Lattice Boltzmann Modeling of the Apparent Viscosity of Thinning–Elastic Fluids in Porous Media

Abstract

Many non-Newtonian fluids, including polymers, exhibit both shear-thinning and viscoelastic rheological properties. A lattice Boltzmann (LB) model is developed for simulation of the flow of thinning–elastic fluids through porous media. This model applies the Oldroyd-B constitutive equation and the Carreau model, respectively, to account for the viscoelastic and shear-thinning behaviors of the thinning–elastic fluid in porous media. Both rheological features are captured well by this model and are verified against analytical solutions. The thinning-then-thickening viscosity curve of the thinning–elastic fluid observed in experiments is reproduced by the present pore-scale simulations. In addition to the traditional extensional theory, we propose other important mechanisms for the increase in apparent viscosity of viscoelastic fluids at higher shear rates. The mechanisms proposed include the reduction in conductivity due to stagnant fluid, the compressed effective flow region, and larger energy dissipations caused by the viscoelastic instability. We find that the viscoelastic thickening effect is more prominent in porous geometries with a large pore–throat ratio.

Article Highlights

• A lattice Boltzmann model is developed to predict the flow behavior of fluids with both viscoelastic and shear-thinning properties

• The thinning-then-thickening apparent viscosity curve of the thinning-elastic fluid observed experimentally in porous media is predicted in the pore-scale models

• Pore-scale mechanisms for the thickening behavior of viscoelastic fluids in porous media at higher shear rates are proposed

Appendices

Appendix 1: Benchmarks

We present three benchmarks for the numerical model: (1) We validate the shear-thinning part of the model by removing the viscoelastic stress terms; (2) the viscoelastic part of the model is validated by imposing a constant solvent viscosity $\eta_{s}$; and (3) the combined rheological properties of the model are validated. Parameters used for these benchmarks are listed in Tables 2 and 3.

Table 2 Parameters used for the first two benchmarks

Table 3 Parameters used for the comparison between the micromodel experiment and LB simulation

Figure 11a describes a two-dimensional Poiseuille flow between two parallel plates with a gap $d$, which has exact theoretical solutions for both Oldroyd-B viscoelastic flow and power-law shear-thinning flow. Figure 11b describes a two-dimensional pore-throat geometry, based on which we made a glass micromodel to conduct experiments to validate our simulation of thinning-elastic fluid flow.

Fig. 11

Schematics of the benchmark problems: (a) Poiseuille flow in a two-dimensional straight channel; (b) flow in a pore–throat channel; and (c) the micromodel platform for analyzing flow behavior

For the first case, the power-law fluid [described by Eq. 4(a)] flowing in the two-dimensional straight channel (Fig. 11a) driven by pressure gradient ${\varvec{F}}$ is considered. The theoretical solution (Robson 2003) for the steady-state velocity is calculated by

$$u\left( y \right) = \frac{n}{n + 1}\left( {\frac{F}{{K_{{{\text{pow}}}} }}} \right)^{\frac{1}{n}} \left[ {\left( \frac{d}{2} \right)^{{\frac{1}{n} + 1}} - \left| {y - \frac{d}{2}} \right|^{{\frac{1}{n} + 1}} } \right].$$

(A1)

The power index $n$ is varied from $0.5$ to $0.9$ in our simulations. As illustrated in Fig. 12, the simulated cross-sectional velocity profiles match well with the theoretical profiles for a wide range of $n$.

Fig. 12

Cross-sectional velocity profiles of the power-law viscoplastic fluids with different power indexes in a two-dimensional straight channel

For the second case, the flow of the Oldroyd-B viscoelastic fluid in the two-dimensional straight channel (Fig. 11a) driven by pressure gradient ${\varvec{F}}$ is considered. Theoretical solutions (Zou et al. 2014) for the viscoelastic shear stress component $\sigma_{xy}$ and normal stress component $\sigma_{xx}$ at the steady state are given by

$$u_{{{\text{max}}}} = Fd^{2} /8\eta_{0} ,$$

(A2a)

$$u\left( y \right) = 0.4u_{{{\text{max}}}} \frac{y}{d}\left( {1 - \frac{y}{d}} \right),$$

(A2b)

$$\sigma_{xy} = \left( {1 - \beta } \right)\eta_{0} \frac{{{\text{d}}u}}{{{\text{d}}y}},$$

(A2c)

$$\sigma_{xx} = 2\lambda \eta_{0} \left( {1 - \beta } \right)\left( {\frac{{{\text{d}}u}}{{{\text{d}}y}}} \right)^{2}.$$

(A2d)

We test two different relaxation times λ = 0.5 and λ = 1 in our simulations, corresponding to Wi = 0.5 and Wi = 1, respectively. The numerical cross-sectional stress profiles are compared with theoretical solutions. As is shown in Fig. 13, the numerical results agree well with the theoretical solutions at different relaxation times.

Fig. 13

Cross-sectional stress profiles of the Oldroyd-B viscoelastic fluids with different relaxation times in a two-dimensional straight channel: (a) the viscoelastic shear stress component σ_xy; (b) the viscoelastic normal stress component σ_xx

The accuracy of the method is also studied by varying the lattice resolution for the gap from N = 12 to N = 200. The error E of an arbitrary variable ξ between the simulation result ξ_LB and the theoretical solution ξ_theory is defined as $E = \sqrt {\frac{1}{N}\mathop \sum \limits_{{k = 1}}^{N} \left[ {\xi _{{{\text{LB}}}} \left( {y_{k} } \right) - \xi _{{{\text{theory}}}} \left( {y_{k} } \right)} \right]^{2} }$. Figure 14 presents the errors of velocity and two stress components for λ = 0.5 (Wi = 0.5) and λ = 1 (Wi = 1). As is seen, the errors decrease with increasing resolutions for both cases. When the resolution is increased to N = 50, all the errors drop below 0.01.

Fig. 14

The errors of velocity and two stress components with respect to the lattice resolution for (a) λ = 0.5 (Wi = 0.5); (b) λ = 1 (Wi = 1)

For the third case, we consider the flow of a thinning–elastic fluid through the pore–throat channel (Fig. 11b), comparing the simulation result with our micromodel experiment. The fluid we used is an aqueous solution that contains 0.3wt% polyethylene oxide (~ 10,000 Mw PEO from Sigma-Aldrich), 5wt% polyethylene glycol (~ 8,000,000 Mw PEG from Sigma-Aldrich), and 2wt% sodium chloride (NaCl). The solution was filtered with a 1.2-µm filter paper and vacuumed for 30 min to get rid of invisible mixed bubbles. Our rheology tests using the Advanced Rheometric Expansion System Low Shear-1 (ARES LS-1) showed that fluid exhibits both shear-thinning and viscoelastic features at room temperature: The steady shear test showed its bulk shear viscosity follows the Carreau model with parameters listed in Table 3 and the dynamic frequency sweep test showed its viscoelastic relaxation time is 0.025 s.

The experimental platform is shown in Fig. 11c. The etched-glass micromodel was horizontally mounted in an aluminum holder. A Hamilton syringe (750 series, 500 μl) was connected to the inlet of the micromodel, and the outlet was open to the atmosphere. The pressure was measured at the inlet by the LabSmith pressure sensor (uPS0250-T116). We controlled the fluid injection rates by the Harvard Apparatus 2000 syringe pump.

Similar to the numerical procedure, we first injected a Newtonian fluid (50wt.% glycerin solution, 5.6cp) into the micromodel at different flow rates (from 10 μL/hr to 600 μL/hr) as a reference to obtain the apparent viscosity of the non-Newtonian fluid. In the Newtonian case, the measured pressure showed a perfect linear relationship with the flow rate (P_N/Q_N = 0.156 kPa hr/μL). Then, we cleaned and dried the chip and started to inject the PEO in PEG solution from 2 μL/hr to 150 μL/hr corresponding to the shear rate varying from 0.79 s⁻¹ to 59 s⁻¹. We kept recording the pressure P_p at each flow rate Q_p until it reached a steady state. By comparing these data with the reference Newtonian case, the apparent viscosities of the PEO in PEG solution at different shear rates were obtained. The above process was repeated twice to ensure the repeatability of the experiments.

In the simulations, since it is not feasible to cover the full length of the channel, we apply periodic boundary conditions and pressure gradient (body force F) to drive the fluid. The pressure gradient is varied from 2 kPa/m to 500 kPa/m, and the time step is set at 2.5 × 10⁻⁹s to ensure the upper bound of the LB relaxation time τ in Eq. (5) to be close to 2.

As shown in Fig. 15, our simulation results and experimental data match well with each other, and both of them show the thickening behavior of the thinning–elastic fluid at higher shear rates in the pore–throat channel.

Fig. 15

Comparison of the apparent viscosity of the PEO in PEG solution in a pore–throat channel obtained by our micromodel experiments and the lattice Boltzmann simulations

Therefore, these benchmarks verify the capabilities of our model to describe the non-Newtonian flow with both viscoelastic and shear-thinning features.

Appendix 2: Convergence tests

For all cases, the convergence criterion is selected as the relative change in flow rate between 10000 time steps is less than 10⁻⁸. In Fig. 16, we present several examples of the convergence tests for both the purely Oldroyd-B viscoelastic fluid and the thinning–elastic fluid flowing in the asymmetric porous geometry. In these examples, the viscoelastic relaxation time is constant at λ = 0.01s, but the pressure gradient varies from 100 kPa/m to 2000 kPa/m. The flow rate evolution curves all show good convergence at the end of the calculations, which demonstrates this convergence criterion is accurate enough to ensure steady-state flow conditions.

Fig. 16

Convergence tests for (a) the purely Oldroyd-B viscoelastic fluid and (b) the thinning–elastic fluid flowing in the asymmetric porous geometry