On-line Optimization-Based Simulators for Fractured and Non-fractured Reservoirs Page: 17 of 188
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family of methods in reservoir simulation is due to the belief that they do not yield
locally conservative solutions. In this paper, proofs are provided regarding the
local mass conservative aspects of the CVFE, the CVM, and the FEM methods.
In 1.2 the governing equations of multiphase oil reservoir simulation are fur-
ther discussed. In 1.3, the upstream weighting of properties is explained in the
context of the finite difference method. The discretized finite element and control
volume finite element formulations are discussed in 1.5. The proposed control
volume method is derived in 1.6. Flux continuity and mass conservation prop-
erties of the above three methods are discussed in 1.7 and 1.8, respectively. In
@1.9, numerical experiments are conducted using the CVFE and CVM to show
the significance of flux continuity.
1.2 Governing equations
We consider a bounded polygonal domain Q in R2 with boundary F. The governing
equations are obtained by substituting (1.2) into (1.1).
0 = -V - kihVri + + ql. (1.3)
The permeability tensor k is symmetric positive definite . The initial conditions
are prescribed phase potential and saturation distributions in the domain Q
yP(Qt = 0) = Pio(Q), S1(Qt = 0) = S10(Q). (1.4)
The boundary condition is
v - r = 0, (1.5)
where n is the unit outward normal on F. Phase potential yo is defined as
'Pi= Pi + piz.
where P is the phase pressure, g is the gravitational acceleration constant, gc is a
conversion constant and z is the elevation. To emphasize the basic ideas, we will
consider only cases where z = 0, that is co = P. Phase pressures are related by
PC(S. ) = P. - P. (1.6)
and phase saturations are coupled by
Sn + Sw = 1. (1.7)
Equations (1.3) to (1.7) form the complete two-phase flow problem.
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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/m1/17/: accessed May 22, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.