On-line Optimization-Based Simulators for Fractured and Non-fractured Reservoirs Page: 16 of 188
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After (1.2) is substituted into (1.1) and discretized by the numerical method of
choice, averaged k, pi, i, and kri between discretized members need to be deter-
mined. To ensure flux continuity between these members the harmonic average of
k is used . In general, pl and l are weak functions of phase pressure; therefore,
the arithmetic average is used. The choice of kri is very important in ensuring that
the numerical solution converges to the real solution. It has been argued that use
of any type of average for kri is inappropriate and that upstream implementation
is essential . Use of asymmetric weighting functions for k, are possible , but
that does not guarantee physically meaningful phase saturation values .
Typically, finite difference is the method of choice for discretization in reser-
voir simulation applications. However, the finite element family of methods are
more suitable for problems with complex geometrical domains. The finite ele-
ment method (FEM) gained popularity in the field of oil reservoir simulation in
the late 1960's [6, 7]. Langsrud  and Dalen  developed the most essential
features required for the success of this family of methods in multiphase flow
applications-mass-lumping and upstream weighting of kri. These features are
still part of current practice [10, 11, 12, 13]. The mass-lumping scheme is nec-
essary to prevent oscillating phase saturation values. The upstream weighted k,
is determined by a potential-based upstream weighting method. In this method,
the flux portion of (1.1) is first rearranged to resemble the finite difference for-
mulation, and then by comparing pairs of potential values, the upstream kri is
The control volume finite element method (CVFE) for oil reservoir simulation
was introduced by Forsyth. The upstream implementation strategy for k, in
the CVFE is the potential-based method as in FEM. This has been adopted by
other researches [14, 15, 16, 17, 18]. In this method, shapes of the triangular
elements should be such that positive transmissibilities are guaranteed.
To distinguish from the CVFE, the method proposed here is called the control
volume method (CVM). The only difference between the CVM and the CVFE is
the upstream weighting scheme. Unlike the CVFE, the upstream direction in the
CVM is determined by the flux direction across each control volume boundary
in each triangle. This type of upstream weighting implementation is considered
flux-based. This idea was first developed by Prakash in single-phase solute
transport applications. Forsyth recognized this concept for multiphase flow
but applied the potential-based method. A box method using a similar flux-based
upstream weighting was proposed by Huber and Helmig[20, equation (18)]. In the
box method, the flux direction is determined by the summation of all the fluxes
exchanged between two control volumes across multiple triangular or quadrilateral
One of the main objectives of this paper is to highlight the differences between
flux- and potential-based upstream weighting methods. It is clearly shown that
the flux-based implementation leads to flux continuous solutions, and eliminates
limitations imposed on the shapes of the elements.
One of the main reasons for the lack of widespread use of the finite element
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Deo, Milind D. On-line Optimization-Based Simulators for Fractured and Non-fractured Reservoirs, report, August 31, 2005; Utah. (https://digital.library.unt.edu/ark:/67531/metadc877059/m1/16/: accessed May 19, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.