On-line Optimization-Based Simulators for Fractured and Non-fractured Reservoirs Page: 20 of 188
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in (1.3) are omitted for simplicity. The transmissibility is defined as
Ti k=-kxxD a k+ k' kL k L2 . (1.12)
Dx8 x Dy yXD Dx Dy Dy!
The potential-based upstream operator is defined by
up~~i() if cpi>
up(ij) = . .f '> (1.13)
Notice that when the two flux terms of (1.11) are considered separately there is
no significant physical meaning associated with either one of them. It is arranged
so only to resemble (1.10).
A positive transmissibility condition is necessary to guarantee Tis > 0 .
Negative transmissibilities are physically unrealistic and also produce unaccept-
able saturation values.
1.6 The Control Volume Method
In this section, we derive the CVM from a finite element point of view with a
focus on the explict expression for local fluid flux.
The basic concept of the CVM is to use the fluid potential values on T for
flux calculation; the flux so obtained is then used for mass balance on B. Take
any triangle t E T as an example; after establishing the flux direction in the
triangle, the fluid exchanged between the three control volumes in the triangle
can be calculated.
1.6.1 Control volume formulation
The residual function for a control volume b2 E B with boundary F' is obtained
by taking the integrated form of (1.1):
F=0= V D(pP) dx
ib~ Dt -(1.14)
= j v - n ds + D(pS) dx
Here n is the unit outward normal on F'. Note that the subscript 1 and the source
term in (1.3) are omitted for simplicity.
1.6.2 Control volume discretization
During the process of computation, it is difficult to evaluate (1.14) because a
control volume is usually distributed accross several triangular elements as shown
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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/m1/20/: accessed May 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.