Numerical identification of a viscoelastic substitute model for heterogeneous poroelastic media by a reduced order homogenization approach R. Jänicke, F. Larsson, K. Runesson, H. Steeb Comput. Methods Appl. Mech. Engrg. 298, 108-120, 2016 Ruhr-Universität Bochum Mechanics – Continuum Mechanics Universitätsstraße 150 D-44780 Bochum Phone: Fax: E-Mail: +49 (0)234 32 22485 +49 (0)234 32 14229 ralf.jaenicke@rub.de Mechanics – Continuum Mechanics Ruhr-Universität Bochum Alle Rechte vorbehalten, insbesondere das Recht auf Übersetzung, Vervielfältigung sowie Verbreitung. Ohne Genehmigung des Autors ist es nicht gestattet, diesen Bericht ganz oder teilweise zu kommerziellen Zwecken zu vervielfältigen oder zu verbreiten. Mechanics – Continuum Mechanics Ruhr-Universität Bochum Numerical identification of a viscoelastic substitute model for heterogeneous poroelastic media by a reduced order homogenization approach Ralf Jänicke1 , Fredrik Larsson2 , Kenneth Runesson2 and Holger Steeb1 1 2 Institute of Mechanics, Ruhr-University Bochum, D-44780 Bochum, Germany Department of Applied Mechanics, Chalmers University of Technology, S-41296 Gothenburg, Sweden Address all correspondence to Ralf Jänicke, E-mail: ralf.jaenicke@rub.de Date: April 10, 2015 Submitted to Computer Methods in Applied Mechanics and Engineering [2015] Abstract The paper deals with the computational homogenization of pressure diffusion processes in a poroelastic medium. The underlying physical phenomena are of interest for the interpretation of seismic data with applications in hydrocarbon production and geothermal energy. Pressure diffusion is assumed to take place on a length scale much smaller than the observer scale. Thus, the macroscopic observer is not able to measure the properties of the poroelastic medium directly but notices an intrinsic viscous attenuation. Under these circumstances, the macro-scale can be interpreted as a single-phase solid with (apparent) viscoelastic properties. In this paper, we establish a numerical upscaling procedure based on a volume averaging concept. This enables us to identify the material properties of the viscoelastic substitute model in a numerically efficient manner. For this purpose, the poroelastic medium on the small scale is modeled in terms of the momentum balance of the biphasic mixture and a coupled diffusion equation. We approximate the poroelastic pressure field on the small scale by a linear combination of pressure modes forming a reduced orthogonal basis and being identified by a Proper Orthogonal Decomposition (POD) technique. From the superposition principle, the evaluation of the poroelastic continuity equation results in a proper identification of the evolution equations defining the apparent viscoelastic model. In comparison to the nested FE2 solution schemes, the reduced order approach only requires a small amount of “off-line” precomputations and, therefore, causes very low numerical costs. The proposed method is validated for the simple setup of a layered porous rock with alternating water- and gas-saturated zones. 2 1 1 Introduction Introduction Porous rocks saturated by one or several fluids show a distinct attenuation of elastic waves, in particular at low seismic frequencies (f < 100 Hz) [4, 14, 16, 18]. Attenuation is caused by dissipative processes on a mesoscopic length scale much smaller than the observer scale but much larger than the scale of discrete grains and pore channels. In particular, diffusion mechanisms in rocks with spatially varying fluid properties (patchy saturation) including long-range fracture networks, see [1, 20, 25] and citations therein, are of enormous scientific and economic interest for applications in hydrocarbon production and geothermal energy. The observed attenuation, therefore, can be used for the interpretation of seismic data in terms of inferring pore fluid saturation or quality of reservoir rocks. In order to gain a deeper understanding of the physical processes in the saturated porous rock and their impact on the observed attenuation, we introduce a multi-scale modeling approach in the subsequent sections. 1.1 Seismic attenuation in porous rocks Fluid transport and pore pressure diffusion in fluid-saturated porous reservoir rocks take place at multiple scales, see Fig. 1. In particular, we distinguish between the short range fluid flow on the micro-level between neighbouring pores and the long range pressure diffusion on the meso-level. Typical length scales for sandstones are λ ∼ 1 mm (micro) and l ∼ 1 m (meso) compared to L > 100 m on the observer (macro) level. Seismic waves (f < 100 Hz) travelling through the porous rock lead, depending on the solid frame and fluid stiffness, to a locally reduced pore space. The resulting pore pressure gradients induce pressure diffusion and, consequently, part of the elastic energy of the wave is dissipated. By contrast, short range processes on the pore scale, such as squirt flow between micro-cracks, see [13], become active at high frequencies (f > 1 KHz). In this contribution, the transition from the meso- to the observer (macro) level is of particular interest. For this purpose, we model the fluid-saturated porous eetive geologial model on maro-sale → visoelasti heterogeneous meso-sale → poroelasti PSfrag replaements disrete grain sale on miro-sale L l λ Figure 1: Multi-scale representation of the fluid-saturated porous rock. We distinguish between the pore scale (micro), the heterogeneous meso-scale and the observer scale (macro). Separation of scales is assumed (λ ≪ l ≪ L). rock on the meso-level as a biphasic poroelastic medium with spatially varying material properties. Since inertia forces due to waves with low seismic frequencies are much smaller than the viscous forces due to wave induced fluid flow and, moreover, the wave lengths related to seismic frequencies are much larger than the length characteristic for the mesoscopic diffusion processes, the meso-structure undergoes an apparent transient loading. We, therefore, ignore inertia effects and use Biot’s quasi-static equations of linear consolidation for the poroelastic modeling, see [2] and, for related applications, [12, 21]. It is important to remark that the pressure diffusion can not be directly observed on the macro-scale. Hence, it is considered as a local phenomenon from the observer’s viewpoint. In other words, the underlying 1.2 Selective and reduced order computational homogenization 3 diffusion length is considered to be much smaller than the characteristic length of the macro-scale (l ≪ L). Since, nevertheless, attenuation is detected, the overall medium has to be interpreted as an one-phasic solid material (Cauchy continuum) exhibiting apparent viscoelastic properties. Thus, the upscaling is identified as a selective scale transition from a heterogeneous poroelastic to a homogeneous viscoelastic model. The identification of the appropriate viscoelastic representation, however, is a non-trivial task even for the very simple 1D model of a layered rock [3, 29]. The so-called White model considers a medium with alternating water- and gas-saturated layers. For symmetry reasons, the considerations are restricted to the undrained unit cell depicted in Fig. 2 a), see [22]. Analyzing the observed attenuation in terms of PSfrag replaements fc White model ε̄ C1 η1 a) saturation gas water C0 ε̄ White PSfrag replaements ε̄ log10 1/Q [℄ 3P rheology d √ ∝ 1/ f ∝f ∝ 1/f 3P l gas ε̄ b) log10 f [Hz℄ Figure 2: a) White’s model of a 1d layered medium compared to the 3-parameter (3P) Maxwell-Zener rheology. b) Frequency dependency of the loss factor 1/Q at low and high frequencies. the frequency-dependent dispersion relation, one finds that the inverse quality or loss factor, respectively, √ ˙ ε̄) ˙ increases with f at low frequencies. At high frequencies, it is proportional to 1/ f . 1/Q = −Im/Re (σ̄/ It is easy to prove that, for the Maxwell-Zener model with one single viscous chain (3-parameter or 3P rheology), the inverse quality factor depends on 1/f , see Fig. 2 b). Obviously, the 3P model is not able to describe the observed behaviour, and a more sophisticated viscoelastic description is required even for this simple 1D case. In this paper, we propose to derive the substitute model from a computational homogenization framework. 1.2 Selective and reduced order computational homogenization of poroelastic media The homogenization procedure introduced in the subsequent sections is based on the volume averaging technique assuming separation of scales (l ≪ L). For this purpose, we define a volume element of finite size that is considered to be representative for the entire heterogeneous structure on the small scale (Representative Volume Element – RVE). The RVE is substituted by one material point of the (viscoelastic) macro-scale. For practical applications, the implemented volume elements are considerably smaller than a RVE, and, therefore, they are called Statistical Volume Elements (SVE), see [17]. Doing so, the macroscopic field quantities are computed as volume averages of their mesoscopic counterparts over the underlying RVE or SVE, respectively. In particular, it is assumed that the macro-homogeneity condition is satisfied [8, 11]. That is, the stress power of the small scale equals, in volume averaged sense, the stress power observed in the particular macroscopic material point. A comprehensive overview of standard first-order homogenization problems is found in [15] and citations therein. In the targeted poro-to-viscoelastic upscaling problem, however, not all variables required in poroelasticity, such as the pore pressure or further quantities related to the fluid constituent, have their one-by-one representation in viscoelasticity. Consequently, we coin the adopted approach “selective homogeniza- 4 2 Poroelastic meso-scale tion”. The evaluation of the macro-homogeneity condition for the poroelastic upscaling problem and the formulation of various boundary conditions on the SVE level based on additional kinematic assumptions has been recently addressed in [8, 19]. It has been shown that the SVE size strongly influences the predicted apparent properties depending on the chosen boundary conditions. Similar properties have also been observed for standard homogenization problems, see for example [9]. However, the so far established volume averaging of the apparent properties from time-dependent mesoscopic computations bears two principal deficiencies: First, each loading scenario requires the time-dependent solution of an individual initial boundary value problem in the sense of the FE2 methodology. This leads to immense computational costs and, therefore, is restricted to rather small and artificial problems. Second, the straight-forward volume averaging approach allows for a prediction of the apparent viscoelastic properties in a heuristic manner only. So far, it has not been possible to derive the constitutive relations of a viscoelastic substitute model. The present contribution, therefore, intends to considerably enhance the known procedure by making use of the linearity of the poroelastic formulation. Deriving a reduced basis for the meso-scale problem, we obtain the system response upon applying the superposition principle. Moreover, the reduced basis approach allows for a direct identification of the apparent viscoelastic model and its material properties. This, consequently, allows for a numerically highly efficient solution of timedependent boundary value problems on the large scale as the result of arbitrary loading histories on the mesoscopic RVE. The computational costs are dramatically reduced, since the full mesoscopic resolution will be executed once and for all as “off-line” precomputations, which is in sharp contrast to the nested solution scheme of the straight-forward FE2 approach. Similar approaches have been successfully established for elasto-viscoplastic and viscoelastic compounds, see [6, 7, 24, 26], to name only a few. The paper is organized as follows: First, Biot’s formulation of linear poroelasticity will be recalled. Second, we introduce an order reduction method for the poroelastic meso-scale problem. A reduced basis for the pressure distribution will be derived by means of a Proper Orthogonal Decomposition (POD) and the viscoelastic evolution equations will be examined. Finally, we validate the proposed methodology in terms of the White problem under different loading conditions. Throughout this manuscript, vector and tensor quantities are written as bold types. Simple and double contractions read a · b = ai bi , A : B = Aij Bij , taking into account the Einstein sum convention. The multiple scales of the problem make it necessary to distinguish between mesoscopic quantities ⋄ and their macroscopic counterparts ⋄ := h⋄i. The average over the volume V✷ occupied by a mesoscopic RVE is R ¯ computed as h⋄i := V1✷ V✷ ⋄ dv. Restricting ourselves to a geometrically linear framework under small strain conditions, we assume partial and material time derivatives to coincide and compute gradients with respect to the current meso- or macroscopic position vector x or x̄, respectively. Our study concerns a biphasic mixture on the meso-scale. Constituents are the solid skeleton ϕs and the pore fluid ϕf following the notation introduced in [28]. We adopt the sign convention for stresses from continuum mechanics: Compression of a volume leads to a negative stress or, equivalently, to a positive pressure response. 2 Poroelastic meso-scale We adopt a continuum mechanics description of the heterogeneous meso-scale represented as a biphasic mixture of a linear-elastic porous medium saturated with a viscous pore fluid, for example water, gas or oil. Thus, we introduce the volume fraction occupied by the fluid phase nf = φ = dv f /dv. The volume fraction occupied by the solid phase (grains, skeleton) computes accordingly as ns = 1 − nf . We assume that the fluid-saturated poroelastic medium is described by Biot’s quasi-static equations of (linear) consolidation [2, 23], for which the primary variables are the solid phase displacement u and the pore pressure p (u-p-formulation see [30]). In addition, we introduce the linear strain tensor ε = (u ⊗ ∇)sym , the velocity of the fluid constituent vf and the seepage velocity w = vf − u̇. 5 Biot’s equations of linear consolidation are expressed as a coupled system of partial differential equations σ·∇ = 0 ∀ x ∈ V✷ , (1) ∇ · (φ w) + Φ̇ = 0 ∀ x ∈ V✷ , (2) including Dirichlet and Neumann boundary conditions for the primary variables u and p, u = u∗ ∀ x ∈ ∂D V✷ , t := σ · n = t∗ ∀ x ∈ ∂N V✷ , (3) p = p∗ ∀ x ∈ ∂D V✷ , q := φ w · n = q ∗ ∀ x ∈ ∂N V✷ , (4) n representing the outwards normal vector on the surface ∂V✷ . Inertia effects are ignored due to the fact that the intertia forces at seismic frequencies are much smaller than the internal and external forces of the mixture. Moreover, the wavelength of seismic waves is much larger than the RVE size. The poroelastic medium, therefore, undergoes a quasi-transient loading with negligible accelerations. The total stress tensor is introduced as σ = 2 G εdev + 3 K εsph −α p I . | {z } | {z } =:σ eff (ε) (5) =:σ p (p) In order to distinguish between instantaneous response of the dry solid frame and the additional stress response due to the fluid constituent, we have split the total stress into the effective stress σ eff (ε) and the stress σ p (p). For simplicity reasons, we introduce the abbreviation σ eff = C : ε with the 4th order tensor C representing the stiffness of the dry solid frame. The seepage velocity is assumed to depend linearly on the pressure gradient, and we use Darcy’s law as φw = − ks ∇p. ηf R (6) The storage function Φ, that represents the volume of fluid accumulated within a unit of bulk volume, is defined as Φ = φ + α∇ · u+ p . M (7) It is important to remark that, in the linear case, we identify φ as the initial porosity φ = φ0 = φ(t = 0) =const and, therefore, it is used as a material parameter. All material constants are specified in Tab. 1. ks φ G, K ρs , ρf K s, K f ηf R α 1/M intrinsic permeability porosity elastic moduli of dry frame (shear, bulk) partial density (solid, fluid) bulk modulus (solid grains, pore fluid) effective dynamic viscosity (pore fluid) = 1 − K/K s = φ/K f + (α − φ)/K s Table 1: Poroelastic material parameters and definitions. 3 Identification of an effective viscoelastic substitute model After having recalled Biot’s theory of linear consolidation, we now investigate the upscaling properties from a heterogeneous poroelastic meso-scale towards a homogeneous viscoelastic macro-scale. External loading will lead to pressure gradients on the meso-level and, consequently, to pore pressure diffusion. 6 3 Identification of an effective viscoelastic substitute model Assuming scale separation, we require that the observable scale is much larger than the scale of mesoscopic heterogeneities. Hence, the diffusion processes evolve over mesoscopic length scales. The diffusion phenomena are local and can not be observed directly at the macro-level. In the context of numerical homogenization the locality finds its representation in the definition of a RVE (or, for practical purposes, a SVE) that is chosen significantly larger than the diffusion length. As a consequence, the substitute medium reduces to a Cauchy medium with viscoelastic properties. In the following, an appropriate computational homogenization scheme will be established. The apparent properties of this viscoelastic substitute medium will be identified from mesoscopic simulations. To this end, we assume the mesoscopic fields to be perfectly periodic in the primary variables u and p. The SVE is considered to represent the unit cell of this periodic structure. For a comprehensive discussion of various boundary settings including Kinematic Uniform Boundary Conditions (KUBC), Stress Uniform Boundary conditions (SUBC) and undrained boundary conditions, see [8]. replaements 3.1PSfrag Variational form of the homogenization problem x− (3) x+ (3) x+ (2) x + ∂V − ∂V l x+ (1) x− (1) V x− (3) x− (2) l Figure 3: RVE in 2D under meso-periodicity conditions with image boundary ∂V✷+ and mirror boundary ∂V✷− . Following the notation presented in [10], we establish the strong form of the kinematically driven poroelastic periodic problem as JuK(x) = ε̄ · JxK, t+ + t− = 0, (8) JpK(x) = 0, q + + q − = 0, (9) where we introduced the “jump operator” J⋄K(x) := ⋄(x+ ) − ⋄(x− ) for all x ∈ ∂V✷+ . Thus, the system undergoes a kinematic loading that depends linearly on the macroscopic strain ε̄. The sub-scale pressure is not controlled by a macroscopic pressure gradient (locality of pressure diffusion), and Eq. (9) enforces the mass conservation of the fluid phase (no net outflux). The presented homogenization approach is, therefore, called selective. The macroscopic stress response computes as σ̄ = hσi . (10) We now reformulate Eqs. (1) and (2) in their weak forms making use of the variational format presented in [10]. Hence, we seek solutions in the trial spaces U✷ and P✷ of admissible displacements and pore pressure fields that are sufficiently regular in V✷ . We furthermore introduce the corresponding trial spaces of self-equilibrated fluxes T✷ and W✷ that are sufficiently regular on ∂V✷+ . We write the equations for 3.2 Approximation of mesoscopic field quantities 7 finding u, p, t, q ∈ U✷ × P✷ × T✷ × W✷ as au (u, δu) + bu (p, δu) − cu (t, δu) = 0, (11) −ap (p, δp) + bp (u̇, δp) + mp (ṗ, δp) + cp (q, δp) = 0, (12) −cu (δt, u) = −cu (δt, ε̄ · x), cp (δq, p) = 0, (13) (14) which hold for any admissible test functions δu, δp, δt, δq ∈ U✷ × P✷ × T✷ × W✷ . Here, we used for the momentum balance * + * + au (u, δu) = (C : ε(u)) : (δu ⊗ ∇) , | {z } bu (p, δu) = cu (t, u) = 1 V✷ t · JuK da, ∂V✷+ and for the continuity equation ap (p, δp) = hφ w(∇p) · ∇δpi , 1 ṗ δp , mp (ṗ, δp) = M 3.2 = −bp (δu, p), (15) =σp (p) =σ eff (ε(u)) Z −α p I : (δu ⊗ ∇) | {z } 1 cu (t, ε̄ · x) = V✷ Z t ⊗ JxK da : ε̄, (16) bp (u̇, δp) = hα ∇ · u̇ δpi = −bu (δp, u̇), Z 1 cp (q, p) = q JpK da. V✷ (17) ∂V✷+ (18) ∂V✷+ Approximation of mesoscopic field quantities In order to identify the effective viscoelastic model emerging from the poroelastic meso-scale we now expand the pore pressure field p(x, t) using spatial pressure modes pa (x) and time-dependent mode activity parameters ξa (t). Similar approaches can be found in literature for the upscaling of elastoviscoplastic and viscoelastic compounds, see [6, 24, 26], to name only a few. We assume, for practical applications, the sum to be reduced to a finite number N of elements. We write p(x, t) ≈ N X ξa (t) pa (x), (19) a=1 P whereby the identity N a=1 ξa pa = 0 is satisfied only by the trivial solution ξa = 0, a = 1, 2, . . . , N . In other words, the pressure modes form a linearly independent basis of the space P✷ of scalar functions comprising all possible pressure distributions inside V✷ . For the subsequent derivations we suppose the pressure modes pa to be known. The identification of these modes will be addressed later. The mode activity parameters ξa (t) control the temporal evolution of the pore pressure state in the poroelastic medium. It is important to remark that the variables ξa (t), in the absence of any dependency on the local position x, can be understood as macroscopic quantities representing the internal variables of the macroscopic viscoelastic substitute medium. Thus, the current state of the poroelastic medium depends on the overall strain ε̄ as well as on the internal variables ξa . We now may expand other mesoscopic field quantities accordingly, namely ε(x) and σ(x). To this end, we make use of the linearity of the underlying poroelastic medium and apply the superposition principle. 8 3 Identification of an effective viscoelastic substitute model We write ε(x, ε̄, ξ) = E0 (x) : ε̄(t) + N X ξa (t) εa (x) and (20) a=1 σ(x, ε̄, ξ) = C(x) : E0 (x) : ε̄(t) + N X ξa (t) σ a (x), (21) a=1 where we introduced the 4th rank strain localization tensor E0 as well as the mode-dependent fields εa and σ a = C : εa − α pa I. The resulting fields depend linearly on the driving variables ε̄ and ξa . Hereby, the quantities associated with the localization tensor represent the instantaneous response of the dry linear-elastic solid skeleton under kinematic loading at zero mode activity (ξa = 0, a = 1, 2, . . . , N , that is p(x) = 0). To compute the particular strain and stress fields, we solve for ui and ti , i = 1, 2, . . . , 6, from au (ui , δu) − cu (ti , δu) −cu (δt, ui ) = 0, (22) = −cu (δt, Bi · x). (23) The Bi represent the 6 members of the irreducible orthonormal basis of the symmetric strain tensor ε̄ (orthotropic case). The localization tensor is computed as 0 E (x) = 6 X i=1 εi (x) ⊗ Bi , (24) where εi = (ui ⊗ ∇)sym . For more information concerning the derivation of the localization quantities see for example [5]. The strain fields εa , representing the mode basis for Eq. (20), can now be computed by solving N linearelastic eigenstress problems corresponding to the unit loading ξa = 1, a = 1, 2, . . . , N , with hεa i = 0. Thus, for known pa , we solve for ua and ta , a = 1, 2, . . . , N , from au (ua , δu) − cu (ta , δu) = cu (δt, ua ) = −bu (pa , δu), (25) 0. (26) Finally, the total stress response of the RVE can be calculated as the volume average of the superimposed local stress field by means of Eqs. (10) and (21). 3.3 Evolution of internal variables The decompositions Eqs. (19)–(21) can now be used to evaluate the continuity equation (12). After integration by parts, we may rewrite Eq. (12) as ! ! ! N N N N N N X X X X X X p 0 ˙ p p = 0 ξ˙b pb , δξa pa ξ˙b ∇ · ub , δξa pa + m ξb pb , δξa pa + b U : ε̄ + a a=1 b=1 b=1 a=1 b=1 a=1 with U0 = I : E0 and, taking into account that ξa represent macroscopic internal variables, N X a, b=1 i h δξa ap (pb , pa ) ξb + [bp (ub , pa ) + mp (pb , pa )] ξ˙b = − N X a=1 ˙ pa , δξa bp U0 : ε̄, (27) 3.4 Mode identification 9 for all admissible test functions δξa , a = 1, 2, . . . , N . Substituting the test function δu by ub in Eq. (15)2 as well as in Eq. (25), we obtain the identity bp (ub , pa ) = −bu (pa , ub ) = au (ua , ub ), (28) which is crucial in order to prove the symmetry of the final system of ODE’s for ξa , as discussed below. More compact, we may introduce the vector ξˆ = [ξ1 , ξ2 , . . . , ξN ]T and write Eq. (27) in matrix-vector form h i ˙ (29) = δ ξˆT B̂ ˆε̄˙ , δ ξˆT Â ξˆ + M̂ ξ̂ whereby the matrix entries are, for a, b = 1, 2, . . . , N , and, for i = 1, 2 . . . , 6, s k p ∇pa · ∇pb , Aab := a (pa , pb ) = ηf R Bai := −bp (Ûi0 , pa ) u p Mab := a (ua , ub ) + m (pb , pa ) E D = − α pa Ûi0 , = 1 εb : C : εa + pa pb . M (30) (31) (32) Hereby, ε̄ˆ˙i and Ûi0 are the vector representations of the second order tensors ε̄˙ and U0 . Since the test functions δξa are arbitrary, we derive from Eq. (29) the evolution equation for the mode activity coefficients as ˙ M̂ ξˆ + Â ξˆ = B̂ ε̄ˆ˙, ˆ = 0) = 0. ξ(t (33) Thus, the evolution of the pressure modes depends linearly on the mode activity and the average strain rate. It is important to remark that the matrices M̂ and  are symmetric and, therefore, can be diagonalized. In other words, it is possible to introduce a shift of mode activity variables {ξa } → {χa } such that the evolution equation (33) can be decomposed into the set of independent equations χ̇a + µa χa = bai ε̄˙i , χa (t = 0) = 0, a = 1, 2, . . . , N, (34) where µa and bai are the pertinent generalized eigenvalues and modal participation factors, respectively. These equations exhibit a structure that is closely related to a generalized Maxwell-Zener model, see Fig. 4. If the latter model is expressed in terms of the elastic strain εae = ε − εav of the ath Maxwell chain, the pertinent evolution equation becomes ε̇ae + Ca a ε ηa e = ε̇, (35) where Ca and ηa represent the stiffness and the viscosity parameter, respectively, of the particular Maxwell chain. Obviously, this is the special case of Eq. (34) that is obtained if we consider the 1D case and set µa = Cηaa and ba = 1. 3.4 Mode identification It remains to specify the pressure modes pa (x) introduced in Eq. (19). We apply the Karhunen-Loève decomposition, also known in the literature as Proper Orthogonal Decomposition (POD), see [24], for example. The procedure is as follows: Executing transient training computations on the RVE level following certain specific loading paths, we generate a finite number of S snapshots p̂k (x) of the local pressure field, k = 1, 2, . . . , S. Considering a cubic RVE in a 3D setting, these loading paths could 10 3 Identification of an effective viscoelastic substitute model PSfrag replaements Ca εa e ηa ε − εa e C0 Figure 4: Generalized Maxwell-Zener model as the rheological representation of the evolution equation (35). be 6 time-dependent experiments undergoing ε̄i (t) = γ(t) Bi , i = 1, 2, . . . , 6. The scalar stimulation function γ(t) may, for example, prescribe a stress-relaxation test or a frequency sweep. In any case it has to be ensured that the loading function includes all relevant frequency contributions or relaxation times, respectively. It is, consequently, required that the loading phase in the stress relaxation case is sufficiently fast. Moreover, it must be ensured that all the pressure states at all relevant frequencies or, respectively, relaxation times are represented by the chosen snapshots. The snapshots are then used to generate the correlation matrix gkl = hp̂k (x) p̂l (x)i , k, l = 1, 2, . . . , S. (36) We solve the eigenvalue problem (gkl − λ δkl ) vl = 0 and arrange the resulting eigenvalues λk in decreasing order. It can be observed that the eigenvalues become small very fast, see the exemplary situation for the White model in Fig. 5. It turns out that reducing the basis to the N members, for which λa > 1e-5 λ1 , a = 1, 2, . . . , N , leads to a highly accurate prediction of the apparent properties with a reasonable small number N < S. Thus, the remaining N basis modes are to be computed as pa (x) = S X vka p̂k (x), a = 1, 2, . . . , N. (37) k=1 Due to the orthonormality of the eigenvectors vka , the pressure modes are orthogonal as well, and it holds λa , if a = b, hpa (x) pb (x)i = (38) 0, otherwise. PSfrag replaements 1e-00 |λk /λ1 | [℄ 1e-04 1e-08 1e-12 1e-16 0 10 20 30 40 50 k [℄ Figure 5: Eigenvalues in decreasing order resulting from a POD of 50 snapshots taken from transient computations of the White model (1D). 11 rock ks φ G K Ks Kf ηf R matrix [mD] [–] [GPa] [GPa] [GPa] [GPa] [mPa s] water-saturated 100 0.1 15.8 16.2 36.0 2.3 3 gas-saturated 1000 0.2 8.8 9.6 36.0 0.022 0.01 Table 2: Poroelastic material parameters for water and gas saturation (1 mD ≈ 1e-15 m2 ). 4 Numerical example: White model for layered poroelastic media We now want to validate the adopted homogenization approach numerically. Without restricting the generality of the method, we study the apparent viscoelastic properties of the White model, which is a specially designed 1D poroelastic medium, and which is used as a standard representation for layered media in Geoscience, see for example [3, 21, 29]. The poroelastic unit cell consists of a water-saturated layer surrounded by symmetrically arranged gas-saturated layers, see Fig. 2 a). Undrained boundary conditions are used and coincide with periodic boundary conditions for symmetry reasons. The chosen material parameters are given in Tab. 2. For a kinematically controlled transient consolidation experiment, we observe a pronounced pressure gradient between the water-saturated and the gas-saturated layer, see Fig. 6 a). From this computation with full resolution, we generate 50 snapshots representing the pore pressure distribution at different time steps during the transient consolidation test. The POD is now used to identify the pressure modes forming the reduced basis of the problem. For this purpose, we only consider the largest eigenvalues λa satisfying the condition 1e-5 λ1 < λa < λ1 and ignore the remaining eigenvalues λk ≤1e-5 λ1 . For the given example, this condition results in 5 pertinent eigenvalues λa and the corresponding pressure modes pa , a = 2, 3, . . . , 6. We include an additional pressure mode p1 (x) = const in order to match precisely the equilibrium pressure distribution (∇p = 0, if t → ∞). The corresponding linear-elastic eigenstress problems due to the N = 5 + 1 = 6 pressure modes as well as the stress response due to external loading ε̄i = Bi , i = 1, 2, . . . , 6, under zero mode activity are solved in order to compute the reduced system matrices in Eqs. (30) – (32). After having solved, for this 1D case, 1 PSfrag replaements PSfrag replaements 0 -1e08 a =3 2e07 a =5 0 [Pa℄ 6e07 a =1 1e08 a =4 pa p̂k [Pa℄ a =6 tր 3e08 1e08 −1 a) −0.6 −0.2 x 0.2 [m℄ 0.6 1 −1 b) a =2 −0.6 −0.2 x 0.2 0.6 1 [m℄ Figure 6: Selection of snapshots p̂k (x) of the pressure field p(x) during the fully-resolved computation of a stress relaxation test of the layered unit cell, l = 2 m, d = 0.5 m, see Fig. 2. b) Pressure modes computed by the POD (a =2, . . . , 6) out of 50 snapshots of the stress relaxation problem. The decrease of the corresponding eigenvalues is given in Fig. 5. The constant pressure mode a = 1 is included and represents the equilibrium state ∇p(x) = 0 for t → ∞. 12 5 Discussion PSfrag replaements transient stress relaxation test, 1 + N = 7 linear-elastic boundary value and eigenstress problems, respectively, as well as 1 low-dimensional eigenvalue problem, we now have all ingredients at hand to establish the evolution equation (33). Thus, the apparent viscoelastic substitute model is successfully identified, and we can now validate our reduced order model. Hereby, time-dependent initial value problems on the mesoscopic level with full resolution serve as reference computations. We start with the consolidation test in Fig. 7 a), where the effective stress σ̄ is displayed. Sfrag replaements The strain loading condition is given as ε̄(t) = t, 0 ≤ t ≤ 0.02 s, and is kept constant for t > 0.02 s. We find an excellent agreement between the reduced order solutions and the reference computations if 4 or 6 pressure modes are considered. By contrast, the effective model is not able to map the relaxation character of the experiment if only the constant pressure mode (a = 1) is used. 1e06 -5.0e08 referene 8e05 [Pa℄ 0.3 -4.0e08 4e05 2e05 σ̄ 0.2 6e05 σ̄p [Pa℄ -4.5e08 0.4 -3.5e08 -3.0e08 6 modes 0.01 t [s℄ 1 mode 0 referene 1 mode 4 modes 6 modes a) 4 modes -2e05 0 0.1 b) 1 2 3 t 4 [s℄ 5 6 7 Figure 7: Evaluation of the stress response predicted by the order reduction method as compared to the fully resolved simulation of a) a stress relaxation test and b) an excitation by the second derivative of the Gaussian function. As a second loading scenario we investigate a kinematic activation by the second derivative of the Gaussian function, see Fig. 7 b). Please note that the kinematic input signal is perfectly symmetric. Due to the dispersive properties of the poroelastic RVE, this symmetry is lost for the effective stress response σ̄ of the reference solution. In Fig. 7 b), only the stress contribution σ̄ p := hσ p i is displayed for the sake of a clear visualization. Again, we observe a very good agreement between the reduced order model and the reference solution if 6 pressure modes are taken into account. By contrast, use of the constant pressure mode (N = 1), only, results in a dispersion-free stress response. Similar numerical experiments can be evaluated for various loading conditions. In all investigated cases the viscoelastic model accounting for 6 modes results in an excellent agreement with the reference computations as long as the contributing frequencies of the loading signal are included in the chosen set of snapshots. Finally, we would like to discuss attenuation in the frequency domain. For this purpose, the inverse quality factor 1/Q observed for the given unit cell is displayed in Fig. 8. The reduced model that exclusively uses the constant pressure mode (N = 1) is dispersion-free (no attenuation) and is, therefore, ignored. Again, we find that the reduced order model including 6 internal variables matches the attenuation behaviour of the layered model with a very high accuracy. 5 Discussion A new computational homogenization approach has been established substituting a heterogeneous poroelastic meso-structure by a homogeneous, one-phasic solid with apparent viscoelastic properties. For this purpose, the pore pressure diffusion at low seismic frequencies (f < 100 Hz) is considered as a local PSfrag replaements 13 1e-01 1/Q [℄ referene 6 modes 1e-02 2 modes 1e-03 1e00 1e01 1e02 1e03 f [Hz℄ Figure 8: Frequency dependence of the inverse quality factor predicted by the reduced order model as compared to the fully resolved solution. process. In other words, pressure diffusion can not be observed macroscopically but in terms of the intrinsic attenuation behaviour, only. Moreover, inertia forces are considered to be negligible compared to the viscous forces in the fluid phase and are, therefore, ignored. The resulting space-time dependent pore pressure fields under transient loading conditions are approximated by a finite-dimensional linear combination of basic pressure modes. The correlated mode activity parameters serve as the macroscopic internal variables of the homogenized viscoelastic model. Evaluating the continuity equation allows for a proper derivation of the equation set controlling the evolution of the internal variables. The method is completed by the identification of the underlying pressure modes that form the reduced basis of the problem. For this purpose, we successfully made use of the Karhunen-Loève decomposition belonging to the family of Proper Orthogonal Decompositions. The proposed reduced order method offers several advantages compared to the averaging techniques that are currently available in the literature. First, numerical efficiency: The existent straight-forward homogenization approaches [8, 14, 21] require time-dependent computations with full mesostructural resolution for each individual loading history. Whereas this might be feasible for investigations on one single SVE, the solution of macroscopic boundary value problems accounting for a time- and spacedependent deformation field is restricted to artificially small and, therefore, academic problems due to numerical costs. By contrast, the new reduced order method enhances the volume averaging approach by the explicit derivation of the effective constitutive model. Thus, the computation of the homogeneous macro-scale boundary value problem reduces to a single-scale FE task and does not require the nested solution of boundary value problems on two scales. Simulations with full mesoscopic resolution must be executed only once in a preprocessing step and comprise for a 3D problem a) 6 transient computations on the RVE scale generating the S snapshots for the POD, b) 1 solution of a S-dimensional eigenvalue problem (POD), c) 6 linear-elastic computations at zero mode activity and d) the solution of N + 1 linear-elastic eigenstress problems at zero overall strain. The second advantage of the proposed order reduction method is the natural outcome of the macroscopic constitutive relation. To the authors’ best knowledge, this is the first time that the constitutive equations defining the viscoelastic substitute for pressure diffusion processes in porous rocks have been derived directly from mesostructural computations. The proposed methodology has, finally, been validated for the 1D case of a layered poroelastic medium with alternating water and gas saturation, referred to as White model. 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