GRAVITY CURRENTS OF HERSCHEL-BULKLEY FLUIDS IN A POROUS MEDIUM: ANALYTICAL MODELS AND EXPERIMENTS

The aim of this paper is to study gravity currents of rheologically complex fluids in porous media. Two-dimensional motion of a shear-thinning/thickening fluid with a yield stress (described by the Herschel-Bulkley model) in a narrow fracture slumping under gravity is investigated analytically, numerically and experimentally, extending the work by King & Woods [1], Longo et al. [2] and Ciriello et al. [3]. A modified version of Darcy's law is assumed to describe the motion of the gravity current, whose volume is taken to vary as a monomial function of time. A closed-form self-similar solution is derived for a fluid volume that scales as time squared. This solution reveals that for injection rates scaling at a rate slower than time, the current will eventually evolve to consist entirely of plug flow (ignoring any three-dimensional disturbances caused by fluid injection). Therefore gravity currents formed either by a finite release of fluid, or by a constant fluid injection rate must eventually drop below the yield stress and consist completely of plugged fluid. For a generic increase of the volume of the gravity current with time, the propagation is studied numerically with an ad-hoc code. The availability of a closed-form self-similar solution allows an immediate verification of the model and a validation of the experimental set-up. For a discharge rate linearly increasing in time, the profile of the current is linear and the velocity of the front of the current is constant. Two series of experiments have been successfully completed: i) direct simulation of the current propagation in an artificial porous medium composed of a homogeneous layer of glass beads, and ii) indirect simulation in an Hele-Shaw analogue model. The fluid used in the experiments is a neutralized mixture of deionized water and Carbopol 980, added with ink for an easy visualization. The rheometric parameters have been measured with a shear stress controlled rheometer with parallel plates. A specific effort has been devoted to measuring the yield stress, which is a well-known difficult task. A series of dynamic tests, creeping and relaxing tests, and direct measurement tests with a tilting plane have been performed for achieving a consistent validation of the yield stress measurements. A last critical aspect is the wall slip which affects many polymer fluids flows, especially if a yield stress is present. While slip does not affect the flow in the direct model, it modifies the measurements in the rheometer and the Hele-Shaw cell tests. To quantify the slip effect, the plates in the rheometer have been roughened with sand paper, and the internal walls of the Hele-Shaw cell have been roughened with transparent anti-slippery ribbons for stairs. The overall conclusion is that the theoretical model is well interpreting the flow of the current in the intermediate asymptotic regime, far from the injection section (where three dimensional effects are quite evident) and in the limit of the thin current approximation. Moreover, the rheological behavior of the HB fluid in the porous medium is well described by the present model.