Abstract

The jupiter fluid problem can be defined as the computation of negative total compressibilities during mathematical simulation of the behavior of a hydrocarbon reservoir. This condition will cause diagonal dominance to be lost in the matrix of coefficients of the finite difference equations which describe reservoir performance. The problem can be avoided by employing mathematical techniques which calculate reservoir pressures with a high degree of precision while maintaining an accurate material balance computation. Some of the reservoir simulation models in current use do not have this capability, however, and a technique is presented for overcoming the jupiter fluid problem which occurs in these models.

Introduction

A jupiter fluid may be defined as a hypothetical gas-liquid system which has the peculiar property of expanding when subjected to increased pressure. Although no such fluid can actually exist in a hydrocarbon reservoir, the mathematical simulation of an oil reservoir can generate equations which imply a negative system compressibility at pressures slightly below the bubble point. This problem often arises in simulation models which do not maintain a high degree of accuracy in the pressure calculations. These inaccuracies are frequently due to the use of a computational technique which is inadequate for the specific problem being solved.

The jupiter fluid problem can usually be avoided by reducing the time step size or by using a more accurate computational method for calculating pressure. However, the problem is still troublesome for users of some types of simulation models. The technique described below has been found to be effective in modifying these types of mathematical models so that stability of the calculation can be maintained at pressures near the bubble point.

PRESSURE EQUATION

Pressure distribution within a hydrocarbon reservoir may be described by the differential equation

Equation (1) (Available in Full Paper).

The transmissibilities and potentials for each phase p, and the flow term. q', are defined by

Equation (2) (Available in Full Paper).

Equation (3) (Available in Full Paper).

and

Equation (4) (Available in Full Paper).

for two-dimensional simulation. In equation (3), P is. the oil phase pressure, p is average density, and Pc is capillary pressure (which is positive between oil and gas and negative between oil and ater), The symbols z and g represent depth below a reference datum and gravitational acceleration, respectively. The subscript gf denotes free gas.

Mathematical representation of reservoir performance is accomplished by writing equation (1) (with appropriate boundary conditions) in finite difference form for each mesh point in the reference grid. For example, the equations required for a two-dimensional simulation, implicit in pressure, may be represented by

Equation (5) (Available in Full Paper).

where the superscript denotes the n + 1 time level, and the subscripts i and j identify the x and y coordinate positions, respectively.

If straight forward finite difference approximations are used (without consideration of the jupiter fluid problem) the matrix elements of equation (5) may be written

Equation (6) (Available in Full Paper).

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