1
A017 THE DEVELOPMENT OF UPSCALING FOR
POROELASTICITY AND MACRO-ANISOTROPY
MODELING OF THE BIOT PROBLEM
A.Kh. Pergament, M.Yu. Zaslavsky
Keldysh Institute for Applied Mathematics. Russia, Moscow, Miusskaya sq., 4
Abstract
This paper focuses on blocks upscaling for 3D poroelastic problem in fractured media.
In the beginning the set of upscaling procedures for determining the coefficients or
poroelastic equations is examined. It is established that the P. King procedure [14] and others
may be obtained by developing the specific finite difference schemes for each cell of the grid,
each difference scheme being in accordance with the kind of upscaling procedure. As a result in
poroelastic theory the diagonal effective tensors of permeability and elastic moduli may be
constructed. But it is evident that the small but inhomogeneous block of locally isotropic
medium may be transformed into the anisotropic one by adequate upscaling. Thus the offdiagonal components of tensors mentioned above should be considered.
Using the difference schemes for the problems with discontinuous coefficients it is
possible to construct the set of homogenization algorithms. In this article the support operator
method by A.A. Samarskii and others [10] has been used. The main idea is to construct the
algorithm that approximates the energy integral ∫ k (∇u ) 2 dV . Its approximation is a quadratic
form. The matrix of the form is the result of homogenizating the permeability coefficients. The
symmetric matrix of the form may be developed under the condition of matching the energy
integral for elements of some functional subspace. For scalar elliptic equation it is possible to use
the linear span of basis constructed in [3], [7]. Its functions have the same properties as the exact
solution. After that the gradient vectors and divergence operators definition is realized in
accordance with Ostrogradsky–Gauss theorem. First the averaging algorithm by S. Moskow and
others [3] has been considered for determining the full tensors of permeability in scalar elliptic
problem. Then the special basis vectors for poroelasticity in the porous fractured formation were
constructed and the homogenization algorithms were generalized for such kind of problems.
As known, all natural layers are characterized by some fractured structure. Barenblatt and
others [13] formulated the concept of the systems with double porosity and double permeability,
i.e. in each cell of the medium there are two values of pressure: pore pressure and pressure in the
fractures. It is supposed that there is a flow between pores and fractures. Since permeability in
fractures is very large, the pressure in fractures becomes stationary much faster, than in pores.
That's why its evolution may be described by a quasistatic equation. Besides it is possible to
assume that the volume of fractures is negligibly small. Thus we may neglect the amount of fluid
content in fractures. As a result the model obtained is distinguished from the standard Biot model
[15] for porous media by the equation for pressure in fractures. At the same time the
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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generalization of Hook's law in Biot model retains its form. Further we will consider uncoupled
variant of the given model proposed by V.N. Nikolaevsky [4].
The most widespread methods for solution such kind of problems in medium with
piecewise-constant coefficients are projective methods, for example a finite element or
superelement method [6], [7], [8], [9]. All methods of such kind guarantee the strong
convergence.
The poroelastic problem is multiscale one. The scale of a porous collector is much less
than actual scales. Usually grids for the problem solution are conforming to the medium structure
and the scales of cells in collector are comparable with the cell scale of surrounding domain. As
a result it is necessary to solve the ill-conditioned algebraic systems. Thus it needs to consider
the elastic problem using larger cells than the characteristic scale of a collector. But these cells
are of complicated structure. As a result it needs to calculate effective coefficients in these cells
so that the mathematical model obtained would take into account properties of the initial media.
The algorithm for calculation of effective elastic moduli tensor for a stationary elastic problem
was constructed by authors in [1]. In poroelastic problems the construction of algorithms is more
complicated because it is necessary to create the algorithm of effective source term calculation
[2].
In the present work the algorithm of determining effective values for the given problem is
constructed and estimations for convergence rate are obtained. The mathematical model of
poroelastic problem in fractured media describes the evolution of stress–strain state and the
change of two pressures, in fractures and in pores. Since permeability in fractures is much
greater than permeability in pores, the governing system for uncoupled model is the following:
⎛k
⎞ α
−∇ ⋅ ⎜ 1 ∇p1 ⎟ + ( p1 − p2 ) = 0 ,
(1)
⎝µ
⎠ µ
⎛k
⎞ α
∂p
β 2 − ∇ ⋅ ⎜ 2 ∇p2 ⎟ + ( p2 − p1 ) = 0 ,
(2)
∂t
⎝µ
⎠ µ
σ ij , j = 0 ;
(3)
σ ij = Λ ijkl ε kl − ζ p2δ ij ,
1
2
ε kl = (uk ,l + ul ,k ) .
(4)
(5)
Here k1 and k2 are permeabilities in fractures and in pores, p1 and p2 are pressures in
fractures and in pores, µ is fluid viscosity, β is compressibility of the fluid, α describes the
flow between pores and fractures, σ and ε are stress and strain tensors respectively, Λ is
elastic moduli tensor, u is displacement, ζ is friability. We will consider the elastic problem in
the domain Ω . At the same time the filtration processes are supposed to take place in a narrow
domain Ω ' ⊂ Ω . On the bottom border Γ′ the kinematic boundary conditions are u |Γ′ = 0 , and
top and lateral borders of Ω represent free boundary σ ij n j |Γ′′ = 0 . The medium is supposed to be
isotropic Λ ijkl = λδ ijδ kl + µ (δ ik δ jl + δ ilδ jk ) but inhomogeneous.
Let’s assume that the Lame coefficients λ and µ are greater than zero and that filtration
processes take place in an area narrow enough. In this domain the one-phase filtration flow will
be considered.
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First we consider the algorithms mentioned above for the scalar elliptic equation of onephase filtration:
∇ ⋅W = f ,
(6)
W = − k ∇p
(7)
in 2D domain Ω with boundary conditions p ∂Ω = 0 . The generalization for 3D is
straightforward. Usually, as in the most of applied programs, the upscaling algorithm by P. King
is implemented. Having used the analogy between Darcy law and Ohm law, he constructed the
specific electric circuits and calculated the effective conductivity. As a result the diagonal tensor
of permeability was obtained. It is easy to establish that King’s effective tensor may be found
from the solution of certain finite-difference problem [5]. Namely, let’s consider the case where
the cell is divided into 4 equal subcells with their own coefficients. For determining k x the
boundary value problem shown on Fig. 1 should be solved. All unknowns are placed into centers
of subcells. Using the simplest finite difference scheme on this grid,
p − p3
p −0
⎧ p2 − p1
− k01 1
− k31 1
= 0,
⎪k12 h
hx / 2
hy
x
⎪
⎪ 1 − p2
p − p1
p − p4
− k12 2
− k42 2
= 0,
⎪k25
hx / 2
hx
hy
⎪
⎨
⎪k p4 − p3 − k p3 − 0 + k p1 − p3 = 0,
03
31
⎪ 34 hx
hx / 2
hy
⎪
⎪k 1 − p4 − k p4 − p3 + k p2 − p4 = 0,
34
42
⎪ 45 h / 2
hx
hy
x
⎩
the finite-difference solution may be found. Here kij are corresponding harmonic averages of
coefficient. Finally, k x is equal to the total flux through the left boundary.
Fig. 1. Boundary value problem for k x calculation
In the same way k y may be determined. But, as mentioned above, it is possible to
determine just diagonal effective tensor using this algorithm. Similar attempts to find offdiagonal components using the same method lead to the nonsymmetric tensor.
The algorithm for calculating symmetric tensor was proposed by S. Moskow and others
[3]. They constructed the averaged tensor on Lebedev grid. Let’s consider the simplest Cartesian
grid: S = {(ih1 ; jh2 )}iN, j =1 . Lebedev’s P-grid is defined as a subgrid of S with even the i+j numbers,
R-grid consists of remaining nodes of S (Fig. 2).
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.
Fig. 2. Lebedev grid
Let’s assume that there exists the prevailing direction n of coefficient change. It means
that k = k (nr ). In this case it is possible to establish [3], [7] that the solution may be locally
approximated in domain D by the function from the linear subspace
nr ds ⎫
⎧
L1 ( D) = span ⎨u0 = 1, u1 = mr, u2 = ∫
⎬
0
k⎭
⎩
where m is perpendicular to n .
We define the effective tensor Σij in R-grid cell H using energy matching for functions
from L1 ( H ) . This means the identity of continuous energy, which may be calculated
analytically, and discrete one. Namely, since the unit gives us a zero gradient, we obtain
α
β
α
β
H
∫ ku,i u, j dV = mes( H )Σij u ,i u , j , α , β = 1, 2 .
H
α
Here mes ( H ) is area of the cell H and u ,i are central difference approximations of
gradient in R-cell of Lebedev’s grid. For example,
uα (2) − uα (4)
α
u ,1 =
.
h1
Now it is possible to construct the finite difference scheme for the problem
divhW = f P ,
(6)
Wi H = −ΣijH u , Hj ,
divhW =
W1 (6) − W1 (8) W2 (5) − W2 (7)
+
h1
h2
(7)
(8)
1
fdV and ΣijH is calculated with energy matching condition.
mes ( H ) ∫H
In this case it is possible to prove that the solution uh of the finite difference problem converges
weakly (with respect to energy inner product) to the boundary value problem solution u , and
convergence is of the first order. Namely, it is possible to establish the estimate
| (u − uh , v) |≤ Ch
that is valid for all functions v under condition that u and v may be locally approximated by
functions from L1 ( D) . Here (⋅; ⋅) is energy inner product.
where f P =
In this work the ideas of support operator method [10] are suggested and realized for
finite difference approximation on Cartesian grids nonconforming to the interfaces. For scalar
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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elliptic problem this method is based on defining the gradient and divergence operators in
accordance with integral identity
(8)
∫ k∇p∇rdV + ∫ r∇ ⋅ k∇pdV = 0 ,
Ω
Ω
which should be valid for any functions p and r . The matrix of quadratic form for energy
integral approximation is determined under the assumption of energy matching condition for
functions that describe the solution features. Then the energy integral may be written as a
quadratic form defined on edges of the grid. The corresponding matrix may be treated as a result
of homogenization. Then, having used the support operator method we have developed the
expressions for divergence operator as a conjugate one to the gradient. Further this method will
be described for poroelastic problem.
As mentioned above, the elastic problem contains singular source term. In fact, the stress
tensor term ζ p2 is equal to zero outside collector and is non-zero inside it. Thus the upscaling
algorithm for the source term should be also developed. Further we suggest that the unknowns
are placed at grid nodes.
Let's assume that interfaces are smooth enough to be approximated by a straight line in
each cell of the grid, and also that Lame coefficients vary locally along one direction only:
λ = λ (nr ), µ = µ (nr ) . Fractures in the porous medium being not open, it is possible to assert that
σ ij n j and ui , j m j are continuous under considering the elastic problem. Here n and m are
normal and tangential to the interfaces vectors, respectively. Thus the forces applied to the
interface as well as displacements are continuous. As a consequence, the corresponding
components of the distortion tensor are continuous. Further we demand that these quantities are
smooth enough to be locally approximated by constants in some norm.
As known, if k1 and k2 are piecewise-continuous and ζ is constant through the collector
then the solution obtained may be locally approximated by constants in collector. Thus, the
expression ζ p2 varies only along one direction in each cell.
Let's define the linear span for each cell Ξ :
nr
nr
⎧ 1
ds 2
ds
L2 (Ξ) = span ⎨U = m ∫ , U = n ∫
,
µ
λ + 2µ
0
0
⎩
nr
U 3 = m(mr ) − n ∫
0
⎫
λ ds
, U 4 = −m(nr ) + n(mr ), U 5 = m, U 6 = n ⎬ .
λ + 2µ
⎭
Due to the assumptions stated above the exact solution of elastic problem may be
approximated by a vector from the linear manifold M (Ξ ) = L2 (Ξ) + f in each cell where vector
nr ζ p ds
2
0 λ +2µ
f = n∫
is determined by a source term in the equations of elasticity (3)–(5).
Our aim is to approximate integrals
∫
H
σ ij (U )ε ij (V )dS for each cell and all vectors U
and V . Indeed, having used the finite-difference analogue of integral identity
∫ σ ij (U )ε ij (V )dS + ∫ σ ij, j (U )vi dS = 0,
Ω
Ω
(9)
it is possible in this case to obtain the divergence operator as the conjugate one to the gradient
operator. Let’s approximate each integral by the expression ∑ φ Sφσ h,ij ,φ (U )ε h,ij ,φ (V ) where the
summation is performed over all vertices of the cell. Here ε h,ij ,φ (V ) is the natural approximation
of strain tensor that uses two finite differences connected with the vertex φ of the cell. The
approximation of stress tensor has the form
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Ξ,φ
σ h,ij ,φ (U ) = Θijkl
ε h,kl ,φ (U ) − M ijΞ,φ
Ξ,φ
where Θijkl
is effective tensor of elastic moduli and M ijΞ,φ is effective anisotropic analogue of
pressure. The tensor form of the last quantity is connected with the fact that Pascal law is not
valid for the anisotropic formation.
Ξ,φ
We define six components of effective elastic moduli tensor Θijkl
by equating continuous
and discrete energy in vertex φ of cell Ξ for all vectors from L2 (Ξ) . Since displacements U 5
and U 6 represent body movement as a rigid one, and U 4 is rotation around the z -axis, the
Ξ,φ
corresponding deformation is zero. Thus, we obtain six equations for six components Θijkl
:
∫Λ
Ξ,φ
ε (U α )ε kl (U β )dS = mes (Ξ)Θijkl
ε h,ij ,φ (U α )ε h, kl ,φ (U β ) , α , β = 1, 2, 3 .
ijkl ij
(10)
Ξ
The effective pressure tensor is defined by conditions
α
α
∫ (ζ p2ε ii (U ) − Λijklε ij (f )ε kl (U ))dS =
Ξ
Ξ,φ
= mes (Ξ)( M ijΞ,φ ε h,kl ,φ (U α ) − Θijkl
ε h,ij ,φ (f )ε h, kl ,φ (U α )) .
The further description of approximating stress tensor divergence is straightforward.
Fig. 3. Distribution of the Lame coefficients (upper is λ , lower is µ ) on the left and
error in L2 norm on the right
Let the filtration part in narrow collector be approximated by support operator method
[10], [11], [12] on curvilinear grid M that is conformed to interfaces. In this case it is possible to
prove some results concerning the convergence of the algorithm constructed. Let pi , U may be
approximated for each moment of time by functions from L1 (Ξ1 ) and M (Ξ) , respectively. Let
h1 and h be characteristic scales of cells for grids in Ω′ and Ω accordingly, τ be timestep and
the problem be considered on the time segment [0; T ] .
By using these assumptions it is possible [2] to establish strong convergence for a
filtrational part and weak one for the elastic problem. Namely, there is constant C such that
max | (U − U h , δ V ) |≤ Ch , max || p1 − ph1 ,1 ||L2,h ( Ω′) ≤ Ch1 + Cτ , max || p2 − ph1 ,2 ||L2,h ( Ω′) ≤ Ch1 + Cτ .
t
t
1
t
1
Here (⋅; ⋅) is energy inner product, || ⋅ ||L2,h1 is finite difference analogue of L2 norm.
At first the algorithm constructed was tested on the 2D elastic problem with zero
kinematic boundary conditions. The distribution of Lame coefficients and error in L2 norm are
shown on Fig. 3. Fig. 4 shows the distribution of x -component of displacement vector. The
algorithm constructed has been generalized for the 3D case. The following problem has been
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chosen as a test. In the 3D domain [0;10]2 × [2;8] ( km3 ) the collector of τ -like shape with
height of 6.25 km and the characteristic cross-section size 0.25 km has been considered.
Permeabilities in fractures and pores were equal to 1 Darcy and 0.1 Darcy accordingly,
compressibility β = 2.5 ⋅10−5 bar. The Lame coefficients in the collector λ2 = 3.4 ⋅1010 Pa,
µ1 = 1.33 ⋅109 Pa, in the surrounding medium λ2 = 5.678 ⋅1010 Pa, µ 2 = 1.292 ⋅1010 Pa. The
coefficient α was considered to be equal to 10−18 m 2 , viscosity µ = 4 ⋅10−4 Pa ⋅ sec . Initial
conditions were geostatic: p1 = p2 = 0 ; boundary conditions: zero flux for p1 everywhere and
x
∈ [ 106tsec ; 2π + 106tsec ] where the flux was equal to
for p2 except a part of upper boundary 1km
x
25 ⋅105 (1 + 0.05sin(2π ( 1km
− 106tsec ))) Pa. For elastic part on lower boundary the displacement was
equal to 0 and remaining part of the boundary was free from action of forces.
Fig. 4. Distribution of displacement along the x -axis
Fig. 5. Pressure in pores and displacement along the x -axis
z
The section of the solution by a plane 1km
= 5 at the moment of time of 2 months is
shown on Fig. 5. The left part is a pressure in pores, the right one is distribution of displacement
along the x axis.
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Calculations have shown that the flow along a descending branch of a collector is
essentially one-dimensional, representing diffusion along it. The maximal values of
displacement, as one would expect, are near to boundary and collector branch examined. Such an
accurate representation of collector structure is achieved by interpolation of the solution inside
cells by means of vectors from M (Ξ) .
Acknowledgements: this approach was influenced by the interaction with S. Asvadurov,
V. Druskin, T. Habashy.
This project was realized under financial support of RFBR (grant №03-01-00641).
LITERATURE
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9th
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004