Non-Cubic Law Fracture Flow Phenomena

ABSTRACT: Hydraulic fracture and fracture flow simulations are almost exclusively based on Poiseuille flow (the cubic law). A key assumption embedded in our models is that flow is steady, inertial effects are negligible, the fracture surfaces are parallel and the rate of change of the fracture aperture does not impact flow. However, in many practical applications most and sometimes all of these assumptions are violated. In this paper, we will discuss various phenomena that may be observed in experiments and the newly derived Reduced Dimension Fracture Flow (RDFF) model which are not captured by simulations based on the cubic law. These include (1): fracture fluid flow induced by seismic waves, (2) phase shifts between pressure, aperture, and flux fields, (3) non-Darcian flow in the direction of positive pressure gradients, (4) negative near well bore fluid pressures induced by high velocity radial flow. 1 INTRODUCTION Modern large-scale simulations of fracture flow and hydraulic fracture operations are based almost exclusively on the cubic law (also known as Poiseuille flow). The cubic law is popular because it provides a constitutive relationship between the fluid flux and the fluid pressure, but this model is accompanied by a number of limiting physical assumptions. The cubic law is the analytical solution to flow through rigid parallel plates and makes the assumptions: a) that flow is steady, b) inertial effects are negligible, c) fracture surfaces may be locally approximated using parallel plates, d) that flow is laminar, and e) that the rock mass is impermeable. In many applications of the cubic law, most and sometimes all of these assumptions are violated. The recently derived Reduced Dimension Fracture Flow (RDFF) model (Gee and Gracie, 2022b) provides an alternative to the cubic law. The RDFF model is derived from the Navier-Stokes equations by integrating over the fracture aperture and making simplifying assumptions about the flow behaviour to generate a model that, at the cost of increased model complexity, is not subject to the same physical limitations as the cubic law. The RDFF model is capable of capturing inertial and transient flow behaviours, and has been shown to conserve energy in fractures of varying aperture where the cubic law does not.