Oil & Gas Science and Technology Ð Rev. IFP, Vol. 56 (2001), No. 6, pp. 531-544 Copyright © 2001, Éditions Technip
Poromechanics: From Linear to Nonlinear Poroelasticity and Poroviscoelasticity
E. Bemer1, M. Boutéca1, O. Vincké1, N. Hoteit2 and O. Ozanam2
1 Institut français du pétrole, 1 et 4, avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex - France 2 ANDRA, Parc de la Croix-Blanche, 92298 Châtenay-Malabry Cedex - France e-mail: [email protected] - [email protected] - [email protected] - [email protected] - [email protected]
Résumé — Poromécanique : de la poroélasticité linéaire à la poroélasticité non linéaire et la poroviscoélasticité — Compte tenu des répercussions sur les calculs de productivité et de réserves, une modélisation fiable du comportement des roches est essentielle en ingénierie de réservoir. Cet article étudie plusieurs aspects du comportement poroélastique des roches dans le cadre de la mécanique des milieux poreux saturés définie par Biot. Les lois de comportement de la poroélasticité linéaire et non linéaire sont tout d’abord établies à partir d’une décomposition fondamentale de l’état de contraintes, qui permet de relier de façon claire les modèles linéaire et non linéaire. Les notions de contrainte effective et de compressibilité sont abordées. Une loi de comportement incrémentale linéaire est définie à partir de la loi de comportement non linéaire proposée en introduisant des propriétés élastiques tangentes. Ces coefficients sont par nature exprimés en fonction des déformations et de la pression de pore, mais l’on établit ici des expressions explicites en fonction des contraintes et de la pression de pore. Des essais effectués sur un grès de réservoir illustrent ces différents points. Enfin, une loi de comportement de la poroviscoélasticité est présentée et appliquée à des données expérimentales provenant d’une argile. Mots-clés : mécanique des roches, poroélasticité, poroélasticité non linéaire, propriétés tangentes, poroviscoélasticité, contrainte effective, grès, argile.
Abstract — Poromechanics: From Linear to Nonlinear Poroelasticity and Poroviscoelasticity — Due to the impact on productivity and oil in place estimates, reliable modeling of rock behavior is essential in reservoir engineering. This paper examines several aspects of rock poroelastic behavior within the framework of Biot’s mechanics of fluid saturated porous solids. Constitutive laws of linear and nonlinear poroelasticity are first determined from a fundamental stress decomposition, which allows to clearly connect linear and nonlinear models. Concept of effective stress and rock compressibility are considered. Linear incremental stress-strain relations are derived from the proposed nonlinear constitutive law by defining tangent elastic properties. These characteristics are naturally functions of strains and pore pressure, but explicit expressions as functions of stresses and pore pressure are established herein. Experiments performed on a reservoir sandstone illustrate these points. A constitutive law of poroviscoelasticity is finally presented and applied to experimental data obtained on clay. Keywords: rock mechanics, poroelasticity, nonlinear poroelasticity, tangent properties, poroviscoelasticity, effective stress, sandstone, shale. 532 Oil & Gas Science and Technology Ð Rev. IFP, Vol. 56 (2001), No. 6
NOMENCLATURE γ poroviscoelastic characteristic which connects the relaxed and instantaneous shear moduli,
Latin letters 111=+ GG∞ 0 γ b Biot coefficient η shear viscosity bt tangent Biot coefficient η fl fluid dynamic viscosity κ D nonlinear characteristic o poroviscoelastic characteristic which connects the relaxed F nonlinear characteristic and instantaneous drained bulk moduli, 111 Fo free energy of the theory of elasticity =+ ∞ 0 κ KKooo G shear modulus ρ fl o fluid density Gt tangent shear modulus σ= Cauchy’s stresses H nonlinear potential tr σ σ = I = tr ε = ε +ε + ε mean stress 1 = 11 22 33 3 first strain invariant σÐ σ 1 = = = + pp = effective stress of Terzaghi I = ε ε + ε ε + ε ε Ð ε ε Ð ε ε Ð ε ε 2 11 22 22 33 33 11 12 21 23 32 31 13 ζ bulk viscosity. second strain invariant ε) I3 = det (= third strain invariant Superscripts k intrinsic permeability e elastic part
Kfl fluid bulk modulus v viscous part 0 instantaneous characteristic Ko drained bulk modulus t ∞ relaxed characteristic. K o tangent drained bulk modulus
Ks bulk modulus of the solid matrix m fluid mass variation INTRODUCTION M Biot modulus Predicted production and restitution rates rely on the accurate p =Ðσ mean pressure c representation of reservoir poromechanical behavior. Most pp pore pressure engineering reservoir models use rock compressibility as a 3 constant mechanical key parameter. Yet, various meanings q = ss: with s=−σσ1 deviatoric stress are still attached to this concept. Moreover rock properties 2 can be stress-dependent and/or time-dependent. This paper Vb bulk volume studies rock poroelastic behavior within the framework of Biot’s mechanics of fluid saturated porous solids. Vp pore volume w relative flow vector of fluid mass. Constitutive laws of linear and nonlinear poroelasticity are first determined from a fundamental stress decomposition, which allows to clearly connect linear and nonlinear models. Greek letters The physical meaning of Biot’s and Terzaghi’s effective β coefficient of proportionality between the viscous stresses is clarified. The definition of formation compressi- porosity and the volumetric viscous strain, φv = βεv bility and expressions of Zimmerman’s compressibilities in terms of linear poroelastic properties are recalled. In order to ε = strains derive linear incremental stress-strain relations from the ε = tr ε= volumetric strain proposed nonlinear constitutive law, tangent elastic properties depending on strains and pore pressure are defined p p εε=+ 1 strains produced by σσ=+p 1 and expressed as explicit functions of stresses and pore 3K p s pressure. Experiments performed on reservoir sandstone 2 illustrate these points. A constitutive law of porovisco- εε==−ee: with e ε1 deviatoric strain d 3 elasticity is then presented and applied to experimental data obtained on clay. V φ= p eulerian porosity The models proposed in this paper are derived for an Vb isotropic homogeneous porous material, saturated with a E Bemer et al. / Poromechanics: From Linear to Nonlinear Poroelasticity and Poroviscoelasticity 533 viscous compressible fluid, under the hypothesis of small Hence, both in linear and nonlinear poroelasticity, the perturbations (Coussy, 1994; Charlez, 1991). Only isothermal solid matrix is supposed isotropic and linearly elastic with transformations are considered. bulk modulus Ks and shear modulus Gs. Under this assumption, elementary loading state [S] leads to a deformation state: 1 FUNDAMENTAL STRESS DECOMPOSITION []S p p ε =− 1 (2) Consider a representative elementary volume of a saturated 3Ks bulk porous material submitted to stresses σ and pore Indeed, loading [S] involves no change of the (eulerian) = []S []S ε porosity. Hence εε= , where s denotes the strain pressure pp. The stress decomposition defined in Figure1 s allows to derive the constitutive law from the behavior of the of the solid matrix. dry material (i.e. without any fluid in the porous space) and Thus, to the stress decomposition defined in Figure 1 the behavior of the matrix (solid and unconnected porosity) corresponds the following strain decomposition: (Biot, 1973). The total loading state [T] is simply described p p by the superposition of two elementary loading states: εε=− 1 3K (3) [B] (for bulk) stresses σÐ and zero pore pressure; s = ε σÐ 1 where denotes the strains produced by =. [S] (for solid matrix) stresses Ð pp = and pore pressure pp σσ=−p 1 p 2 LINEAR POROELASTICITY =+ (1) pppp0 Ð where σ= amounts to the effective stress of Terzaghi. Note 2.1 Mechanical Part of the Linear Constitutive Law that the stress convention of the mechanics of continuous and Biot’s Effective Stress media is adopted. Hence Ð p 1 denotes a compressive stress. p = In addition to the assumption of isotropic and linear elastic Component [B] involves no fluid pressure. Hence, from a behavior of the solid matrix, the law of linear poroelasticity mechanical point of view, it corresponds to the loading of the relies on the following assumption: the equivalent dry dry material by a stress field σσ=+p 1, whereas p material is isotropic and linearly elastic with bulk modulus K component [S] corresponds to the hydrostatic loading of the o and shear modulus G. Deformation state ε is then given by: solid matrix with pressure pp. Van der Knapp brought out that elastic porous media 2G σεε=− K 12+ G submitted to isotropic loading exhibit a partial linear o 3 (4) behavior with regard to the solid matrix compressibility (Van der Knapp, 1959). Biot extended this work to the general Let σσ=+ case by introducing the concept of semilinearity (Biot, 1973), ' bpp 1 which stipulates that the solid matrix strains depend linearly with: on stresses and pore pressure, whereas the strains due to the effective stresses involve nonlinear modifications of local Ko b =−1 (5) geometries, such as changes in contact areas, crack closure... Ks
- pp 1 σ σ (compressive stress)
pp 0 pp pp =+0 pp
pp 0 pp
[T] [B] [S]
Figure 1 Fundamental stress decomposition (Note that, according to the stress convention of the mechanics of continuous media, compressive stresses are negative while pressures are positive). 534 Oil & Gas Science and Technology Ð Rev. IFP, Vol. 56 (2001), No. 6
The mechanical part of the linear constitutive law is ()− φσ= σ zero, we have 1 os . Taking Equation (5) into obtained by introducing Equations (1), (3) and (5), into account, Equation (11) becomes: Equation (4): m[]B b ε= 2G ρ fl (12) σεε=− K 12+−Gbp 1 (6) o op3 Equations (10) and (12) lead to: 2G σεε' =− K 12+ G m 11 o (7) =+b εφ − p (13) 3 ρ fl o p o KKfl s σσ=+ The hydraulic part of the linear constitutive law is b is the well-known Biot coefficient and ' bpp 1 is the effective stress of linear elasticity. obtained by introducing Equation (3) into Equation (13): m p Note the difference between Equations (4) and (7): Biot’s =+b ε p σ' ε ρ fl M (14) effective stress is the stress that produces total strains =, o σ while stress that amounts to Terzaghi’s effective stress where M is Biot modulus: only produces strains ε . For an incompressible solid matrix − φφ σσ' = 1 b o o (i.e. 1/Ks = 0), we simply have . Boutéca and = + (15) Guéguen (1999) clearly show the different roles played by M KKs fl Terzaghi’s and Biot’s effective stresses from experiments performed on Tavel limestone for which b = 0.65. 2.3 Rock Compressibility Formation compressibility is defined as the in situ pore 2.2 Hydraulic Part of the Linear Constitutive Law volume strains that follows changes in reservoir pore pressure: The variation in fluid mass content m per unit of initial 1 ∆V volume (taken positive for an incoming fluid) can be C = p formation ∆ expressed as (Coussy, 1994): Vp p p Thus, formation compressibility depends on the produc- =+()ερfl φ− ρfl φ (8) m 1 o o tion induced changes in effective stress state. These changes are usually described by the reservoir stress path, defined as where φ is the (eulerian) porosity, ε = tr ε and subscript o = the change in effective minimum horizontal stress divided by denotes the reference state. the change in effective vertical stress from initial reservoir After linearization, we get: conditions. m ρρfl − fl Zimmerman’s compressibilities describe the volumetric = φ o ++−φε φ φ (9) ρ fl o ρ fl oo response of a sandstone to applied isotropic stresses o o σ=− pc 1 and pore pressure pp (Zimmerman, 1991): Loading [S] leads to no porosity change. Taking Equation 11∂V ∂V (2) into account, Equation (9) leads to: C =− b C =− p bc ∂ pc ∂ Vb pc p Vp pc p []S ρρfl − fl p pp m = φ o −=φφp 11 − o o o p p (10) ∂ ∂V ρ fl ρ fl 11Vb p o o KKKs fl s C = C = bp ∂ pp ∂ Vb p p Vp p p pccp where Kfl is the fluid bulk modulus. As loading [B] keeps pore pressure constant, m[B] only The so-defined porous rock compressibilities can be expressed as a function of linear poroelastic properties K results from the change in pore volume. Taking the relation o and b (Boutéca, 1992). ()11− φε=−() φε+− φ φinto account, we obtain: ooso Equation (6) yields:
[]B p b m =−−εφε() (11) ε=−c + p 1 os p ρ fl KooK o From which we get: The assumptions of linear behavior of the dry material and εσ= εσ= 1 b the solid matrix yield:Ko and sssK , where C ==and C σ bc bp s is the mean stress in the solid matrix. As pore pressure is Ko Ko E Bemer et al. / Poromechanics: From Linear to Nonlinear Poroelasticity and Poroviscoelasticity 535
Expressions of Cpc and Cpp are derived from the change in where Ko is the drained bulk modulus, G is the shear pore volume: modulus and D, F, N, are nonlinear parameters. Introducing Equations (17) and (18) into Equation (16) ∆V φφ− ()b − φε p = o +=ε o + ε leads to the nonlinear stress-strain relation associated with φ φ Vp o o loading [B], which is to be compared to Equation (4): Taking Equations (4), (1) and (6) into account, we finally 2G ∂H σεε=− K 12++G obtain: o 3 ∂ε (20) ∆V b bb−−φ ()1 p =−p + o p The mechanical part of the nonlinear constitutive law is φ c φ p VpooK ooK then given by: From which we get: 2G ∂H σεε=− K 12+−Gbp 1 + op ∂ε (21) b C bb−−φ ()1 3 C ==bp and C =o pc φφ pp φ ooK o ooK and is to be compared to Equation (6). Note that for an incompressible solid matrix (i.e. b = 1), 3.2 Hydraulic Part of the Nonlinear Constitutive Law we simply have: 11Due to the assumption of linear behavior of the solid matrix CC==and CC == bc bp K pc pp φ K (see Paragraph 1), elementary variation in fluid mass content o oo m[S] is still given by Equation (10): The actual stress path followed by the reservoir can be m[]S 11 significantly different from the one predicted by the isotropic =−φ p (22) ρ fl o p loading or uniaxial strain boundary models (Bévillon, 2000). o KKfl s Reservoir simulations conducted for weakly cemented The expression of elementary variation in fluid mass sandstone show a significant increase in production rate content m[B] includes an additional nonlinear term. Indeed, when changing the formation compressibility from a high the introduction of Equation (20) into Equation (11) yields: stress path value to a low value (Ruisten et al., 1996). Moreover, for highly compacting reservoir, the bulk com- m[]B 1 ∂H =−b ε tr ρ fl ∂ε (23) pressibility depends on stresses and pore pressure (see o 3Ks Paragraph 3). The concept of rock compressibility is thus to be handled cautiously. Equations (22) and (23) lead to Equation (24), which is to be compared to Equation (13):
m 11 1∂H 3 NONLINEAR POROELASTICITY =+b εφ − p − tr (24) ρ fl o p ∂ε o KKfl s 3Ks 3.1 Mechanical Part of the Nonlinear Constitutive Law The hydraulic part of the nonlinear constitutive law is From a mechanical point of view, loading [B] corresponds to obtained by introducing Equation (3) into Equation (24): the loading of the dry material. Stresses σ and strains ε are thence related through the free energy F of the theory of m p p 1 ∂H o =+b ε − tr elasticity: ρ fl ∂ε (25) o MK3 s ∂F σ = o and is to be compared to Equation (14). ∂ε (16) where Fo is a function of the three strain invariants: 3.3 Additional Hypothesis ==εε III123tr and, . Biot adopted an expression derived from the second-order elasticity theory (Biot, 1963). The free We make herein the reasonable assumption that there is no effect of pure distortion on the variation in fluid mass energy Fo is then the sum of the classical potential of linear elasticity W and a nonlinear potential H: content. Introducing Equation (19) into Equation (24) yields: =+ FWHo (17) m 11 =+b εφ − p ρ fl o KK p 1 4G o fl s WK=+ ε 2 − 2GI (18) 2 o 3 2 1 DN+ 2DN− − + 2F ε 2 + R =−+()εε3 +−() ε + HD33 I23 I FI 2 I 3 NI 3(19) 33Ks 6 536 Oil & Gas Science and Technology Ð Rev. IFP, Vol. 56 (2001), No. 6
=−=−εε2 2 where the invariant RI26262 I2 accounts 3.4 Definition of Tangent Properties for the distortion since it becomes zero for εε= 1 . Hence, the above hypothesis implies: N = 2D. Nonlinear elastic behavior is often handled by deriving linear The nonlinear constitutive law is then written: incremental stress-strain relations and introducing tangent elastic properties depending on the actual state (see Eq. (29) 2G ∂H σεε=− K 12++G and Fig. 2). o 3 ∂ε (26a) 3.4.1 Tangent Drained Bulk Modulus and Tangent Biot or Coefficient 2G ∂H σεε=− K 12+−Gbp 1 + Differentiating Equations (28a) and (27) yields: op ∂ε (26b) 3 + σε=+DF2 ε dK o 2 d (29) 3 m DF+ 2 11 =−b εεφ2 +− p (27) ρ fl 3KKKo p dm DF+ 2 11 o s fl s =−b 2 εεφ d +− dp (30) ρ fl 3K o KK p where o s fl s ε 3 Equations (29) and (30) allow to derive explicit expressions HD=+−() FDI()ε − 3 I 3 23 of the tangent drained bulk modulus and the tangent Biot coefficient as functions of the volumetric strain ε : When considering the principal directions, Equation (26a) gives: DF+ 2 KKt =+2 ε (31) o o 3 DF+ 2 σε=+ ε2 Ko (28a) 3 DF+ 2 bbt =−2 ε (32) 3K 2G s σεε=− K + 2G (28b) 11 o 3 11 Note that we have: 2 2 2 ++DF()εεεεεεεε22++++[]() t 11 22 33 22 33 11 22 33 Ko bt =−1 (33) σσ−= Ks 22 11 (28c) ()εε− −−() εε2 −−2 () εεε − 22GFD22 11 []22 11 22 11 33 3.4.2 Tangent Shear Modulus for Axisymmetric Loading Deriving an explicit expression of the tangent shear modulus Other principal stresses are simply derived by cyclic is a more complex issue, which involves the deviatoric stress permutation. ε q and the deviatoric strain d: =−()σσ2 +−() σσ2 +−() σσ2 q []11 22 22 33 33 11 2/ (34a)
1 εεεεεεε=−2 []()2 +−()2 +−()2 d 3 11 22 22 33 33 11 (34b)
t Ð ε εÐ K 0 Note that we have q = q and d = d. However, a simple expression of Gt can be obtained under σ the assumption of axisymmetric loading, which remains a rather general case. Taking for example axis 1 as axis of ε ε σ σ symmetry (i.e. 22 = 33) and assuming that | 11| > | 22|, Equations (34a) and (34b) read: 2 q =−σσand ε =() εε − 22 11 d 3 22 11 ε Equation (28c) gives then: Figure 2 9 qG=+−3 εε() FD2 (35) Definition of a tangent drained bulk modulus. dd4 E Bemer et al. / Poromechanics: From Linear to Nonlinear Poroelasticity and Poroviscoelasticity 537
−=−σ Differentiating Equation (35) finally yields: ppcp tr is an “effective stress” in that it relates the FD− drained bulk modulus and Biot coefficient to stresses and dq=+33 G εε d pore pressure (Boutéca and Guéguen, 1999). 2 dd (36) which allows to derive an explicit expression of the tangent 3.5.2 Expression of Gt shear modulus as a function of the deviatoric strain ε : d In the case of an axisymmetric loading, Equation (37) defines FD− the tangent shear modulus as a function of ε . As ε = 0 when t =+ ε d d GG3 d (37) ε 2 q = 0, solving Equation (35) for d gives: −+GG2 +() FDq − 3.4.3 Linear Incremental Constitutive Law ε = 2 (42) for Axisymmetric Loading d 3 FD− A simple calculation shows that under the assumption of axisymmetric loading, the nonlinear constitutive law can be Introducing Equation (42) into Equation (37) yields: written in the following linear incremental form: Gqt ()=+− G2 () F Dq (43) 2G t σεε=−t + t dK o dGd12 (38a) 3 It follows from Equation (43) that F Ð D > 0.
dm 11 =+bdt εφ − dp 3.6 Application to Experimental Results ρ fl o p (38b) o KKfl s Cyclic loading was performed on a 19.8% porosity reservoir t t t where tangent properties Ko, G and b , are given by Equa- sandstone, which contains mainly quartz (70%), the tions (31), (32) and (37). remaining part including feldspar, mica and clay minerals. The experimental setup consists of a pressure cell and two 3.5 Expression of Tangent Properties as Functions volumetric pumps (Laurent et al., 1993). The sample is of Stresses and Pore Pressure coated with impermeable materials, which allows application of pore pressure and isotropic confining pressure. 3.5.1 Expression of Kt and bt Nine experiments were conducted on the same sample. o Figures 3a and 3b show the applied stress paths (Boutéca et Equation (31) defines the tangent drained bulk modulus as a al., 1998). Experiment 1 is a loading cycle. The loading Ð Ð Ð function of ε. As ε = 0 when σ = 0 , solving Equation (28a) procedure consists of an increase of the confining pressure at Ð for ε gives: constant pore pressure, followed by a loading sequence DF+ 2 repeated several times: confining pressure is first maintained −+KK2 +4 σ constant while pore pressure is increased, then confining oo 3 ε = pressure is increased at constant pore pressure. The DF+ 2 (39) 2 unloading procedure is performed along the same stress path. 3 Experiments 2 to 5 are loading cycles conducted at constant Introducing Equation (39) into Equation (31) yields: pore pressure, loading and unloading following the same stress path. Experiment 6 is a loading cycle similar to cycle 1. DF+ 2 In experiment 7, confining and pore pressures are increased KKt ()σσ=+2 4 o o 3 following the same load path. In experiment 8, pore pressure σ is decreased while confining pressure is maintained constant. Denoting by pc the mean pressure (i.e. pc = Ð ) and taking Equation (1) into account, we finally get: And, in experiment 9, confining pressure is decreased at constant pore pressure. Stress-strain curve corresponding to experiment 1 is DF+ 2 (40) Kpt ()− p=− K2 4 ()pp− plotted in Figure 4. Despite a slight hysteresis, the reservoir o cp o 3 cp sandstone shows a nonlinear elastic behavior. It follows from Equation (40) that D + 2F < 0. Introducing Equation (40) into Equation (33) and taking 3.6.1 Checking of the Semilinearity Assumption Equation (5) into account give: According to the definition of semilinearity, the solid matrix strains depend linearly on stresses and pore pressure. DF+ 2 bpt ()− p =−11() −b2 − 4 ()pp− (41) Experiment 7 allows to check this hypothesis. The stress path cp 2 cp 3 Ks followed in this experiment involves no effective stress σÐ. 538 Oil & Gas Science and Technology Ð Rev. IFP, Vol. 56 (2001), No. 6
60 1.5 Experiment 1 50
40 1
30 Experiments 2, 3, 4 and 5 20 0.5 Pore pressure (MPa) Pore pressure (MPa) 10
0 0 0820 40 60 00 20 40 60 Confining pressure (MPa) Confining pressure (MPa)
Figure 3a Stress paths of experiments 1 to 5.
60 120
50 100
40 80 Experiment 7
30 60
20 40 Experiment 8 Pore pressure (MPa) Pore pressure (MPa) 10 20 Experiment 9 Experiment 6 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Confining pressure (MPa) Confining pressure (MPa)
Figure 3b Stress paths of experiments 6 to 9.
70 120 pp = 55 MPa 60 100 50 pp = 37 MPa 80 40 (MPa)
p
p 60 (MPa) c 30 p pp = 19 MPa = ∆
c 20 p 40 ∆ Experiment 7 10 pp = 1 MPa 20 Experiment 1 Semilinear model 0 0 -0.006 -0.005 -0.004 -0.003 -0.002 -0.001 0 -0.004 -0.003 -0.002 -0.001 0 ∆ε ∆ε
Figure 4 Figure 5 Experiment 1. Stress-strain curve (despite a slight hysteresis Simulation of experiment 7. Stress-strain curve (checking of the reservoir sandstone shows a nonlinear elastic behavior). the semilinearity assumption: linear behavior of the matrix). E Bemer et al. / Poromechanics: From Linear to Nonlinear Poroelasticity and Poroviscoelasticity 539
∆ε ∆ According to Equation (2), should then be a linear where m = m Ð mo. ∆ function of pp: Taking Equations (31) and (32) into account, we finally ∆ get: p p ∆ε=− (44) K DF+ 2 s ∆∆∆=−t o − o εε− ()2 pKppco()c p (47a) ∆ε ε ε ∆ o 3 where = Ð o and pp = pp Ð pp (index o denoting the initial state). Figure 5 shows that it is actually the case. One can then ∆m DF+ 2 =−bpt ()o po ∆∆εε− ()2 (47b) consider that Biot’s semilinear theory should apply. Equa- ρ fl c p o 3Ks tion (44) gives: Ks = 33 505 MPa.
3.6.2 Simulation of Loading Conducted t ()o − o t o − o whereKpo c pp and bp()c pp are given by Equa- at Constant Pore Pressure tions (40) and (41). o Loading conducted at constant pore pressure pp = pp allows a Equations (47a) and (47b) depend on three material simple identification of the semilinear model described by characteristics: Ko, (D + 2F)/3 and Ks. Parameters Ko and Equations (1), (3), (27) and (28). The volumetric strain (D + 2F)/3 are derived from Equation (47a) and experiment 9, variations are then directly related to the mean effective whereas Ks has been derived from experiment 7 (see Table 1). σÐ o stress = Ð(pc Ð pp). So that loading and unloading stages The data of experiment 9 are plotted in Figures 6 and 7 may be compared, the strain variations are calculated with together with the theoretical results computed from respect to the strain corresponding to the lowest confining Equations (47a) and (47b). These curves show that, in pressure (this reference state being denoted by index o). experiment 9, the rock is actually behaving as a semilinear Writing Equation (28a) for actual and reference states and porous material according to Biot’s theory. subtracting the two obtained equations yield: DF+ 2 DF+ 2 ∆∆pK=− +2 εε − ()∆ ε2 TABLE 1 co 3 o 3 (45) Material characteristics derived from experiments 7 and 9 ∆ o ∆ε ε ε Ðε Ðε where pc = pc Ð pc and = Ð o = Ð o. Parameter Value Experimental data Similarly, Equation (27) leads to: Ko 920 MPa Experiment 9 ∆m DF+ 2 DF+ 2 =−b 2 εε ∆∆− () ε2 Ð(D + 2F)/3 301 350 MPa Experiment 9 ρ fl o (46) o 3Ks 3Ks Ks 33 505 MPa Experiment 7
140 0
120 Experiment 9 Experiment 9 Semilinear model -0.003 Semilinear model 100
80 fl -0.006 o / ρ (MPa) c 60 m p ∆
∆ -0.009 40 -0.012 20
0 -0.015 -0.02 -0.015 -0.01 -0.005 0 -0.02 -0.015 -0.01 -0.005 0 ∆ε ∆ε
Figure 6 Figure 7 Simulation of experiment 9. Stress-strain relation (checking Simulation of experiment 9. Fluid mass variation as a of the semilinear behavior of the rock using the first function of strain (checking of the semilinear behavior of the constitutive equation). rock using the second constitutive equation). 540 Oil & Gas Science and Technology Ð Rev. IFP, Vol. 56 (2001), No. 6
3.6.3 Effect of Cycling 12 000 In laboratory experiments, an evolution of the measured rock properties is often observed when the sample undergoes 10 000 cyclic solicitations. Cycling finally ends up in a non variable 8000 rock sample, from which accurate and reproducible measure- ments can be obtained. 6000 (MPa) t Figure 8 shows the data of experiments 1, 6 and 9, plotted o K together with the semilinear model identified from experi- 4000 ment 9. The experimental curves appear to gradually move Experiment 9 2000 towards the theoretical trend derived from experiment 9. We Semilinear model believe that the behavior exhibited in experiment 9 is the 0 intrinsic behavior of the rock and that this intrinsic behavior 0 20 6040 80 100 120 was obtained resorting to cycling (Boutéca et al., 1998). pc - pp (MPa)
Figure 10 Variation law of the tangent drained bulk modulus. 140
120 Experiment 1 Experiment 6 100 Experiment 9 Semilinear model 3.6.4 Tangent Bulk Modulus 80
(MPa) Experimental data confirm that the tangent bulk modulus c 60 p
∆ varies according to the pressure difference pcÐpp as indicated 40 by Equation (40) (Boutéca et al., 1994). This is illustrated in t Figure 9, where estimations of Ko computed from experi- 20 ment 6 are plotted as a function of the confining pressure for
0 different pore pressure values and as a function of pcÐpp. -0.02 -0.015 -0.01 -0.005 0 Figure 10 shows the variation law of the tangent drained ∆ε bulk modulus derived from Equation (40) and which Figure 8 represents the rock intrinsic behavior (see Paragraph 3.6.3). The estimations of Kt computed from experiment 9 have Stress-strain relations for experiments 1, 6 and 9 (effect of o cycling). been plotted for comparison.
10 000 10 000
8000 8000
6000 6000 (MPa) (MPa) t t o 4000 o 4000 K K
2000 pp = 1 MPa 2000 pp = 1 MPa pp = 51 MPa pp = 51 MPa 0 0 02550 75 100 125 02040 60 pc (MPa) pc - pp (MPa)
Figure 9
Tangent drained bulk modulus as a function of pc-pp (experiment 6). E Bemer et al. / Poromechanics: From Linear to Nonlinear Poroelasticity and Poroviscoelasticity 541
4 POROVISCOELASTICITY 4.2 Instantaneous and Relaxed Characteristics
Further details on poroviscoelastic modeling can be found in Consider an experiment in which the following stress and Coussy (1994). fluid pressure history is imposed:
σσ−=o ∆ σY pp−=∆ pY 4.1 Constitutive Law o
Actual strain and variation in fluid mass content are defined where Y(t) is the Heaviside step function that reads Y(t) = 0 if with respect to a reference relaxed state (i.e. a state of t < 0 and Y(t) = 1 if t ≥ 0. thermodynamic equilibrium where no evolution of the state Let ε 0 and m0 be the instantaneous response. When variables occur). They can be decomposed in elastic parts considering finite steps∆σ and ∆p, Equation (51) shows that (superscript e), immediately recovered through unloading, the viscous strain rate can not be infinite and thus that the and viscous parts (superscript v), not instantaneously viscous strain is continuous. Hence the instantaneous viscous recovered through unloading: strain is zero and the instantaneous response is purely elastic. mme Equations (50) yield then: εε=+ev ε =+φv ρρfl fl (48) o o 2 G 0 ∆∆σεε=−0 0000+− φv v ρfl Ko 12Gbp1 where = m / o appears as the viscous porosity. Note that 3 φv is a lagrangian porosity (i.e. defined with respect to the 2 G 0 m0 initial volume). ∆σεε=− K 0 0000012+−GbM1 3 ρ fl (52) Adopting a second-order expression for the free energy o 0 leads to Equation (49) and constitutive law (50), where K , m0 K0, G0, b0 and M0, are the material instantaneous ∆pM=−+000 bε o ρ fl characteristics (with respect to viscous phenomena and not o hydraulic diffusion). 0 0 0 0 0 Characteristics K , Ko, G , b and M , can therefore φβεvv= (49) actually be interpreted as instantaneous material properties. Let ε ∞ and m∞ be the asymptotic response (i.e. obtained 2 G 0 σ=+ σo K 0 − ()εε− vv12+−G 0 () εε after an infinite time). This response corresponds to a new o 3 (50a) relaxed thermodynamic state, for which strain rates are zero. −−0 () Equations (50) and (51) lead then to: bp po 1 2 G ∞ ∆∆σεε=− ∞ ∞∞∞∞+− 2 G 0 Ko 12Gbp1 σ=+ σovv K 0 − ()εε− 12+−G 0 () εε 3 3 2 G ∞ m∞ (50b) ∆σεε=− K ∞ ∞∞∞∞∞12+−GbM1(53) m 3 ρ fl −−bM00 βε v 1 o ρ fl o m∞ ∆pM=−+∞∞∞ bε ρ fl o m =+00 −()εε −v +−βε v ppo M b fl (50c) ρ ∞ ∞ ∞ ∞ o 2 with K = K o + M (b ) When adopting a second-order expression for the viscous dissipative potential as well, we get the following comple- 111111=+ =+ ∞∞00κγ mentary evolution law: KKooo GG (54) 2 γ ∞ 0 β ()ββ− ∞ ()− o σβ+=++− σo β κ εγεvv+ b b 11 bb pp11oo 12 =+ = + 3 ∞∞00κ κ (51) KoooKMM o 2 η +−ζ εηεúúvv+ 12 3 ∞ ∞ ∞, ∞ ∞ Characteristics K , K o , G b and M are the material where η is the shear viscosity and ζ the bulk viscosity. relaxed properties. Equations (54) show that parameters β, κ Equations (50) and (51) define a Zener material. and γ , connect instantaneous and relaxed characteristics. 542 Oil & Gas Science and Technology Ð Rev. IFP, Vol. 56 (2001), No. 6
4.3 Application to Experimental Results m pp−= M00 − b()εε −v +− βεv (58c) oz zρ fl z Uniaxial strain tests including drained and undrained loading o cycles were performed on a shale. Undrained unloading allows a simple application of Zener model. 4 γ 4 η σσβ−+o ()pp − =+ κ εζv ++ εú v (59) Consider an initial equilibrium state defined by: zz oo 3 z 3 z
σσ==∀
2.4 0.04
Upstream Downstream 2.3 0.03
2.2 Pore pressure equilibrium 0.02 Axial strain equilibrium Axial strain (%) 2.1 0.01 Pore pressure (MPa)
2 0 0510 15 20 0510 15 20 Time (h) Time (h)
Figure 11 Illustration of the almost instantaneous nature of the pore pressure variation in undrained condition.
which gives: 0.05 ∞ −C t −=∞∞ −−() −0 C 0 τ ppppoo ppe v (62) 0.04 where 0.03 Axial strain equilibrium 0 p Ð po = p (t = 0) Ð po is the instantaneous pressure variation, 0.02
B∞ Axial strain (%) pp∞ −=−∆σ o C ∞ 0.01 Zener's model is the asymptotic pressure variation. Experimental data The differential equations that govern the evolution of 0 01020 3040 pore pressure as a function of time is then: Time (h) dε ()B∞ 2 ετ+=−z A∞ ∆σ Figure 12 zvdt C ∞ (63) Simulation of the axial strain evolution during an undrained ∞ unloading (checking of Zener proviscoelastic model). 0 ∞ −C t ∞ B B ∞ 0 τ +−C ()ppe− 0 C v C 0 C ∞
Experimental data show that the pore pressure equilibrium Experimental data showing the axial strain evolution time is small when compared to the strain equilibrium time during an undrained unloading are plotted in Figure 12 (see Fig. 11). Thus, the variation in pore pressure can be together with the theoretical results computed from Equation considered instantaneous with respect to the strain evolution, (64). Zener poroviscoelastic model appears to represent which amounts to disregard the expressions including the satisfactorily shale time-dependent behavior. exponential term in Equations (62) and (63). We finally get: −≅∞ − ppoo p p CONCLUSION − t ∞ τ εε≅+00() εε −1 − e v (64) zz zz Reliable modeling of rock behavior is essential in reservoir engineering due to the impact on productivity and oil in place where estimates. This paper has attempted to review the main εε0 == is the instantaneous axial strain, zz()t 0 poroelastic models. Constitutive laws for linear and nonlinear poroelasticity, ()B∞ 2 εσ∞∞=− ∆ is the asymptotic axial strain. and poroviscoelasticity, have been presented and illustrated z A ∞ C by experimental applications. The concepts of effective stress 544 Oil & Gas Science and Technology Ð Rev. IFP, Vol. 56 (2001), No. 6 and compressibility, and their various meanings, have been Boutéca, M.J. (1992) Elements of Poroelasticity for Reservoir considered. Explicit expressions of tangent properties as Engineering. Rev. IFP, 47, 4, 479-490. functions of stresses and pore pressure have been derived for Boutéca M.J., Bary D., Piau, J.M., Kessler, N., Boisson, M., Fourmaintraux, D. (1994) Contribution of Poroelasticity to rocks presenting a nonlinear behavior. Reservoir Engineering: Lab Experiments, Application to Core We hope that this document provides the fundamental Decompression and Implication in HP-HT Reservoirs Depletion. theoretical laws needed to represent the poroelastic behavior Paper SPE/ISRM 28093. of most reservoir and cap rocks. Boutéca, M.J., Bary, D., Fourmaintraux D. (1998) Does the Seasoning Procedure Lead to Intrinsic Properties? Paper SPE/ISRM 47264. Boutéca, M.J., Guéguen, Y. (1999) Mechanical Properties of ACKNOWLEDGEMENTS Rocks: Pore pressure and Scale Effects, Oil & Gas Science and Technology Ð Rev. IFP, 54, 6, 703-714. The authors would like to thank Schlumberger for supporting Charlez, Ph. A. (1991) Rock Mechanics - Theoretical Funda- the theoretical research program on sandstone nonlinear mentals, Volume 1, Éditions Technip, Paris. behavior and the ANDRA for supporting the research Coussy, O. (1994) Mechanics of Porous Continua, John Wiley program on shale time-dependent behavior. and Sons Ltd., Chichester. Laurent, J., Boutéca, M.J., Sarda, J.P., Bary, D. (1993) Pore Pressure Influence in the Poroelastic Behavior of Rocks: Experimental Studies and Results. SPE Formation, Evaluation, REFERENCES Transactions, 117-122. Bévillon, D. (2000) Couplage d’un modèle de gisement et d’un Ruisten, H., Teufel, L.W., Rhett, D. (1996) Influence of modèle mécanique – Application à l’étude de la compaction des Reservoir Stress Path on Deformation and Permeability of réservoirs pétroliers et de la subsidence associée. PhD Thesis, Weakly Cemented Sandstone Reservoirs. Paper SPE 36535. université des Sciences et Technologies de Lille. Van der Knaap, W. (1959) Nonlinear Behavior of Elastic Porous Biot, M.A. (1963) Incremental Elastic Coefficients of an Isotropic Media. Pet. Trans. AIME 216, 179-187. Medium in Finite Strain. Appl. Sci. Res., 12, Section A, 151-167. Zimmerman, R. W. (1991) Compressibility of Sandstones, Biot, M.A. (1973) Nonlinear and Semilinear Rheology of Porous Elsevier Science Publishers BV, New York. Solids. J. Geoph. Res., 78, 23, 4924-4937. Final manuscript received in October 2001