JOURNALOF GEOPHYSICALRESEARCH,
VOL. 90,NO. B8,PAGES6817-6828,
JULY 10,1985
A Numerical Model of Compaction-DrivenGroundwater Flow and
Heat Transferand Its Applicationto the Paleohydrology
of Intracratonic SedimentaryBasins
CRAIG M. BETHKE
Hydro•Ieolo•7y
Pro,tram,Department
of Geolo•Iy,Universityof Illinois,Urbana
A newnumericalmethodallowscalculationof compaction-driven
groundwater
flow and associated
heattransfer
in evolving
sedimentary
basins.
The modelis formulated
in Lagrangian
coordinates
and
considers
two-dimensional
flow in heterogeneous,
anisotropic,
and accreting
domains.Boththe continuityof thedeforming
mediumandaquathermal
pressuring
areexplicitlytakeninto account.
A calculationof compaction-driven
flowduringevolution
of an idealized
intracratonic
sedimentary
basinincluding a basalaquiferpredictsslowgroundwater
movement
overlong time periods.Fluidsin shallow
sediments
tendto moveupwardtowardthesedimentation
surface,
anddeeper
fluidsmovelaterally.
The
hydraulic
potentialgradientwithdepthreverses
itselfnearthebasalaquifer,andfluidsin thisareahavea
tendency
to migrateobliquely
intostratigraphically
lowersediments.
Onlysmallexcess
pressures
develop,suggesting
thatintracratonic
basins
arenotsubject
to overpressuring
duringtheirevolutions.
Owing
to the smallfluidvelocities,
heattransferis conduction-dominated,
andthe geothermal
gradientis not
disturbed.
Variational
studies
showthatexcess
hydraulic
potentials,
but notfluidvelocities,
depend
on
assumptions
of permeability
and that bothexcess
potentialsand velocities
scalewith sedimentation
rate.
Aquathermal
pressuring
is foundto account
for < 1% of theexcess
potentials
developed
duringcompaction.Theseresultscast doubt on roles of compaction-driven
flow within intracratonicbasinsin
processes
of secondary
petroleum
migration,osmoticconcentration
of sedimentary
brines,andformation
of Mississippi
Valley-typeore deposits.
Resultsmightalsobe combinedwith chemicalmodelsto investigatethe relationshipof compactionflow to cementationin sediments.
INTRODUCTION
importance
of transportby groundwaters
to sedimentdiage-
Compaction-driven groundwater flow, the movement of
fluidsdue to collapseof pore volumeduringsedimentburial
withinevolvingsedimentary
basins,
is thoughtto bean important mechanismin geologicalprocesses
as diverseas petroleum migration,generationof overpressures,
concentration
of subsurfacebrines,formation of ore deposits,and cementation and porosity enhancementin sedimentaryrocks.
Compaction-driven
flowis a drivingforcein the poorlyunderstoodphenomenon
of petroleummigration,especially
for primary migrationfrom sourcerocksinto carrier beds[Athy,
1930; Bonharn,1980]. It is not clear whethercompactiondrivenor gravity-driven
groundwaterflow drivessecondary
migrationwithin carrierbeds[Rich, 1921; Athy, 1930; Toth,
nesis.
Despite the potential importance of compaction-driven
groundwaterflow to geologicalproblems,this processhas receivedlittle quantitativeevaluationon a basin-widescale,and
its trueimportance
hasbeendifficultto estimate.
Sharp[1976]
and SharpandDomenico[1976] developeda one-dimensional
model of vertical compaction-drivenflow and heat transfer
which they appliedto relativelyrapidly subsidingbasins.
Their analysis,however,doesnot addressthe possibilityof
lateralflow, whichmay be more significantthan verticalflow
in many sedimentarybasins[Magara, 1976]. In addition,
aquathermalpressuringis not taken into account.Cathiesand
Smith[1983] useda two-dimensional
model to predictthe
of overpressured
basins,
1980]. Sediment compactionis also believed to cause over- thermaleffectsof suddendewaterings
to generpressuringin basins[Dickinson,1953; Rubey and Hubbert, butit is not clearwhethertheirmethodis applicable
flow, and they only brieflydiscuss
1959;Bredehoeft
andHanshaw,1968],and thisprocess
maybe al studiesof compaction
augmentedby aquathermalpressuring,
definedas pore pres- their calculationtechnique.
This paper presentsa two-dimensionalanalysis of
surecreatedby thermalexpansionof pore fluidsrelativeto the
sediment
matrixduringburial[Barker,1972].High porepres- compaction-drivengroundwater flow and associated heat
anisotropic,and accretingmedia,
suregradientsfoundin overpressured
basinsare proposedto transferin heterogeneous,
concentratesubsurface
sedimentarybrinesby reverseosmosis applied to basin-widegeologicalproblems.The model is esacrossshaleystrata [Graf, 1982].MississippiValley-typeore peciallyusefulin evaluatingtheorieswhich dependon lateral
depositsare thoughtto be formedby dischargeof subsurface transportby groundwaters,suchas petroleummigrationand
The modelalsodiffersfrompreviousmodelsby its
brinesfrombasinsby compaction-driven
flow.Thesedeposits oregenesis.
may haveformedslowlyduringbasinevolution[Noble, 1963; formulation in a convenientLagrangianreferenceframe and
JacksonandBeales,1967;Dozy,1970]or suddenlyfrom flow by explicitly treating aquathermalpressuringand the contiout of overpressured
basins[Sharp, 1978; Cathiesand Smith, nuity of the deforming medium. These latter features allow
1983]. Finally, cementationand dissolutionprocesses
within more accurateand completemodeling of compactionflow. A
sedimentary
rockshavebeenrelatedto chemicaltransportby
fluidsredistributed
by compaction[Hayes,1979].Sibleyand
Blatt [1976] and Wood and Surdarn[1979] also discussthe
samplecalculation of compaction-drivenflow within an idealized intracratonic basin both givesrepresentativemodel resultsand providesgeneralinsightsinto thistype of hydrologic
regime.
SOLUTION PROCEDURE
Copyright1985by theAmericanGeophysical
Union.
Paper number 4B5062.
0148-0227/85/004B-5062505.00
6817
The numericaltechniqueof modelingcompaction-driven
flowis basedon approximate
solutions
to threecoupledpar-
6818
BETHKE: COMPACTION-DRIVENGROUNDWATER FLOW AND HEAT TRANSFER
tial differential equationsdescribingmedium continuity,fluid
flow, and heat transfer.These equationsare written and solved
in a Lagrangian referenceframe which remains fixed with respectto the subsidingmedium but movesthrough space.
Differential equations in deforming media may also be
derived in other referenceframes [Welty et al., 1976, p. 31].
Sharp [1976] used an Eulerian referenceframe 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 instantaneousspatial derivatives, the measurementof which is independentof any
temporal variation in the location of the observationpoint
[see Cooper, 1966]. In addition, both Lagrangian and Eulerian formulations require a moving boundary condition,
either at the sedimentation
Az "+ • = Az"
of the medium.
In order to evaluate the coupled differential equations,
which have no known analyticalsolution,the techniquesolves
decoupled or iteratively coupled finite difference approximations. These approximate equationscontain variablesonly
at discrete points in time and space and are amenable to
numerical solution by linear algebraic or iterative methods.
Subscriptsin finite differenceequations,by convention,representpositionin the spatialdomain. Superscriptsgive pointsin
time, either the previousand current time levelsn 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.
is usedin (2) and (3) to calculateAz at the (n + 0) time level.
Adding (2) and (3) gives the velocity differencebetween
neighboringnodal points:
V:m"J+
• --V:m'f
--2At•
[L' • _-•;,45 _li,j
+•
(4)
Specifyingporosityin this equation,typicallyas a functionof
burial depth or burial depth and pore pressure, defines
medium compaction and the driving force for compaction
flow.
Inasmuchas porosityat the unknown time leveldependson
the settlingvelocityover the time step (i.e., on burial depth at
the new time level), equation(4) is solvediteratively at each
nodal point. Several back substitutionsare usually sufficient
to converge to consistentvalues of velocity, porosity, and
nodal block height. These values,then, allow solution of the
fluid flow equation.
Fluid Flow Equation
A fluid flow equationwritten in termsof hydraulicpotential
describesmovement of a single-phasepore fluid of constant
compositionthrough a medium undergoingpore collapseand
temperaturevariation.An equationof statefor a slightlycompressiblefluid
Medium Continuity Equation
1
- •p = •8P -- o•8T
Subsidence
and compactionin evolvingsedimentary
basins
require that strata at different depths subsideat different velocities,much as coils of a springmove at varyingspeedsas
the springis compressed.
The relationshipof settlingvelocity
to compactionof the medium is given
8
1
From the definitionsof hydraulic potential, density,and
(1)
which givessettlingvelocity over the domain as a function of
the time rate of pore collapse. Porosity is defined here as
effectiveporosity [Bear, 1972, p. 43] at ambient conditions.
Equation (1) may be written in a finite differenceform to
give velocity differencesfor the current time step between a
node and the nodal block boundary below it and betweenthe
boundary and the node immediatelyunderneath:
Az n+O
•)zmi,j+
1 -- •)zmi,j+
1/2 •
8P • 80 + pg 8z
(6)
1
p
8p =-- 8m ---- 8V
(7)
V
8•p
8-•
V•m
--(1--•p)8t
Uzmi,j+
1/2 • •)zmi,j--
[Domenico and Palciauskas,1979; also Lewis and Randall,
1961, p. 25] providesthe basisfor a simplederivationof the
flow equation.This is a weakly nonlinearequationin P and
porosity,
[Cooper, 1966], wheremathematicalsymbolsare listed separately. Assumingincompressible
rock grains,this reducesto the
continuity equation
1
(5)
P
T.
8
• v•,•- Azet(Az)
8
(• - •)
(1
surface or the basement contact.
The Lagrangian formulation, however, is most concise becauseexceptat the sedimentationsurfaceno rock grainsmove
across the boundaries between nodal blocks, the elemental
volumes of the solution procedure.The Lagrangian solution
also minimizes "bookkeeping"sincenodal blocks always represent the same subdivision
Sincethe heightsof the nodal blockschangewith compaction,
the expression
n+ 1 -
2(1 - rh
'r i,j .+0•
;
Azi,j+1n+O •i,j+l
2(1- 05i,j+
•,,+o)
n
-- •i,j+l
-
(1 - q•)
(8)
respectively.
Also,from the potentialform of Darcy'slaw,
in an arbitrary curvilinear direction s,
(2) 8m=
[•x(pk•A•
•xx)AX+
•zz
(pk•A=
•zz)Az]
8t (9)
At"
n+l
8V=4•SV•+V•Sc
V
[Muskat, 1937,pp. 725-726].
Substituting(6), (7), (8), and (9) into (5), multiplying by V =
n
At n
(3)
&V•,andtakingthederivatives
with respectto time,
BETHKE'
COMPACTION-DRIVEN
GROUNDWATER
FLOWANDHEATTRANSFER
6819
Thisis similarto the diffusionequationin hydraulicpotential
with additional"forcing"termsdescribing
ratesof changein
fluid potentialenergywith burial, collapseof pore volume,
and thermalexpansion
of the porefluid.When the potential
energy,pore collapse,and thermalexpansiontermsare providedby solutionof the mediumcontinuityand heat transfer
equations,hydraulicpotentialis the only unknownvariable.
Using centraldifferenceapproximationsand the notation of
Stone[1968], equation(10)convertsto a finite differenceform
n+l
Bi,j • i,j+1
n+l
+1
+ Di,j • i- 1,j
+ Ei,j • i,jn
n+l
+ F•,•• +•,;•+• + H•,;• •,•_•
= Q•,•
(11)
wherethecoe•cients
B•,•,D•,;,E•,•,F•,•,Hi,j,andQ•,;aregiven
in Appendix1. A pentadiagonal
matrixequationis produced
by writing(11) at eachnodalpointwhilescanning
the finite
difference
gridalongthe i orj direction.Thismatrixequation
may be solvedfor hydraulicpotential at the new time level
usingeitherdirector iterativetechniques.
Calculationsin this
paper were made usinga band matrix direct method.
Heaf Transfer Equafion
The heat transferequation is used to solve for the basin
temperaturedistributionresultingfrom heat conduction,heat
Fig. 1. Technique of mapping an irregular basin cross section
(upper left) with a finite differencegrid (lower right). The basin x
directionis curvilinearalongstratigraphic
time linesand mapsto the
finite difference i direction.
In the caseof no pore collapseor internal heat sources,this
reducesto the convection-diffusion
equation.
This equationmay also be expressed
as a finite difference
approximation
n+l
Bi,./T/,.•+
•n+l+ Di,./T/_•,jn+l + E•,.•T:
•,.•
n+l
n+l
+ F•,jT•+•,j + H•,j T•,j_•
= Q•,j (15)
transferby advectinggroundwaters,
and internal energy wherethe coefficients
are givenin Appendix3. Equation(15)
sources,assuminglocal thermalequilibriumamongthe pore
fluid and rock grains.Dispersiveheat transfermay also be
producesa pentadiagonalcoefficientmatrix when written at
eachnodal point.
consideredby modifyingcoe•cients of the heat conduction
term [DybbsandSch•eifzer,197•; Rubin,1974].The energy Numerical Solution
usedin compression
of porefluidsandrockgrainsat geologiThe finite difference
equationsare appliedby mappinga
callyreasonable
sedimentation
ratesis negligible
comparedto finite differencegrid onto a crosssectionthrougha basin
normalheatflow(seeAppendix2) andmaybe ignored.
(Figure1). In orderto considerbasinscomposedof strataof
For a Lagrangianelementalvolume,whichalwayscontains
the samerock grains,the time rate of changein overallenthalpyequalsthe sumof the ratesof changedue to conduction, advection,andinternalenergysources,
8t- 8Hi]
8t 8Hi]
+0H
8Hi
c
a
The left sideof (12) expandsto separatethe enthalpies
of
in unfractured
-8t (Hw+
(
=p
=
vertical, linear coordinate. This choice of coordinates also
tendsto keepthex directioncoincident
with themajoraxisof
the permeability
ellipsoid,whichis usuallyalongstratigraphy
fluid and grains
8t
varying thicknesses,the basin cross section is described in
terms of a curvilinear coordinate [see Thompson,1982;
Mastin,1982]x alongstratigraphic
timelines(i.e.,connecting
rock grainswhichweredepositedcontemporaneously)
and a
coordinateacrossstratigraphy,
z. Owingto the muchgreater
breadth than depth of sedimentarybasins,the z direction,
whichis formallyorthogonalto x, may be approximatedas a
+
8T
phw• 8•
- c3 + _
Enthalpy changesdue to conductiveand advectiveeffects
arederivedby Stallman
•1963].Substituting
into(12),
at =a•
ax
•[p•Cw+p•(l_•)C•]ST
8(KxAx•
ST)
sediments.
The x coordinatemapsto the finite differencei direction,
and z to finitedifference
j. Integralvaluesof i andj are nodal
pointsat whichsolutions
to the finitedifference
equationsare
obtained.Each nodal point representsthe propertiesof a
nodal block (Figure2). Most variables,includingporosity,
hydraulicpotential,and temperature,vary with time but not
with positionin the nodal block.Settlingvelocity,however,
varieswith positionandnot with timeovera timestep.
The solution proceedsby steppingfrom a time level of
known valuesto a time level of unknown values,where the
time step is the time differencebetweenlevels.Sincethe finite
difference
equations
(4),(11),and(15)aredecoupled,
theymay
be solvedcyclicallyin this order for the unknown valuesat the
- phw
8x
Az
8t
(•8Q
xAx+•8Q•
)+Qu (1-•)
ph
w• 8•
new time level.A secondpassthroughthe cyclemay be used
to reducenumericalerrorswhichare introducedby thedecoupledsolution,if necessary.
In thiscase,the equationsare iteratively coupled.Once the three equationsare solved,the solu-
(14)
6820
Z
BETHKE: COMPACTION-DRIVEN GROUNDWATER FLOW AND HEAT TRANSFER
X
tAZij
.•Ax,
ii,j'
ß
.
=
AX
ß AZ
Vbi,
j i
i,j
Fig. 2. Relationship of finite differencenodal points (dots) to an
associatedquadrilateral 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 stepuntil the exact valuescan be computed.
Constraints on the size of the time step, which may be
dynamically chosenduring the course of the calculation, include a maximum proportional increase over the previous
time step, a maximum estimatedchangein settlingvelocity,
potential, and temperature,and the requirementthat no fluid
particle move farther than the dimensionsof its nodal block
during one time step.The latter expressesan empirical stability requirementof the heat transferequation [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
equalszero and one, the finite differenceequationsreduceto
explicit and fully implicit forms, respectively[Peaceman,1977,
p. 66]. A value of one half minimizes local truncation errors
and, basedon Von Neumann's analysis[seeRichtmyer,1957;
Ames, 1977], representsthe lower limit of unconditional stability in solutionsof the diffusion equation [Peaceman,1977,
pp. 69-74]. In solutions to the compaction flow problem,
however, small oscillationsmay appear in the numerical resultsif 0 is set near this limit, especiallywhen initiating pore
collapse from hydrostatic conditions. Oscillations occur becausechangesin the rate of pore volume collapseor temperature variation represent"errors" to solutionsof the diffusion
equation in the senseof Von Neumann's analysiswhich 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 dampensuchoscillationsat only a small increasein trun-
(i.e., approach lithostatic), they tend to counteract sediment
compaction. This effect has been observed in a number of
rapidly subsidingbasinscomposedof fine-grainedsediments
[Dickinson,1953; Thomeerand Bottema, 1961; Rogers, 1966;
Magara, 1975]. The resultingstatein which lesscompactionis
observedthan might be expectedat a given burial depth is
termed "compactiondisequilibrium"[Magara, 1975] and is
interpretedto 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] proposethat the stateof compaction of overpressuredsedimentscan be estimated by a
corresponding-state
model. They define an effectivedepth Ze
to representthe depth in a hydrostaticallypressuredsediment
at which effectivestressequals the effectivestresson a sediment in question. Effective stressis defined as the difference
betweenweight per unit area of overlyingsedimentsand pore
pressure[Jaeger and Cook, 1976, p. 219]. Predicted porosity
at depth z is calculatedas the normal or "equilibrium"porosity at depthze.For constantPwand Psm,
Zeis given:
Sedimentswith pore pressuresgreaterthan hydrostatic,then,
will haveeffectivedepthsshallowerthan actualburial depths.
A similarmodelwasemployedby Sharp[1976] and Sharpand
Domenico[1976] in modeling one-dimensionalcompactiondriven flow.
In practice,useof a compactiondisequilibriummodelwith
the numerical techniquealready presentedcan slow convergenceof the algorithm by introducing feedbackbetween the
fluid flow and medium continuity equations. In this case,
modification of the fluid flow equation to describeexplicitly
the changein pore volume due to variation in hydraulic potential will assurerapid convergence.Since porosity in the
corresponding-statemodel is a function of effective depth
alone,
0(/)
use different
solution
0(/) OZe
0(/)
1
[
c•t-- (•Z
e c3t- (•Z
e l)zm
-- (Psm
-- Pw)g
•
This result may be substitutedinto (10) to obtain a modified
fluid flow equation
1 c• p x
Ax
P
cation error.
Accuracy of the numerical techniqueis inferred from tests
showing excellent agreement between numerical results and
analytical solutions[Carslaw and Jaeger, 1959] to simplified
problems, small finite differenceresiduals,and checksfor conservationof rock and fluid massand total energy.Nonetheless,it is impossibleto check simultaneouslyall aspectsof a
complicatednumerical simulation techniquewhich may be
applied to a variety of problems.The best evaluation of the
model may come from comparisonwith future modelswhich
(•o - •Os½)
(,Os,,,-pw)g
Ze--Z•
L (1-•b)02
e{-(/)tiP{]
l)zm
"3'
(/)L OW
Someof the generalityof the previousdevelopmentis lost
sinceCgCk/CgZe
must be known to evaluatethis equation,while
no assumption
of theformof a porositymodelhadbeenmade
to this point. This is usuallya minor restriction.If an exponentialporosityeffectivedepthmodel FRubeyand Hubbert,
1959]
methods.
•b = •boexp (-bze)
CompactionDisequilibrium
If excesspore pressuresin evolving basinsbecomesignificant in comparisonto confining pressuresof basin sediments
is used, then
O(/)/OZ
e -'- -- b•)
BETHKE' COMPACTION-DRIVENGROUNDWATERFLOW AND HEAT TRANSFER
Differential Compaction
TABLE 1. Data Assumedin Compaction-Driven Flow Calculation
Sedimentarybasinsgenerallycontain a variety of rock types
intermixed on a finer scalethan the practical sizesof nodal
blocks. In this case, each block is divided into volume frac-
tions occupiedby each rock type. The volume fractions allow
calculationof bulk propertiessuchas porosity,permeability,
heat capacity,etc.
If the various rock typescompactdifferentiallywith burial,
the volume fraction of each will vary [Perrier and Quiblier,
1974]. Defining volume fractionsand porositiesof each rock
type at a referencecondition, such as the state of compaction
as measuredin a basin, solvesthis problem. Given reference
values, the current volume fractions as a function of current
porositiesfor a rock type I are
X, =
x/ref(1__•/ref) gbref
6821
Parameter
P
Pscexp [fl(P - 0.1)- a(T - 25)] kg/m3
Psc
0.001kg/m3
(P in MPa, T in øC)
•
5 x 10 -½ øC-•
•
4.3 x 10-'• MPa -1
#
1.0 cP
Pr
0.0027kg/m3
Cw
Cr
4.2 J/g øC
0.84 J/g øC
•b
0.5 exp [-(0.5
0.6 exp [-(0.6
-13 q- 2(/)
-19 + 8(/)
log k,,
kx/k:
(16)
where
Vb
ref EIX!(l -- •)l)
Vb ElX/ref(1
__
Sincethe unknownsXt appear on both sidesof (16), the equation may be solved iteratively. Calculating the ratio of bulk
volumesusingpreviousvalues,then calculatingeach Xt, and
finally applyingthe accelerator
X• = X•/y•X•
however,seemsto give excellentresultsin only one pass.
Boundary Conditions
Dirichlet and Neumann boundary conditions [Garabedian,
1964, pp. 227-228] to the fluid flow and heat transferequations may be specifiedalong the bottom edgeand sidesof the
domain using standardfinite differencetechniques[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 consideration.
An expanding layer of nodal blocks accommodatesthe
boundary at the sedimentationsurface,which moves with respect to the Lagrangian solution grid. This layer, which is
initially arbitrarily thin, grows to accept sedimentdeposition
until it reachesa target thickness,at which point a new layer
of nodesis created.Valuesof hydraulicpotential and temperature in thesenodal blocksmay be set at boundary valuesor,
more accurately,treated as unknowns.In the latter case,fluid
mass and total energy introducedinto the uppermostnodal
blocks by sedimentationmust be taken into account in (10)
and (14) and their finite differencecounterparts,(1l) and (15).
In addition, since nodal points remain in the center of these
expandingblocks,they subsideat only half the velocity of the
medium, and all values of V:min (11) and (15) should be
halved.
The medium continuity equation requireseither one or two
boundary conditions,dependingon whether sedimentationis
in equilibriumwith subsidence.
If equilibriumis maintainedso
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 basementsubsidencevelocity 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
Value
Kx = K:
2.5
sands
10.0
shales
x 10-5)z]
sands(z in cm)
x 10-5)z]
shales(z in cm)
sands,log m2
shales,log me
1.67 W/m øC
Basementheat flow
63 mW/m 2
0
0.65
subsidencevelocity should be specified.The secondboundary
condition might be estimated,for example,based on isostatic
adjustmentor tectonictheoriesof basin evolution.
COMPACTION-DRIVEN
FLOW IN INTRACRATONIC
BASINS
Intracratonic basins are broad, shallow, slowly subsiding
basins associated with continental interiors [Sleep et al.,
1980]. Examplesin 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] depositedin the course of more than 100 m.y. of
continuous subsidence and are generally not extensively
faulted [Sleep et al., 1980]. Sedimentson continentalplatforms, in general, are approximately one-half shalesand onehalf sandsand carbonates[Ronov, 1968].
A calculation of compaction-drivenflow 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,
pressuresand solution compositionswithin computer codes,
were assumedto be constant,for simplicity. The basin basement was assumedto be impermeable and to supply a constant heat flux. The center of the basin was chosenas a symmetry plane, and the arch at the edgeof the basinwas kept at
hydrostatic potential and a constant temperature gradient.
Pressureand temperature at the sedimentationsurfacewere
held at 0.1 MPa (1 atm) and 25øC. Sedimentation was in
equilibrium with subsidenceso that the sedimentationsurface
remained
at constant
elevation.
Five hundred
meters of sand
were presentat the initiation of subsidence,
and initial temperatureswere set at a conductiveprofile basedon the basement
heat flux.
Five kilometers of uncompactedsedimentswere continuously depositedin the center of the basin, and progressively
lesstoward the edge, over the course of 100 m.y. This translates to a sedimentationrate of 0.005 cm/yr and an approximate subsidencerate of 0.003 cm/yr. For comparison,Schwab
[1976] and Nalivkin [1976] report subsidencerates of 0.0010.003 cm/yr for intracratonic basinsin general.These overlying strata were one-half shaley sedimentsand one-half sands
beforecompactionand were assumedto be interlayeredvertically on a scalefiner than the nodal point spacings.Shalesand
sandscompacteddifferentially(Figure 3), and both had anisotropic permeabilities which decreased with compaction
6822
BETHKE' COMPACTION-DRIVENGROUNDWATERFLOW AND HEAT TRANSFER
0.2
0.4
0.6
t-
5o M.Y.
Max. Vz =0.(X]33cm/yr
Max. Vzm=0.0050cm/yr
Fig. 3. Assumed
compaction
as a functionof depthfor sandsand
sandstones
(solidline)andshalysediments
andshales(dashedline).
(Figure4). Compactionassumptions
are consistent
with data
compiledby Perrier and Quiblier[1974] and Hanor [1979, p.
140]. Someof the sandsare proxiesfor carbonatesediments,
which by availabledata [Halley and Schmoker,1983] seemto
compactin a mannersimilarto sands.Assumed
permeabilities
agree with measuredvalues for sands from intracratonic
basins[Buschbach
andBond,1974;Beckeret al., 1978] and for
shalesundergoingburial [Neglia, 1979,p. 582]. Becauseof the
interlayeringof the overlyingsediments,
overallpermeabilities
used in the finite differencecalculation were computed using a
parallel conductanceanalog in the x directionand a series
analogin the z direction[Halek andSvec,1979,p. 46].
One passthroughthe solutioncyclewas made for eachof
the 235 time stepstaken in the calculation.Another calculation in which two passeswere made at each time step gave
nearly identicalresults.Forty-five interior nodal pointswere
used at the onset of subsidence,and 285 were present as the
last sediment was deposited.The calculation which made a
singlepassper time stepused10.4CPU seconds,
32K words
of array area, and 126K wordsmaximumtotal memoryon a
t-- lOO M.Y.
Max. Vx = 0.2060cm/yr
Max. Vz = 0.0056cm/yr
Max. Vzm= 0.0050cm/yr
fkm
50km
I
Fig. 5. Compaction-drivenflow calculationwithin a basin cross
section,as explainedin the text. The crosssectionand velocityvectors
are plotted with a 30:1 vertical exaggeration.Fluid velocitiesare
shown relative to the subsidingmedium. 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 vx and vz have been
converted from curvilinear
to Cartesian coordinates.
CRAY-1 computer.The sameproblem executedin 64.1 CPU
seconds on a smaller CDC CYBER-175 time-sharing computer.
Calculation
ated by the sameamountas the crosssection,so the vectors
accuratelyreflectthe directionof flow. The lowertwo rowsof
nodesshow the location of the initial sandlayer. Arrows along
Results
Results of the calculation are shown in Figure 5 for basin
crosssectionsbeforebasin subsidence,after 50 m.y., and after
the bottom of the cross section show subsidence velocities of
the basement relative to fixed elevation at the same vertical
scaleas the fluid velocityvectors.Contour linesare equipoten100 m.y. of subsidence.
Each crosssectionis drawn with a
30:1 vertical exaggeration.Arrows within the crosssections tials spacedat 0.001 MPa (0.01 atm) intervalsat 50 m.y. and
show horizontal and vertical componentsof the fluid average 0.002 MPa (0.02 atm) at 100 m.y.
Results
showa tendency
for fluidsto flowlaterallytoward
true velocitiesrelative to the medium (insteadof fixed elevathe edge of the basin,as predictedby Magara [1976]. This
tion). Vertical componentsof the velocityvectorsare exaggerlateral flow is due to the assumptionsof anisotropic permeabilitiesand of interlayeredstrata with contrastingper-12
z
meabilities.Deep lateral flow persistsover a rangeof geologi-13
cally reasonablepermeabilityvaluesas theseassumptions
are
-14
relaxed. Vertical flow would become dominant in the deep
basin as the ratio of vertical to lateral overall permeabilities
exceedsthe ratio of the lengthsof vertical to lateral pathways
-16
to the surface.Unlike deeper fluids, fluids within approxix
C• -17
matelythe upperkilometerof sedimentstend to move verti-
o
-18
cally to the surface.
An effectof the basalaquiferis to causethe potentialgradi-
-19
ent with depth to reverseitself, forming a wedgeof greater
hydraulicpotentialtowardthe centerof the basinat moderate
depth.Fluid abovethiswedgemovesupwardtowardthe surFig. 4. Assumed
permeabilities
in the x and z directions
for sands face, while fluid below the wedge tends to migrate obliquely
acrosstime linesinto stratigraphicallylower units.
and sandstones
and claysand shalesas a functionof compaction.
-20
0.2
O4
06
BETlIKE: COMPACTION-DRIVEN
GROUNDWATER FLOW AND HEAT TRANSFER
c•2tI)
1 c•tI)
c2z2
•c c•t
6823
Ao l)zm
where
,40:- (1--qb)
c•z
+ qbo•
Tz'zqb,Bpg
Fig. 6. Resultant temperaturedistribution of the flow calculation
shownin Figure 5. Isotherms(solidlines)are drawn at 25øC intervals.
The geothermal gradient is not perturbed by either sedimentationor
fluid advection.
Ao is a grouping of forcing terms which causesthe compaction flow equation to differ from the diffusionequation.
For constantAo this equation is a counterpartin hydraulic
potentialto the heat flow equationfor a mediumwith internal
heat production[Carslaw and Jaeger, 1959, p. 130, equation
(1)] whichhas an analyticalsolutionfor 0 < z < Zmax
[Carslaw
and Jaeger,1959,equation(7) and Figure 20]. For dimension-
lesstimes,Kt/Zmax
2, greaterthanaboutone,whicharequickly
Fluid fluxes along lateral flow paths increase toward the
margin of the basin.This is becausecompaction-drivenflow is
additive along a flow path, with the volume flux at a point
along the path nearly equal to the rate of pore volume collapsefor the entire path up to that point. Owing to the small
sedimentationrate, however,flow is very slow. The maximum
true fluid velocitiesdo not exceedapproximately 2 mm/yr,
although suchcreepingflows can be significantover long geologic time periods.
Another result of the calculation is that only about 0.01
MPa (0.1 atm) of excesspotential is formed during basin compaction. Predictedexcesspotentialsmight be increasedby reducing assumedpermeabilities,but it is generally difficult to
maintain high potential gradients such as those observedin
the U.S. Gulf Coast [Dickinson,1953] using geologicallyreasonable permeability values in a basin of this configuration
and sedimentation
rate.
in which sedimentation
occurred
over
only 10 m.y., suggestingthat the thermal effectsdue to sedimentation and compaction-drivenflow in intracratonic basins
are negligible.
Variational
permeabilityand potential.By Darcy'slaw, then,fluid velocity
shouldscalewith v:,• but be independentof permeabilitybecause any permeability increasewill be matched by a proportionaldecreasein potentialgradient.
These scaling relationshipsare supported by variational
studiesin which calculations such as the one already presented (Figures 5 and 6) were repeated using differing permeabilities
and sedimentation
rates. Results of variational
studiesare shown in Figures 7 and 8 as the maximum hydraulicpotentialsand maximumfluid velocitiespresentafter
the final time step in calculationsmade consideringcompactiondisequilibrium(squares).Resultsof calculationsmade
ignoringcompactiondisequilibriumare representedby solid
lines,for reference.An arrow in each plot identifiesthe original calculation.
The resulting temperaturedistribution is shown in Figure 6.
Isotherms, shown by dark lines, remain horizontal and describe a steady state conductive profile set by basement heat
flow. Possiblethermal effectsof the moving boundary at the
sedimentation surface [Sharp and Domenico, 1976] (see also
Ockendonand Hodgkins [1975]) and advecting pore fluids
[Stallman, 1963; Bredehoeftand Papadopulos,1965; Smith and
Chapman,1983] are not observeddue to relatively small sedimentation rates and low fluid velocity componentsnormal to
the isotherms.An identical temperaturedistribution was obtained in a calculation
reachedin compaction flow problems,Carslaw and Jaeger's
solutionappliedto (10) predictsdirect scalingbetweenVzmand
hydraulic potential at all points and inversescalingbetween
Studies
Variational studiesof compaction-drivenflow systemsshow
that in contrastto gravity-drivenflow regimes,fluid velocityis
usuallyindependentof permeabilityand that permeabilityand
excesshydraulic potential are inversely related. This "backward" scalingarisesbecausealthoughdriving forcesin gravity
flow systemsare external and set by potentiometric boundaries, compaction flow systemsare driven by internal processes.Studiesalso show that both fluid velocity and excess
potential scaleproportionally with sedimentationrate.
Behavior of compaction-drivenflow systemscan be predicted by writing (10) in one dimension for the case in which
permeabilityand viscosityare constantand porosity and temperatureare functionsof depth alone,
In most results,as predicted,fluid velocity is unaffectedby
permeabilitybut scaleslinearly with sedimentationrate, and
hydraulic potential scaleswith both variables.Simulations
made under conditionsof low enough permeabilityor great
enoughsedimentationrate to developsignificantexcesshydraulic potentials,however,give resultswhich deviatefrom
predictedscalingrelations.Deviation is due to the dependence
of porosity under disequilibriumconditionson equivalent
depthZerather than on depthalone,as assumedin the above
analysis.Fluid potentialsand velocitiesunder conditionsof
compactiondisequilibriumare less than those predictedby
scaling relations becausepore fluids are compressedmore
slowly and more pore fluid is carried to depth than under
equilibrium conditions.
The logical product of further permeabilityreduction or
sedimentationrate increaseis the approachof pore pressures
to the lithostaticlimit at which pore fluids support the weight
of the saturatedoverburden[Dickinson,1953;Rubeyand Hubbert, 1959]. Becausefluid pressures
under theseconditionsare
fixed at a lithostatic gradient, fluid velocity alone can scale
with permeabilityand must be independentof sedimentation
rate. These alternate scaling relations may be important in
rapidly subsidingbasinssuchas the U.S. Gulf Coast but are
not predictedin intracratonicbasins.
The flow calculation shown in Figures 5 and 6 was also
recalculatedwhile ignoring thermal expansion of the pore
fluid in order to assessthe importanceof aquathermalpressuring. Excesspotential only decreasedby 0.7% in the recalculation, suggestingthat thermal expansionof pore fluids is of
6824
BETHra•' COMPACTION-DRIVENGROUNDWATERFLOW AND HEAT TRANSF•
ston Basin [Dow, 1974] and Denver Basin [Clayton and Swet-
land,1980].Thenature
ofthismigration
isproblematic,
partly
becausebuoyancyis of lessenedeffectin lateral migration and
previous studieshave not eliminated compaction-drivenflow
as a possible driving force. Becauseboth elevated temperatures and time are thought to be required for organic maturation 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 cessationof
compaction. Considering this time constraint and maximum
fluid velocitiesof onlyseveralmillimetersper year(or kilome-
terspermillionyearstpredicted
for compaction-driven
flowas
•
0.2
o
D
O
[]
D
well as the probability that hydrocarbon phaseswill move
more slowly than wetting fluids due to capillarity effects,this
processdoesnot seemto be a likely mechanismfor driving the
long-rangelateral migration observedin intracratonic basins.
A possibleexceptionis oil generatedin early stagesof basin
evolution.
Development of only small excesspotentials in calculation
resultsalso castsdoubt on the possibilityof forming subsurface brines by compaction-drivenflow in intracratonic basins.
In order to concentratesedimentarybrinesby reverseosmosis
0.0
as proposedby Bredehoeftet al. [1963, 1964] and Graf et al.
I
I
I
I
I
I
I
I
[1965, 1966], a pore pressuregradient must be maintained
xO.0001
xO.001
xO.01
xO.1
Xl
x10
x100
x1000
acrossshaleswhich serveas semipermeablemembranes[Graf,
1982]. Calculationsby Graf [1982] for high-efficiency
memPermeability
Fig. 7. Results of a parametric study in which all permeability branes indicate that a positive excesspressuregradient with
0.1
_
[]
assumptionsin the calculationalready presented(Figures 5 and 6)
wcrcsystematically
varied.The originalcalculationis identifiedby
arrows. Squaresshow maximum excesspotentials(in mcgapascals)
2
-
1
-
3
-
andmaximum
fluidtrucvelocities
(in centimeters
peryear)observed '•'
at the end of simulations.Solid lines show trends of resultsof simulations in which compactiondisequilibriumwas ignored.Resultsshow
"backward"scalingin which assumptionsof permeabilityaffectdevelopment of excesspressuresbut not fluid velocities.This observed
scalingbreaksdown underconditionsof very low permeabilitydue to
effectsof compactiondisequilibrium.
'•'
o
only limited importancein generatingexcesspressureswithin
slowly subsiding basins. This is consistent with Barker's
[1972]interpretation
that aqu0.thermal
pressuring
shouldbe
operativeonly in sedimentswhich are hydraulicallyisolated
over the geologictime periodsconsidered.
DISCUSSION
Resultsof this study show that compaction-drivengroundwater flow within intracratonic basins is a process
characterized
by slowfluid velocities
and smallexcess
pressures.Fluid velocitiesin compactionflow systemsare further
shown
tobeindependent
ofpermeability.
Thisresult
points
to
a fundamental
difference
between
compaction
flowandgravity flow systems.Compactionflow is by naturelimited 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 meteoricrecharge.
Predicted fluid velocitiespose a constraint on the relationship of compactionflow to secondarypetroleummigration in
intracratonic basins. Although compaction flow has gained
much acceptanceas a driving force for primary migration
[Athy, 1930;Ma•;ara,1980],secondary
migrationhas often
been accountedfor by buoyancy or gravity-driven groundwater flow [Munn, 1909; Rich, 1921; Toth, 1980]. Remarkable
amounts of lateral secondarymigration have been observedin
intracratonic basins,including more than 150 km in the Willi-
-1
-2
-3
-
i,
xo.o1
i
i
i
xo.1
Xl
xlo
Sedimentation
i
XlOO
i
i
XlOOO
x lOOOO
Rate
Fig. 8. Resultsof a parametricstudy in sedimentationrate, in the
same format as Figure 7. Although sedimentationrate is varied, the
total amount of sedimentdepositedis identical in each calculation.
The original calculation (Figures 5 and 6) is identified by arrows.
Resultsshowthat both excesspressures
and fluid velocities
scale
directly with assumptionsof sedimentationrate except under conditionsof compactiondisequilibriumcausedby very rapid deposition.
BETHKE' COMPACTION-DRIVEN GROUNDWATER FLOW AND HEAT TRANSFER
depth of about 5 MPa/km (50 atm/km) is required to counteract osmoticpressure.Although low-efficiencymembranesmay
require lessergradients[Phillips, 1983; Graf, 1983], this value
is about three orders of magnitude greater than pressure
gradientspredictedin this study,indicating that compactionis
not a driving force for reverseosmosisin this type of basin.
Gravity-driven groundwater flow is an attractive alternate explanation for osmotic origins of subsurfacebrines [Bredehoeft
et al., 1963;Phillips, 1983; Graf, 1983].
Results of this study also help constrain theories of the
genesisof MississippiValley-type ore deposits.This type of
epigenetic ore deposit is commonly found near margins of
sedimentarybasins [Andersonand Macqueen, 1982], and several authors (Noble [1963], Jacksonand Beales [1967], Dozy
[1970], and a review by Cathles [1981, pp. 448-450]) have
suggestedthat ore mineralsare precipitatedfrom fluids driven
from thesebasinsby compactionflow. Becausethesedeposits
are thought to form at elevated temperatures and shallow
depths [Anderson and Macqueen, 1982], Cathles and Smith
[1983] note that the requirementof heat transport from basin
to depositprovidesa constrainton compactionflow theories
of ore genesis.The result that heat is not carried toward basin
margins by uninterrupted compaction-drivenflow, also predicted by Cathles and Smith [1983], indicates that simple
basindewateringcannot explain the origin of thesedeposits.A
variant of this hypothesisin which deposits are formed by
sudden dewaterings of overpressuredbasins [Sharp, 1978;
Cathles and Smith, 1983] also does not seem applicable to
intracratonicbasins,which by calculationresultsdo not develop significantexcesspotentials.While this variant may be
operative in more rapidly subsidingbasinssuch as the Ouachita Basin studied by Sharp [1978], it would not explain
occurrencessuch 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 suggestedas a mechanismfor the genesisof MississippiValleytype oresby Siebenthal[1915], may provide an explanationof
depositorigin in thesecases[Garvenand Freeze, 1984a,b].
Evaluation of the contributionof compaction-drivenflow to
sedimentcementationrequiresthe combination of flow calculations and chemical modeling of cement precipitation and
dissolutionreactions[Hayes, 1979; Wood and Surdam,1979].
While such calculationsare beyond the scope of this paper,
small fluid velocitiesmay limit the importanceof cementation
processeswhich are dependenton compaction-drivenflow in
intracratonic basins.These processeswould at least have to
occur very slowly.
Finally, the fact that intracratonic basins do not develop
significantexcesspotentials,especiallycomparedto those observedin rapidly subsiding,shaly basinssuchas the U.S. Gulf
Coast, raises the possibility that compaction-drivenflow is
subordinate to gravity-driven flow even during basin compaction. While evolving basinswould have no elevation heads
acrosssubaqueoussedimentationsurfaces,subaerealsurfaces
and erosional hiatuses can provide elevation heads which
might be capable of overwhelmingcompaction-drivenflow.
Quantitative study of interaction of compaction heads with
elevationheadsin deep sedimentswould be very enlightening
in this respect.
Calculations show that compaction-driven groundwater
flow in intracratonic basinsis characterizedby fluid velocities
6825
of only millimetersper year and very small excesshydraulic
potentials. Compaction-drivenfluids move too slowly to be
effectivein altering basin temperature distributions. Lack of
significantexcesspotentials during compactionflow suggests
that intracratonic basins are hydrologically different during
their evolutionsthan rapidly subsidingbasins,suchas the U.S.
Gulf Coast, where overpressuringis common.
Parametric exploration of the resultsshowsthat, contrary
to the more familiar systematicsof gravity-driven groundwater regimes,excesspotentials and not fluid velocities are
affected by variations of medium permeability. Both excess
potentials and fluid velocitiesare found to scalelinearly with
sedimentationrate, and aquathermal pressuringaccountsfor
< 1% of excesspotentials.
Results argue against a causal relationship between
compaction-drivengroundwater flow in intracratonic basins
and secondarypetroleum migration, osmoticconcentrationof
brines,and formation of MississippiValley-type ore deposits.
The amount of cementation and porosity enhancementattributable to migrating fluids may also be limited by slow
rates of fluid movement, and this possibilitycould be investigated by combining chemical calculations with the results of
this study.Small excesspotentialsfurther raise the possibility
that gravity-driven groundwater flow may overwhelm compaction flow, evenin activelysubsidingintracratonicbasins.
APPENDIX
1
Coefficientsof the finite differenceapproximation (11) to the
fluid flow equation (10) are
Bi,j -- 0 T•i.j
+,/2n+1
Di,j= 0 Txi_•/2.•
n+1
Fi,j = 0 T•,,+
•/:.jn+
1
Hid-' 0 rz,d_l/2
n+1
Fi'j
Atn (B+D+F+
Eij=
--p•It)zm)i,j
n--(1--0)[r:,.j+,/•
Qid= -- Fi,j
Atn(ap
n
,- •/2
,jn(Oi-1,jn__(•)i,jn)
'(Oi,J+In-- Oi,J
n)-•-Tx'
•- Txi
+•/2,jn((•)i+
1,jn-- (•)i,j
n)•- T•i.j_
u2n(c•)i,j
- 1n-- •i,jn)]
1
+
•_
n
Pi,j
n
n[Oxid(Pi+1,j -- Pi- 1,jn)+ Q•,.jn(pi,j+
1
Pi,jln)]
+(1--•,jn+O)
At
n(•n+
•__
•n)•,j
•
n+l
__(•n+O
•n+O•n)i,ji,j
= (4
•
n
-- i,j
K"+o
The flow transmissibilitycoefficientsT• and • give the fluid
fluxes between neighboringnodal blocks per unit potential
differencebetween nodal points. These may be calculated by
harmonic averagingas
n+l
Txi• •/2,j
__
2
(•,•" + •
•+1A •+1kxi,
in+l k , •,•n+l
A•,•"+• k•.•"+• Axi••,•+ A•.•.•"+• k•.••,•"+• Axi,•
6826
BETHKE: COMPACTION-DRIVENGROUNDWATERFLOW AND HEAT TRANSFER
n+l
2
__
tzi,j+1/2
-- [Wi,j+1/2ri,jn-4-(1 - W•,j+
1/=)T/j+
ln]Ugi,j+l/2}
(•i,jn
n)
h n+øA I
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, comprcssivcwork done on a volume of material may bc expressed
Oxi•l/2.
j = (,ncwn)i.j
Oxi•l/2.jn+O
Uzcj•/2
= (pnCwn)i,j
Qzi,j•/2n+O
The upscaleweighting coefficientsmay be set at [Peaceman,
1977, p. 66]
where P is confining pressure [Denbigh, 1971, p. 14]. Assuming a ]ithostaticaHypressuredcolumn, which would do the
greatestamount of work,
P = P• +
[R•bey and H•bber•, 19•9]. Also, the volume changeduc to
compressionof fluid and rock grainsmay bc written
J• = -
The thermal transmissibilitiesrx and r, are the thermal
fluxesbetweenneighboringnodal blocksper unit temperature
differencebetween nodal points. As with the flow transmissibilities,thesemay be calculatedby harmonicaveragingas
n+l
Combining these relations and integrating over a sediment '•xi+1/2,j
column of depth z.... the rate of energy addition to the
columnby compressirework per unit area of a basinis given
a,ot- +
2Axi4n+1Axiß•4n+1Kx.
',in+1Kxi+•4n+1
AXi,j n+1 K,,.
1 Axi'}' 1 ,j + Ax'_+! ,j n+1 Kx,_
1 Axi,j
r,jn+
! ,j n+
n+l
TZ'j+ 1/2
•,
_
2A=.i
At.i+ K•.jn+
1K•.i 1n+1
Using appropriate valuesfor fluid (Table 1) and mineral compressibilities[Birch, 1966] and assuming5- and 10-km-deep
columnsand a very high subsidencerate of 1 cm/yr, energy
used in compressionwould amount to less than 0.25 and 1.0
mW/m=, respectively.
This is a smallfractionof the average
terrestrialheat flow of about 60 mW/m2 [Lee and Uyeda,
1965]. Becausebasins are observedto subsidemore slowly
than 1 cm/yr [Schwab, 1976], work of compressionmay be
safelyignored in compactionflow calculations.
APPENDIX 3
Coefficientsof the finite differenceapproximation(15) to the
heat flow equation (14) are
Azi,jKzi.jn+l Azi,j+1n+l -It'Azi.j+
• Kzhj+
tn+l Azi,j n+l
NOTATION
Ax, A•
area of a plane through an elementalvolume
normalto a specified
direction(L2).
Ao groupingof forcingtermsin fluid flow equation
(M/Lt2).
B finitedifference
coefficient
for lowernode(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-sidenodes,respectively
(Dt/M or
B.• =
E/tT).
g acceleration
of gravity(L/t2).
H finitedifference
coefficient
for uppernode(Dt/M
F•,•=
H•.•= O(rz
"+• + W U•)•,•_•/•
' m,•
n+1+ Zx
' •.•n+1
Eij= •i,j
At" O(z•,•+
• n+1+ Zx
,,+
+
kx,k, intrinsicpermeability
in indicateddirection(L2).
+ [(1 Fi,J
Q'J=
Kx, K,
n
at"
thermal conductivityin indicateddirection
(E/tLT).
m fluid masscontainedin an elementalvolume(M).
n superscriptfor current time stepor known time
- ( - 0)
' [•Zi,j+
1/2
n(•,J ._ •,jn) + •X' l/2,j
+
t
,
or E/tT).
hw,hr fluid and rock grain enthalpies(E/M).
Hw, H, H t net fluid, rock grain, and total enthalpiesfor an
elementalvolume (E).
i column subscript.
j row subscript.
I subscriptfor a rock type.
n(
,in
_
level.
n + 1 superscriptfor unknown time level.
n + 0 superscriptfor averagedtime level.
t, - 1/2
jn
_
• .n+ (1 -- • +,/2,)• +, ,jn]Vx.+l/2,j
P porefluidpressure
(M/Lt2).
q fluid volume flux (L/t).
Q finite differencecoefficientfor nodesat the
knowntimelevel(L3/tor E/t).
BETHKE: COMPACTION-DRIVEN GROUNDWATER FLOW AND HEAT TRANSFER
Qa internal heat sourcefor elemental volume (E/t).
Q,,, Qz total volume fluxesacrossan elemental volume
in indicateddirection(L3/t).
ref superscriptindicatingarbitrary referencecondition of differentialcompaction.
s arbitrary curvilinear direction (L).
sc subscriptdenoting surfaceconditions.
t time (t).
T temperature (T).
T,,, T: flow transmissibilitiesin indicated direction
(L4t/M).
Ux, U:
finite differenceadvectiveheat transfercoefficients(E/t T).
v,,, vz fluid true velocitiesin indicated direction and
relative to the medium (L/t).
Vzm settlingvelocityof 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 volumeof an element(L3).
V• rock grain volume contained in an elemental
volume(L3).
w mechanicalwork of compression(E).
W finite differenceupscaleweightingcoefficient.
x distancealong stratigraphictime lines (L).
X volume fraction of a rock type within an elemental
volume.
z depth (L).
ze effectivedepth (L).
• isobariccoefficientof thermal expansionfor pore
fluid at ambientP and T (T-1).
/• isothermalcoefficientof compressibilityfor pore
fluidat ambientP and T (Lt2/M).
6827
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., Aquathermalpressuring--Roleof temperaturein development of abnormal-pressurezones,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)
stratigraphyin Indiana, Indiana Geol.Surv.Bull., 57, 1978.
Birch, F., Compressibility;elasticconstants,in Handbookof Physical
Constants,Mere. 97, edited by S. P. Clark, Jr., pp. 95-173, Geological Societyof America,Boulder,Colo., 1966.
Bonham, L. C., Migration of hydrocarbonsin compactingbasins,Am.
Assoc.Pet. Geol. Bull., 64, 549-567, 1980.
Bredehoeft, J. D., and B. B. Hanshaw, On the maintenance of anomalous fluid pressures,I, Thick sedimentarysequences,Geol. Soc. Am.
Bull., 79, 1097-1106, 1968.
Bredehoeft,J. D., and I. S. Papadopulos, Rates of vertical groundwater movementestimatedfrom 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, Possible 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, Possible 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
gasin 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,Conductionof Heat in Solids,2nd ed.,
University Press,Oxford, 1959.
Cathles, L. M., Fluid flow and genesisof 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 MississippiValley-type lead-zinc depositsand their implications for episodic basin dewatering and deposit genesis,Econ.
Geol., 78, 983-1002, 1983.
/•r coefficientof compressibilityof rock grains
Chapman, R. E., Clays with abnormal interstitial fluid pressures,Am.
(Lt•-/M).
F finitedifference
coefficient
(L4t•-/Mor E/T).
Clayton, J. L., and P. J. Swetland,Petroleum generationand migra-
Ax, Az
At
0
dimensionsof an elemental volume (L).
size of a time step (t).
weightingcoefficientfor time averaging.
•c diffusivitytermin fluidflowequation(L2/t).
/• fluid dynamic viscosity(M/Lt).
P, Pr, Psm densityof fluid, rock grains,and fluid-saturated
medium,respectively
(M/L3).
Zx,z: thermal transmissibilities
in indicateddirection
(E/tT).
•b effectiveporosity at ambient conditions.
ß hydraulic potential, product of Hubbert's potential [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 ResearchCompany, ARCO Oil
and Gas Company, and National ScienceFoundation Graduate Fellowship Program. Karolyn Roberts typed the manuscript, and Pat
Bremsethhelpeddraft the figures.
REFERENCES
Ames, W. F., Numerical Methodsfor Partial Differential Equations,
2nd ed., Barnes and Noble, New York, 1977.
Anderson,G. M., and R. W. Macqueen, Ore deposit models, 6, Mis-
Assoc. Pet. Geol. Bull., 56, 790-795, 1972.
tion in Denver Basin, Am. Assoc. Pet. Geol. Bull., 64, 1613-1633,
1980.
Cooper, H. H., Jr., The equation of groundwaterflow in fixed and
deforming coordinates,J. Geophys.Res., 71, 4785-4790, 1966.
Denbigh, K., The Principlesof ChemicalEquilibrium,3rd ed., CambridgeUniversity Press,New York, 1971.
Dickinson, G., Geological aspectsof abnormal reservoirpressuresin
Gulf Coast Louisiana, Am. Assoc. Pet. Geol. Bull., 37, 410-432,
1953.
Domenico, P. A., and V. V. Palciauskas,Thermal expansionof 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-rockdata to
exploration in Williston Basin, Am. Assoc. Pet. Geol. Bull., 58,
1253-1262, 1974.
Dozy, J. J., A geologicmodel for the genesisof the lead-zincores of
the MississippiValley, U.S.A., Trans. Inst. Min. Metall., Sect.B, 79,
163-170, 1970.
Dybbs, A., and S. Schweitzer,Conservationequationsfor nonisothermal flow in porousmedia, 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 genesisof stratabound ore deposits, 1,
Mathematicaland numericalmodel}Am. J. Sci.,284, 1085-1124,
1984a.
Garven, G., and R. A. Freeze, Theoretical analysis of the role of
groundwater flow in the genesisof 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 subsurfacebrines, Geochim.Cosmochim.Acta, 46, 14311448, 1982.
Graf, D. L., Chemical osmosis, reverse chemical osmosis, and the
origin of subsurfacebrines (reply to a comment by F. M. Phillips),
Geochim. Cosmochim.Acta, 47, 1333, 1983.
6828
BETHKE: COMPACTION-DRIVENGROUNDWATER FLOW AND HEAT TRANSFER
Graf, D. L., I. Friedman, and W. F. Meents, The origin of saline
formationwaters,II, Isotopicfractionationby shalemicroporesystems,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,GroundwaterHydraulics,Elsevier,New York,
1979.
migration and accumulationof oil and gas, Econ. Geol., 16, 347371, 1921.
Richtmyer, R. D., Difference Methods for Initial-Value Problems,
Wiley-Interscience,New York, 1957.
Rogers,L., Shale-densitylog helpsdetect overpressures,
Oil Gas J.,
64, 126-130, 1966.
Ronov, A. B., Probable changesin the composition of sea water
during the course of geological time, Sedimentology,10, 25-43,
1968.
Halley, R. B., and J. W. Schmoker,High-porosityCenozoiccarbbnate
rocksof southFlorida: Progressive
lossof porositywith depth,Am. Rubey,W. W., and M. K. Hubbert, Role of fluid pressurein mechan-
Assoc.Pet. Geol. Bull., 67, 191-200, 1983.
ics of overthrustfaulting, Geol.Soc.Am. Bull., 70, 167-206, 1959.
Rubin, H., Heat dispersioneffecton thermal convectionin a porous
chemistryof HydrothermalOre Deposits,2nd ed., editedby H. L.
medium layer, J. Hydrol., 21, 173-185, 1974.
Barnes,pp. 137-172, John Wiley, New York, 1979.
Schwab,F. L., Modern and ancientsedimentarybasins:Comparative
Hayes, J. B., Sandstonediagenesis--Thehole truth, Spec.Publ. Soc.
accumulationrates,Geology,4, 723-727, 1976.
Econ. Paleontol.Mineral., 26, 127-139, 1979.
Sharp,J. M., Jr., Momentum and energybalanceequationsfor comHeyl, A. V., and W. S. West, Outlying mineral occurrences
related to
pacting sediments,Math. Geol.,8, 305-322, 1976.
the Upper MississippiValley Mineral District, Wisconsin,Iowa, Sharp, J. M., Jr., Energy and momentum transport model of the
Illinois, and Minnesota, Econ.Geol.,77, 1803-1817, 1982.
Ouachita Basin and its possibleimpact on formation of economic
Heyl, A. V., A. F. Agnew,E. J. Lyons,and C. H. Behre,The geology
mineral deposits,Econ. Geol., 73, 1057-1068, 1978.
of the Upper MississippiValley zinc-leaddistrict, U.S. Geol.Surv. Sharp,J. M., Jr., Permeabilitycontrolson aquathermalpressuring,
Prof Pap., 309, 1959.
Am. Assoc.Pet. Geol. Bull., 67, 2057-2061, 1983.
Hubbert, M. K., The theory of groundwatermotion, J. Geol., 48, Sharp, J. M., Jr., and P. A. Domenico, Energy transport in thick
785-944, 1940.
sequencesof compacting sediment,Geol. Soc. Am. Bull., 87, 390Jackson,S. A., and F. W. Beales,An aspectof sedimentarybasin
4O0, 1976.
evolution:The concentration
of Mississippi
Valley-typeoresduring Sibley, D. F., and H. Blatt, Intergranular pressuresolution and
the late stagesof diagenesis,
Bull. Can. Pet. Geol.,15, 393-433, 1967.
cementationof the Tuscaloosaorthoquartzite, J. Sediment.Petrol.,
Jaeger,J. C., and N. G. W. Cook, Fundamentalsof Rock Mechanics,
46, 881-896, 1976.
2nd ed., John Wiley, New York, 1976.
Siebenthal,C. E., Origin of the zinc and lead depositsof the Joplin
Kingston, D. R., C. P. Dishroon, and P. A. Williams, Global basin
region, U.S. Geol.Surv.Bull., 606, 1915.
classificationsystem,Am. Asssoc.Pet. Geol. Bull., 67, 2175-2193, Sleep,N.H., J. A. Nunn, and L. Chou, Platform basins,Annu. Rev.
Hanor, J. S., The sedimentarygenesisof hydrothermalfluids,in Geo-
1983.
Kyle, J. R., Geology of the Pine Point lead-zinc district, in Handbook
of Strata-Bound
andStratiformOre Deposits,
vol. 9, editedby 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, editedby W. H. K. Lee,
pp. 87-190, AGU, Washington,D.C., 1965.
Lewis,G. N., and M. Randall,Thermodynamics,
2nd ed., revisedby
K. S. Pitzer and L. Brewer,McGraw-Hill, New York, 1961.
Magara, K., Reevaluationof montmorillonitedehydrationas causeof
abnormal pressureand hydrocarbonmigration, Am. Assoc.Pet.
Geol. Bull., 59, 292-302, 1975.
Magara, K., Water expulsion from clastic sediments during
compaction--Directionsand volumes,Am. Assoc.Pet. Geol. Bull.,
60, 543-553, 1976.
Earth Planet. Sci., 8, 17-34, 1980.
Smith,L., and D. S. Chapman,On the thermal effectsof groundwater
flow, 1, Regionalscalesystems,
J. Geophys.Res.,88, 593-608, 1983.
Stallman,R. W., Computationof ground-watervelocityfrom temperature data, Methods of Collectingand Interpreting Ground-Water
Data, edited by R. Bentall, U.S. Geol. Surv. Water SupplyPap.,
1544-H, 36-46, 1963.
Stone,H. L., Iterative solution of implicit approximationsof multidimensionalpartial differentialequations,SIAM J. Numer. Anal., 5,
530-558, 1968.
Thomeer, J. H. M. A., and J. A. Bottema, Increasingoccurrenceof
abnormallyhigh reservoirpressures
in boreholes,and drilling problems resulting therefrom,Am. Assoc.Pet. Geol. Bull., 45, 1721-1730,
1961.
Thompson,J. F., General curvilinearcoordinatesystems,in Numerical Grid Generation,edited by J. F. Thompson,pp. 1-30, North-
Magara, K., Agentsfor primary hydrocarbonmigration: A review,in
Problemsof PetroleumMigration,editedby W. H. Robertsand R. J.
Holland, New York, 1982.
Cordell,pp. 33-45, AmericanAssociationof PetroleumGeologists, Tissot, B. P., and D. H. Welte, PetroleumFormation and Occurrence,
Tulsa, Okla., 1980.
Springer-Verlag,New York, 1978.
Mastin, C. W., Error induced by coordinate systems,in Numerical Torrance, K. E., Comparisonof finite-differencecomputationsof naGrid Generation,edited by J. F. Thompson, pp. 31-40, Northtural convection, J. Res. Natl. Bur. Stand., Sect. B 72B, 281-301,
Holland, New York, 1982.
1968.
Munn, M. J., Studiesin the applicationof the anticlinaltheoryof oil Toth, J., Cross-formationalgravity-flow of groundwater: A mechaand gas accumulation,Econ.Geol.,4, 141-157, 1909.
nism of the transport and accumulationof petroleum,in Problems
Muskat, M., The Flow of Homogeneous
Fluids Through Porous
of PetroleumMigration, edited by W. H. Roberts and R. J. Cordell,
Media, McGraw-Hill, New York, 1937.
pp. 121-167, AmericanAssociationof PetroleumGeologists,Tulsa,
Nalivkin, V. D., Dynamicsof the developmentof the Russianplatform structures,Tectonophysics,
36, 247-262, 1976.
Neglia, S., Migration of fluidsin sedimentarybasins,Am. Assoc.Pet.
Geol. Bull., 63, 573-597, 1979.
Okla., 1980.
Waples,D. W., Time and temperaturein petroleumformation:Application of Lopatin's method to petroleum exploration, Am. Assoc.
Pet. Geol. Bull., 64, 916-926, 1980.
Noble, E. A., Formation of ore depositsby water of compaction, Welty, J. R., C. E. Wicks, and R. E. Wilson, Fundamentalsof MomenEcon. Geol.,58, 1145-1156, 1963.
Ockendon,J. R., and W. R. Hodgkins,MovingBoundaryProblemsin
Heat Flow andDiffusion,UniversityPress,Oxford, 1975.
Peaceman, D. W., Fundamentalsof Numerical ReservoirSimulation,
tum,Heat, and Mass Transfer,John Wiley, New York, 1976.
Wood, J. R., and R. C. Surdam, Application of convective-diffusive
models to diageneticprocesses,Spec. Publ. Soc. Econ. Paleontol.
Mineral., 26, 243-250, 1979.
Elsevier, New York, 1977.
Perrier,R., and J. Quiblier,Thicknesschangesin sedimentary
layers
during compactionhistory: Methods for quantitative evaluation,
Am. Assoc.Pet. Geol. Bull., 58, 507-520, 1974.
Phillips, F. M., Comment on "Chemical osmosis,reversechemical
osmosis,and the origin of subsurfacebrines"by Donald L. Graf,
Geochim.Cosmochim.Acta, 47, 1331, 1983.
Rich, J. L., Moving undergroundwater as a primary causeof the
C. M. Bethke, HydrogeologyProgram, Department of Geology,
245 Natural History Building, University of Illinois, Urbana, IL
61801.
(ReceivedApril 3, 1984;
revisedMarch 13, 1985;
acceptedApril 2, 1985.)