Advances in the Use of Convolution Methods in Well Test Analysis
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This thesis formalizes and extends a prior work in the effort to create explicit solutions to directly compute the effects of wellbore storage and phase redistribution (phase redistribution is treated as a special case of the wellbore storage problem and is of secondary priority in this work). The objectives of this work are to: derive approximate solutions in the Laplace domain that can be inverted directly to the real domain; validate these approximate solutions against the exact solutions for wellbore storage; develop correlations to improve approximate solutions which do not perform well in their original form; develop schemes to "deconvolve" the effects of wellbore storage using either a direct "inversion" to remove these effects or one of the approximations to determine the undistorted solution as a root-solving problem. The key element of this work is the development of the approximate solutions for the wellbore storage distortion case in the Laplace domain (i.e., part of the first objective). As a starting point, we retrace the work of SPE 21826 and we note that these solutions hinge on the use of approximations for the constant rate (undistorted) solution (psD). In this work we utilize three scenarios to approximate the psD(tD) function for the purpose of the Laplace transform formulation: the "constant" psD(tD) case which considers psD(tD) to be constant; the "linear" psD(tD) case which considers psD(tD) to be defined by a linear relationship of tD; and the "quadratic" psD(tD) case which considers psD(tD) to be defined by a quadratic relationship of tD. As in SPE 21826, each of these solutions has been recast and compared to the exact solution for cases of effects of wellbore storage and phase redistribution (the second objective). The development of correlations to improve the derived explicit (real domain) solutions has proven problematic, for example, the simplest case is that of the "constant" psD(tD), where the solution is given as: PwCD(tD)=PsD(tD)［1-exp［-tD/PsD(tD)C¬¬_D］］(“constant”p_s(tD) case) The most interesting aspect of this result is that it is exact at very early and very late times, but has errors as high as 15.6 percent in terms of pwCD(tD) and as high as 25.9 percent in terms of pwCD'(tD). The goal is to ensure errors less than 1.5 percent for pwCD(tD). This led to the effort to develop correlations (the third objective) for an "additive" error term (), where this function would be in terms of the variable [tD/(psD(tD)CD)], which appears to be a unique correlation variable for wellbore storage cases. Several correlations are presented in this work. The last goal of this work is to provide a wellbore storage "deconvolution" scheme (the fourth objective) which uses a permutation of the methodology used to derive the approximate pwCD(tD) solutions in order to derive the psD(tD) function in terms of the pwCD(tD) solutions, and/or uses the pwCD(tD) approximate solutions as "root solutions" to solve for the input psD(tD) function. Demonstrative cases are provided for this "deconvolution" process.
Kio, Adaiyibo Emmanuel (2017). Advances in the Use of Convolution Methods in Well Test Analysis. Master's thesis, Texas A & M University. Available electronically from