ИСТИНА

With the development of hydraulic fracturing for stimulation of oil and gas production, modeling of non-Newtonian fluid flows in fractures is becoming increasingly important. Only a few studies in the open literature are devoted to two-dimensional flows of Bingham fluids1,2. Comprehensive analysis of interactions of Bingham particle-laden fluids in a narrow slot has not been carried out yet. The aim of the present study is to analyze various physical phenomena accompanying multiphase flow in a Hele-Shaw cell, such as combined effect of gravitational slumping and viscous fingering, particle settling and granular packing. We consider a flow of a mixture of non-Newtonian particle-laden fluids in a Hele-Shaw cell. The suspension flow is described within the framework of the two-fluid approach3, where the two interpenetrating continua are assumed to be a carrier medium (composed of several immiscible fluids), and a dispersed continuum (large non-Brownian particles with the diameter much smaller than the fracture width). In the lubrication approximation, 3D governing equations are reduced to a system of 2D width-averaged equations, including hyperbolic equations for transport of fluids and particles coupled with a quasi-linear elliptic equation for the pressure. Governing parameters involve the Bingham number (Bn), which is the ratio of the typical yield stress to the shear stress, the Buoyancy number, the Stokes-to-Froude number ratio determining the particle settling velocity, as well as density and viscosity ratios of the fluids. Governing equations are solved numerically using flux-limiting scheme for hyperbolic equations and the BiCGStab method with ILU(2) preconditioner for the pressure equation. The model and its numerical implementation are validated against available lab data on Newtonian viscous fingering in a slot4 and also against some unpublished lab data on channelling of Bingham suspensions in a slot. Parametric study of different injection sequences is carried out. It is found that in the absence of fingering no unyielded zones are created, and Bingham fluids behave similarly to Newtonian ones. In the presence of fingers, Bingham fluid becomes essentially unyielded due to decrease in local pressure gradient. Yield stress mitigates gravitational slumping, therefore the slumping rate of a Bingham fluid is always lower than that of a Newtonian fluid with the same viscosity and density. Increase in the Bingham number leads to the increase in finger shielding effect: smaller fingers of the displacing fluid left behind tend to stop completely (Fig. 1). Simulations of fluid injection under the conditions when both viscous fingering and gravitational slumping occurs showed that slumping damps fingering.