Fast and Slow Compaction in Sedimentary Basins Andrew C....

Fast and Slow Compaction in Sedimentary Basins

Andrew C Fowler Xin-She Yang

SIAM Journal on Applied Mathematics Vol 59 No 1 (Sep - Oct 1998) pp 365-385

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SIAM J APPL MATH 1998 Society for Industrial and Applied Mathematics Vol 59 No 1 pp 365-385

FAST AND SLOW COMPACTION IN SEDIMENTARY BASINS

Abstract A mathematical model of compaction in sedimentary basins is presented and analyzed Compaction occurs when accumulating sediments compact under their own weight expelling pore water in the process If sedimentation is rapid or the permeability is low then high pore pressures can result a phenomenon which is of importance in oil drilling operations Here we show that one-dimensional compaction can be described in its sirnplest forrn by a nonlinear diffusion equation controlled principally by a dimensionless parameter A which is the ratio of the hydraulic conductivity to the sedimentation rate Large X corresponds to very permeable sediments or slow sedimentation a situation which we term fast compaction since the rapid pore water expulsion allows the pore water pressure to equilibrate to a hydrostatic value On the other hand small X corresponds to slow compaction and the pore pressure is in excess above the hydrostatic value and more nearly equal to the overburden value We provide analytic and numerical results for both large and srnall A using also the assumption that the permeability is a strong function of porosity In particular we can derive Athys law (that porosity decreases exponentially with depth) when X gtgt 1

Key words compaction sedimentary basins abnorrnal pore pressure

AMS subject classifications 35C20 35R35 76305 86A60 86A99

PII SO036139996287370

1 Introduction Sedimentary basins such as those in the North Sea or the Gulf of Mexico form when waterborne sediments in shallow seas are deposited over periods of tens of millions of years The resulting sediments which may for example be sands or river muds washed down from land then compact under their own weight causing a reduction of porosity (and hence the expulsion of pore water) and eventually (as depth and thus pressure and temperature increase) cementation reactions occur causing a transformation from a granular aggregate to rocks such as shales or sandstones

Sedimentary basins are prime locations for tlie forniation of liydrocarbons and are thus important in the oil industry One particular problem which affects drilling operations is the occasional occurrence of abnormally high pore fluid pressures which if encountered suddenly can cause drill hole collapse and consequent failure of the drilling operation An understanding of how such high pore pressures occur is therefore of some industrial as well as scientific interest (Bredehoeft and Hanshaw (1968) Bishop (1979)) Furthermore the variation of porosity with depth is a source of information for geologists who are concerned with understanding the burial and subsidence histories of sedimentary basins (Smith (1971) Lerche (1990)) Compaction models which describe these processes are thus of practical interest

The basic model of compaction is rather analogous to the process of soil consolidation The sediments act as a compressible porous matrix so mass conservation of pore fluid together with Darcys law leads to an equation of the general type q5t + V q = 0 q oc -Vp where 4 is porosity q is fluid flux and p is fluid pressure This model must be supplemented by a constatutzve law relating pore pressure p to porosity Q

and in soil mechanics this takes (most simply) the form of the normal consolzdataon

Received by the editors August 22 199G accepted for publication (in revised form) April 25 1997 published electronically October 29 1998

httpww~siamorgjournalssiap59-128737~html tCentre for Industrial and Applied Mathematics Mathematical Institute Oxford University 24-

29 St Giles Oxford OX13LB UK (fow1erQmathsoxacuk)

366 ANDREJI C FOTVLER AND XIN-SIIE YANG

l l r~e which relates o to the effective pressure p defined by

where P is the overburden pressure Terzaghis principle of effective stress (Terzaghi (1943)) states that for soils it is tlie effective pressure (inore generally the effective stress) which controls the deformation ancl the extension of this principle to compacting sediments is termed -4th7js laul after work by Athy (1930) who measured porosity-depth profiles and proposed that o = q5(pe) In many cases an exponential decrease of porosity with depth occurred and we will sometimes equivocally refer to this kind of profile as an Athy profile

It can be seen that with this constitutive assumption the basic compaction model will essentially comprise a nonlinear diffusion equation and our purpose here is to provide analytic and numerical solutions which provide signposts to the kind of behavior that can be expected in more realistic models

Two principal kinds of realism are of concern although we do not analyze them in this paper The first is cliagenesis which is effectively a dewatering reaction which occurs when sniectite (a water-rich clay mineral) clissolws to form quartz in solution together with free water the quartz subsequently precipitating as illite (a water-free clay-) Effectively the smectite-illite reaction acts as a source for pore water and can thus enhance (or even be a primary cause of) abnormal pore pressures though to what extent is unknown

Thc other complication concerns the assurned rheology Even for soils the rela- tionsliip Q = 4(pe) is irreversible exhibiting hysteresis and incorporation of this into basin loadingunloacling histories complicates the model conceptually Furthermore as pressure increases pressure solution occurs as precipitation and dissolution depend on tlie local grain-to-grain pressure This leads to an effective creep of tlie solid grains Also calcite precipitation at grain junctions causes cementation and thus stiffening of the solid matrix These effects ancl those of cliagenesis are not considered in this paper

Early studies of compaction by Athy (1930) and Heclberg (1936) have more re- cently been followed in work by Gibson (1958) see also Gibson England and Hussey (1967) and Gibson Schiffman ancl Cargill (1981) Other recent models of compaction include tliose of Srnitli (19711 Keith and Rimstidt (1985) TllTangen (19921 Shi and 1Iang (1986) and Luo and Vasseur (1992) Some of this (and other) work was reviewed by Auclet ancl Fowler (1992) whose formalism we follow here All of tlie above papers however ignore the complications of diagenesis and realistic rheology Ricke and Chilingarian (1974) offer a comprehensive review of sedimentary compaction

Audet and Fowler (1992) provitlc a general discussion of how diagenesis antl to some extent exotic rheology can be included in compaction models Because the structure of such models is very complicated their approach was to develop analytic insight into simpler models first and they were able to find reasonable approximations for large X and large time and also for small X where X is a dimensionless number which measures the ratio of the 11ydrtulic conductivity of the sediments to the sedimentation rate Here we extend ancl improve those results for smaller times which are of nlore geological significance

The term cornpaction has been ~nuch used in the geophysical literature mainly to describe the extraction of magma from source regions in tlle earths nlantle This work has been reviewed by Fowler (1990) and was based on original papers by Scott and Stevenson (1984) LIcKenzie (1984) and Fowler (1985) The principal difference

367 FAST AYU SLOW COMPACTION IN SEUIlENTAR BASINS

01 Basin basement z = b( t )

F I G 1 One-dimensional cornpacling sedv~nentary bnszn The coordinnle z is directed upuards

between those models and the present one is in the rheology of the solid matrix The solid grains in tlhe mantle at depths of 100-200 km respond to a differcrltial pressure between overburden and pore pressures by grairi creep This leads to a very different model from the one proposed here which essentially considers the matrix to deform elasticallv Pressure solution crcclp rrlight be rnorr analogous to the viscous compaction of the riitntle and this has been analyzcd by Angevine and Turcotte (1983)

2 Model equations Wc considcr the solid matrix to behave as an elastic solid and specificall) so that the porosity q5 is a function of effwtive pressure p The riiodel describes the one-dimensional flow of both solid and liquid phases and is based on the framework developed by Audet and Fowler (1992) For a one-dimensional basin b( t ) lt z lt h( t ) where h(t)is the ocean floor and b( t ) is the bascment rock as shown in Fig 1the governing model equations for one-tlimensional compaction can be written as follows

h1ass conser rlatzon

D n ~ c ys law

Force balance

- -

368 ANDRETV C FOWLER AND XIN-SHE YANG

Constitutive relation

In these equakions 7 ~ and us are tlie velocities of fluid and solid iilatrix k and p are the matrix permeability and tlie liquid viscosity o3 is tlie vertical component of the stress tensor and g is the gravitatioilal acceleration

We call relate 03 to the effective pressure and pore pressure as follows First of all we may modify Terzaghis relation (1)by writing

a relationship due to Skempton (1960) who suggested that although for soils (L might be srrlall due to a low grain-to-grain interfacial contact area this would not necessarily be the case for a more compacted rocklike rnatrix Further disc~lssiorl of the effective pressure is given by Bear anti Bachmat (1990)

In conditions of uniaxial strain where the only nonzero strain rate is aUDz where U is vertical strain the nonzero components of the eflective stress tensor

are the diagonal components given for an elastic medium by

mliere G is the shear modulus Now from (3)we have

and d d ts is a material time deriIvative following the solid matrix thus

where

is the dilation Thus

au 1 d A (14) az - I a dt-

--I-= constant1-a

369 FAST AND SLOW COIblPACTION IN SEDIMENTARY BASINS

following the solid matrix that is it is constant in time for each solid matrix element Since each solid element originates a t the surface where conditions are assumed to he uniform we can also assume that this expression is constant also in space and equal to A say then if G is constant and

(which may depend on d) we find

and (5) becomes using (8)

21 Boundary conditions These are five equations for five unknown variables one for porosity 4 two for velocities us u l and two for effective pressure p and pore pressure p The system is of fourth order so we will require boundary conditions on I L ~ u ~ ~ ~ in addition we assume h(t) is known but h(t) is not which is therefore described by a further boundary condition The natural boundary conditio~lsare the kinematic boundary conditions at z = h

and a kinematic condition at z = IL

where r n is the sedimentation rate a t z= h Also at z = h

where po is the overburden pressure eg due to ocean depth 4 is the value at tlie top of basin during sedimentation Equation (20) gives h and then we have four conditions for u us p p as required

The choice of do will normally follow from the constitutive relation p = p(qh) for example if we take p = 0 at z = h The value of A then follows from a normal stress balance since we require also -a = po which implies

a t z = h For example the reasonable assumption p = 0a = 0 implies A = 0 and thus A = 1 - 40 (everywhere)

22 Nondimensionalization A natural depth scale to choose is that over which Q changes significantly Since p = p(q5) we can equivalently define a pressure scale over which Q changes significantly To be specific define a pressure scale [p] by writing

370 ANDREIV C FOWLER AND XIN-SHE YANG

where b] is such that 6 varies by O(1) when 4 does Since the variation of p is determin~d by (18) we can equivalently choose a depth scale d by putting

Here we assume that GIK is constant which may be a reasonable assumption Let h ~ fbe a typical value of the (positive) sedii~ientation rate We now scale the variables by mriting

Time variables are substituted into the equations which then become on dropping the astcrisks for further convenience

where

The boundary conditions take the same form as in (19)-(21) c~xcept that p = 0 at z = h Tle add the first two equation5 of inass conservation together and integrate frorli 0 to 2thus

By using Darcys lam me obtain

371 FAST AND SLOW COICIPACTION IN SEDIICIENTART BASINS

The boundary conditions can thus be urritten in the form

23 Excess pore pressure The hydrostatic pressure at z is defined as

The overburden pressure (strictly the norrnal stress) a t z is defined as

The excess pore pressure or abnormal overpressure peX is defined as

which is the pressure in excess of the hydrostatic pressure By using these definitions and employing the force balance equation (29) the

dimensionless differential forms of the above definitions are

24 A general nonlinear diffusion equation By using (29) (32) and (33) (27) reduces to a nonlinear diffusion equation for 4

The boundary conditions are then

Since in practice (qh) lt 0 we see that (42) is a nonlinear diffusion equation valid in the domain h lt z lt h where h is unknown and is determined by the extra boundary condition in (43) The problem is thus one of free boundary type

372 Ah UREII C b OLKH AND XIN-SHk 152hG

25 D e t e r m i n a t i o n of m o d e l pa ramete rs Of the parameters appearing in (d2) and (d3) r arld (1 arc 0(1)constants which are cssentiallj- fixed as material properties The important parameter which controls corllpaction is the compaction number A Me estirlrate its size using observations givcn by other authors (Smith (1971) Sharp (1976) Sharp and Donienico (1976) Eberl and Hower (1976) Bethkr ant1 Corljct (1988) Lerche (1990) Audet and Fowler (1992)) For exa~ltple if n-e take (I - 1 lim kn - 1 x 10 111~ p - 26 x 103 kg g - 10rns-5 pi - 1 x 10 kg I ~ ~ - 1 x lo- K m-- ri2 - 300 111 Map - 1 x 10-- m s then X x 1 arrdi-L

1 = 063 Typical values of the uncorrrpacted pernleability kn are given by Freeze and Cherry (1979) Thi perniealjilitj- is proportional to the scluarc of the grain size with a typical proportionality factor of 1 0 which allows for tort~lositj- and cor~striction of the 1)oie spare Llarint clays (particle sizt less than 2 jlrir) havc pernieabilities in tlie range 10-0 m2 silts (particle size 2-60 ~rir) have pcrrirca)ilitics 10-1 lo- i l l2 while srntls (60 p r n 2 mm) 11rvc permcrbilitics 10-OV1 irl C~enientcd clay forms shale cemented sand fornls sandstone and these have somewhat lower li(r~ncnbilities than the corresponding uncelliented 1nrtrix ITT(see that L wide range of pern~cabilities between 1 0 ~ m2 and 10-m2 can occur so values of X inay lie in the range 10- 10 rrlues of X which ire either smrll or large are therefore of interest although that of large X is thc marc likcly This is also the nrore ir~tercsting rase mathematically An initial porositj- of q$ = 05 at the top of t l ~ e I-msin is used 115- other authors (Smith (1971) Sharp (1976) Bcthke and Corbet (1988) A~tdct and Fowler (1992))

26 Simplification of the nonlirlear diffusion equa t ion There is 110 loss of gcirerality ill choosing i = 0 so that a - 0 dcnotcs the Iiascillent Hlienipton (19GO) silggcstcd that a is snrall and in what follows urc lakc n = 0witlroirt expecting that tlris choice will have a major effect on the solutions Based on the work of S ~ l l i t l ~(1971) Sharp (1976) and Audet and Fowler (1992)) we adopt the following (on~tit~ltivcfilrlctions

(44) j= ( (do - d)-

(16) I ~ I = 1

it11 rllcsc sinrplitic-ations the ~iorlli~iear can l)c mrittt11 ill a diffi~sion equation for o coiripac-t fornr as

7 1 1 ~analysis of this nlodel forrns the suljjcct of the rest of this paper 111 addition the prol~lem is also solved nul~lerically on a ~iormalized grid Z= a l i ( t ) by using the predictor-corrector ilnplicit finite-cii8ercrlce rrlcthod presented hy Alecli and Norburp (1982)to nraki a c~orrlparison with tlie obtainetl analytic solutions

FAST AND SLOW COMPACTION IN SEDIMENTARY BASINS 373

3 Analysis We expect that values of X will usually lie in the range 1 0 - ~ - - 1 0 ~ Since X is the controlling parameter which characterizes the compaction behavior we can expect that X = 1 defines a transition between slow sedimentation (fast compaction) X gtgt 1 and fast sedimentation (slow compaction) X ltlt 1 and that the evolution features of fast and slow compaction also may be quite different

31 Slow compaction (A lt lt 1)For X ltlt 1 z 1 and t - 1 (47) implies that aqbat M 0 SO with o = oo on z = h then o = do and 6 M 1 for z gt 0 Furthermore we see that h M t The boundary condition at the base is not satisfied and a boundary layer is necessary there For sufficiently small tirnes we can take I) M near z = 0 as well so that (47) may be approximated (uniformly) as

with appropriate boundary conditions for the basal boundary layer being

where the latter represents the matching condition outside the boundary layer in which z - X12 The solution can be easily obtained by a standard Laplace transformation method (Carslaw and Jaeger (1959)) as

We see that the assumption that 4 is close to is self-consistent for t ltlt l X 1 and in particular for times (of interest) of O(1) In fact expansion of (53) with 17 = z2(X1t)ll O(1) and X1t small shows that Q = The solution + ~ [ ( X l t ) ~ ~ ] indicates that compaction develops only in a small range near the basin basement with a thickness growing with JXt When a = 0 we are in the case discussed by Audet and Fowler (1992) with a similarity solution (their equation (526))

Audet and Fowler (1992) gave a slightly different result for this case by putting Q - z = on z = 0 While their result is asyrnptotically equivalent to (53) for X1t ltlt 1 it is likely to be less accurate for larger times a fact which is confirmed by numerical integration as shown in Fig 2

The dimensionless overburden hydrostatic and excess pore pressures satisfy re- spectively

ANDREW C FOWLER AND XIN-SHE YANG

Porosity

F I G 2 Comparzsor~ of sol~ations of (47)--(49)in the v ic ini ty of t he basement z = 0 i n t e rms of t he normalized height Z = z h ( t ) for X = 001 T h e solid lanes represer~t a direct numerical sol7ation whale the dashed lznes are the solutions of ( 5 0 ) - - ( 5 3 ) Aude t a r ~ d Fowlers further approzimat ior~ (1992 eq (5 26) ) i s s h o w r ~ as the dotted profiles It can be seen that the dotted profile deviates from the correct sol7atzon at larger values o f t

It follows that

and hence for X t ltlt 1we have the leading-order solution p = (1- 4 0 ) ( h- 2 ) The other terms are only small corrections The excess pressure develops proportionally to hasill thickness

A co~riparison of the above solution with the nuirlerical results is plotted in Fig 3 It call be seen that the agreement is very good and that for X ltlt Ioverpressure is essentially proportio~ial to hasin thickness

32 Fast compaction (A gtgt 1) For large values of A we assurlre expansio~is of the form

375 FAST AND SLOW COMPACTION IN SEDIMENT4RY BASINS

pressure

FIG3 Hydrostatzc pore and overburden pressures at t = 5 uszng values r = 063 and X = 001 Solzd lines correspond to numerical results the dashed line for p is calculated from the solution (57) The numerical and analytical results are indistanguishable

Substituting the above exparlsions into (47)-(49) and equating the coefficients of powers of 1X we have

where i0= (q5(0)q50)hnd we have in (61) anticipated the result in (65) below The boundary conditions become the following

At z = h(O)

376 ANDREW C FOWLER AND XIN-SHE YANG

(63)

with

on z = h(O) where again we use (60) to anticipate (G5) Integrating equatiorl (60) and using the boundary cordition (63) we have

and hence

Its solutiorl is

(67) ($0) = ~ o e ( o ) - ~ )

This leading-order solution corresponds to equilibrium cornpaction to which the porosity curve will tend when t +m The exponentially decreasing dependence of porosity on depth was found by 4thy (1930) by fitting observed data of Paleozoic shales from Kansas and Oklahoma Athys porosity curv represents compactiorl equilibriurrl attained over a very long time span Hedbergs (1936) porosity curve for the Tertiary shales in Venezuela is similar to Atllys curve

lJsirlg (67) (61) becomes

1ntegrat)ing the above equation using (63) we have

Using (69) (64) and (67) we obtain a relation for A( )

Integrating this equation we have

Clearly if t is large then e ~ ~ [ - h ( ~ ) ] ltlt 1 we thus have

(72) h(O) zz 1- do

-If t is srrrall then exp[-h()] 1 and

(73) h(0) zz 1

377 FAST AND SLOW COMPACTION IN SEDIMENTARY BASINS

33 Strongly varying permeability m gtgt 1 With go = ($(0)40)nwe see that as decreases Lo car1 decrease dramatically if m is relatively large The perturbatiorl exparlsions in (58) and (59) are valid only if X i gtgt 1 and we can therefore define a critical porosity $ when X i = 1 With = ($40)mwe have

and with values = 05 rn = 8 X = 100 this critical value is q5 = 028 As the slowly compacting layer thickens we see that the perturbation solutiorl is valid until q 5 ( O ) = $ which is when

with the same values of m and A and using (71) this occurs at t = t where

For t gt t the solution above can be expected to apply for z gt h(O) - I2

Porosity

FIG 4 Comparison of the large X solutson (67) with numerical results (solsd) i n terms of a normalized depth Z = z lh For t = t given by (76) the asy7nptotic result ss accurate but for t gt t it becomes invalid when 4 lt 4 = 028 for the values used here ofX = 100 do = 05 m = 8

A comparison of the solutiorl with related numerical results is presented in Fig 4 The corrlparisorl clearly shows that Athys exponential porosity-depth relation (Athy


(1930)) is valid only in the range of 0 058dkm in sllch sedimentary basins where -

the parameter X gtgt 1 If d = 1 krn then the range is 0-580 m i1rc now extend the analysis for X gtgt 1 to deal with this situation

4 Thick layer sediments with q5 lt q5 and t gt t Note that from its definition a ltlt 1 if X gtgt 1 so we must formally assume m gtgt 1 in ordcr to have q5 of order 1 Tlms we now consider a limit iri wllich m is large T4c write the equations 111 t ~ r m sof q5 defined by (74) as follows

Now if gt (o)Vs exponentially large and therefore neglecting at in

(77)

using the boundary condition (78) at z = gtgt 1)h R-e still have (since ( 4 1 ~ ) ~

from which Qf R= -ho and an improved approximation to (79) is thcrcforc using this in (77)

This approximation however becomes invalid when h - z = II and specifically we dcfinc lt and Q in the transition region near h - z = II bv

lnmz = h n - - - +- m m

from which it follows by a irlatclling principle (Hinch 1991) that Q - lt as lt + ac Q satisfies the cquatiorl


where we define

and 9 will bc dcfincd below Kcglecting terms of O(lm) in (83) we find

(85) K - h49 = e ( l -q5m)2(9c- 1)

where

and 9 increases rrlonotonically frorn 9 as J + -xto O(J) as J i +m The value of qmmust now be found by matching to the solution below z = h -ll

41 Prescription of h Before finding this solution we can find h by comparing (85) to (81) We write the latter equation in terms of 9 and J to obtain

and in order tlrat this irlatches to (851 we require (using the definition of K in (IG)) that

Solution below the transition layer Ve write the equation for 9 (83) in terms of z It is


01 02 03 04 Porosity

F I G 5 C o m p a r i s o r ~ of asy7riptotic solut ions ( 8 1 ) and (94) (dashed l ines) and r~urrierical results (solzd lznes) for tarnes t = 2 and t = 5 taking X = 100

and at leading order

a hyperbolic equation Tlle loss of the highest derivative means that only one boundary condition can be satisfied and because the characteristics of (91) move upward the appropriate condition to satisfy is that a t z = 0 wllich is 9 = m It seems that this condition is not correctly ordered thus warranting consideration of a further basal boundary layer but we show that by solvirlg (91) together with 9 = m on z = 0we obtain a uniformly valid solutiorl below z = h -n

lye suppose that the initial data for (91) are

(92) Q = Qb( r ) when z = 0 t = 7

where if h = + Alnm at t = to (zt ) then

and we choose llrh(7) in order that Q = m at z = 0 The solutiorl is easily found to 11c

1+ rnz 3 = 111 I

and llrh = 0 This satisfies the boundary condition on z = 0 moreover we see that (i)dz2)e = 0 so that the diffusiori term in (90) is of order 1rn2 so long as 9is

F4ST AND SIOW COMPACTION IN SEDIIIENTAR BASINS 381

of order 1 From (94) this is true for 2 N 0(1) so that (94) is (very) accurate away from the base Near z = 0 however 9 N O(m) SO the diffusion term is of order 1 there and riot negligible Since this is only true for z O( l m) the implication is that the solution (94) is uniformly accurate to O( l m) for 9 in z lt h - IIwith less accuracy near the base This appears to be borne out in Figs 5 6 and 8 below

Porosity

F I G 6 Cornparison of asyrr~ptotic solutions ( 8 1 ) and (94) (dashed lines) and nvn~er ica l results (solid lines) with different values of m t = 2 in all plots X = 100 and the curves correspond t o rtalues m = 81624

Mre can now finally obtain 9 by matching (94) with (85) as z - h - IT and lt i -oo In fact putting z = h - IT - Alnm + A (94) becomes

from which wrc require

dm( h- rI) Ivm = ln [(l - h j~( t - to)] + 41)

This completes the asymptotic solution Comparisons of the approximate solution derived abovc with the nunlcrical solu-

tion are shown in Figs 5 anti 6 For a value X = 100 Fig 5 shows the comparison at times t = 2 and t = 5 while Fig 6 cornpares approxinlate and exact solutions at t = 2 for increasing values of m The accuracy increases with m as we expcct Figure 7 shows computed and approximate values of h(t)

ANDREMiC FOVVLER 4ND XIK-SHE YANG

F I G 7 Conaparison of upprosamate h ( t )solutaonfrom ( 8 9 ) and ( 9 6 ) (dashed line) and numerical res~i l ts (solid lzr~e) X = 100

-hen X gtgt 1 and uJ gt dsubstituting (80) for o into (56) we have

This equation with the boundary condition p = 0 at thc top z= h ( t ) givcs p = 0 for the leading-order solution This means excess pressure does not occur for short tirnes or in thc top region where h - z lt II This region is clearly shown in Fig 8 For larger times the approximate sol~ltioil suggests that h ltlt 0 whence

which shows that the excess pore pressure cievelops at largc times even if X gtgt 1 The cornparison of the rlurnerical results for the pore prcssure with that calculated

from the asymptotic solutions (dashed lincs) is shown in Fig 8 Tlle overpressure only develops in the lower region while the pore pressure rc~nains hydrostatic in the top region within a depth of order from tlie surface

5 Conclusions In the absence o i cliagenesis and temperature effects tlie gen- eralized one-dimensional rnociel of compaction given by Audet and Fowler (1992) reduces t o a nonlinear diffusion equation in a domain wit11 a moving boundary When scaled this rnodel depends primarily on onc dimensionless parameter A which is the ratio of the sedimentation tirne scale to the Darcy flow time scale Thus X gtgt 1 if


pressure

FIG8 Hydrostatic pore and overburden pressures at t = 5 X = 100 Dashed lines are computed by using (81) and ( 9 4 )

sedimentation is very slow while X ltlt 1 if it is very fast Realistically both limits are possible depending principally on the permeability In addition strong variability of the permeability through the exponent m complicates the solution method

In particular we find that in the limit X ltlt 1 (slow compaction) the model can be simply analyzed by means of a boundary layer analysis a t the sediment base Essentially sediment is added so fast that the porosity remains virgin except near the base where compaction occurs The pore pressure is then essentially lithostatic that is excess pore pressures exist over the whole domain

The more interesting (and probably more relevant) case is when X gtgt 1 (fast compaction) For sufficiently small times (and thus also basin thicknesses) the porosity profile is exponential with depth and the pore pressure has relaxed to a hydrostatic value However because of the large exponent m in the permeability law = (gigjo) we find that even if X gtgt 1 the product X i may become small a t sufficiently large depths In this case there is a critical depth such that when the basin thickness ex- ceeds it the porosity profile consists of an upper part near the surface where X i gtgt 1 and the exponential profile is attained and a lower part where Aamp ltlt 1 and the porosity is higher than equilibrium Straightforward asymptotic methods are difficult to implement because the limit m gtgt 1 implies exponential asymptotics but we use a hybrid method which appears to correspond accurately to numerical computations

384 ANIIREJV C FOTVLER AND XIN-SHE Y4PG

The rnethods presented in this paper palre the path for the analysis of compaction in sedimentary basins when more ~ornplicat~ed loading histories are studictl and also when morc realistic phenomena are inclutlrd such as diagenesis or state-dcpendent rlieology (Schofield and JVroth (1968))

C I ANGEVINE (1983) Poroszty reductzon by p r c s s ~ ~ r e -1 theorttzcal4 S D D 1 TURCOITE s o l ~ ~ t z o n - model for quartz arenites Geol Soc Amer Bull 94 pp 1129 1124

L F ATHY (1930) Denszty porosaty and conapactior~ of sedimentary rocks Amer Ass Petrol Geol Bull 14 pp 1 -~22

D AI ~ ~ I ~ I I E TAND A C FOWI~EII(1992) -4 mathe7r~atical model joy compaction zn sedznentary basins Ccophys J Internat 110 pp 577--590

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nat 110 pp 601-613

SIAM J APPL MATH 1998 Society for Industrial and Applied Mathematics Vol 59 No 1 pp 365-385


Abstract A mathematical model of compaction in sedimentary basins is presented and analyzed Compaction occurs when accumulating sediments compact under their own weight expelling pore water in the process If sedimentation is rapid or the permeability is low then high pore pressures can result a phenomenon which is of importance in oil drilling operations Here we show that one-dimensional compaction can be described in its sirnplest forrn by a nonlinear diffusion equation controlled principally by a dimensionless parameter A which is the ratio of the hydraulic conductivity to the sedimentation rate Large X corresponds to very permeable sediments or slow sedimentation a situation which we term fast compaction since the rapid pore water expulsion allows the pore water pressure to equilibrate to a hydrostatic value On the other hand small X corresponds to slow compaction and the pore pressure is in excess above the hydrostatic value and more nearly equal to the overburden value We provide analytic and numerical results for both large and srnall A using also the assumption that the permeability is a strong function of porosity In particular we can derive Athys law (that porosity decreases exponentially with depth) when X gtgt 1

Key words compaction sedimentary basins abnorrnal pore pressure

AMS subject classifications 35C20 35R35 76305 86A60 86A99

PII SO036139996287370

1 Introduction Sedimentary basins such as those in the North Sea or the Gulf of Mexico form when waterborne sediments in shallow seas are deposited over periods of tens of millions of years The resulting sediments which may for example be sands or river muds washed down from land then compact under their own weight causing a reduction of porosity (and hence the expulsion of pore water) and eventually (as depth and thus pressure and temperature increase) cementation reactions occur causing a transformation from a granular aggregate to rocks such as shales or sandstones

Sedimentary basins are prime locations for tlie forniation of liydrocarbons and are thus important in the oil industry One particular problem which affects drilling operations is the occasional occurrence of abnormally high pore fluid pressures which if encountered suddenly can cause drill hole collapse and consequent failure of the drilling operation An understanding of how such high pore pressures occur is therefore of some industrial as well as scientific interest (Bredehoeft and Hanshaw (1968) Bishop (1979)) Furthermore the variation of porosity with depth is a source of information for geologists who are concerned with understanding the burial and subsidence histories of sedimentary basins (Smith (1971) Lerche (1990)) Compaction models which describe these processes are thus of practical interest

The basic model of compaction is rather analogous to the process of soil consolidation The sediments act as a compressible porous matrix so mass conservation of pore fluid together with Darcys law leads to an equation of the general type q5t + V q = 0 q oc -Vp where 4 is porosity q is fluid flux and p is fluid pressure This model must be supplemented by a constatutzve law relating pore pressure p to porosity Q

and in soil mechanics this takes (most simply) the form of the normal consolzdataon

Received by the editors August 22 199G accepted for publication (in revised form) April 25 1997 published electronically October 29 1998

httpww~siamorgjournalssiap59-128737~html tCentre for Industrial and Applied Mathematics Mathematical Institute Oxford University 24-

29 St Giles Oxford OX13LB UK (fow1erQmathsoxacuk)
















D n ~ c ys law

Force balance

- -










where


au 1 d A (14) az - I a dt-

--I-= constant1-a
















where

















(44) j= ( (do - d)-

(16) I ~ I = 1












Porosity


It follows that





pressure




At z = h(O)


(63)

with


and hence

Its solutiorl is

(67) ($0) = ~ o e ( o ) - ~ )







(72) h(O) zz 1- do


(73) h(0) zz 1






Porosity








(77)




lnmz = h n - - - +- m m



where we define


(85) K - h49 = e ( l -q5m)2(9c- 1)

where











(92) Q = Qb( r ) when z = 0 t = 7



1+ rnz 3 = 111 I




Porosity















pressure




































nat 110 pp 601-613
















D n ~ c ys law

Force balance

- -










where


au 1 d A (14) az - I a dt-

--I-= constant1-a
















where

















(44) j= ( (do - d)-

(16) I ~ I = 1












Porosity


It follows that





pressure




At z = h(O)


(63)

with


and hence

Its solutiorl is

(67) ($0) = ~ o e ( o ) - ~ )







(72) h(O) zz 1- do


(73) h(0) zz 1






Porosity








(77)




lnmz = h n - - - +- m m



where we define


(85) K - h49 = e ( l -q5m)2(9c- 1)

where











(92) Q = Qb( r ) when z = 0 t = 7



1+ rnz 3 = 111 I




Porosity















pressure




































nat 110 pp 601-613

- -










where


au 1 d A (14) az - I a dt-

--I-= constant1-a
















where

















(44) j= ( (do - d)-

(16) I ~ I = 1












Porosity


It follows that





pressure




At z = h(O)


(63)

with


and hence

Its solutiorl is

(67) ($0) = ~ o e ( o ) - ~ )







(72) h(O) zz 1- do


(73) h(0) zz 1






Porosity








(77)




lnmz = h n - - - +- m m



where we define


(85) K - h49 = e ( l -q5m)2(9c- 1)

where











(92) Q = Qb( r ) when z = 0 t = 7



1+ rnz 3 = 111 I




Porosity















pressure




































nat 110 pp 601-613
















where

















(44) j= ( (do - d)-

(16) I ~ I = 1












Porosity


It follows that





pressure




At z = h(O)


(63)

with


and hence

Its solutiorl is

(67) ($0) = ~ o e ( o ) - ~ )







(72) h(O) zz 1- do


(73) h(0) zz 1






Porosity








(77)




lnmz = h n - - - +- m m



where we define


(85) K - h49 = e ( l -q5m)2(9c- 1)

where











(92) Q = Qb( r ) when z = 0 t = 7



1+ rnz 3 = 111 I




Porosity















pressure




































nat 110 pp 601-613















(44) j= ( (do - d)-

(16) I ~ I = 1












Porosity


It follows that





pressure




At z = h(O)


(63)

with


and hence

Its solutiorl is

(67) ($0) = ~ o e ( o ) - ~ )







(72) h(O) zz 1- do


(73) h(0) zz 1






Porosity








(77)




lnmz = h n - - - +- m m



where we define


(85) K - h49 = e ( l -q5m)2(9c- 1)

where











(92) Q = Qb( r ) when z = 0 t = 7



1+ rnz 3 = 111 I




Porosity















pressure




































nat 110 pp 601-613










Porosity


It follows that





pressure




At z = h(O)


(63)

with


and hence

Its solutiorl is

(67) ($0) = ~ o e ( o ) - ~ )







(72) h(O) zz 1- do


(73) h(0) zz 1






Porosity








(77)




lnmz = h n - - - +- m m



where we define


(85) K - h49 = e ( l -q5m)2(9c- 1)

where











(92) Q = Qb( r ) when z = 0 t = 7



1+ rnz 3 = 111 I




Porosity















pressure




































nat 110 pp 601-613


pressure




At z = h(O)


(63)

with


and hence

Its solutiorl is

(67) ($0) = ~ o e ( o ) - ~ )







(72) h(O) zz 1- do


(73) h(0) zz 1






Porosity








(77)




lnmz = h n - - - +- m m



where we define


(85) K - h49 = e ( l -q5m)2(9c- 1)

where











(92) Q = Qb( r ) when z = 0 t = 7



1+ rnz 3 = 111 I




Porosity















pressure




































nat 110 pp 601-613


(63)

with


and hence

Its solutiorl is

(67) ($0) = ~ o e ( o ) - ~ )







(72) h(O) zz 1- do


(73) h(0) zz 1






Porosity








(77)




lnmz = h n - - - +- m m



where we define


(85) K - h49 = e ( l -q5m)2(9c- 1)

where











(92) Q = Qb( r ) when z = 0 t = 7



1+ rnz 3 = 111 I




Porosity















pressure




































nat 110 pp 601-613






Porosity








(77)




lnmz = h n - - - +- m m



where we define


(85) K - h49 = e ( l -q5m)2(9c- 1)

where











(92) Q = Qb( r ) when z = 0 t = 7



1+ rnz 3 = 111 I




Porosity















pressure




































nat 110 pp 601-613






(77)




lnmz = h n - - - +- m m



where we define


(85) K - h49 = e ( l -q5m)2(9c- 1)

where











(92) Q = Qb( r ) when z = 0 t = 7



1+ rnz 3 = 111 I




Porosity















pressure




































nat 110 pp 601-613


where we define


(85) K - h49 = e ( l -q5m)2(9c- 1)

where











(92) Q = Qb( r ) when z = 0 t = 7



1+ rnz 3 = 111 I




Porosity















pressure




































nat 110 pp 601-613







(92) Q = Qb( r ) when z = 0 t = 7



1+ rnz 3 = 111 I




Porosity















pressure




































nat 110 pp 601-613



Porosity















pressure




































nat 110 pp 601-613









pressure




































nat 110 pp 601-613
































nat 110 pp 601-613

Fast and Slow Compaction in Sedimentary Basins Andrew C....

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