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Collaborating Authors
Results
Analysis of Injection/Falloff Data From Horizontal Wells
Boughrara, Amina A. (Schlumberger) | Reynolds, Albert C. (University of Tulsa)
Summary We have constructed new approximate analytical solutions for injection and falloff pressure response that include thermal effects that arise when flooding a reservoir with water that has a temperature considerably lower than that of the reservoir. We have developed an optimization code based on the Levenberg-Marquardt algorithm and coupled it with our new approximate analytical solutions to obtain a procedure for data analysis where our approximate analytical solutions are used as the forward model in the nonlinear regression. We demonstrate that we can generate estimates of absolute permeabilities, the well skin factor, the length of the well and relative premeabilities by matching data to analytical solutions by minimization of a weighted least squares objective function. The relative permeability curves are constructed assuming a power law parametrization. In the horizontal well case, we show that the absolute permeabilities in the three principal directions can be resolved separately, provided the duration of the test is sufficiently long. Introduction Heat transfer must occur whenever a temperature difference exists in a medium or between media. When cold water is injected into a hot reservoir, the formation around the water injector will cool down to the temperature of the injected water. This creates a cold water bank around the injector that expands with time into the reservoir. Similar to the saturation front, the temperature front will also propagate in the reservoir. Both the solid and fluid phases contribute to the heat transfer. The heat exchange in the reservoir occurs mainly through three processes: convective heat transfer between injected fluid and solid matrix, heat conduction (vertical and horizontal conduction), and heat transfer by radiation. The last mechanism is not considered to be important in porous media and, therefore, is usually neglected when the gas phase is not involved.
Summary 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. Introduction 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).
- South America > Brazil (1.00)
- North America (0.68)
- Europe > Norway > Norwegian Sea (0.24)
- Water & Waste Management > Water Management > Lifecycle > Disposal/Injection (1.00)
- Energy > Oil & Gas > Upstream (1.00)
- Well Drilling > Drilling Operations > Directional drilling (1.00)
- Reservoir Description and Dynamics > Improved and Enhanced Recovery > Waterflooding (1.00)
- Reservoir Description and Dynamics > Formation Evaluation & Management > Pressure transient analysis (1.00)
- Reservoir Description and Dynamics > Formation Evaluation & Management > Drillstem/well testing (1.00)
Rate Superposition for Generating Pressure Falloff Solutions
Peres, Alvaro M. (Petrobras S.A.) | Boughrara, Amina A. (U. of Tulsa) | Reynolds, Albert C. (U. of Tulsa)
Summary Although the Thompson-Reynolds steady-state theory has proved useful for explaining the relation between reservoir physics and the pressure/pressure derivative response for both injection and falloff tests, until now, we have been unable to apply this method to construct analytical solutions for the falloff response. In this work, we remedy this deficiency by constructing approximate analytical solutions for the pressure falloff response subsequent to water injection at a vertical or horizontal well. 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 falloff solution can be written as the sum of the single-phase falloff solution based on oil properties at initial water saturation plus a multiphase flow term, which reflects the deviation of the total mobility (in the region contacted by injected water) from oil mobility at initial water saturation. The multiphase term is presented as an integral in the vertical well case and a sum of one to three integrals in the horizontal well case. For the purpose of constructing an accurate estimate of the falloff multiphase pressure change term, one can use a series of 1D Buckley-Leverett solutions (one for each integral in the multiphase term) and assume that, throughout the falloff period, the total mobility profile in the reservoir is equal to the total mobility profile that existed at the instant of shut-in. Evaluation of each integral in the multiphase term requires the 1D mobility profile constructed from the Buckley-Leverett solution and a corresponding 1D flow rate profile during falloff. For linear single-phase flow, it is shown that rate superposition applies and we use this concept in a reasonable but ad hoc way to estimate the rate profiles needed to compute the multiphase pressure term. It is shown that even in cases where falloff data allow one to accurately estimate the properties of the oil zone, knowledge of the multiphase term is critical in order to obtain an accurate estimate of the mechanical skin factor.