Approximate Analytical Solutions for the Pressure Response at a Water-Injection Well
- Amina A. Boughrara (U. of Tulsa) | Alvaro Marco Peres (Petrobras S.A.) | Shi Chen (U. of Tulsa) | Augusto A.V. Machado (Petrobras) | Albert Coburn Reynolds (U. of Tulsa)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 2007
- Document Type
- Journal Paper
- 19 - 34
- 2007. Society of Petroleum Engineers
- 5.1.2 Faults and Fracture Characterisation, 2.2.2 Perforating, 5.1.1 Exploration, Development, Structural Geology, 5.3.2 Multiphase Flow, 5.6.4 Drillstem/Well Testing, 5.5 Reservoir Simulation, 5.2 Reservoir Fluid Dynamics, 5.6.3 Pressure Transient Testing, 5.8.8 Gas-condensate reservoirs, 6.5.2 Water use, produced water discharge and disposal, 5.3.4 Reduction of Residual Oil Saturation, 5.2.1 Phase Behavior and PVT Measurements
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In this paper, we construct approximate analytical solutions for the injection wellbore pressure at vertical and horizontal water injection wells using the Thompson- Reynolds steady-state theory. The solutions are based on adding to the single- phase solution, a two- phase term which represents the existence of the two-phase zone and the movement of the water front. We first present the solutions for an isotropic reservoir and then show that we can obtain the solution to an anisotropic problem by introducing a coordinate transformation to convert an anisotropic system to an equivalent isotropic system.
The analytical solutions provide insight into the behavior of injectivity tests at horizontal and 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.
We consider water injection at a constant rate through a vertical or horizontal well into a homogeneous oil reservoir above bubblepoint pressure. We provide approximate analytical solutions for the injection pressure change at the injection well under isothermal conditions. Wellbore storage effects are not considered.
In past work (Peres and Reynolds 2003), 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, it 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 k = kz and for a horizontal well for the case where the well's axis is not equidistant from the top and bottom boundaries and the permeability field is anisotropic. The solutions are based on adding to the single-phase solution, a two- phase 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 two-phase flow component of the analytical solution. The accuracy of results generated from approximate solutions are checked by comparing them to solutions generated from a black-oil simulator (IMEX 2000).
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