Abstract The first and fifth authors have previously used the Thompson-Reynolds steady-state theory to derive solutions for the pressure response at a water injection well. In the vertical well case, their solutions assumed a complete-penetration well; in the horizontal well case, they assumed that the well is equidistant from the top and bottom of the formation. Here, we construct approximate analytical pressure solutions for the restricted-entry vertical well case and for a horizontal well for the case where the well's axis is not equidistant from the top and bottom boundaries. The solutions are based on adding to the single-phase solution, a multiphase term which represents the existence of the two-phase zone and the movement of the water front. We present models for the movement of water based on a combination of Buckley-Leverett equations that allow us to accurately compute the multiphase flow component of the analytical solution. By comparison with a finite-difference simulator using grid refinement and a hybrid grid, it is shown that our multiphase flow solutions are accurate.
The analytical solutions provide insight into the behavior of injectivity tests at horizontal an vertical wells. For example, for a restricted-entry case, it is shown that the pressure derivative may be negative throughout an injection test even when the duration of the test exceeds ten or more days. We also show that for a well near a fault, the ratio of slopes reflected by derivative data will not in general be equal to two but is given by a formula that involves the mobility ratio. In the restricted-entry vertical well case, we provide the equations for three flow regimes.
Introduction We consider water injection at a constant rate through a vertical or horizontal well into a homogeneous oil reservoir above bubble point pressure. We provide approximate analytical solutions for the injection pressure change at the injection well. Wellbore storage effects are not considered.
In past work, we have used a steady-state theory to derive solutions for the pressure response at a water injection well. In the vertical well case, the solution assumed a complete-penetration well; in the horizontal well case, they assumed that the well is equidistant from the top and bottom of the formation and that the formation is isotropic kz=k. Here, we construct approximate analytical pressure solution for the restricted-entry vertical well case for kkz and for a horizontal well for the case where the well's axis is not equidistant from the top and bottom boundaries. The solutions are based on adding to the single-phase solution, a multiphase term which represents the existence of the two-phase zone and the movement of the water front. We present models for the movement of water based on a combination of Buckley-Leverett equations that allow us to accurately approximate the multiphase flow component of the analytical solution. This paper considers only the injection solution. A companion paper presents analytical solutions for the falloff pressure response.
The accuracy of results generated from approximate solutions are checked by comparing them to solutions generated from a black oil simulator. For the horizontal well problem, we present only the case where kz=k. Conceptually, the procedure presented here applies to the anisotropic horizontal well case, but for the anisotropic case, we have been unable to confirm the accuracy of the analytical solution. At the date of this writing, it is unclear as to whether there is a theoretical error, a programming error in our solution and that have have chosen inappropriate grids for generating accurate solutions with the reservoir simulator.