Mathematical Modeling of Multi-Phase Flow in Porous Media for Environmental and Petroleum Engineering

Abstract

The relevance of multi-phase flow modeling in porous media has increased in importance in environmental engineering as well as petroleum engineering as it has a direct influence on the effectiveness of hydrocarbon recovery, ground water remediation, the prediction of contaminant transport, and the large-scale, CO 2, sequestration project. The flow of immiscible fluids e.g. water, oil and gas in a single phase is multi-phase flow and intrinsically complicated due to the combination of pore-scale heterogeneity, nonlinear capillary forces, wettability effects, and dynamic phase interactions. These complexities are important to be captured through mathematical modeling, and to develop predictive tools that can be employed in the management of the utilization of resources in a sustainable way. In this paper, a detailed mathematical model is presented such that it incorporates the law of Darcy with the law of multi phase mass conservation and provides a detailed constitutive equation of relative permeability and capillary pressure and other non-Darcy flow corrections with inertial effects in the high velocity regime. The model states a fractional-order differentiation formulations to extend the limitations of the classical approaches to model anomalous diffusion and memory-dependent transport typical in the case of heterogeneous reservoirs and aquifers. Adaptive meshing and finite element discretization of the coupled nonlinear equations are solved numerically, to solve nonlinear equations with complex boundary conditions that are made stable and accurate. Studies with numerical models demonstrate the time and space behavior of saturation fields, displacement fronts and breakthrough times in heterogeneous fields, and indicate the scale of the influences of pore-scale variability, capillary hysteresis, and changes in wettability on the system dynamics on a large scale. Findings reveal that long-tail breakthrough and front lagging development is more effectively modeled using fractional-order formulations in contrast to classical formulations. The results motivate the incorporation of heterogeneity and non-local effects in predictive simulations and provide a practical framework to support optimization of enhanced oil recovery plans, enhanced CO 2 storage security, and the reduction of transport of contaminants in the subsurface environment. Comprehensively, this literature adds a sound, scaleable, and physically coherent framework to the modelling of multi-phase flow, including the gap between theoretical models and practical applications to the environment and energy engineering.