ices report 17-08 a multiscale fixed stress split ... · coupled flow and poromechanics in deep...
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ICES REPORT 17-08
April 2017
A multiscale fixed stress split iterative scheme for coupledflow and poromechanics in deep subsurface reservoirs
by
Saumik Dana, Benjamin Ganis, Mary. F. Wheeler
The Institute for Computational Engineering and SciencesThe University of Texas at AustinAustin, Texas 78712
Reference: Saumik Dana, Benjamin Ganis, Mary. F. Wheeler, "A multiscale fixed stress split iterative scheme forcoupled flow and poromechanics in deep subsurface reservoirs," ICES REPORT 17-08, The Institute forComputational Engineering and Sciences, The University of Texas at Austin, April 2017.
A multiscale fixed stress split iterative scheme for coupled flow
and poromechanics in deep subsurface reservoirs
Saumik Danaa, Benjamin Ganisa, Mary. F. Wheelera
aCenter for Subsurface Modeling, Institute for Computational Engineering and Sciences, UTAustin, Austin, TX 78712
Abstract
In coupled flow and poromechanics phenomena representing hydrocarbon production
or CO2 sequestration in deep subsurface reservoirs, the spatial domain in which fluid
flow occurs is usually much smaller than the spatial domain over which significant
deformation occurs. The typical approach is to either impose an overburden pressure
directly on the reservoir thus treating it as a coupled problem domain or to model
flow on a huge domain with zero permeability cells to mimic the no flow boundary
condition on the interface of the reservoir and the surrounding rock. The former
approach precludes a study of land subsidence or uplift and further does not mimic the
true effect of the overburden on stress sensitive reservoirs whereas the latter approach
has huge computational costs. In order to address these challenges, we augment the
fixed-stress split iterative scheme with upscaling and downscaling operators to enable
modeling flow and mechanics on overlapping nonmatching hexahedral grids. Flow
is solved on a finer mesh using a multipoint flux mixed finite element method and
mechanics is solved on a coarse mesh using a conforming Galerkin method. The
multiscale operators are constructed using a procedure that involves singular value
decompositions, a surface intersections algorithm and Delaunay triangulations. We
numerically demonstrate the convergence of the augmented scheme using the classical
Mandel’s problem solution.
Email addresses: [email protected] (Saumik Dana), [email protected] (Mary. F.Wheeler)
coupled problem
non-pay
reservoir
1000 ft-30000 ft
100 ft-1000 ft
Solve mechanics on coarse mesh
Solve flow on fine mesh
upscale
downscale
1000 ft-30000 ftfree surface
1000 ft-30000 ft
x
yz
g
Figure 1: Our multi-scale approach allows us to spatially decouple the flow and mechanics domains
with different finite element discretizations and impose boundary conditions individually on flow and
mechanics to more accurately represent deep subsurface activity. Typical dimensions are provided
for the sake of clarity.
Keywords: Fixed-stress split iterative scheme, Overlapping nonmatching
hexahedral grids, Upscaling and downscaling, Singular value decompositions,
Surface intersections, Delaunay triangulations, Mandel’s problem
1. Introduction
Solution schemes for coupled deformation-diffusion phenomena in porous media
can be broadly classified into fully coupled, loosely coupled and iteratively coupled
schemes. In a fully coupled scheme, the flow and mechanics equations are solved
2
simultaneously at each time step (Phillips and Wheeler [22], Jha and Juanes [17]).
The fully coupled approach is unconditionally stable, but requires careful implemen-
tation with substantial local memory requirements and specialized linear solvers. In
a loosely coupled scheme, the coupling between flow and mechanics is resolved only
after a certain number of flow time steps (Minkoff et al. [21]). Such a scheme is
only conditionally stable and requires a priori knowledge of the desired frequency of
geomechanical updates leading to an accurate solution.
Iteratively coupled schemes are those in which an operator splitting strategy (see
Armero and Simo [2], Schrefler et al. [23]) is used to split the coupled problem into
flow and mechanics subproblems. At each time step, either the flow or mechanics
problem is solved first, then the other problem is solved using the previous iterate
solution information alternatively (Wheeler and Gai [26], Mikelic and Wheeler [19],
Mikelic et al. [20]). This sequential procedure is iterated until the solution converges
to an acceptable tolerance. Carefully crafted convergence criteria lend solutions as ac-
curate as that obtained using a fully coupled approach. Iteratively coupled algorithms
have inherent advantages compared to fully coupled schemes from the standpoint of
customization, software reuse and code modularity (see Felippa et al. [9]). A numer-
ical comparison of the three techniques can be found in Dean et al. [8]. Kim et al.
[18] studied the properties of a few operator splitting strategies for the coupled flow
and poromechanics problem and recommended the fixed-stress split strategy where
the flow problem is solved first while freezing the total mean stress. Later, Mikelic
and Wheeler [19] rigorously proved the convergence of the fixed-stress split scheme
using the principle of contraction mapping with appropriately chosen metrics.
Previous attempts at solving the multi-scale problem include the works of Dean
et al. [8], Gai et al. [12], Ita and Malekzadeh [16] and Florez et al. [10]. Gai et al. [12]
reformulated an iterative sequential scheme as a special case of a fully coupled ap-
proach and implemented the algorithm on overlapping nonmatching rectilinear grids
but avoided 3D intersection calculations instead evaluating the displacement-pressure
3
Flow Loop
Coupling Iteration
Upscaling
Downscaling
Mechanics Solve
Augmented Scheme
Flow Loop
Mechanics Solve
Coupling Iteration
Existing Scheme
Figure 2: Comparison of the existing fixed-stress split scheme with the augmented fixed-stress split
scheme.
coupling submatrices using a midpoint integration rule. Florez et al. [10] implemented
a procedure in which a saddle-point system with mortar spaces on nonmatching in-
terfaces of a decomposed geomechanics domain is solved by applying a balancing
Neumann-Neumann preconditioner. It involved subdomain to mortar and mortar to
subdomain projections, Lagrange multiplier solve and parallel subdomain solves at
each time step with computationally expensive subdomain solves.
As shown in Figure 1, our multi-scale approach allows us to spatially decouple the
flow and mechanics domains with different discretizations thus giving us the option of
imposing more accurate boundary conditions on each subproblem. In this work, we
augment the operator splitting scheme of Mikelic and Wheeler [19] with multiscale
operators as shown in Figure 2 and further demonstrate the numerical convergence
of the augmented scheme using the Mandel’s problem solution as a benchmark. The
constructions of the multiscale operators are performed only once during the pre-
processing step thus avoiding the expense of the mortar based method of Florez et al.
[10]. To the best of our knowledge, this is the first time the concepts of discrete
geometry are being used to solve the coupled flow and poromechanics problem on
nonmatching distorted hexahedral grids. The paper is structured as follows: Section
2 presents the finite element formulation for flow and mechanics. Section 3 presents
4
the augmented solution algorithm. Section 4 presents the details about the singular
value decompositions, the surface intersections algorithm and Delaunay triangulations
used in constructing the operators. In Section 5, we numerically demonstrate the
convergence of the augmented scheme using the analytical solution to the classical
Mandel’s problem. Finally, in Section 6, we present our conclusions and discuss scope
for future work.
1.1. Preliminaries
Given a bounded, convex domain Ω ⊂ R3, Pj (Ω) denotes the set of restriction
of polynomials of total degree not greater than j to Ω. The set of square integrable
functions in Ω is L2(Ω) ≡f :
∫Ω|f |2 < ∞
with the inner product (v, w)Ω :=
∫Ωvw ∀ v, w ∈ L2(Ω). The Sobolev space of degree k consists of functions that possess
square integrable derivatives through order k i.e. Hk(Ω) ≡w : w ∈ L2(Ω), ∂αw ∈
L2(Ω), α ≤ k
. A natural space used in mixed formulations of second order PDEs is
H(div; Ω) ≡v : v ∈ (L2(Ω))3,∇ · v ∈ L2(Ω)
with the inner product 〈v,w〉Ω :=
∫Ω
v ·w ∀ v,w ∈ H(div; Ω).
2. Model equations and discretization
Let Ωflow ⊂ R3 be the flow domain with boundary ∂Ωflow = ΓflowD ∪ ΓflowN where
ΓflowD is Dirichlet boundary and ΓflowN is Neumann boundary. The mass conservation
equation (2.0.1) for coupled single phase flow (see Gai [11]) with the Darcy law (2.0.2)
for slightly compressible fluid (2.0.3) with boundary conditions (2.0.4) and initial
conditions (2.0.5) is
∂(φ∗ρ)
∂t+∇ · z = q (2.0.1)
z = −Kρ
µ(∇p− ρg∇d) (2.0.2)
ρ = ρ0ecf p (2.0.3)
p = g on ΓflowD × (0, T ], u · n = 0 on ΓflowN × (0, T ] (2.0.4)
5
p(x, 0) = p0(x) , ρ(x, 0) = ρ0(x) , φ(x, 0) = φ0(x) ∀x ∈ Ωflow (2.0.5)
where p : Ωflow × (0, T ] → R is the fluid pressure, z : Ωflow × (0, T ] → R3 is the
fluid flux, ρ is the fluid density, φ is the porosity, φ∗ = φ(1 + εv) is the so called
fluid fraction, εv is the volumetric strain, n is the unit outward normal on ΓflowN , q
is the source or sink term, K is the uniformly symmetric positive definite absolute
permeability tensor, µ is the fluid viscosity, cf is the fluid compressibility, d is the
depth and T > 0 is the time interval.
Let Ωporo ⊂ R3 be the poroelasticity domain with boundary ∂Ωporo = ΓporoD ∪ΓporoN
where ΓporoD is Dirichlet boundary and ΓporoN is Neumann boundary. Linear momentum
balance for the porous solid in the quasi-static limit of interest (2.0.6) (see Biot [4])
with small strain assumption (2.0.7) with boundary conditions (2.0.8) and initial
conditions (2.0.9) is
∇ · (σ0 + Dε− α(p− p0)I) +
f︷ ︸︸ ︷ρφg + ρr(1− φ)g = 0 (2.0.6)
ε(us) =1
2(∇us + (∇us)T ) (2.0.7)
us · n1 = 0 on ΓporoD × [0, T ], σTn2 = t on ΓporoN × [0, T ] (2.0.8)
p(x, 0) = p0(x), φ(x, 0) = φ0(x) ∀x ∈ Ωporo (2.0.9)
where us : Ωporo × [0, T ] → R3 is the solid displacement, ρr is the rock density, G is
the shear modulus, ν is the Poisson’s ratio, n1 is the unit outward normal to ΓporoD , n2
is the unit outward normal to ΓporoN , α is the Biot parameter, Kb is the bulk modulus
of the skeleton, f is body force per unit volume, t is the traction boundary condition,
ε is the strain tensor, σ0 is the in situ stress, D is the fourth order elasticity tensor
and I the is second order identity tensor. For x ∈ Ωporo, G = G(x), ν = ν(x) and
α = α(x) may have jump discontinuities.
2.1. Mixed formulation for single phase flow coupled with poroelasticity
Let T flowh be finite element partition of Ωflow comprising of distorted hexahedral
elements E. A locally mass conservative mixed formulation with enhanced BDDF1
6
spaces Vh ×Wh (see Appendix A) is employed. The problem statement is : Find
zh ∈ Vh, ph ∈ Wh such that ∀v ∈ Vh and ∀w ∈ Wh
〈µρ
K−1zh,v〉E − (ph,∇ · v)E = −(p,v · n)∂E/ΓflowD− (g,v · n)∂E∩ΓflowD
+ 〈ρg∇d,v〉E
(2.1.1)(
(φ∗ρ)n+1 − (φ∗ρ)n
∆t, w
)
E
+ (∇ · zh, w)E = (q, w)E
(2.1.2)
All the above terms are evaluated at time level n+1 unless explicitly stated otherwise.
Let v and w denote the bases for Vh and Wh respectively. (2.1.1), (2.1.2) are
linearized and recast as A B
−∆tBT C
δZ
k−1h
δpk−1h
=
−R1
−R2
(2.1.3)
for the kth Newton iteration where submatrices A, B, C and the nonlinear residuals
R1 and R2 are
Aij = 〈 µ
ρk−1K−1vi,vj〉E, Bij = −(wi,∇ · vj)E, Cij =
(((φ∗ρ)k−1cf + ρk−1∂φ
∗
∂p
)wi, wj
)
R1 = 〈 µ
ρk−1K−1zk−1
h ,v〉E − (pk−1h ,∇ · v)E + (pk−1,v · n)∂E/ΓflowD
+ (g,v · n)∂E∩ΓflowD
−〈ρk−1g∇d,v〉E
R2 =
((φ∗ρ)k−1 − (φ∗ρ)n, w
)
E
+ ∆t
[(∇ · zk−1
h , w)E − (q, w)E
]
(2.1.4)
where the term ∂φ∗
∂pis evaluated in equation (3.1.2) in section 3. A trapezoidal quadra-
ture rule (Ingram et al. [15]) is used to evaluate the term 〈 µρk−1 K
−1uk−1h ,v〉E as
〈 µ
ρk−1K−1uk−1
h ,v〉E =1
8
8∑
i=1
µ
ρk−1K−1(ri)JE(ri)u
k−1h (ri) · v(ri) (2.1.5)
where ri = (xi, yi, zi)T , i = 1, ..., 8 are vertices of E and JE = det(DFE) where DFE
is Jacobian of mapping FE. The details of the finite element mapping are given in
7
Appendix B. Ingram et al. [15] formulated a scheme in which quadrature rule (2.1.5) is
used to reduce system of equations (2.1.3) to a cell centered pressure stencil. Pressure
pk−1h in each element E is coupled with pressures in all elements that share a vertex
with E, i.e. a 27 point stencil is obtained. The resulting algebraic system is solved
for δpk−1h and the kth Newton iterate is obtained as
pkh = pk−1h + δpk−1
h (2.1.6)
2.2. Conforming Galerkin formulation for poroelasticity
Let T poroh be finite element partition of Ωporo comprising of distorted hexahedral
elements E. The problem statement is: Find ush ∈ Qh such that ∀q ∈ Qh
∫
E
ε(q) : Dε(ush) = −∫
E
ε(q) : σ0 +
∫
E
ε(q) : α(ph − p0)I +
∫
E
q · f +
∫
∂E∩ΓporoN
q · t
(2.2.1)
where Qh ≡ q ∈ (H1(E))3 : (q · n)ΓporoD= 0. Weak form (2.2.1) eventually leads
to the following system of equations for the nodal displacements Us
KUs = F (2.2.2)
K =
∫
E
BTDB
F = −∫
E
BTσ0 +
∫
E
BTα(ph − p0)I +
∫
E
NT f +
∫
∂E∩ΓporoN
NT t (2.2.3)
where N is the shape function matrix, B is the strain-displacement interpolation
matrix, K is refered to as the global stiffness matrix and F is refered to as the global
force vector. To simplify the computations, (2.2.2) is recast in compact engineering
notation (see Hughes [14]) wherein stresses σ, strains ε and identity tensor I are
represented as vectors and fourth order tensor D is represented as a second order
tensor. The matrices N and B are also recast appropriately.
8
3. Operator splitting algorithm
New time step
New fixed− stress iteration m
New flow iteration k
Solve for δpk−1,m keeping σv fixed
noConverged?
Solve for um
yes
Upscale pk,m, ρk,m
φ∗m = φ0 + α(ǫmv − ǫv0) +(1− α)(α− φ0)
Kb(pk,m − p0)
Downscale φ∗m
noyesConverged?
φ∗k,m = φ∗k−1,m
+
(α(1 + ǫv)− φ∗k−1,m
Kb
)δpk−1,m
Figure 3: Augmented fixed-stress split iterative scheme for poroelasticity coupled with single phase
flow.
The augmented fixed-stress split iterative scheme decouples the flow system and
mechanics system solving them sequentially at each time step. A flowchart showing
the framework for poroelasticity coupled with single phase flow is given in Figure 3.
9
3.1. Fluid fraction update during flow solve
Flow is solved first by freezing the total mean stress i.e. δσv = 0 (see Kim et al.
[18], Mikelic and Wheeler [19]). According to Geertsma [13], the relative porosity
variation in a deformable porous medium is approximated as
δφ
φ=
[1
φ
(αKb︷ ︸︸ ︷
1
Kb
− 1
Ks
)− 1
Kb
](δσv + δp
)(3.1.1)
where Kb and Ks represent the bulk modulus of the framework and solid grains
respectively and α = 1− KbKs
is the Biot’s constant (Biot [4]). Imposing the constraint
δσv = 0 in (3.1.1) results in
δφ =(α− φ)
Kb
δp→ φk = φk−1 +(α− φk−1)
Kb
δpk−1
where k refers to the Newton iteration for the flow solve. Using the relation φ∗ =
φ(1 + εv), we get
φ∗k
= φ∗k−1
+
[α(1 + εv)− φ∗k−1
Kb︸ ︷︷ ︸∂φ∗∂p
]δpk−1 (3.1.2)
Hence the fixed stress constraint provides us with a framework in which the fluid
fraction evolves with both the volumetric strain and the pore pressure during the
flow solve.
3.2. Fluid fraction update during mechanics solve
We now work on providing an expression for the fluid fraction update during the
mechanics solve. Invoking (3.1.1) and using the relation δσv = Kbδεv −αδp (see Biot
[4]), we get
δφ = (α− φ)δεv +(α− φ)(1− α)
Kb
δp (3.2.1)
Following the arguments of Coussy [7], linear poroelasticity consists in setting the
tangent properties (α − φ) and (α−φ)(1−α)Kb
in (3.2.1) as constants. Hence (3.2.1) can
10
be integrated in the form
φ− φ0 = (α− φ0)(εv − εv0) +(α− φ0)(1− α)
Kb
(p− p0)
with the relation φ∗ = φ(1 + εv) to obtain
φ∗ = φ0 + α(εv − εv0) +(α− φ0)(1− α)
Kb
(p− p0) +O(ε2v)
≈ φ0 + α(εv − εv0) +(α− φ0)(1− α)
Kb
(p− p0) (3.2.2)
where the O(ε2v) terms are neglected in lieu of the small strain assumption (2.0.7).
The details of the upscaling and downscaling procedure are presented next.
4. Upscaling and downscaling operators
The basic strategy of the multi-scale approach to the coupled problem is to up-
scale pore pressure and bulk density from fine scale fluid flow domain to the coarse
scale geomechanics domain and conversely downscale porosity from coarse scale ge-
omechanical domain to the fine scale fluid flow domain.
4.1. Upscaling pore pressure and bulk density
Let E represent the intersection polyhedron of a distorted hexahedral flow element
Eflow with a distorted hexehedral mechanics element Eporo. Let IEporo ≡ E : E ≡Eporo ∩ Eflow ∀Eflow ∈ T flowh represent the (possibly incomplete) partition of any
Eporo ∈ T poroh . Let pEflow
represent the cell-centered flow solution for pore pressure
at flow element Eflow such that
pE = pEflow
if E ∈ Eflow
After flow solve, the pore pressures are upscaled onto the mechanics grid via the
forcing term∫Eporo
BTαphI in (2.2.3) as follows
∫
EporoBTαphI =
∫
EporoBTα
[ ∑
E∈IEporo
Meas(E)
Meas(Eporo)pE
]I (4.1.1)
11
Also, the bulk densities are upscaled onto the mechanics grid via the forcing term∫Eporo
NT f in (2.2.3) as follows
∫
EporoNT f =
∫
EporoNTρr(1− φ)g +
∫
EporoNT
[ ∑
E∈IEporo
Meas(E)
Meas(Eporo)ρEφE
]g
(4.1.2)
In essence, the upscaled pore pressures and bulk densities on Eporo are local volume
averages over IEporo of the information obtained from the flow solve.
4.2. Downscaling porosity
Let IEflow ≡ E : E ≡ Eflow∩Eporo ∀Eporo ∈ T poroh represent the partition of any
Eflow ∈ T flowh . Let φEporo
be cell-centered mechanics solution for porosity at Eporo
such that
φE = φEporo
if E ∈ Eporo
Let Wh ≡ P0(T flowh ) represent the space of constants defined on T flowh . The L2
projection of porosity φ(x), x ∈ Ωflow onto Wh is obtained as
Pφ(x) =
∑E∈IEflow
φEMeas(E)
Meas(Eflow)∀x ∈ Eflow ∈ T flowh (4.2.1)
where P is the L2 projector. Details of the derivation of (4.2.1) are given in Appendix
C.
4.3. Constructing the operators
• The first step in constructing the multiscale operators is to obtain the equations
of the faces at the geometrically non-disjoint flow and mechanics elements. The
details of the process are given in Appendix D.
• The next step is to design an algorithm that uses the equations of the element
faces to obtain points on the periphery of the intersection polyhedron. Algo-
rithm 1 constructs the intersection polyhedron E ≡ Eflow ∩ Eporo where Eflow
12
Eporo
Eflow
E
a bc
d
e
fg
h
Figure 4: E ≡ abcdefgh = Eporo ∩ Eflow.
a bc
d
e
fg
h
a bc
d
e
fg
h
Figure 5: Solid circles representing points a and e are the vertices of Eflow inside Eporo. Hollow
circles representing points b, c, d, e, f , g and h are the end points of the curve traces obtained using
the surface intersections algorithm. The arrows represent the direction of the curve traces .
and Eporo represent the flow and mechanics elements respectively. A depiction
of E is provided in Figure 4. The algorithm proceeds by tracing the curves on
the intersection of the faces of Eflow and Eporo and is based on the predictor-
corrector approach of Bajaj et al. [3]. The starting points for the curve traces
are the set of vertices of Eflow inside Eporo. The predictor gives a second order
Taylor approximant to the trace of the intersection curves and the corrector
refines the approximant to points on the intersection curves using the Newton
method. As shown in Figure 5, the algorithm proceeds from points a and e
and traces the intersection curves to arrive at points b, c, d, e, f , g and h on
the periphery of the intersection polyhedron. Red arrows represent the curve
13
traces on the intersection of faces of Eflow. Blue arrows represent the curve
traces on the intersection of the faces of Eflow with faces of Eporo. Green arrows
represent the curve traces on the intersection of faces of Eporo. It is important
to note that Figures 4 and 5 are only a depiction of an intersection polyhedron
and that there is no restriction whatsoever that it be 8 - noded. The details of
our implementation of the predictor-corrector scheme are given in Appendix E.
• The final step is to use the set of points obtained on the periphery of the in-
tersection polyhedron to determine the measure of the intersection polyhedron.
We use a library code TetGen (Si. [24]) for this purpose. The library code takes
as input the coordinates of the set of points and decomposes the polyhedron
into multiple 3 simplices or tetrahedra, a process refered to as Delaunay tetrahe-
dralization. Let D be Delaunay tetrahedralization of E consisting of tetrahedra
T ∈ D such that E =⋃T∈D T . Denoting v0, v1, v2 and v3 as position vectors
of the vertices of T and (v1 − v0), (v2 − v0) and (v3 − v0) as columns of the
3× 3 matrix X T , we get
Meas(E) =∑
T∈D
Meas(T ) =∑
T∈D
1
3!det(X T )
14
Algorithm 1 Constructing Eif Eflow ∩ Eporo ← ∅ then
E ← ∅ . If Eflow and Eporo are geometrically disjoint, E is a null set
else if Eflow ⊂ Eporo then
E ← Eflow . If Eflow is inside Eporo, E ≡ Eflow
else . If Eflow and Eporo intersect
N ← ∅ . Initialize the set of points N on periphery of Efor i = 1, .., 6 do . Loop over six faces of Eflow
q ∈ A ∩ Sflowi . A is set of vertices of Eflow inside Eporo, Sflowi is the ith
face of Eflow, q is the starting point for the curve trace
for j = 1, .., 6 do . Loop over six faces of Eflow
while q ∈ Eflow ∧ q ∈ Eporo do
q← TRACE (q,Sflowi ,Sflowj ) . Sflowj is the jth face of Eflow
Q1 ← q . Final curve trace on the intersection of Sflowi with Sflowj
for k = 1, .., 6 do . Loop over six faces of Eporo
while q ∈ Eflow ∧ q ∈ Eporo do
q← TRACE (q,Sflowj ,Sporok ) . Sporok is the kth face of Eporo
Q2 ← q . Final curve trace on the intersection of Sflowj with Sporok
for l = 1, .., 6 do . Loop over six faces of Eporo
while q ∈ Eflow ∧ q ∈ Eporo do
q← TRACE (q,Sporok ,Sporol ) . Sporol is the lth face of Eporo
Q3 ← q . Final curve trace on the intersection of Sporok with
Sporol
N ← N ∪ (Q1 ⊕Q2 ⊕Q3) . Union of the curve traces
VE ← A⊕N . Union of A and N
15
5. Numerical results
The augmented scheme is implemented in the in-house parallel reservoir simu-
lator IPARS (Integrated Parallel Accurate Reservoir Simulator) at the Center for
Subsurface Modeling.
5.1. Mandel’s problem
2a
2b xy
2F
2F
p = 0σxx = 0σxy = 0
F
u · n = 0
u · n = 0
Figure 6: Circles indicate rollers and solid black boxes indicate rigid frictionless plates. The biaxial
symmetry of the problem allows us to replicate the problem by only modeling a quarter of the
domain as indicated by the red dotted line.
Solve flow on fine mesh
Solve mechanics on coarse meshUpscale
pore pressure
Downscaleporosity
Figure 7: Solution methodology
The analytical solution provided by Abousleiman et al. [1] to the Mandel’s problem
with compressible fluid and solid components serves as a benchmark for validation
of coupled flow and poroelasticity codes. The flow and mechanics domains, although
16
identical, have different finite element discretizations, with mechanics being resolved
on a coarser mesh. For the sake of clarity, we write the governing equations of the
elliptic-parabolic quasi-static Biot system applicable to the Mandel’s problem here as
follows
∇ · (σ0 + D(
1
2(∇us + (∇us)T )
)− α(p− p0)I) = 0 (5.1.1)
∂
∂t
(1
Mp+∇ · (αus)
)+∇ ·
(− k
µ∇p)
= 0 (5.1.2)
where (5.1.1) is the usual linear momentum balance for the solid phase with the small
strain assumption (see (2.0.6)-(2.0.9)) in the absence of gravity and (5.1.2) is obtained
by linearizing (see Gai [11]) (2.0.1) for one dimensional flow with gravity turned off.
The quantity M ≡(φ0cf + (α−φ0)(1−α)
Kb
)−1
in (5.1.2) is refered to as the Biot modulus
(see Biot and Willis [5]).
As shown in Figure 6, an infinitely long rectangular isotropic specimen is sand-
wiched between rigid, frictionless plates. The lateral sides are free from normal and
shear stress and pore pressure. At t = 0+, a force intensity of 2F N/m is applied to
the rigid plates. The initial and boundary conditions are
σxx|t=0 = σxy|t=0 = σyy|t=0 = 0, p|t=0 = 0 ∀x, y
σxx|x=±a = σxy|x=±a = σyx|y=±b = 0,
(∫ a
−aσyydx
)
y=±b= −2F ∀t
p|x=±a = 0,
(u · n
)
y=±b= 0 ∀t
where n is unit outward normal to the boundary. Plane strain condition is applicable
i.e. εzz = 0. Given the biaxial symmetry of the problem, only a quarter of the domain
needs to be modeled as shown in Figure 6. Following the approach of Mikelic et al.
[20], the boundary conditions are recast as
σxx|x=a = σxy|x=a = σyx|y=b = 0,
(us · n
)
y=b
= Uanalyticaly (b) ∀t (5.1.3)
(us · n
)
x=0
=
(us · n
)
y=0
= 0, p|x=a = 0 ∀t
17
Parameter Quantity Value
a x dimension 100m
b y dimension 10m
E Young’s Modulus 5.94× 109 Pa
ν Poisson’s ratio 0.2
νu Undrained Poisson’s ratio 0.3846
α Biot parameter 0.8
k Permeability 100md
B Skempton coefficient 0.8333
cf Fluid compressibility 3.03× 10−10 Pa−1
φ0 Initial porosity 0.2
µ Fluid viscosity 1.0 cp
ρ0 Reference fluid density 62.4 lbm/ft3
F Point load intensity 5.94× 108N/m
Table 1: Parameters for Mandel’s problem
(u · n
)
x=0
=
(u · n
)
y=0
=
(u · n
)
y=b
= 0 ∀t
where Uanalyticaly (b) in (5.1.3) is analytical solution for the y displacement at y = b.
We solve the system (5.1.2)-(5.1.1) using the augmented solution scheme on rec-
tilinear nonmatching grids as shown in Figure 7 and show its convergence by mea-
suring the upscaled pressure solution error. We employ the parameters given in
Table 1 and keep the refinement level r ≡max
E∈T flowh
diam(E)
maxE∈T poro
h
diam(E)fixed. Since the Biot
modulus 1M
is bounded below by a positive constant (as the initial porosity field
φ0 is strictly positive), optimality should be achieved when the pressure solution
error is measured in the L∞(L2) norm (see Phillips and Wheeler [22]). The er-
ror norm is computed using the midpoint quadrature rule: ‖ 1M
(p − ph)‖L∞(L2) ≡
18
T flowh T poroh ‖ 1M
(p− ph)‖L∞(L2) Rate
15× 15 6× 6 0.459× 10−1 -
20× 20 8× 8 0.339× 10−1 1.053
25× 25 10× 10 0.265× 10−1 1.104
30× 30 12× 12 0.213× 10−1 1.198
Table 2: Order of convergence of pore pressure solution using the augmented scheme for the Mandel’s
problem.
max0<τ≤T
( ∑E∈T poroh
|E|(p(τ,me)−ph(me)
M
)2) 1
2
where me is the center of mass of element E
and T is the total time. To minimize the effects of the error produced by time dis-
cretization, a small time step of 1 × 10−3 sec is chosen. As shown in Table 2, we
observe first order convergence for the pore pressure solution with the augmented
scheme for r = 2.5. A fractional value of r ensures the cardinality |E| of the set
E of intersection polyhedra is non-zero i.e. the number of instances of intersecting
flow and mechanics elements is not zero. It is important to note that there is no
restriction posed on the value of r and we choose a value of 2.5 only for the sake of
convenience.
We then compare the upscaled pore pressure solution at the cell-center closest
to the origin of the quarter domain with the analytical solution for all the above
combinations of T flowh , T poroh . The reason for choosing the cell-center closest to the
origin of the quarter domain is that the classical non-monotonic pore pressure re-
sponse, which we intend to replicate in our numerical model, is expected only near
the central region of the specimen (see Abousleiman et al. [1]). We also compare the
computed x-displacement at the free end x = a with the analytical solution. The total
simulation time is 50000 sec with a time step of 10 sec. According to the analytical
solution,
• At the instant of loading, a uniform pressure rise of ∆p(x, y, 0+) = FB(1+νu)3a
19
should be observed.
• After the initial outward movement of ux(a, y, 0+) = Fνu
2G, the side boundaries
will contract toward the center and its final state should be ux(a, y,∞) = Fν2G
.
The pore pressure and displacement solutions are non-dimensionalized by multiplying
with ( aF
) and (2GF
) respectively. As shown in Figures 8 and 9, we observe an excellent
match with the expected results for all the above combinations.
0 1 2 3 4 5
x 104
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time (seconds)
aP F
Analytical
Numerical,T poroh ≡ 6× 6, T flow
h ≡ 15× 15
0 1 2 3 4 5
x 104
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time (seconds)
aP F
Analytical
Numerical,T poroh ≡ 8× 8, T flow
h ≡ 20× 20
0 1 2 3 4 5
x 104
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time (seconds)
aP F
Analytical
Numerical,T poroh ≡ 10× 10, T flow
h ≡ 25× 25
0 1 2 3 4 5
x 104
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Time (seconds)
aP F
Analytical
Numerical,T poroh ≡ 12× 12, T flow
h ≡ 30× 30
Figure 8: Non-monotonic pore pressure response at the cell-center closest to the origin for the
Mandel’s problem with nonmatching grids. limt→0+
(aP (xc,yc,t)
F
)= B(1+νu)
3 = 0.3846 where xc, yc are
coordinates of cell-center closest to the origin.
20
0 1 2 3 4 5
x 104
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
0.36
0.38
0.4
Time (seconds)
2Gux
F
Analytical
Numerical,T poroh ≡ 6× 6, T flow
h ≡ 15× 15
0 1 2 3 4 5
x 104
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
0.36
0.38
0.4
Time (seconds)
2Gux
F
Analytical
Numerical,T poroh ≡ 8× 8, T flow
h ≡ 20× 20
0 1 2 3 4 5
x 104
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
0.36
0.38
0.4
Time (seconds)
2Gux
F
Analytical
Numerical,T poroh ≡ 10× 10, T flow
h ≡ 25× 25
0 1 2 3 4 5
x 104
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
0.36
0.38
0.4
Time (seconds)
2Gux
F
Analytical
Numerical,T poroh ≡ 12× 12, T flow
h ≡ 30× 30
Figure 9: Displacement response at the free end for the Mandel’s problem with nonmatching grids.
limt→0+
(2Gux(a,y,t)
F
)= νu = 0.3846 and lim
t→∞
(2Gux(a,y,t)
F
)= ν = 0.2.
21
5.2. Single well in an infinite confined aquifer
Figure 10: Schematic for single well in an infinite confined aquifer problem from Verruijt [25]
The problem considered here is that of flow to a rate specified production well in
a confined compressible aquifer of thickness H as shown in Figure 10. The analytical
solution for the vertical displacement of the upper surface is given as (see Verruijt
[25])
w =αQµ
4πk(Kb + 4G/3)E1(r2/4ct) (5.2.1)
where α is the Biot constant, Q is the production rate, µ is the fluid viscosity, k is the
fluid permeability, Kb is the drained bulk modulus, G is the shear modulus, r is the
radial coordinate measured from the center of the well, t is the total time and E1(x)
is the exponential integral E1(x) ≡∞∫x
exp(−t)t
dt and c is the diffusivity coefficient given
by c = k(Kb+4G/3)
µ
(α2+
(φ0cf+(α−φ0)(1−α)/Kb
)(Kb+4G/3)
) where φ0 is the initial porosity and
cf is the fluid compressibility. The underlying assumptions in the development of
(5.2.1) are that there are no horizontal deformations in the aquifer and that the total
vertical stress remains constant during the development of the hydrological process.
We employ the parameters given in Table 3. The flow grid is at a depth of 800 ft
with the mechanics grid extending all the way to the traction free surface. The lateral
extents of both the flow and mechanics domains are 8000 ft. As shown in Figure 11,
22
Parameter Quantity Value
E Young’s Modulus 3.4474× 109 Pa
ν Poisson’s ratio 0.2
α Biot parameter 0.8
cf Fluid compressibility 1.45× 10−8 Pa−1
φ0 Initial porosity 0.2
k Permeability 100md
µ Fluid viscosity 1.0 cp
Q Injection rate 10STB/day
H Aquifer thickness 200 ft
Ωflow Flow domain (800 ft, 0 ft, 0 ft) To (1000 ft, 8000 ft, 8000 ft)
Ωporo Mechanics domain (0 ft, 0 ft, 0 ft) To (1000 ft, 8000 ft, 8000 ft)
T flowh Flow grid 10× 200× 200
T poroh Mechanics grid 20× 100× 100
∆t Time step 0.1 day
Table 3: Parameters for single well in an infinite confined acquifer problem
we get an excellent match for the computed vertical displacement with the analytical
solution thus validating our multi-scale scheme in the presence of wells.
23
0 5 10 15 20−5
−4
−3
−2
−1
0
−ux/(α
qH)
r/H
AnalyticalNumerical
Figure 11: Comparison of the computed vertical displacement with the analytical solution for the
case of rate specified production well in an infinite confined aquifer at t = 10 days. r is the radial
coordinate measured from the center of the well and H is the aquifer thickness. q = Qµ4πkH(Kb+4G/3) .
24
6. Conclusions and Outlook
We sucessfully implemented a procedure that enables the fixed-stress split itera-
tive sequential strategy to model flow and poromechanics on differing length scales.
The procedure uses the geometry of the finite element triangulations of the flow and
poromechanics subdomains to construct upscaling and downscaling operators. We
numerically demonstrated the convergence of the augmented scheme using the ana-
lytical solution to the classical Mandel’s problem. We also validated the multi-scale
scheme for a problem which involves a rate specified well with an overburden.
In this work, we considered coupled single phase flow with poromechanics and
further investigations on coupled two-phase flow as well as coupled compositional flow
with poromechanics will follow as future work. Another area of active investigation
is the construction of multi-scale operators with heterogeneity in properties at both
the fine scale and the coarse scale factored in.
Acknowledgements
The first author Saumik Dana would like to thank Gurpreet Singh (Research
Associate at the Center for Subsurface Modeling) for his invaluable suggestions during
the course of the preparation of this document.
Appendix A. Enhanced BDDF1 spaces
For the sake of clarity, we provide a brief description of the mixed finite element
spaces used in the flow model. Let V∗h×Wh be the lowest order BDDF1 MFE spaces
on hexahedra (see Brezzi et al. [6]). On the reference unit cube these spaces are
defined as
V∗(E) = P1(E) + r0 curl(0, 0, xyz)T + r1 curl(0, 0, xy2)T + s0 curl(xyz, 0, 0)T
+ s1 curl(yz2, 0, 0)T + t0 curl(0, xyz, 0)T + t1 curl(0, x
2z, 0)T
W (E) = P0(E)
25
FE
v11
v12v13
v21
v23v22
v31
v33
v32
v41
v42
v43
v51
v52
v53
v61
v63
v62
v72
v71
v73
v81
v82
v83
v11
v12v13 v22 v23
v21
v33
v31
v32v42
v41
v43
v51
v52
v53 v61v62
v63
v71
v72v73
v81v82
v83
Figure A.12: Degrees of freedom and basis functions for the enhanced BDDF1 velocity space on
hexahedra.
with the following properties
∇ · V∗(E) = W (E), and ∀v ∈ V∗(E), ∀e ⊂ ∂E, v · ne ∈ P1(e)
The multipoint flux approximation procedure requires on each face one velocity degree
of freedom to be associated with each vertex. Since the BDDF1 space V∗h has only
three degrees of freedom per face, it is augmented with six more degrees of freedom
(one extra degree of freedom per face). Since the constant divergence, the linear
independence of the shape functions and the continuity of the normal component
across the element faces are to be preserved, six curl terms are added (Ingram et al.
[15]). Let Vh ×Wh be the enhanced BDDF1 spaces on hexahedra. On the reference
unit cube these spaces are
V(E) = V∗(E) + r2 curl(0, 0, x2z)T + r3 curl(0, 0, x
2yz)T + s2 curl(xy2, 0, 0)T
+ s3 curl(xy2z2, 0, 0)T + t2 curl(0, yz
2, 0)T + t3 curl(0, xyz2, 0)T
W (E) = P0(E)
with the following properties
∇ · V(E) = W (E), and ∀v ∈ V(E), ∀e ⊂ ∂E, v · ne ∈ Q1(e)
where Q1 is the space of bilinear functions. Since dimQ1(e) = 4, the dimension of
V(E) is 24 as shown in Figure A.12.
26
Appendix B. Finite element mapping
r2
r3r4
r5 r6
r7r8
r1
FE
r1 r2
r3r4
r5r6
r7r8
Figure B.13: Trilinear mapping FE : E → E for 8 noded distorted hexahedral elements. The faces
of E can be non-planar.
Let Th be finite element partition of Ω ⊂ R3 consisting of distorted hexahedral
elements E where h = maxE∈Th diam(E). Let ri, i = 1, .., 8 be the vertices of E. Now
consider a reference cube E with vertices r1 = [0 0 0]T , r2 = [1 0 0]T , r3 = [1 1 0]T ,
r4 = [0 1 0]T , r5 = [0 0 1]T , r6 = [1 0 1]T , r7 = [1 1 1]T and r8 = [0 1 1]T as
shown in Figure B.13. Let x = (x, y, z) ∈ E and x = (x, y, z) ∈ E. The function
FE(x) : E → E is
FE(x) = r1(1− x)(1− y)(1− z) + r2x(1− y)(1− z) + r3xy(1− z) + r4(1− x)y(1− z)
+r5(1− x)(1− y)z + r6x(1− y)z + r7xyz + r8(1− x)yz
Denote Jacobian matrix by DFE and let JE = det(DFE). Defining rij ≡ ri − rj, we
have
DFE(x) =
r21 + (r34 − r21)y + (r65 − r21)z + ((r21 − r34)− (r65 − r78))yz;
r41 + (r34 − r21)x+ (r85 − r41)z + ((r21 − r34)− (r65 − r78))xz;
r51 + (r65 − r21)x+ (r85 − r41)y + ((r21 − r34)− (r65 − r78))xy
3×3
Denote inverse mapping by F−1E , its Jacobian matrix by DF−1
E and let JF−1E
=
det(DF−1E ) such that
DF−1E (x) = (DFE)−1(x); JF−1
E(x) = (JE)−1(x)
27
Let φ(x) be any function defined on E and φ(x) be its corresponding definition on
E. Then we have
∇φ = (DF−1E )T (x) ∇φ = (DFE)−T (x) ∇φ (B.1)
Appendix C. Downscaling porosity
Let Wh ≡ P0(T flowh ) represent the space of constants defined on T flowh . Let P be
the L2 projection of porosity φ(x), x ∈ Ωflow onto Wh. Define P by
∫
Ωflow
(φ(x)− (Pφ)(x)
)w = 0 ∀w ∈ Wh (C.1)
Let φ′(x′), x′ ∈ Eflow denote the restriction of φ(x), x ∈ Ωflow to Eflow ∈ T flowh . Let
w′ ∈ Wh denote the restriction of w ∈ Wh to Eflow. We rewrite (C.1) as
∑
E∈T flowh
∫
E
φ′(x′)w′ −∫
Ωflow(Pφ)(x)w = 0 (C.2)
Let IEflow = E : E = Eflow ∩ Eporo ∀Eporo ∈ T poroh represent the partition of any
Eflow ∈ T flowh . Let φEporo
be cell-centered mechanics solution for porosity at Eporo
such that
φE = φEporo
if E ∈ Eporo
Then φ′(x′) is defined by discontinuous piecewise constants over IEflow as follows
φ′(x′) = φE ∀x′ ∈ E ∈ IEflow (C.3)
From (C.3), noting that C.2 is satisfied for w′ = 1 ∈ Wh, we get
∑
E∈T flowh
∑
E∈IEflow
∫
EφE −
∫
Ωflow(Pφ)(x)w = 0 (C.4)
Since Pφ ∈ Wh, we can write it in terms of discontinuous piecewise constants as
Pφ(x) = γEflow ∀x ∈ Eflow ∈ T flowh (C.5)
28
Substituting (C.5) in (C.4), and again noting that w = 1 ∈ Wh, we get
∑
E∈T flowh
∑
E∈IEflow
∫
EφE −
∑
E∈T flowh
∫
E
γEflow
= 0
or∑
E∈T flowh
[ ∑
E∈IEflowφEMeas(E)− γEflowMeas(Eflow)
]= 0
from which we finally get γEflow
as
γEflow
=
∑E∈IEflow
φEMeas(E)
Meas(Eflow)∀Eflow ∈ T flowh (C.6)
Appendix D. Obtaining equations of the element faces
a
b
c
de
f
g
h
a
e
h
dae
f
b
e
h
f
g
b
f
g
c
a
d
b
c
h
d
g
c
Figure D.14: A representation of hexahedral element E ≡ abcdefgh with its six faces aehd, abfe,
ehgf , bcgf , cdhg and adcb. The coordinate information of the four vertices of each of the faces is
used to obtain its equation.
Let S(x) = 0, x ≡ (x, y, z) ∈ e be the equation of face e of element E with its
vertices vi ≡ (xi, yi, zi), i = 1, 2, 3, 4. A representation of E with its faces is provided
29
in Figure D.14. Define S(x) by a trilinear as
S(x) =[xyz xy yz xz x y z 1
]c8×1 (D.1)
where c8×1 is the vector of coefficients to be determined. Since the equation S(x) = 0
is satisfied at each of the four vertices defining the face, we get the system of equations
M4×8︷ ︸︸ ︷
x1y1z1 x1y1 y1z1 x1z1 x1 y1 z1 1
x2y2z2 x2y2 y2z2 x2z2 x2 y2 z2 1
x3y3z3 x3y3 y3z3 x3z3 x3 y3 z3 1
x4y4z4 x4y4 y4z4 x4z4 x4 y4 z4 1
c8×1 =
0
0
0
0
4×1
for c. The objective is to determine c ∈ Null(M). First, we get the SVD of M as
M4×8 = U4×4σ4×8VT8×8 (D.2)
where σ = diag(σ1, .., σr) is diagonal matrix of singular values of M and the columns
of U and V are left and right singular vectors of M respectively. Since the nullspace
of M is spanned by right singular vectors corresponding to the vanishing singular
values of M, we express c as
c8×1 =[V[:, r + 1] . . . V[:, 8]
]8×(8−r)
κ(8−r)×1 (D.3)
where κ is the vector of coefficients and r is rank of M. The objective now is to
determine κ. First, using (D.1), we obtain an expression for the gradient ∇S(x) of
S(x) as
∇S(x) =
H(x,y,z)3×8︷ ︸︸ ︷
yz y 0 z 1 0 0 0
xz x z 0 0 1 0 0
xy 0 y x 0 0 1 0
[V[:, r + 1] . . . V[:, 8]
]8×(8−r)
κ(8−r)×1
(D.4)
30
Let S(x) be corresponding definition on face e of reference element E of S(x) on face
e of actual element E. Then, from (B.1),
∇S(x) = (DFE)−T (x) ∇S(e) (D.5)
where ∇S(e) can be either[1 0 0
]T,[0 1 0
]Tor[0 0 1
]Tdepending on whether
e is normal to x, y or z axis. Equating (D.4) and (D.5) for all four vertices of e ∈ E,
we get the following system of equations for κ(8−r)×1
H(x1, y1, z1)
H(x2, y2, z2)
H(x3, y3, z3)
H(x4, y4, z4)
12×8
[V[:, r + 1] . . . V[:, 8]
]8×(8−r)
κ(8−r)×1 = B12×1 (D.6)
where B is obtained as
B[(i− 1) ∗ 3 + 1→ i ∗ 3, 1] = (DFE)−T (vi) ∇S(e)
where vi, i = 1, 2, 3, 4 on e ∈ E is the corresponding definition of vi, i = 1, 2, 3, 4 on
e ∈ E. The solution κ of (D.6) is substituted into (D.3) to obtain c, which is then
substituted into (D.1) to obtain the polynomial expression of S(x).
Appendix E. Tracing surface intersections
As shown in Figure E.15, a second order Taylor approximant is used as a predictor
to the trace of the intersection curve of surfaces S1 and S2 with the trace being stored
in qp. Next, a corrector scheme is implemented that refines the estimate of qp to a
point qc on the intersection curve. The schemes are elucidated in sections Appendix
E.1 and Appendix E.2 respectively.
31
S1
S2
q
qp
qc
Figure E.15: qp is the predictor to the trace of S1 ∩ S2 represented by red solid line. qc is the
corrector to qp.
Appendix E.1. Predictor scheme
The intersection curve r(s) of surfaces S1 and S2 with initial point q is expressed
as a second order Taylor interpolant qp(s) with arc length parameter s as
r(s) = r(0) + sr′(0) +s2
2!r′′(0) + e(s) ≡ q + sr′(q) +
s2
2!r′′(q)
︸ ︷︷ ︸qp(s)
+e(s)
where r(0) ≡ q is initial point for curve tracing, r′(0) ≡ r′(q) is unit tangent to
curve at initial point, r′′(0) ≡ r′′(q) is curvature at initial point and e(s) = O(s3) is
error of the quadratic interpolant to r at s = 0. We assume s = 0.1, a value small
enough to make qp(s) an accurate estimate of r(s) i.e. |e(s)| << |p(s)|. As long as
q is not singular on S1 or on S2, the surface gradients ∇S1(q) and S2(q) are linearly
independent [Bajaj et al. [3]] and the unit tangent vector r′(q) is obtained as
r′(q) =∇S1(q)×∇S2(q)
‖∇S1(q)×∇S2(q)‖ (E.1)
It follows that r′(q) is perpendicular to both the surface gradients such that ∇S1(q) ·r′(q) = ∇S2(q) ·r′(q) = 0 implying the vectors ∇S1(q), ∇S2(q) and r′(q) are linearly
independent. Any vector in R3 can be expressed as linear combination of these three
as dim(R3) = 3. In particular,
r′′(q) = αr′(q) + β∇S1(q) + γ∇S2(q) (E.2)
32
The points on the curve r(s) are defined as solutions of Sj(x, y, z) = Sj(r(s)) = 0,
j = 1, 2. The Taylor expansion of Sj(r(s)), j = 1, 2 with q ≡ r′(0) is
Sj(r(s)) = Sj(q) + s∇Sj(q) · r′(q) +s2
2!
[∇Sj(q) · r′′(q) + r′(q) ·HSj(q) · r′(q)
]
where HSj(q) is the Hessian of the surface Sj evaluated at q. Since the intersection
curve satisfies Sj(r(s)) ≡ 0, j = 1, 2, the coefficient of each power of s in Sj(r(s))
must be zero. We already know that the coefficient of s in Sj(r(s)) is zero i.e.
∇Sj(q) · r′(q) = 0, j = 1, 2. Equating the coefficient of s2 in Sj(r(s)) to zero, we get
∇Sj(q) · r′′(q) = −r′(q) ·HSj(q) · r′(q)
∇Sj(q) ·(αr′(q) + β∇S1(q) + γ∇S2(q)
)= −r′(q) ·HSj(q) · r′(q)
and noting again that ∇Sj(q) · r′(q) = 0, j = 1, 2, we get the following system of
equations for β and γ∇S1(q) · ∇S1(q) ∇S1(q) · ∇S2(q)
∇S2(q) · ∇S1(q) ∇S2(q) · ∇S2(q)
βγ
= −
r′(q) ·HS1(q) · r′(q)
r′(q) ·HS2(q) · r′(q)
(E.3)
Solution of (E.3) and the choice α = 0 leads to a unique vector r′′(q) in (E.2). The
second order interpolant qp(s) is finally obtained as
qp(s) = q + 0.1∇S1(q)×∇S2(q)
‖∇S1(q)×∇S2(q)‖ +0.01
2!
[β∇S1(q) + γ∇S2(q)
](E.4)
Appendix E.2. Corrector scheme
Given the quadratic interpolant to the curve at qp in (E.4), we refine its estimate
to a point on the curve by generating a sequence of points q1, q2, · · · → qc with
q0 = qp. The Newton method for the solution of Sj(r(s)) = 0, j = 1, 2 at r(s) = qk
where k is the iteration number is
∇Sj(qk) ·∆k︷ ︸︸ ︷
(qk+1 − qk) = −Sj(qk), k = 0, 1, ..., n (E.1)
Expressing ∆k as a linear combination of r′(qk), ∇S1(qk) and ∇S2(qk) as
∆k = ςkr′(qk) + ϑk∇S1(qk) + ϕk∇S2(qk) (E.2)
33
and substituting in (E.1) results in
∇Sj(qk) ·(ςkr′(qk) + ϑk∇S1(qk) + ϕk∇S2(qk)
)= −Sj(qk), k = 0, 1, ..., n
The choice ςk = 0 leads to the following system of equations for ϑk and ϕk∇S1(qk) · ∇S1(qk) ∇S1(qk) · ∇S2(qk)
∇S2(qk) · ∇S1(qk) ∇S2(qk) · ∇S2(qk)
ϑkϕk
= −
S1(qk)
S2(qk)
(E.3)
Solution of (E.3) is alongwith (E.2) is used to obtain ∆k. The Newton method (E.1)
is iterated until a convergence criterion is met as shown in Algorithm 2.
Algorithm 2 Predictor-Corrector scheme
function TRACE (q,S1,S2) β
γ
←
∇S1(q) · ∇S1(q) ∇S1(q) · ∇S2(q)
∇S2(q) · ∇S1(q) ∇S2(q) · ∇S2(q)
−1 −r′(q) ·HS1(q) r′(q)
−r′(q) ·HS2(q) r′(q)
qp ← q + 0.1 ∇S1(q)×∇S2(q)‖∇S1(q)×∇S2(q)‖ + 0.01
2
[β∇S1(q) + γ∇S2(q)
]. Second order
approximant
k ← 0
∆0 ← q0 ← qp . Initial guess to the Newton method is the second order
approximant
while (‖∆k‖ > 10−6‖qk‖) do . Newton Loop ϑk(qk)
ϕk(qk)
←
∇S1(qk) · ∇S1(qk) ∇S1(qk) · ∇S2(qk)
∇S2(qk) · ∇S1(qk) ∇S2(qk) · ∇S2(qk)
−1 −S1(qk)
−S2(qk)
∆k ← ϑk(qk)∇S1(qk) + ϕk(qk)∇S2(qk)
qk+1 ← qk + ∆k
k ← k + 1
qc ← qk+1
q← qc
34
References
[1] Y. Abousleiman, A.H.D. Cheng, L. Cui, E. Detournay, and J.C. Roegiers. Man-
del’s problem revisited. Geotechnique, 46(2):187–195, 1996.
[2] F. Armero and J. C. Simo. A new unconditionally stable fractional step method
for non-linear coupled thermomechanical problems. International Journal for
Numerical Methods in Engineering, 35(4):737–766, 1992.
[3] C.L. Bajaj, C.M. Hoffmann, R.E. Lynch, and J.E.H. Hopcroft. Tracing surface
intersections. Computer Aided Geometric Design, 5(4):285–307, 1988.
[4] M.A. Biot. General theory of three dimensional consolidation. Journal of Applied
Physics, 12:155–164, 1941.
[5] M.A. Biot and D.G. Willis. The elastic coefficients of the theory of consolidation.
Journal of Applied Mechanics, 24:594–601, 1957.
[6] F. Brezzi, J. Douglas, R. Duran, and M. Fortin. Mixed finite elements for second
order elliptic problems in three variables. Numerische Mathematik, 51(2):237–
250, 1987.
[7] O. Coussy. Poromechanics. Wiley, 2nd ed edition, 2004.
[8] R.H. Dean, X. Gai, C.M. Stone, and S.E. Minkoff. A comparison of techniques
for coupling porous flow with geomechanics. SPE Journal, 11(1):132–140, 2006.
[9] C.A. Felippa, K.C. Park, and Charbel Farhat. Partitioned analysis of coupled
mechanical systems. Computer Methods in Applied Mechanics and Engineering,
190(24):3247–3270, 2001.
[10] H. Florez, M.F. Wheeler, A.A. Rodriguez, M. Palomino, and E. Jorge. Domain
decomposition methods applied to coupled flow-geomechanics reservoir simula-
tion. In SPE Reservoir Simulation Symposium - The Woodlands, Texas, USA
(2011-02-21).
35
[11] X. Gai. A Coupled Geomechanics and Reservoir Flow Model in Parallel Com-
puters. PhD thesis, The University of Texas at Austin, 2004.
[12] X. Gai, S. Sun, M.F. Wheeler, and H. Klie. A timestepping scheme for coupled
reservoir flow and geomechanics on nonmatching grids. In SPE Annual Technical
Conference and Exhibition - (2005.10.9-2005.10.12), 2005.
[13] J. Geertsma. The effect of fluid pressure decline on volumetric changes of porous
rocks. SPE, 210:331–340, 1957.
[14] T.J.R. Hughes. The Finite Element Method: Linear Static and Dynamic Finite
Element Analysis. Dover Civil and Mechanical Engineering. Dover Publications,
2000.
[15] R. Ingram, M.F. Wheeler, and I. Yotov. A multipoint flux mixed finite element
method on hexahedra. SIAM Journal of Numerical Analysis, 48(4):1281–1312,
2010.
[16] J. Ita and F. Malekzadeh. A true poroelastic up and downscaling scheme for
multi-scale coupled simulation. In Society of Petroleum Engineers SPE Reservoir
Simulation Symposium - Houston, Texas, USA (2015-02-23).
[17] B. Jha and R. Juanes. A locally conservative finite element framework for the
simulation of coupled flow and reservoir geomechanics. Acta Geotechnica, 2(3):
139–153, 2007.
[18] J. Kim, H.A. Tchelepi, and R. Juanes. Stability, accuracy and efficiency of
sequential methods for coupled flow and geomechanics. SPE Journal, 16(2):
249–262, 2011.
[19] A. Mikelic and M.F. Wheeler. Convergence of iterative coupling for coupled flow
and geomechanics. Computational Geosciences, 17(3):455–461, 2013.
36
[20] A. Mikelic, B. Wang, and M.F. Wheeler. Numerical convergence study of iterative
coupling for coupled flow and geomechanics. In 13th European conference on the
Mathematics of Oil recovery, Biarritz, France, 2012.
[21] S.E. Minkoff, C.M. Stone, S. Bryant, M. Peszynska, and M.F. Wheeler. Cou-
pled fluid flow and geomechanical deformation modeling. Journal of Petroleum
Science and Engineering, 38:37–56, 2003.
[22] P.J. Phillips and M.F. Wheeler. A coupling of mixed and continuous galerkin
finite element methods for poroelasticity ii: the discrete-in-time case. Computa-
tional Geosciences, 11(2):145–158, 2007.
[23] B.A. Schrefler, L. Simoni, and E. Turska. Standard staggered and staggered
newton schemes in thermo-hydro-mechanical problems. Computer Methods in
Applied Mechanics and Engineering, 144(1-2):93–109, 1997.
[24] Hang Si. Tetgen, a delaunay-based quality tetrahedral mesh generator. ACM
Trans. on Mathematical Software, 41(2), 2015.
[25] A. Verruijt. Theory and Problems of Poroelasticity. Delft University of Technol-
ogy, 2013.
[26] M.F. Wheeler and X. Gai. Iteratively coupled mixed and galerkin finite element
methods for poro-elasticity. Numerical Methods for Partial Differential Equa-
tions, 23(4):785–797, 2007.
37