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 nonvolumeaverage 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 Signorinitype 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.
In the past decades, the understanding of water seepage flow in unsaturated, fractured rocks has been investigated by many researchers. Wang and Narasimhan [
In these conventional models, the macroscale continuum and dualcontinuum concepts [
Furthermore, it has been revealed by mounting evidence that the presence of fractures in unsaturated fractured rock formations can enhance the permeability of rock masses and that preferential and fast flow is associated with fractures [
In this study, the PVI method is extended by developing a systematic approach on modeling variably saturated water flow in the heterogeneous fracture networks and eliminating the singularity on the Signorinitype conditions, where the equivalent PVI formulations of the variably saturated water flow seepage problems in the fracture networks are proposed and a finite element procedure is also set up. To evaluate the unsaturated seepage flow in the heterogeneous fracture network, the following assumptions are made: the rock matrix is treated as impermeable; the fracture is nondeformable and the fracture system is under isothermal conditions; the fluid is essentially incompressible and it is assumed that the fluid flow obeys Darcy’s law. In contrast to the volumeaveraged model, our seepage analysis is established on the basis of fracture segments and interconnections between fractures. The organization of the paper is as follows. In Section
The variably saturated groundwater flow in a discrete fracture network, shown in Figure
Variably saturated flow movement in the fracture network.
Local coordinates of the fracture
These two apertures are substantially different in general cases, except for smooth fractures. The cubic law, as the simplest approach to describe the fracturedominated flow, was deduced on the basis of the fact that fracture was bounded by smooth and parallel plates. In reality, all fractures have rough walls and variable apertures, and fluid flow moving through a real fracture will take a tortuous path. Obviously, the deviation between real hydraulic properties and cubic law is expected [
As shown in Figure
Mass conservation at fracture intersection
Constitutive relationships between the pressure head
One approach to establish the related constitutive relationships for either single fractures [
Nevertheless, the physical meaning of the relevant parameters in these relationships cannot always be easily understood for fractures, because the micro geometry in the fracture is twodimensional while that of pores in porous media is threedimensional [
Initial and boundary conditions are also needed to solve (
The seepage face
On the rain infiltration surfaces
In nature, the conditions shown in (
In addition, the saturatedunsaturated water flow in fracture networks should satisfy the following conditions:
The initial condition:
The pressure head boundary condition
The flux boundary condition
where
Richards’ equation for describing variably saturated flow in the fracture networks is nonlinear and the solution is further complicated due to the singularity of seepage face and rainfall boundary conditions, as shown in (
By employing integration by parts, (
Inserting (
Suppose that the function of
Hence,
Similarly,
Thus, (
Supposing
Equation (
Taking
Herein, (
As the term
From (
Particularly, inserting
From (
The equivalence relation between the PDE and PVI formulations is presented in the above proof. It is important to note that the PVI formulation as shown in (
When the finite element method and a backward time difference scheme are applied to (
Equation (
Four typical examples are presented to investigate the performance of the proposed algorithm. In the first and second examples, the van Genuchten [
Constitutive relationships for four examples in Section
The first example validates the proposed method with respect to the semianalytical solution derived by Warrick et al. [
Simulated and analytical water table positions at various times.
According to the transient analysis through the onedimensional infiltration fracture, comparisons of the numerical predictions and Warrick’s solutions at three elapsed times are shown in Figure
Example
Verification on saturated steady seepage problems.
Homogenous dam
Nonhomogenous dam
Figure
The third example involves groundwater infiltrations in two typical 1.0 m × 1.0 m squares with simple twodimensional fracture networks consisting of vertical and horizontal fractures, as shown in Figure
Numerical generations of two fracture systems: (a) fracture sets with mean trace length of 2 m and spacing of 0.005 m and (b) fracture sets with mean trace length of 0.2 m and spacing of 0.01 m.
Case
Case
Different methods exist to computationally generate a fracture network [
To investigate how the extent and density of fractures affect the variably saturated flow infiltration processes in the rock square, two types of fracture systems are mimicked and their related statistical parameters are given in Table
Parameters of fractures and probability model.
Dip (°)  Aperture (10^{−4} m)  Mean trace length (m)  Mean spacing (m)  

Mean values  Variance  Probability model  
Case 

180  0.1  0.0  Normal  2.0  0.005 
90  0.5  0.5  Normal  2.0  0.005 
Case 

180  0.1  0.0  Normal  0.2  0.010 
90  0.5  0.5  Normal  0.2  0.010 
Figure
Variation of water seepage in the two fracture systems at times 2 s and 10 s.
Case
Case
The seepage patterns are distinguishable in appearance but share some common features. Flow generally proceeds in a manner of narrow fingers in the early stage. Several flow paths can develop from localized infiltration. As indicated by Pruess [
Comparing cases
The connectivity of the fracture networks is an essential feature controlling the flow movement in these impermeable geological fractured media. In order to quantify the connectivity in these heterogeneous fracture networks with fracture geometrical properties, the concept of geological entropy [
An example to characterize such fracture network system is shown in Figure
Parameters of the geological entropy for connected and unconnected systems.
Subdomain  1  2  3  4  5  6  7  8  9 

Connected system  


Fracture  0  0.03  0  0.03  0.06  0.03  0  0.03  0 
Rock mass  1  0.97  1  0.97  0.94  0.97  1  0.97  1 

0  0.135  0  0.135  0.227  0.135  0  0.135  0 

0.085  

0.098  

0.867  
Unconnected system  


Fracture  0.03  0.015  0.03  0.015  0  0.015  0.03  0.015  0.03 
Rock mass  0.97  0.985  0.97  0.985  1  0.985  0.97  0.985  0.97 

0.135  0.078  0.135  0.078  0  0.078  0.135  0.078  0.135 

0.095  

0.098  

0.969 
Local and relative entropy calculation for connected and unconnected systems.
Connected (
Unconnected (
Herein, the relative entropy
Variation of
The fourth example is to investigate the effect of rainfall infiltration on a fractured rock slope with two sets of random distribution fractures, where the rapid generation of higher hydraulic pressure within a short possible time may become one of the critical factors on slope stability. Four types of discrete fracture networks are employed and their geometrical parameters are given in Table
Parameters of fractures and probability model.
Dip (°)  Aperture (10^{−4} m)  Mean trace length (m)  Mean spacing (m)  

Mean values  Variance  Probability model  
Case 
0.1  Normal  0.2  
15  0.3  2.0  
135  0.1  0.8  
Case 

30  0.3  2.0  
135  0.1  0.8  
Case 

45  0.3  2.0  
135  0.1  0.8  
Case 

60  0.3  2.0  
135  0.1  0.8 
Numerical realizations of two fracture systems in a fractured rock slope under rainfall condition: (a) dip of 15 degrees; (b) dip of 30 degrees; (c) dip of 45 degrees; (d) dip of 60 degrees.
Case
Case
Case
Case
The left and bottom boundaries are impermeable and the right boundary under the initial water table is always treated as a water pressure head boundary. The infiltration boundaries are imposed on the other boundaries along the slope. Due to the high gradient of the slope, it is reasonable to assume that no runoff would be produced on the infiltration boundary, which creates a zero value of
The normalized flow rate distributions inside the rock slope after 10 and 20 hours of water infiltration are shown in Figure
The normalized flow rate distributions in the slope at times (a) 10 hr and (b) 20 hr.
Case
Case
Case
Case
Evolution of the total net flow into the unsaturated domain.
As the permeability of matrix is extremely low and can be neglected, the fractures would have a great impact on the partially saturated water seepage flow. In order to understand the water seepage behavior in such heterogeneous fracture systems, a numerical approach based on the PVI formulations of variably saturated water flow in a discrete fracture network has been proposed.
In consideration of the essential distinction in geometrical characteristic between the twodimensional fracture and threedimensional porous media, the constitutive relationships including water pressure head and relative permeability as functions of saturation, compared with the conventional van Genuchten’s model, are employed. While modeling of unsaturated flow in complex fracture systems is difficult and uncertain, the seepage surface and infiltration boundaries are unified as a complementary condition of Signorinitype formulation. Through the equivalence proof between the PDE and PVI formulations of the variably saturated water flow seepage problems in the fracture networks, the difficulty in solving such problems with boundary nonlinearity is reduced. In addition, the corresponding numerical finite element method is presented in detail.
During the analysis of saturatedunsaturated water seepage simulation, the validity of the proposed procedure is demonstrated by the match between the predicted results of the proposed PVI method and analytical/numerical solutions. The model calculations in complicated fracture systems also suggest that water seepage may proceed by means of fast preferential flow paths along partial fractures due to the inhomogeneity of spatial distribution, which has a great disparity with the volumeaveraged mode. Simultaneously, the variably saturated seepage behavior is significantly sensitive to the geometrical characteristics of the fracture system. The fast flow pathways are not only dominated by the spatial distribution of fracture aperture, but also strongly dependent on the connectivity, density, and orientation. Actually, the water flow through complicated fracture networks is also affected by the fracture density and connectivity. In particular, the relative entropy
Pressure head, L
Elevation heads, L
Gravity acceleration, LT^{−2}
Darcy flow velocity, LT^{−1}
Flux rate, L^{2}T^{−1}
Effective saturation
Residual saturation
Satiated saturation
Shape function
Iteration step
Error tolerance
Time step
Time increment, T
Hydraulic aperture of fracture segment
Relative permeability
Water saturation
Rainfall intensity, LT^{−1}
Ponding depth, L
Critical flow rate, L^{2}T^{−1}
Prescribed water head, L
Prescribed flow rate, L^{2}T^{−1}
Saturated hydraulic conductivity, LT^{−1}
Dynamic viscosity of water, L^{2}T
Equivalent hydraulic aperture, L
Critical water pressure head, L
Specific moisture capacity, L^{−1}.
All data used for the plots were generated using the authors’ own computer codes. The data are available upon request.
The authors declare that they have no conflicts of interest regarding the publication of this paper.
The financial supports from the National Natural Science Foundation of China (nos. 51709207, 41762020, 51679173, and 51604195) and Natural Science Foundation of Hubei Province (no. 2015CFA142) are gratefully acknowledged.