Compensating Mechanisms That Minimize Flux Variability Through Unsaturated Fractures

Fast flow in fractures and macropores is a major cause of discrepancy between measurements and unsaturated flow models. Most models treat preferential flow as diffuse Darcy-Richards flow, so it is important to ascertain whether the mechanisms of unsaturated fracture flow accord with Darcy's law. The key issue is whether water flux is directly proportional to driving force with a proportionality factor, the hydraulic conductivity (K), that is independent of flux and force. We consider flow in a partially water-filled fracture with continuously supplied (e.g. ponded) water, responding to a change in driving force such as a change in tilt angle with respect to gravity. Four general flow modes, alone or in combination, can account for the dominant portion of the flow for these conditions, as shown by the experimental studies of Su and others (1999) and Tokunaga and Wan (1997). (1) Film flow occurs within a sheet or film that contacts a wall of the fracture. (2) Connected rivulet flow occurs when a rivulet that bridges across the fracture aperture by capillary force is consistently connected across the domain of interest from the inflow point to the outflow point. (3) Snapping rivulet flow occurs if the rivulet sometimes but not always extends continuously across the domain. (4) Pulsating-blob flow occurs in isolated blobs that bridge across the fracture aperture and move across the domain of interest without ever extending completely between the inflow and outflow points. Where fractures are large enough that the air-water interfaces are free to change shape or position in response to an externally applied change, each flow mode has its own characteristic relation between force and flow rate. This contrasts with the air-water interfaces commonly visualized in fine-textured media, in which the interface is constrained to a particular shape and position by capillarity and adsorption, so that the consistent geometry of the effective flow conduits leads to Darcian flow. In the four unsaturated-fracture flow modes described above, the air-water interfaces are likely to change in ways that oppose the externally applied change. For the film, continuous rivulet, and snapping rivulet modes, increased driving force causes the film or rivulet that constitutes the flow channel to become thinner, and hence less conductive. For pulsating-blob flow, additional mechanisms are likely to be active, but K-compensating effects are still likely to dominate. For a given increase in force, these compensating mechanisms cause the increase in flux to be less than Darcy's law would predict, a pattern referred to here as sub-Darcian behavior. In other words, effective K decreases as force increases. Sub-Darcian behavior may explain field and lab observations of unsaturated-zone flow velocities that seem to be independent of scale and minimally dependent on medium. For the purpose of flow-rate prediction in fractured media, sub-Darcian behavior is generally favorable because the fluxes predicted by a suitable model would be less sensitive to errors arising from unknown conditions than they would be for Darcian flow.