Analytical solutions for the flow of Carreau and Cross fluids in circular pipes and thin slits

Sochi, Taha

doi:10.1007/s00397-015-0863-x

Cited by 73 publications

(71 citation statements)

References 32 publications

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“…Moukhtari and B. Lecampion 2.5) where the dimensionless apparent width-averaged viscosity for parallel plates flow Γ (τ w , n, µ ∞ /µ o ,γ c ) depends non-linearly on the shear stress at the fracture wall τ w = w 2 ∂p ∂x as well as the rheological parameters of the Carreau model. This dimensionless apparent viscosity requires the solution of the rheological equation at the fracture wall in order to obtain the wall shear rateγ w = τ w /µ(γ w ) for a given value of wall shear stress (Sochi 2014(Sochi , 2015 (see appendix A for details). It is interesting to note that an analytical solution exists for the lubrication of a power-law rheology as well as for the Ellis rheological model (Myers 2005).…”

Section: Problem Formulationmentioning

confidence: 99%

“…The solution for the uni-dimensional pressure-driven flow of a Carreau fluid between parallel plate can be solved semi-analytically (Sochi 2015). The dimensionless apparent viscosity Γ Ω 2 ∂Π ∂ξ , α, n, µ ∞ /µ o is obtained from the dimensionless shear stress at the…”

Section: Fe Moukhtari and B Lecampionmentioning

confidence: 99%

“…where I(n,γ w , α, µ ∞ /µ o ) is an analytical function derived by Sochi (2015) which depends on the rheological parameters (n, µ ∞ /µ o , α) and the dimensionless wall shear rateγ w :…”

Section: Fe Moukhtari and B Lecampionmentioning

confidence: 99%

“…The differences between these models can be clearly seen in figure 1 while the corresponding rheological parameters are listed in table 1. In order to best fit the experimental data, the power-law index may be adjusted independently from the Carreau index (see examples in Sochi (2015); Myers (2005) Table 1. Rheological parameters of a HPG and VES fluids (for a given temperature) for the power-law, Carreau and Ellis models corresponding to the experimental data of figure 1.…”

Section: Introductionmentioning

confidence: 99%

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A semi-infinite hydraulic fracture driven by a shear-thinning fluid

Moukhtari

Lecampion

2018

J. Fluid Mech.

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We use the Carreau rheological model which properly account for the shear-thinning behavior between the low and high shear rates Newtonian limits to investigate the problem of a semi-infinite hydraulic fracture propagating at a constant velocity in an impermeable linearly elastic material. We show that the solution depends on four dimensionless parameters: a dimensionless toughness (function of the fracture velocity, confining stress, material and fluid parameters), a dimensionless transition shear stress (related to both fluid and material behaviour), the fluid shear thinning index and the ratio between the high and low shear rate viscosities. We solve the complete problem numerically combining a Gauss-Chebyshev method for the discretization of the elasticity equation, the quasi-static fracture propagation condition and a finite difference scheme for the widthaveraged lubrication flow. The solution exhibits a complex structure with up to four distinct asymptotic regions as one moves away from the fracture tip: a region governed by the classical linear elastic fracture mechanics behaviour near the tip, a high shear rate viscosity asymptotic and power-law asymptotic region in the intermediate field and a low shear rate viscosity asymptotic far away from the fracture tip. The occurrence and order of magnitude of the extent of these different viscous asymptotic regions are estimated analytically. Our results also quantify how shear thinning drastically reduces the size of the fluid lag compared to a Newtonian fluid. We also investigate simpler rheological models (power-law, Ellis) and establish the small domain where they can properly reproduce the response obtained with the complete rheology.

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Section: Problem Formulationmentioning

confidence: 99%

Section: Fe Moukhtari and B Lecampionmentioning

confidence: 99%

Section: Fe Moukhtari and B Lecampionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A semi-infinite hydraulic fracture driven by a shear-thinning fluid

Moukhtari

Lecampion

2018

J. Fluid Mech.

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“…When including extensional rheology effects, it is expected that regions with large variation in pore diameter such as pore throats would be significantly impacted by higher flow velocities, while the stagnant regions become even more stagnant. It might be useful to consider the case of the capillary tube bundle and sphere packs as reference cases for which exact analytical solutions can be computed for certain rheological models like Carreau fluids using the Weissenberg-Rabinowitsch-Mooney-Schofield integral method (Sochi 2015). But also a 3D porous rock should be considered as this work clearly showed that capillary bundles and sphere packs are only limiting cases for flow fields of sandstone rock.…”

Section: Relationship Between Flow Velocity and Shear Rate In Sandstomentioning

confidence: 95%

Shear Rate Determination from Pore-Scale Flow Fields

Berg

Wunnik

2017

Transp Porous Med

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Aqueous solutions with polymer additives often used to improve the macroscopic sweep efficiency in oil recovery typically exhibit non-Newtonian rheology. In order to predict the Darcy-scale effective viscosity μ eff required for practical applications often, semiempirical correlations such as the Cannella or Blake-Kozeny correlation are employed. These correlations employ an empirical constant ("C-factor") that varies over three orders of magnitude with explicit dependency on porosity, permeability, fluid rheology and other parameters. The exact reasons for this dependency are not very well understood. The semi-empirical correlations are derived under the assumption that the porous media can be approximated by a capillary bundle for which exact analytical solutions exist. The effective viscosity μ eff (v Darcy ) as a function of flow velocity is then approximated by a cross-sectional average of the local flow field resulting in a linear relationship between shear rate γ and flow velocity. Only with such a linear relationship, the effective viscosity can be expressed as a function of an average flow rate instead of an average shear rate. The local flow field, however, does in general not exhibit such a linear relationship. Particularly for capillary tubes, the velocity is maximum at the center, while the shear rate is maximum at the tube wall indicating that shear rate and flow velocity are rather anti-correlated. The local flow field for a sphere pack is somewhat more compatible with a linear relationship. However, as hydrodynamic flow simulations (using Newtonian fluids for simplicity) performed directly on pore-scale resolved digital images suggest, flow fields for sandstone rock fall between the two limiting cases of capillary tubes and sphere packs and do in general not exhibit a linear relationship between shear rate and flow velocity. This indicates that some of the shortcomings of the semi-empirical correlations originate from the approximation of the shear rate by a linear relationship with the flow velocity which is not very well compatible with flow fields from direct hydrodynamic calculations. The study also indicates that flow fields in 3D rock are not very well represented by capillary tubes.

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A continuous finite element framework for the pressure Poisson equation allowing non‐Newtonian and compressible flow behavior

Pacheco

Steinbach

2020

Numerical Methods in Fluids

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Computing pressure fields from given flow velocities is a task frequently arising in engineering, biomedical, and scientific computing applications. The so‐called pressure Poisson equation (PPE) derived from the balance of linear momentum provides an attractive framework for such a task. However, the PPE increases the regularity requirements on the pressure and velocity spaces, thereby imposing theoretical and practical challenges for its application. In order to stay within a Lagrangian finite element framework, it is common practice to completely neglect the influence of viscosity and compressibility when computing the pressure, which limits the practical applicability of the pressure Poisson method. In this context, we present a mixed finite element framework which enables the use of this popular technique with generalized Newtonian fluids and compressible flows, while allowing standard finite element spaces to be employed for the unknowns and the given data. This is attained through the use of appropriate vector calculus identities and simple projections of certain flow quantities. In the compressible case, the mixed formulation also includes an additional equation for retrieving the density field from the given velocities so that the pressure can be accurately determined. The potential of this new approach is showcased through numerical examples.

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Analytical solutions for the flow of Carreau and Cross fluids in circular pipes and thin slits

Abstract: In this paper, analytical expressions correlating the volumetric flow rate to In all the investigated cases, the three methods agree very well. The agreement with the variational method also lends more support to this method and to the variational principle which the method is based upon.

Cited by 73 publications

References 32 publications

A semi-infinite hydraulic fracture driven by a shear-thinning fluid

A semi-infinite hydraulic fracture driven by a shear-thinning fluid

Shear Rate Determination from Pore-Scale Flow Fields

A continuous finite element framework for the pressure Poisson equation allowing non‐Newtonian and compressible flow behavior

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