# A NUMERICAL STUDY OF SHALE GAS FLOW IN TIGHT POROUS MEDIA THROUGH NONLINEAR TRANSPORT MODELS

A NUMERICAL STUDY OF SHALE GAS FLOW IN TIGHT POROUS MEDIA THROUGH NONLINEAR TRANSPORT MODELS. PhD thesis, King Fahd University of Petroleum and Minerals.

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IftikharPhDThesisMay31-2016.pdf - Accepted Version
Flow through tight porous media, especially shale gas flow, is an important and fast developing discipline. At the current times, little is known about the governing equations that describe these systems. Here, we explore a range of transport equations both conventional and fractional transport equations that describe the transport systems in tight porous media, such as in shale rocks. First, we report upon the development and application of new conventional transport model for gas flow in tight porous media. The main goal was the construction of a new realistic advection-diffusion transport model with nonlinear advection and diffusion terms, $U_a$ and $D_a$ respectively. $U_a (p,p_x)$ is a function of pressure $p(x,t)$ and the pressure gradient $\partial p/\partial x$, and $D_a (p)$ is a function of the pressure $p$. This model is based upon mass balance, and momentum balance equations, and consideration of the adsorption of the gas in the tight porous media. The model includes turbulence effects through the inclusion of a Forchchiemer's quadratic term, and it also includes the four flow regimes, that is, slip flow, Knudsen diffusion flow, transition flow, and free continuous flow, through Hagen-Poiseuille-type equation formulated in terms of a local Knudsen number, $K_n$. The transport equation contains compressibility coefficient, $\zeta (p)$, for each model parameter which are functions of the pressure. Thus the new model is a transient advection-diffusion partial differential equation for the pressure field, $p(x,t)$, with highly nonlinear and pressure dependent coefficients, apparent diffusivity $D_a(p)$, and apparent advection velocity $U_a(p,p_x)$. These features give the new model a high degree of realism, and it incorporates previous models as limiting cases. The model is first developed for three-dimensional porous media, although in application we solve the simpler 1-dimensional case for which we have also developed an implicit finite volume solver containing a flux limiter for increased stability. The solver is validated by matching the simulation results against previous model results, and also against some exact solutions in limiting cases (such as steady state problems). The effectiveness of the model and solver are demonstrated by applying it to determine rock properties of shale core samples using the available data sets. From sixteen different models that were analyzed, the model in which all the parameters were pressure dependent produces the smallest relative error of the order of $O(10^{-5})$. Its estimate of rock properties, such as the porosity $\phi$ and the permeability $K$, are more realistic than previous models. Sensitivity analysis on the model parameters show that different parameters are critical under different conditions, and that all model parameters should be retained for general applications. The model is moderately sensitive to the turbulence parameters for the cases considered. Including the turbulence term greatly improves the estimates for the shale rock characteristics such as porosity and permeability to within realistic values, indicating the importance of including turbulence effects in such types of transport models. Finally, the ability of new model to simulate pressure distributions, $p$, over a period of time is demonstrated with application to a shale rock core sample. In a second part of this study, we explore fractional transport models. Fractional derivatives contain a certain type of memory retention, and are typical of non-Gaussian processes. Several different types of fractional derivatives exist (Caputo, Riemann-Liouville, Hilfer), and from these a large number of possible transport models could be posed. Here, we have considered several cases, the most important of which was the transport model containing the Hilfer derivative of fractional order $0 < \alpha < 1$, and type $0 \leq \beta \leq 1$. The Hilfer fractional derivative is essentially an interpolation between the Caputo derivative, $\beta = 0$, and the R-L derivative, $\beta = 1$. This transport model for the pressure field $p(x,t)$ was solved using the Variation Iteration Method, VIM. A parametric study of the numerical solutions for $p(x,t)$ shows that the solution converges exponentially fast for a wide range of $\alpha$. Similar types of studies were carried out for other linear and non-linear fractional transport models, such as: with a time-fractional Caputo derivative in a linear model; with a time-fractional Caputo derivative in a non-linear model. These studies lay the foundations for application to fractional shale gas transport models.