Complex wells such as horizontal wells or multi-lateral wells are increasingly used in reservoir engineering to improve the oil recovery. A widely used semi-analytical method to study the behavior of this kind of wells, such as production performances or transient pressure responses is the source function method. However, the source method is only a particular approach to this problem, and it can be improved using boundary integral techniques.
In this paper, a mathematical model, based on the potential theory, is established for transient problem. A boundary integral equation (BIE) is introduced using the single layer heat potential. The finite (pressure drop along the well length) conductivity can be taken into account in this model. So, the wellbore flow model is adequately coupled with the semianalytical reservoir model.
The collocation and Galerkin methods are particularly studied to solve the BIE. It is shown that the Galerkin method is more efficient than the collocation method or the line source method, which can be deduced from the collocation method. Some applications of this model are also presented.