A Semi-Analytic Method for Thermal Coupling of Reservoir and Overburden

A semianalytic method of handling the energy balance solution in the overburden has been developed. The method results in a single overburden energy coupling equation applicable at the reservoir - overburden boundary. This feature facilitates simultaneous solution of the pressure and energy equations in the reservoir. Test problems show the semianalytic method to compare favorably with the fully finite-difference technique. Both reservoir - overburden boundary temperature and the important factor of beat transfer into the overburden are accurately predicted by the method. predicted by the method. The semianalytic procedure may have considerable usefulness in the solution of reservoir problems. The method can definitely be applied in thermal simulation programs. Other anticipated applications are to an aquifer underlying a reservoir, an overlying gas cap, or to increasing definition in the vicinity of a wellbore. In each of these applications, the semianalytic procedure is expected to be considerably faster than the finite - difference solution. Introduction In thermal reservoir simulation programs, the material balances are solved over the reservoir; the energy balance is solved over the underburden-reservoir-overburden system. Hence, to avoid excess storage and computation time requirements, the material and energy balances are generally not solved simultaneously. In many cases it would be desirable to solve simultaneously the material balances (or equivalently, the pressure equation obtained by combining the material balances to eliminate saturation) and the energy balance. Problems with solution convergence and the treatment of mass transfer terms could be avoided in this way. This paper describes a semianalytic method of handling the energy balance solution in the overburden (and underburden). This solution results in a single overburden energy coupling equation that can be solved easily in conjunction with the reservoir pressure and energy equations. pressure and energy equations. The paper presents the mathematical development of the method and shows results for several test problems. The problems are similar to those encountered in thermal recovery processes. They compare the semianalytic method with the fully finite-difference procedure and, in one of the problems, with a completely analytic procedure and, in one of the problems, with a completely analytic solution. THEORY GOVERNING EQUATION AND BOUNDARY CONDITIONS Consider the overburden (and underburden) to be a homogeneous medium. The energy balance for the overburden can then be written as follows: (1) where is the density, C is the heat capacity, and khx, khy, khz, are the thermal conductivities in the x, y, z directions, respectively. The initial and boundary conditions for Eq. 1 are (2) Here we assume 0 x a, 0 y b to be the lateral extent of the reservoir and overburden, and 0 z to be the vertical extent of the overburden. SPEJ P. 439