Abstract
Izbash's law, which has a simpler form and exhibits narrower variations of coefficients than Forchheimer's law, has been successfully used to describe nonlinear flow in rough fractures. However, the meaning of its coefficients and its relationship with Forchheimer's law are poorly understood. Based on mathematical derivations, we identified the relationship between the cubic law, Forchheimer's law, and Izbash's law. An improved, dimensionally consistent Izbash's law which can be treated as the modified cubic law was formulated by incorporating a correction coefficient mj. The value of this coefficient depends on fluid properties, fracture geometries, and flow rate, and can be used to estimate the nonlinear flow state at given flow rates, in a similar fashion to Forchheimer number (Fq). A critical mj value (mcri) of 1.1 was determined that indicates whether the nonlinear effect can be ignored. Because mj is flow-rate-dependent, it can serve to describe the local flow behaviors, whereas the Izbash's exponent m is generally treated as a constant. At the maximum flow rate for a given pressure gradient versus flow rate curve, the value of m is approximately equal to that of m_I indicating that m represents the maximum nonlinear deviation. To validate the improved Izbash's law and the relevant findings, results from four experimental studies were utilized, and numerical simulations were performed. The result of this validation suggests the rationality of the analysis. The multiscale roughness effects on the coefficients were also investigated by selecting two different sampling intervals to construct the fracture surface. It was demonstrated that secondary roughness can enhance the development of nonlinear flow and increase the value of m, while its effects are strongly dependent on internal geometrical properties.
NSTL