Open fractures are commonly modeled as parallel plates in fluid flow simulations. This notion is the basis of the cubic law that relates flow rate to pressure difference and aperture of fractures (Equation 1).

Equation 1. Equation relating flow rate (Q) and pressure (P) through the cubic law (the cube of aperture, 2B) and viscosity.

The flow rate per unit pressure gradient is related to the aperture as plotted in the diagram (dotted line in Figure 1). Here '2B' is known as the hydraulic aperture. Results from laboratory experiments (solid lines with full circles) verified this relationship for extended values of apertures (Cook et al. 1990). Here, fracture surfaces are assumed to be perfectly smooth and frictionless. Modifications to this mathematical relationship have been made in order to compensate for the fact that natural fracture surfaces almost always have a certain degree of roughness.

Figure 1. Flow rate through a fracture in laboratory experiment (solid line with full circles) and using 'cubic law' (dotted line). From Cook et al. (1990).

Flow through natural fractures is a very difficult problem in application due to a high degree of sensitivity of the aperture or flow rate to the remote stress perpendicular to fractures and fluid pressure within fractures as shown in the flow rate-effective stress relationship (Figure 2) established in laboratory experiments (Myer, 1991). An additional complexity arises from the fact that stresses and apertures are always perturbed around well bores in the subsurface.

Figure 2. Flow per unit gradient as a function of effective stress using a natural fracture in laboratory. From Myer (1991).