Theoretical and Numerical Simulation of Herschel-Bulkley Fluid Flow in Propped Fractures
Authors: Ouyang L. , Zhu D. , Hill A.D. 
Published: May 2016

Abstract
The flow of non-Newtonian fluids in porous media is important in many applications, such as polymer processing, heavy oil flow, and gel cleanup in propped fractures. Residual polymer gel in propped fractures results in low fracture conductivity and short effective fracture length, sometimes causing severe productivity impairment of a hydraulically fractured well. But non-Newtonian fluid flow behavior in porous media is difficult to be described and modeled. The Kozeny-Carman equation, a traditional permeability-porosity relationship, has been popularly used in porous media flow models. However, this relationship is not suitable for non-Newtonian fluid flow in porous media.
The aim of this work is to use a combination of 3D finite volume simulation and analytical calculations to develop a comprehensive model of Herschel-Bulkley non-Newtonian fluid flow through porous media. We present the mathematical model development, and then modify the model based on numerical simulation results. In the simulations, we developed a micro pore-scale model to mimic the real porous structure. The correlation of pressure gradient and superficial velocity was investigated under the influence of primary parameters, such as yield stress, power law index, and consistency index. We also considered the effect of proppant packing arrangement and proppant diameter. The Herschel-Bulkley model was used with an appropriate modification proposed by Papanastasiou to avoid the discontinuity of the apparent viscosity and numerical difficulties.
The result of the new model indicates that yield stress has a significant impact on non-Newtonian fluid flow through porous media, and the pressure gradient strongly depends on pore structure. The analytical expression reveals the physical principles for flow velocity in porous media. The variation trends of the threshold pressure gradient versus different influence factors are presented. By Computational Fluid Dynamics (CFD), we obtained a detailed view of the flow streamlines, the velocity field, and the pressure distribution in porous media. Numerical calculation results show that, in the center of the throats of porous media, the increasing yield stress widens the central plug-like flow region, and the increasing power law index sharpens the velocity profile. The new model can be readily applied to provide a clear guide to selection of fracture fluid, and can be easily incorporated into any existing reservoir simulators.

Introduction
To create a long and highly conductive hydraulic fracture in unconventional gas reservoirs, it is necessary to pump a large amount of viscous fluid with proppant deep into the fracture. The highly viscous fluid ensure effective proppant transportation and the proppant is used to create large fracture widths after the hydraulic fracture closes. But the fracture fluid, incorporating the complex non-Newtonian rheology, damages the proppant pack. In additional, as water leaks off into the matrix, some gel is dehydrated, forming a filter cake on the fracture face. The filter cake can have much higher concentration than the original fracture fluid, so the filter cake has a large yield stress. The residual gel can be difficult to clean up, decreasing initial gas production rate and reducing gas reserves. However, the porous media flow behavior of non-Newtonian fluid with complicated rheology is not clearly understood. Therefore, it is important to model and numerically simulate non-Newtonian fluid flow at the pore scale to obtain a better physical understanding of the mechanical process of gel cleanup.
Chase and Dachavijit (2003) modified Ergun's equation to include the effect of yield stress. One parameter in their model needs to be empirically determined by experimental measure with yield stress fluids. Their model results show that the yield stress has an important effect on the flow rate and cake growth rate.