Modeling of Gravity-Imbibition and Gravity-Drainage Processes: Analytic and Numerical Solutions

Abstract

Summary A matrix/fracture exchange model for a fractured reservoir simulator isdescribed. Oil/water imbibition is obtained from a diffusion equation withwater saturation as the dependent variable. Gas/oil gravity drainage andimbibition are calculated by taking into account the vertical saturationdistribution in the matrix blocks. Introduction In most simulators intended for naturally fractured reservoirs, the fractureand matrix systems are considered to be two overlapping media. Flow between thetwo is described in various ways by means of source and sink terms. Thedescription of the matrix/fracture interaction is a key point in the modeling of dual-porosity systems. In this paper, the modeling of oil/water imbibition is based on thediffusion equation approach of Beckner et al. The effect of gravity isincorporated through a modification of the boundary conditions imposed. Analytical and numerical solutions are presented, and computed results arecompared with experimental data. presented, and computed results are comparedwith experimental data. Gas/oil gravity drainage and imbibition are calculatedby taking into consideration the vertical saturation distribution in thematrix. The principles for the implementation of the proposed methods in areservoir simulator are described. The following limitations and assumptions apply.The models presented are valid only for two-phase oil/water and gas/oilsystems.Matrix blocks within a grid cell are identical and box-shaped withdimensions L, L, and L.For oil/water systems, capillary continuity exists inside a grid cellbetween vertically stacked matrix blocks.The two phases in the fracture system are gravity segregated.Analytical solutions can be obtained only in the oil/water case and onlyif the water level in the fracture system rises with a constant velocity andthe diffusion coefficient is constant.The matrix-block gas and oil are at capillary/gravitationalequilibrium. Flow Equations Dual-porosity reservoirs are modeled by the continuum approach, where thefracture and matrix systems are considered to be two overlapping continuousmedia. The basic equations for isothermal fluid flow in porous media aretransformed to a system of ordinary differential equations by means of theintegral finite-difference method (see Pruess and Bodvarsson). In case of adual-porosity, single-permeability reservoir composed of a continuous fracturesystem containing discontinuous matrix blocks, the following equations areobtained for each component (1 =o, g, or w) and the kth grid cell in thereservoir. Fracture equation: (1) Matrix equation: (2) where mi = (3) (4) The summation is over all phases--i.e., =o, g, and w. The sum over Index isover all grid cells adjacent to grid-cell number k. Hence Index k refers to theboundary between the grid cells k and . The individual terms of Eqs. 1 and 2 describe the transport of Component ithrough Phase by various mechanisms. For further details regarding theequations and their derivation, see Bech. Water/Oil Imbibition The formulations of the matrix/fracture exchange term in manydouble-porosity simulators suffer from two limitations:imbibition from newly contacted matrix-block face as the fracture water level advances is not described andsaturation gradients within the matrix blocks are notmodeled. These factors can be taken into consideration by modeling theimbibition as a diffusion process. The matrix-block water saturationdistribution is determined from (5) (6) In the derivation of Eqs. 5 and 6, it is assumed that the flow is 2D andthat the fluid and the rock are incompressible. It is also assumed thatoil-phase pressure gradients and gravity terms are negligible. The diffusion equation (Eq. 5) has been solved with the boundary condition S= S at the part of the matrix-block surface that is submerged in water and Sw =S, elsewhere on the surface (see Fig. 1). Initially, the matrix-block watersaturation is S everywhere. The boundary condition (7) where S corresponds to zero capillary pressure at the surface, implies thatinstantaneous imbibition occurs at the matrix/fracture in-terface. A delayedimbibition can be introduced by the boundar conditions: (8) (9) and S = S (z) is the ultimate matrix-block water saturation at the height z, which may be equal to or less than 1 - Sor. Here t = t (z) denotes the time when zwf = z; i.e., t (z) is the time whenimbibition starts at the height z. is an inverse time constant. The problem as presented in Eqs. 5 through 9 must be solved numerically. This is done by using central differences and applymg a Newton-Raphsonalgorithm in solving the nonlinear algebraic equations.