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.