Betancourt Sanchez, Alvaro David (2017-08). Analytical Considerations of the Stretched Exponential/Power Law Exponential Relations Used for Time-Rate Decline Curve Analysis. Master's Thesis.
Thesis
Since their introduction to the petroleum literature in 2008/2009, the stretched exponential (SE) and the power-law exponential (PLE) relations have become the "conservative" standard for time-rate (decline curve) analysis of well performance data from low/ultra-low permeability reservoirs. The origins of the SE relation can be traced to Rudolf Kohlraush circa 1854; however, its use as an expression for time-rate production data analysis essentially began in 2009 with Valk?. The PLE time-rate relation, on the other hand, was derived empirically (and independently) from observations of well performance data by Jones in 1942 and by Ilk and Blasingame in 2008. To date, there is no "proof" of the SE/PLE from a theoretical basis; however, there are many references devoted to the characteristic functions which have been proposed or derived from the given form of these relations. In this work, we attempt to provide analytical and semi-analytical bases/relations for the SE/PLE functional form to deliver insight on its mathematical behavior and to offer an understanding of its performance as a production forecasting tool. Our first approach consists on approximating the SE/PLE relation by a truncated sum-of-exponentials providing that the SE/PLE model behave as a linear superposition of exponentially decaying functions. For completeness, we extend this work to approximate the hyperbolic and modified hyperbolic time-rate relations with this mentioned sum-of-exponentials function. Next, we develop numerical approximations of the SE/PLE relation in the Laplace domain using three approaches -- using the Taylor series expansion, the Laguerre quadrature, and applying the methodology proposed by Blasingame which transform a piecewise power-law function into the Laplace domain. Our goal in this section is to use the Laplace transform and the convolution identity to resolve the SE/PLE decline model in the Laplace domain with the perspective that there may be some sort of diagnostic capability or another sort of mathematical identity, which in the process, may arise. The last part of this work is devoted to "reverse engineer" the flowing bottomhole pressure required to generate a specific SE/PLE case using numerical reservoir simulation. By means of the pseudosteady-state flow equation, the material balance equation, and a prescribed time-rate model (for our case, the SE/PLE, and the hyperbolic time-rate relations), we are able to get a mathematical expression for the flowing bottomhole pressure as a function of time. This mathematical model will be compared against well performance obtained from numerical reservoir simulation. This part of the "reverse engineering" approach provides a "proof of concept" of the validity of the SE/PLE time-rate relation and corroborates the derived functional form of the flowing bottomhole pressure.
Since their introduction to the petroleum literature in 2008/2009, the stretched exponential (SE) and the power-law exponential (PLE) relations have become the "conservative" standard for time-rate (decline curve) analysis of well performance data from low/ultra-low permeability reservoirs. The origins of the SE relation can be traced to Rudolf Kohlraush circa 1854; however, its use as an expression for time-rate production data analysis essentially began in 2009 with Valk?. The PLE time-rate relation, on the other hand, was derived empirically (and independently) from observations of well performance data by Jones in 1942 and by Ilk and Blasingame in 2008. To date, there is no "proof" of the SE/PLE from a theoretical basis; however, there are many references devoted to the characteristic functions which have been proposed or derived from the given form of these relations.
In this work, we attempt to provide analytical and semi-analytical bases/relations for the SE/PLE functional form to deliver insight on its mathematical behavior and to offer an understanding of its performance as a production forecasting tool. Our first approach consists on approximating the SE/PLE relation by a truncated sum-of-exponentials providing that the SE/PLE model behave as a linear superposition of exponentially decaying functions. For completeness, we extend this work to approximate the hyperbolic and modified hyperbolic time-rate relations with this mentioned sum-of-exponentials function.
Next, we develop numerical approximations of the SE/PLE relation in the Laplace domain using three approaches -- using the Taylor series expansion, the Laguerre quadrature, and applying the methodology proposed by Blasingame which transform a piecewise power-law function into the Laplace domain. Our goal in this section is to use the Laplace transform and the convolution identity to resolve the SE/PLE decline model in the Laplace domain with the perspective that there may be some sort of diagnostic capability or another sort of mathematical identity, which in the process, may arise.
The last part of this work is devoted to "reverse engineer" the flowing bottomhole pressure required to generate a specific SE/PLE case using numerical reservoir simulation. By means of the pseudosteady-state flow equation, the material balance equation, and a prescribed time-rate model (for our case, the SE/PLE, and the hyperbolic time-rate relations), we are able to get a mathematical expression for the flowing bottomhole pressure as a function of time. This mathematical model will be compared against well performance obtained from numerical reservoir simulation. This part of the "reverse engineering" approach provides a "proof of concept" of the validity of the SE/PLE time-rate relation and corroborates the derived functional form of the flowing bottomhole pressure.