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Geophys. J . Inf. (1995) 122,283-298 Mathematical modelling of gravitational compaction and clay dehydration in thick sediment layers D. Marc Audet University of Oxford, Deparfment of Earth Sciences, Parks Road, Oxford OX1 3PR, U K Accepted 1995 February 8. Received 1995 January 24; in original form 1994 October 5 SUMMARY Gravitational compaction is an important process in sedimentary basins which controls the reduction of porosity with burial depth and the development of high pore fluid pressures. Often, sediments contain hydrated, expandable clay (smectite) that can undergo a transition to a dehydrated, non-expandable clay (illite). During dehydration, structural water bound within the sheet layers of smectite is released into the pore space, which can increase the pore pressure and influence geological processes such as solute transport, hydrocarbon migration and hydrothermal fracturing. A multicomponent continuum mechanics model that accounts for Darcy’s Law, Terzaghi’s principle of effective stress and a thermally activated dehydration reaction is derived and solved numerically. A closed-form solution is available in the limiting case of hydrostatic pore pressure and no dehydration. The results show that excess pore-pressure development is controlled by the sedimentation parameter, the dimensionless ratio of the hydraulic conductivity to the sedimentation rate. For relatively impermeable sediments, chemically released water can increase the excess pore pressure by as much as 30 per cent, and the excess pressure persists over geological time-scales. The pressure contribution due to the excess pore water is important provided that the dehydration goes to completion at sufficient burial depth, which depends in part on the activation energy of the reaction. If sediments overlie a permeable basement, fluid can flow out of the sediments and relieve pore pressure throughout the sedimentary column. Key words: compaction, layered media, sedimentation. 1 INTRODUCTION This paper presents a continuum mechanics model for the l - D gravitational compaction of a sediment layer whose thickness increases with time. The sediment grains undergo a thermally activated dehydration reaction that releases water, which increases the pore fluid pressure above normal hydrostatic values. The original contribution of this work is the derivation and numerical solution of a mathematical model that describes the coupling of compaction with concurrent sediment dehydration, along with the associated effects on pore pressure and porosity. High pore pressures in sedimentary basins influence many geophysical processes, such as the tectonic deformation of the Earth’s crust, the transport of solutes in groundwater flow, and the migration of hydrocarbons in petroleum reservoirs. The mathematical model is based on the continuum theory of mixtures (Atkin & Craine 1976) and consists of a series of mass balances for the solid components and the pore fluid. The overall momentum balance is quasi-static, and the fluid flow obeys Darcy’s Law. The sediment mechanical behaviour is assumed to follow Terzaghi’s principle of effective stress (Terzaghi 1943). The constitutive laws for the sediment rheology and permeability are adopted from soil mechanics theory. The equations lead to a non-linear parabolic equation for the porosity as a function of time and depth, which can be solved using a finite-difference method and a Newton-Raphson scheme. Since the thickness of the sediment layer is determined as part of the solution, the model represents a moving boundary value problem (Crank 1984). Since the study of compaction is multidisciplinary, it is useful to review the geological jargon used to describe mechanical and chemical alteration of sediments. Diugenesis refers to the sum of all physical, chemical and biological processes that alter sediments after they are deposited. Some of these processes, compaction and cementation, convert loose. soil-like material into hard rock-like material. 283 284 D. M . Audet This hardening process is called lithification or consolidation. Consolidation is often used in geotechnical engineering and implies the reduction of pore space by mechanical loading. Lirhification, a geological term, refers to the combined effects of mechanical loading and cementation. Znduration refers to the hardening of rock material by heat, pressure and cementation, that is, a more advanced state of lithification. In the extreme case, indurated sediments undergo metamorphism, that is, substantial changes to the mineral structure of the original rock material caused by high temperatures and pressures. In geology, the distinction between diagenesis and metamorphism is clear and the two subdisciplines rarely overlap. This paper concerns two specific diagenetic processes: compaction and clay dehydration. Compaction. the reduction of pore space by mechanical effects, increases sediment strength. In this paper, the role of clay dehydration is simply to generate additional pore fluid. It is assumed that dehydration does not affect the sediment mechanical properties. An example of clay dehydration is the conversion of smectite to illite. Smectite is an expansive clay that can swell and retain excess water. Under suitable conditions, smectite transforms to illite, and the excess water is liberated into the pore fluid. The dehydration mechanism used in this paper is only an analogue for the real process. Geochemically realistic reaction mechanisms are much more complicated. This paper is organized as follows. The background literature concerning gravitational compaction and the modelling of smectite-illite transformation is reviewed in Section 2. The geological system is described in Section 3, and the coupled compaction-dehydration model is derived in Section 4. Details of the numerical solution scheme are shown in Section 5. Section 6 summarizes the choice of parameters in the model, Results are presented in Section 7 and discussed in Section 8. Conclusions are given in Section 9. 2 BACKGROUND The roles of fluid flow and pore pressure in soils and rocks are important for understanding tectonic deformation of sediments, especially near faults and thrust zones. The interaction of pore pressure and fracturing of rocks relates to earthquake mechanisms. These geophysical applications have been reviewed by Mase & Smith (1987). On a regional scale pertaining to hydrology, fluid flow in sediments is important for understanding groundwater flow and solute transport, as recently reviewed by Mangold & Tasang (1991). The work in this paper pertains specifically to sedimentary basins. The gravitational compaction of sediments has received much attention in geotechnical engineering. One of the first 1-D compaction models was by Gibson (1958), who proposed a linear model for the compaction of thin layers of clays (fine-grained sediments consisting of clay minerals). Gibson’s (1958) model is in terms of an evolution equation for the void ratio. The linear model assumes that the sediment mechanical properties do not depend on the compaction, that is, the void ratio or the porosity. Subsequent models were fully non-linear and were solved numerically (Gibson, England & Hussey 1967; Gibson, Schiffman & Cargill 1981). The original linear model was extended by Gibson & Sills (1990) to account for active fluid removal from the base of the sediment layer. One of the important ideas from geotechnical engineering is Terzaghi’s principle of effective stress, which states that the strength of the sediments depends only on the difference between the total stress and the pore pressure, which is known as the effective stress. For a given material, the effective stress depends only on the porosity (or the void ratio). Terzaghi’s principle forms the basis of soil mechanics theory. The above 1-D models have been used to study the development of excessively high pore pressures in sedimentary basins. Bredehoeft & Hanshaw (1968) and Hanshaw & Bredehoeft (1968) used a linear compaction model to study the pore pressure in thick shale layers. Smith (1971) derived a non-linear model assuming a two-phase continuum model along with the effective stress concept. A further investigation using linear models was made by Bishop (1979). Bethke (1986) presented a compaction model that included a variety of effects, including water release due to clay hydration. However, since Bethke’s model does not incorporate Terzaghi’s principle of effective stress, the mechanics of compaction are not treated rigorously. Recently, a number of authors have shown renewed interest in I-D compaction models, namely Audet & Fowler (1992); Audet & McConnell (1992); Wangen (1992); Luo & Vasseur (1992). The paper by Audet & Fowler gives a brief review of the literature, poroelasticity (Biot 1941), and critical state theory (Chen & Mizuno 1990). The compaction model gives a porosity evolution equation similar to that of the model by Smith (1971). The main extension is that the equations are put into dimensionless form, which allows the compaction process to be characterized by the sedimentation parameter, denoted by A , which is the ratio of the hydraulic conductivity of the sediments to the sedimentation rate. Asymptotic solutions for large and small As are given along with numerical results. A detailed parameter study is presented in Audet & McConnell (1992). Audet & Fowler (1992) formulated a simplified model for clay dehydration but did not actually solve it. The purpose of this paper is to extend and solve the coupled compaction-dehydration model from Audet & Fowler (1992) and to present results pertinent to geophysical applications. Using a void ratio formulation like that of Gibson (1958), Wangen (1992) included a heat equation in the compaction model and studied the relative importance of non-linear conduction versus advection. Wangen scaled his equations and derived the sedimentation parameter, which he denotes by A(). The coupled effects of kerogen maturation and hydrocarbon migration on compaction are included (Wangen 1993). England et al. (1987) studied a similar problem in the context of petroleum entrapment in oil reservoirs. Shi & Wang (1986) formulated a 1-D compaction model in which the governing equation is in terms of the pore pressure instead of either the porosity or the void ratio. 4 similar model was used by Palciauskas & Domenico (1989) and by Luo & Vasseur (1992). Shi & Wang (1986) and Luo & Vasseur (1992) studied the relative importance of aquathermal pressuring, that is, the development of excess pore pressure due to the thermal expansion of the pore fluid. Both papers seem to agree that compaction Gravitational compaction and clay dehydration disequilibrium is the main cause of overpressuring, and the thermal effects on fluid density are less important. However, it is not likely that everyone will ever agree on this issue; see the discussions by Miller & Luk (1993) and Luo & Vasseur (1 993). The focus of this paper is the transition of smectite to illite and the concurrent release of water at depth. The release of extra pore water may lead to excess pore pressure (Powers 1967; Burst 1969). For this reason, there has been interest in understanding the kinetics of the smectite-illite transition. There are two hypotheses concerning illitization; first, smectite converts to illite by a series, either parallel or sequential, of dissolution-precipitation reactions; second, illitization takes place as a solid reaction due to ionic substitution of aluminium cations for silicon atoms. Assuming a first-order dissolution-precipitation model, Eberl & Hower (1976) estimated an effective activation energy of 82.0 kJ mol-' for the overall reaction. This value is within the range of activation energies measured for a variety of mineral dissolution reactions (Lasaga 1984). Bethke & Altaner (1986) used a Monte Carlo method to study the solid-state transition of smectite to illite. Preliminary results suggest that the Monte Carlo simulations may be able to predict more realistic illitization rates, but. as pointed out by Bethke & Altaner. this does not preclude dissolution-precipitation mechanisms. Field studies in the Gulf Coast region of the United States (Freed & Peacor 1989) and in the North Sea (Pearson & Small 1988) suggest that illitization occurs at burial depths between 1.5 and 3.5 km, corresponding to temperatures between 69 "C and W) I 1 5 1 1 Depositional Surface z = h(t) I" Compacting Sediment Layer 285 116 "C. However, the field data show appreciable variablity and further theoretical and observational studies are required. 3 DESCRIPTION OF T H E MODEL The idealized geological system being modelled is depicted in Fig. 1. A basin consists of a basement composed of relatively incompressible rocks. Overlying the basement, there is a layer of compressible sediments. The top of the sediment layer is called the depositional surface. A space coordinate, z , is defined such that z = 0 is the basement and z = h ( t ) is the depositional surface. The thickness of the sediment layer is h ( t ) , where r is time. For convenience, the burial depth d is defined by d = h ( r ) - z . Sediments are added to the layer at the rate V ( t ) , which means that h ( t ) increases with t. Gravity acts in the negative z direction. A heat flux, Q, enters the sediment layer at z = 0 and sets up a temperature gradient across the layer. A fluid flux, &), can enter or leave the sedimentary column through the basement. If the basement is impermeable, &,=O. As the sediment layer thickness increases with time, the sediments compact under their own weight, which leads to a reduction in the pore space of the sediments, which causes the pore fluid to migrate upward towards the depositional surface. This process is called gravitational compaction. If the sediments have low enough permeability, then the fluid flow through the sediment layer is hindered and the pore pressure increases above hydrostatic. Sediments with excess pore pressure resulting from hindered fluid flow are said to be in a state of compaction disequilibrium, since the excess pore pressure can dissipate, leading to further compaction. The solid portions of the sediments are assumed to contain a clay such as smectite which can change chemically to a non-expandable clay such as illite. As part of this clay-to-clay transition, water that is bound in the structure of the smectite is released and increases the amount of fluid within the pore space of the sediments. Under lowpermeability conditions, this excess pore water can lead to higher than normal pore fluid pressures. One of the aims of the model in this paper is to quantify how much excess pore pressure results from fluid release at depth. 4 MATHEMATICAL FORMULATION 4.1 Geochemistry 19 Basement Figure 1. Diagram of a compacting sediment laycr overlying an incompressible basement. The basement is at z = 0 and the depositional surface is at z = h(r). where h ( t ) is the sediment thickness. The sediment deposition rate is V ( f ) . The depth coordinate d is measured from the depositional surface. A heat flux, Qo, enters the sediment laycr at z = O and establishes thc geothermal gradient. T h e basement fluid flux, 6,. may either seep into or out of the layer. Gravity, denoted by g , acts in the - z direction. The sediments are modelled as a two-phase system of solids and pore fluid. The solid phase consists of three components: a large-grained, coarse material: a fine-grained hydrated clay; and a fine-grained dehydrated clay. The pore fluid is assumed to be pure water, even though, in reality, pore fluid contains dissolved mineral species. At this point, the model does not consider the liquid side of the geochemistry, and it ignores the effects of any dissolved cations. The hydrate undergoes a simple dehydration reaction given by H ~ " D + ~ w . (1) where H, D and W represent hydrated clay, dehydrated clay and water, respectively. It is assumed that n moles of W are D. M. Audet 286 liberated for every mole of D formed. The coarse material is assumed to be inert and it does not participate in the reaction. The overall reaction rate kr, for the process is assumed to follow an Arrhenius temperature dependence: k,, = Ea v exp - RT' where E , is the activation energy for the dehydration process, v is the frequency (pre-exponential) factor, R is the gas constant and T is the absolute temperature. The following kinetic model is postulated. Let [HI, [D] and [W] denote the molar concentrations of hydrate, dehydrate and water respectively. Assuming a first-order reaction mechanism, the rates of formation of each species are d - - [HI = dt + d I d - [D] = + - - [W] = + k,,[H]. dt n dt (3) For a unit volume of sediment material, the volume fraction of each of the four species, water, hydrate, dehydrate and coarse, are denoted by &, c $ ~ ,+d and dC,respectively. By definition, the volume fractions satisfy the condition 4,' 4 h + 4 d S 4 c = 1, 4.2 Mass conservation The following dimensionless variables, shown as quantities with overbars, are defined: The time- and length-scale factors are taken so that y l = Y / F where V, is the nominal value of the sediment deposition rate. The temperature-scale TI is the temperature at the depositional interface. The dimensionless reaction rate lrx is defined by where kf, is the reaction rate at TI and E,, = E,/(RT,) is the scaled activation energy. Hatted symbols denote dimensionless functions such as the reaction rates, the effective stress, the permeability and the pore fluid viscosity. Assuming that pw, ph, pd and pc are constant, scaling eqs (9)-( 12) gives (4) and the bulk density of the mixture, p, is given by P = 4wPw + 4hPh + 4 d P d + ~ c P c , (14) (5) where pw, phr pd and pc are the densities of the four components. The molar concentrations can be written in terms of the volume fractions as follows: =Ph4hlMh, LD1 =Pd4dlMd, LW1 =Pw4wlMw, (6) where Mh, Md and M, are the molecular weights of H, D and W respectively. The rate at which mass of H is transformed is r, (kgm-3s-') and this is balanced by the production of D at a rate r, and of W at rate r,. The reaction rates are thus rh = krxPhdhr rd = (Md/Mh)rh. rw= (nMw/Mh)rh. (7) For a closed system, conservation of mass implies that rh = r, + r,, (8) which is equivalent to Mh = Md + nM,. The model assumes that the motion is only in one dimension, the z direction, and that the three solid phases move with velocity u, and that the pore fluid moves with velocity u,. Thus, the 1-D mass conservation equations for the four components are (9) where a , and a2 are defined by The water denisty pw depends both on temperature and pressure. However, Luo & Vasseur (1993) showed that this dependence is not a major control on overpressure development, so it is not considered in this paper. The specific molar volumes of components W, D and H are qr W =-M W M d qr,,=--, Pw Pd Mh 7" h ---. Ph Thus a , = nVwlYh and a2 = Td/Vh represent ratios of specific molar volumes. The constant R,, = k:x9, the reaction parameter, is the ratio of the sedimentation time-scale to the reaction time-scale. The following result will be useful later. Combining eqs (14), (15) and (16) gives a y[l - at a 4 w I +azy [ (-l~ W > ~ =, I-(I - a2)Rrxkrx4h, (19) which can be combined with eq. (13) and then integrated ro give where F, = [4wVw,z=o, a function of i only, is the flux of Gravitational compaction and day dehydration water prescribed at the basement. Eq. (20) uses the boundary condition 6, = 0 at Z = 0 (see Section 4.7). 4.3 Evolution equation for sediment thickness Define m ( t ) to be the solid mass per unit area of the sedimentary column at time t: 287 used to calculate h(?)at the first iteration of a time-step. For each subsequent Newton-Raphson iteration, eq. (29) is used to update h(7).The use of (29) helps to maintain the accuracy of the overall mass balance of the solid phase throughout the time-stepping history of the numerical simulation. 4.4 Momentum conservation The momentum balance in the where q5s= defined by ~ - c # I ~ P545 = 4hPh + 4dPd + 4cPc. (22) Differentiating rn with respect to t gives where ( 4 , ~ ~means ) ’ &p5 evaluated at z = h ( t ) . Using eqs (10)-(12) and the definition of u2 (eq. 17) gives a -[ at a P , ~ S+] i)z [ P S ~ ? = ~ ,- (]r h - ‘d) = - ( P h Substituting eq. (24) into eq. (23), integrating, using u, = 0 at z = 0, and scaling leads to where V(T) = ri?/[(pS4J1V,] represents the dimensionless sedimentation rate based on the flux of solids deposited on the column, and 4; means 41T evaluated at z = h ( t ) , and so on for 4; and 6;. The constants Ap, yh and yd are defined by +A, Instead of integrating eq. (25) directly, it is more accurate to calculate h(t)_from an integral formulation. Using the definition for V ( i )from eq. (25), JI: Lr ri? dt = ( + s p s ) ’ v ) . T where nvis the vertical stress, g is gravitational acceleration and p is the bulk density given by eq. (5). Darcy’s law is where k is the permeability, p is the fluid viscosity and pf is the pore fluid pressure. The effective stress, denoted by uc, is - PduZ)krx4h. (24) m(t) = z direction is and p s is the average solid density From soil mechanics theory, a: is a function of the porosity. For many types of fine-grained sediments, the effective stress correlates with the porosity according to where u , ( ~ is a reference value of the effective stress l &) is the void ratio, el,,,, is (usually 100 kPa), e = + w / ( the void ratio of the sediments when u: = glo0and Cc is the compression index (Burland 1990). Correlations such as eq. (33) are sometimes called compressibility laws. Since lithology and mineral composition control sediment compressibility, the mechanical parameters C, and el,,,, should depend explicitly on 4d and 4=. However, the relationship between the mechanical parameters and the solid-phase composition is not well established, and for this reason C, and el(xlare assumed to be constant. Let u1be the stress scale. Then, the dimensionless vertical stress, pore pressure and effective stress are u,= u,a,, V ( i ) dT. (27) p f= ^I ff= ;( T I U ” . In keeping with Audet & Fowler (1992), the stress-scale is taken to be Eq. (21) and Z = z / h ( t ) (see Section 5) gives m ( t ) = K(7)Y JT: u1= ApgY. +,p, d Z . (34) (28) The vertical stress, u,, and the pore pressure, pf, can be eliminated from Darcy’s law by using eqs (30) and (32): In the numerical implementation (see Section S ) , eq. (25) is where eq. (4) was used to eliminate 4= from the Combining eqs (27) and (28) gives D.M . Audet 288 formulation. The quantity J(4,) is the volume-averaged flux of water relative to the moving matrix, and it depends primarily on 4,. The dimensionless group A , the sedimentation parameter, is given by where k , is the permeability at z = h ( t ) , p I is the fluid is the hydraulic viscosity at TI, and K~ = k,p,g/pl conductivity, which has units of velocity. Therefore, A represents the ratio of the permeability-controlled fluid velocity versus the sediment deposition rate. For large values of A , the sediments are highly permeable or deposited at slow rates, and pore pressures tend to be hydrostatic. Conversely, relatively impermeable sediments deposited at fast rates (small As) lead to high pore pressures. The dimensionless permeability is taken to be 4.5 Exact solution A + + 00 The limiting case of large A represents a normally pressured sediment layer for which u, and u , are zero. For the case of F;, = R,, = yh = yd = 0, an impermeable basement, no dehydration, homogeneous solid composition, one can obtain a closed-form expression for 2 as a function of e: (43) Eq. (43) is useful because it connects geotechnical, laboratory-scale parameters (C, and e,,,,) to observable field-scale data (4w versus d ) . It was derived earlier in Audet (1995) and used t o study the compaction trends of normally pressured, pelagic sediments in the Pacific Ocean. 4.6 Non-linear heat conduction with = 4;, C#I~ = The geothermal gradient is assumed to be due primarily to non-linear, quasi-steady heat conduction resulting from a specified basement heat flux. Since the Peclet number for this system tends to be small, and the dimensionless viscosity is [ = exp q(+111 I)]. and 17 are constants. If u ,is the effective stress at 4w= 4;, then where ,. CY a: gv=-= WI exp I &I) el - e ~ P T=I+Q,, df 7 . k,,, (44) The dimensionless, bulk thermal conductivity of the sediment mixture is taken to be (39) where P =C,/ln10, using the notation of Audet & McConnell (1992), and e , = +1/(1 Substituting eq. (20) into eq. ( 3 9 , multiplying by (1 - &,), differentiating with respect to Z and using eq. (19) gives Eq. (40) is a non-linear diffusion equation for 4, with a combination of point and distributed source terms. Using eqs (20) and (35). V 5 and V , can be written in terms of the volumetric flux J: based on the work of Woodside & Messner (1961). and k;h and k:h are the thermal conductivities of the solid and fluid phases respectively. The dimensionless basal heat flux, Q,), is defined by where q,, is the unscaled geothermal heat flux (in mW m-*) at Z= 0. In this study, q,, = 37.9 mW m -', which corresponds to Q,, = 0.02. In practice, this yields a temperature gradient of 29-31 "C km-', in keeping with observations (Deming & Chapman 1989). Heat flow in sedimentary basins is an important topic and there is an extensive body of literature devoted to it, of which Deming & Chapman (1989) and Gallagher (1987) are only two representative works. An overview of the topic is given by Furlong & Chapman (1987) from a global, geophysical point of view. 4.7 Boundary conditions At z = h ( t ) , values of 4,, 4 h , 4dare specified. At z = 0, the prescribed velocity boundary conditions are u, = 0 and 4,uW = &), both of which are satisfied identically if 6,= J ( 4,) at z = 0 is satisfied, which represents a non-linear boundary condition for I$,.,. For initial conditions, at t = 0, all the volume-fraction profiles are uniform. Because of eq. Crauitutional compaction and clay dehydration (33), specifying c $ ~ at the depositional surface (4;) is equivalent to specifying the effective stress (cT,). 5 289 the volume fractions are rescaled by NUMERICAL SCHEME The following coordinate transformation is used: (47) This has the advantage of fixing the moving boundary, the depositional surface, to Z = 1, which simplifies the numerical treatment. From the chain rule, a dz - l a hdZ’ ._- a’ _1 _a’ a i)z2 h’dZ’’ dl - ij at’ ~h a h aZ’ (48) where h = dhldi. Since there is no difference between t and t ’ except when taking partial derivatives of functions that depend on both t’ and Z , there is no need to distinguish between t and t‘ in the rest of the paper. Even though the moving boundary is now fixed at Z = 1, the sediment thickness appears as a time-dependent coefficient in the relevant equations, and the evolution equation for h ( f) still has to be solved. The coupled compaction-dehydration model is a system of partial differential cquations consisting of a non-linear parabolic equation for 4w (eq. 40) and three non-linear first-order hyperbolic equations for 4,,, +d, +c (eqs 14, 15, 16). In addition, there are two quadratures, one for t% (eq. 29) (see comments in Section 4.3) and one for 7 (eq. 44). The solution procedure begins by solving eq. (40) for 4w assuming suitable guesses for the other unknowns (&, 4d, 4c,T, h,.A t a given time-step, the unknowns are evaluated at the previous time-step. The non-linear parabolic equation is put into discrete form using second-order, centred finite-difference approximations for the Z derivatives, and a second-order, backward difference for the T derivative (Smith 1985). The time-steps are constant and the space mesh is uniform. The non-linear algebraic equations are solved using a Newton-Raphson scheme. After obtaining 4wat the current time-step. t% and 7 are updated using eqs (29) and (44), respectively, and the solid-phase velocity comes from eq. (41). Using secondorder finite-difference approximations for the space and time derivatives (forward in Z , backward in T), the hyperbolic equations are integrated from Z = 1 to Z = 0 in order to obtain the current values of (bh, (bdr +c. Owing to numerical dispersion and dissipation in the solution of the hyperbolic equations, the sum of the volume fractions d o not sum t o unity, that is, The notation (.), means some quantity evaluated at the kth step of the iteration procedure. Note that S, varies with 2. Using the rescaled volume fractions for the solid components, and the current temperature profile, bWis recalculated by resolving eq. (40), followed by resolving the relevant equations for T, h, 4,,. +d, 4c. The changes in these five unknowns are monitored until the differences between any two iterations are sufficiently small. Let r, be the residual of one of the five unknowns at the ith grid point. For each unknown, the iterations were deemed to have converged when llr, 5 2 X lo-’. As the iterations proceed, the renormalization factor S, -+ 1 over all mesh points. The renormalization step does not affect the accuracy of the numerical scheme because it is used only to prevent numerical noise from the solution of the hyperbolic equations from feeding back into the non-linear parabolic equation. Test cases using various mesh sizes were made in order to check the convergence of the numerical scheme. The numerical model was verified by comparison with asymptotic cases given in an earlier paper (Audet & Fowler 1992). Typically, the space mesh was AZ=0.01 (100 grid points) and the time-step was AT= 0.025. In a few cases (Section 7.5), up to 400 grid points (AT = 0.0025) were used in order to obtain good accuracy on the overall solid mass balance on the sediment layer (eq. 29). The numerical scheme was implemented in FORTRAN using doubleprecision arithmetic on a Sun S P A R C station ipx. 6 PARAMETER VALUES The kinetics of the dehydration reaction are controlled by two parameters, E , and v. Values for E , range from 40 to 80 kJ mol for mineral dissolution reactions (Lasaga 1984). In this study, E,, = 60 kJ mo1-l was used as a representative value. The pre-exponential is more difficult to quantify because it depends on many factors, such as local fluid-phase chemistry, the details of which are beyond the scope of this paper. In order to assess the scope of the model, it is postulated that the reaction goes to completion when R,,k^,, = Sy l ) , and that this happens when T = T,, some specified temperature. Under these conditions, v is given by 1 E,, Try Y = -exp e , The volume fractions must be corrected to prevent the solution from deteriorating. The correction scheme used consists of renormalizing the volume fractions of the solid components. It is assumed that 4,, is correct since the non-linear parabolic equation is not prone to the numerical problems associated with the hyperbolic equations. Defining the renormalization factor Sk for the kth iteration: 3 (54) and R,, is & R,, = exp E()( - 1 ). The advantage of this choice of parameter is that R,, is independent of the time-scale. The specific values o f the parameters are given in Table 1. After examining the effects of temperature-dependent 290 D. M . Audet Table 1. Summary of model parameters Parameter Nominal Value Reference 1,050 kg/m3 Hamilton (1976) 2,740 kg/m3 Lambe & Whitman (1979) 2,840 kg/m3 ,I 2,700 kg/m3 I, 770 g/mol 5-10 Deer et al. (1962, p. 213) (using KyAl~(Si~-yA1,)02~(OH)4 with 1 < y < 1.5) Freed & Peacor (1989) 18 g/mol 2.02 W/m K Sharp & Domenico (1976, Table 1) 0.631 W/m K I, 60 kJ/mol Lasaga (1984) 1.62 MPa Burland (1990) 0.5 I, 500 m/Ma 100 m 0.2 Ma 14 lo-" m/s 37.9 mW/mz Olsen (1960) I1 Deming & Chapman (1989) 300 K viscosity on the solutions, it was concluded that viscosity effects were comparable to the effects of varying the permeability for md 5 k , 5 10F2md (note: 1 md = lO-'m s-' = lo-'' m2). The effects of a, which controls the sensitivity of permeability to porosity, were studied in Audet & McConnell (1992). From these considerations, it was decided to use a constant viscosity with 7 = 0 (see eq. 38) since the main objective of the paper is to study the effects of clay dehydration on pore pressure. Recent in-depth studies (Luo & Vasseur 1992; Wangen 1992) have examined the role of temperature-dependent fluid density and viscosity on overpressuring. T h e effect of yh and yd (eq. 26) were tested. For y h = +0.25, the overall effects on the volume fractions and the pore pressure were comparable to the effects of varying a, (eq. 17; results in Section 7.4). Using the density values given in Table 1, typical values of the density ratios are yh = 0.0242 and yd = 0.0848, both of which are smaller than the value of 0.25 used in the test cases. Since the density ratios seemed to be of secondary importance, the values yh = yd = 0 are used for all the cases presented in Section 7. 7 RESULTS 7.1 The effects of the sedimentation parameter: A This section considers the simple case in which there is no dehydration ( R r x= 0) and the basement is impermeable (4)= 0). The sedimentation rate is constant throughout the deposition history [V(T)= 11, which means that, for equal time spans, the same mass of solid material has accumulated in the sediment layer. Fig. 2 shows the porosity (water volume fraction, 4w) and the pore pressure (pf)as a function of burial depth (d) for 0.5 5 A 5 10. The porosity profiles (Fig. 2a) begin with 4w=0.5 at d=O and the porosity decreases to 0.2 < 4,,, < 0.3 at the basement depth of 37 < d< 43. Each of the profiles corresponds t o T = 50, and in each case the same amount of solid mass has accumulated between the basement and the depositional surface. For reference, the porosity profile for normally pressured sediments (eq. 43) is shown. A s A increases, the porosity approaches the normally pressured profile. The Gravitational compaction and clay dehydration 291 and the lithostatic pressure is -50 -- A = 0.5 1 1 1 1 , 1 1 1 1 ~ 1 1 1 1 where 4; is c $ ~ at d=O, the depositional surface. The lithostatic pressure represents the pressure that the pore fluid would obtain if the sediments were not compacted. In this study, the variation of A is interpreted as a change in sediment permeability. As A decreases, the sediments are less permeable, which hinders the expulsion of pore fluid by compaction, and the sediments retain a higher porosity compared to the normally pressured situation. Since each profile represents the same amount of solid mass, the total thickness (basement depth) of the sediment decreases with increasing A since the average porosity decreases with increasing A. A more detailed study of porosity evolution is given in Audet & McConnell (1992), in which the effects of time, rheology and permeability are examined. 7.2 The effcts of stoichiometry: n 0.0 10.0 20.0 30.0 40.0 50.0 60.0 Pf Figure 2. The porosity, & (a), and the pore pressue, jif (b), as functions of burial depth, 2, at ?=SO for four values of the sedimentation parameter, A =0.5. 1, 2 and 10. The dashed line in (a) is the porosity versus depth profile for normally pressured sediments (eq. 43). In (b), the hydrostatic pressure is denoted by phydro and the lithostatic pressure is F,,,,,", as defined by eqs (56) and (57) in the text. corresponding pore-pressure profiles (Fig. 2b) reflect the dependence on A. The case of A = 10 is near hydrostatic, and the case of A = 0.5 is closer to the lithostatic profile. The hydrostatic pressure is This section looks at the effects of dehydration for the case of R,, =0.01 with A = 1 and &,=0. For E,,= 24 (corresponding to E;,= 60 kJ mol-'), R,, = 0.01 corresponds to 7;.x= 100°C. Assuming a , + a, = 1, the sediments undergo no overall specific volume change, which means that the integral term in eq. (40) drops out. The number of moles of water released per mole of hydrate formed, n , is varied such that n = 2.5 ( a , = 0.14198), n = 5 ( a , = 0.26910) and n = 10 ( a , = 0.48722). The values of u , are based on Md = 770, M, = 18, c $ ~ = 4c= 27oO kg m-3 and 4w= 1050 kg rK3, with a , = 1 - a , by assumption. The initial composition of the sediments is +h = 0.2, 4d= 0.0, $c = 0.3 and c$w = 0.5. These values are based on the composition of North Sea shales given in Table 1 of Pearson & Small (1988). Referring to Table 1 of Pearson & Small (1988), sample P1 has about 26 wt per cent of fraction less than 2 p m , of which at least 83 per cent is smectite. Since 83 per cent of 26 per cent is about 22 per cent, 4hwas set to 0.2. Since it is not known what size fraction of the sediments undergoes dehydration, using 2 p m to identify the hydrated clay fraction is not very precise. Even though the initial value of 4, does affect the amount of excess pore pressure predicted by the model, this detail does not alter the overall conclusions derived from the model. Figure 3(a) shows the volume fractions of the hydrate phase (H), the dehydrate (D) and the water (W) phase as a function of burial depth for 5 = 50. The profile of 4wversus d for no dehydration is also shown (dashed curve). As n increases, more water is released during the reaction, and 4,,, increases for a given burial depth. This is due to the reaction term u , R , , ~ ^ , ,in~ ~eq. (40). The 4h versus 2 profiles do not vary much with n. This is because the evolution equation for eq. (14), does not depend on a , or a,. In contrast, the c $ ~profiles vary with n , because eq. (15) depends on u 2 . Since u , = 1 - a , . as n increases, a , increases and a , decreases, which is why, at a given depth, 4ddecreases as n increases. The corresponding excess pore-pressure 6, - jihydrC,) profiles are shown in Fig. 3(b). Results are shown for A = 1 and 10. The results show that, for A = 1, a water-releasing +,, D. M . A d e t 292 parameters are used: a , = 0.25: a , = 0.75, corresponding approximately to n = 5 ; h = 1: F;, = O ; E,,= 24. Results are shown for T = SO. A temperature, T,,, is picked and R,, is chosen so that the reaction source term is O(1) at that ,, #, and #,, as a function of temperature. Fig. 4(a) shows # 2 for R,, = 0.05, 0.01 and 0.002 corresponding t o T,, = 70, 100 and 130 "C, respectively. The volume-fraction profiles for the hydrate and dehydrate phases (Fig. 4a) show that the dehydration reaction goes to completion at greater depth with decreasing Rrx. By decreasing R,,, the pre-exponential 0.0 0.2 0.1 4w 0.3 or 4d 0.4 or 0.6 0.5 4 h 0.0 03 0.2 0.1 d 0.4 4 w or 4 d or - 50 X=l I 1 1 1 1 1 1 1 I l l 0.6 0.5 d'h - 1 Pf - P h y d r o Figure 3. (a) The volume fractions of hydrate (&), dehydrate (4d) and water (4,) as a function of burial depth d a t f = 50 and A = 1. The dashed curve shows r # ~for~ t h e case of no dehydration. The profiles are shown for three valucs of the stoichiometric coefficient, n = 2.5, 5, 10. (b) shows the excess pore pressure, P; - flhydrC,. as a function of 2 for the three values of n , for A = 1 and 10. reaction can increase excess pore pressure by about 30 per cent at the basement compared to the case of no water release (dashed curves). The extra amount of overpressuring due to dehydration increases as n increases. 7.3 The effect of the reaction parameter: R , In this section, R,, is varied while keeping the activation energy constant. This is equivalent to varying the pre-exponential factor v independently of E,. The following 50 ( c ) 0.002 1 1 1 1 1 1 1 1 I I I I I Gravitational compaction and clay dehydration factor decreases, which means that higher temperatures are required for the reaction t o go to completion, which, in this system, are found at greater burial depths. As the reaction takes place at greater depth, the porosity (&) is perturbed further away from the reference profile corresponding to no dehydration. The change in porosity is reflected in the excess pore pressure (Fig. 4b). A s pore fluid is released at deeper levels (Rrx = 0.002), the excess pore fluid increases by about 10 per cent with respect to the case of R,, = 0.01. d 7.4 293 The effects of sediment expansion and contraction: a2 The purpose of this section is to examine how the integral source term affects the porosity and the excess pore pressure. The integral source term is important if there is a net change in the specific volume of the sediments during dehydration. The stoichiometric coefficients are chosen such that a , = 0.25 and a2 = 0.6, 0.7.5 and 0.90, corresponding to relative specific volume changes (i.e. AY"/Yh= u 1 + a2 - 1, with AY"=nVw+ Vd- Yh) of -0.15, 0.0 and +0.15 respectively. Fig. 5(a) shows the porosity profiles for A = 1 and 4, = 0 at T = 50 for the three values of a2. The kinetic parameters are R,, = 0.01 and E,, = 24. The results show that when the reaction products shrink during dehydration (a2 =0.6), that is, the combined specific volume of the dehydrate component and the extra, released water is less than the specific volume of the hydrate component, the porosity increases relative to the isovolume case ( a 2 = 0.75). and the excess pore pressure (Fig. Sb) increases by about 5 per cent near the basement. Because n'Vw+ 'Vd is less than 'Vh when a , + a , < 1, extra void space is created by the dehydration reaction. Water originally present in the sediments fills the extra void space. The extra void space increases 4,.,, which decreases g:. causing p , to increase in order to balance the overburden, w,,. In contrast, when the reaction products expand (a2 = 0.90), the porosity decreases relative to the isovolume case and the excess pore pressure decreases by about 5 per cent. 50 0.2 0.4 0.3 0.5 7.5 Dissipation of excess pore pressure 4w The previous sections examined how chemically released pore fluid affected the excess pore pressure in a thick layer of low-permeability sediments. This section cxamines how long the excess pore pressure will be maintained if sedimentation ceases and the pore pressure dissipates to its hydrostatic value. Dehydration is omitted since the primary control on overpressuring is through A, and the basement is impermeable (4,= 0). Consider a sedimentary layer with A = 1 that has been uniformly deposited for O S T S S O . For ?>SO, sedi= O . Fig. 6(a) shows the mentation ceases, that is, evolution of the porosity as a function of depth for T = 5 0 , 75, 100 and 12.5. As T gets larger, the porosity profile approaches the normally pressured equilibrium profile (dashed curve) denoted by T = +x. The excess porepressure profiles (Fig. 6b) show a similar trend. After the first time interval (7 = 75), about 50 per cent of the excess pore pressure has dissipated. After each subsequent time interval, the excess pore pressure decays by about SO per cent, which is reminiscent of the exponential decay associated with parabolic equations such as eq. (40). As the pore pressure approaches hydrostatic (zero excess pore pressure), both the sediment and the fluid velocities go to zero (Fig. 6c). T h e pore fluid velocity is monotonic at T = SO. After sedimentation stops, u, near the depositional surface (0 < d< 20) decreases faster than near the basement region. A t T = 75, V , reaches a maximum value near d = 15. v(T) 50 0.0 5.0 10.0 15.0 pf - p h y d r o Figure 5. The effects of specific volume change on porosity (a) and excess pore pressure (b). The dashed curves arc the case of no dehydration, and the profiles are for T = 50 and h = 1. The profiles are for uz = 0.60, 0.75 and 0.90. D . M . Aiidet 294 d - I - 50 0.2 0.3 4W 0.4 0.5 50 0.0 5.0- 10.0 15.0 50 I - I 1 1 1 1 I I 1 normally pressured sediment layer having the same amount of solid mass as the porosity profiles shown in Fig. 2(a). As T increases, h(T)approaches a uniform value. This is clear for A = 1, 2 and 10. For A = 0.5, the asymptotic limit is reached for some time greater than 350. The final asymptotic values calculated from solving eq. (40) agree to within 0.2 per cent of the theoretical value, determined by integrating the hydrostatic porosity-depth profile (eq. 43). The agreement improves as the resolution of the Z mesh is increased, so the agreement between h(T)and h, for large 7values is limited by the accuracy of the numerical solution. The time needed to approach h, depends on A. For normally pressured sediments, A = 10, the column relaxes quickly, and for smaller As, relaxation times increase, which reflects the effects of relative decreases in the sediment permeability. pf - phydro Figure 6. The relaxation of porosity, excess pore pressure and phase velocities as a function of time following the end of active sedimentation. (a) Shows & w versus 2 for T=50 (end of Sedimentation), 75, 100 and 12.5, and part (b) shows the corresponding profiles of - P;lydr,,versus ;% The solid velocity (17,) and fluid velocity (6,) profiles are shown in part (c). The dashed 4,,, profile in (a) is the normally pressured profile (eq. 43). The case corresponds to A = 1. R,, = 0, fil = 0. In contrast, the velocity of the solids remains monotonic with depth. Figure 7 shows the sediment thickness, h(T)as a function of time (?>SO) for different values of the Sedimentation parameter, A = 0.5, 1, 2 and 10. In each case, the sediment layer has been uniformly deposited for 0 5 Ts 50, followed by no sedimentation for T>50. The dashed horizontal line (labelled h,) represents the theoretical thickness of a 7.6 The effects of basement fluid flux: F, All the previous cases have assumed an impermeable basement (6) = 0). This section examines the role of pore < 0) fluid either seeping into (Cl > 0 ) or draining out of (6) the sediments at the basement for the case of no dehydration (Rrx = 0). Figure 8(a) shows the porosity profiles (A = 1 and T = 50) for different values of For &,>O, water seeps into the sediments and the porosity increases relative to the case with an impermeable basement ( f i ) = O ) . For 6,<0, the sediments are draining at the basement and the sediments are more compacted. For Fo< -0.05, it is possible for the sediments to be overconsolidated, that is, at a given burial depth, the porosity is less than that given by the normally pressured porosity profile (eq. 43). The pore-pressure profiles (Fig. 8b) show an increase in pore pressure due to fluid seeping into the sediments Grauitationul compaction und clay dehydration 205 &(t) 50.0 100.0 150.0 200.0 - 250.0 300.0 350.0 t Figure 7. The sediment thickness, &(T) as a function of time following the end of active sedimentation at T = SO. for four values of the Sedimentation parameter, A = 0.5, 1. 2 and 10. The horizontal. dashed line, denoted by i xis, the thickness of the normally pressured sediment column which the curves approach as T - t +=. (4)>0)and a decrease in pt for drainage (F;,<O). The overconsolidated region in Fig. 8(a) corresponds to driving the pore pressure below the hydrostatic value. which occurs for 6)< -0.05. Figure 8(c) shows the pore fluid velocity as a function of depth. The solid-phase velocity is shown for 6) = id.10. These results show that the basement flux condition affects the fluid velocity but has little effect on the solid-phase velocity. 8 DISCUSSION The sedimentation parameter, A, is the most important parameter in the I-D compaction model because it controls whether or not the pore pressure is greater than hydrostatic. This is clear from Fig. 2, and consistent with results from other similar models (Wangen 1992: Luo & Vasseur 1992; Bethke 1986; Smith 1971). The sedimentation rate is an important control, but only relative to the hydraulic conductivity of the sediments. If the sediment permeability is sufficiently high, then the pore pressure will essentially be hydrostatic regardless of the sedimentation rate. A larger problem concerns the constitutive laws used for the effective stress and the permeability. Permeability measurements are often done on sediment samples several centimetres in length, and yet the model in this paper is concerned with compaction trends on the length-scale of kilometres. The same can be said for the effective stress. More work is required concerning how to scale up laboratory data in order to use it in macro-scale models for basin analysis. Attention has been given to the role of temperature effects on overpressure development. One effect is aquathermal pressuring, that is, excess pore-pressure development due to the thermal expansion of pore fluid. Related to this is temperature-dependent viscosity. A second effect is the role of chemically released pore fluid at depth, an example of which is water released due to the transformation of smectite to i l k . The results in this paper address these problems. Chemically released water is probably more significant then thermal expansion of pore fluid. The results of Fig. 3 are important because they quantify when fluid release is important. To have an appreciable increase in excess pore pressure, the dchydration mechanism has to release at least 5-10 moles of water per mole of reaction product when d,, = 0.2 initially. Reactions with n < 5 will not influence pore-pressure development to any significant degree. Even though sediment dehydration may increase pore pressure, the effects are significant only if A = O(1). Chemically released water in highly permeable sediments will not cause overpressuring, especially when A > 10, as shown in Fig. 3(b). The geochemical model used in this paper is very simple and ignores the pore fluid chemistry. As shown in Fig. 4, for a given value of the activation energy, the pre-exponential factor (i.e. R J determines at what depth the dehydration reaction goes to completion. which in turn influences the enhancement of the pore pressure. If this model is to be quantitatively useful for studying real sedimentation systems, then the geochemical details, that is the kinetics and the thermodynamics of the relevant reactions, have to be included in the mechanical compaction model. With a more robust geochemical formulation, a coupled compactiondehydration model may be used to discern the effective activation energy of the rate controlling reaction from the ~ c#J,, versus depth data. In addition. the depth slope of $ J or range over which the reaction goes to completion can provide some information about the pre-exponential factor D.M . Audet 296 Figure 8. (Continued.) 50 f 0.0 1 1 ~ 1 10.0 1 1 1 11 1 ~ 20.0 1 1 .\. 1 30.0 ~ 1 1 1 40.0 1 1 1 50.0 Pf Figure 8. The porosity (a) and the pore-pressure (b) profiles at T = 50 and A = 1 for different basement seepage conditions. Values of 4,> 0 correspond to water entering the sediments at the basement and 4)<0 corresponds to water draining out of the sediments at the basement. The normally pressured q5w and jic profiles are denoted by dashed lines. (c) shows the fluid-phase velocity (V,) versus 2 for -0.1 5 4 ) s +0.1 (solid lines). For comparison, the solid-phase velocity profiles (V,) are shown for 6,= k0.1 (dotted lines). in terms of the concentrations of the cations that control the mineral dissolution rates, which in turn control the overall reaction rate. The results of Fig. 5 illustrate that the exact details of the specific volume changes between the reactants and the products do not have a controlling effect on the pore pressure, at least for overall volume changes less than k15 per cent. This further supports the idea that the thermal expansion of the pore fluid exerts only a minor control on overpressuring, in keeping with the more detailed study by Luo & Vasseur (1992). In addition to understanding the mechanisms leading to excess pore pressure, it is important to understand the time-scale over which the excess pressure can be maintained in a geological system. The results from Fig. 6 provide some insight. For a length-scale d;p= loom, the profiles in Fig. 6(a) correspond to about 4 k m of sediments. For V,, = 500 m Ma-', the relevant time-scale is .T = 0.2 Ma. Fig. 6(b) shows that excess pore pressure will be maintained from T = 50 to T = 125, which can be interpreted as a time interval of 15 Ma for A = 1. Thus, the excess pore pressure is maintained longer than it took to accumulate the sediment column, which took i= 50, that is, about 10 Ma. The results from Fig. 7 demonstrate that the time for excess pressure dissipation scales with A. For the case of A = 0.5, dissipation will take about twice as long, say of the order of 30Ma. Likewise, for A = 10, the excess pore pressure, which is not much to begin with, will dissipate in less than 1.5 Ma. These estimates are important because they give some insight into how long high pore fluid pressures are capable of causing oil migration in sedimentary basins. This has implications for the timing of petroleum migration into reservoir traps. In a more general context, in order to understand how fluids move in any sedimentary environments, it is important to model the pore pressure rigorously, which is why the fundamental equations given in this paper are important. To date, many of the 1-D compaction models have assumed an impermeable basement. The results shown in Fig. 8 are the first attempt at studying the effects of a pore fluid flux at the base of a thick sedimentary sequence. One relevant application is the common situation in which a thick layer of low-permeability sediments (e.g. shales or mudstones) overlies a highly permeable material such as a sandstone layer. If the sandstone layer is not hydrodynami- Gravitational compaction and clay dehydration cally isolated, then it is possible for fluid to seep out of the shale layer and into the sandstone, which would relieve excess pore pressure as shown in Fig. 8(b). Thus, in addition to A, the overpressuring is influenced by the basement water flux boundary condition, which up to now has not been considered in most basin fluid-flow models published to date. e,, 9 CONCLUSIONS This paper develops a 1-D model for the gravitational compaction of thick sediment layers in which the sediments undergo a dehydration reaction that releases pore fluid at depth. Using a finite-difference method to solve the governing partial differential equations, a series of case studies illustrates the effects of the model parameters on the porosity and pore-pressure profiles. This work presents three original accomplishments: a self-consistent set of equations that describes the dehydration reaction; a study of the time-scale over which excess pore pressure dissipates after sedimentation stops; a study of the effects of pore fluid flux at the basement. A dehydration reaction, such as smectite transforming to illite, can increase overpressuring by about 30 per cent provided that the sediments are relatively impermeable, that is, A = O( l), and that sufficient water is released, n 2 5. The exact details of the density changes of the components are not crucial to the conclusions. However, the geochemical details that govern the pre-exponential factor of the kinetic rate law are important because they control the depth at which the dehydration goes to completion, which controls the enhancement of the overpressuring by the released fluid. For similar reasons, the activation energy of the ratecontrolling step is also important. The results show that, for a 4 km thick sediment layer, with K~ = lop" m s p ' , excess pore pressure can exist over the course of 1.5 Ma, provided that A = ql).The excess pore pressure dissipates more quickly for large A , as expected since this is interpreted as sediments of higher permeability. The basement fluid-flux boundary condition can also control the pressure profile in the sediments. If the pore fluid drains out of the sediment layer at the basement, the pore pressure in the bottom half of the layer is reduced, sometimes to subhydrostatic values, which means that it is possible for the,sediments to be overconsolidated. The model can be improved by improving the scale-up of constitutive laws for the permeability and the effective stress. The coupled dehydration-compaction model can be made more realistic by adding more kinetic and thermodynamic details. Further studies in these areas will help to improve the understanding of pore pressure and fluid flow in sedimentary systems. ACKNOWLEDGMENTS This work was supported by the Natural Environment Research Council through grant No. Dl/G1/189/01, along with continuing support from Amoco (UK) Exploration Company. The author thanks Prof. J. D. C. McConnell and Dr A. C. Fowler for useful discussions throughout the course of this work. T h e author is especially grateful to M. Wangen and B. KrooR for reviewing the manuscript. 297 REFERENCES Atkin, R.J. & Craine, R.E.. 1976. Continuum theories of mixtures, basic theory and historical development, Q. J . Mech. appl. Math., 29, 207-244. Audet, D.M., 1995. Modelling of porosity evolution and mechanical compaction of calcareous sediments, Sedimentology, 42, 355-373. Audet, D.M. & Fowler, A.C.. 1992. A mathematical model for compaction in sedimentary basins, Geophys. J . Int., 110, 577-590. Audet, D.M. & McConnell. J.D.C., 1992. 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