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
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+49 (0)234 32 22485
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ralf.jaenicke@rub.de
Mechanics – Continuum Mechanics
Ruhr-Universität Bochum
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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. Based on a series of 50 snapshots
we have found that the homogenized constitutive model predicts the apparent behaviour in a highly
accurate manner if the reduced basis consists of 6 pressure modes. Obviously, this number of internal
variables is easily manageable. It is, nevertheless, noticeable that as many as 6 chains in the corresponding
rheological model are needed for this rather simple 1D problem.
14
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The established method offers a wide field for ongoing research. We would like to mention the obvious
extension to higher-dimensional applications. Additionally, the method is intended to undergo further
enhancements towards the modeling of pressure diffusion in fractured rocks. Due to the extreme aspect
ratios of these fractures, a hybrid-dimensional description of the problem is required [27] and will be
included in the reduced order model in future activities.
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