1
B026 NON-UPWIND MONOTONICITY BASED FINITE
VOLUME SCHEMES FOR HYPERBOLIC
CONSERVATION LAWS IN POROUS MEDIA
Michael G Edwards
Civil and Computational Engineering Centre, University of Wales Swansea,
Singleton Park, Swansea SA2 8PP, UK
Abstract
The focus of this paper is on the development of Mathematically robust convective flow
approximation schemes that offer the benefits of an upwind formalism without actually
upwinding. Development of robust schemes that remove upwind dependence would represent a
significant step forward leading to a fundamental simplification of current methods.
Introduction
Standard reservoir simulation schemes employ single-point upstream weighting [1] (a first order
upwind scheme) for approximation of the convective fluxes when multiple phases or
components are present. The definition of the upstream schemes depends upon the direction of
the local phase velocities. Higher order upwind approximations have also been developed [2-7].
The most successful high resolution schemes for systems of hyperbolic conservation laws in
terms of actual front resolution, depend upon a characteristic decomposition of the system. The
decomposition identifies characteristic wave components of a system enabling upwind
approximations to be applied with minimum dissipation. A survey of schemes is given in [8].
For reservoir simulation mathematically more robust higher order Godunov schemes that are
monotonicity preserving have also been developed for reservoir simulation [2, 7]. These schemes
require a characteristic decomposition when applied to hyperbolic systems with multiple phases
or components present. The decomposition leads to optimal upwind schemes where upwind
directions are resolved according to characteristic wave components. However the
decomposition adds further complexity compared to the standard scheme and additional
computation is required to account for the decomposition matrices. Special treatment of
stagnation points or “sonic points” where eigenvalues change sign is also required [2].
This paper presents novel robust schemes for reservoir simulation that permit reconstruction of
stable lower order and higher order monotonicity preserving approximations while avoiding
dependence upon both upwinding and characteristic decomposition. The formulation can also be
used to simplify the higher order Godunov scheme locally in the presence of stagnation points
(or if equal eigenvalues are detected), the details are beyond the scope of this paper. Here a new
simplified scheme for reservoir simulation is presented that has no upwind (or upstream)
dependence.
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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The schemes are formulated within a locally conservative finite volume framework and avoid
dependence upon a characteristic decomposition by employing a form of Local Lax-Friedrichs
(LLF) based flux [9]. Such schemes permit the construction of lower and higher order
approximations without recourse to characteristic decomposition. This is achieved by using the
local maximum eigenvalue of the hyperbolic system within the definition of the LLF flux.
Flow Equations
Without loss of generality in terms of the numerical methods applicability, the schemes
presented here are illustrated with respect to two phase and three component two phase flow
models, with unit porosity and where capillary pressure and dispersion are neglected. The
conservation equations over Ω are written as
∂S
∂F
∂G
∫Ω ∂t dτ + ∫Ω ( ∂x + ∂y )dτ = M p
1
where
F = (V1 , CV1 )T , G = (V2 , CV2 )T
are the fluxes, (V1 ,V2 ) is the phase velocity vector (defined below), and S = ( S , SC )T ,
C = ( S , C )T are the vectors of conservative and primitive variables respectively, S is the miscible
phase saturation and C the component concentration in the miscible phase, M p is a specified
phase flow rate. The system considered here is comprised of an aqueous phase (water and
polymer concentration) together with an oil phase. Phase quantities bear suffices a and o
respectively. The velocities are defined via Darcy’s law, where for the aqueous phase velocity
Va = (V1, V2 ) = − λa K (∇φ + ρ a g∇h )
In this work, the phase velocity is expressed in terms of total velocity VT = Va + Vo with
Va = f ( VT − gλo ∆ρK∇h )
2
where f = f ( S ) is the fractional flow, λa , λo are the phase mobilities and K is the permeability
tensor. The saturation of the remaining phase is deduced from the volume balance equation,
where saturations sum to unity. The incompressible flow condition leads to
∫ ∇ • VT dτ = M
3
which together with Eq. 2 are used to determine pressure and velocity, for further details of flow
equations see [1].
For the initial value problem (IVP) field data is prescribed. For initial boundary value problems
(IBVP), considered here in two-dimensions, an initial flow field is prescribed together with flux
or pressure boundary values. Zero normal flow is imposed on solid walls.
Locally Conservative Upwind Approximation
First we shall consider schemes in one spatial dimension on a computational grid with discrete
nodes xi = i∆x and time t n = n∆t . Before moving to systems we briefly recall that
for a scalar equation of the form
∂s ∂f ( s )
+
=0
∂t
∂x
4
3
the standard explicit first order upwind scheme can be written as
sin +1 = sin −
∆t n
( f i +1 / 2 − f in+1 / 2 )
∆x
5
where the approximate flux is defined by
f in+1 / 2 =
(
1
( f in+1 + f in ) − λi +1 / 2 ( sin+1 − sin )
2
)
6
where
⎧⎪( f n −
λi +1 / 2 = ⎨ i +1
⎪⎩
f in ) /( sin+1 − sin )
∂f / ∂s
( sin+1 − sin ) > ε
( sin+1 − sin ) ≤ ε
This definition of wave speed ensures that shocks are captured with precision for any finite jump
in si with λi +1 / 2 assuming the Rankine-Hugoniot shock speed across a mesh interval. In this
form the first order scheme appears as a central scheme and is comprised of a central difference
in flux with a central difference of a specific diffusion term. The scheme can be seen in its
original upwind form by noting that for a positive wave speed the flux uses data to the left and
reduces to f in+1 / 2 = f ( sin ) , otherwise f in+1 / 2 = f ( sin+1 ) and the flux uses data to the right as with
single point upstream weighting. While the definition of Eq's. 5, 6 does not require any explicit
sign dependence in the scheme, the upwind directions are clearly detected. This explicit scheme
is the most fundamental scheme for scalar conservation laws in one dimension and is stable,
locally conservative and monotonicity preserving subject to a maximum CFL condition of unity.
This scheme also requires an entropy fix to disperse expansion shocks [8]. In order to apply such
a scheme to a system of hyperbolic conservation laws of the form of Eq 1 the system is first
decomposed into characteristic form. First we consider the system in one dimension.
Decomposition is performed via the transformation
∆S = R ∆W
7
where R is the matrix of right eigen-vectors of the system Jacobian matrix A = ∂F / ∂S and the
matrix of eigenvalues Λ is defined via
Λ = R −1 AR
8
and ∆S, ∆W represent the respective conservative and characteristic variable increments. The
upwind scheme is in effect applied to each characteristic wave component and the discrete
system is recomposed into conservation form. The first order scheme for a system can be written
as
S in +1 = S in −
∆t n
(fi +1 / 2 − fin+1 / 2 )
∆x
9
with approximate flux defined by
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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fin+1 / 2 =
(
1 n
(fi +1 + fin ) − R Λ i +1 / 2 R −1 (S in+1 − Sin )
2
)
10
provided the definition of the discrete eigenvalues Λ i +1 / 2 have an appropriate generalization of
Eq. 6 such that conservation and exact shock speed are maintained. The CFL condition now
applies with respect to the maximum eigenvalue of the system.
Locally Conservative Non-Upwind Approximation in One Dimension
The appearance of the matrix of eigenvectors Ri +1 / 2 in the system flux approximation is a
consequence of upwinding on each characteristic component. This leads to an optimal diffusive
operator in terms of numerical diffusion provided the shock jump criteria can be satisfied for the
system in hand. If the matrix of eigenvalues is proportional to the unit matrix (equal eigenvalues)
then the dependency of the discrete flux on the matrix of eigenvectors is removed, which
essentially leaves a much simpler coefficient of artificial diffusion. The scheme can be
simplified in this way without appearing to violate the crucial monotonicity preserving property
of the scheme, subject to the CFL condition, by replacing the diagonal matrix of absolute eigenvalues Λ i +1 / 2 with the matrix
Λ LFi +1 / 2 = max[x L , x R ] λij+1 / 2 I
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j
and the maximum absolute eigen-value of the system, (here max( ∂Va / ∂s , Va / s ) I ) over the
local interval [xi , xi +1 ] is used. The discrete flux then reduces to
fin+1 / 2 =
(
1 n
(fi +1 + fin ) − Λ LFi +1 / 2 (Sin+1 − Sin )
2
)
12
The flux of Eq. 12 is called the Local Lax-Friedrich flux LLF. The non-upwind LLF scheme is
then defined be Eq's. 12 and 9. Note that the scheme is still locally conservative. Extension to
higher order accuracy is discussed in the next section. Since all eigenvalues in the diffusion
operator are replaced by their maximum modulus, it follows that the price to be paid for this
simplification is extra numerical diffusion.
Higher Order Schemes Without Upwinding
In the scalar case a higher order approximation is applied to the conservation variable. When an
upwind scheme is applied to a system the higher order approximation is typically introduced
wave by wave and applied to the characteristic variables, followed by recomposition to the
conservative variables. In contrast, when using the discrete LLF flux Eq. 12, the extension to
higher order accuracy is simpler to achieve since the non-upwind system formulations have no
dependency upon characteristic variables in this definition of flux. Consequently a higher order
approximation can be introduced for the left and right states respectively and be expressed
directly in terms of the conservative variables. Alternatively the higher order expansions can also
be applied to other sets of variables such as primitive or characteristic variables. In this work the
primitive variables are selected.
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In one dimension the scheme is expressed as a two-step process. First the higher order states are
defined using a MUSCL formalism [10]. Higher order left and right hand side states are obtained
by expansions about the states L and R, viz
1
S L = Si + PΦ ( ri++1 / 2 )(Cin+1 − Cin )
2
1
S R = Si +1 − PΦ ( ri−+1 / 2 )(Cin+1 − Cin )
2
13
where Φ ( ri++1 / 2 ) and Φ ( ri−+1 / 2 ) are flux limiters, or in this case slope limiters [10] and P is the
transformation matrix between conservative and primitive variables. The slope limiters are
functions of adjacent discrete gradients where
ri++1 / 2 = (Cin − Cin−1 ) /(Cin+1 − Cin ),
ri−+1 / 2 = (Cin+ 2 − Cin+1 ) /(Cin+1 − Cin )
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The slope limiters constrain the expansions to ensure that the higher order data remains
monotonic. Details can be found in [10, 7 and 11]. The above flux of Eq. 12 is now applied to
the higher order data so that the local Riemann problem is resolved with a higher order flux of
the form
fin+1 / 2 =
1
((f (S L ) + f (S R )) − Λ LFi +1 / 2 (S R − S L ) )
2
15
The flux of Eq. 15 is now used to integrate the system via Eq. 9. The higher order spatial scheme
as defined by Eq's 13-15 and 9 which uses first order forward-Euler time stepping. The CFL
condition of the scheme is dependent on the choice of limiter, typically an upper limit of 1/2 is
selected. Extension to higher order time accuracy is achieved either by using the scheme of [6] or
with the second or third order monotonicity preserving Runge-Kutta schemes of [12]. Note that
the first order flux is recovered if the limiters are set to zero.
Approximate Riemann Solvers Without Upwinding in Two Dimensions
The higher dimensional extension of the above scheme is based on a generalization of the one
dimensional discrete flux. The flow equations Eq. 1 are integrated in space and time over a
discrete control-volume Ω CV with surface δΩ CV by direct use of the Gauss divergence theorem
applied to yield a surface integral of divergence
∫Ω CV (S(t + ∆t ) − S(t + ∆t ))dτ = − ∫∆t ∫δΩ CV (Fdy − Gdx )dt + ∫∆t M p dt
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The discrete scheme is developed from Eq 16, space only permits a brief summary here. The
two-dimensional normal flux ℑ = ( F∆y − G∆x ) / ( ∆y , − ∆x ) is approximated in an analogous
form to Eq 15. Higher order states are defined using a MUSCL based formalism [10], which
essentially follows the above 1-D principle applied relative to each control-volume face. The
higher order left and right hand side states are obtained by Taylor expansions and as in 1-D the
expansions are constrained with slope limiters to ensure that the higher order data remains
monotonic. For a structured grid limiting is performed in the co-ordinate directions. The limiting
is based on the Fromm limiter Φ ( r ) = min(2,2r, (1 + r ) / 2) . Note as before, that the first order
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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flux is recovered if the limiters are set to zero. Formal time integration is achieved as above,
employing either [6] or [12], without disturbing monotonicity subject to a reduced theoretical
CFL limit in 2-D (as with standard schemes), although here up to 1/2 is used. Discrete
approximations developed for general unstructured grids comprised of quadrilateral and-or
triangular cells will be presented in a future report.
RESULTS
One Dimension
The schemes are first compared for gravity segregation with reverse flow in one-dimension.
Initial data consists of a vertical aqueous phase column above an oil column of equal volume
(95% oil saturation) , separated by a diaphram which is removed at time t=0. The heavier
miscible phase moves downwards, creating a shock followed by a constant state, contact
discontinuity (due to polymer) and rarefaction wave. The oil moves upwards creating a shock at
the opposite end of the rarefaction. The computed and exact water saturation solutions are shown
in Fig’s 1-3 at time 1.25, (exact solution solid line, computed using 100 nodes square symbol)
with the prescribed initial data S,C = 0.05, 0.1, x < 0.5 , S,C = 1.0, 0.7 otherwise.
The first order scheme result is shown in Fig.1 and completely smears the solution, failing to
detect the contact, constant state and shock caused by the miscible phase. The higher order LLF
scheme and higher order Godunov (HOG) scheme results are shown in Fig’s 2 and 3
respectively. The comparison for water saturation shows that both the higher order schemes
resolve the two shocks and contact discontinuity induced by the polymer concentration
discontinuity. While the results of Fig. 2 indicate additional diffusion that is inherent in the LLF
scheme in the region of the contact when compared to HOG, the non-upwind scheme achieves a
considerable improvement in front resolution compared to the first order scheme.
Two Dimensions
The reverse flow problem is next considered in 2-D in the presence of a shale barrier with solid
walls at the sides and top boundaries, and pressure specified on the lower boundary, the
boundaries and initial interface is shown in Fig 4. All saturation contours are shown at the same
output time, where the shock due to the upward moving oil phase is just being reflected from the
top wall. The higher order scheme results are; HOG in Fig.5, HOG with LLF at sign change Fig.
6, and full LLF in Fig.7. The higher order schemes are each able to detect the shocks, expansion
about the shale corner and the weaker linear contact wave. As in 1-D shocks are again well
resolved, the sensitive linear contact wave is more spread. The simplified HOG scheme result
shown in Fig 6 produces a similar result to HOG, with a slight increase of spread in the contact
due to the local additional diffusion. The higher order LLF scheme result, Fig. 7, shows that the
non-upwind scheme is able to capture the key flow features with similar resolution. A
comparison the first order scheme result Fig. 8, shows that the base scheme fails to detect the
contact and constant state as they pass the shale, and provides poor resolution of the lower shock
and the slow moving shock directly above the shale. The non-upwind higher order LLF scheme
Fig. 7 achieves a considerable improvement in front resolution compared to the first order
scheme and detects all discontinuities that are present in the HOG solution.
Conclusions
A novel formalism is presented for hyperbolic conservation laws in reservoir simulation. The
scheme has the following properties:
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1) Upwinding and characteristic decomposition are circumvented, leading to a fundamental
simplification of current methods.
2) A sound mathematical basis for stability and monotonicity is retained while avoiding both
upwinding and characteristic decomposition.
3) Local conservation is maintained and the new formulation offers consistency in IMPES mode
(or sequentially implicit mode), with respect to sign change in wave velocity and mass balance.
4) The new formulation can also be used to simplify the higher order Godunov scheme in the
presence of sign change in wave velocity or in occurrence of degenerate equal eigenvalues.
Results are presented for gravity driven flows involving two phase flow and three component
two-phase flow, where the wave direction changes sign (reverse flow). Results obtained with the
new schemes indicate a competitive comparison with the higher order Godunov scheme. Finally
it is noted that the new formulation offers the benefits of being directly applicable to other
systems of hyperbolic conservation laws without requiring a characteristic decomposition.
Acknowledgement: Support of EPSRC grant GR/S70968/01 is gratefully acknowledged.
References
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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1.2
1
0.8
1st order
Exact
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
Fig 1 1st order versus exact solution
1.2
1
0.8
LLF HO
0.6
Exact
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
Fig 2 higher order LLF versus exact solution
1.2
1
0.8
HOG
Exact
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Fig 3 higher order Godunov versus exact
solution
1.2
9
Water+polymer
Fluid interface
Shale barrier
Oil
Fig 4: Initial Conditions
9th
Fig 5: Higher Order Godunov
Fig 6: Higher Order Godunov (LLF)
Fig 7: Higher Order LLF (non-upwind)
Fig 8: First Order
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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