MATRIX HETEROGENEITY EFFECTS ON FLUID TRANSPORT IN POROUS MEDIUM: USING PERTURBATION THEORY

Abstract

In this dissertation the processes of transport and reaction in heterogeneous porous medium is upscaled from pore scale to the scale of interest (e.g., core sample or reservoir scale) and new governing equations describing gas transport and reaction including the effects of heterogeneity is introduced.

New upscaled equations for the first time described observations have been made earlier in the laboratory regarding storage and transport of gas in tight formations, such as gas trapping mechanism in gas release experiments and gas threshold effect in gas uptake, gas loading effects, adsorbed phase transport and nonlinearity effects of gas sorption kinetics on diffusive transports. New upscaled governing equations provide basis for further experimental works to quantify and distinguish effects of local heterogeneities on transport and storage in tight formations.\\

Matrix heterogeneity effects on fluid transport in porous medium is investigated using an upscaling approach based on small perturbation theory.

The approach allows us investigate the heterogeneity effects in spectral domain in the presence of non-equilibrium sorption with random partition coefficient. The work is motivated by the fact that (1) the porous medium is heterogeneous, i.e., it has a significant degree of spatial variability, and shows a complex structure at larger scales; (2) there always exists a lack of knowledge of the detailed local structure of these spatial variations; and (3) difficulties appear in obtaining sufficient data related to spatial and temporal distributions of mass and momentum variables, dictated by these large-scale variations. Here, the heterogeneity of the matrix is introduced using random porosity or permeability fields that holds the assumption of first and second orders of stationarity, i.e., constant mean and variance, and possess a well defined gaussian correlation function. All transport and kinetics parameters and dependent variables (e.g., free and adsorbed gas concentration) are also affected by the matrix heterogeneity and represented by their average and perturbations around the mean (arithmetic average) values.

First, fundamental investigation is performed on shale and coal samples with simplified gas transport and adsorption kinetics. The former is simplified by considering the free phase transport only in the micro-pores and the latter by assuming linear non-equilibrium sorption kinetics. Substituting the perturbed variables and coefficients into the basic governing equations leads to the set of stochastic partial differential equations including mean and perturbation equations. Mean equations are essentially upscaled new governing equations that includes the auto- and cross-correlations between different perturbed quantities. The auto- and cross-correlations could be found solving perturbation equations in the Laplace-Fourier domain and back-transforming them to the time-space domain.

It is found that upscaled deterministic gas mass balance includes new sink and source terms into the governing equations related to the local heterogeneity. Heterogeneity affects the gas transport and adsorption significantly through macro-transport and macro-kinetics terms. Macro-transport depends on P$\acute{e}$clet number and interestingly persist at the diffusive limit, while micro-kinetics is related to the modified Thiele modulus. Heterogeneity retards gas release from the matrix and influences the ultimate gas recovery adversely. Both effects are directly related to the amount of initial gas adsorbed and the level of heterogeneity introduced to the system by the porosity variance. Next, the effects of heterogeneity on gas transport and adsorption in the presence of earlier ignored adsorbed phase transport and non-linear sorption kinetics investigated. The heterogenous porosity field leads to a significant improvement in adsorbed-phase transport when non-linear sorption kinetics is considered in very low permeability porous media. We theoretically observe new transport effects in the presence of adsorbed-phase. Furthermore, it is shown that the conventional Langmuir isotherms are not representing the sorption behavior of the gas correctly in the heterogeneous formations where the nonlinearity in sorption kinetics acts as trapping mechanism for adsorbed phase and surface diffusion decreases the time needed to reach the saturation pressure. Finally we investigated the effects of heterogeneity introduced by random permeability field on advection- reaction problem, where one step nonlinear reaction takes place. The results show rich nonlinear interplay between the existing mechanisms. The effect of heterogeneity on steady planar reaction wave is also investigated. It is revealed that the planar reaction wave is intrinsically unstable. The later is anticipated to expand due to development of non--uniform velocity field along the reaction wave.