Extended Semianalytic Method for Increasing and Decreasing Boundary Temperatures
Authors: Weinstein H.G. 
Published: December 2007

Abstract
A semianalytic method developed earlier couples the overburden energy balance solution to reservoir equations by a single differential equation applicable at the reservoir/overburden boundary. The semi-analytic method is extended in this work to allow temperatures at the reservoir/overburden boundary to decrease, as well as increase, with time. Computer calculations on several test problems show a close agreement of the semianalytic method with the fully finite-difference solution. Both reservoir/overburden boundary temperature and heat flux into the overburden are accurately calculated. Because of its extended generality, the semi-analytic procedure should be quite useful in solving reservoir problems. It is expected that in addition to being useful in thermal simulation programs, it will also be applicable to aquifer, programs, it will also be applicable to aquifer, gas-cap, pseudorelative-permeability, and wellbore problems. The method is faster and requires problems. The method is faster and requires significantly less computer storage than the finite-difference solution.

Introduction
Thermal reservoir simulation programs, to avoid excessive storage and computation-time requirements, generally do not solve the material balance and energy balance simultaneously, Instead, there an two separate solution steps. First, the material balances are solved over the reservoir; then, with updated pressures and saturations, the energy balance is solved over the underburden/reservoir/ overburden system. However, problems with solution convergence and the treatment of mass transfer terms could be avoided by simultaneously solving the energy balance and the material balances (or equivalently, the pressure equation obtained by combining the material balances to eliminate saturation).
A recent paper showed how variational methods could be applied to eliminate the energy-balance solution in the overburden, and thus avoid the problems enumerated above. Included in the model problems enumerated above. Included in the model were a three-dimensional variational principle and an overburden temperature approximation proportional to the solution of the one-dimensional heat conduction equation. However, only the case of a monotonicallyincreasingreservoir/overburden boundary temperature was treated. Using a variational principle complementary to Weinstein's, Chase and O'Dell considered the flow of heat both into and out of the overburden. Their variational equation was one-dimensional, and the assumed overburden temperature function was a one-dimensional cubic polynomial, chosen because of its simplicity. Because the variational model was one-dimensional, no account could be taken of conduction parallel to the reservoir/overburden boundary. Thus, their results are restricted to situations where convection parallel to the reservoir dominates conduction. Chase and O'Dell derived two coupled nonlinear differential equations for the two free parameters of their model. An analytic solution was obtained for increasing boundary temperatures; however, the two equations had to be integrated numerically for decreasing boundary temperatures. To calculate their heat loss vector they had to perform an inner iteration with respect to both perform an inner iteration with respect to both temperature and time. Solving the parameter equations and solving for the heat flux vector were both time-consuming, leading to only a "moderate" savings in computation time over the fully finite-difference model. On the test problems they studied, the model showed increasing errors as the simulation proceeded through the soak and backflow periods. Presumably, these errors would continue to periods. Presumably, these errors would continue to grow if additional heat-flow reversals were invoked. The model to be described here has alleviated the shortcomings in Chase and O'Dell's procedure. This paper describes a generalized semianalytic method paper describes a generalized 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. The coupling equation pressure and energy equations. The coupling equation is general, whether reservoir/overburden boundary temperature increases or decreases with time, or increases at some boundary locations while decreasing at others. The paper presents the mathematical development of the extension of the original method to increasing and decreasing temperature problems.
SPEJ
P. 152