Tracer dispersion in a hydraulic fracture with porous walls

This study presents an analytical expression for the dispersion of a tracer transporting through a hydraulic fracture with porous walls. Three different geometrical models for the hydraulic fracture, including rectangular, triangular, and elliptical models, are applied to evaluate the role of the hydraulic fracture geometry on the tracer dispersion coefficient. It is revealed that the average tracer dispersion coefficients for all hydraulic fracture geometries with porous walls are smaller than those with non-porous walls. However, the magnitudes of the average tracer dispersion coefficients in hydraulic fractures with both non-porous and porous walls follow an order of Triangular > Elliptical> Rectangular geometries. The analysis recognizes three distinct regimes of diffusion-dominated, transition, and advection-dominated for each hydraulic fracture geometry. In the diffusion-dominated regime, the advection is not important for the tracer transport and the ratios of the average dispersion coefficients in hydraulic fractures with porous walls to those in hydraulic fractures with non-porous walls are unity (R = 1). In the transition regime, the ratios depend on the Peclet number and they vary in the range of 0.3 < R < 1. The magnitudes of the ratios follow an order of Rectangular> Elliptical> Triangular hydraulic fracture geometries in the transition regime. The average tracer dispersion coefficients in the hydraulic fractures with porous walls are 0.3 times smaller than those with non-porous walls within the advection-dominated regime (R = 0.3). Therefore, it is crucial to consider the mass transfer of a tracer from the hydraulic fractures into the matrix in derivation of the dispersion coefficient within the transition and advection-dominated regimes for all hydraulic fracture geometries. (C) 2019 Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.