ices report 17-08 a multiscale fixed stress split ... · coupled flow and poromechanics in deep...

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


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


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


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


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


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


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


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


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


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

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