Knowledge of porosity and saturation‐dependent thermal conductivities is necessary to investigate heat and water transfer in natural porous media such as rocks and soils. Thermal conductivity in a porous medium is affected by the complicated relationship between the topology and geometry of the pore space and the solid matrix. However, as water content increases from completely dry to fully saturated, the effect of the liquid phase on thermal conductivity may increase substantially. Although various methods have been proposed to model the porosity and saturation dependence of thermal conductivity, most are empirical or quasiphysical. In this study, we present a theoretical upscaling framework from percolation theory and the effective‐medium approximation, which is called percolation‐based effective‐medium approximation (P‐EMA). The proposed model predicts the thermal conductivity in porous media from endmember properties (e.g., air, solid matrix, and saturating fluid thermal conductivities), a scaling exponent, and a percolation threshold. In order to evaluate our porosity and saturation‐dependent models, we compare our theory with 193 porosity‐dependent thermal conductivity measurements and 25 saturation‐dependent thermal conductivity data sets and find excellent match. We also find values for the scaling exponent different than the universal value of 2, in insulator‐conductor systems, and also different from 0.76, the exponent in conductor‐superconductor mixtures, in three dimensions. These results indicate that the thermal conductivity under fully and partially saturated conditions conforms to nonuniversal behavior. This means the value of the scaling exponent changes from medium to medium and depends not only on structural and geometrical properties of the medium but also characteristics (e.g., wetting or nonwetting) of the saturating fluid.

The porosity dependence of the formation factor for geologic media is examined from the perspective of universal scaling laws from percolation and effective medium theories. Over much of the range of observed porosity, the expected percolation scaling is observed, but the values of the numerical prefactor do not conform to the simple predictions from percolation theory. Combining effective medium and percolation theories produces a numerical prefactor whose value depends on both the threshold porosity and the porosity above which the formation factor crosses from percolation to effective medium scaling. This change allows extraction of a numerical value of the prefactor, which is reasonably close to experimental values. Subsequent evaluation of the porosity dependence of the formation factor shows that difficulties in prior comparisons of theory and experiment are largely removed when percolation scaling is allowed to transition to effective medium scaling far above the percolation threshold.

Quantifying fluid flow through porous media hinges on the description of permeability, a property of considerable importance in many fields ranging from oil and gas exploration to hydrology. A common building block for modeling porous media permeability is consideration of fluid flow through tubes with circular cross section described by Poiseuille's law in which flow discharge is proportional to the fourth power of the tube's radius. In most natural porous media, pores are neither cylindrical nor smooth; they often have an irregular cross section and rough surfaces. This study presents a theoretical scaling of Poiseuille's approximation for flow in pores with irregular rough cross section quantified by a surface fractal dimension D_s2. The flow rate is a function of the average pore radius to the power 2(3‐D_s2) instead of 4 in the original Poiseuille's law. Values of D_s2 range from 1 to 2, hence, the power in the modified Poiseuille's approximation varies between 4 and 2, indicating that flow rate decreases as pore surface roughness (and surface fractal dimension D_s2) increases. We also proposed pore length‐radius relations for isotropic and anisotropic fractal porous media. The new theoretical derivations are compared with standard approximations and with experimental values of relative permeability. The new approach results in substantially improved prediction of relative permeability of natural porous media relative to the original Poiseuille equation.

Upscaling pore pressure dependence of shale gas permeability is of great importance and interest in the investigation of gas production in unconventional reservoirs. In this study, we apply the Effective Medium Approximation, an upscaling technique from statistical physics, and modify the Doyen model for unconventional rocks. We develop an upscaling model to estimate the pore pressure‐dependent gas permeability from pore throat size distribution, pore connectivity, tortuosity, porosity, and gas characteristics. We compare our adapted model with six data sets: three experiments, one pore‐network model, and two lattice‐Boltzmann simulations. Results showed that the proposed model estimated the gas permeability within a factor of 3 of the measurements/simulations in all data sets except the Eagle Ford experiment for which we discuss plausible sources of discrepancies.