A novel well-test for in situ estimation of relative permeabilities was recently proposed. This test consists of three periods,

  • injection of water into an oil reservoir,

  • a falloff test and

  • a producing period.

The producing period is critical as it yields production data that reflects changes in sandface mobility and is thus highly sensitive to the parameters used to model relative permeability curves. A major shortcoming of the data analysis procedure proposed in previous work was that it required a numerical reservoir simulator for matching pressure data. In this work, based on Buckley-Leverett theory and ideas from a front tracking method, we derive an approximate analytical solution for the pressure response during an injection/falloff/production test (IFPT) by applying on the Thompson-Reynolds steady-state theory. By matching data to the analytical solution by minimization of a weighted least squares objective function, we generate estimates of absolute permeability, relative permeabilities and the well skin factor. We show the method can be applied with either power law models or B-splines.


Beginning at least as early as 1952 (Verigin1), various analytical or approximate analytical solutions for the pressure response during injection and/or falloff tests have been presented in the literature. In general, these solutions require a model for generating the saturation distribution as a function of time (piston displacement of Buckley-Leverett) during injection and assume that the front does not move during falloff so that the falloff solution is equivalent to solving a multicomposite single-phase reservoir problem where the pressure at the end of the injection solution period provides the initial condition for the falloff problem. Abbaszadeh and Kamal2 show that the pressure solution at the end of injection can itself be obtained from an equivalent multicomposite problem, whereas, Bratvold and Horne3 assume the validity of the Boltzman transform to generate the injection solution. Most solutions available

consider only injection at a constant rate followed by a falloff test, the major exception being the work of Levitan4 who was able to generate a solution for multirate tests. Most solutions, including all those mentioned above, consider only one-dimensional radial flow problems, the major exceptions being Thambynayagam5 who presented injection/falloff solutions for a vertical restricted-entry well assuming piston displacement recent papers6,7,8 which used the Thompson-Reynolds steady-state theory9,10 to generate approximate analytical solutions for complete penetration and restricted-entry vertical wells as well as for horizontal wells.

Peres and Reynolds6 showed that incorporation of the skin factor using the infinitesimally thin skin is invalid during the injection period; for example, for one-dimensional radial flow problems, they showed that a skin zone with a width of a few inches can have a pronounced effect on the injection pressure and its derivative for several hours. For a damaged well and an unfavorable mobility ratio, they showed that the injection pressure derivative may be negative for a considerable period of time. Because of this result, we use a finite-radius skin zone11 for the injection/falloff/production test (IFPT) considered here.

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