Pressure Distributions in Rectangular Reservoirs

Abstract

EARLOUGHER JR., ROBERT C., MARATHON OIL CO., LITTLETON, COLO. JUNIOR MEMBER AIME RAMEY JR., H.J., STANFORD U., STANFORD, CALIF. MILLER, F.G., STANDARD OIL CO. OF CALIFORNIA, SAN FRANCISCO, CALIF. MUELLER, T.D., STANDARD OIL CO. OF CALIFORNIA, SAN FRANCISCO, CALIF. MEMBERSAIME Abstract There are many studies of flow in radial systems that can be used to interpret unsteady reservoir flow problems. Although solutions for systems of infinite extent can be used to generate solutions for finite flow systems by super-position, application is tedious. In this paper a step is made toward simplifying calculations of such solutions for finite flow systems. Superposition is used to produce a tabulation of the dimensionless pressure drop function at several locations within a bounded square that has a well at its center. The square system provides a useful building block that may be used to generate flow behavior for any rectangular shape whose sides are in integral ratios. Values of the tabulated dimensionless pressure drop function are simply added to obtain the dimensionless pressure drop function for the desired rectangular system. The rectangular system may contain any number of wells producing at any rates. Furthermore, the outer boundaries of the rectangular system may be closed (no-flow) or they may be at constant pressure. Mixed conditions also may be considered. Tables of the dimensionless pressure drop function for the square system are prevented and various applications of the technique are illustrated. Introduction In 1949 van Everdingen and Hurst published solutions for the problem of water influx into a cylindrical reservoir. Since this problem is mathematically identical with the depletion of a cylindrical reservoir with a well at the origin, the van Everdingen-Burst solution may be used to study the depletion problem. In their analysis, they assumed that the fluid had a small, constant compressibility such that flow was governed by the diffusivity equation (1) For a constant production rate q starting at time zero, van Everdingen and Hurst showed that the unsteady pressure distribution for both finite and infinite systems could be expressed in terms of a dimensionless pressure (2) (3) (4) where rw = wellbore radius (reservoir radius for influx)pD = dimensionless pressure at rD at tD. Tabulations of the dimensionless pressure drop for a unit value of rD were provided by van Everdingen and Hurst, and later by Chatas. Others also presented values in graphical or tabular form. If the radius of the well becomes vanishingly small, rW 0, the line source solution may be used for Eq. 2 when infinite systems are considered. (5) where -Ei(-x) is the well known exponential integral. If the argument of the exponential integral is small enough, (6) Eqs. 5 and 6 are excellent approximations for Eq. 2 under certain conditions. In 1954, Matthews, Brons and Hazebroek demonstrated that solutions such as Eq. 5 can be superposed to generate the behavior of bounded geometric shapes; i.e., the behavior of a bounded single-well system can be calculated by adding together the pressure disturbances caused by the appropriate array of an infinite number of wells producing from an infinite system. These wells are referred to as image wells. Matthews, Brons and Hazebroek considered systems containing a single well producing at a constant rate. JPT P. 199ˆ