Experimental evidence of scale-dependent hydraulic conductivity for fully developed turbulent flow in a single fracture

Summary

This study uses laboratory experiments to investigate scale dependent hydraulic conductivity of fully developed turbulent flow in a single fracture under different fracture surface roughness, fracture apertures, and hydraulic gradients. The hydraulic conductivity for fully developed turbulent flow is defined as K = V²/J, where V and J are absolute values of the flow velocity and the hydraulic gradient, respectively. Three different surface roughness (fine, medium, and coarse), three different fracture apertures (1.0 mm, 2.0 mm, and 2.5 mm), and five different hydraulic gradients have been tested. Experimental evidence shows that K values generally increase with scale in a linear fashion. Surface roughness and fracture apertures appear to have the most significant influence upon the scale-dependency of K, which is less sensitive to hydraulic gradients. In general, a higher hydraulic gradient will lead to a lower K value at a given scale. The scale-dependency of K might be a manifestation of two-dimensional torturous flow within a rough surface fracture.

Introduction

Hydraulic conductivity (K) describes how easy a geologic medium can transmit groundwater, and is one of the fundamental parameters for investigating groundwater movement. Values of K for saturated groundwater flow can be measured from a variety of methods such as grain size analysis, permeameter tests, slug tests, pumping tests, flow meter tests, and geophysical methods (Neuman and Di Federico, 2003). Whether K is scale-dependent or not has been intensively debated among hydrologists for several decades (Clauser, 1992, Neuman, 1994, Rovey, 1994, Rovey and Cherkauer, 1995, Tidwell and Wilson, 1997, Tidwell and Wilson, 1999, Butler and Healey, 1998a, Butler and Healey, 1998b, Schulze-Makuch and Cherkauer, 1998, Schulze-Makuch et al., 1999, Hsieh, 1998, Zlotnik et al., 2000, Nilsson et al., 2001, Hunt, 2003, Neuman and Di Federico, 2003, Illman, 2006).

One suggestion is that K varies with scale of measurement. For instance, Clauser (1992) showed that K values for various granitic rocks increased with scale over several orders of magnitude until a tested volume of rock was reached beyond which K remained constant. Carrera (1993) claimed that scale effects on K might also affect solute transport modeling. Neuman (1994) proposed a universal exponential relationship of horizontal hydraulic conductivity to scale of measurement for all types of geologic media. Rovey, 1994, Rovey and Cherkauer, 1995 analyzed scale behavior of K for various geologic units and concluded that small-scale measurement of K did not average to regional values. Instead, mean K increased with measurement scale up to a critical distance, beyond which a constant regional value prevailed. This finding agreed with the conclusions of Clauser (1992). Illman and Neuman, 2001, Illman and Neuman, 2003 showed a strong permeability scale effect in unsaturated fractured tuff through comparison of results from single- and cross-hole pneumatic injection tests. Recently, Illman (2006) showed a field evidence of directional permeability scale effect from cross-hole pneumatic injection tests conducted in unsaturated fractured tuff at a field site in Arizona, USA. Illman and Tartakovsky (2006) have reported a hydraulic conductivity and specific storage scale effect in fractured granite in Switzerland. There are many other evidences showing the scale-dependency of K under a wide range of geological environments (Schulze-Makuch and Cherkauer, 1998, Schulze-Makuch et al., 1999, Tidwell and Wilson, 1997, Tidwell and Wilson, 1999, Nilsson et al., 2001, Neuman and Di Federico, 2003).

The other suggestion is that K is independent of scale of measurement. An intuitive argument against the scaling of K being dependent on heterogeneity is that the observed behavior is actually caused by different methods of measurement (Butler and Healey, 1998a, Butler and Healey, 1998b). Butler and Healey (1998a) argued that incomplete well development and low-permeability skins surrounding the well screens often resulted in artificially lower K values in slug tests, while those factors had less influence on K values measured from pumping tests. Zlotnik et al. (2000) thoroughly analyzed different aquifer test techniques in several field sites and pointed out that scale effect in K was not readily apparent, a point also supported by Hsieh (1998). Numerical modeling by Sanchez-Vila et al., 1996, Meier et al., 1998 showed that there would be no scale dependence in two-dimensional flow systems when K could be represented as a multilognormal random field.

As far as we know, there is no experimental study investigating scale-dependency of K associated with turbulent groundwater movement in a single fracture. Fractured aquifers are vital groundwater resources in many parts of the world. Turbulent flow is often observed in fractured aquifers because of relatively high Reynolds numbers of flow in fractures (Berkowitz, 2002). The objective of this study is to see if K is scale dependent or not under fully developed turbulent flow conditions in controlled laboratory experiments.

Conventionally, K is a concept used in describing laminar Darcian flow $\overrightarrow{q}=-K\nabla h$, where $\overrightarrow{q}$ and ∇h are the specific discharge and the hydraulic gradient, respectively. Under turbulent flow condition, a nonlinear relation exists between $\overrightarrow{q}$ and ∇h, sometimes via a power function as $|q^{n-1}|\overrightarrow{q}=-K\nabla h$, where ∣·∣ is the absolute value sign and n is the power index (Qian et al., 2005, Wen et al., 2006). If n = 1, flow is Darcian; if 1 < n < 2, flow is partially developed turbulent, and if n = 2, flow is fully developed turbulent. For partially developed turbulent flow, it is possible that both n and K vary with scale. However, for fully developed turbulent flow, n equals 2 and K becomes the only variable describing ease of flow. The K used in above nonlinear function is termed the hydraulic conductivity for fully developed turbulent flow in this study. It is worthwhile to point out that such a definition of K does not apply to other forms of turbulent or non-Darcian flow such as the Forchheimer equation where $\overrightarrow{q}$ and ∇h are connected via a second-order polynomial function (Wu, 2002). For the Forchheimer type of flow, two parameters are needed to describe the ease of water movement. Under the fully developed turbulent flow condition, the linear term of the Forchheimer equation is negligible, and the Forchheimer equation becomes the same as the power-law equation with n = 2.

This study is an extension of Qian et al. (2005) which focused on non-Darcian flow in a single fracture under different conditions of fracture apertures and surface roughness. The emphasis of this work is to experimentally investigate turbulent groundwater flow in a single fracture to test scale dependency of K under different conditions of surface roughness, fracture apertures, and hydraulic gradients.

It is worthwhile to point out that the scale dependence discussed in this article is slightly different from the conventional view of scale dependence reported in previous studies (e.g. Neuman, 1994, Illman, 2006). Traditionally, hydrologists often think of scale dependence as ranging over core, laboratory, and field scales spanning several orders of magnitude, and the scale dependence perhaps arising from increasing correlation scale of heterogeneity. In this paper, the heterogeneity scale constituted by the fine, medium, and coarse surface roughness is of the same order, and the hydraulic conductivity is derived from a presumed empirical relation. Because the heterogeneity scale is of the same order, one probably will not observe a scale dependence feature as strong as that observed in studies involving scales over many orders of magnitude (e.g. Illman and Neuman, 2001, Illman and Neuman, 2003, Illman, 2006).