Non-Darcy effects in fracture flows of a yield stress fluid

Abstract

We study non-inertial flows of single-phase yield stress fluids along uneven/rough-walled channels, e.g. approximating a fracture, with two main objectives. First, we re-examine the usual approaches to providing a (nonlinear) Darcy-type flow law and show that significant errors arise due to self-selection of the flowing region/fouling of the walls. This is a new type of non-Darcy effect not previously explored in depth. Second, we study the details of flow as the limiting pressure gradient is approached, deriving approximate expressions for the limiting pressure gradient valid over a range of different geometries. Our approach is computational, solving the two-dimensional Stokes problem along the fracture, then upscaling. The computations also reveal interesting features of the flow for more complex fracture geometries, providing hints about how to extend Darcy-type approaches effectively.

Flow of yield stress and Carreau fluids through rough‐walled rock fractures: Prediction and experiments

“…Indeed, drilling fluids are often designed so as to have a yield stress in order to prevent cutting from settling when circulation stops [ Lavrov , ]. Also, a number of fracturing fluids used in hydraulic fracturing exhibit a yield stress designed to enhance proppant transport [ Talon et al ., ; Roustaei et al ., ] and present shear‐thinning behavior [ Lavrov , ; Perkowska et al ., ].…”

“…Many complex fluids used in industrial applications exhibit yield stress behaviour, e.g., polymer solutions, waxy crude oils, volcanic lavas, emulsions, colloid suspensions, foams, etc. [ Coussot , ; Dimitriou and McKinley , ; Roustaei et al ., ; Talon et al ., ; Lavrov , ; Coussot , ]. Common examples of yield stress shear‐thinning fluids are the slurries or cement grouts injected to reinforce soils, the heavy oils, or the drilling fluids injected into rocks for the reinforcement of wells [ Lavrov , ; Coussot , ].…”

“…More recently, Roustaei et al . [] numerically investigated the 2‐D flow of a yield stress fluid along an uneven fracture, showing that important Darcy‐type flow law lead to important errors in the case of short fractures due to self‐selection of the flowing region and the existence of fouling layers of unyielded fluid.…”

“…Also, pore‐network methods do not lead to analytical ‐u expressions, which is particularly aggravating in applications involving a wide range of injection flow rates. An important difference between Newtonian and complex fluids is the coupling of geometrical and rheological parameters in the flow law [ Roustaei et al ., ]. Indeed, a nontoxic method of porosimetry has been proposed, which is based on the injection of yield stress fluids through porous media and takes advantage of the mentioned coupling [ Ambari et al ., ; Malvault , ; Oukhlef et al .…”

Pressure‐driven displacement flows of yield stress fluids: Viscosity ratio effects

“…The majority of non‐Newtonian displacement studies have analyzed pattern formations in the Hele‐Shaw geometry, while considering shear‐thinning, shear‐thickening, yield stress, elasticity, and elasto‐visco‐plastic properties . There have been relatively fewer non‐Newtonian research works in other flow geometries, perhaps with the exception of yield stress displacement flows in capillary tubes, capillary plane channels, 2D channels, channels with uneven walls, horizontal pipes, vertical tubes, and symmetric and asymmetric tubes …”