Viscoelastic surfactant (VES) based fracturing fluids have shown distinct rheological characteristics from conventional polymer based fluids. A nearly constant shear stress plateau is commonly observed in the shear stress versus shear strain rate curve of the VES fracturing fluids. In this paper, a piecewise power law model is first proposed to describe the rheology of these fluids. A solution of Poiseuille flow with the piecewise rheology is then derived and implemented into the PKN hydraulic fracturing model. The effect of rheology on the fracturing process is evaluated numerically by using an explicit moving mesh algorithm. The numerical solutions with the piecewise rheology are then compared with those from power law models with parameters deduced from different shear strain rate ranges. The results indicate that the conventional "averaged" power law model may be inadequate to yield an accurate prediction for the fracture geometry due to the limited shear strain rate range sampled.
1. INTRODUCTION
Hydraulic fracturing is a strongly coupled process involving fracture initiation/propagation, rock mass deformation, fluid flow and solid particle transport in a fracture, fluid flow in a porous medium and heat transfer. While fluid rheology is one of the most critical pieces of information for the successful design of hydraulic fractures, in classical hydraulic fracturing models, the rheology of fluids has traditionally been kept simple in order to accommodate the difficulties arising from the strong coupling between fluid flow and rock deformation. A Newtonian or a power law rheology model is generally assumed for conventional fracturing fluids such as polymer based gels. In a Newtonian model, the shear stress t behaves linearly proportional to the shear strain rate, i.e., t=¿ ¿ , where ¿ is the fluid viscosity. In a power law model, t=K¿n , where K is the consistency parameter and n is the power index. As fracturing fluids evolve towards what the petroleum industry calls "lean fracturing fluids", these models may often fail to describe the complex behavior of the fluids. For viscoelastic surfactant (VES) based fluids, which form the base of many lean fracturing fluids, a nearly constant stress plateau in shear stress versus shear strain rate curve is commonly observed (see Figure 1 showing the rheology of a model fracturing fluid - fluid XE). One may wonder if the presence of this nearly constant stress plateau in the fluid rheology has any
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implications for hydraulic fracturing design, in particular, for the geometry of hydraulic fractures. It is common practice in the petroleum industry to sample only limited data, typically within the shear strain rate range from 25 s-1 to 100 s-1, in order to then fit a power law model to approximate the rheology of the fracturing fluids. Such an approximation has been validated for typical guar based polymer fluids. As for VES fluids, the presence of the stress plateau may, however, result in the corresponding power index n to become nearly zero. Not only will this nearly zero index n cause numerical difficulties for most hydraulic fracturing simulators, but also it is doubtful that such an approximation will capture the behavior of VES fluids in a hydraulic fracturing context.
