On-line Optimization-Based Simulators for Fractured and Non-fractured Reservoirs Page: 24 of 188
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Figure 1.5: A schematic representation of the applicable upstream nodes and flux
where b,,m = fb dx. Other fluxes can be calculated in the same way and are
f = f = PupAup L - (k + kx (Yc - yij)
- (kyX + kyy+) (xzi - xc) ,
fj,- -fkT - pupAup - (kxx + kxy )(Ye - Yjk)
- kyx + y 0' )~ (xjk - x,)](15b
fki -- --f= - pupAup - (kx + kx (Ye - ysk)
- (kYx+ kyy+) (xk - x,)].
Note that the first equality in (1.25) states the fact that any flux flowing out of
one control volume equals the flux flowing into another control volume through
their common boundary within a triangular element. Thus, the CVM is a flux
continuous numerical method.
1.6.4 Upstream weighting
The upstream weighted properties in the CVM are determined by the flux-based
upstream weighting scheme (unlike the CVFE where the discretized equations are
reformulated to resemble the finite difference formulation and a potential-based
approach is used).
The concept of flux-based upstream weighting in the CVM is better explained
by examples. Figure 1.5 shows an example triangle with constant flux across the
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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/m1/24/: accessed May 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.