3D Hydraulic Conductivity Tensor Calculator for Jointed Rock Masses

Hydraulic conductivity in jointed rock masses is primarily governed by the geometrical characteristics of discontinuities, including dip and dip direction, spacing, length, aperture, and the degree of interconnectivity between different discontinuity sets. Although analytical methods based on these geometrical properties are relatively simple, they can still provide reasonable approximations of hydraulic conductivity and help illustrate the influence of each discontinuity set on the overall hydraulic behaviour of the rock mass.

Based on the cubic law, the hydraulic aperture of rock discontinuities has the most significant influence on the hydraulic conductivity of jointed rocks (Snow, 1969).

Quantifying the hydraulic conductivity of jointed rocks is vital for many fields of ground engineering. This parameter is difficult to determine and there are several methods to estimate it. Numerous studies have attempted to quantify the hydraulic conductivity of jointed rocks by laboratory and field tests, experimental relationships, analytical methods and numerical simulations. Lugeon (1933) presented a constant head field test that (packer test or Lugeon test) takes place in an isolated section of a borehole. By assuming isotropic and homogenous condition, one Lugeon is equivalent to.

Barton (2002) proposed an empirical relationship between Lugeon value of jointed rock and Q-value. This relationship is based on correlation between Lugeon value, Q-value and seismic velocity which recorded for various dam foundations.

Long (1983) conducted comprehensive research to apply Discrete Fracture Network (DFN) for investigating the hydraulic conductivity of jointed rocks. In DFN method, fracture network is generated based on stochastic procedure using the probabilistic density functions of fracture parameters which are determined by joint mapping (Min, Jing et al. 2004). A considerable amount of literature has been published on applying DFN method to study hydraulic properties of jointed rocks (Rouleau and Gale, 1987; Cacus et al., 1990; Panda and Kulatilake, 1999; Min et.al, 2004; Baghbanan et.al, 2008; Lee et.al, 2011).

Unlike DFN modelling, most of analytical methods consider geometrical properties of rock joints as deterministic parameters. These methods combine geometrical properties (orientation, length, spacing and hydraulic aperture) of rock joints along with Darcy flow conditions and cubic law concept to obtain the hydraulic conductivity of jointed rocks. The assumption on the length of rock joints (Finite or infinite joint length) is the main difference between these methods. Most of the analytical methods are quite easy to understand and provide a tool to conduct sensitivity analyse relatively fast.

Snow (1969) introduced the first comprehensive analytical method for hydraulic conductivity of jointed rocks. Rock joints are considered as infinite plane; therefore, they extend all over the considered volume of rock mass (objective volume) and intersect its boundary.

However, rock joints, except bedding planes, have finite persistence. In the other hand, hydraulic contribution of joints on hydraulic conductivity of whole network is not same. For example, joint number ‘’ intercepts the model boundary but not the other joints, and joint number b does not affect the hydraulic conductivity of network. Generally, joints with more intersection with other fractures have higher contribution on the hydraulic conductivity of joint network (Rouleau and Gale 1985). In Snow’s method, rock joints are considered as infinite plane within the objective volume of rock mass. Therefore, it overestimates the permeability of rock mass and its results can be considered as upper bound values for hydraulic conductivity.

In the new analytical method introduced in this research, rock discontinuities were considered to have finite persistence (Zoorabadi et al., 2012a, b). This method provides a formulation to estimate the number of joints from each joint sets which exist inside the objective volume of rock.

Similar to the all analytical methods, the formulations of the new proposed method are based on some assumptions. These assumptions are listed as follows:

1.      The jointed rock can be simulated as a homogenous and anisotropic porous medium.

2.      The water flow through discontinuities is linear and the cubic law is valid.

3.      All discontinuities have a planar shape.

4.      Rock discontinuities are represented as circular discs.

5.      All discontinuities can be grouped as several sets inside the objective volume.

6.      The centres of disc shaped discontinuities are distributed in space randomly and independently.

7.      Locations of discontinuity sets are equally probable.

The global coordinate system is supposed such that the x-axis points toward the east, y-axis points north, and z-axis in upward direction.

Further details of the analytical method are provided in the following document. I have also developed a calculation tool that allows users to input the geometrical characteristics of joints for a given project, with the option to define joint aperture directly or estimate hydraulic aperture for each discontinuity set using in-situ stress data. The tool calculates the 3D hydraulic conductivity tensor, the principal hydraulic conductivity components, and their corresponding trend and plunge. It can be used for either a single discontinuity set or a discontinuity network consisting of up to 10 sets.