Partial differential equations are ubiquitous in petroleum engineering. In this thesis, we begin by introducing a generalization to Darcy's law which includes the effects of fluid inertia. We continue by using the generalization of Daryc's law to derive the hyperbolic diffusion equation (a generalization of the parabolic diffusion equation) which takes into account a finite propagation speed for pressure propagation in the fluid. We develop the mathematical theory used to solve the diffusion equations with various boundary conditions, which include the theory of Sturm-Liouville problems, eigenfunction expansions, and Laplace transformations. Further, we introduce the application of a nonsingular Hankel transform method for finding the solution to the diffusion equations with nonzero and nonconstant initial and boundary conditions. It will be shown that the Hankel transform method developed herein proves to be a more straightforward and less time consuming computation than those found in the literature. To show the application of the parabolic diffusion equation in the industry, we proceed to derive the solution to the pressure pulse decay method as well as the well-known GRI crushed core permeability method. After derivation of the solutions, we show that the results obtained have excellent agreement with the data that can be found from sources for the pressure pulse decay method and the actual crushed core experiments from the GRI. To provide further insight, we investigate the pressure behavior inside the crushed core sample and core samples as the pressure response moves from transient to steady-state. This type of analysis has not been discussed in existing literature.
Partial differential equations are ubiquitous in petroleum engineering. In this thesis, we begin by introducing a generalization to Darcy's law which includes the effects of fluid inertia. We continue by using the generalization of Daryc's law to derive the hyperbolic diffusion equation (a generalization of the parabolic diffusion equation) which takes into account a finite propagation speed for pressure propagation in the fluid. We develop the mathematical theory used to solve the diffusion equations with various boundary conditions, which include the theory of Sturm-Liouville problems, eigenfunction expansions, and Laplace transformations. Further, we introduce the application of a nonsingular Hankel transform method for finding the solution to the diffusion equations with nonzero and nonconstant initial and boundary conditions. It will be shown that the Hankel transform method developed herein proves to be a more straightforward and less time consuming computation than those found in the literature.
To show the application of the parabolic diffusion equation in the industry, we proceed to derive the solution to the pressure pulse decay method as well as the well-known GRI crushed core permeability method. After derivation of the solutions, we show that the results obtained have excellent agreement with the data that can be found from sources for the pressure pulse decay method and the actual crushed core experiments from the GRI. To provide further insight, we investigate the pressure behavior inside the crushed core sample and core samples as the pressure response moves from transient to steady-state. This type of analysis has not been discussed in existing literature.