A Numerical Model of Compaction&Driven Groundwater … · to the small fluid velocities, ......

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 90, NO. B8, PAGES 6817-6828, JULY 10, 1985

A Numerical Model of Compaction-Driven Groundwater Flow and Heat Transfer and Its Application to the Paleohydrology

of Intracratonic Sedimentary Basins

CRAIG M. BETHKE

Hydro•Ieolo•7y Pro,tram, Department of Geolo•Iy, University of Illinois, Urbana

A new numerical method allows calculation of compaction-driven groundwater flow and associated heat transfer in evolving sedimentary basins. The model is formulated in Lagrangian coordinates and considers two-dimensional flow in heterogeneous, anisotropic, and accreting domains. Both the continuity of the deforming medium and aquathermal pressuring are explicitly taken into account. A calculation of compaction-driven flow during evolution of an idealized intracratonic sedimentary basin including a basal aquifer predicts slow groundwater movement over long time periods. Fluids in shallow sediments tend to move upward toward the sedimentation surface, and deeper fluids move laterally. The hydraulic potential gradient with depth reverses itself near the basal aquifer, and fluids in this area have a tendency to migrate obliquely into stratigraphically lower sediments. Only small excess pressures develop, suggesting that intracratonic basins are not subject to overpressuring during their evolutions. Owing to the small fluid velocities, heat transfer is conduction-dominated, and the geothermal gradient is not disturbed. Variational studies show that excess hydraulic potentials, but not fluid velocities, depend on assumptions of permeability and that both excess potentials and velocities scale with sedimentation rate. Aquathermal pressuring is found to account for < 1% of the excess potentials developed during compaction. These results cast doubt on roles of compaction-driven flow within intracratonic basins in processes of secondary petroleum migration, osmotic concentration of sedimentary brines, and formation of Mississippi Valley-type ore deposits. Results might also be combined with chemical models to investi- gate the relationship of compaction flow to cementation in sediments.

INTRODUCTION

Compaction-driven groundwater flow, the movement of fluids due to collapse of pore volume during sediment burial within evolving sedimentary basins, is thought to be an important mechanism in geological processes as diverse as petroleum migration, generation of overpressures, concentration of subsurface brines, formation of ore deposits, and cementation and porosity enhancement in sedimentary rocks. Compaction-driven flow is a driving force in the poorly under- stood phenomenon of petroleum migration, especially for primary migration from source rocks into carrier beds [Athy, 1930; Bonharn, 1980]. It is not clear whether compaction- driven or gravity-driven groundwater flow drives secondary migration within carrier beds [Rich, 1921; Athy, 1930; Toth, 1980]. Sediment compaction is also believed to cause overpressuring in basins [Dickinson, 1953; Rubey and Hubbert, 1959; Bredehoeft and Hanshaw, 1968], and this process may be augmented by aquathermal pressuring, defined as pore pressure created by thermal expansion of pore fluids relative to the sediment matrix during burial [Barker, 1972]. High pore pressure gradients found in overpressured basins are proposed to concentrate subsurface sedimentary brines by reverse osmosis across shaley strata [Graf, 1982]. Mississippi Valley-type ore deposits are thought to be formed by discharge of subsurface brines from basins by compaction-driven flow. These deposits may have formed slowly during basin evolution [Noble, 1963; Jackson and Beales, 1967; Dozy, 1970] or suddenly from flow out of overpressured basins [Sharp, 1978; Cathies and Smith, 1983]. Finally, cementation and dissolution processes within sedimentary rocks have been related to chemical transport by fluids redistributed by compaction [Hayes, 1979]. Sibley and Blatt [1976] and Wood and Surdarn [1979] also discuss the

importance of transport by groundwaters to sediment diagenesis.

Despite the potential importance of compaction-driven groundwater flow to geological problems, this process has received little quantitative evaluation on a basin-wide scale, and its true importance has been difficult to estimate. Sharp [1976] and Sharp and Domenico [1976] developed a one-dimensional model of vertical compaction-driven flow and heat transfer which they applied to relatively rapidly subsiding basins. Their analysis, however, does not address the possibility of lateral flow, which may be more significant than vertical flow in many sedimentary basins [Magara, 1976]. In addition, aquathermal pressuring is not taken into account. Cathies and Smith [1983] used a two-dimensional model to predict the thermal effects of sudden dewaterings of overpressured basins, but it is not clear whether their method is applicable to general studies of compaction flow, and they only briefly discuss their calculation technique.

This paper presents a two-dimensional analysis of compaction-driven groundwater flow and associated heat transfer in heterogeneous, anisotropic, and accreting media, applied to basin-wide geological problems. The model is especially useful in evaluating theories which depend on lateral transport by groundwaters, such as petroleum migration and ore genesis. The model also differs from previous models by its formulation in a convenient Lagrangian reference frame and by explicitly treating aquathermal pressuring and the continuity of the deforming medium. These latter features allow more accurate and complete modeling of compaction flow. A sample calculation of compaction-driven flow within an idealized intracratonic basin both gives representative model results and provides general insights into this type of hydrologic regime.

Copyright 1985 by the American Geophysical Union. SOLUTION PROCEDURE

Paper number 4B5062. 0148-0227/85/004B-5062505.00

The numerical technique of modeling compaction-driven flow is based on approximate solutions to three coupled par-

6817

6818 BETHKE: COMPACTION-DRIVEN GROUNDWATER FLOW AND HEAT TRANSFER

tial differential equations describing medium continuity, fluid flow, and heat transfer. These equations are written and solved in a Lagrangian reference frame which remains fixed with respect to the subsiding medium but moves through space.

Differential equations in deforming media may also be derived in other reference frames [Welty et al., 1976, p. 31]. Sharp [1976] used an Eulerian reference frame which remains fixed in space and moves with respect to the medium. Choice of a reference frame is mathematically arbitrary because Darcy's and Fourier's laws require instantaneous spatial derivatives, the measurement of which is independent of any temporal variation in the location of the observation point [see Cooper, 1966]. In addition, both Lagrangian and Eu- lerian formulations require a moving boundary condition, either at the sedimentation surface or the basement contact.

The Lagrangian formulation, however, is most concise because except at the sedimentation surface no rock grains move across the boundaries between nodal blocks, the elemental volumes of the solution procedure. The Lagrangian solution also minimizes "bookkeeping" since nodal blocks always represent the same subdivision of the medium.

In order to evaluate the coupled differential equations, which have no known analytical solution, the technique solves decoupled or iteratively coupled finite difference approximations. These approximate equations contain variables only at discrete points in time and space and are amenable to numerical solution by linear algebraic or iterative methods. Subscripts in finite difference equations, by convention, represent position in the spatial domain. Superscripts give points in time, either the previous and current time levels n and (n + 1) or an intermediate or average time level (n + 0). Numerical considerations in choosing the position of the intermediate time level are discussed in a later section.

Medium Continuity Equation

Subsidence and compaction in evolving sedimentary basins require that strata at different depths subside at different velocities, much as coils of a spring move at varying speeds as the spring is compressed. The relationship of settling velocity to compaction of the medium is given

8 1 8

• v•,• - Az et (Az)

[Cooper, 1966], where mathematical symbols are listed separ- ately. Assuming incompressible rock grains, this reduces to the continuity equation

8 1 8•p

8-• V•m --(1 -- •p) 8t (1) which gives settling velocity over the domain as a function of the time rate of pore collapse. Porosity is defined here as effective porosity [Bear, 1972, p. 43] at ambient conditions.

Equation (1) may be written in a finite difference form to give velocity differences for the current time step between a node and the nodal block boundary below it and between the boundary and the node immediately underneath:

Uzmi,j+ 1/2 • •)zmi,j -- Az n+O n+ 1 n -

2(1 - rh .+0• At" 'r i,j ;

•)zmi,j+ 1 -- •)zmi,j+ 1/2 • Azi,j+ 1 n+O

2(1 - 05i,j+ •,,+o)

n+l n

•i,j+l -- •i,j+l At n

Since the heights of the nodal blocks change with compaction, the expression

(• - •) Az "+ • = Az" (1

is used in (2) and (3) to calculate Az at the (n + 0) time level. Adding (2) and (3) gives the velocity difference between

neighboring nodal points:

V:m"J+ • -- V:m'f -- 2At• [L' • _- •;, 45 _li,j +•

(4)

Specifying porosity in this equation, typically as a function of burial depth or burial depth and pore pressure, defines medium compaction and the driving force for compaction flow.

Inasmuch as porosity at the unknown time level depends on the settling velocity over the time step (i.e., on burial depth at the new time level), equation (4) is solved iteratively at each nodal point. Several back substitutions are usually sufficient to converge to consistent values of velocity, porosity, and nodal block height. These values, then, allow solution of the fluid flow equation.

Fluid Flow Equation

A fluid flow equation written in terms of hydraulic potential describes movement of a single-phase pore fluid of constant composition through a medium undergoing pore collapse and temperature variation. An equation of state for a slightly com- pressible fluid

1 - •p = •8P -- o•8T (5) P

[Domenico and Palciauskas, 1979; also Lewis and Randall, 1961, p. 25] provides the basis for a simple derivation of the flow equation. This is a weakly nonlinear equation in P and T.

From the definitions of hydraulic potential, density, and porosity,

8P • 80 + pg 8z (6)

1 p 8p =-- 8m ---- 8V (7)

V V

8V=4•SV•+V•Sc - (1 - q•)

- (8)

respectively. Also, from the potential form of Darcy's law,

in an arbitrary curvilinear direction s,

(2) 8m= [•x (pk•A• •xx)AX+ •zz (pk•A= •zz)Az] 8t (9) [Muskat, 1937, pp. 725-726].

Substituting (6), (7), (8), and (9) into (5), multiplying by V = (3) & V•, and taking the derivatives with respect to time,

BETHKE' COMPACTION-DRIVEN GROUNDWATER FLOW AND HEAT TRANSFER 6819

This is similar to the diffusion equation in hydraulic potential with additional "forcing" terms describing rates of change in fluid potential energy with burial, collapse of pore volume, and thermal expansion of the pore fluid. When the potential energy, pore collapse, and thermal expansion terms are pro- vided by solution of the medium continuity and heat transfer equations, hydraulic potential is the only unknown variable.

Using central difference approximations and the notation of Stone [1968], equation (10)converts to a finite difference form

n+l n+l +1 B i,j • i,j + 1 + D i,j • i - 1 ,j + E i,j • i,j n

n+l + F•,• • + • ,;• + • + H•,; • •,• _ • = Q •,• (11)

where the coe•cients B•,•, D•,;, E•,•, F•,•, Hi,j, and Q•,; are given in Appendix 1. A pentadiagonal matrix equation is produced by writing (11) at each nodal point while scanning the finite difference grid along the i or j direction. This matrix equation may be solved for hydraulic potential at the new time level using either direct or iterative techniques. Calculations in this paper were made using a band matrix direct method.

Heaf Transfer Equafion

The heat transfer equation is used to solve for the basin temperature distribution resulting from heat conduction, heat transfer by advecting groundwaters, and internal energy sources, assuming local thermal equilibrium among the pore fluid and rock grains. Dispersive heat transfer may also be considered by modifying coe•cients of the heat conduction term [Dybbs and Sch•eifzer, 197•; Rubin, 1974]. The energy used in compression of pore fluids and rock grains at geologically reasonable sedimentation rates is negligible compared to normal heat flow (see Appendix 2) and may be ignored.

For a Lagrangian elemental volume, which always contains the same rock grains, the time rate of change in overall enthalpy equals the sum of the rates of change due to conduction, advection, and internal energy sources,

8Hi 8Hi] 8Hi] 8t- 8t +0H c a

The left side of (12) expands to separate the enthalpies of fluid and grains

8t -8t (Hw +

( =p 8T phw• 8•

= + - c3 + _ Enthalpy changes due to conductive and advective effects

are derived by Stallman •1963]. Substituting into (12),

•[p•Cw+p•(l_•)C•]ST 8 ( ST) at =a• KxAx• ax

( 8Q x 8Q• ) ph w • 8• (14) - phw • 8x Ax + • Az + Qu (1-•) 8t

Fig. 1. Technique of mapping an irregular basin cross section (upper left) with a finite difference grid (lower right). The basin x direction is curvilinear along stratigraphic time lines and maps to the finite difference i direction.

In the case of no pore collapse or internal heat sources, this reduces to the convection-diffusion equation.

This equation may also be expressed as a finite difference approximation

n+l jn+l T: n+l B i,./ T/,.•+ • + D i,./ T/_ •, + E•,.• •,.• n+l n+l

+ F•,j T• + •,j + H•,j T•,j_ • = Q•,j (15)

where the coefficients are given in Appendix 3. Equation (15) produces a pentadiagonal coefficient matrix when written at each nodal point.

Numerical Solution

The finite difference equations are applied by mapping a finite difference grid onto a cross section through a basin (Figure 1). In order to consider basins composed of strata of varying thicknesses, the basin cross section is described in terms of a curvilinear coordinate [see Thompson, 1982; Mastin, 1982] x along stratigraphic time lines (i.e., connecting rock grains which were deposited contemporaneously) and a coordinate across stratigraphy, z. Owing to the much greater breadth than depth of sedimentary basins, the z direction, which is formally orthogonal to x, may be approximated as a vertical, linear coordinate. This choice of coordinates also

tends to keep the x direction coincident with the major axis of the permeability ellipsoid, which is usually along stratigraphy in unfractured sediments.

The x coordinate maps to the finite difference i direction, and z to finite difference j. Integral values of i and j are nodal points at which solutions to the finite difference equations are obtained. Each nodal point represents the properties of a nodal block (Figure 2). Most variables, including porosity, hydraulic potential, and temperature, vary with time but not with position in the nodal block. Settling velocity, however, varies with position and not with time over a time step.

The solution proceeds by stepping from a time level of known values to a time level of unknown values, where the time step is the time difference between levels. Since the finite difference equations (4), (11), and (15) are decoupled, they may be solved cyclically in this order for the unknown values at the new time level. A second pass through the cycle may be used to reduce numerical errors which are introduced by the decoupled solution, if necessary. In this case, the equations are iteratively coupled. Once the three equations are solved, the solu-


X Z

tAZij

.•Ax, i i,j ' ß .

= AX ß AZ Vbi, j i i,j Fig. 2. Relationship of finite difference nodal points (dots) to an

associated quadrilateral nodal block (shaded area), drawn to a great vertical exaggeration.

tion begins another time step to a further time level. At the beginning of a new time step, variables at the unknown time level are estimated from the rate of change over the previous time step until the exact values can be computed.

Constraints on the size of the time step, which may be dynamically chosen during the course of the calculation, in- clude a maximum proportional increase over the previous time step, a maximum estimated change in settling velocity, potential, and temperature, and the requirement that no fluid particle move farther than the dimensions of its nodal block during one time step. The latter expresses an empirical stability requirement of the heat transfer equation [Torrance, 1968].

Choice of the intermediate time level in the finite difference

equations, set at (n + 0) where 0 varies from zero to one, controls stability of this type of numerical solution. When 0 equals zero and one, the finite difference equations reduce to explicit and fully implicit forms, respectively [Peaceman, 1977, p. 66]. A value of one half minimizes local truncation errors and, based on Von Neumann's analysis [see Richtmyer, 1957; Ames, 1977], represents the lower limit of unconditional stability in solutions of the diffusion equation [Peaceman, 1977, pp. 69-74]. In solutions to the compaction flow problem, however, small oscillations may appear in the numerical results if 0 is set near this limit, especially when initiating pore collapse from hydrostatic conditions. Oscillations occur because changes in the rate of pore volume collapse or temperature variation represent "errors" to solutions of the diffusion equation in the sense of Von Neumann's analysis which are neither amplified nor damped at the limiting value of 0 of one half. Weighting the solution forward in time (0 of 0.6 to 0.7) will dampen such oscillations at only a small increase in truncation error.

Accuracy of the numerical technique is inferred from tests showing excellent agreement between numerical results and analytical solutions [Carslaw and Jaeger, 1959] to simplified problems, small finite difference residuals, and checks for conservation of rock and fluid mass and total energy. Nonethe- less, it is impossible to check simultaneously all aspects of a complicated numerical simulation technique which may be applied to a variety of problems. The best evaluation of the model may come from comparison with future models which use different solution methods.

Compaction Disequilibrium

If excess pore pressures in evolving basins become significant in comparison to confining pressures of basin sediments

(i.e., approach lithostatic), they tend to counteract sediment compaction. This effect has been observed in a number of rapidly subsiding basins composed of fine-grained sediments [Dickinson, 1953; Thomeer and Bottema, 1961; Rogers, 1966; Magara, 1975]. The resulting state in which less compaction is observed than might be expected at a given burial depth is termed "compaction disequilibrium" [Magara, 1975] and is interpreted to be the result of the inability of a sedimentary column to expel pore fluids quickly enough to allow normal compaction [Dickinson, 1953]. Rubey and Hubbert [1959, pp. 174-175] and Chapman [1972] propose that the state of compaction of overpressured sediments can be estimated by a corresponding-state model. They define an effective depth Ze to represent the depth in a hydrostatically pressured sediment at which effective stress equals the effective stress on a sediment in question. Effective stress is defined as the difference between weight per unit area of overlying sediments and pore pressure [Jaeger and Cook, 1976, p. 219]. Predicted porosity at depth z is calculated as the normal or "equilibrium" porosity at depth z e. For constant Pw and Psm, Ze is given:

Ze--Z• (•o - •Os½)

(,Os,,,- pw)g

Sediments with pore pressures greater than hydrostatic, then, will have effective depths shallower than actual burial depths. A similar model was employed by Sharp [1976] and Sharp and Domenico [1976] in modeling one-dimensional compaction- driven flow.

In practice, use of a compaction disequilibrium model with the numerical technique already presented can slow convergence of the algorithm by introducing feedback between the fluid flow and medium continuity equations. In this case, modification of the fluid flow equation to describe explicitly the change in pore volume due to variation in hydraulic potential will assure rapid convergence. Since porosity in the corresponding-state model is a function of effective depth alone,

[ 0(/) 0(/) O Z e 0(/) 1

c•t -- (•Z e c3t - (•Z e l)zm -- (Psm -- Pw)g • This result may be substituted into (10) to obtain a modified fluid flow equation

1 c• p x Ax P

L (1 -•b) 02 e {- (/)tiP{] l)zm "3' (/)L OW Some of the generality of the previous development is lost

since CgCk/CgZe must be known to evaluate this equation, while no assumption of the form of a porosity model had been made to this point. This is usually a minor restriction. If an ex- ponential porosity effective depth model FRubey and Hubbert, 1959]

•b = •bo exp (-bze)

is used, then

O(/)/OZ e -'- -- b•)


Differential Compaction

Sedimentary basins generally contain a variety of rock types intermixed on a finer scale than the practical sizes of nodal blocks. In this case, each block is divided into volume fractions occupied by each rock type. The volume fractions allow calculation of bulk properties such as porosity, permeability, heat capacity, etc.

If the various rock types compact differentially with burial, the volume fraction of each will vary [Perrier and Quiblier, 1974]. Defining volume fractions and porosities of each rock type at a reference condition, such as the state of compaction as measured in a basin, solves this problem. Given reference values, the current volume fractions as a function of current porosities for a rock type I are

x/ref(1 __ •/ref) gbref X, = (16)

where

Vb ref EIX!( l -- •)l) Vb ElX/ref(1 __

Since the unknowns Xt appear on both sides of (16), the equation may be solved iteratively. Calculating the ratio of bulk volumes using previous values, then calculating each Xt, and finally applying the accelerator

X• = X•/y•X•

however, seems to give excellent results in only one pass.

Boundary Conditions

Dirichlet and Neumann boundary conditions [Garabedian, 1964, pp. 227-228] to the fluid flow and heat transfer equations may be specified along the bottom edge and sides of the domain using standard finite difference techniques [Carnahan et al., 1969, p. 462]. The moving boundary at the sedimentation surface and the boundary conditions to the medium continuity equation, however, require special con- sideration.

An expanding layer of nodal blocks accommodates the boundary at the sedimentation surface, which moves with respect to the Lagrangian solution grid. This layer, which is initially arbitrarily thin, grows to accept sediment deposition until it reaches a target thickness, at which point a new layer of nodes is created. Values of hydraulic potential and temperature in these nodal blocks may be set at boundary values or, more accurately, treated as unknowns. In the latter case, fluid mass and total energy introduced into the uppermost nodal blocks by sedimentation must be taken into account in (10) and (14) and their finite difference counterparts, (1 l) and (15). In addition, since nodal points remain in the center of these expanding blocks, they subside at only half the velocity of the medium, and all values of V:m in (11) and (15) should be halved.

The medium continuity equation requires either one or two boundary conditions, depending on whether sedimentation is in equilibrium with subsidence. If equilibrium is maintained so that the sedimentation surface is at fixed elevation, either the surface sedimentation rate, which is the volume of uncompacted sediment crossing the sedimentation surface per unit time, or the basement subsidence velocity suffices. These values may be taken from stratigraphic data. When sedimentation does not match subsidence, the sedimentation surface is free to move, and both a sedimentation rate and a

TABLE 1. Data Assumed in Compaction-Driven Flow Calculation

Parameter Value

P Psc exp [fl(P - 0.1) - a(T - 25)] kg/m 3 (P in MPa, T in øC)

Psc 0.001 kg/m 3 • 5 x 10 -½ øC-• • 4.3 x 10 -'• MPa -1 # 1.0 cP Pr 0.0027 kg/m3 C w 4.2 J/g øC Cr 0.84 J/g øC •b 0.5 exp [-(0.5 x 10-5)z] sands (z in cm)

0.6 exp [-(0.6 x 10-5)z] shales (z in cm) log k,, -13 q- 2(/) sands, log m 2

-19 + 8(/) shales, log m e kx/k: 2.5 sands

10.0 shales

Kx = K: 1.67 W/m øC Basement heat flow 63 mW/m 2 0 0.65

subsidence velocity should be specified. The second boundary condition might be estimated, for example, based on isostatic adjustment or tectonic theories of basin evolution.

COMPACTION-DRIVEN FLOW IN INTRACRATONIC BASINS

Intracratonic basins are broad, shallow, slowly subsiding basins associated with continental interiors [Sleep et al., 1980]. Examples in North America are the Michigan, Illinois, and Williston basins. These basins typically contain a basal sand layer overlain by mostly marine sediments [Kinaston et al., 1983] deposited in the course of more than 100 m.y. of continuous subsidence and are generally not extensively faulted [Sleep et al., 1980]. Sediments on continental plat- forms, in general, are approximately one-half shales and one- half sands and carbonates [Ronov, 1968].

A calculation of compaction-driven flow in a symmetrical, 400-km-wide (200 km half width) basin cross section was made using the data in Table 1. Values of/4 •z, and/• and Cw and C,, which are often tabulated at various temperatures, pressures and solution compositions within computer codes, were assumed to be constant, for simplicity. The basin basement was assumed to be impermeable and to supply a constant heat flux. The center of the basin was chosen as a sym- metry plane, and the arch at the edge of the basin was kept at hydrostatic potential and a constant temperature gradient. Pressure and temperature at the sedimentation surface were held at 0.1 MPa (1 atm) and 25øC. Sedimentation was in equilibrium with subsidence so that the sedimentation surface remained at constant elevation. Five hundred meters of sand

were present at the initiation of subsidence, and initial temperatures were set at a conductive profile based on the basement heat flux.

Five kilometers of uncompacted sediments were continu- ously deposited in the center of the basin, and progressively less toward the edge, over the course of 100 m.y. This trans- lates to a sedimentation rate of 0.005 cm/yr and an approximate subsidence rate of 0.003 cm/yr. For comparison, Schwab [1976] and Nalivkin [1976] report subsidence rates of 0.001- 0.003 cm/yr for intracratonic basins in general. These overlying strata were one-half shaley sediments and one-half sands before compaction and were assumed to be interlayered verti- cally on a scale finer than the nodal point spacings. Shales and sands compacted differentially (Figure 3), and both had anisotropic permeabilities which decreased with compaction

6822 BETHKE' COMPACTION-DRIVEN GROUNDWATER FLOW AND HEAT TRANSFER

0.2 0.4 0.6

Fig. 3. Assumed compaction as a function of depth for sands and sandstones (solid line) and shaly sediments and shales (dashed line).

(Figure 4). Compaction assumptions are consistent with data compiled by Perrier and Quiblier [1974] and Hanor [1979, p. 140]. Some of the sands are proxies for carbonate sediments, which by available data [Halley and Schmoker, 1983] seem to compact in a manner similar to sands. Assumed permeabilities agree with measured values for sands from intracratonic basins [Buschbach and Bond, 1974; Becker et al., 1978] and for shales undergoing burial [Neglia, 1979, p. 582]. Because of the interlayering of the overlying sediments, overall permeabilities used in the finite difference calculation were computed using a parallel conductance analog in the x direction and a series analog in the z direction [Halek and Svec, 1979, p. 46].

One pass through the solution cycle was made for each of the 235 time steps taken in the calculation. Another calculation in which two passes were made at each time step gave nearly identical results. Forty-five interior nodal points were used at the onset of subsidence, and 285 were present as the last sediment was deposited. The calculation which made a single pass per time step used 10.4 CPU seconds, 32K words of array area, and 126K words maximum total memory on a CRAY-1 computer. The same problem executed in 64.1 CPU seconds on a smaller CDC CYBER-175 time-sharing computer.

Calculation Results

Results of the calculation are shown in Figure 5 for basin cross sections before basin subsidence, after 50 m.y., and after 100 m.y. of subsidence. Each cross section is drawn with a 30:1 vertical exaggeration. Arrows within the cross sections show horizontal and vertical components of the fluid average true velocities relative to the medium (instead of fixed elevation). Vertical components of the velocity vectors are exagger-

-12 z

-13

-14

-16

C• -17 o

-18

-19

-20 0.2 O4 06

Fig. 4. Assumed permeabilities in the x and z directions for sands and sandstones and clays and shales as a function of compaction.

x

t- 5o M.Y. Max. Vz =0.(X]33 cm/yr Max. Vzm = 0.0050 cm/yr

t-- lOO M.Y.

fkm 50km I

Max. Vx = 0.2060 cm/yr Max. Vz = 0.0056 cm/yr Max. Vzm = 0.0050 cm/yr

Fig. 5. Compaction-driven flow calculation within a basin cross section, as explained in the text. The cross section and velocity vectors are plotted with a 30:1 vertical exaggeration. Fluid velocities are shown relative to the subsiding medium. Equipotentials (solid lines) are drawn at 0.001 MPa (0.01 atm) intervals in the 50-m.y. results, and 0.002 MPa (0.02 atm) at 100 m.y. Values of v x and v z have been converted from curvilinear to Cartesian coordinates.

ated by the same amount as the cross section, so the vectors accurately reflect the direction of flow. The lower two rows of nodes show the location of the initial sand layer. Arrows along the bottom of the cross section show subsidence velocities of the basement relative to fixed elevation at the same vertical scale as the fluid velocity vectors. Contour lines are equipotentials spaced at 0.001 MPa (0.01 atm) intervals at 50 m.y. and 0.002 MPa (0.02 atm) at 100 m.y.

Results show a tendency for fluids to flow laterally toward the edge of the basin, as predicted by Magara [1976]. This lateral flow is due to the assumptions of anisotropic permeabilities and of interlayered strata with contrasting permeabilities. Deep lateral flow persists over a range of geologically reasonable permeability values as these assumptions are relaxed. Vertical flow would become dominant in the deep basin as the ratio of vertical to lateral overall permeabilities exceeds the ratio of the lengths of vertical to lateral pathways to the surface. Unlike deeper fluids, fluids within approximately the upper kilometer of sediments tend to move verti- cally to the surface.

An effect of the basal aquifer is to cause the potential gradient with depth to reverse itself, forming a wedge of greater hydraulic potential toward the center of the basin at moderate depth. Fluid above this wedge moves upward toward the surface, while fluid below the wedge tends to migrate obliquely across time lines into stratigraphically lower units.

BETlIKE: COMPACTION-DRIVEN GROUNDWATER FLOW AND HEAT TRANSFER 6823

Fig. 6. Resultant temperature distribution of the flow calculation shown in Figure 5. Isotherms (solid lines) are drawn at 25øC intervals. The geothermal gradient is not perturbed by either sedimentation or fluid advection.

Fluid fluxes along lateral flow paths increase toward the margin of the basin. This is because compaction-driven flow is additive along a flow path, with the volume flux at a point along the path nearly equal to the rate of pore volume collapse for the entire path up to that point. Owing to the small sedimentation rate, however, flow is very slow. The maximum true fluid velocities do not exceed approximately 2 mm/yr, although such creeping flows can be significant over long geologic time periods.

Another result of the calculation is that only about 0.01 MPa (0.1 atm) of excess potential is formed during basin compaction. Predicted excess potentials might be increased by re- ducing assumed permeabilities, but it is generally difficult to maintain high potential gradients such as those observed in the U.S. Gulf Coast [Dickinson, 1953] using geologically reasonable permeability values in a basin of this configuration and sedimentation rate.

The resulting temperature distribution is shown in Figure 6. Isotherms, shown by dark lines, remain horizontal and describe a steady state conductive profile set by basement heat flow. Possible thermal effects of the moving boundary at the sedimentation surface [Sharp and Domenico, 1976] (see also Ockendon and Hodgkins [1975]) and advecting pore fluids [Stallman, 1963; Bredehoeft and Papadopulos, 1965; Smith and Chapman, 1983] are not observed due to relatively small sedimentation rates and low fluid velocity components normal to the isotherms. An identical temperature distribution was obtained in a calculation in which sedimentation occurred over

only 10 m.y., suggesting that the thermal effects due to sedimentation and compaction-driven flow in intracratonic basins are negligible.

Variational Studies

Variational studies of compaction-driven flow systems show that in contrast to gravity-driven flow regimes, fluid velocity is usually independent of permeability and that permeability and excess hydraulic potential are inversely related. This "backward" scaling arises because although driving forces in gravity flow systems are external and set by potentiometric boundaries, compaction flow systems are driven by internal processes. Studies also show that both fluid velocity and excess potential scale proportionally with sedimentation rate.

Behavior of compaction-driven flow systems can be predicted by writing (10) in one dimension for the case in which permeability and viscosity are constant and porosity and temperature are functions of depth alone,

where

c• 2 tI) 1 c•tI)

c2z 2 •c c•t Ao l)zm

,40: - (1 -- qb) c•z + qbo• Tz'z- qb,Bpg Ao is a grouping of forcing terms which causes the com-

paction flow equation to differ from the diffusion equation. For constant Ao this equation is a counterpart in hydraulic potential to the heat flow equation for a medium with internal heat production [Carslaw and Jaeger, 1959, p. 130, equation (1)] which has an analytical solution for 0 < z < Zmax [Carslaw and Jaeger, 1959, equation (7) and Figure 20]. For dimension- less times, Kt/Zmax 2, greater than about one, which are quickly reached in compaction flow problems, Carslaw and Jaeger's solution applied to (10) predicts direct scaling between Vzm and hydraulic potential at all points and inverse scaling between permeability and potential. By Darcy's law, then, fluid velocity should scale with v:,• but be independent of permeability because any permeability increase will be matched by a proportional decrease in potential gradient.

These scaling relationships are supported by variational studies in which calculations such as the one already presented (Figures 5 and 6) were repeated using differing permeabilities and sedimentation rates. Results of variational

studies are shown in Figures 7 and 8 as the maximum hydraulic potentials and maximum fluid velocities present after the final time step in calculations made considering compaction disequilibrium (squares). Results of calculations made ignoring compaction disequilibrium are represented by solid lines, for reference. An arrow in each plot identifies the original calculation.

In most results, as predicted, fluid velocity is unaffected by permeability but scales linearly with sedimentation rate, and hydraulic potential scales with both variables. Simulations made under conditions of low enough permeability or great enough sedimentation rate to develop significant excess hydraulic potentials, however, give results which deviate from predicted scaling relations. Deviation is due to the dependence of porosity under disequilibrium conditions on equivalent depth Ze rather than on depth alone, as assumed in the above analysis. Fluid potentials and velocities under conditions of compaction disequilibrium are less than those predicted by scaling relations because pore fluids are compressed more slowly and more pore fluid is carried to depth than under equilibrium conditions.

The logical product of further permeability reduction or sedimentation rate increase is the approach of pore pressures to the lithostatic limit at which pore fluids support the weight of the saturated overburden [Dickinson, 1953; Rubey and Hub- bert, 1959]. Because fluid pressures under these conditions are fixed at a lithostatic gradient, fluid velocity alone can scale with permeability and must be independent of sedimentation rate. These alternate scaling relations may be important in rapidly subsiding basins such as the U.S. Gulf Coast but are not predicted in intracratonic basins.

The flow calculation shown in Figures 5 and 6 was also recalculated while ignoring thermal expansion of the pore fluid in order to assess the importance of aquathermal pressuring. Excess potential only decreased by 0.7% in the recalcula- tion, suggesting that thermal expansion of pore fluids is of

6824 BETHra•' COMPACTION-DRIVEN GROUNDWATER FLOW AND HEAT TRANSF•

0.2

0.1

0.0

• o D O [] D

_ []

I I I I I I I I

xO.0001 xO.001 xO.01 xO.1 Xl x10 x100 x1000

Permeability Fig. 7. Results of a parametric study in which all permeability

assumptions in the calculation already presented (Figures 5 and 6) wcrc systematically varied. The original calculation is identified by arrows. Squares show maximum excess potentials (in mcgapascals) and maximum fluid truc velocities (in centimeters per year) observed '•' at the end of simulations. Solid lines show trends of results of simulations in which compaction disequilibrium was ignored. Results show '•' "backward" scaling in which assumptions of permeability affect development of excess pressures but not fluid velocities. This observed scaling breaks down under conditions of very low permeability due to

o effects of compaction disequilibrium.

only limited importance in generating excess pressures within slowly subsiding basins. This is consistent with Barker's [1972] interpretation that aqu0.thermal pressuring should be operative only in sediments which are hydraulically isolated over the geologic time periods considered.

DISCUSSION

Results of this study show that compaction-driven groundwater flow within intracratonic basins is a process characterized by slow fluid velocities and small excess pressures. Fluid velocities in compaction flow systems are further shown to be independent of permeability. This result points to a fundamental difference between compaction flow and gravity flow systems. Compaction flow is by nature limited by the volume of pore fluid carried to depth in a basin. Gravity flow, on the other hand, is only limited by medium permeability and meteoric recharge.

Predicted fluid velocities pose a constraint on the relationship of compaction flow to secondary petroleum migration in intracratonic basins. Although compaction flow has gained much acceptance as a driving force for primary migration [Athy, 1930; Ma•;ara, 1980], secondary migration has often been accounted for by buoyancy or gravity-driven groundwater flow [Munn, 1909; Rich, 1921; Toth, 1980]. Remarkable amounts of lateral secondary migration have been observed in intracratonic basins, including more than 150 km in the Willi-

ston Basin [Dow, 1974] and Denver Basin [Clayton and Swet- land, 1980]. The nature of this migration is problematic, partly because buoyancy is of lessened effect in lateral migration and previous studies have not eliminated compaction-driven flow as a possible driving force. Because both elevated temperatures and time are thought to be required for organic matu- ration before a source bed may produce petroleum [Tissot and Welte, 1978, pp. 194-200; Waples, 1980], petroleum migration by compaction flow would be limited to the time period between the onset of generation and the cessation of compaction. Considering this time constraint and maximum fluid velocities of only several millimeters per year (or kilometers per million yearst predicted for compaction-driven flow as well as the probability that hydrocarbon phases will move more slowly than wetting fluids due to capillarity effects, this process does not seem to be a likely mechanism for driving the long-range lateral migration observed in intracratonic basins. A possible exception is oil generated in early stages of basin evolution.

Development of only small excess potentials in calculation results also casts doubt on the possibility of forming subsurface brines by compaction-driven flow in intracratonic basins. In order to concentrate sedimentary brines by reverse osmosis as proposed by Bredehoeft et al. [1963, 1964] and Graf et al. [1965, 1966], a pore pressure gradient must be maintained across shales which serve as semipermeable membranes [Graf, 1982]. Calculations by Graf [1982] for high-efficiency membranes indicate that a positive excess pressure gradient with

2 -

1 -

3 -

-1

-2

-3 -

i, i i i i i i

xo.o1 xo.1 Xl xlo XlOO XlOOO x lOOOO

Sedimentation Rate

Fig. 8. Results of a parametric study in sedimentation rate, in the same format as Figure 7. Although sedimentation rate is varied, the total amount of sediment deposited is identical in each calculation. The original calculation (Figures 5 and 6) is identified by arrows. Results show that both excess pressures and fluid velocities scale directly with assumptions of sedimentation rate except under conditions of compaction disequilibrium caused by very rapid deposition.


depth of about 5 MPa/km (50 atm/km) is required to counteract osmotic pressure. Although low-efficiency membranes may require lesser gradients [Phillips, 1983; Graf, 1983], this value is about three orders of magnitude greater than pressure gradients predicted in this study, indicating that compaction is not a driving force for reverse osmosis in this type of basin. Gravity-driven groundwater flow is an attractive alternate explanation for osmotic origins of subsurface brines [Bredehoeft et al., 1963;Phillips, 1983; Graf, 1983].

Results of this study also help constrain theories of the genesis of Mississippi Valley-type ore deposits. This type of epigenetic ore deposit is commonly found near margins of sedimentary basins [Anderson and Macqueen, 1982], and several authors (Noble [1963], Jackson and Beales [1967], Dozy [1970], and a review by Cathles [1981, pp. 448-450]) have suggested that ore minerals are precipitated from fluids driven from these basins by compaction flow. Because these deposits are thought to form at elevated temperatures and shallow depths [Anderson and Macqueen, 1982], Cathles and Smith [1983] note that the requirement of heat transport from basin to deposit provides a constraint on compaction flow theories of ore genesis. The result that heat is not carried toward basin margins by uninterrupted compaction-driven flow, also predicted by Cathles and Smith [1983], indicates that simple basin dewatering cannot explain the origin of these deposits. A variant of this hypothesis in which deposits are formed by sudden dewaterings of overpressured basins [Sharp, 1978; Cathles and Smith, 1983] also does not seem applicable to intracratonic basins, which by calculation results do not develop significant excess potentials. While this variant may be operative in more rapidly subsiding basins such as the Oua- chita Basin studied by Sharp [1978], it would not explain occurrences such as the Upper Mississippi Valley District [Heyl et al., 1959; Heyl and West, 1982] or the Pine Point District [Kyle, 1981] which are proximal to intracratonic basins. Gravity-driven groundwater flow, perhaps first suggested as a mechanism for the genesis of Mississippi Valley- type ores by Siebenthal [1915], may provide an explanation of deposit origin in these cases [Garven and Freeze, 1984a, b].

Evaluation of the contribution of compaction-driven flow to sediment cementation requires the combination of flow calculations and chemical modeling of cement precipitation and dissolution reactions [Hayes, 1979; Wood and Surdam, 1979]. While such calculations are beyond the scope of this paper, small fluid velocities may limit the importance of cementation processes which are dependent on compaction-driven flow in intracratonic basins. These processes would at least have to occur very slowly.

Finally, the fact that intracratonic basins do not develop significant excess potentials, especially compared to those observed in rapidly subsiding, shaly basins such as the U.S. Gulf Coast, raises the possibility that compaction-driven flow is subordinate to gravity-driven flow even during basin compaction. While evolving basins would have no elevation heads across subaqueous sedimentation surfaces, subaereal surfaces and erosional hiatuses can provide elevation heads which might be capable of overwhelming compaction-driven flow. Quantitative study of interaction of compaction heads with elevation heads in deep sediments would be very enlightening in this respect.

Calculations show that compaction-driven groundwater flow in intracratonic basins is characterized by fluid velocities

of only millimeters per year and very small excess hydraulic potentials. Compaction-driven fluids move too slowly to be effective in altering basin temperature distributions. Lack of significant excess potentials during compaction flow suggests that intracratonic basins are hydrologically different during their evolutions than rapidly subsiding basins, such as the U.S. Gulf Coast, where overpressuring is common.

Parametric exploration of the results shows that, contrary to the more familiar systematics of gravity-driven groundwater regimes, excess potentials and not fluid velocities are affected by variations of medium permeability. Both excess potentials and fluid velocities are found to scale linearly with sedimentation rate, and aquathermal pressuring accounts for < 1% of excess potentials.

Results argue against a causal relationship between compaction-driven groundwater flow in intracratonic basins and secondary petroleum migration, osmotic concentration of brines, and formation of Mississippi Valley-type ore deposits. The amount of cementation and porosity enhancement at- tributable to migrating fluids may also be limited by slow rates of fluid movement, and this possibility could be investi- gated by combining chemical calculations with the results of this study. Small excess potentials further raise the possibility that gravity-driven groundwater flow may overwhelm compaction flow, even in actively subsiding intracratonic basins.

APPENDIX 1

Coefficients of the finite difference approximation (11) to the fluid flow equation (10) are

Bi, j -- 0 T•i.j + ,/2 n+ 1

Di,j = 0 Txi_•/2.• n+ 1

Fi, j = 0 T•,,+ •/:.jn+ 1

Hid -' 0 rz,d_l/2 n+ 1

Fi'j (B + D + F + Eij = At n

Fi,j (ap -- p•It)zm)i,j n -- (1 -- 0) [r:,.j+,/• Qid = -- At n n

'(Oi,J + In -- Oi,J n) -•- Tx' •/2 jn(Oi- 1, i,j ) ,- , jn__(•) n

•- Txi + •/2,jn((•)i+ 1,j n -- (•)i,j n) •- T•i.j_ u2n(c•)i,j - 1 n -- •i,jn)] 1

•_ n n

+ n [Oxid (Pi+ 1,j -- Pi- 1,j n) + Q•,.jn(pi,j+ 1 Pi,j

Pi,j- ln)] + (1 -- •,jn+O) At n (•n+ • __ •n)•,j • n+l • n

__ (•n+O •n+O •n)i,j i,j -- i,j

= (4 K"+o

The flow transmissibility coefficients T• and • give the fluid fluxes between neighboring nodal blocks per unit potential difference between nodal points. These may be calculated by harmonic averaging as

2 n+l __

Txi • •/2,j (•,•" + •

n+l k n+l •+1 A • +1 kxi, i , •,• A•,• k•.• "+ • k•.• "+ • Axi,• "+ • "+ • Axi• •,• + A•.•.• •,•


2 n+l __

tzi,j+ 1/2 n n) (•i,j

2 Az. Az.s , , •, _ 1 •, _ 1

APP•mx

Work done by compression on a sedimentary column during burial is small compared to normal terrestrial heat flow. In general, comprcssivc work done on a volume of ma- terial may bc expressed

where P is confining pressure [Denbigh, 1971, p. 14]. As- suming a ]ithostaticaHy pressured column, which would do the greatest amount of work,

P = P• +

[R•bey and H•bber•, 19•9]. Also, the volume change duc to compression of fluid and rock grains may bc written

J• = -

Combining these relations and integrating over a sediment column of depth z .... the rate of energy addition to the column by compressire work per unit area of a basin is given

a, ot- + Using appropriate values for fluid (Table 1) and mineral com- pressibilities [Birch, 1966] and assuming 5- and 10-km-deep columns and a very high subsidence rate of 1 cm/yr, energy used in compression would amount to less than 0.25 and 1.0 mW/m =, respectively. This is a small fraction of the average terrestrial heat flow of about 60 mW/m 2 [Lee and Uyeda, 1965]. Because basins are observed to subside more slowly than 1 cm/yr [Schwab, 1976], work of compression may be safely ignored in compaction flow calculations.

APPENDIX 3

Coefficients of the finite difference approximation (15) to the heat flow equation (14) are

B.• =

F•,• =

H•.• = O(rz "+ • + W U•)•,•_•/•

•i,j n + 1 + Zx ' n + 1 + Zx ' n + 1 Eij= At" O(z•,•+ • ,- m,• ,+ •.•

+

+ [(1 -

Fi,J n Q'J = at" - ( - 0)

._ n(• J •,jn) + •X' l/2,j ' [•Zi,j+ 1/2 ,

n ,in + ( t , t, - 1/2

_ jn _

• .n + (1 -- • +,/2,) • +, ,jn] Vx.+l/2,j

-- [Wi,j+ 1/2 ri,j n -4- (1 - W•,j+ 1/=)T/j+ ln]Ugi,j+l/2} h n+øA I

Oxi•l/2. j = (,n cwn)i.j Oxi•l/2.jn+O

Uzcj•/2 = (pn Cwn)i,j Qzi,j•/2n+O The upscale weighting coefficients may be set at [Peaceman, 1977, p. 66]

The thermal transmissibilities rx and r, are the thermal fluxes between neighboring nodal blocks per unit temperature difference between nodal points. As with the flow transmissibilities, these may be calculated by harmonic averaging as

n+l

'•xi+ 1/2,j

2A n+ 1 n+ 1 Kx. in+ 1 K n+ 1 xi4 Axi ß • 4 ', xi + • 4 A n+ 1 K,,. jn+ 1 Ax i + Ax ' n+ 1 Kx,_ n+ 1 Axi, j Xi,j r, '}' 1 ,j _+ ! ,j ! ,j

n+l

TZ' j+ 1/2 •, _

2A=.i A t.i+ K•.jn+ 1 K•.i 1 n+ 1 n+l Azi,j+ n+l n+l Azi, n+l Azi,j Kzi.j 1 -It' Azi.j+ • Kzhj+ t j

NOTATION

Ax, A• area of a plane through an elemental volume normal to a specified direction (L2).

Ao grouping of forcing terms in fluid flow equation (M/Lt2).

B finite difference coefficient for lower node (Dt/M or E/tT).

Cw, Cr fluid and rock grain heat capacities (E/MT). D, E, F finite difference coefficients for left-side, center,

and right-side nodes, respectively (Dt/M or E/tT).

g acceleration of gravity (L/t2). H finite difference coefficient for upper node (Dt/M

or E/tT). hw, hr fluid and rock grain enthalpies (E/M).

Hw, H, H t net fluid, rock grain, and total enthalpies for an elemental volume (E).

i column subscript. j row subscript. I subscript for a rock type.

kx, k, intrinsic permeability in indicated direction (L2). Kx, K, thermal conductivity in indicated direction

(E/tLT). m fluid mass contained in an elemental volume (M). n superscript for current time step or known time

level.

n + 1 superscript for unknown time level. n + 0 superscript for averaged time level.

P pore fluid pressure (M/Lt2). q fluid volume flux (L/t). Q finite difference coefficient for nodes at the

known time level (L3/t or E/t).

BETHKE: COMPACTION-DRIVEN GROUNDWATER FLOW AND HEAT TRANSFER 6827

Qa internal heat source for elemental volume (E/t). Q,,, Qz total volume fluxes across an elemental volume

in indicated direction (L3/t). ref superscript indicating arbitrary reference con-

dition of differential compaction. s arbitrary curvilinear direction (L).

sc subscript denoting surface conditions. t time (t).

T temperature (T). T,,, T: flow transmissibilities in indicated direction

(L4t/M). Ux, U: finite difference advective heat transfer coef-

ficients (E/t T). v,,, v z fluid true velocities in indicated direction and

relative to the medium (L/t). Vzm settling velocity of a point in the medium relative

to a point at fixed elevation (L/t). V fluid volume contained in an elemental volume

(La). V• bulk volume of an element (L3). V• rock grain volume contained in an elemental

volume (L3). w mechanical work of compression (E). W finite difference upscale weighting coefficient. x distance along stratigraphic time lines (L).

X volume fraction of a rock type within an elemental volume.

z depth (L). z e effective depth (L). • isobaric coefficient of thermal expansion for pore

fluid at ambient P and T (T-1). /• isothermal coefficient of compressibility for pore

fluid at ambient P and T (Lt2/M). /•r coefficient of compressibility of rock grains

(Lt•-/M). F finite difference coefficient (L4t•-/M or E/T).

Ax, Az dimensions of an elemental volume (L). At size of a time step (t). 0 weighting coefficient for time averaging. •c diffusivity term in fluid flow equation (L2/t). /• fluid dynamic viscosity (M/Lt).

P, Pr, Psm density of fluid, rock grains, and fluid-saturated medium, respectively (M/L3).

Zx, z: thermal transmissibilities in indicated direction (E/tT).

•b effective porosity at ambient conditions. ß hydraulic potential, product of Hubbert's poten-

tial [Hubbert, 1940] and fluid density, equal to P -- pgz (M/Lt2).

L, length; t, time; M, mass; T, temperature; E, energy.

Acknowledgments. I would like to thank Larry Cathles, Turgay Ertekin, and Albert Hsui for help with numerical modeling techniques and Hubert Barnes, Joan Crockett, Pat Domenico, Grant Garven, Ken Green, Jim Kirkpatrick, Jack Sharp, and Rudy Slingerland for important discussions. Dave Converse, Ramon Espino, Lori Filipek, Leslie Smith, and John Ziagos reviewed the manuscript. This work was supported by Exxon Production Research Company, ARCO Oil and Gas Company, and National Science Foundation Graduate Fel- lowship Program. Karolyn Roberts typed the manuscript, and Pat Bremseth helped draft the figures.

REFERENCES

Ames, W. F., Numerical Methods for Partial Differential Equations, 2nd ed., Barnes and Noble, New York, 1977.

Anderson, G. M., and R. W. Macqueen, Ore deposit models, 6, Mis-

sissippi Valley-type lead-zinc deposits, Geosci. Can., 9, 108-117, 1982.

Athy, L. F., Compaction and oil migration, Am. Assoc. Pet. Geol. Bull., 14, 25-35, 1930.

Barker, C., Aquathermal pressuring--Role of temperature in development of abnormal-pressure zones, Am. Assoc. Pet. Geol. Bull., 56, 2068-2071, 1972.

Bear, J., Dynamics of Fluids in Porous Media, Elsevier, New York, 1972.

Becker, L. E., A. J. Hreta, and T. A. Dawson, Pre-Knox (Cambrian) stratigraphy in Indiana, Indiana Geol. Surv. Bull., 57, 1978.

Birch, F., Compressibility; elastic constants, in Handbook of Physical Constants, Mere. 97, edited by S. P. Clark, Jr., pp. 95-173, Geologi- cal Society of America, Boulder, Colo., 1966.

Bonham, L. C., Migration of hydrocarbons in compacting basins, Am. Assoc. Pet. Geol. Bull., 64, 549-567, 1980.

Bredehoeft, J. D., and B. B. Hanshaw, On the maintenance of anoma- lous fluid pressures, I, Thick sedimentary sequences, Geol. Soc. Am. Bull., 79, 1097-1106, 1968.

Bredehoeft, J. D., and I. S. Papadopulos, Rates of vertical groundwater movement estimated from the earth's thermal profile, Water Resour. Res., 1, 325-328, 1965.

Bredehoeft, J. D., C. R. Blyth, W. A. White, and G. B. Maxey, Possi- ble mechanism for concentration of brines in subsurface forma-

tions, Am. Assoc. Pet. Geol. Bull., 47, 257-269, 1963. Bredehoeft, J. D., C. R. Blyth, W. A. White, and G. B. Maxey, Possi-

ble mechanism for concentration of brines in subsurface forma-

tions, Reply, Am. Assoc. Pet. Geol. Bull., 48, 236-238, 1964. Buschbach, T. C., and D.C. Bond, Underground storage of natural

gas in Illinois, Ill. Pet., 101, 71 pp., 1974. Carnahan, B., H. A. Luther, and J. O. Wilkes, Applied Numerical

Methods, John Wiley, New York, 1969. Carslaw, H. S., and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed.,

University Press, Oxford, 1959. Cathles, L. M., Fluid flow and genesis of hydrothermal ore deposits,

Econ. Geol., 75th Anniv. Vol., 424-457, 1981. Cathles, L. M., and A. T. Smith, Thermal constraints on the forma-

tion of Mississippi Valley-type lead-zinc deposits and their impli- cations for episodic basin dewatering and deposit genesis, Econ. Geol., 78, 983-1002, 1983.

Chapman, R. E., Clays with abnormal interstitial fluid pressures, Am. Assoc. Pet. Geol. Bull., 56, 790-795, 1972.

Clayton, J. L., and P. J. Swetland, Petroleum generation and migration in Denver Basin, Am. Assoc. Pet. Geol. Bull., 64, 1613-1633, 1980.

Cooper, H. H., Jr., The equation of groundwater flow in fixed and deforming coordinates, J. Geophys. Res., 71, 4785-4790, 1966.

Denbigh, K., The Principles of Chemical Equilibrium, 3rd ed., Cam- bridge University Press, New York, 1971.

Dickinson, G., Geological aspects of abnormal reservoir pressures in Gulf Coast Louisiana, Am. Assoc. Pet. Geol. Bull., 37, 410-432, 1953.

Domenico, P. A., and V. V. Palciauskas, Thermal expansion of fluids and fracture initiation in compacting sediments, Geol. Soc. Am. Bull., Part II, 90, 953-979, 1979.

Dow, W. G., Application of oil-correlation and source-rock data to exploration in Williston Basin, Am. Assoc. Pet. Geol. Bull., 58, 1253-1262, 1974.

Dozy, J. J., A geologic model for the genesis of the lead-zinc ores of the Mississippi Valley, U.S.A., Trans. Inst. Min. Metall., Sect. B, 79, 163-170, 1970.

Dybbs, A., and S. Schweitzer, Conservation equations for nonisother- mal flow in porous media, J. Hydrol., 20, 171-180, 1973.

Garabedian, P. R., Partial Differential Equations, John Wiley, New York, 1964.

Garven, G., and R. A. Freeze, 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, 1984a.

Garven, G., and R. A. Freeze, Theoretical analysis of the role of groundwater flow in the genesis of stratabound ore deposits, 2, Quantitative results, Am. J. Sci., 284, 1125-1174, 1984b.

Graf, D. L., Chemical osmosis, reverse chemical osmosis, and the origin of subsurface brines, Geochim. Cosmochim. Acta, 46, 1431- 1448, 1982.

Graf, D. L., Chemical osmosis, reverse chemical osmosis, and the origin of subsurface brines (reply to a comment by F. M. Phillips), Geochim. Cosmochim. Acta, 47, 1333, 1983.


Graf, D. L., I. Friedman, and W. F. Meents, The origin of saline formation waters, II, Isotopic fractionation by shale micropore systems, Circ. Ill. State Geol. Surv., 393, 32 pp., 1965.

Graf, D. L., W. F. Meents, I. Friedman, and N. F. Shimp, The origin of saline formation waters, III, Calcium chloride waters, Circ. Ill. State Geol. Surv., 397, 60 pp., 1966.

Halek, V., and J. Svec, Groundwater Hydraulics, Elsevier, New York, 1979.

Halley, R. B., and J. W. Schmoker, High-porosity Cenozoic carbbnate rocks of south Florida: Progressive loss of porosity with depth, Am. Assoc. Pet. Geol. Bull., 67, 191-200, 1983.

Hanor, J. S., The sedimentary genesis of hydrothermal fluids, in Geo- chemistry of Hydrothermal Ore Deposits, 2nd ed., edited by H. L. Barnes, pp. 137-172, John Wiley, New York, 1979.

Hayes, J. B., Sandstone diagenesis--The hole truth, Spec. Publ. Soc. Econ. Paleontol. Mineral., 26, 127-139, 1979.

Heyl, A. V., and W. S. West, Outlying mineral occurrences related to the Upper Mississippi Valley Mineral District, Wisconsin, Iowa, Illinois, and Minnesota, Econ. Geol., 77, 1803-1817, 1982.

Heyl, A. V., A. F. Agnew, E. J. Lyons, and C. H. Behre, The geology of the Upper Mississippi Valley zinc-lead district, U.S. Geol. Surv. Prof Pap., 309, 1959.

Hubbert, M. K., The theory of groundwater motion, J. Geol., 48, 785-944, 1940.

Jackson, S. A., and F. W. Beales, An aspect of sedimentary basin evolution: The concentration of Mississippi Valley-type ores during the late stages of diagenesis, Bull. Can. Pet. Geol., 15, 393-433, 1967.

Jaeger, J. C., and N. G. W. Cook, Fundamentals of Rock Mechanics, 2nd ed., John Wiley, New York, 1976.

Kingston, D. R., C. P. Dishroon, and P. A. Williams, Global basin classification system, Am. Asssoc. Pet. Geol. Bull., 67, 2175-2193, 1983.

Kyle, J. R., Geology of the Pine Point lead-zinc district, in Handbook of Strata-Bound and Stratiform Ore Deposits, vol. 9, edited by K. H. Wolf, pp. 643-741, Elsevier, New York, 1981.

Lee, W. H. K., and S. Uyeda, Review of Heat flow data, In Terrestrial Heat Flow, Geophys. Monogr. Set., vol. 8, edited by W. H. K. Lee, pp. 87-190, AGU, Washington, D.C., 1965.

Lewis, G. N., and M. Randall, Thermodynamics, 2nd ed., revised by K. S. Pitzer and L. Brewer, McGraw-Hill, New York, 1961.

Magara, K., Reevaluation of montmorillonite dehydration as cause of abnormal pressure and hydrocarbon migration, Am. Assoc. Pet. Geol. Bull., 59, 292-302, 1975.

Magara, K., Water expulsion from clastic sediments during compaction--Directions and volumes, Am. Assoc. Pet. Geol. Bull., 60, 543-553, 1976.

Magara, K., Agents for primary hydrocarbon migration: A review, in Problems of Petroleum Migration, edited by W. H. Roberts and R. J. Cordell, pp. 33-45, American Association of Petroleum Geologists, Tulsa, Okla., 1980.

Mastin, C. W., Error induced by coordinate systems, in Numerical Grid Generation, edited by J. F. Thompson, pp. 31-40, North- Holland, New York, 1982.

Munn, M. J., Studies in the application of the anticlinal theory of oil and gas accumulation, Econ. Geol., 4, 141-157, 1909.

Muskat, M., The Flow of Homogeneous Fluids Through Porous Media, McGraw-Hill, New York, 1937.

Nalivkin, V. D., Dynamics of the development of the Russian platform structures, Tectonophysics, 36, 247-262, 1976.

Neglia, S., Migration of fluids in sedimentary basins, Am. Assoc. Pet. Geol. Bull., 63, 573-597, 1979.

Noble, E. A., Formation of ore deposits by water of compaction, Econ. Geol.,58, 1145-1156, 1963.

Ockendon, J. R., and W. R. Hodgkins, Moving Boundary Problems in Heat Flow and Diffusion, University Press, Oxford, 1975.

Peaceman, D. W., Fundamentals of Numerical Reservoir Simulation, Elsevier, New York, 1977.

Perrier, R., and J. Quiblier, Thickness changes in sedimentary layers during compaction history: Methods for quantitative evaluation, Am. Assoc. Pet. Geol. Bull., 58, 507-520, 1974.

Phillips, F. M., Comment on "Chemical osmosis, reverse chemical osmosis, and the origin of subsurface brines" by Donald L. Graf, Geochim. Cosmochim. Acta, 47, 1331, 1983.

Rich, J. L., Moving underground water as a primary cause of the

migration and accumulation of oil and gas, Econ. Geol., 16, 347- 371, 1921.

Richtmyer, R. D., Difference Methods for Initial-Value Problems, Wiley-Interscience, New York, 1957.

Rogers, L., Shale-density log helps detect overpressures, Oil Gas J., 64, 126-130, 1966.

Ronov, A. B., Probable changes in the composition of sea water during the course of geological time, Sedimentology, 10, 25-43, 1968.

Rubey, W. W., and M. K. Hubbert, Role of fluid pressure in mechanics of overthrust faulting, Geol. Soc. Am. Bull., 70, 167-206, 1959.

Rubin, H., Heat dispersion effect on thermal convection in a porous medium layer, J. Hydrol., 21, 173-185, 1974.

Schwab, F. L., Modern and ancient sedimentary basins: Comparative accumulation rates, Geology, 4, 723-727, 1976.

Sharp, J. M., Jr., Momentum and energy balance equations for compacting sediments, Math. Geol., 8, 305-322, 1976.

Sharp, J. M., Jr., Energy and momentum transport model of the Ouachita Basin and its possible impact on formation of economic mineral deposits, Econ. Geol., 73, 1057-1068, 1978.

Sharp, J. M., Jr., Permeability controls on aquathermal pressuring, Am. Assoc. Pet. Geol. Bull., 67, 2057-2061, 1983.

Sharp, J. M., Jr., and P. A. Domenico, Energy transport in thick sequences of compacting sediment, Geol. Soc. Am. Bull., 87, 390- 4O0, 1976.

Sibley, D. F., and H. Blatt, Intergranular pressure solution and cementation of the Tuscaloosa orthoquartzite, J. Sediment. Petrol., 46, 881-896, 1976.

Siebenthal, C. E., Origin of the zinc and lead deposits of the Joplin region, U.S. Geol. Surv. Bull., 606, 1915.

Sleep, N.H., J. A. Nunn, and L. Chou, Platform basins, Annu. Rev. Earth Planet. Sci., 8, 17-34, 1980.

Smith, L., and D. S. Chapman, On the thermal effects of groundwater flow, 1, Regional scale systems, J. Geophys. Res., 88, 593-608, 1983.

Stallman, R. W., Computation of ground-water velocity from temperature data, Methods of Collecting and Interpreting Ground-Water Data, edited by R. Bentall, U.S. Geol. Surv. Water Supply Pap., 1544-H, 36-46, 1963.

Stone, H. L., Iterative solution of implicit approximations of multidi- mensional partial differential equations, SIAM J. Numer. Anal., 5, 530-558, 1968.

Thomeer, J. H. M. A., and J. A. Bottema, Increasing occurrence of abnormally high reservoir pressures in boreholes, and drilling problems resulting therefrom, Am. Assoc. Pet. Geol. Bull., 45, 1721-1730, 1961.

Thompson, J. F., General curvilinear coordinate systems, in Numeri- cal Grid Generation, edited by J. F. Thompson, pp. 1-30, North- Holland, New York, 1982.

Tissot, B. P., and D. H. Welte, Petroleum Formation and Occurrence, Springer-Verlag, New York, 1978.

Torrance, K. E., Comparison of finite-difference computations of natural convection, J. Res. Natl. Bur. Stand., Sect. B 72B, 281-301, 1968.

Toth, J., Cross-formational gravity-flow of groundwater: A mechanism of the transport and accumulation of petroleum, in Problems of Petroleum Migration, edited by W. H. Roberts and R. J. Cordell, pp. 121-167, American Association of Petroleum Geologists, Tulsa, Okla., 1980.

Waples, D. W., Time and temperature in petroleum formation: Appli- cation of Lopatin's method to petroleum exploration, Am. Assoc. Pet. Geol. Bull., 64, 916-926, 1980.

Welty, J. R., C. E. Wicks, and R. E. Wilson, Fundamentals of Momen- tum, Heat, and Mass Transfer, John Wiley, New York, 1976.

Wood, J. R., and R. C. Surdam, Application of convective-diffusive models to diagenetic processes, Spec. Publ. Soc. Econ. Paleontol. Mineral., 26, 243-250, 1979.

C. M. Bethke, Hydrogeology Program, Department of Geology, 245 Natural History Building, University of Illinois, Urbana, IL 61801.

(Received April 3, 1984; revised March 13, 1985; accepted April 2, 1985.)

A Numerical Model of Compaction&Driven Groundwater … · to the small fluid velocities, ......

Documents

Transcript of A Numerical Model of Compaction&Driven Groundwater … · to the small fluid velocities, ......