We consider the problem of the control of a displacement front in a porous medium, via flow-rate partition in a well. We assume that the flow is potential and that the displacement is at a unit mobility ratio. These assumptions are for mathematical convenience only and can be relaxed. They allow, however, significant insight into the problem. The specific question we address is how to partition the flow rate within the injection well, so that the induced displacement front can be steered according to pre-determined dynamics.
When the reservoir is homogeneous and isotropic, we derive an integral equation in an analytical form, the solution of which determines the desired injection rate profile. We provide illustrative applications. A similar approach applies for an anisotropic or a heterogeneous system, except that the kernel in the integral equation must be determined numerically. This can be obtained by repeated calculations of the Green's function in a heterogeneous system or a modified two well system.
For the solution of the integral (Fredholm) equation, a regularization technique is necessary. However, it is found that numerical instabilities do develop, even with the use of regularization, for later times, when tranverse cross-flow is large. Conversely, the instabilities diminish with a more stratified structure.
The results find applications to the rapidly emerging field of smart wells and the optimization of displacement problems in oil reservoirs using flow rate control.