Abstract
The second-order, nonlinear, partial-differential equation representing the transient radial flow of gas-condensate fluids in reservoirs has been integrated by using finite-difference equations and electronic computers. Effect was given to pressure-dependent permeability, viscosity, and compressibility and to distance-dependent permeability. The influence of a second-degree velocity term in the Darcy equation was investigated. Implicit methods were used and practical, convergent solutions were obtained with material balance to less than 6 x 10 for recovery of one-half the reserve at constant flow rate. Integration results provide the productive period of a reservoir for a given constant rate and the fraction of the fluid initially in place that can be recovered in that period. The properties of a lean and a rich fluid are represented in a set of integrations designed to demonstrate the effect of different constant-recovery rates and significant variables emphasized one at a time.
Introduction
A method is needed for computing the availability or reserve of a gas in a formation, making use of all technical and engineering information that is pertinent. As a step in this direction, a program for computing transient linear flow was developed that utilizes principles of earlier finite-difference computing for a similar purpose and gives effect to significant pressure- and distance-dependent properties of reservoirs and their contents. The present paper pertains to transient, radial flow of gas-condensate fluids. In the partial-differential equation of flow, effect is given to the variables viscosity and compressibility, and also to permeability. Included in the study is the quadratic form of the Darcy equation of flow that has been the subject of field tests and that has been applied to a gas with constant properties in transient flow computing. Inclusion in the differential equation of variable coefficients to represent properties greatly complicates the higher derivatives of the equation. Because it was impractical to make a required improvement in certain of the finite-difference derivatives by Taylor-series expansion, five-term derivatives were used in the implicit computing. Related methods were developed that can improve general facility in manipulating finite-difference equations.
BASIC EQUATIONS
PARTIAL DIFFERENTIAL EQUATION
The basic partial differential equation for transient radial flow in the direction of decreasing radius is
(1)
where r radius, Lv = apparent velocity along radius, L/tp = density, M/L3t = time, tphi = porosity, dimensionless.
In this equation, only the porosity is regarded constant. The density is
(2)
where p = pressure, m/LtM = molecular weight of fluid, mT = constant temperature of fluid, Tz(p)= pressure- dependent compressibilityfactor, dimensionlessR = gas constant, mL2/t2T.
SPEJ
P. 141ˆ
A parabolic differential equation with unbounded piecewise constant delay
