In reservoir simulation, the question is not whether, but how and how much. The complexity of the questions being asked, and the amount and reliability of the data available, must determine the sophistication of the system to be used. Introduction Webster defines "simulate" as "to assume the appearance of without the reality". Simulation of petroleum reservoir performance refers to the construction and operation of a model whose behavior assumes the appearance of actual reservoir behavior. The model itself is either physical (for example, a laboratory sandpack) or mathematical. A mathematical model is simply a set of equations that, subject to certain assumptions, describes the physical processes active in the reservoir. Although the model itself obviously lacks the reality of the oil or gas field, the behavior of a valid model simulates (assumes the appearance of) that of the field. The purpose of simulation is to estimate field performance (e.g., oil recovery) under a variety of performance (e.g., oil recovery) under a variety of producing schemes. Whereas the field can be produced producing schemes. Whereas the field can be produced only once and at considerable expense - a model can be produced or "run" many times at low expense over a short period of time. Observation of model performance under different producing conditions, performance under different producing conditions, then, aids in selecting an optimum set of producing conditions for the reservoir. More specifically, with reservoir simulation the following are possible.We can determine the performance of an oil field under water injection or gas injection, or under natural depletion.We can judge the advisability of flank waterflooding as opposed to pattern waterflooding.We can determine the effects of well location and spacing.We can estimate the effect of producing rate on recovery.We can calculate the total gas field deliverability for a given number of wells at certain specified locations.We can estimate the lease-line drainage in heterogeneous oil or gas fields. The tools of reservoir simulation range from the intuition and judgment of the engineer to complex mathematical models requiring use of digital computers. The question is not whether to simulate but rather which tool or method to use. There is no general answer to the question as to when the computerized mathematical model should be employed. After some preliminary discussion of the nature of mathematical models and sources of error, some valid and invalid model applications will be illustrated here with specific examples. It should be noted that this discussion is restricted to models for multidimensional, single or multiphase flow in reservoirs. These models apply to dry gas reservoirs and to oil reservoirs undergoing natural depletion or pressure maintenance (such as natural water drive, waterflood or gas injection). The Mathematical Model In 1959 Douglas, Peaceman and Rachford proposed the "Leap-Frog" and "Simultaneous Solution" methods for solving two-dimensional, two-phase flow problems. In 1960, Stone and Sheldon described an problems. In 1960, Stone and Sheldon described an "Implicit Pressure-Explicit Saturation" method. JPT P. 1391