien­ z Depth relative to absolute elevation, such as sea level ce Foundation grants EAR 85-52649 and EAR 86-01178, (m) and the generosity of Amoco Production Company, Exxon ex Coefficient of thermal expansion for a groundwater Corp., Texaco USA, and Shell Oil Company. (OC-I) (3 Compressibility of a groundwater (Pa- I) \1 Gradient operator (a/ox, %y, %z) \1 . Divergence operator (cJ/ox, a/ay, o/az) \12 Laplacian operator (cJ2/ax2, a2/oyl, o2/az2j Glossary of Variables J( Thermal conductivity of a fluid saturated sediment Ao Rate of oil generation within a sediment (kg/m3 s) aim s 0C) Ar Rate of reaction of a groundwater per unit volume P Dynamic viscosity of a groundwater (kg/m s) (moles/m3 s) Po Dynamic viscosity of an oil phase (kg/m s) C Solute concentration in a groundwater (moles/m 3 of e Groundwater density (kg/m3) water) eo Density of an oil phase (kg/m3) Ctq Concentration of a solute at chemical equilibrium Qr Density of the sediment grains (kg/m3) (moles/m3 of water) ~ Sediment porosity Cf Heat capacity of a groundwater a/kg 0C) 4> Hydraulic potential of a groundwater (Pa) Cr Heat capacity of the sediment grains a/kg 0C) 4>bdy Hydraulic potential along the water table (Pa) Dim Coefficient of dispersion along dimension m result­ 4>, Hydraulic potential arising from sediment compac­ ing from flow along I (m2/s) tion (Pa) o Dispersion tensor (m2/s) 4>, Hydraulic potential arising from topographic relief g Acceleration of gravity (m/s2) (Pa)

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