Tracer dispersion due to non-Newtonian fluid flows in hydraulic fractures with different geometries and porous walls

Abstract

Tracer transport in hydraulic fractures is substantially affected by the rock matrix surrounding them. Nevertheless, there is not yet an expression in literature to address the relationship among the transport coefficients, porous walls, hydraulic fracture geometries, and non-Newtonian fluid properties. Therefore, the tracer dispersion due to non-Newtonian fluid flows in hydraulic fractures with different geometries and porous walls is mathematically derived and studied in the current work. Rectangular, triangular, and elliptical models and power-law model are considered here to describe the hydraulic fracture geometry and the non-Newtonian fluid rheology, respectively. The results reveal that as the flow behavior index grows, the coefficient of the shear dispersion term in the case of the shear thinning and Newtonian fluids follows an order of triangular > elliptical > rectangular and in the case of the shear thickening while the same order is followed at the beginning, it turns first to triangular > rectangular > elliptical, then to rectangular > triangular > elliptical, and finally to rectangular > elliptical > triangular. The coefficient of the shear dispersion term increases sharply for each geometrical model with the flow behavior index when the fluid is shear thinning while it first increases slightly and then flattens when the fluid is shear thickening. However, the coefficient of the shear dispersion term for porous walls is lower than the one for nonporous walls. It is also found that the average tracer dispersion coefficients in rectangular, triangular, and elliptical nonporous-walled and porous-walled hydraulic fractures and their ratio dictate that the hydraulic fracture-rock matrix communication through the continuity of the tracer concentration and the mass flux needs to be considered for determination of the tracer dispersion during the transition for intermediate Peclet numbers and the advection-dominated tracer transport for large Peclet numbers. This study along with its findings can pave the way for future investigations on the tracer dispersion in a network of hydraulic fractures with rough and porous walls.