aa r X i v : . [ phy s i c s . g e o - ph ] M a r Nonlinear Viscoelastic Compaction in Sedimentary Basins
Xin-She Yang
Department of Fuel and Energy and Applied Mathematics, University of Leeds, LEEDS LS2 9JT, UK
Abstract
In the mathematical modelling of sediment com-paction and porous media flow, the rheologicalbehaviour of sediments is typically modelled interms of a nonlinear relationship between effectivepressure p e and porosity φ , that is p e = p e ( φ ).The compaction law is essentially a poroelasticone. However, viscous compaction due to pressuresolution becomes important at larger depths andcauses this relationship to become more akin toa viscous rheology. A generalised viscoelasticcompaction model of Maxwell type is formulated,and different styles of nonlinear behaviour areasymptotically analysed and compared in thispaper. Citation detail : X. S. Yang, Nonlinear viscoelasticcompaction in sedimentary basins,
Nonlinear Pro-cesses in Geophysics , , 1-7 (2000). Drilling mud is used in well-bores drilled for oil ex-ploration to maintain the integrity and safety of thehole. The mud density must be sufficient to preventcollapse of the hole, but not so high that hydrofrac-turing of the surrounding rock occurs. Both theseeffects depend on the pore fluid pressure in the rock,and drilling problems occur in regions where abnor-mal pore pressure or overpressuring occurs, such asin the sedimentary basins of the North Sea, wherepore pressure increases downward faster than hy-drostatic pressure. Such overpressuring can affectoil-drilling rates substantially and even cause se-rious blowouts during drilling. Therefore, an in-dustrially important objective is to predict over-pressuring before drilling and to identify its pre-cursors during drilling. Another related objectiveis to predict reservoir quality and hydrocarbon mi-gration. An essential step in the achievement ofsuch objectives is the scientific understanding oftheir mechanisms and the evolutionary history ofpost-depositional sediments such as shales. Shales and other fine-grained compressible rocksare considered to be the source rocks for muchpetroleum found in sandstones and carbonates. Atdeposition, sediments such as shales and sands typ-ically have porosities of about 0 . ∼ .
75 (Lerche,1990). When sediments are drilled at a depth of say5000 m, porosities are typically 0 . ∼ .
2. Thusan enormous amount of water has escaped from thesediments during their deposition and later evolu-tion. Because of the fluid escape, the grain-to-graincontact pressure must increase to support the over-lying sediment weight. Dynamical fluid escape de-pends lithologically on the permeability behavior ofthe evolving sediments. As fluid escape proceeds,the porosity decreases, so the permeability becomessmaller, leading to an ever-increasing delay in ex-tracting the residual fluids. The addition of over-burden sediments is compensated by an increaseof excess pressure in the retained fluids. Thusoverpressure develops from such a non-equilibriumcompaction environment (Audet and Fowler, 1992;Fowler and Yang, 1998). A rapidly accumulat-ing basin is unable to expel pore fluids sufficientlyrapidly due to the weight of overburden rock. Thedevelopment of overpressuring retards compaction,resulting in a higher porosity, a higher permeabil-ity and a higher thermal conductivity than are nor-mal for a given depth, which in turn changes thestructural and stratigraphic shaping of sedimentaryunits and provides a potential for hydrocarbon mi-gration. Therefore, the determination of the mech-anism of the dynamical evolution of escape of fluidsand the timing of oil and gas migration out of suchfine-grained rocks is a major problem. The funda-mental understanding of mechanical and physico-chemical properties of these rocks in the earth’scrust has important applications in petrology, sed-imentology, soil mechanics, oil and gas engineeringand other geophysical research areas.Compaction is the process of volume reductionvia pore-water expulsion within sediments due tothe increasing weight of overburden load. The re-quirement of its occurrence is not only the applica-tion of an overburden load but also the expulsion ofpore water. The extent of compaction is strongly1nfluenced by the burial history and the lithologyof sediments. The freshly deposited loosely packedsediments tend to evolve as an open system, towarda closely packed grain framework during the ini-tial stages of burial compaction and this is accom-plished by the processes of grain slippage, rotation,bending and brittle fracturing. Such reorientationprocesses are collectively referred to as mechanicalcompaction , which generally takes place in the first1 - 2 km of burial. After this initial porosity loss,further porosity reduction is accomplished by theprocess of chemical compaction such as pressure so-lution (Fowler and Yang, 1999).Despite the importance of compaction and di-agenesis for geological problems, the literature ofquantitative modelling is not a huge one, thoughthe processes have received much attention in theliterature, and the mechanism leading to pressuresolution is still poorly understood. The effect ofgravitational compaction was reviewed by Hed-berg (1936) who suggested that an interdisciplinarystudy involving soil mechanics, geochemistry, geo-physics and geology is needed for a full understand-ing of the gravitational compaction process. Morecomprehensive and recent reviews on the subjectof compaction of argillaceous sediments were madeby Rieke and Chilingarian (1974) and Fowler andYang (1998).Compaction is a density-driven flow in a porousmedium, a fascinating multidisciplinary topic thathas attracted attention from scientists with differ-ent expertise for a long time. Holzbecher (1998)provides an up-to-date comprehensive review ofthe previous work and state-of-art numerical meth-ods and softwares for modeling density-driven flowand transport in porous media where the constantporosity is used. Here, we will mainly model howporosity changes with time and depth, rather thanusing a constant density; thus an appropriate com-paction relation is vitally important.Nonlinear compaction models have been formu-lated in two ways. The early and most models usedelastic or poroelastic rheology, and the compactionrelation is of Athy’s type p e = p e ( φ ) (Gibson, Eng-land & Hussey, 1967; Smith, 1971; Sharp, 1976;Wangen, 1992; Audet and Fowler, 1992; Fowlerand Yang, 1998). More recent models use a viscousrheology with a compaction relation p e = − ξ ∇ . u s (Angevine and Turcotte, 1983; Birchwood and Tur-cotte, 1994; Fowler and Yang, 1999). The poroe-lastic models are valid for mechanical compactionwhile the viscous models mainly describe the chem-ical compaction such as pressure solutions. Morerecently, efforts have been made to give a more re- alistic visco-elastic model (Revil, 1999; Fowler andYang, 1999). Fowler and Yang (1999) use a viscouslaw p e = − ξ ∇ . u s to model compaction due to pres-sure solution, while Revil (1999) uses a poro-visco-plastic model, with a relationship between poros-ity strain and effective stress, to study pressure so-lution mechanism and its applications. However,there is no viscoelastic model which has been for-mulated to analyse the compaction problem on abasin scale, and most of the conventional studiesare mainly numerical simulations. Obviously morework has yet to be done. This paper aims at provid-ing a unified approach to the compaction relationby using a visco-poroelastic relation of Maxwelltype. The nonlinear partial differential equationsare then analysed by using asymptotic methods andthe analytical solutions are compared with numer-ical simulations. For the convenience of investigating the effectof sediment compaction, we will assume a singlespecies only. The sediments act as a compress-ible porous matrix, so that mass conservation ofpore fluid together with Darcy’s law leads to theequations:Conservation of mass, ∂∂t (1 − φ ) + ∇ · [(1 − φ ) u s ] = 0 , (1) ∂φ∂t + ∇ · ( φ u l ) = 0 , (2)Darcy’s law, φ ( u l − u s ) = − kµ ( ∇ p l + ρ l g j ) , (3)Force balance, ∇ · σ e − ∇ [ p l ] − ρg j = , (4)where u l and u s are the velocities of the fluid andsolid matrix, k and µ are the matrix permeabilityand the liquid viscosity, ρ l and ρ s are the densi-ties of the fluid and solid matrix, σ e is the effec-tive stress, p e is the effective pressure, j is the unitvector pointing vertically upwards, p l is the porepressure, and g is the gravitational acceleration. Inaddition, a rheological compaction law is needed tocomplete this model.2 .1 Poroelasticity and Viscous Com-paction The compaction law is a relationship between ef-fective pressure p e and strain rate ˙ e or porosity φ .The common approach in soil mechanics and sed-iment compaction is to model this generally non-linear behaviour as poroelastic, that is to say, arelationship of Athy’s law of the form p e = p e ( φ ),which is derived by fitting real data for sediments.Athy’s poroelasticity law is also a simplified form ofCritical State Theory (Schofield and Wroth, 1986).A common relation describing poroelasticity is Dp e Dt = − K ∇ · u s , DDt = ∂∂t + u s · ∇ , (5)and equation (12) can be rewritten as11 − φ D (1 − φ ) Dt = −∇ · u s , (6)combining with the previous equation, we have p e = p e ( φ ) , (7)which is Athy’s law for poroelasticity. A typicalform of this constitutive relation (Smith, 1971; Au-det and Fowler, 1992; Fowler and Yang, 1998) is p e = ln( φ /φ ) − ( φ − φ ) . (8)However, this poroelastic compaction law is onlyvalid for sediment compaction in the upper andshallow region, where compaction occurs due topurely mechanical movements such as grain slidingand packing rearrangement. In the deeper region,mechanical compaction is gradually replaced bychemical compaction due to stress-enhanced flowalong grain boundaries from the grain contact ar-eas to the free pore, where the pressure is essen-tially the pore pressure. A typical process of suchchemical compaction in sediment is pressure solu-tion, whose rheological behavior is usually viscous,so that it is sometimes called viscous pressure so-lution or viscous creep.The mathematical formulation of compactionlaws for pressure solution is to derive the creep ratein terms of concentrations, grain size and geometry(usually spherical or cylindrical packings), effectivestress, grain boundary diffusion. This allows us toinclude the detailed reaction-transport process in arelation between strain rate and effective stress, al-though further simplifications are usually assumedsuch as steady-state dissolution and local reprecip-itation along the grain boundary. Rutter’s creep law (1976) is widely used˙ e = A k c wD gb ρ s ¯ d σ, (9)where σ is the effective normal stress across thegrain contacts, A k is a constant, c is the equilib-rium concentration (of quartz) in pore fluid, ρ, ¯ d are the density and (averaged) grain diameter (ofquartz). D gb is the diffusivity of the solute in wateralong grain boundaries with a thickness w . Notethat p e = − σ and ˙ e kk = ∇ · u s . With this, (9)becomes the following compaction law p e = − ξ ∇ . u s . (10)This was first used by Birchwood and Turcotte(1994) to study pressure solution in sedimentarybasins by presenting a unified approach to geopres-suring, low permeability zone formation and sec-ondary porosity generation. Following the discussions of elastic compaction(Fowler and Yang 1998) and viscous compaction(Fowler and Yang, 1999), we can generalise theabove relations to a viscoelastic compaction law ofMaxwell type ∇ . u s = − K Dp e Dt − ξ p e . (11)Equivalently, we would anticipate that a viscoelas-tic rheology holds for the medium, involving mate-rial derivatives of tensors, and some care is neededto ensure that the resulting stress-strain relationbe invariant under the coordinate transformation.This is not always guaranteed due to the com-plexity of the rheological relations (Bird, Arm-strong & Hassager 1977). Fortunately, for one-dimensional flow, which is always irrotational , theequation is invariant and all the different equationsin corotational and codeformational frames degen-erate into the same form (Yang, 1997). In the one-dimensional case we discuss below, we can take thisfor granted. The 1-D model equations become ∂ (1 − φ ) ∂t + ∂∂z [(1 − φ ) u s ] = 0 , (12) ∂φ∂t + ∂ ( φu l ) ∂z = 0 , (13) φ ( u l − u s ) = k ( φ ) µ [ − G ∂p e ∂z − ( ρ s − ρ l )(1 − φ ) g ] , (14) ∂u s ∂z = − K Dp e Dt − ξ p e , DDt = ∂∂t + u s ∂∂z , (15)3here G = 1 + 4 η/ ξ is a constant describing theproperties of the sediments, and η is the viscosity ofthe medium. By assuming that the densities ρ s and ρ l are constants, we can see that only the densitydifference ρ s − ρ l is important to the flow evolution.Thus, the compactional flow is essentially density-driven flow in a porous medium (Holzbecher, 1998). Let the length-scale d is defined by d = { ξ ˙ m s G ( ρ s − ρ l ) g } / , (16)so that the dimensionless pressure is p = Gp e / ( ρ s − ρ l ) gd = O (1). Meanwhile, we scale z by d , u s by˙ m s , t by d/ ˙ m s , permeability k by k . By writing k ( φ ) = k k ∗ , z = dz ∗ , ..., and dropping the aster-isks, we have − ∂φ∂t + ∂∂z [(1 − φ ) u s ] = 0 , (17) ∂φ∂t + ∂ ( φu l ) ∂z = 0 , (18) φ ( u l − u s ) = λk ( φ )[ − ∂p∂z − (1 − φ )] , (19) ∂u s ∂z = − φ (1 − φ ) DpDt − p, DDt = ∂∂t + u s ∂∂z , (20)where λ = k ( ρ s − ρ l ) gµ ˙ m s . (21)In the above derivation, we have used the require-ment that the poroelastic case (8) result in the limitas the viscous rheology vanishes.Adding (17) and (18) and integrating, we have u s = − φ ( u l − u s ) , (22)where u = φ ( u l − u s ) is the Darcy flow velocity. Byusing the equation (6), we have ∂φ∂t = ∂∂z [(1 − φ ) u s ] , (23) u s = λ ( φφ ) m [ − ∂p∂z − (1 − φ )] . (24)1(1 − φ ) DφDt = − φ (1 − φ ) DpDt − p. (25)The constitutive relation for permeability k ( φ ) isnonlinear, and of the form, k ( φ ) = ( φφ ) m , m = 8 . (26) The boundary conditions at z = 0 are ∂p∂z − (1 − φ ) = 0 (or equivalently , u s = 0) , (27) φ = φ , p = 0 , (28)˙ h = ˙ m ( t ) + λ ( φφ ) m [ ∂p∂z − (1 − φ )] at z = h ( t ) , (29)which is a moving boundary problem.We estimate these parameters by using valuestaken from observations. From the typical valuesof ρ l ∼ kg m − , ρ s ∼ . × kg m − , k ∼ − −− − m , µ ∼ − N s m , ξ ∼ × Ns m − , ˙ m s ∼
300 m Ma − = 1 × − m s − , g ∼
10m s − , G ∼ , d ∼ λ ≈ . ∼ λ , which governs theevolution of the fluid flow and porosity in sedimen-tary basins, is the ratio between the permeabilityand the sedimentation rate. Since the nondimensional parameter λ ≈ . ∼ λ ≪ λ ≫ λ = 1 defines a tran-sition between slow compaction ( λ <<
1) and fastcompaction ( λ >> λ ≪ ) If λ ≪ z ∼ t ∼ p ∼
1, then u s ≪ ∂φ∂t ≈
0, and it follows that φ ≈ φ and Dφ/Dt ≈ ∂φ∂t . Thus ∂φ∂t ≈ − λ (1 − φ ) ∂ p∂z , (30) u s ≈ λ [ − ∂p∂z − (1 − φ )] , (31)1(1 − φ ) ∂φ∂t ≈ − φ (1 − φ ) ∂p∂t − p, (32)which can be rewritten approximately as ∂p∂t = λ ′ ∂ p∂z + (1 − φ ) φ p, λ ′ = (1 − φ ) λφ , (33)with the boundary conditions ∂p∂z ≈ − φ , on z = 0 , (34)4 .2 0.25 0.3 0.35 0.4 0.45 0.500.050.10.150.20.25 Porosity Z Figure 1: Comparison of numerical solutions (solidcurves) with asymptotic solution (39) for λ = 0 . t = 3 ,
5. Where Z = z/h ( t ) is the scaled height. p → , z → ∞ , (35)This is in fact equivalent to the case of conductionin a semi-infinite space with a constant flux at z =0. The solution in this case can be approximatelyexpressed as p ≈ (1 − φ ) √ λ ′ t ierfc( ζ )+ p λ ′ φ exp[ − (1 − φ ) z √ λ ′ φ ] , (36)where ζ = z √ λ ′ t , (37)and ierfc( ζ ) = 1 √ π e − ζ − ζ erfc( ζ ) . (38)This gives the approximate solution of φ as φ ≈ φ − φ √ λ ′ t ierfc( ζ ) − φ √ λ ′ φ (1 − φ ) te − (1 − φ z √ λ ′ φ , (39)thus compaction essentially occurs in a boundarylayer near the bottom with a thickness of the orderof √ λ ′ . The comparison of this approximate solu-tion (39) (dashed curves) with numerical solutions(solid curves) is shown in Figure 1 for the values of λ = 0 . , t = 5. The approximate solution is accu-rate when ( φ/φ ) m ≪ t ∼ /m √ λ ∼ φ ( z = 0) = 1 − O ( √ λt ). Theagreement is clearly shown in the figure. λ ≫ ) Either viscous or poroelastic fast compaction, ismore complicated and interesting in contrast tothe simple structure of the boundary layer for slowcompaction. Since λ ≫ k = ( φ/φ ) m , m = 8,the governing equations are also highly nonlinear.However, we use these features and pursue asymp-totic analysis to seek appropriate asymptotic solu-tions. For the case λ ≫
1, we rewrite (25) as ∂φ∂t + λ ( φφ ) m [ − ∂p∂z − (1 − φ )] ∂φ∂z = − (1 − φ ) p − φ (1 − φ ) { ∂p∂t + λ ( φφ ) m [ − ∂p∂z − (1 − φ )] ∂p∂z } , (40)By using the perturbations φ = φ (0) + 1 λ φ (1) + ..., p = p (0) + 1 λ p (1) + ..., (41)the leading order equation becomes ∂φ (0) ∂z = − φ (0) (1 − φ (0) ) ∂p (0) ∂z , (42)whose integration gives p (0) = ln( φ /φ (0) ) − ( φ − φ (0) ) , (43)which is the Athy-type relation and is exactly thesame form as in Smith (1971) and Fowler and Yang(1998). From (24) the equation of leading order isthus (1 − φ (0) ) φ (0) ∂φ (0) ∂z − (1 − φ (0) ) = 0 , (44)or ∂φ (0) ∂z − φ (0) = 0 . (45)The boundary condition φ (0) = φ gives φ (0) = φ e − ( h − z ) , (46)which decreases with depth h − z exponentially.This solution is the same as the equilibrium solu-tion in the poroelastic case, and thus the top regionof viscoelastic compaction is essentially poroelasticand the viscous effect is only of secondary impor-tance in this region. However, as φ decreases, theterm λ ( φ/φ ) m may become very small due to the5arge exponent m = 8. The relation λ ( φ/φ ) m = 1defines a critical value of φ in the transition region φ ∗ = φ e − ln λm . (47)In fact, the above solutions are only valid when φ (0) > φ ∗ and h − z < Π = (ln λ ) /m . Since φ ∼ φ ∗ , we define φ = φ ∗ e ψ − ln mm , z = h − Π + η − ln mm , (48) u s = Wm , p = p ∗ − Pm , (49)where p ∗ = ln( φ /φ ∗ ) − ( φ − φ ∗ ). By changing vari-ables to ( t, η ) via ∂ t → ∂ t − m ˙ h∂ η , ∂ z → m∂ η , andassuming m ≫
1, we have the equation of leadingorder ˙ hφ ∗ ψ η + (1 − φ ∗ ) W η = 0 , (50) W = e ψ [ P η − (1 − φ ∗ )] , (51)˙ hφ ∗ ψ η = φ ∗ ˙ h (1 − φ ∗ ) P η + (1 − φ ∗ ) p ∗ . (52)Thus W = W ∗ − ˙ hφ ∗ (1 − φ ∗ ) ψ, (53) ψ η − − φ ∗ ) p ∗ ˙ hφ ∗ + ( ψ ∞ − ψ ) e − ψ , (54) ψ ∞ = (1 − φ ∗ ) W ∗ ˙ hφ ∗ , (55)whose solution can be written as a quadrature.In the limit η → −∞ , P η →
0, equation (52)shows that the dominant term is the viscous term(1 − φ ∗ ) p ∗ so that compaction will gradually shiftfrom viscoelastic to purely viscous behavior. Thishas important geophysical implication for com-paction in sedimentary basins, since the purely vis-cous mechanism may be responsible for overpres-suring and mineralized seals in oil-reservoir andhydrocarbon basins. Furthermore, In order to de-termine ˙ h , we require (53) and (55) to match thesolution below the transition layer. In the region below the transition layer, the poros-ity φ < φ ∗ is usually very small, while the effectivepressure p is increasing and p ∼ p ∗ = O (1). Rewrit-ing (25) as ∂u s ∂z = − φ (1 − φ ) [ ∂p∂t + u s ∂p∂z ] − p, (56) From (49) and (52), we know that p changes slowly,which implies ∂p∂t ∼ ∂p∂z ≪ φ ( ∂p∂t + u s ∂p∂z ) ≪ p .We then have approximately p ≈ − ∂u s ∂z , (57)which implies that compaction is now essentiallypurely viscous. Thus we get ∂φ∂t = ∂∂z [(1 − φ ) u s ] , (58) u s = λ ( φφ ) m [ ∂ u s ∂z − (1 − φ )] , (59)which are the equations solved by Fowler and Yang(1999) when Ξ = 1 for purely viscous compaction.Following the same solution procedure given byFowler and Yang (1999), we can expect to get thesame solutions. Thus, we only write down here thesolution for ˙ h ˙ h = ˙ m s [ (1 − φ )(1 − φ ∗ ) + φ ∗ ψ ∞ m (1 − φ ∗ ) ] − p ∗ γm ln m, (60)where γ = p ∗ (1 − φ ∗ ) ˙ m s φ ∗ (1 − φ ) . This essentially completesthe solution procedure. Figure 2 shows the compar-ison of numerical results with the above obtainedasymptotic solutions (46) and (55) in the poroelas-tic and transition region. The present model of viscoelastic flow and nonlin-ear compaction in sedimentary basins uses a rheo-logical relation which incorporates both poroelasticand viscous effects in the 1-D compacting frame.Based on the frame invariance of irrotational fea-ture of the 1-D flow, a generalised viscoelastic com-paction relation of Maxwell type has been formu-lated. The nondimensional model equations aremainly controlled by one parameter λ , which is theratio of the hydraulic conductivity to the sedimen-tation rate. Following a similar asymptotic analysisgiven by Fowler and Yang (1998), we have been ableto obtain the approximate solutions for either slowcompaction ( λ ≪
1) or fast compaction ( λ ≫ λ ≫
1, and the solution gives a nearly ex-ponential profile of porosity versus depth, whichimplies that compaction in the top region is es-sentially poroelastic and its profile is virtually atequilibrium.The numerical simulations and asymptotic anal-ysis have shown that porosity-depth profile is6 Z poroelastictransition Figure 2: Numerical results (solid curves) with λ = 100 , t = 10. The asymptotic solutions (46)and (55) are also plotted as a comparison (dashed).Profile in the top region is nearly exponential fol-lows by a transition to pure viscous compactionwhere porosity is nearly uniform.nearly exponential followed by a a transition froma poroelastic to a viscoelastic regime. This is be-cause the large exponent m in the permeability law k = ( φ/φ ) m ; even if λ ≫
1, the product λk maybecome small at sufficiently large depths. In thiscase, the porosity profile consists of an upper partnear the surface where λk ≫ λk ≪
1, andthe porosity is higher than equilibrium, which ap-pears to correspond accurately to numerical com-putations. Below this transition region, porosity isusually uniformly small and compaction is essen-tially purely viscous. From the definition of excesspore pressure p ex = R zh [ p + (1 − φ )] dz , we knowthat the sudden switch from poroelastic to viscouscompaction means a quick decease of porosity φ ,which leads to a sudden increase of p ex . Therefore,the transition is often associated with a jump toa high pore pressure and low permeability regionwhere a mineralized seal may be formed. This con-clusion is consistent with the earlier work (Birch-wood and Turcotte, 1994). As viscous compactionproceeds, porosity and permeability may becomeso small that fluid gets trapped below this region,and compaction virtually stops.Further work shall focus on more realistic andcorrect formulation of rheology. In a recent work on pressure solution and its application to somefield problems such as land subsidence associ-ated with fluid withdrawal from undercompactedaquifers, Revil (1999) suggests a Voigt-type poro-visco-plastic rheological behavior to characterizepressure solution and to applications to some fieldproblems including equilibrium and disequilibriumcompactions and subsidence. Naturally, more workis needed to incorporate a Voigt-type rheology ap-plied to compaction in addition to the presentMaxwell-type law. Acknowledgements : The author thanks the ref-erees for their very helpful comments and very in-structive suggestions. I also would like to thankProf. Andrew C Fowler for his very helpful direc-tion on viscous compaction.
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