A Simple Method For Predicting Cap And Base Rock Heat Losses In' Thermal Reservoir Simulators
Authors: Vinsome P.K.W. , Westerveld J. 
Published: March 2010

Abstract
A fast simple method is given for calculating heat losses to cap and base rock in thermal reservoir simulators. The method can handle backflow of heat and cyclic temperature variations. Test comparisons against analytic results are given. The method gives remarkably accurate results on the test problems, considering its simplicity.

Introduction
In thermal reservoir simulations, it is necessary to calculate the heat loss from the reservoir to the surrounding environment, usually to the cap and base rock. A finite-difference calculation can become expensive, especially on computer storage, as the temperature profile can extend large distances into the overburden and underburden, requiring large numbers of grid blocks for its description. However, high accuracy in the solution of the heat loss calculations is not usually required in reservoir simulators owing to the uncertainty in cap and base rock thermal parameters. This paper describes an efficient semi-analytical method for the case where the one-dimensional linear heat conduction equation is adequate for the heat loss calculations. Semi-analytical methods have been used previously (1,2,3), and a study of the results given in these papers leads us to the following tentative conclusions:Conduction within the cap rock rapidly wipes out any sharp differences in temperature. Thus, the temperature in the cap rock varies smoothly, even for rather erratic changes at the interface.Longitudinal heat conduction in the cap rock can usually be neglected, because the Peclet number is normally high for reservoir floods(2).The most important physical features of the heat loss process are good descriptions of the boundary conditions at the interface and conservation of cap rock energy. The tail of the temperature distribution contains little energy and is relatively unimportant.
These points suggest that the temperature profile into the cap or base rock can be adequately approximated by any reasonably flexible function containing a few parameters. If such a function could be made to fit the thermal diffusivity equation with appropriate boundary conditions, and the calculation of the parameters could be done in a simple fast fashion, the method would become extremely attractive further justification for such an approach is that high accuracy is not required, as the coefficient of thermal conduction of the cap and base rock is rarely known precisely.

The Method
We choose as the fitting function for the temperature profile into the cap or base rock:
(Equation Available In Full Paper)
θ is the temperature at the interface between the reservoir and the cap or base rock (the zero level has been defined at the initial interface temperature); p and q are the fitting parameters yet to be determined; d may be interpreted as the diffusion length and should be of order √kt where t is the time measured from the instant at which the interface temperature first begins to change. The deepest penetrating term in (1) is z2e−z/d, which has its maximum at z = 2d. Hence we choose for the diffusion length: