Page 163 - Principles of Applied Reservoir Simulation 2E

148 Principles of Applied Reservoir Simulation

changes, it is necessary to calculate derivatives, either numerically or analyti-
cally, of the flow equation terms. The derivatives are stored in a matrix called
the acceleration matrix or the Jacobian. The Newton-Raphson technique leads
to a matrix equation J • $X = R that equates the product of the acceleration
matrix /and a column vector Xof changes to the primary unknown variables
6
to the column vector of residuals R. It is solved by matrix algebra to yield the
changes to the primary unknown variables $X. These changes are added to the
value of the primary unknown variables at the beginning of the iteration. If the
changes are less than a specified tolerance, the iterative Newton-Raphson
technique is considered complete and the simulator proceeds to the next timestep.
The three primary unknown variables for an oil-water-gas system are oil-
phase pressure, water saturation, and either gas saturation or solution GOR. The
choice of the third variable depends on whether the block contains free gas,
which depends, in turn, on whether the block pressure is above or below bubble
point pressure. Naturally, the choice of unknowns is different for a gas-water
system or a water only-system. The discussion presented here applies to the most
general three-phase case.
A simpler procedure is the IMplicit Pressure-Explicit Saturation (IMPES)
procedure. It is much like the Newton-Raphson technique except that flow
coefficients are not updated in an iterative process. The Newton-Raphson
technique is known as a fully implicit technique because all primary variables
are calculated at the same time; that is, primary variables at the new time level
are determined simultaneously. By contrast, the IMPES procedure solves for
pressure at the new time level using saturations at the old time level, and then
uses the pressures at the new time level to explicitly calculate saturations at the
new time level. WINB4D, the program provided with this book, is an implemen-
tation of a noniterative IMPES formulation [Fanchi, et al., 1982; Fanchi, et al.,
1987]. The formulation is outlined in Chapters 27 and 32. A variation of this
technique is to iteratively substitute the new time level estimates of primary
variables in the calculation of coefficients for the flow equations. The iterative
IMPES technique takes longer to run than the non-iterative technique, but
generates less material balance error [Ammer and Brummert, 1991],