Stochastic continuum representation of fractured rock permeability as an alternative to the REV and fracture network concepts

Laboratory and field measurements of hydraulic conductivity (or permeability) in porous media represent averages over many pores. Such measurements are insensitive to small variations in the volume of the porous medium affected by a hydraulic conductivity test. This justifies treating the hydraulic conductivity of porous media as a continuous function of space which varies smoothly enough to be analyzed by standard methods of differential calculus. Laboratory and field measurements of hydraulic conductivity in fractured rocks represent volumes that are often intersected by only a few fractures. Therefore, these measurements tend to be erratic and sensitive to the volume of rock sampled by the test. For this erratic behavior and sensitivity to be acceptably small, the test would have to be conducted on a relatively large scale at which the sample corresponds to a Representative Elementary Volume or REV. However, there is generally no guarantee that an REV can be defined for a given rock mass. When an REV can be defined, it is often so large as to render the measurement of its hydraulic conductivity impractical. To study questions related to the existence and properties of REV’s and/or to eliminate the need for their use, some investigators have relied on discrete models of fracture networks. These models require detailed deterministic and/or statistical information about the geometry of fractures and the spatial distribution of their apertures which is difficult to obtain. In addition, there is growing laboratory and field evidence that the manner in which such models translate data about fracture geometry into hydraulic and transport properties of the rock is open to serious questions. This paper describes an alternative to both the classical continuum concept based on an REV and the discrete fracture network approach. The proposed alternative places less emphasis on fracture geometric data than on the results of hydraulic tests conducted on scales at which testing is practical with available technology. Such tests usually sample rock volumes smaller than an REV and must therefore be analyzed statistically. However, they are generally easier to conduct than measuring all the geometric parameters that are required for the construction of reliable discrete fracture network models. There is evidence that hydraulic test data collected on appropriate sub-REV scales correlate well with logs obtained by selected borehole geophysical devices and with subsurface geotomographic images yet their correlation with geometric parameters such as fracture density are tenuous at best. The paper demonstrates that such hydraulic test data are amenable to quantitative analysis by treating them as the realization of a stochastic process defined over a continuum. The nature of this stochastic process on scales smaller and/or larger than the scale of measurement can be studied by means of deconvolution and/or spatial averaging techniques. In this way, data obtained on different scales can be analyzed jointly within a unified conceptual framework so as to reinforce each other. The concept and mathematics are illustrated with data obtained from fractured granites near Oracle, Arizona. The paper concludes by describing how the continuum concept can be used to analyze the spread of dissolved constituents in fractured rocks.