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)
References
ANDERSON, M. P. (1984): Movement of contaminants in groundwater: Groundwater transport, advection and dispersion. - In: Studies in Geophysics: Groundwater Contamination, National Research Council, Washington, 37-45.
AZIZ, K., BORlES, S. A. & COMBARNOUS, M. A. (1973): The influence of natural convection in gas, oil and water reservoirs. - J. Can. Pet. Technol., 12, 41-47.
BEAR, J. (1972): Dynamics of fluids in porous media. - Elsevier, New York, 764 pp.
- (1979): Hydraulics of groundwater. - McGraw Hill, New York, 569 pp.
BEGG, S. H. & CARTER, R. R. (1987): Assigning effective values to simulator grid-block parameters for heterogeneous reservoirs. - Soc. Pet. Eng. Paper 16754, p. 601-611.
- & KING, P. R. (1985): Modelling the effects of shales on reservoir performance: calculation of effective vertical permeability. - Soc. Pet. Eng. Paper 13529, p. 31-338.
BETHKE, C. M. (1985): A numerical model of compactiondriven groundwater flow and heat transfer and its application to the paleohydrology of intracratonic sedimentary basins. - J. Geophys. Res., 90, 6817-6828.
- (1986a): Hydrologic constraints on the genesis of the
Modeling subsurface flow in sedimentary basins 151
Upper Mississippi Valley mineral district from Illinois basin brines. - Econ. Geol., 81, 233-249.
- (1986b): Inverse hydrologic analysis of the distribution and origin of Gulf Coast-type geopressured zones. - J. Geophys. Res., 91, 6535-6545.
- & CORBET, T. E (1988): Linear and nonlinear solutions for onedimensional compaction flow in sedimentary basins. - Water Resour. Res., 24, 461-467.
- , HARRISON, W. J., UPSON, C. & ALTANER, S. P. (1988): Supercomputer analysis of sedimentary basins. -Science, 239, 261-267.
BJ0RLYKKE, K. (1984): Formation of secondary porosity: How important is it? - In: MacDonald, D. A. & Surdam, R. C. (eds.) Clastic Diagenesis, Amer. Assoc. Pet. Geol.. Tulsa, 277-286.
BLACKWELL, R. J. RAYNE, J. R. & TERRY, W. M. (1959): Factors influencing the efficiency of miscible displacement. - Trans. Soc. Pet. Eng. AIME, 216, 1-8.
BLANCHARD, P. E. & SHARP, J. M., Jr. (1985): Possible free convection in thick Gulf Coast sandstone sequences. -Trans. Southwest Section Am. Assoc. Pet. Geol., 6-12.
BoDNER, D. P., BLANCHARD, P. E. & SHARP, J. M., Jr. (1985): Variations in Gulf Coast heat flow created by groundwater flow. - Gulf Coast Assoc. Geol. Soc. Trans., 35, 19-28.
BRADLEY, J. S. (1985): Safe disposal of toxic radioactive liquid wastes. - Geology, 13, 328-329.
BREDE HOEFT, J. D. & HANSHAW, B. B. (1968): On the maintenance of anomalous fluid pressures, I., thick sedimenary sequences. - Geol. Soc. Amer. Bull., 79, 1097-1106.
- , DJEVANSHIR, R. D. & BELTIZ, K. R. (1988): Lateral fluid flow in a compacting sand-shale sequence: south Caspian basin. - Am. Assoc. Pet. Geol. Bull., 72, 416-424.
- , NEUZIL, C. E. & MILLY, P. C. D. (1983): Regional flow in the Dakota aquifer: a study of the role of confining layers. - U. S. Geol. Surv. Water Supply Paper, 2237, 1-45.
- , BLYTH, C. R., WHITE, W. A. & MAYEY, G. B. (1963): Possible mechanism for concentration of brines in subsurface formations. - Am. Assoc. Pet. Geol. Bull., 47, 257-269.
CARPENTER, A. B. TROUT, M. L. & PICKETT, E. E. (1974): Preliminary report on the origin and chemical evolution of lead- and zinc-rich oil field brines in central Mississippi. - Econ. Geol., 69, 1191-1206.
CHUOKE, R. L., VAN MEURS, P. & VAN DER POEL, C. (1959): The instability of slow, immiscible, viscous liquid-liquid displacements in permeable media. - Trans. Soc. Pet. Eng. AIME, 216, 188-194.
CLAYTON, J. L. & SWETLAND, P. J. (1980): Petroleum generation and migration in Denver basin. - Am. Assoc. Pet. Geol. Bull., 64, 1613-1633.
CLAYTON, R. N., FRIEDMAN, I., GRAF, D. L., MAYEDA, T. K. MEENTS, W. E & SHIMP, N. E (1966): The origin of saline formation waters 1. Isotopic composition. - J. Geophys. Res., 71, 3869-3882.
COATS, K. H., NIELSEN, R. L., TERHUNE, M. H. & WEBER, A. G. (1967): Simulation of three-dimensional, two-
phase flow in oil and gas reservoirs. - Soc. Petro Eng. J., 7, 377-388.
COMBARNOUS, M. H. & BORIES, S. A. (1975): Hydrothermal convection in saturated porous media. - Adv. Hydrosci., 10, 231-307.
DARTON, H. H. (1909): Geology and underground waters of South Dakota. - U. S. Geol. Surv. Water Supply Pap., 227, 156 pp.
DAVIS, R. W. (1987): Analysis of hydrodynamic factors in petroleum migration and entrapment. - Am. Assoc. Pet. Geol. Bull., 71, 643-649.
DAVIS, S. H., ROSEN BLAT, S., WOOD, J. R. & HEWITT, T. A. (1985): Convective fluid flow & diagenetic patterns in domed sheets. - Am. J. Sci., 285, 207-223.
DICKEY, P. A., GEORGE, G. 0. & BARKER, C. (1987): Relationships among oils and water compositions in Niger delta. - Am. Assoc. Pet. Geol. Bull., 71, 1319-1328.
DICKINSON, G. (1953): Geologic aspects of abnormal reservoir pressures in Gulf Coast Louisiana. - Am. Assoc. Pet. Geol. Bull .. 37. 410-432.
DOLlGEZ, B., BESSIS, E, BURRUS, J., UNGERER, P. & CHENET, P. Y. (1986): Integrated numerical simulation of the sedimentation, heat transfer, hydrocarbon formation and fluid migration in a sedimentary basin: the Themis model. - In: Burrus, J. (ed.), Thermal modeling in sedimentary basins, Editions Technip, Paris, p. 173-195.
DOMENICO, P. A. & PALCIAUSKAS, V. V. (1973): Theoretical analysis of forced convective heat transfer in regional ground-water flow. - Geol. Soc. Am. Bull., 84, 3803-3814.
- & PALCIAUSKAS, V. V. (1979): Thermal expansion of fluid and fracture initiation in compacting sediments: Summary. - Geol. Soc. Am. Bull., 90, 518-520.
- & ROBBINS, G. A. (1985): The displacement of connate water from aquifers. - Geol. Soc. Am. Bull., 96, 328-335.
Dow, W. G. (1974): Application of oil-correlation and source-rock data to exploration in Williston basin. -Am. Assoc. Geol. Bull., 58, 1253-1262.
DOYLE, M., LAWRENCE, D., SNELSON, S. & HORSFIELD, W. (1988): Computer simulation of basin stratigraphy (abs.). - Terra Cognita, 8, 21.
ENGLAND, W. A., MACKENZIE, A. S., MANN, D. M. & QUIGLEY, T. M. (1987): The movement and entrapment of petroleum fluids in the subsurface. - J. Geol. Soc. Lond., 144, 327- 347.
ESLINGER, E. V., SAVIN, S. M. & YEH, H.-W. (1979): Oxygen isotope geothermometry of diagenetically altered shales. - In: Scholle, P. A. & Schluger, P. R. (eds.) Aspects of Diagenesis, Soc. Econ. Paleont. Mineral. Spec. Pub., 26, 113-124.
ESSAID, H. I. (1988): A multilayered sharp interface model of coupled freshwater and saltwater flow in coastal systems: Model application. - Water Resourc. Res., submitted.
FOGG, G. E. (1986): Groundwater flow and sand body interconnectedness in a thick multiple aquifer system. -Water Resour. Res., 22, 679-694.
FREEZE, R. A. & WITHERSPOON, P. A. (1966): Theoretical analysis of regional groundwater flow, 1., analytical and
152 CRAIG M. BETHKE
numerical solutions to the mathematical ~odel. -Water Resour. Res., 2, 641-656.
- & WITHERSPOON, P. A. (1967): Theoretical analysis of regional groundwater flow, 2., Effect of water-table configuration and subsurface permeability variation. -Water Resour. Res., 3, 623-634.
GALLOWAY, W. E. (1979): Diagenetic control of reservoir quality in arc-derived sandstones: implications for petroleum exploration. - In: Scholle, P. A. & Schluger, P. R. (eds.) Aspects of Diagenesis, Soc. Econ. Paleont. Mineral. Spec. Pub., 26, 251-262.
- (1984): Hydrogeologic regimes of sandstone diagenesis. -In: MacDonald, D. A. & Surdam, R. C. (eds.) Clastic Diagenesis, Am. Assoc. Pet. Geol. Memoirs, 37, 3-13.
GARVEN, G. (1985): The role of regional fluid flow in the genesis of the Pine Point deposit, Western Canada sedimentary basin. - Econ. Geol., 80, 307-324.
- (1986): The role of regional fluid flow in the genesis of the Pine Point deposit, Western Canada sedimentary basin - a reply. - Econ. Geol., 81, 1015-1020.
- (1988): A hydrogeologic model for the formation of the giant oil sands deposits of the Western Canada Sedimentary Basin. - Am. j. Sci., in press.
- & FREEZE, R. A. (1984a): Theoretical analysis of the role of groundwater flow in the genesis of stratabound ore deposits, 1. Mathematical and numerical model. - Am. J. Sci., 284, 1085-1124.
- - (1984b): Theoretical analysis of the role of groundwater flow in the genesis of stratabound ore deposits, 2. Quantitative results. - Am. j. Sci., 284, 1125-1174.
GIBSON, R. E. (1958): The progress of consolidation in a clay layer increasing in thickness with time. -Geotechnique, 8, 171-182.
GRAF, D. L. (1982): Chemical osmosis, reverse chemical osmosis, and the origin of subsurface brines. - Geochim. Cosmochim. Acta, 46, 1431-1448 ..
HALDORSEN, H. H. (1986): Simulator parameter assignment and the problem of scale in reservoir engineering. - In: Lake, L. W. & Carroll, H. B., Jr. (eds.), Reservoir Characterization, Academic Press, New York, 293-340.
HAQ, B. u., HARDENBOL,j. & VAIL, P. R. (1987): Chronology of fluctuating sea levels since the Triassic. - Science, 235, 1156-1167.
HANOR, j. S. (1987a): Kilometre-scale thermohaline overturn of pore waters in the Louisiana Gulf Coast. -Nature, 327, 501-503.
- (1987b): History of thought on the origin of subsurface sedimentary brines. - History of Geophysics, American Geophysical Union, 3, 81-91.
HARRISON, W. j. & BETHKE, c. M. (1986): Paleohydrologic analysis of interacting meteoric and compactional flow regimes in the U. S. Gulf Coast (abs.). - Geol. Soc. Am. Abstr. Programs, 18, 630.
HAY, R. L. (1966): Zeolites and zeolitic reactions in sedimentary rocks. - Geol. Soc. Am. Spec. Pag. 85, 130 pp.
HEARN, P. P., jr., SUTTER, j. F. & BELKIN, H. E. (1987): Evidence for late-Paleozoic brine migration in Cambrian carbonate rocks of the central and southern Appalachians: Implications for Mississippi Valley-type sulfide
mineralization. - Geochim. Cosmochim. Acta, 51, 1323-1334.
HELGESON, H. c., BROWN, T. H., NIGRINI, A. & JONES, T. A. (1970): Calculation of mass transfer in geochemical processes involving aqueous solutions. - Geochim. Cosmochim. Acta, 34, 569-592.
HILL, S. (1952): Channeling in packed columns. - Chern. Eng. Sci., 1, 247-253.
HITCHON, B. (1984): Geothermal gradients, hydrodynamics, and hydrocarbon occurrences, Alberta, Canada. - Am. Assoc. Pet. Geol. Bull., 68, 713-743.
HUBBERT, M. K. (1940): The theory of ground-water motion. - J. Geol., 48, 785-944.
- (1953): Entrapment of petroleum under hydrodynamic conditions. - Am. Assoc. Pet. Geol. Bull., 37, 1954-2026.
JENNINGS, j. B. (1987): Capillary pressure techniques: Application to exploration and development geology. -Am. Assoc. Pet. Geol. Bull., 71, 1196-1209.
KEEN, C. E. (1983): Salt diapirs and thermal maturity: Scotian basin. - Bull. Can. Pet. Geol., 31, 101-108.
KIRALY, L. (1975): Rapport sur l'etat actuel des connaissances dans Ie domaine des caracteres physiques des roches karstiques. - Int. Union Geol. Sci., Ser. B., 3, 53-67.
LAND, L. S. & DUTTON, S. P. (1979): Reply: cementation of sandstones. - J. Sed. PetroL, 49, 1359-1361.
- & PREZBINDOWSKI, D. R. (1981): The origin and c\olution of saline formation water, Lower Cretaceous carbonates, south-central Texas. - j. Hydrol., 54, 51-74.
LAPWOOD, E. R: (1984): Convection of a fluid in a porous medium. - Pre. Cambridge Phil. Soc., 44, 508-521.
LEHNER, F. K., MARSAL, D., HERMANS, L., & VAN KUYK, A. (1987): A model of secondary hydrocarbon migration as a buoyancy-driven separate phase flow. - In: Doligez, B. (ed.), Migration of Hydrocarbons in Sedimentary Basins, Editions Technip, Paris, 457-471.
LEVANDOWSKI, D. w., KALEY, M. E., SILVERMAN, S. R. & SMALLEY, R. G. (1973): Cementation in the Lyons sandstone and its role in oil accumulation, Denver basin, Colorado. - Amer. Assoc. Pet. Geol. Bull., 57, 2217-2244.
LEWAN, M. D. (1985): Evaluation of petroleum generation by hydrous pyrolysis experimentation. - Philos. Trans. R. Soc. London, A., 315, 123-134.
LICHTNER, P. C. (1985): Continuum model for simultaneous chemical reactions and mass transport in hydrothermal systems. - Geochim. Cosmochim. Acta, 49, 779-800.
- (1988): The quasi-stationary state approximation to coupled mass transport and fluid-rock interaction in a porous medium. - Geochim. Cosmochim. Acta, 52, 143-165.
MARCHER, M. V. & STEARNS, R. G. (1962): Tuscaloosa formtion in Tennessee. - Geol. Soc. Am. Bull., 73, 1365-1386.
MATHERON, G. & DE MARSILY, G. (1980): Is transport in porous media always diffusive? A counterexample. -Water Resour. Res., 16, 901-917.
MCCABE, c., VANDER VOO, R., PEACOR, D. R., SCOTESE, C R. & FFREEMAN, R. (1983): Diagenetic magnetite carries
Modeling subsurface flow in sedimentary basins 153
ancient yet secondary remanence in some Paleozoic sedimentary carbonates. - Geology, 11,221-223.
MEISLER, H., LEAHY, P. P. & KNOBEL, L. L. (1985): Effect of eustatic sea-level changes on saltwater-freshwater relations in the northern Atlantic coastal plain. - U. S. Geol. Surv. Water-Supply Pap., 2255, 28 pp.
MERCER,J. w., PINDER, G. F. & DONALDSON, I. G. (1975): a system at Wairakei, New Zealand. - J. Geophys. Res., 80, 2608 - 2621.
NAESER, C. W. (1979): Thermal history of sedimentary basins: fission-track dating of subsurface rocks. - In: Scholle, P. A. & Schluger, P. R. (eds.) Aspects of Diagenesis, Soc. Econ. Paleont. Mineral. Spec. Pub., 26, 109-112.
NEUZIL, C. E. (1986): Groundwater flow in low-permeabilityenvironments. - Water Resour. Res., 22, 1163-1195.
- & POLLOCK, D. W. (1983): Erosional unloading and fluid pressures in hydraulically tight rocks. - J. Geol., 91, 179- 193.
NIELD, D. A. (1968): Onset of thermohaline convection in a porous medium. - Water Resour. Res., 4, 553-560.
NUNN, J. A. & SASSEN, R. (1986): The framework of hydrocarbon generation and migration, Gulf of Mexico continental slope. - Gulf Coast Assoc. Geol. Soc. Trans., 36, 257-262.
OHLE, E. L. (1951): The influence of permeability on ore distribution in limestone and dolomite. - Econ. Geol., 46, 667-706.
- (1980): Some considerations in determining the origin of ore deposits of the Mississippi Valley type, Part II. -Econ. Geol., 75, 161-172.
PALCIAUSKAS, V. V. & DOMENICO, P. A. (1976): Solution chemistry, mass transfer, and the approach to chemical equilibrium in porous carbonate rocks and sediments. -Geol. Soc. Am. Bull., 87, 207-214.
PEACEMAN, D. W. (1977): Fundamentals of numerical reservoir simulation. - Elsevier, New York, 176 pp.
- & RACHFORD, H. H. (1962): Numerical calculation of multidimensional miscible displacement. - Soc. Petro Eng. J., 2, 328-339.
PEARSON, F. J. & WHITE, D. E. (1967): Carbon 14 ages and flow rates of water in Carrizo sand, Atascosa County, Texas. - Water Resour. Res., 3, 251-261.
PHILIP, J. R. (1982): Free convection at small Rayleigh number in porous cavities of rectangular, elliptical, triangular, and other cross sections. - Int. J. Heat Mass Transfer, 25, 1503- 1509.
PRATS, M. (1966): The effect of horizontal fluid flow on thermally induced convection currents in porous mediums. - J. Geophys. Res., 71, 4835-4838.
RABINOWICZ, M., DANDURAND, J.-L., JAKUBOWSKI, M., SCHorr, J. & CASSAN, J.-P. (1985): Convection in a North Sea oil reservoir: inferences on diagenesis and hydrocarbon migration. - Earth Planet. Sci. Lett., 74, 387-404.
RANGANATHAN, V. & HANOR, J. S. (1987): A numerical model for the formation of saline waters due to diffusion of dissolved NaCI in subsiding sedimentary basins with evaporites. - J. Hydrol., 92, 97-120.
RUBIN, H. (1974): Heat dispersion effect on thermal convec-
tion in a porous medium layer. - J. Hydrol., 21, 173-185.
SANFORD, R. F. (1982): Preliminary model of regional Mesozoic groundwater flow and uranium deposition in the Colorado plateau. - Geology, 10, 348-352.
SENGER, R. K. & FOGG, G. E. (1987): Regional underpressuring in deep brine aquifers, Palo Duro basin, Texas, 1., Effects of hydrostratigraphy and topography. -Water Resour. Res., 23, 1481-1493.
- , KREITLER, C. W. & FOGG, G. E. (1987): Regional underpressuring in deep brine aquifers, Palo Duro basin, Texas, 2., The effect of Cenozoic basin development. -Water Resour. Res., 23, 1494-1504.
SIBLEY, D. F. & BLATf, H. (1976): Intergranular pressure solution and cementation of the Tuscarora orthoquartzite, J. Sed. PetroL, 46, 881-896.
SILLIMAN, S. E., KONIKOW, L. F. & VOSS, c.1. (1987): Labortory investigation of longitudinal dispersion in anisotropic porous media. - Water Resour. Res., 23, 2145-2151.
SMITH, L. & SCHWARTZ, F. W. (1984): An analysis of the influence of fracture geometry on mass transport in fractured media. - Water Resour. Res., 20, 1241-1252.
SVERJENSKY, D. A. (1986): Genesis of Mississippi Valley-type lead-zinc deposits. - Ann. Rev. Earth Planet. Sci., 14, 177-199.
TETZLAFF, D. M. (1988): SEDSIM: a simulation model of clastic sedimentary processes. - Ph. D. Diss., Stanford University, in prep.
TISSOT, B. P. & WELTE, D. H. (1984): Petroleum formation and occurrence, 2nd. ed. - Springer-Verlag, Berlin, 699 pp.
T6TH, J. (1962): A theory of groundwater motion in small basins in central Alberta, Canada. - J. Geophys. Res., 67, 4375-4387.
- (1963): A theoretical analysis of groundwater flow in small drainage basins. - J. Geophys. Res., 68, 4795-4812.
- (1978): Gravity-induced cross-formational flow of formation fluids, Red Earth region, Alberta, Canada: Analysis, patterns, evolution. - Water Resour. Res., 14, 805-843.
- & CORBET, T. (1986): Post-paleocene evolution of regional groundwater flow systems and their relation to petroleum accumulations, Taber area, southern Alberta, Canada. - Bull. Can. Pet. Geol., 34, 339-363.
- & MILLAR, R. F. (1983): Possible effects of erosional changes of the topographic relief on pore pressures at depth. - Water Resour. Res., 19, 1585-1597.
UNGERER, P., BEHAR, F. & DISCAMPS, D. (1981): Tentative calculation of the overall volume expansion of organic matter during hydrocarbon genesis from geochemistry data. Implications for primary migration. - In: Advances in Organic Geochemistry, John Wiley, New York, 129-135.
- , CHENET, P. Y., MORETfI, I., CHIARELLI, A. & OUOIN, J. L. (1986): Modeling oil formation and migration in the southern part of the Suez rift, Egypt. - Org. Geochem., 10, 247-260.
- , DOLlGEZ, B., CHENET, P. Y., BURRUS, J., BESSIS, F., LAFARGUE, E., GIROIR, G., HEUM, O. & EGGEN, S.
154 CRAIG M. BETHKE
(1987): A 2-D model of basin scale petroleum migration by two-phase fluid flow: Application to some case studies. - In: Doligez, B. (ed.), Migration of Hydrocarbons in Sedimentary Basins, Editions Technip, Paris, 415-456.
WALDSCHMIDT, W. A. (1941): Cementing materials in sandstones and their probable influence on migration and accumulation of oil and gas. - Amer. Assoc. Pet. Geol. Bull., 25, 1839-1879.
WEBER, K. J. (1982): Influence of common sedimentary structures on fluid flow in reservoir models. - J. Pet. Technol., 34, 665-672.
- (1986): How heterogeneity affects oil recovery. - In: Lake, L. W. & Carroll, H. B., Jr. (eds.), Reservoir Characterization, Academic Press, pp. 487-544.
WHEATCRAFT, S. W. & TYLER, S. W. (1988): An explanation of scale-dependent dispersivity in heterogeneous aquifers using concepts of fractal geometry. - Water Resour. Res., 24, 566-578.
WILLET, S. D. & CHAPMAN, D. S. (1987): Temperatures, fluid flow and the thermal history of the Uinta basin. - In: Doligez, B. (ed.), Migration of Hydrocarbons in Sedimentary Basins, Editions Technip, Paris, 533-551.
WOOD, J. R. (1986): Thermal mass transfer in systems containing quartz and calcite. - In: Gautier, D. L. (ed.), Roles of Organic Matter in Sediment Diagenesis, Soc. Econ. Paleontol. Mineral. Spec. Publ. 38, 169-180.
- & HEWETT, T. A. (1982): Fluid convection and mass transfer in porous sandstones, a theoretical approach. -Geochim. Cosmochim. Acta, 46, 1707-1713.
- & HEWETT, T. A. (1984): Reservoir diagenesis and convective fluid flow. - In: McDonald, D. A. and Surdam, R. C. (eds.), Clastic Diagenesis, Am. Assoc. Pet. Geol. Memoir, 37, p. 99-110.
- & HEWETT, T. A. (1986): Forced fluid flow and diagenesis in porous reservoirs - controls on the spatial distribution. - In: Gautier, D. L. (ed.), Roles of Organic Matter in Sediment Diagenesis, Soc. Econ. Paleontol. Mineral. Spec. Publ. 38, 181-187.
- & SURDAM, R. C. (1979): Application of convectivediffusion models to diagenetic processes. - In: Scholle, P. A. & Schluger, P. R. (eds.) Aspects of Diagenesis, Soc. Econ. Paleont. Mineral. Spec. Pub., 26, 243-250.
WOODBURY, A. D. & SMITH, L. (1987): Simultaneous inversion of hydrogeologic and thermal data, 1, theory and application using hydraulic head data. - Water Resour. Res., 23, 1586-1606.