A New Method for Determining the Non-Linear Effective Pressure

The physical properties (for example, permeability k) of linear elastic materials usually obey a simple effective pressure law (EPL), p_eff= p_c-αp_f (p_eff: effective pressure; p_c: confining pressure; p_f: pore fluid pressure), where α is a constant, often taken to be equal to 1 in the well-known Terzaghi' law, p_eff=p_c-p_f. However, non-linear EPL's, p_eff=p_c-α_s(p_f, p_c)p_f, where the secant coefficient α_s(p_f, p_c) is a function of p_c and p_f, should be expected in non-linear elastic rocks [Robin, 1978] and have been previously reported for permeability in low-permeability sandstones [Li et al, 2009, 2014]. A new method for experimentally determining non-linear EPL's for permeability was tested on low-permeability sandstones from reservoirs in China. The permeability of these low-permeability sandstones was measured while simultaneously cycling p_f and p_c (with 0 ≤ p_f < p_c). Based on the analysis of the experimental data using the Response Surface Method [Box and Draper, 1987], a contour map of permeability was drawn in the plane [p_f, p_c], from which the secant coefficient α_s(p_f, p_c) and the effective pressure p_eff(p_f, p_c) were calculated. We found that α_s(p_f, p_c) varied in the entire theoretically allowed range, φ ≤ α_s(p_f, p_c) ≤ 1, where φ is the porosity. It is most interesting that α_s(p_f, p_c) could be approximately described as a decreasing function α_s(p_c-p_f) of Terzaghi’s differential pressure. Moreover, the non-linear EPL determined using the new method allowed a better estimation of the pressure dependence of permeability, k(p_eff), than classic Terzaghi' law, k(p_c-p_f).