Fluid Flow in the Subsurface

Darcy’s Law was discovered by Henry Darcy (1803–1858) based on experimental observations of steady-state water flow through sand columns. It states that water flux in saturated porous media is linearly proportional to hydraulic gradient. For low-permeability porous media, however, Darcy’s law is not adequate because of the strong fluid–solid interaction that results in non-linear flux-gradient relationships. This chapter presents a new phenomenological relationship between water flux and hydraulic gradient, or a generalized Darcy’s law. The traditional form of Darcy’s law and two other generalizations for low-permeability media, proposed by other researchers, are shown to be special cases of the generalization. The consistency between the generalization and experimental observations from different sources is demonstrated. The generalized Darcy’s law and its variations are used to attack several key technical issues facing the geoscience community, including the relative importance of diffusion in the excavation damaged zone for a shale repository of high-level nuclear waste, the accurate measurement of relative permeability for multiphase flow in a low-permeability porous medium, non-Darcian flow behavior during imbibition of fracturing fluids into a shale gas reservoir, and formation of the pressure seal in shale formations.

The Darcy-Buckingham law extends Darcy’s law from single phase flow in the subsurface to multiphase flow and was discovered by Edgar Buckingham (1867–1940). It is valid under the local-equilibrium condition that, however, does not always hold especially when fingering flow occurs. This chapter is devoted to generalizing the Darcy-Buckingham law by relaxing the local-equilibrium condition. The new development is based on an optimality principle that water flow in unsaturated media self-organizes in such a way that the resistance to water flow is minimized. Unlike the traditional form of the Darcy-Buckingham law, the relative permeability in the generalization is also a function of water flux, which is a direct result of the non-equilibrium flow behavior. The generalized Darcy-Buckingham law has been shown to be consistent with laboratory observations and field data for gravitational fingering flow in unsaturated soils. It is also the theoretical foundation for the active fracture model, the key constitutive relationship for modeling flow and transport in the unsaturated zone of Yucca Mountain, Nevada, that is the national geological disposal site for high-level nuclear waste in USA.

Hooke’s law was discovered by Robert Hooke (1635–1703) and states that during elastic deformation of a solid body the strain is linearly proportional to the stress. While Hooke’s law was initially developed based on experimental observations for relatively homogeneous materials, natural rocks generally contain small-scale deformation heterogeneities (such as the existence of micro-cracks). To address this issue, this chapter introduces the two-part Hooke model that was developed by conceptually dividing a natural rock into hard and soft parts that are subject to different degrees of mechanical deformation under a given stress. Remarkable consistency between the model and observations from different sources, for both rock matrix and fractures, has been achieved. The usefulness of the model in dealing with several engineering problems is also demonstrated, including coal-permeability changes caused by swelling under different stress conditions, coupled hydro-mechanical processes associated with CO₂ geological sequestration in a fractured formation, and evolution of the excavation damage zone related to nuclear waste disposal in a shale formation.

This chapter summarizes key results from previous chapters. While these chapters document the important progresses in generalizing several physical laws related to subsurface fluid flow and mechanical deformation, they may or may not provide the final forms of the generalizations. Thus, this book should be viewed as a starting point for revisiting these physical laws and for stimulating further research activities in the related areas.