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

Grundwasserbewegungen in sedimentaren Becken, die von dem topographischen Relief, konvektionsbedingtem Auftrieb, Sedimentkompaktion, isostatischen Ausgleichsbe­wegungen in Folge von Erosion und v~n ~mbinati~n~n dieser Kriifte gesteuert werden, konnen mit Hilfe quanutattv modellierender Techniken beschrieben werden. In diesen Modellen kann man die Auswirkungen des Transports von Warme und gel osten Stoffen, Petroleum-Migration und die chemische Interaktion zwischen Wasser und dem grund­wasserleitenden Gestein beriicksichtigen.

Die Genauigkeit der Modell-Voraussagen ist allerdings begrenzt wegen der Schwierigkeit, hydrologische Ei­genschaften von Sedimenten in einem regionalen Rahmen vorauszusagen, dem Schatzen vergangener Bedingungen und dem Problem der Abschatzung von Wechselwirkungen physikalischer und chemischer Prozesse in geologischen Zeitriiumen. Fortschritte fUr das Modellieren von Becken werden mit der Integration hydrologischer Forschungsan­strengungen in benachbarte Fachebiete wie Sedimentologie, Gesteinsmechanik und Geochemie zunehmen.

Abstract

Groundwater flows that arise in sedimentary basins from the effects of topographic relief, buoyant convection, sedi­ment compaction, erosional unloading, and combinations of these driving forces can be described using quantitative modeling techniques. Models can be constructed to consider the effects of heat and solute transport, petroleum migra­tion, and the chemical interaction of water and rocks. The accuracy of model predictions, however, is limited by the difficulty of predicting hydrologic properties of sediments on regional dimensions, estimating past conditions such as topographic relief, and knowledge of how physical and che­mical processes interact over gelogic time scales. Progress in basin modeling will accelerate as hydrologic research efforts are better integrated with those of other specialities such as sedimentology, rock mechanics, and geochemistry.

Resume

n est possible, par l'utilisation de techniques quantitatives de modelisation, de decrire les mouvements des eaux souter-

*) Author's address: Dr. C. M. BETHKE, Department of Geology, 1301 West Green Street, University of Illinois, Urbana, Illinois 61801, USA.

raines qui se manifestent dans les bassins sedimentaires, et qui resultent du relief topographique, de la convection, de la compaction des sediments, de la decharge due a l'erosio? et de la combinaison de ces divers facteurs. Dans ces modeles, on peut prendre en con~ideration les effets d~s tr:'nsferts d,e chaleur et de matieres dlssoutes lors de la migration du pe­trole et ceux de I'interaction chimique de I'eau avec les roches. Toutefois la precision des previsions que I'on peut en deduire est limitee par la difficulte d'estimer a I'echelle regia­nale les proprietes hydrologiques des sediments, de reconsti­tuer les conditions anciennes, et de connaitre de quelle maniere les processus physiques et chimiques interferent a I'echelle des temps geologiques. La modelisation des bassins progressera dans la mesure ou la recherche hydrogeologique ser mieux integree a celles d'autres disciplines telles que la se­dimentologie, la mecanique des roches et la geochimie.

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1. Introduction

In recent years the interest in describing quantita­tively the subsurface movement of fluids in sedime~lt­ary basins and the effects of such movements has In­

creased rapidly. Fluids in basins, whether groundwa­ters, hydrocarbons, or gases, are mobile over geologic

no CRAIG M. BETHKE

time periods. Moving fluids create and localize econo­mic resources in sedimentary basins, including petro­leum and gas reservoirs and metallic ores. The increas­ing interest in quantifying subsurface flow stems from escalating costs of locating new resources as well as the need to isolate radioactive and chemical wastes from the biosphere for long periods of time.

There is a strong need in petroleum geology to pre­dict the effects of hydrocarbon migration. Oil reser­voirs sometimes are found more than 150 kilometres from source rocks; others are separated vertically from their sources by kilometers of overpressured shale. The distribution of mature source beds is rou­tinely determined during basin exploration, but the distance and even direction that oil migrates after be­ing released from source rocks are commonly un­known. Efficient mineral exploration also requires knowledge of present and past hydrologic conditions. Sedimentary brines migrate for hundreds of kilo-' metres from deep strata onto basin margins where they precipitate the metallic ores of Mississippi Val­ley-type deposits. Oxidizing surface waters infiltrate basins and leach uranium and other elements, and then precipitate ores in roll-front deposits as they encounter reducing conditions at depth. Promising exploration targets could be identified more accurate­ly in each of these cases with knowledge of the pre­sent or past groundwater hydrology of the basin be­ing explored.

Understanding present-day hydrologic conditions is also of economic and societal concern. Predicting subsurface fluid pressures can be critically important when oil wells are drilled, especially in provinces where overpressures can blowout wells. Safely dispos­ing of persistent toxins and radioactive elements with half-lives of geologic duration requires knowledge of the rates and directions of transport by subsurface fluids.

Quantitative modeling techniques combined with the results of observational and experimental studies have proved successful in analyzing flow and trans­port in sedimentary basins on natural time and dist­ance scales (BETHKE et al. 1988). The purpose of this paper is to examine the variety of models that have been applied to analyze basin processes and consider some of the principal uncertainties in applying these models to study groundwater hydrology in the pre­sent and geologic past.

2. Flow driven by topographic relief

Topographic relief along basin surfaces can drive groundwater through deep strata. The flow is driven by variation in the potential energy of groundwater

along the water table. Given adequate rainfall, the water table forms a subdued replica of the land sur­face so that to a first approximation topography de­scribes the drive for flow. DARTON (1909), in his stu­dy of the Dakota aquifer system in North America, was among the first to recognize the role of topogra­phy in causing groundwater flow on regional scales. The isotopic compositions of sedimentary brines (e. g., CLAYTON et al. 1966) provide chemical eviden­ce that meteoric water circulates deeply in many basins.

2.1 Mathematical model

The subsurface flow field can be predicted accord­ing to a continuum model for any subsurface perme­ability distribution and water table configuration. Ignoring the effects of varying fluid density, ground­water flows according to Darcy's law

Ie, a~ q, :;;: --;;:81 (1)

where q{ is specific discharge (the volumetric flow rate per unit area) in an arbitrary direction I, p, is fluid viscosity, and k{ is directional permeability. Herein the directions of the coordinate axes are assumed to corespond to the principal permeability values; in this case permeability is described as a' vector (or, more accurately, a diagonal tensor) rather than a full tensor quantity. 4> is hydraulic potential

~ :;;: P - pgz (2)

the mechanical energy per unit volume of ground­water (HUBBERT 1940). P and p are fluid pressure and density, g is the acceleration of gravity, and z is depth relative to a fixed datum. Hydraulic potential ac­counts for the work of compression and elevation performed in moving a groundwater of constant den­sity along a flow path; the vector - \7~ is the driving force per volume of groundwater.

Given sufficient time, flow will adjust to a steady state for a given water table configuration. Combin­ing Darcy's law with conservation of mass gives the equation for steady groundwater flow

in three dimensions, where x and yare horizontal di­rections. If the medium can be assumed to be homo­genous and viscosity constant, this equation can be simplified to give

Ie a2~ + Ie a2

c1> + Ie a2c1> c: 0 (3 b)

s az2 u ay2 • az2

Modeling subsurface flow in sedimentary basins 131

and to V2c1> =0 (3c)

if permeability everywhere is isotropic. Equation (3c) is Laplace's equation which has many known solu­tions 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 water table which is at known potential and a strati­graphic contact above an aquitard, such as an evapo­rite bed or crystalline basement rocks, which is taken as a barrier to flow. Side boundaries are generally taken as groundwater divides.

2.2 Subsurface flow fields

Solutions for the subsurface flow field can be ob­tained analytically by symbolic manipulation for basins with simple boundaries and permeability structures, or by numerical methods when the flow system is more complex (FREEZE & WITHERSPOON 1966). The assumption of constant density can be relaxed in obtaining numerical solutions (see Section 3). Fig. 2 shows calculated flow systems for a variety of hypothetical water table configurations and permeability distributions. Equipotentials (dashed lines) connect points of equal hydraulic potential. In a well open to flow at a point intersecting an equipo­tential, water will rise to the elevation at which the equipotential meets the top surface.

Flow fields (a-d) in Fig. 2 are analytical solutions (T6TH 1962, 1963, 1978) that illustrate the salient fea­tures of flow systems arising from topographic relief. In each case, groundwater recharges the flow system at high elevation, and thus high potential. Ground­water everywhere moves toward regions of lower fluid potential, eventually discharging at low elevation. As a consequence, subsurface fluids in recharge areas are underpressured with respect to a column of water ex­tending to the surface, whereas fluids in discharge areas are overpressured. In Fig. 2(b), a layer at depth of relatively low permeability enhances the effect of topography in creating overpressured and underpress­ured regions. In a system with irregular relief (Fig. 2c), flow systems develop over a range of scales from local to regional. When the irregularity of the relief is large compared to overall relief and to the thickness of the flow system (Fig. 2d), the local flow systems can overwhelm regional flow.

Flow fields 2(e) and 2(£), which assume irregular water tables and permeability distributions, were ob­tained numerically (FREEZE & WITHERSPOON 1967). Fig. 2(e) shows that regional scale flows can occur

r~ z

<l> = P-pgz

Fig. 1. Continuum model of groundwater flow in a sedi­mentary basin. Potential distribution is given mathematic­ally as the solution to a boundary value problem that ac­counts for relief on the water table and sediment permeabi­lity. The flow field is then determined from Darcy's law.

even in the presence of highly irregular relief, given a permeable aquifer at depth. In Fig. 2(£), the flow pattern reflects the effects of a discontinuity in the distribution of an aquifer. Here, an aquifer that spans just part of the domain causes recharge and discharge in areas that would not be expected from the water table configuration.

Fig. 3 shows the evolution of groundwater flow sys­tems in two North American basins calculated using continuum models. Flow in the Western Canada Sedi­mentary Basin probably occurred as a basin-wide re­gional system (a) immediately aher the western basin margin was uplihed as the Canadian Rocky Moun­tains developed in the late Tertiary (GARVEN 1988). As the basin surface was eroded, flow evolved into the more localized regimes (b) present today. Plots (c-d) show how Cenozoic erosion has produced under­pressured conditions at depth in the Palo Duro basin (SENGER & FOGG 1987, SENGER et al. 1987). Under­pressures in the present day (d) have arisen from change in the water table configuration due to dissec­tion of the Pecos River Valley and the High Plains. The underpressures have reversed the direction of flow across Permian evaporites that make up a region­al aquitard system. The reversal is significant because present-day flow across the evaporites, which have been considered as sites for disposing of radioactive waste, moves away from the biosphere.

These examples illustrate the importance of recon­structing topographic relief in simulating past groundwater flows. Past relief can sometimes be esti­mated sedimentologically from the grain sizes of sedi­ments shed from uplifted areas, and from the distri­bution of facies in the strata formed as these sedi­ments are deposited (e. g., MARCHER & STEARNS 1962). Erosion rates can be inferred from the sedi­ment loads of rivers draining exposed areas, or from 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 help reconstruct erosional history. In addition, mea­surements of thermal maturity or shale compaction can be used to estimate past burial depths of preserv­ed sediments. The accuracy of models of past basin flows will depend in large part on the extent to which such geological techniques are integrated into paleo­hydrologic stu'dies.

3. Flow driven by buoyancy

Variations in fluid density create buoyant drives for fluid flow in the subsurface. Groundwater density varies mostly because of thermal expansion and chan­ges in salinity. Pressure exerts only a small effect on groundwater, but provides a major control on the densities of hydrocarbon and gas phases. When groundwater density varies, specific discharge is given

by k, (ap az) (4) q,:: --; at - pg ai

This relationship differs from equation (1) in that the work of compressing and elevating groundwater along

a flow path are considered separately rather than in combined form within a hydraulic potential func­tion. In fact, there is no scalar hydraulic potential function applicable to flows of varying density (HUB­BEKf 1940). By equation (4), flow will occur in a verti­cal cross-section given any pressure distribution in response to a lateral density gradient. In other words, there is no pressure distribution that can balance a la­teral density variation so that neither horizontal or vertical flow will occur.

Free convection is the continual overturn of fluid by buoyant forces. Slow fluid circulation by convec­tion provides an attractive explanation for the degree to which basin sediments are altered diagenetically (WOOD & HEWETT 1982, 1984). Considering the quantity of cement and amount of secondary porosi­ty observed in the subsurface, pore fluids in some rocks appear to have been replaced many thousands of times (e. g., SIBLEY & BLATT 1976, LAND & DUT­TON 1979). These quantities could be accounted for either by regional flows or fluids recirculating locally.

There is some field evidence indicating that fluids in reservoir rocks convect. RABINOWICZ et al. (1985)

WEST

WEST

400m~ 40km

Modeling subsurface flow in sedimentary basins

(a)

Texas

10

Ikm~ 50 km

133

EAST

I EAST iOklahoma

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 from the North Sea seem to reflect the effects of con­vective circulation. Convection also may have driven oil migration within the field. AZIZ et al. (1973) found thermal gradients within a French oil field suggesting that reservoir fluids convect actively and redistribute heat.

3.1 Thermal convection

Free convection can arise spontaneously from thermal gradients in a basin. When cool fluids overlie warmer fluids, the system will convect freely given sufficient permeability and aquifer thickness. The cri­terion for the instability of a thermal stratification is the filtration Rayleigh number

kga.p2C,HI1T Ra = ......:..........:....~--

.... K

where ex and QCr are the coefficient of thermal expansion and volumetric heat capacity for the pore fluid, flT is the temperature difference across an aquifer of thickness H, and x is thermal conductivity (LAPWOOD 1948).

Convection is expected for R" > - 40; stratifica­tions are stable at smaller values because ascending fluids cool by conduction too rapidly to maintain their buoyancy. Convective flow fields can be found analytically as 'solutions to boundary-value problems (PHILIP 1982) for many configurations and boundary conditions, or numerically (MERCER et al. 1975) in the general case.

The critical Rayleigh number (about 40) is rather large given the ranges of sediment permeability and aquifer thicknesses likely to occur in basins. By equa­tion (5), a 100-m thick aquifer under a normal geo­thermal gradient requires a permeability of about 2 darcys (2x10-12 m2) to satisfy the Rayleigh criterion; a kilometre-thick aquifer requires about 20 milli­darcys (2xl0-14 m2). Aquifers with large Rayleigh numbers may occur, for example, in sandstone fair­ways in basins at continental margins (BLANCHARD & SHARP 1985).

Determining Rayleigh numbers representative of the subsurface is complicated by heterogeneity in the permeability structures of aquifers. A few percent shale or silt partings in an otherwise homogeneous sandstone can cause strong anisotropy in permeabili­ty (see section 8.2; Fig. 11). Anisotropy works to de­crease R" significantly relative to an isotropic medium of the same horizontal permeability (COMBARNOUS & BORIES 1975). BEGG & CARTER (1987), for example, found that the vertical permeability of a re­servoir in a fluvial sandstone formation was about

three orders of magnitude smaller than the horizon­tal value.

There is no stability criterion when a lateral temperature gradient is superimposed on the vertical geothermal gradient (WOOD & HEWETT 1982); systems where lateral thermal variations are maintain­ed will convect at any value of R". Flow rates when R" is small, however, may be too slow to be signifi­cant even on geologic time scales. Lateral temperature gradients occur due to variations in basement heat flux, structures with large thermal conductivities such as salt domes, or thermal conductivity contrasts between a sloping aquifer and surrounding sediments (DAVIS et al. 1985). In the case of a sloping aquifer (Fig. 4a), elongate convection cells can develop. These cells may redistribute hydrocarbons and localize diagenetic alteration in areas of upward and down­ward flow where thermal gradients along flow paths are steepest. The flow rates predicted for such cells, however, are commonly quite slow, depending on per­meability and the lateral temperature gradient.

There has been little work to determine the extent to which convection persists in the presence of regio­nal flow systems. PRATS (1966) pointed out that a la­teral flow across the domain does not effect the Ray­leigh criteria for convective stability. His analysis, however, does not consider the effects of thermal dis­persion. Dispersion works for hydrodynamic stabili­ty and against formation of convective cells (RUBIN, 1974) because the process increases the apparent ther­mal conductivity of the medium. Since the rate of thermal dispersion varies with flow velocity, superim­posing regional flow might eliminate the possibility of convection, or overwhelm the relatively slow con­vective flow rates so that coexisting convection is less significant.

3.2 Thermohaline convection

Thermohaline convection results when variations in both solute concentration and temperature affect density. Solutes can retard convective circulation when concentrations are highest in warmer fluids, or enhance it when cooler fluids are most saline. Fig. 4(b) shows convective flow near a salt dome. Flow may rise along the dome due to heat conducted from depth by the salt (KEEN 1983), or descend in response to salt dissolving into the groundwater.

HANOR (1987a) found evidence of complex convec­tive patterns surrounding piercement salt domes in the Gulf of Mexico Basin in Louisiana. Flow in this system is driven in part by salt dissolution at shallow depths, which causes warm flows that have ascended 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 me­ters per year.

Thermohaline effects might also be important to the formation and migration of sedimentary brines. Fig. 4(c) shows how seawater concentrated into bit­terns during deposition of evaporite beds might infil­trate into deep strata due to its greater density relative to normal seawater. CARPENTER et al. (1974), on the basis of the panitioning of chlorine and bromine dur­ing the formation of evaporite minerals, envisioned such a process as a possible origin of saline formation waters.

Fig. 4{d) shows brines that have pooled by density stratification into deep strata. Thermal expansion off­sets the increase in density due to salinity sufficiently that warm saline brines pooled at depth may be ei­ther more or less dense that shallower, fresher fluids.

In the latter case, the stratification maintains a dyna­mic stability if upwelling fluids cool too quickly to maintain their buoyancy. The sum of the thermal and solute Rayleigh numbers describes dynamic stability in thermohaline systems (NIELD 1968). Given a sufficient driving force such as a regional flow system set up by topographic relief, the hydraulic gradient in deep aquifers can overwhelm buoyant forces and dis­place deep brines into shallow strata. Such a process is apparently responsible for forming the lead-zinc ores of the Mississippi Valley-type deposits (GARvEN & FREEZE 1984a & b, BETHKE 1986a).

4. Transient flow systems in evolving basins

As basins evolve, groundwater flow systems can develop that are significantly out of equilibrium with

136 CRAIG M. BETHKE

the top?graphy of the .l~nd surface. Transient flow sys­tems arIse from deposltlon of sediments that increases t?e load on deep strata, and from rapid surficial ero­SIO~ that decreases confining stress at depth. Once a basin ceases to evolve at a geologically rapid rate, the disequilibrium hydraulic potentials dissipate toward the distribution that reflects the basin's topography and permeability structure.

4.1 Effects of rapid sedimentation

Large excess pressures develop in shaly basins un­dergoing rapid sedimentation (DICKINSON 1953). The overpressures pose a substantial risk to drillers and may play roles in localizing petroleum reservoirs and ?re deposits. Overp.ressures are commonplace today In the Gulf of MeXICO, North Sea, and Niger Delta basins.

A deep flow regime results in basins during sedi­mentation from the fluid displaced as strata compact under the weight of the accumulating sediments, and to a lesser extent from dehydration of clay minerals (GALLOWAY 1984, BREDEHOEFT et al. 1988). Over­pressures, fluid pressures greatly in excess of hydrosta­tic, are likely to develop in shaly basins where burial proceeds at rates of at least 0.1-1 mm/yr (BETHKE 1986b). In this case, burial proceeds so rapidly that enough fluids cannot escape to allow the sediments to compact fully. The pore fluid, then, becomes over­pressured as it bears part of the overburden weight that would normally be supported by the sediment.

Although the flow rates resulting from compaction are rather slow, generally centimetres or tens of centi­metres per year, the large potential gradients in over­pressured basins seem certain to block meteoric water from circulating into deep sediments. Only small ex­cess potentials, on the other hand, are expected in ba­sins that subside slowly or that contain deep aquifer systems (BETHKE 1985, 1986a).

The effects of sedimentation on pressure evolution in basin sediments can be calculated as the solution to a moving boundary problem. The problem can be solved analytically in the vertical dimension (GIBSON 1958, BREDEHOEFT & HANSHAW 1968). In two di­mensions, the fluid pressure distribution P(x, ~ z, t) can be calculated by numerical solution (BETHKE 1985, BETHKE & CORBET 1988). In the reference frame of the sediments, flow through a deforming medium is described by

(6) ~p ~ = :. [~ [~: 1I + :. [~ [~~ Il + :. [~ [~: - PIII- (1 ~ ~) ~~

In .this equation, t is time, cf> is sediment porosity, and ~ IS flu.id compressibility. The top boundary in the sl~ulatl?n mov~ to accept sedimentation. Porosity varIes With effective stress, which is the difference be­tween the stress exerted by the weight of overlying sediments and the fluid pressure.

Using this technique, DOLIGEZ et al. (1986) and BETHKE et al. (1988) simulated development of over­pressures in the Viking graben of the North Sea basin and the Gulf of Mexico basin. Fig. 5 shows the evolu­tion of fluid pressures calculated for a north-south cross-section through the Gulf of Mexico basin. Fluid pressures are represented as average gradients, the ratio of pressure to burial depth. A hydrostatic gradient is about 10 MPa/km; lithostatic is about 23 MPa/km. The calculation shows that the expanse of overpress­ured sediments increased over the past 30 m. y. to the present maximum. Many sediments became over­pressured within the past 2 m. y.

The phenomenon of young overpressures in older sediments poses special difficulty in modeling be­cause the hydrologic properties of sediments must be ~stimat.ed when effective stress is decreasing as well as I?cre~slng. Fo~ exam~le, because sediment compac­~Ion IS largely Irreversible, microfractures can develop In shales that develop significant overpressures, de­pending on the confining stress field (DOMENICO & PALCIAUSKAS 1979). UNGERER et al. (1987), in their model of compaction-driven flow, assumed that the permeabilities of sediments that develop overpress­ures increase due to microfracturing, assuming that lateral stress constitutes a fixed fraction of vertical load.

4.2 Effects of erosion

Flow pattern~ that. are in disequilibrium with pre­s~nt topographic. rehef can develop in basins being dissected by erosIOn (T6TH & CORBET 1986). Two effects are significant in strata with sufficiently small permeabilities. First, subsurface flow patterns can remain partly adjusted to a previous higher position of the water table (T6TH & MIllAR 1983). This effect produces pressures in excess of an equilibrium gra­dient from the current land surface. On the other hand, un?e~p~e~ures ca? develop in deep strata due to the diminishing weight of overlying sediments and, to a lesser extent, the thermal contraction of pore fluids (NEUZIL & POLLOCK 1983). On the basis of available estimates. of material properties, underpres­sur~s seem more hkely than overpressures in eroding basinS (NEUZIL 1986). The origin and persistence of 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 expected to drive flow into deep strata and away from the biosphere (BRADLEY 1985), and because of the possible effects of such environments on petroleum migration.

Flow patterns in eroding basins can be calculated as solutions to an initial-boundary value problem. The scheme is similar to that applied in basins accepting sedimentation (equation 6), except that the upper boundary subsides during erosion. Fig. 6a shows the transient flow pattern in a basin containing a 540 m­thick aquitard of small permeability (10-7 to 10-8 dar­cys; 10-19 to 10-20 m2) and an anisotropy of 10-1

(THOMAS CORBET, unpublished data). The basin has undergone erosion for 5 m. y. at rates between zero (at left boundary) and 0.1 mm/yr (right). Strata expand somewhat as they are unloaded, so that the apparent erosion rate is slightly less than these values. In the simulation, sediment porosity increases at about 10% of typical compaction rates during burial in basins to account for the fact that sediment compaction is only partly reversible (NEUZIL 1986). The calculation is

also representative of basins with more permeable aquitards undergoing faster erosion.

Hydraulic potential, contoured in MPa relative to surface potential at the right boundary, is at a local minimum within the aquitard due to its expansion and to a lesser extent the thermal contraction of pore fluids. The transient flow regime (a) can be compared to the pattern in equilibrium with the land surface (Fig. 6b). The equilibrium pattern was calculated from the topographic relief and permeability distri­bution (see section 2.1), ignoring the effects of ero­sion. Given sufficient time, the transient regime would approach this steady state. From the calcula­tions, erosion has generated hydraulic potentials as much as 0.6 MPa (60 m of hydraulic head) less than those expected at steady state. This class of models is limited, however, by poor understanding of the phy­sical properties of sediments as they are unloaded over geologic time. In particular, the degree to which sediment porosity rebounds and to which fracture permeability develops during unloading is poorly known.

138 CRAIG M. BETHKE

5 m.y. surface --------------------------------------p;;;;~t-l

(b) Fig. 6. Transient flow system (a) resulting from erosional dis­section of basin surface (THOMAS CORBET, unpublished data). Equipotentials (MPa) are contoured relative to surface potential at right of figure. Comparison to the steady-state flow system (b) shows that the transient system has not equi­librated with the land surface.

5. Incursion of fresh water into compacting basins

Each of the processes discussed to this point is like­ly to drive subsurface flow under certain conditions, but relatively little work has been aimed at determin­ing how these processes interact. Consider the hydro­logic regime near the margin of a basin accepting ra­pid sedimentation. Sediment compaction within the basin drives fluids landward. Topographic relief on the coastal plain, however, drives an opposing flow system of fresh water basinward (GALLOWAY 1984). The basinward flow converges with the compaction­flow system and discharges vertically. Flow in the freshwater regime can be relatively rapid and extend for considerable distances subsea.

As fresh waters infiltrate basin strata, they cause sig­nificant chemical changes in the subsurface. Some carbonate cements are associated with meteoric flows, and the infiltration of dilute pore fluids can cause se­condary porosity by dissolving silicate grains (BJ0R. LYKKE 1984). Oxidizing flow systems form ore bodies by precipitating uranium minerals as flow encounters reducing conditions at depth (SANFORD 1982). Infil­trating fresh water can degrade petroleum by leaching the more soluble hydrocarbon compounds and by

bacterial attack. Further, the presence of unusually dilute pore water alters the electrical conductivity of deep sediments, complicating interpretation of the re­sistivity logs used to locate reservoirs during petro­leum exploration (DICKEY et al. 1987).

Flows resulting from the combined effects of com­paction and topographic relief can be modeled by solving the equation of flow in a deforming medium (6) subject to a boundary condition reflecting the to­pography of the coastal plain (Fig. 1). The overall so­lution 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 compac­tion, is given as the solution to an initial-boundary value governed by

.!.. (~ acz, e ) + .!.. (~ acz, e 1 + .L (~ acz, c ) az ,... az ay J1 a1l az,... a: (7b)

( acz,c 1 1 ~ = <I>~ at + pgv"" + (1 - <1» at

subject to the upper boundary condition 4>Wy = o. Equation (7b) is (6) recast in terms of hydraulic poten­tial.

The solution to the combined problem is

(7c)

which can be seen to satisfy the equation of flow in a deforming medium (equation 7b) as well as the boun­dary conditions reflecting topographic relief. By Dar­cy's law (equation 1), discharges predicted by solving (7a) and (7b) are also additive to give the overall flow rates. The decoupled solution (7a-c) is strictly valid when hydraulic potential does not affect permeabili­ty. In practice, the procedure is a good approximation because large potentials generally arise from sediment compaction in strata too impermeable to host signi­ficant topographic flow.

Fig. 7 shows interaction of the flow regimes set up by sediment compaction and relief on the coastal plain of the Texas Gulf Coast (HARRISON & BETHKE 1986, BETHKE et al. 1988). The extent of the meteoric flow is shown before and after a drop in sea level oc­curring 31 m. y. ago during the Oligocene (HAQ et al. 1987). The drop increased the hydraulic potential rela­tive to sea level of groundwater along the coastal

Modeling subsurface flow in sedimentary basins 139

plain, and exposed more of the coastal plain to mete­oric precipitation. Fresh water invaded more basal strata in the Oligocene than in the present day be­cause the strata were less deeply buried and wide­spread geopressures had yet to develop. At the low stand of sea level, fresh water infiltrates into deeper strata and farther basinward along coastal aquifers.

There is field evidence that changes in sea level in the geologic past have affected present-day flow sys­tems in coastal aquifers. MEISLER et al. (1985) found that pore fluids in the Atlantic Coastal Plain are di­luted by fresh water 100 kilometres offshore of New Jersey. The authors interpreted the distal freshwater occurrences as remnants of low stands of sea level from the Pleistocene. ESSAID (1988) observed that flow in the coastal aquifers of Monterey Bay, Califor­nia, is still adjusting to Pleistocene fluctuations in the sea level.

6. Hydrologic transport

Many problems in basin hydrology are ultimately concerned with the ability of groundwater flow sys­tems to transport dissolved mineral mass, thermal energy, or hydrocarbons within basins. Like ground­water flow, the effects of hydrologic transport pheno­mena can be studied as boundary value problems for­mulated using continuum theory.

2kmL 100 km

6.1 Solute transport

Moving groundwaters redistribute their dissolved load in the subsurface by advection, diffusion, and hydrodynamic dispersion. The components carried in solution are the raw materials for precipitating dia­genetic minerals and metallic ores. For this reason, understanding transport processes in basins is basic to constructing quantitative models of sediment diage­nesis (WOOD & SURDAM 1979) and effective explora­tion strategies for ore deposits (OHLE 1951, 1980).

The distribution and transport of a solute through a groundwater flow system is described by the equa­tion

:t (cf>C) = V'(cf>D VC) - v·(qC) + cf>A, (8)

where C is solute concentration; Ar is the rate of reac­tion with the medium per volume of fluid, where positive values denote dissolution reactions. The tensor

describes the combined processes of molecular diffu­sion and hydrodynamic dispersion.

(b)

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 pro­cess. The process results primarily from the branch­ing and joining of flow paths and variations in mi­croscopic flow rates as the fluid moves around rock grains and through heterogeneities in the medium. Dispersion occurs both along and transverse to the direction of bulk flow, but at different rates. Hence the process is described by a tensor that contains co­efficients for transport in each direction resulting from the component of flow along each axis (BEAR 1979).

To date there has been little work on determining values for the coefficients of the dispersion tensor for 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) represents dispersion as a Fickian process, that is in the form of Fick's law of diffusion, although detailed 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 sea­water. The brines are believed to be the residual bitterns from precipitating evaporite beds (CARPEN. TER et al. 1974), groundwaters that have dissolved eva­porite minerals in the subsurface (LAND & PREZBIN­DOWSKI 1981), or waters concentrated by membrane filtration (BREDEHOEFT et al. 1963, GRAF 1982). Groundwaters in marine sediments that are more di­lute than seawater are sometimes explained as con­taining the water released as clay minerals dehydrate, but more commonly are attributed to infilrating me­teoric 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 meteo­ric recharge along a portion of the boundary (DOME­NICO & ROBBINS 1985). At steady state the infiltrat­ing fresh water increases in concentration along its flow paths by dispersive mixing, forming a broad range of salinities.

RANGANATHAN & HANOR (1987) studied the ef­fects of a basal salt layer on the salinity distribution in a basin undergoing sedimentation. Their results show that diffusion over geologic time can significantly af­fect groundwater salinity in overlying strata. GARVEN & FREEZE 1984a, b) used transport theory to show that metals leached from sediments deep in basins can

be carried by advection into shallow strata to form ore deposits.

6.2 Heat transfer

Basin thermal budgets are dominated by heat con­ducted into the sedimentary pile from the underlying crystalline crust. Basement heat flow ultimately moves to the surface by conduction, which is perhaps most common, or through the advection of ground­water. When advective transfer dominates, discharge areas are warmed and recharge areas cooled relative to a conductive gradient (e. g., DOMENICO & PALCIAUS­KAS 1973, HITCHON 1984).

~ __ -- 2500

~ ____ --------------2~----~

~~['----~ 50km

(0)

100km

Fig. 8. Calculated effects of mass and heat transport by groundwater flows in sedimentary basins. Plot (a) is a map view showing the steady-state salinity distribution in an aquifer containing a connate brine and being recharged by meteoric water (DOMENICO & ROBBINS 1985). Contours show solute concentration in mg/l, and the hatched region is the recharge area. Plot (b) shows the calculated effects of a Mesozoic groundwater flow system on the subsurface temperature distribution along a cross-section through the Illinois basin (BETHKE 1986a). Contours give temperature in °C.

Modeling subsurface flow in sedimentary basins 141

Assuming that groundwater and sediment maintain thermal equilibrium locally, heat transfer by conduc­tion and advection is described by

Here, Cf and Cr are fluid and rock heat capacities, Qr

is density of the rock grains, x is thermal conducti­vity (modified to account for thermal dispersion), and hw is fluid enthalpy. Variants of this equation can also describe cases in which fluid and rock do not equilibrate thermally (COMBARNOUS & BORIES 1975). Equation (9) can be solved for the temperature field T(x, y, z, t) in a basin where groundwater discharge q is known from solution of the flow equations (e. g., DOMENICO & PALCIAUSKAS 1973).

Heat transfer theory has been applied to study how ores form in sedimentary basins. Mississippi Valley­type lead-zinc deposits occur in shallow sediments on basin margins. On the basis of fluid inclusion studies and isotopic analyses, the ores are believed to have precipitated from sedimentary brines at temperatures ( - 50 to 200°C) characteristic of deep strata (SVER­JENSKY 1986). GARVEN & FREEZE (1984a, b) and GARVEN (1985) showed that groundwater flow driven by topographic relief in some basins is capable of transporting brines from deep strata to the sites of de­position. In the resulting hydrothermal system, the brines move rapidly enough to avoid complete cool­ing by conduction to the surface.

Fig. 8(b) shows the temperature distribution in the Illinois basin resulting from such a hydrothermal sys­tem. Flow is driven by uplift of the Pascola arch in the southern basin (BETHKE 1986a). In the calcula­tion, groundwater flowing through basal Paleozoic aquifers entrains heat conducted into the basin from below. The groundwater advects the heat along flow paths, creating a significant thermal anomaly in the discharge area. A regional hydrothermal system of this type probably formed the ores of the lead-zinc district on the northern margin of the basin.

Equation (9) can sometimes be used to infer groundwater flow in basins with known temperature distributions. In this case, the flow field q is the unknown variable. For example, WILLET & CHAPMAN (1987) used temperature measurements from oil wells to constrain groundwater flow rates and the permeability structure of the Uinta basin. WOODBURY & SMITH (1987) considered mathe­matical techniques to automatically find the most satisfactory flow regime by inverting field data for temperature and hydraulic head.

6.3 Petroleum migration

It is clear from the distribution of source rocks and petroleum reservoirs that oil can migrate for remark­able 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 Creta­ceous and Early Tertiary source rocks through thick overpressured shale sections to present reservoirs, a process that continues today (NUNN & SASSEN 1986).

Discharge of an oil phase along I through a carrier bed is described by

__ k,e k, (ap. _ az) q,. - IL. al P.g al (10)

where Po and Jlo are the oil-phase density and viscosi­ty (PEACEMAN 1977). The buoyant drive on the oil phase arises from the lesser value of Qo relative to the density of water. Equation (10) differs from the flow law for a single phase (equation 4) by accounting for the pressure on the oil phase Po and by the factor kro, the relative permeability of the medium to oil.

The oq-phase pressure is the sum of the water and capillary pressures

(11)

Capillary pressure increases with oil saturation So in a given rock, and with decreasing apertures of the pore throats for rocks in general. By equations (10-11), oil will seek areas where capillary pressures are small; hence there is a strong drive for oil to migrate through rocks with broad pore openings. Relative permeability increases from zero at small saturation toward one as as oil saturates most of the pore volume.

Like groundwater flow, petroleum migration in a sedimentary basin can be modeled as a boundary va­lue problem. The governing equation may be repre­sented

1...( s) = ..!..[pokrokz (ap. lj + ..!..[pok,.kl/ (ap. lj at cjlPo. az IL. az all IL. all

+..!..[P.kr.k6(

apO + lj+A az IL. az P.g •

(12)

where Ao is the local rate of petroleum generation. Because capillary pressure and relative permeability vary sharply with saturation, equation (12) is non­linear in its coefficients and must be solved numeric­ally. A variety of solution techniques have been de­veloped in the petroleum industry to facilitate simu­lation of multiphase flow in oil reservoirs (e. g., PEACEMAN 1977).

142 CRAIG M. BETHKE

Much effort is expended during petroleum explora­tion to delineate the extent of mature source rocks in basins; attempts at quantifying distances or even di­rections of secondary migration are less common. The theory of the kinetics of petroleum generation is relatively well established and calibrated (e. g.; TISSOT & WELTE 1984, LEWAN 1985), and the properties of evolving oils can be estimated (UNGERER et a1. 1981, ENGLAND et al. 1987). These theories can be combin­ed with the techniques already presented for calculat­ing flow and transport to describe in an integrated fashion the oil generation, migration, and evolution of groundwater flow systems in basins.

DOLIGEZ et a1. (1986) modeled pressure evolution, generation, and migration along a cross section through the Viking graben of the North Sea basin; UNGERER et a1. (1986) applied similar techniques to study migration in the Suez Rift. Fig. 9 shows an ex­ample calculation of petroleum migration during ba­sin development (LEHNER et a1. 1987). The calcula­tion accounts for secondary migration and reservoir development due to buoyancy in a hypothetical car­rier bed. The thickness of the oil column at each point in the domain is tracked through time. As the bed undergoes burial, oil is generated at rates deter­mined by the temperature and thermal history of underlying source rocks. The calculation shown does not account for groundwater flow through the bed or variation in capillary pressure, but the technique might be readily generalized.

Considerable uncertainty remains, however, about the nature of migration. From equations (10-11), petroleum moves in response to buoyancy, the hydro­dynamic drive of groundwater flow, and gradients in capillary pressure resulting from heterogeneous distri­butions of porosity or grain sizes (HUBBERT 1953). Although the relative magnitudes of these factors can be calculated (e. g., DAVIS 1987, JENNINGS 1987), little is known about how they interact. Buoyancy, for ex­ample, might provide the dominant driving force for migration, but hydrodynamic flow may be required to sweep oil from the structural irregularities of car­rier beds to traps of commercial size. In addition, he­terogeneity in the capillary properties of a bed may control whether oil can migrate for long distances through carrier beds without being dissipated as irre­ducible saturation (see Section 7.4). These questions might be answered by combining statistical descrip­tions of irregularities and heterogeneities within car­rier beds with detailed modeling studies.

7. Hydrologic properties on regional scales

It is broadly recognized that the hydrologic proper­ties of porous media vary with the scale on which

they are observed (BEAR 1972). Thus, measurements made on small samples may describe poorly the beha­vior of the rock unit sampled in the vicinity of a well. The local behavior, in turn, may not represent pro­perties of the same rock on the regional scale of inter­est in basin hydrology. Because there is no direct tech­nique for measuring the hydrologic properties needed to model flow on large scales, developing effective methods for inferring regional properties is among the most significant challenges in basin hydrology.

7.1 Regional permeability

Permeability in sedimentary basins tends to in­crease (but can decrease) with breadth of the scale of observation. This phenomenon is attributed to the effects of heterogeneities in the medium. Fracture sets, karst networks, and lenses of coarse grained sedi­ments contribute to the added permeability found on macroscopic scales. Fig. 10 shows the effect of scale on hydraulic conductivity in carbonate aquifers of central Europe. Conductivities determined by labora­tory measurement represent the effects of the prim­ary porosity and microfractures of small samples. Values determined from well tests, which encompass the response of the area near the well-bore, are consi­derably larger because flow also moves through ma­croscopic fracture sets. Conductivities inferred on re­gional scales are about four orders of magnitude lar­ger than the laboratory measurements because of the effect of regional karst networks in the aquifers.

Scale effects can also be significant in aquitards. Large volumes of low-permeability rocks can be more conductive than small samples because of the pre­sence of joints, fractures, and faults (NEUZIL 1986). BREDEHOEFT et a1. (1983) found that the vertical con­ductivity of the Cretaceous Pierre Shale in South Dakota is as much as a thousand times greater on a regional scale than values suggested by laboratory or in situ tests. Many aquitards, however, fail to develop significant permeability.

Regional permeabilities sometimes can be inferred by matching results of numerical simulations to historical observations of groundwater systems during their exploitation (BREDEHOEFT et a1. 1983), or indirectly by estimating flow rates geochemically in aquifers with known head gradients (PEARSON & WHITE 1967). When these methods cannot be ap­plied, as would be the case in attempts to study flows occurring in the geologic past, techniques are needed to determine regional permeabilities from observable properties of the medium.

Recently, efforts have been made to improve under­standing of the nature of regional permeability by

Modeling subsurface flow in sedimentary basins

(0) (b)

(d)

143

Oil Column (m)

iii >10 t------~ 1-10 ~ .1-1 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

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 pri­marily along left side of figure.

I Karst

Network

t Fracture Sets

Laboratory * .~ Porosity and ~ 1~8~~~~~~~~~~~~~~~~~~~~Microkadures

~ 10-1

~ :t: 10° 101 102 103

Scale of Measurement (m) Fig. 10. Schematic representation of the effect of scale on typical hydraulic conductivities of carbonate rocks to water in cen­tral Europe (GARvEN 1986, after KIRALY 1975).

144 CRAIG M. BETHKE

quantifying the heterogeneities in basin strata. For example, probability distributions of fracture apertu­re, density, and lengths can be estimated by fracture mapping techniques applied at well-bore or outcrop. The distributions can then be used to define Monte Carlo simulations (SMITH & SCHWARTZ 1984) of flow through media with statistically distributed fractures sets. Such simulations give estimates of regional pro­perties directly. Alternatively, heterogeneities arising from facies distributions might be defined using de­terministic models of sediment deposition (e. g.; DOYLE et al. 1988, TETZLAFF 1988).

7.2 Anisotropies of basin strata

Permeability anisotropy of basin strata exerts a strong influence on the direction of subsurface flow.

140

For example, anisotropies assumed in modeling topo­graphy-driven flow can control whether regional or local flow systems develop in a basin with irregular topography (see Fig. 2). Most experimental studies of groundwater flow in natural media have employed sandstones chosen for their homogeneity, such as the Berea in North America and the Bentheim in Eu­rope. Choice of these nearly isotropic sands has serv­ed to de-emphasize the amount to which permeabi­lity varies with flow direction in sedimentary rocks.

Anisotropy in permeability results from microsco­pic factors such as the orientation of mica flakes and variations in grain size among individual laminae in the sediment, as well as from macroscopic heteroge­neities. Heterogeneities include bedding shale part­ings, interlayered formations, and fracture sets. Ani­sotropy, like permeability, varies with the scale of ob-

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

~ =- -=---

:t:: (b) q,

~ ~ 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 more pronounced as system dimensions increase.

For this reason, although the permeability of a la­boratory sample most commonly varies with the di­rection of measurement by less than an order of mag­nitude, sediments typically show greater anisotropies when studied on larger scales. BEGG & CARTER (1987) found good results by simulating production in a reservoir composed of fluvial sandstones with short shale interbeds by assigning an anisotropy (k/kj of 10-). FOGG (1986) estimated anisotropy in the Wilcox aquifer system in east Texas to be about 10-" on a regional scale.

The influence of heterogeneities can be investigated quantitatively by stochastic methods. Fig. 11 shows the calculated effect of the most likely shale length on the anisotrophy of a reservoir sandstone containing just 5 percent shale interbeds (BEGG & KING 1985). In the three-dimensional calculation, the thicknesses, lengths, and breadth were determined stochastically from randomly distributed interbeds. The results show that for likely shale lengths, vertical permeabili­ty is small relative to horizontal values even when shale interbeds make up a small fraction of the forma­tion.

The marked effect of shale length on directional permeability in these calculations suggests that aniso­tropy in basin strata might be better estimated by considering the depositional environment of the for­mation in question. Fig. 12 shows the probability dis­tribution of the lengths of shale interbeds within sandstones deposited in various fluvial and marine environments (WEBER 1982). Because shale length varies so strongly with depositional environment,

100 ----..:::;;:::::~--------.....,

200 400 600

Length of Shale Interbed (m) Fig. 12. Cumulative frequency diagram of the lengths of shale interbeds in sandstone formations deposited in various depositional environments (WEBER 1982).

these data suggest that formation anisotropy might be estimated even in the absence of detailed statistical data on the basis of sedimentological study.

7.3 Roles of faults and fractures

Although faults can dominate the structures of se­dimentary basins and may be clearly visible in seis­mic profiles, their role in affecting subsurface flow is poorly understood. Faults can provide a barrier to la­teral flow or channel flow across stratigraphy. Changes in fluid pressure across faults and discontinuities in the levels of oil-water contacts provide evidence that faults can serve as subsurface seals (WEBER 1986). Be­cause petroleum is retained in faulted reservoirs, seals seem to be able to remain intact over geologic time periods.

On the other hand, faults clearly are capable of channelling flow. Many hydrothermal ores in sedi­mentary basins are deposited in or near fault systems. Diagenetic cements precipitated from groundwaters are found localized along fault planes. In the Gulf Coast basin, faults comprise the most likely migra­tion pathways across the thick shale sections separat­ing deep source beds from reservoirs (NUNN & SA5-SEN 1986); many of the most productive fields here lie near fault systems. BODNER et al. (1985) attributed local thermal anomalies in this basin to the effects of fluids upwelling along growth faults. Of course, faults may be intermittently transmissive, with fractures opening during movement and later becoming sealed diagenetically.

The varying hydrologic properties of faults may be the result of differing stratigraphic settings. Fault pro­perties depend on the plasticity of enclosing rocks, the compressive stress, the amount of gouge derived primarily from shales, and the extent to which aqui­fers become juxtaposed with aquitards. Thus fault hydraulics can be expected to be related to facies dis­tributions and the sealing capacity determined in part by shale abundance. Better predictive techniques might arise from study of the relationship of hydro­logic conditions, diagenetic alteration, and thermal patterns near faults to stratigraphic and tectonic en­vironment.

7.4 Relative permeability to hydrocarbons

Darcy's law for flow of an immiscible phase (equa­tion 10) relates the mobility of a hydrocarbon phase to its relative permeability kro. The concept of rela­tive permeability follows from the assumption that oil and water phases make up three-dimensional net-

146 CRAIG M. BETHKE

works through the pore space of the rock (BEAR 1972). Each phase moves within its portion of the pore network according to its own effective perme­ability.

The value of kro varies with saturation So, approach­ing zero when oil saturation is too small to maintain an interconnected network. This point is the irreduc­ible oil saturation. Permeability to oil approaches the intrinsic permeability k as the oil phase nears complete saturation. The relationship between kro and So in small samples is routinely measured in the laboratory to provide data for reservoir simulation.

Measured values can be applied to study migration in basins only to the extent that the distribution of oil saturation on the regional scale is analogous to the roughly homogeneous distribution within the labo­ratory sample. The nature of two-phase flow, how­ever, conspires to produce heterogeneous saturations within carrier beds (Fig. 13). There are several effects,

Channeling into heterogeneities

Buoyant segregation

Viscous fingering Fig. 13. Processes that can cause heterogeneities in oil satura­tion within a carrier bed.

the most significant of which is probably differences in the capillary properties of laminae in the carrier bed. These differences arise, for example, from varia­tions in grain size and cement distribution among the laminae. Oil moves preferentially into laminae with large pore openings because of their small capillary pressures.

Even in a perfectly homogeneous carrier bed, how­ever, heterogeneities in saturation can develop from viscous fingering and buoyant segregation. Fronts of one phase displacing another of contrasting viscosity of relative permeability can be unstable, depending on the velocity of displacement (HILL 1952, CHUOKE et al. 1959). An unstable front seeks to lengthen itself through viscous fingering, a common phenomenon in petroleum reservoirs (BLACKWELL et al. 1959, PEACEMAN & RACHFORD 1962). Buoyant segregation works to redistribute less dense hydrocarbon phases 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 distan­ces without being dissipated as the irreducible satura­tion 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 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 compen­sate, engineers use »pseudo-functions« that give rela­tive permeability from saturation averaged over large sections of the reservoir (COATS et al. 1967, HALDOR­SEN 1986). Pseudo-functions are determined by histo­ry matching or simulating flow through heteroge­neous 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 insigni­ficant. For example, DOLIGEZ et al. (1986) assumed a pseudo-function that allows at least some oil move­ment at any saturation in modeling migration in the Viking graben of the North Sea basin (Fig. 14). Fur­ther work is needed to establish a quantitative basis for determining pseudo-functions for carrier beds, because these functions determine migration distan-

Modeling subsurface flow in sedimentary basins 147

1.0

~ \ ...... \ ~ 0.8 \ ~ \ Cb \ ~

0.6 \ ~ \ Oil

~ \~~ ~ 0.4 \ ......:: \

~ , , ,

0.2 , , " ' ....

20 40 60 80 100

Water Saturation (%) Fig. 14. Relative permeability curves for oil and water as functions of fractional water saturation of a sedimentary 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 determina­tion from a small rock sample (COATS et al. 1967).

ces and the amount of oil that reaches reservoirs in migration simulations.

8. Unraveling the diagenetic record

Diagenetic minerals within basin sediments record past thermal, chemical, and hydrologic conditions. Although groundwater flow has long been known to affect sediment diagenesis (e. g., HAY 1966), the past dozen years has seen increasing appreciation for the cumulative impact of slow fluid migration over geo­logic time periods (DAVIS et al. 1985). For example, SIBLEY & BLATI (1976) used cathodoluminescent microscopy to show that the Tuscarora orthoquartzite in the Appalachian basin, which had been viewed pre­viously as an example of pressure welding, was tightly cemented by as much as 40% silica cement introduced by advecting groundwater.

Predicting the distribution of subsurface reactions and the volumes of reactants consumed and products created within basin strata is of considerable import­ance. The type and degree of diagenetic alteration in aquifers can control where petroleum and natural gas accumulate into reservoirs (WALDSCHMIDT 1941, LEVANDOWSKI et al. 1973). Diagenetic alteration

strongly influences sediment permeability and there­fore production from reservoirs (GALWWAY 1979). OHLE (1951) noted a relationship between permeabi­lity structures and the localization of metallic ores in basins. Subsurface reactions further control the ex­tent to which contaminants remain mobile as they migrate through aquifers.

Models of the distribution of diagenetic reactions in the subsurface can be formulated by combining the transport equations already presented with descrip­tion of the equilibrium state or kinetics of the reac­tions considered. The governing equations can be solved analytically for simple cases involving a single reacting component. Examples of such systems in­clude precipitation or dissolution of quartz or calcite in the absence of significant shifts in solution compo­sition. For example, P ALCIAUSKAS & DOMENICO (1976) solved equations describing concurrent trans­port and reaction of a groundwater flowing through a carbonate aquifer under isothermal conditions. Their analysis assumes that carbonate precipitation and dis­solution is described by

(13)

The left side of the equation describes the rate at which the reaction would change solute concentra­tion in the absence of transport. krxn is the rate con­stant that accounts for the chemical kinetics of reac­tion and the mineral surface area per unit mass of groundwater, and C<'q is the equilibrium solute con­centration.

Subsurface alteration in more complicated geoche­mical systems can be modeled by tracing the irrevers­ible reactions that accompany mass transfer, assuming local or partial chemical equilibrium (HELGESON et al. 1970). For example, Fig. 15 traces the diagenetic effects of meteoric water infiltrating a feldspathic sandstone originally saturated with a sedimentary brine (BETHKE et al. 1988). More sophisticated calcu­lations can be applied to determining the distribution of reactions along flow paths (e. g., LICHTNER 1985, 1988).

WOOD & HEWETI (1986) considered the patterns of mineral precipitation and dissolution that would occur in a polythermal system. They assume a single reacting component in a groundwater that remains in local equilibrium with an aquifer. Fig. 16 shows re­sulting patterns in an aquifer with domal upwarps and structural depressions, considering groundwater flows in different directions. The geothermal gradient is assumed to be constant so that the upwarps are cooler than the depressions. Minerals such as quartz,

148 CRAIG M. BETHKE

~ 01 ~ ~ -I 8 I~ 10-2S h~1 ~ 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 temperature, dissolve where fluids descend and preci­pitate where they ascend to cooler conditions. Mine­rals that can show retrograde solubilities such as cal­cite would be likely to react in antithetical patterns (WOOD 1986).

Increasingly, diagenetic patterns are evident in time as well as space. HEARN et al. (1987) used radiometric age determinations of authigenic feldspar over­growths to show that deep sedimentary brines migrat­ed through the Appalachian basin during the Alleg­henian orogeny in the Permian. Absolute ages of dia­genetic alteration have also been estimated by paleomagnetic techniques (McCABE et al. 1983), and temperatures at which alteration occurred can some­times be determined from the stable isotopic compo­sitions of diagenetic minerals (ESLINGER et al. 1979). The timing of thermal events can be inferred by fis­sion-track dating of apatite and other mineral grains (NAESER 1979). Such data provide important con­straints on the nature of past hydrologic regimes.

9. Concluding remarks

It is clear that quantitative models of flow and transport in the subsurface can provide insights to the processes that shape sedimentary basins over geologic time periods, but which may occur too slowly to be observed in the field or laboratory. In many cases, such models are the only available tool for studying basin processes on natural time and distance scales.

Important uncertainties remain, however, in for­mulating and applying hydrologic and paleohydro­logic techniques to basin studies. The permeabilities of basin strata on regional scales are affected by hete­rogeneities such as the distribution of facies and inter­beds and faults and fractures. Few quantitative tech­niques are available, however, for estimating hydro­logic properties over large scales of observation. In paleohydrologic modeling, records of critical vari­ables such as past topographic relief may have been destroyed by erosion. In addition, there are only empirical methods of assessing changes in the

Modeling subsurface flow in sedimentary basins 149

(a) ~

. ... ~ ..... --...

lit::::::··········::::·

~(c)

~ Fluid Cooling

(d)

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 of increasing and decreasing effective stress during basin evolution. There have been few studies of how the various driving forces for fluid flow interact. Much works remains, furthermore, to refine

predictive models of the chemical interactions among groundwaters and rocks so that past flows can be in­ferred from the diagenetic record.

These uncertainties underscore the importance of integrating techniques from a variety of specialties

150 CRAIG M. BETHKE

into hydrologic analysis. For example, sedimentologic study can help in estimating past topographic relief and describing the morphologies of heterogeneities within strata; techniques of rock mechanics might be used to predict the development of fracture perme­ability; and progress in geochemical research will improve the base of thermodynamic and kinetic data and theoretical tools for understanding regional pro­cesses of sediment diagenesis.

Acknowledgements

thank Thomas Corbet, University of Illinois, Grant Garven, Johns Hopkins University and Wendy Harrison, Colorado School of Mines, for sharing their recent calcula­tions. Ken Belitz, John Bredehoeft, John Harbaugh, Charles Kreider, Pat Leahy, Florian Lehner, John Sharp, and Phillipe Ungerer provided interesting discussions and unpublished manuscripts. Joan Apperson drafted the figures. Much of the work described herein was supported by National Scien­ce Foundation grants EAR 85-52649 and EAR 86-01178, and the generosity of Amoco Production Company, Exxon Corp., Texaco USA, and Shell Oil Company.

Glossary of Variables

Ao Rate of oil generation within a sediment (kg/m3 s) Ar Rate of reaction of a groundwater per unit volume

(moles/m3 s) C Solute concentration in a groundwater (moles/m3 of

water) Ctq Concentration of a solute at chemical equilibrium

(moles/m3 of water) Cf Heat capacity of a groundwater a/kg 0C) Cr Heat capacity of the sediment grains a/kg 0C) Dim Coefficient of dispersion along dimension m result­

ing from flow along I (m2/s) o Dispersion tensor (m2/s) g Acceleration of gravity (m/s2)

H Thickness of an aquifer (m) krxn Rate constant for a chemical reaction (S-I) kl Intrinsic permeability of a sediment along I (m2) km Relative permeability of a sediment to oil P Pressure on a groundwater (Pa) P, Capillary pressure on an oil phase (Pa) Po Pressure on an oil phase (Pa) ql Specific discharge in an arbitrary direction I (m3 of

waterlm2 s) 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) R.. Rayleigh number for a thermally stratified ground­

water of constant composition So Oil saturation of a sediment, as a fraction of the pore

volume t Time (s) T Temperature (0C) V,m Velocity at which a sediment subsides relative to abso­

lute elevation (m/s) x, y Lateral distance (m) z Depth relative to absolute elevation, such as sea level

(m) ex Coefficient of thermal expansion for a groundwater

(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 J( Thermal conductivity of a fluid saturated sediment

aim s 0C) P Dynamic viscosity of a groundwater (kg/m s) Po Dynamic viscosity of an oil phase (kg/m s) e Groundwater density (kg/m3)

eo Density of an oil phase (kg/m3) Qr Density of the sediment grains (kg/m3) ~ Sediment porosity 4> Hydraulic potential of a groundwater (Pa) 4>bdy Hydraulic potential along the water table (Pa) 4>, Hydraulic potential arising from sediment compac­

tion (Pa) 4>, Hydraulic potential arising from topographic relief

(Pa)

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