On-line Optimization-Based Simulators for Fractured and Non-fractured Reservoirs Page: 30 of 188
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k k k
bk k k
f ki j7k ki
b, b, b2 bj
i ij j i ij j i ij j
Ok > 02S j
Figure 1.7: Acute, right-angled, and obtuse triangles used in the discussion of the
consequences of applying either the potential- or flux-based upstream condition.
Therefore, a fluid phase is immobile if S < Sir even when there exists a potential
The positive transmissibility condition for a single triangle is that all angles are
equal to or less than r/2 . Consider the acute triangle in Figure 1.7. Relative
values of phase potentials and saturations are indicated in the figure. Assume
that the permeability tensor is identity; then, the flux direction is pointed in the
negative y-direction. The flux flowing out of bj through cij in the y-direction
should be zero because Sj = Sir. However, the actual flux calculated by the
CVFE: fjy,-g = ky(zj - xc) (pA)k i2A 2 ( k - yO) 0.
It is clear that even if the positive transmissibility condition is satisfied, the CVFE
still has unrealistic fluxes. In contrast, the same flux in the CVM is
CVM: =pA) ky (zj - xC) = 0
because, Aj = 0.
Consider the right-angled triangle in Figure 1.7. The potential and saturation
conditions in Figure 1.7 require the flux through the boundary cjk should be
nonzero. This means that the value of Sj should increase. The flux flowing into
bj through cjk in y-direction calculated by the CVFE is
CVFE: - - x) (pA) kX2A
It is clear that fjyj+f = 0 because (xz - x,) = -(X, -xjk). As a result, Sj
remains unchanged. Thus, once again, it is possible to develop physically incorrect
solutions using CVFE. It is for this reason that CVFE is considered a five-point
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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/m1/30/: accessed May 26, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.