1
B004 A FLUX-VECTOR-BASED GREEN ELEMENT
METHOD FOR HIGHLY-HETEROGENEOUS
MEDIA
1
1
2
P. LORINCZI , S.D. HARRIS & L. ELLIOTT
1
Rock Deformation Research, School of Earth Sciences, University of Leeds, Leeds, LS2 9JT, UK
2
Department of Applied Mathematics, University of Leeds, Leeds, LS2 9JT, UK
Abstract
The Green element method (GEM) is a modern technique for solving nonlinear problems encountered in flow in
porous media. It combines the second-order accuracy of the boundary element method with the efficiency and
versatility of the finite element method. The high accuracy of the GEM comes from the direct representation of the
normal fluxes as unknowns. However, the classical GEM procedure, which overcomes the difficulty imposed by a
large number of normal fluxes at each internal node by approximating them in terms of the primary variable, leads
to a diminution of the overall accuracy.
To maintain the second-order accuracy, another approach was proposed, namely the ‘flux-vector-based GEM’,
or ‘q-based’ GEM. According to this approach, only two and three unknown flux components are required for a
node in 2D and 3D domains, respectively. An important advantage of this approach is that the flux vector at each
node is determined directly.
We present a comparison between the results obtained using the classical GEM and those obtained using the ‘qbased’ GEM for problems in heterogeneous media with permeability changes of several orders of magnitude. In
these situations, the classical GEM fails to produce sensible results, whilst our novel development of the ‘q-based’
GEM is in agreement with both the boundary element method and the control volume method. However, to acquire
results to the same degree of accuracy, these two approaches are less efficient than the ‘q-based’ GEM. This new
approach is thus suitable for overcoming the difficulties present when modelling flow in heterogeneous media with
rapid and high-order changes in material parameters. We can therefore apply the ‘q-based’ GEM to flow through
layered sequences, across partially-sealing faults and around wells.
1. Introduction
The Green element method (GEM) is a promising new numerical technique for solving nonlinear
problems encountered in flow in porous media. Introduced by Taigbenu [4], this method is derived from
the boundary element method (BEM) by expressing its domain integrals over the meshes arising from the
finite element method (FEM). By implementing the singular boundary theory in an element-by-element
fashion in such a way that the calculations of a nodal solution require only local support, the GEM is able
to fully accommodate the heterogeneity of the medium parameters.
2. Mathematical formulation of the standard GEM
Let us first consider the steady problem governed by the following second-order partial differential
equation on the two-dimensional domain Λ :
∇ ⋅ ( K∇p ) = − F ( x ) on Λ ,
(1)
where p(x) is the dependent or primary variable (pressure in our case), K(x,p) is the permeability, which,
in general, is a second-order ranked tensor, F(x) is a known internal/external forcing that accounts for
either point or distributed imposed inputs, and x denotes the Cartesian coordinates. Equation (1) is
derived by combining the continuity equation at steady state with the phenomenological law
q = − K ∇p ,
(2)
in which q(x) is the flux (velocity) vector. The specified conditions along the domain boundary Γ = ∂Λ
may be any combination of the following three types:
(i) Dirichlet condition: p = p ( x ) on Γ1 ,
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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(ii) Neumann condition: q n = q n ( x ) on Γ2 ,
(iii) Robin condition: p + α ( x )q n = β ( x ) on Γ3 ,
where Γ1 + Γ2 + Γ3 = Γ , the quantity q n ( x ) represents the flux in the direction of the unit outward vector
nˆ ( x ) normal to the boundary, and α ( x ) and β ( x ) are known coefficients. We first consider an isotropic
medium, with scalar permeability K, where the governing partial differential equation expressed by
equation (1) is given by
∇ ⋅ [ K ( x , p)∇p] = − F ( x ) on Λ ,
(3)
which, according to Taigbenu [4], can be rewritten as
1
∇ 2 p = −∇ ln K ⋅ ∇p − F .
(4)
K
The adjoint equation, based on the Dirac delta function δ ( x , x ' ) , and which applies to an infinitely
extensive domain, defines the free-space Green’s function G(x,x’):
(5)
∇ 2 G = −δ .
1
The solution to equation (5) is G = −
ln( x − x ' ) , where the point x is called the field node and the
2π
point x ' is the source node. Using equations (4) and (5) in Green’s second identity, the integral equation
of the differential equation is obtained:
1
1
ηp( x ' ) = ∫ ( pH n − q n G ) d Γ + ∫ G (∇ ln K ⋅ ∇p + F ) d Λ,
(6)
K
Γ K
Λ
where η ∈ [0,1] is the contact (or corner) coefficient reflecting the domain continuity at a singular point
and H n = − K∇G ⋅ n̂.
The GEM requires that the computational domain be discretised by M suitable polygonal elements that,
collectively, closely represent the shape of the domain, each element consisting of N nodes. Over each
∂p
1
element p, q n = − K
and F are approximated by basis functions whose expressions are in
, lnK,
∂n
K
terms of the coordinates of the vertices of the element, namely the linear interpolation functions Ω j of the
Lagrange family, as, for example:
N
p( x ) ≈ ∑ Ω j ( x ) p j .
j =1
(7)
If two degrees of freedom are retained at each node, which means solving for the pressure and the flux,
then this imposes a considerable demand on computing resources due to the large dimensions of the
matrix of the system obtained. To overcome this difficulty, Taigbenu [4] proposes an alternative approach
based on the following approximation for the normal derivative of pressure:
∂p N ∂Ω j
≈∑
pj .
(8)
∂n j =1 ∂n
By this treatment of the normal derivative of pressure, only the pressure needs to be computed at each
internal node. Unfortunately, this approach leads to a diminution of the overall accuracy.
3. Flux-vector-based GEM
To maintain the second-order accuracy of the GEM, another approach was proposed by Pecher et al. [3],
namely the ‘flux-vector-based’ (or ‘q-based’) GEM. This concept is based on the fact that the maximum
number of linearly independent vectors at any node matches the number of spatial dimensions.
Specifically, at every node shared by more than two elements, each normal flux can be expressed as a
linear combination of two and three other distinct normal fluxes, in 2D and in 3D domains, respectively.
As a result, only two and three unknown flux components are required for a node in 2D and in 3D,
respectively. At the same time, the accuracy of the method remains of second order.
We consider the boundary value problem in which the flux, q(x), is related to the pressure, p(x), via the
law (2) and the pressure in the domain Λ is governed by equation (1), with the boundary conditions any
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combination of the three types specified in Section 2. In order to find the corresponding Green’s function,
equation (1) must be reformulated by two additional modifications.
(i) The coordinate system is rotated such that its axes match with the principal axes of the permeability
tensor.
(ii) The non-zero members Ki(x,p) of the diagonal tensor matrix K with respect to the new coordinate
~
system x are accompanied by an additional set K i = κ iυ ( x , p ), where κ i are constants, i=1,…,d (d =
~
dimensionality of space domain), and υ ( x , p) is a function that relates each component K i to the
corresponding κ i.
A function called the correction on correlated anisotropy is defined as:
d ∂ ⎡
∂p ⎤
~
σ = ∇ ⋅ [( K − K )∇p ] = ∑
(9)
⎢( K i − υκ i )
⎥.
∂x i ⎦
i =1 ∂x i ⎣
Equation (1) is converted into the following form:
L[ p ] = − s on Λ,
where L is a linear differential operator defined as
(10)
∂2
,
(11)
∂x i2
i =1
as κ is a constant diagonal tensor, and the term s(x,p) is the general internal/external source:
⎛ ⎛ 1 ⎞⎞
1
s ≡ ( F + σ ) + ⎜⎜ ∇⎜ ⎟ ⎟⎟ ⋅ q~.
(12)
υ
⎝ ⎝ υ ⎠⎠
~
The flux q~ in equation (12) is based on the permeability K ( x , p) :
∂p
~
q~ = − K∇p = −υκ i
, i = 1,..., d .
(13)
∂xi
The adjoint equation with respect to equation (10) is
L[G ] = −δ
(14)
and the matching free-space Green’s functions can be found in Brebbia and Walker [2].
Since the tensor K is symmetric, the differential operator defined in equation (11) is self-adjoint, so
Green’s second identity can be expressed as
(15)
∫ ( pL[G ] − GL[ p]) d Λ = ∫ [κ ( p∇G − G∇p)] ⋅ nˆ dΓ.
d
L ≡ ∇ ⋅ ( κ∇ ) = ∑ κ i
Λ
Γ
Using equations (10) and (14) in equation (15) reduces this equation to
1 ~
ηp( x ' ) = ( pH − q~ ⋅ nˆ G ) d Γ + sG d Λ,
∫υ
Γ
∫
n
(16)
Λ
d
∂G
~
~
.
where H n = −( K∇G ) ⋅ nˆ = −υ ∑ κ i nˆ i
∂xi
i =1
Equation (16) is the fundamental formula of the flux-vector-based GEM and can be solved using the
standard GEM procedure, except for the approximation of the normal derivative of pressure proposed by
Taigbenu [4]. Instead, the following expansion is imposed for the flux:
d
⎧⎪q~x nˆ x + q~ y nˆ y in 2 D
~
~
ˆ
ˆ
q ⋅ n = ∑ q i ni = ⎨ ~
(17)
~
~
⎪⎩q x nˆ x + q y nˆ y + q z nˆ z in 3 D
i =1
and in this way the normal component q n is replaced by the components q~x and q~y (and q~z in 3D),
which are then individually approximated as
N
N
N
q~x ≈ ∑ Ω i q~x ,i , q~ y ≈ ∑ Ω i q~ y ,i , q~z ≈ ∑ Ω i q~z ,i ,
i =1
i =1
(18)
i =1
where we denote by q~x ,i the value of q~x at node i.
In a two-dimensional domain, only two flux components are independent at each internal node, which
results in three solution quantities, namely p, qx and qy at every internal node. The proper number of
equations, i.e. three, is generated by integrating in turn over three elements surrounding the internal node.
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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Although more equations could be produced in a case when an internal node is surrounded by more than
three elements, their number must be limited to three, because otherwise the equations would be linearly
dependent. For a side source node, where either the pressure or one component of the flux is known, there
will be two unknown quantities, and the proper number of equations, i.e. two, is obtained by integrating
in turn over two elements that share the node. For a corner source node, the equation necessary for the
one unknown at that node is generated from an element to which the node belongs.
In order to investigate the performance of the ‘q-based’ GEM, we first tested it on an example in which
the permeability was continuous. The results obtained by the ‘q-based’ GEM were much more accurate
than those obtained by the classical GEM. We will further focus on situations when a discontinuity in the
permeability of the medium occurs. The examples are in terms of non-dimensional variables.
We consider a steady-state example governed by the Laplace equation, on the rectangular domain
[0,1] × [0,2] with the following prescribed conditions:
p ( x,0) = 1, p( x,2) = 0, 0 ≤ x ≤ 1;
(19)
q x (0, y ) = 0, q x (1, y ) = 0, 0 ≤ y ≤ 2.
(20)
In problems where large discontinuities in permeability are imposed, by using the standard ‘q-based’
GEM, pressure values in excess of unity have been obtained, something that does not correspond with
physical intuition, and the following section describes a technique which extends the ‘q-based’ GEM
approach to such situations.
4. The averaging ‘q-based’ GEM
The reason for the physically incorrect pressure values in the above situations is believed to be due to the
large difference in permeability that exists between the elements surrounding the internal boundaries of
the low-permeability regions. This can be explained by the fact that the definition of the flux incorporates
the value of the permeability. For an internal node, different values of the permeability are used for the
elements sharing it, which leads to a large contrast between the definitions of the flux at the same node.
The idea in the averaging ‘q-based’ GEM that we will now describe relates to satisfying the
conservation of mass at each node. To illustrate this, we will refer to a rectangular grid and consider the
internal node 5 from Figure 1, with the permeabilities of the elements sharing it being K1, K2, K3 and K4.
Figure 1: The permeabilities of the
elements sharing an internal node.
Figure 2: ‘Blow-up’ of four elements that coincide at an
internal source node P.
We will further denote by q xLT and q xLB the left-top and the left-bottom components of qx in the positive
x direction, and by q xRT and q xRB the right-top and the right-bottom components of qx in the negative x
direction, respectively. Similarly, for the y component of the flux, we will denote by q yLB and q yRB the
left-bottom and the right-bottom components of qy in the positive y direction, and by q yLT and q yRT the
left-top and the right-top components of qy in the negative y direction, respectively (see Figure 2).
The standard ‘q-based’ GEM approach would be to take q xLT = q xLB = − q xRT = − q xRB (i.e. a single qx
value at P), but this causes problems when K is not continuous. The components of qx , q xLT and q xLB , for
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example, cannot be equal, since K is discontinuous in the y direction, but ∂p ∂x continuous. The y
component of the flux, qy, would be treated in the same way, by a single value at P, which again leads to
errors, because in the x direction K is discontinuous but ∂p ∂y is continuous.
In what follows, the values of the fluxes q xLB and q xLT are averaged to give q xL = (q xLB + q xLT ) 2 ;
likewise for the fluxes q xRB and q xRT , so that q xR = (q xRB + q xRT ) 2 , leading to q xL = − q xR . The continuity
of these averaged expressions, necessary to maintain conservation of mass, together with a repeat of the
procedure in the y direction, leading to q yB = − q Ty , avoid an incorrect definition for qx at a discontinuity of
K in the y direction, and for qy at a discontinuity of K in the x direction. At the same time the number of
unknowns at each internal source node remains at three.
In detail, instead of using the different value of the permeability for each element, we use an average
between each two neighbouring elements. To express the qx component of the flux at point 5, when
writing the integral equations for elements (1) and (3), we consider for the permeability the average of K1
K + K 3 ∂p
and K3, so this flux is q xL = − 1
. Likewise, the flux value qx at point 5 (directed in the negative
∂x
2
x direction), when writing the integral equations for elements (2) and (4), is taken to have the form
K + K 4 ∂p
q xR = 2
. Similar expressions for the qy component of the flux are considered:
∂x
2
K + K 4 ∂p
K + K 2 ∂p
q yB = − 1
for elements (1) and (2), and q Ty = 3
for elements (3) and (4).
2
∂y
2
∂y
For the side source nodes, this approach can be considered only for one of the two components of the
flux, either qx or qy, because in these situations there are only two elements sharing each of these nodes.
For a node situated on the boundaries parallel with the Ox-axis, such as, for example, point 2 in Figure 1
K + K 2 ∂p
is, qy has a single value, namely 1
(in the negative y direction), while qx has two different
∂y
2
∂p
∂p
values: q xL = − K1
for element (1) and q xR = K 2
for element (2), with q xL = −q xR . For a node
∂x
∂x
situated on the boundaries parallel with the Oy-axis, such as, for example, point 4 in Figure 1 is, qx has a
K1 + K 3 ∂p
single value, namely
(in the negative x direction), while qy has two different values:
2
∂x
∂p
∂p
q yB = − K 1
for element (1), and q Ty = K 3
for element (3), with q yB = − q Ty . We therefore have two
∂y
∂y
unknowns at each side source node.
5. Numerical results
To undertake a thorough investigation of the averaging ‘q-based’ GEM, the examples to which the
process has been applied have been extended to three situations involving discontinuities in permeabilities
of O(1), O( 10 4 ) and O( 10 6 ) at their interfaces and for the intersection of several of these different
permeabilities to occur at a single node. The same domain, the rectangle [0,1] × [0,2], was symmetrically
divided into four regions, so that each of the four permeabilities corresponds to a quarter of the domain.
The same boundary conditions have been applied as in equations (19) and (20). The results have been
compared to those obtained by two other numerical methods, namely the BEM using constant elements
and the control volume method (CVM) over a regular Cartesian grid. As there is a good agreement
between the three sets of solutions in the O(1) discontinuity case, we will present the results only for the
largest permeability contrasts.
Example 1
When the order of the permeability discontinuities is extended to O( 10 4 ) with K1=1, K2= 10 −2 , K3= 10 −1
and K4= 10 −4 then the agreement between the three sets of results can only be maintained by increasing
the number of elements in the BEM and in the CVM. This is in evidence in Figure 3 when the results for
64 × 128 elements in the averaging approach of the ‘q-based’ GEM compare favourably with the BEM
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and with the CVM, but the latter two methods require that their number of elements is increased, namely
to 1536 boundary elements in the BEM and to 360 × 720 elements in the CVM (even then the CVM
solution is not grid-independent). This indicates that, to acquire results to the same degree of accuracy,
the averaging ‘q-based’ GEM is more efficient than both the BEM and the CVM. This is in fact a severe
test example as virtually all of the fluid is forced to pass through the centre point of the domain.
Figure 3: The numerical solution obtained for the
pressure by (a) the averaging approach of the
‘q-based’ GEM, (b) the BEM, and (c) the CVM
with K1=1, K2= 10 −2 , K3= 10 −1 and K4= 10 −4 .
Figure 4: The numerical solution obtained for the
pressure by (a) the averaging approach of the
‘q-based’ GEM, (b) the BEM, and (c) the CVM
with K1=1, K2= 10 2 , K3= 10 −1 and K4= 10 −4 .
Example 2
The second example has abrupt changes of order O( 10 6 ) in the permeability, namely K1=1, K2= 10 2 ,
K3= 10 −1 and K4= 10 −4 . The numerical results obtained for the pressure in Figure 4(a) for 16 × 32
elements in the averaging ‘q-based’ GEM are compared with the BEM results for 768 boundary elements
(Figure 4(b)) and the CVM for 80 × 160 elements (Figure 4(c)). The numerical solution obtained by the
CVM is considered as not being very accurate, due to the graphical errors and also the fact that increasing
the number of elements in the CVM did not lead us to a grid-independent solution.
At this stage it seems appropriate to return to the classical GEM and to see how this method copes with
the discontinuities in the permeability within the above examples. As expected from the discussion in
Section 2, in all of the examples the numerical solutions are extremely inaccurate, with clear errors
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arising at the interfaces between regions of contrasting permeability. However, this inaccuracy is only to
be expected, because of the flux approximations and of the fact that the classical GEM requires a higherorder pressure correction related to a backward difference (see Archer [1]).
For the example involving discontinuities in permeability of O( 10 4 ), the flux vectors for the averaging
‘q-based’ GEM and the BEM are compared in Figure 5. In the BEM the flux values must be derived from
the pressure solution, which leads to a slight difference in the values of the flux between the two methods.
The arrows in both cases display the same orientation, with the fluid trying to avoid the regions of low
permeability and instead passing mainly through the regions of high permeability.
Figure 5: Representation of the flux vectors obtained by
(a) the averaging approach of the ‘q-based’ GEM and (b)
Figure 6: Representation of (a) the pressure and (b)
the flux vectors for faults of thickness 10 −3 and
the BEM with K1=1, K2= 10 −2 , K3= 10 −1 and K4= 10 −4 .
permeability 10 −1 , for 16× 34 elements.
Example 3
In this simple test example, we introduce partially sealing faults to an otherwise homogeneous domain by
including cells of lower permeability within a non-uniform Cartesian grid. We consider the same domain
and the same boundary conditions as before. The faults are represented by two very thin layers, of
thickness 2t f = 10 −3 , and of length 0.75. One is situated in the bottom-right part of the domain, over
[0.25,1] × [0.5 − t f ,0.5 + t f ], and the other one is located over [0,0.75] × [1.5 − t f ,1.5 + t f ]. The
permeability of each fault is 10 −1 , and the permeability of 1 applies for the remainder of the domain. In
this case, the layers allow an amount of fluid to pass through them (Figure 6(b)), and the pressure field is
only slightly influenced by the faults (Figure 6(a)).
Example 4
When the permeability of the two faults is lower, namely 10 −3 , with all the other conditions the same as
in Example 3, the faults have a much more significant influence on the flow. In this situation, as
illustrated in Figure 7, the fluid is mostly forced to avoid the two faults and a pressure difference is
achieved across the faults.
Example 5
The last example presented is the simulation of two wells, situated in the same domain with sealing
boundary conditions, namely zero flux imposed everywhere on the boundary. One well is located at (0.25,
1.75) and is considered as a source (injection) with a strength of 1; the other well, considered as a sink
(production), has a strength of –1 and is located at (0.75, 0.25). The terms representing the integrals of the
source/sink terms were treated as in Archer [1]. Two faults, of thickness t f = 10 −2 and permeability 10 −3 ,
are considered in the domain, located over [0.225,1] × [0.5,0.5 + t f ] and over [0,0.775] × [1.5 − t f ,1.5].
The pressure contours, for 18 × 36 elements, are shown in Figure 8(a), and the flux vectors for the same
case are shown in Figure 8(b).
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Figure 7: Representation of (a) the pressure and (b) the
flux vectors for faults of thickness 10 −3 and permeability
Figure 8: Representation of (a) the pressure and (b)
the flux vectors for the case when there are faults
10 −3 , for 16× 34 elements.
and injection/production wells in the domain.
6. Conclusions
In this study we have illustrated and extended the GEM approach to consider the ‘flux-vector-based’
GEM and modifications that enable large discontinuities in permeability to be treated. The ‘flux-vectorbased’ GEM approach is very suitable for problems where the permeability of the medium is
differentiable throughout the domain. However, when a discontinuity in the permeability of the medium
occurs then that approach is no longer accurate and a different suggested treatment for the flux has been
given. This approach is based on the conservation of mass at each point and has been successfully applied
to cases when a discontinuity of the medium of O( 10 4 ) and O( 10 6 ) occurs across the interfaces of the
subdomains. The results were compared with those obtained by the BEM and the CVM and they were in
agreement. Two physical applications were then simulated using the modified approach of the ‘q-based’
GEM. As seen in Examples 1 and 2 and due to its second-order accuracy, the ‘q-based’ GEM requires a
much smaller number of elements than the BEM and the CVM, in order to acquire grid-independent
results to the same degree of accuracy, which makes it more efficient than both the BEM and CVM in
such cases. Future work will investigate this approach for transient problems and two-phase flow. The
GEM approach and the extensions discussed here can be naturally extended to triangular finite element
grids, over which geological features such as faults and stratigraphic layers can be accurately represented,
and so this is an important advancement in solution techniques for reservoir simulation.
Acknowledgement
Piroska Lorinczi would like to acknowledge the financial support received from the ORS and the Rock
Deformation Research Group, University of Leeds.
References
[1] Archer, R., Computing Flow and Pressure Transients in Heterogeneous Media Using Boundary
Element Methods, Ph.D. dissertation, Stanford University, 2000.
[2] Brebbia, C.A., Walker S., Boundary Element Techniques in Engineering, Newnes-Butterworths,
London, 1980.
[3] Pecher, R., Harris, S.D., Knipe, R.J., Elliott, L., Ingham, D.B., New formulation of the Green element
method to maintain its second-order accuracy in 2D/3D, Int. J. Engineering Analysis with Boundary
Elements, 25, 211-219, 2001.
[4] Taigbenu, A.E, The Green Element Method, Kluwer Academic Publishers, Boston, 1999.