Abstract
Many nonNewtonian fluids, including polymers, exhibit both shearthinning 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 OldroydB constitutive equation and the Carreau model, respectively, to account for the viscoelastic and shearthinning 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 thinningthenthickening viscosity curve of the thinning–elastic fluid observed in experiments is reproduced by the present porescale 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 shearthinning properties

The thinningthenthickening apparent viscosity curve of the thinningelastic fluid observed experimentally in porous media is predicted in the porescale models

Porescale mechanisms for the thickening behavior of viscoelastic fluids in porous media at higher shear rates are proposed
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Acknowledgements
The authors acknowledge the Chemical EOR Industrial Affiliates Project in the Center for Subsurface Energy and the Environment (CSEE) for the financial support of this research and the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing computing resources. The authors note that there are no data sharing issues since all of the numerical information is provided in the tables and figures produced by solving the equations in the paper.
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Appendices
Appendix 1: Benchmarks
We present three benchmarks for the numerical model: (1) We validate the shearthinning 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.
Figure 11a describes a twodimensional Poiseuille flow between two parallel plates with a gap \(d\), which has exact theoretical solutions for both OldroydB viscoelastic flow and powerlaw shearthinning flow. Figure 11b describes a twodimensional porethroat geometry, based on which we made a glass micromodel to conduct experiments to validate our simulation of thinningelastic fluid flow.
For the first case, the powerlaw fluid [described by Eq. 4(a)] flowing in the twodimensional straight channel (Fig. 11a) driven by pressure gradient \({\varvec{F}}\) is considered. The theoretical solution (Robson 2003) for the steadystate velocity is calculated by
The power index \(n\) is varied from \(0.5\) to \(0.9\) in our simulations. As illustrated in Fig. 12, the simulated crosssectional velocity profiles match well with the theoretical profiles for a wide range of \(n\).
For the second case, the flow of the OldroydB viscoelastic fluid in the twodimensional 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
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 crosssectional 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.
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.
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 SigmaAldrich), 5wt% polyethylene glycol (~ 8,000,000 Mw PEG from SigmaAldrich), and 2wt% sodium chloride (NaCl). The solution was filtered with a 1.2µm filter paper and vacuumed for 30 min to get rid of invisible mixed bubbles. Our rheology tests using the Advanced Rheometric Expansion System Low Shear1 (ARES LS1) showed that fluid exhibits both shearthinning 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 etchedglass 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 (uPS0250T116). 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 nonNewtonian 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^{−9}s 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.
Therefore, these benchmarks verify the capabilities of our model to describe the nonNewtonian flow with both viscoelastic and shearthinning 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 OldroydB 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 steadystate flow conditions.
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Xie, C., Balhoff, M.T. Lattice Boltzmann Modeling of the Apparent Viscosity of Thinning–Elastic Fluids in Porous Media. Transp Porous Med (2021). https://doi.org/10.1007/s1124202101544y
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Keywords
 Apparent viscosity
 Shearthinning
 Viscoelastic thickening
 Porous media
 Lattice Boltzmann method