On-line Optimization-Based Simulators for Fractured and Non-fractured Reservoirs Page: 43 of 188
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Start with equation 2.18 and integrate over Vo in Figure 2.2:
Fo= j V.V + dx=0. (2.19)
Note that q is dropped in equation 2.19 since it is more convenient to deal with
point source at global assembly stage. Applying divergence theorem to the flux
term in equation 2.19 gives
Fo = fV - nds + 0Odx
aikj+h' Jb-1- Jc JVo (2.20)
= V nds + V -nds+ V -nds+ Codx
aakj bjkh chki VOt
where n is the outward normal of the corresponding boundary as shown in Figure
fo = fo,aikj + fo,bjkh + fo,chki, (2.21)
fo,aikj = V - nds, (2.22a)
fo,bjkh = V - 9ds, (2.22b)
fo,chkz = V - 1ds, (2.22c)
then equation 2.20 becomes
Fo = fo + dx. (2.23a)
For F1, F2, and F3,
F1 = fi + dx, (2.23b)
F2 = f2 + dx, (2.23c)
F3 = f3 + dx. (2.23d)
fl = fl,ajki + fl,dgkj + fi,eikj, (2.24)
f2 = f2,bhkj + f2,djkg + f2,fgkh, (2.25)
f3 = f3,chki + f3,ejki + f3,fhkg. (2.26)
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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/m1/43/: accessed May 26, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.