A Technique for Improving Material Balance Accuracy in Reservoir Simulation Models
Authors: Merchant A.R. , Arnold M.D. , Harvey A. Herbert 
Published: December 2007

This paper was prepared for the 48th Annual Fall Meeting of the Society of Petroleum Engineers of AIME, to be held in Las Vegas, Nev., Sept. 30-Oct. 3, 1973. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgement of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made.
Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines.

Abstract
One criterion for determining the precision of pressures calculated by a simulation model is the accuracy of the material balance. A high degree of precision in the calculated pressures is a necessary but not sufficient condition for elimination of significant errors in the prediction of fluid migration in the reservoir. prediction of fluid migration in the reservoir. A new procedure, called error matrix technique, has been developed for reduction of material balance errors. This reduction is accomplished by adjusting potential gradients at every grid point in the simulation model so that the point in the simulation model so that the material balance is accurately maintained. Performance of a hypothetical oil reservoir Performance of a hypothetical oil reservoir simulated both by conventional methods and the error matrix technique. The new technique was found to be more accurate than the conventional simulation.

Introduction
A mathematical reservoir simulation model is based on sets of nonlinear partial differential equations that express pressure and saturations as functions of time and position. Two types of error should be considered in evaluating the accuracy of a simulation developed by such a model. One of them is machine round-off error. Accumulation of this error can lead to instability of the calculation scheme. If round-off error were not present, an exact solution of the finite-difference equations would be obtained. However, an exact solution of the difference equations usually is not an exact solution of the differential equations that they represent.
Exact finite-difference representation of the differential equations is not ordinarily achieved because the derivatives are approximated by truncated Taylor's series expansions. The discretization error that is caused by the truncation is another source of inaccuracy that can be anticipated when finite-difference techniques are employed. The magnitude of this error is reflected in the deviation of the calculated pressures from the true ones.
The use of imprecise pressures for saturation computations leads to the calculation of inaccurate saturation values. If these saturations are used to compute the mass (or volume) of fluids in the reservoir, these computations will also be somewhat in error. Therefore, the law of mass conservation provides a basis for evaluating the accuracy of a reservoir simulation model. This paper describes a method for improving the accuracy of the simulation by increasing the precision of the mass balance.