"An Adaptive Nonlinear Solution Scheme for Reservoir Simulation."

                        G. Scott Lett

Numerical reservoir simulation involves solving large, nonlinear systems
of PDE with strongly discontinuous coefficients.  Because of the large
demands on computer memory and CPU, most users must perform simulations
on very coarse grids.  The average properties of the fluids and rocks
must be estimated on these grids.  These coarse grid "effective" 
properties are costly to determine, and risky to use, since their 
optimal values depend on the fluid flow being simulated.  Thus, they 
must be found by trial-and-error techniques, and the more coarse the 
grid, the poorer the results.

This paper describes a numerical reservoir simulator which accepts fine
scale properties and automatically generates multiple levels of coarse
grid rock and fluid properties.  The fine grid properties and the coarse
grid simulation results are used to estimate discretization errors with
multilevel error expansions.  These expansions are local, and identify
areas requiring local grid refinement.  These refinements are added
adaptively by the simulator, and the resulting composite grid equations
are solved by a nonlinear Fast Adaptive Composite (FAC) Grid method, with
a damped Newton algorithm being used on each local grid.  The nonsymmetric
linear system of equations resulting from Newton's method are in turn
solved by a preconditioned Conjugate Gradients-like algorithm.

The scheme is demonstrated by performing fine and coarse grid simulations
of several multiphase reservoirs from around the world.