UNSATURATED FLOW IN FRACTURES WITH ANISOTROPIC VARIABLE APERTURES

 

C. MASCIOPINTO*, M. BENEDINI

 

CNR- National Research Council - Water Research Institute,

Via F. De Blasio 5, 70123 - Bari, Italy

Tel.: 0039 80 5313354 - Fax: 0039 80 5313365 E-mail: irsacm08@area.ba.cnr.it

 

 

ABSTRACT

Vertical movement of water in a fracture plane under unsaturated condition has been investigated by means of a computational code. The relevant conceptual model is based on a 2D approximation of the generalized Darcy law. Portion of the fracture is occupied by wetting and non-wetting phases, respectively, according to the capillary pressure and global accessibility criteria. The finite difference method was used in the computational code to solve the nonlinear set of equations. An internal iteration method was used at every time step to evaluate the nodal value of the pressure head, in agreement with the mass-balance equation and the characteristic functional relationships of the coefficients. The variable apertures were calculated by means a spatial correlation matrix which can be considered also anisotropic. The code results were tested with results known in the literatures for isotropic correlation of fracture apertures. A fingering effect in the pressure head distribution within the fracture plane was noted. The application of the code to a fracture with anisotropic aperture variability showed a reduction of the fingering due to larger values of apertures selected, in the x direction.

 

Keywords: Fractured media; Unsaturated flow; Numerical model.

 

INTRODUCTION

Unsaturated flow is very important to understand the vertical movement of pollutants. An example is the estimation of the effects on deeper aquifers of potential releases of leachate by no-controlled waste landfills. In the Apulia Region (South Italy) this aspect received particular attention because in the carbonatic platform is inserted a fractured aquifer interested by heavy withdrawals for drinking uses. Obviously, the behavior of leachate infiltration derived by landfills depends upon the medium characteristics, which govern fluid flow, fluid interactions with the rock matrix and fracture aperture variability. Moreover, unsaturated hydraulic conductivity is highly dependent upon the moisture content in the system. Capillary theory of moisture retention suggests that since fractures are region at relatively large voids, they will drain under smaller suction head than the surrounding rock matrix. Therefore in steady unsaturated flow through fractured media the moisture will preferentially move through the matrix pore system. Nevertheless the existence of a low permeability mineralized layer or coating at the interface fracture-rock matrix may substantially reduce matrix imbibition and the water flow can be considered fracture dominated. Fractures in the Apulia aquifer are often open and partly filled with calcspar or terra rossa, which is a coating material that include iron (15%) and aluminum oxides (30%), silica (33%) and other minor compounds (Dell'Anna et al., 1985).

Pruess and Tsang (1990) developed a conceptual and numerical model as a two-dimensional heterogeneous porous medium to study unsaturated flow in fractures. Their illustrative examples indicated that relative permeability depend sensitively on the nature and range of spatial correlation between apertures. Studies of individual fractures conducted in the field (Abelin et al., 1994; Shapiro and Nicholas, 1989) and with numerical models (Moreno et al. 1988; Masciopinto et al., 1999), have shown that within a fracture the aperture variability can significantly affect the movement of water and solutes by forming preferential channels. Kwicklis and Healy (1993) used the Pruess and Tsang (1990) approach to study unsaturated flow in a discrete horizontal fracture network with impermeable rock matrix. Abdel-Salam and Chrysikopoulos (1996) demonstrated in their simulations that, for the case of no-moisture exchange with the rock matrix, the moisture follows preferential pathways within fracture and exhibits pronounced fingering effects. Following this latter approach, in this study a comprehensive mathematical model describing moisture flow within a fracture with anisotropic variable aperture is presented. Two stochastic fields based upon exponential variogram models generated the apertures. The finite difference method was used to evaluate the pressure head and potential field at each time. Moreover an impermeable rock matrix has been considered in this first step of the work.

 

MODEL DEVELOPMENT

The mathematical model presented here was proposed to study vertical movement of moisture in a sub-vertical fracture. This study has been carried out under the following hypotheses:

-          validity of the generalized Darcy law;

-          fracture with incompressible rock matrix;

-          variable apertures and hydraulic fracture conductivity based upon incompressible fluid.

 

Under 2D approximation with vertical axis upward and horizontal one in the direction of groundwater flow, the components of water velocities are

 

 

where b is the nodal fracture aperture. The mass-balance equation is given by

 

 

where the suction head is

 

(3)

 

Q takes into account sink or source terms able to describe cross-flow due to drains and pumping or recharge wells. The specific moisture capacity Γ=¶q_w/¶y in the equation (2) refers to the fracture water content.

Writing the equation (2) on the computational cell we have

 

(4)

 

(5)

where n and n-1 stand for present and previously time step, respectively, while the relaxation parameter w_t should be 0 £ w_t £ 1. The superscript k and k1 denote that every time step n, the coefficients K of equation (4), as a function of nodal y values, are evaluated by using

which allows the y and K=K(y) evaluation, at each time step, according to the continuity equation (4) to be developed. Obviously also this relaxation parameter should be selected as 0 £ w_it £ 1 and a tolerance,

 

(6)

 

should be fixed. Particular attention is required to fix the time step size (Masciopinto et al. 1994). Every time steps the van Genuchten retention curve was used to update the water content and Γ=¶q_w/¶y values.

 

 

Fig. 1. Suction head at wetting front as a function of fracture aperture.

 

 

Fig. 2. Temporal distribution of pressure head averaged across the fracture width, at 2 and 3 m downstream from the top of column: a comparison between Abdel-Salam and Chrysikopoulos (1996) (****, ++++) and this work (solid lines).

 

Applying this equation in each grid nodes of a discretized domain a system of equation can be obtained. The system solution can be found after imposing suitable boundary and initial conditions. In particular the boundary conditions can be imposed in form of water content or suction head. The latter is a function of the surface tension between air and water s and the fracture aperture b

 

(7)

 

A simplified description of vertical water movement in the fracture is given by Green-Ampt equation (Bear, 1972) with z'=-z,

 

(8)

 

valid for a one-dimensional water movement and assuming the suction head at wetting front to be constant. This equation suggests that for a surface tension equal to 0.0074 kg/m for water a 20 °C, h_f can be expressed by equation (5) as a function of fracture aperture. Figure 1 shows that in fracture with aperture larger than 50 mm the suction head cannot affect water movement, which can be considered dominated by gravitational forces.

 

IDEALIZED FRACTURE

The hypothetical fracture used here is 8 m long and 4 m wide. The fracture aperture is assumed to be a stationary stochastic variable supposed correlated in anisotropic fashion, over some distance in fracture plane. To generate aperture distributions two exponential covariance models were used. A known aperture generation code (Wiliams and El-Kadi, 1986) was modified for anisotropic covariance matrix. Input parameters for aperture generation were the average aperture, the standard deviation, the sill, the range and the nugget, respectively in x and z direction.

The fracture plane was discretized by using Dx=Dz=0.5 (153 nodes) while the parameters of characteristics functional relationships for the equation coefficients (i.e., van Genuchten equation parameters) were derived by the work of Abdel-Salam and Chrysikopoulos (1996).

 

COMPUTATIONAL CODE VALIDATION

To test the model results the computational code was applied to a fracture plane described by Abdel-Salam and Chrysikopoulos (1996). In order to investigate the effect of variable aperture on unsaturated flow in fracture these authors examined a first column 8 m x 4 m, where there is no moisture exchange with the rock matrix. In Figure 2 are compared the temporal distribution of mean pressure head given by model and results derived by Abdel-Salam and Chrysikopoulos (1996). Obviously the same parameters, boundary and initial conditions were imposed. Although the trends of temporal distribution of pressure heads can be considered very similar, there are some differences on the local

 

a) b)

Fig. 3. Spatial distribution of the pressure head in the vertical fracture after 1 day (isotropic aperture correlation): a) contour map - b) two-dimensional snapshot.

 

a) b)

Fig. 4: Contour map and spatial distribution of pressure head in the fracture after 4 days for isotropic (a) and anisotropic (b) spatial correlation of the apertures.

 

values of the pressure. These differences of pressure head are due to the larger grid size of the domain applied in the current simulation (153 nodes instead of 328).

Also the spatial distribution of the pressure heads in the fracture plane for unsaturated flow after 1-day (Fig. 3) is quite similar to that obtained by the above mentioned authors, when no moisture exchange with the rock matrix is considered. The code results show that at low pressure heads, areas with large apertures do not hold as much moisture as areas with small apertures. That is areas with small aperture dominate the flow causing localized buildup of potential suction heads. Inside of column an area with small aperture completely surrounded by areas with large aperture might be noted. These areas may remain at a high suction head for a long time.

 

ANISOTROPIC SIMULATION

To test its capability the code was applied to unsaturated flow in a fracture with aperture generated with an anisotropic matrix of spatial correlation. To produce significant changes in the stochastic generation, the apertures were calculated by means of different parameters (table 1) in x and z direction.