The Parabolic Variational Inequalities for Variably Saturated Water Flow in Heterogeneous Fracture Networks

Abstract

Fractures are ubiquitous in geological formations and have a substantial influence on water seepage flow in unsaturated fractured rocks. While the matrix permeability is small enough to be ignored during the partially saturated flow process, water seepage in heterogeneous fracture systems may occur in a non-volume-average manner as distinguished from a macroscale continuum model. This paper presents a systematic numerical method which aims to provide a better understanding of the effect of fracture distribution on the water seepage behavior in such media. Based on the partial differential equation (PDE) formulations with a Signorini-type complementary condition on the variably saturated water flow in heterogeneous fracture networks, the equivalent parabolic variational inequality (PVI) formulations are proposed and the related numerical algorithm in the context of the finite element scheme is established. With the application to the continuum porous media, the results of the numerical simulation for onedimensional infiltration fracture are compared to the analytical solutions and good agreements are obtained. From the application to intricate fracture systems, it is found that water seepage flow can move rapidly along preferential pathways in a nonuniform fashion and the variably saturated seepage behavior is intimately related to the geometrical characteristics orientation of fractures.

A Fractal Model for Predicting the Relative Permeability of Rough-Walled Fractures

“…For D T /λ � 0 when λ ⟶ ∞ in which the unsaturated flow behavior in the rough-walled fracture is approximated to that in porous media with uniform pore size, equation ( 20) is reduced to the X-curve equation (1). For D T /λ � 1, equation ( 20) is reduced to the Brooks and Corey model equation (8). For D T /λ � 1.5, equation ( 20) is reduced to the Corey model equation (7).…”

“…Multiphase flow through fractured media is an important process for a number of practical applications, such as seepage control for rock slope [1,2], petroleum reservoirs [3], nuclear waste disposal [4], and geologic storage of carbon dioxide [5]. In order to understand the influence of multiphase flow behavior in fractured media on implementation with these applications, numerical simulations have been applied in fractured media using a continuum method [6,7] or a discrete fracture network model [8,9]. In contrast to the macroscale continuum model, the flow behavior in the fractured rock is mainly governed by the presence of fracture, which can highly improve the permeability of rock masses [10][11][12].…”

Modeling of Transient Flow in Unsaturated Geomaterials for Rainfall-Induced Landslides Using a Novel Spacetime Collocation Method

“…This is also known as the variably saturated flow equation to describe the groundwater flow in saturated and unsaturated zones simultaneously [17,18]. Equation (2) can be rewritten as the three-dimensional form…”

“…However, the influence of matric suction of soil and the soil-water characteristic curve (SWCC) cannot be comprehensively considered during the analysis. The theory of geofluids when rainfall infiltrates vadose zones can be described using either the variably saturated flow equation or the generalized Richards equation [17,18]. Governed by nonlinear physical relationships, the Richards equation is a highly nonlinear equation.…”

Experimental Investigation on Nonlinear Flow Anisotropy Behavior in Fracture Media

“…In the study of rock mass hydrology, discontinuities are usually generalized into two smooth parallel plates, and the famous cubic law is hence obtained through theory and experiment. A variety of correction models [6][7][8][9][10][11] has been proposed to account for fracture roughness, contact, or filling.…”