The engineering approach versus the mathematical approach in developing reservoir simulators

Reservoir simulation in the oil industry has become the standard for solvingreservoir- engineering problems. Reservoir simulation combines physics, mathematics,reservoir engineering, and computer programming to develop a tool for predictinghydrocarbon reservoir performance under various production strategies. The stepsinvolved in the development of a simulator include: derivation of the partial differentialequations (PDE's) describing the recovery process through formulation, discretization ofthe PDE's in space and time to obtain nonlinear algebraic equations, linearization ofresulting algebraic equations, solving the linearized algebraic equations numerically, andfinally validation of the simulator. Developers of simulators relied heavily onmathematics in the first two steps (mathematical approach) to obtain the third step(nonlinear algebraic equations or finite-difference equations). A new approach, thatderives the finite-difference equations without going through the rigor of PDE's anddiscretization, is presented in this paper. The new approach is called the engineeringapproach because it is closer to the engineer's thinking and to the physical meaning of theequations. Both the engineering and mathematical approaches treat boundary conditionswith the same accuracy if the mathematical approach uses second order approximations.The engineering approach is simple and yet general and rigorous. In addition, it results inthe same finite-difference equations for any hydrocarbon recovery process. Because theengineering approach is independent of the mathematical approach, it providesjustification for the use of central differencing in space, and gives implications of the approximations, that are usually used in the mathematical approach, in timediscretization.