Flow through porous media of a shear‐thinning liquid with yield stress

Abstract

Darcy's law for the laminar flow of Newtonian fluids through porous media has been modified to a more general form which will describe the flow through porous media of fluids whose flow behavior can be characterized by the Herschel-Bulkley model. The model covers the flow of homogeneous fluids with a yield value and a power law flow behavior. Experiments in packed beds of sand were carried out with solutions of paraffin wax in two oils and with a crude oil from the Peace River area of Canada. The model fitted the data well.A sensitivity analysis of the fitting parameters showed that the model fit was very sensitive to errors in the flow behavior index, n, of the Herschel-Bulkley model. A comparison of the "n" values calculated from viscometer measurements and from flow measurements agreed well.A more general Reynolds number for flow through porous media, which includes a fluid yield value, was developed. The data were fitted to a Kozeny-Carman type equation using this Reynolds number. The constant in the Kozeny-Carman equation was determined for the two packed beds studied using Newtonian oils.The data could all be represented, within the experimental error, by the relationship f* = 15O/Re*. Since the mean volume to surface diameter of the packing was determined by the measurement of its permeability to a Newtonian oil, assuming C" = 150, the new definition of the Reynolds number allows the direct use of the Kozeny-Carman equation with Herschel-Bulkley type fluids. On a adapt6 la loi de Darcy pour I'Ccoulement laminaire de fluides newtoniens B travers des milieux poreux afin de rendre plus gentral et de dCcrire I'Ccoulement 21 travers des milieux poreux, de fluides obkissant au modtle Herschel-Bulkley . Ce modtle dCcrit I'ecoulement de fluides homogtnes comportant un seuil d'Ccoulement ayant un comportement de type loi de puissance.Des expCriences dans des lits 9 garnissage de sable ont CtC mentes avec des solutions de cire de paraffine dans deux huiles et avec du Ntrole brut provenant de la rkgion de Peace River au Canada. Le modtle reprksente bien ces donntes.Une analyse de sensibilitk des paramttres du modtle montre que le calage du modtle est trts sensible aux erreurs sur I'indice du comportement de I'tcoulernent, n, du modtle Herschel-Bulkley. Une comparaison des valeurs de "n" calcultes a partir des mesures au viscombtre et des mesures d'Ccoulement rCv8le un bon accord.On a dCfini un nouveau nombre de Reynolds pour I'Ccoulement k travers un milieu poreux, qui inclut la notion de seuil d'Ccoulement des fluides. Les donnCes ont CtC reprCsentCes par une Cquation de type Kozeny-Carman en utilisant ce nombre de Reynolds. La constante dans I'Cquation de Kozeny-Carman a CtC dCterminke pour les deux lits a garnissage CtudiCs avec des huiles newtoniennes.Toutes les donnCes ont pu &re reprtsenttes, a I'erreur exptrimentale prbs, par la relation F* = 150/Re*. bisque le diamttre moyen (volume sur surface) du garnissage a it6 dCtermint par la mesure de sa permCabilitC B une huile newtonienne, en supposant C " = 150, la ...

Effective rheology of Bingham fluids in a rough channel

“…The length scales a * allows an expansion in series of the two limit cases, the low (a * < a) and high (a * > a) flow rates regimes where a is the minimal aperture. At large flow rate (large a * ), the flow rate scales linearly as 19 . Black plain line represent numerical integration of the exact expression of Equation (12).…”

“…Both expansion (19) and (25) are applicable for any aperture distribution, as long as the lubrication approximation remains valid. As example, following Frigaard et al [23,24] who considered an aperture varying according to a cosine function.…”

“…We now return to the series expansions of the relation between pressure drop P and flow rate Q. We have introduced the length scale a * in Equation (18) and the expansion for small flow rates-hence, small a * -is given in Equation (19). Combining this expression with the assumption that H a L H , we find…”

“…We combine the two expansions (19) and (25) by splitting up the integrals I k (a, L), defined in Equation (20),…”

“…One of the main objective is to derive an generalized Darcy equation for yield stress fluids relating mean flow rate, the pressure gradient and a critical pressure gradient. Experimentally [2,9,19] they establish various laws which have quite essentially the same structure,…”