EXAMPLES

Table 1 presents data for Barre granite and Bedford limestone taken from laboratory measurements by Coyner [1984]. Coyner's experiments included a series of tests on several types of laboratory scale rock samples at different confining pressures. The values quoted for K and K_s are those for a moderate confining pressure of 10 MPa (values at lower confining pressures were also measured but we avoid using these values because the rocks generally exhibit nonlinear behavior in that region of the parameter space), while the values quoted for K⁽¹⁾ and K_s⁽¹⁾ are at 25 MPa, which is close to the value beyond which the constants cease depending on pressure -- and therefore for which we assume all the cracks were closed. Thus, based on the idea that the pressure behavior is associated with two kinds of porosity in the laboratory samples -- a crack porosity, which is being closed between 10 and 25 MPa, and a residual matrix porosity above 25 MPa, we assume the available data are K, K_s, K⁽¹⁾, K_s⁽¹⁾, K_f, $\phi^{(1)}$, and v⁽²⁾. We find that these data are sufficient to compute all the coefficients. In Table 2, we find in both types of rock that the coefficient a₂₃ is positive and small -- about an order of magnitude smaller than the other matrix elements. Another unusual feature of the results computed using these laboratory data is the occurrence of values larger than unity for B and B[u⁽¹⁾] in Barre granite; similar results were observed by Berryman and Wang [1995] for Chelmsford granite, and in subsequent work for Westerly granite. Also note that $\alpha^{(2)}$ for both rocks is close to unity for these calculations. In this example, seven measurements (together with Poisson's ratio) are sufficient to determine completely the mechanical behavior of the double-porosity model. Having a direct measurement of K eliminates the necessity of assuming a₂₃ = 0, which we have found is sometimes necessary when dealing with field data [Berryman and Wang, 1995].

TABLE 2. Double porosity parameters computed from material properties in Table 1.

Barre Bedford Parameter