Analytical and Numerical Solution of Oil Recovery From Fractured Reservoirs With Empirical Transfer Functions (includes associated papers 25528 and 25818)

Abstract

Summary. An analytical and a numerical solution to Buckley-Leverettflow in a ID fractured porous rock is given. An empirical matrix/fracture transfer function was used. The numerical solution is fast and is adaptable to 3D field problems. The limitations ofempirical transfer functions are discussed. Introduction In the last decade, interest has grown in the use of dual-porosity/dual-permeability numerical simulators to model fluid flow in naturally fractured reservoirs. In practice, sufficiently refining the computational grid to simulate fracture/matrix flow rigorously is very expensive. Recently, several investigators resorted to the use of empirical transferfunctions(through pseudocapillary pressures) to account for thefracture/matrix flow and to scale laboratory or fine-grid single-blockresults to the field dimensions. The main thrust of this paper is to showthe limitations of such empirical transfer functions. However, thetransfer-function approach is computationally very fast and can lead toextremely useful and practical solutions for reservoirs under waterdrive, waterflood, or gas injection. Empirical Matrix/Fracture Transfer Functions When water imbibition is the dominant force for displacing oil from the matrix, the cumulative oil recovery by imbibition from a pieceof rock surrounded by water generally can be approximated by an exponential form as shown: (1) where (2) and (3) A more general and flexible oil recovery function is presented in the discussion section. The usefulness of Eq. 1 is illustrated by fitting it to theimbibition oil recovery data of Mattax and Kyte for alundum and sandstonecores imbibing water from one end or from all sides. Mattaxand Kyte correlated their data using a dimensionless time, given in consistent units by (4) When Eq. 4 is used, the cumulative recovery curve will be of theform (5) where (6) X is a constant equal to the reciprocal of the time required to recover 63% of the recoverable oil. To fit all Mattax and Kyte's data into a single correlation, weredefined Eqs. 4 and 6 in terms of matrix block shape factor, F, as (7) and (8) where (9) Vma = volume of the matrix block, Am = area of a surface areaopen to flow in a given flow direction, d.=distance from theopen surface to the center of the matrix block, and Es = summationover all open surfaces of a matrix block. For example, for arectangle with all sides imbibing, Eq. 9 gives (10) For a cylinder with all sides imbibing, the shape factor reduces to (11) Fig. 1 shows our correlation of Mattax and Kyte's data. Buckley-Loverett Solution The Buckley-Leverett displacement in a fracture surrounded bymatrix rock undergoing imbibition was given by deSwaan: (12) We derived an analytic solution to this problem when fwf = Swf, Laplace transform is used as (13) and (14) where (15) and (16) Eq. 14 is very similar to the solution obtained by deSwaan;however, it is not exactly the same solution. The condition fwf = Swfshould be attainable if oil in the fracture is dispersed as dropletsand entrained in the moving water in the fracture. SPERE P. 219⁁