Approximate Solutions For Unsteady LiquidFlow In Composite Reservoirs
Authors: Ramey H.J. 
Published: March 2010

Abstract
A great deal of interest in composite reservoir flow solutions has been shownby the increasing number of publications dealing with this problem. So far, only a limited number of solutions have appeared. The solutions available donot seem to be of completely general utility. The purpose of this paper is todemonstrate a class of approximate solutions to composite reservoir flowproblems which may be generated readily. These solutions are applicable to welltest analysis and, although approximate, have the advantage of involving simplefunctions only. One interesting result is that secretary approximating formswhich ha""e been presented previously derive readily from this class ofsolutions. Another interesting result is that outer boundary conditions ofeither a closed boundary or a constant pressure boundary may be approximated aslimits of composite reservoir solutions.

INTRODUCTION
THE TERM ‘COMPOSITE’ was originally used in very early studies of heatconduction in solids composed of two different materials; see Carslaw(l) for example. Despite the well-known analogy between heatconduction and laminar fluid flow through porous media, intense interest influid flow through composite solids has occurred recently. Hurst(2)and Mortada(3) analyzed the interference between oil fields in acommon aquifer of two different permeabilities in 1960. Loucks and Guerrero(4) presented an analysis of drawdown and buildup in radialcomposite systems in 1961. Rowan and Clegg(5) presented approximatesolutions in 1962. Van Poollen and associates have presented several papersconcerning radial and linear discontinuities within porous media and theresultant effects upon well test analysis(6,7). Carter(8)presented an analysis of the depletion of a closed composite radial reservoir.and discussed reservoir limit tests for this class of reservoirs. Adams, etal.(9) employed Hurst's approximate result for radial compositereservoirs to interpret pressure buildup tests in a fractured dolomitereservoir. Recently, Wattenbarger and Ramey(10) employed a radialcomposite reservoir model to simulate the skin effect of a finite storagecapacity.
The references given above concerning oil and gas reservoir applications of thecomposite reservoir problem are provided to show the increasing interest inthis class of flow problem. No attempt has been made to provide a completelisting. The bibliographies of the papers cited list other pertinent studies.The main point is that composite reservoir flow problems are significant, andthat the connection between various applications of the solutions is notclearly evident.
One interesting fact is that papers concerning composite reservoir problemshave often raised controversies or debates, or left other evident problemsunsettled. Part of the reason must lie in the complex nature of these problems.Fortunately, the approximate solutions which will be discussed here containonly elementary functions.
The experimental data of Donnelly and Katz(1) were used for themethane – carbon dioxide system and the data of Price and Kobayashi(10) were used for the methane-ethane-propane system. Theresults of the calculations are shown graphically in Figure 6, where again itis seen that the convergence pressure concept has failed to correlate the K-factors for methane in the presence of the non-hydrocarbon component.