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Prof. Prabir Daripa

Mathematics Department, Texas A&M University

Collaborator: Dr. Rohit Mishra

Gamma Technologies, Westmont, IL

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INTERPORE 2025

A proposal to model non-uniform mixing during enhanced oil recovery by polymer flooding

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Outline:

  • Motivation: Chemical EOR by Polymer-Surfactant Flooding
  • Governing Equations of A Newtonian Model for Multi-Component Multi-Phase Flow through Porous Media
  • A Hybrid Method based on Modified Method of Characteristics and Discontinuous Finite Element Method (MMOC-DFEM)
  • Shear Thinning Model
  • Numerical Results using Shear Thinning model
  • Mixing Model and its Effect on Viscosity
  • Conclusions

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Motivation: Chemical EOR by Surfactant-Polymer Flooding

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Surfactant: Mobilizes the oil by reducing capillary pressure

Polymer: Provides favorable mobility ratio

Polymer solution

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Non-Newtonian Model (TransPore v2.0)

  • The components are polymer and surfactant
  • s – Saturation, c – Concentration of polymer,
  • Г – Concentration of surfactant

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[1] P. Daripa and S. Dutta, Modeling and Simulation of Surfactant-Polymer Flooding using a New Hybrid Method, J. Comp. Phys., 335, pp. 249-282, 2017; doi:10.1016/j.jcp.2017.01.038

[2] P. Daripa and R. Mishra, Modeling shear thinning polymer flooding using a dynamic viscosity model, Physics of Fluids, 2023, Vol 35, Issue 4

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Numerical methods

  • Hybrid method:

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[1] P. Daripa and S. Dutta, Modeling and Simulation of Surfactant-Polymer Flooding using a New Hybrid Method,

J. Comp. Phys., 335, pp. 249-282, 2017; doi:10.1016/j.jcp.2017.01.038

[2] P. Daripa and S. Dutta, On the Convergence analysis of a hybrid method for multicomponent transport in porous media,

Appl. Numer. Math., vol. 146, pages 199-220, 2019; doi:10.1016/j.apnum.2019.07.009

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Todd-Longstaff (TL) Mixing Model

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Newtonian Model

  • The components are polymer and surfactant
  • s – Saturation, c – Concentration of polymer,
  • Г – Concentration of surfactant

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P. Daripa and S. Dutta, Modeling and Simulation of Surfactant-Polymer Flooding using a New Hybrid Method,

J. Comp. Phys., 335, pp. 249-282, 2017; doi:10.1016/j.jcp.2017.01.038

 

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Numerical methods

  • Hybrid method:

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[1] P. Daripa and S. Dutta, Modeling and Simulation of Surfactant-Polymer Flooding using a New Hybrid Method,

J. Comp. Phys., 335, pp. 249-282, 2017; doi:10.1016/j.jcp.2017.01.038

[2] P. Daripa and S. Dutta, On the Convergence analysis of a hybrid method for multicomponent transport in porous media,

Appl. Numer. Math., vol. 146, pages 199-220, 2019; doi:10.1016/j.apnum.2019.07.009

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Shear-thinning model

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  • Experimental results from Lindner et. al. [1]

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  • Data chosen dynamically in the code from the curve fit depending on the concentration of polymer in the cell

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[1] A. Lindner, D Bonn, J Meunier, Viscous fingering in a shear-thinning fluid, Physics of Fluids 2000

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  • Higher viscosity at higher IPC
  • lower viscosity at higher shear rate : shear thinning behavior

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Results – Finger width

  • Wider fingers observed at higher injection rates (IR) at water breakthrough.
  • Higher initial polymer concentration (IPC) leads to wider fingers at water breakthrough
  • Not the best criterion to test sweep efficiency due to the competing effects of shear thinning and viscous fingering.

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IR=2; (L) IPC=0.0002; (R) IPC = 0.001

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Results – Cumulative Oil Recovery (COR)

  • Interestingly optimum COR is observed at intermediate Initial Polymer Concentration (IPC) and Injection Rate (IR)
  • At higher IR, COR is inversely proportional to IPC because at higher IR and high IPC, shear thinning effect dominates over the stabilizing effect of increasing IPC resulting in less COR
  • Dynamically changing IR and IPC can lead to true optimization of COR

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Shear-thinning model

 

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[1] P. Daripa and R. Mishra, Modeling shear thinning polymer flooding using a dynamic viscosity model, Physics of Fluids, 2023, Vol 35, Issue 4

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Mixing Model

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Concentration levels

Mixing levels (at the same concentration)

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Viscosity model based on mixing model

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Results: Effective Aqueous Viscosity for different mixing parameter ω

  • Plots show the effective aqueous viscosity versus concentration of polymer
  • All plots begin at pure water viscosity 0.5
  • Viscosity depends on how well the polymer is mixed with water.
  • This shows the subgrid capturing of unmixed and mixed states that the polymer solution might exist in
  • Next step: Spatio-temporal variation of the omega parameter depending on identification of mixed states based on the local conditions of solution

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Permeability dependent mixing factor

  • Solubility of polymer is directly proportional to the capillary pressure
  • Capillary pressure is inversely proportional to permeability
  • A function f(K) is generated such that the mixing parameter is inversely proportional to K
  • Figure shows omega value based on point-wise varying permeability

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log(K)

omega

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  • Experimental results from Lindner et. al. [1]

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  • Data chosen dynamically in the code from the curve fit depending on the concentration of polymer in the cell

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[1] A. Lindner, D Bonn, J Meunier, Viscous fingering in a shear-thinning fluid, Physics of Fluids 2000

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Results without Shear Thinning Effect: Difference in viscosity field plotted (mu (omega = 1)-mu(omega=0))

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Time = 5

Time = 10

Time = 15

Time = 20

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Results with Shear Thinning Effect: Viscosity contours at different mixing parameter values. (t=500 and IR=50000)

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Omega=0

Omega=0.5

Omega=1

Qualitative results

Viscosity

Mobility Ratio

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Qualitative results

Interface between poly-solution and displaced oil highlights different mixing states

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Results with Shear Thinning Effect: Mean Finger Width for different mixing parameter ω

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Results with Shear Thinning Effect: Cumulative Oil Recovered for different mixing parameter ω

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Results: Effective Aqueous Viscosity for different mixing parameter ω

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Results

  • Cases go from omega=0 to omega=1 (first five Xanthane and next five Schizophyllan in each plot)
  • For most case groups the COR reduces with increasing omega
  • Only anomaly is for Xanthane at IR==120000 and IPC=300wppm

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Results

  • Location of maximum viscosity across the x axis is shown at different times
  • Left column: Xanthane; Right column: Schizophyllan
  • A clear distinction can be seen for the anomaly where one inflection point is noticed as compared to two inflection points for most other cases
  • This shows that the accurate viscosity prediction can uncover anomalous behavior in this complex multi-physics problem

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Results

  • After implementing permeability based omega (shown in histogram )
  • The COR for omega=f(K) is as expected between the extreme conditions of completely isolated and completely mixed states.

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Summary:

  • Competing effects of shear thinning and mobility ratio: Higher shear rates cause a decrease in viscosity ratio (ratio of viscosity of aqueous phase over that of oil phase) due to shear thinning which reduces oil recovery from what it would otherwise be.
  • Finger width not the best parameter to assess sweep efficiency due to non-linear effects.
  • Higher initial polymer concentration (IPC) leads to wider fingers at water breakthrough.
  • Wider fingers observed at higher injection rates (IR) at water breakthrough.
  • Cumulative oil recovered is not a monotonic function of IPC and IR demonstrating competing effects of shear thinning and viscous fingering.
  • Todd Longstaff mixing model does not affect the viscosity significantly.

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Summary:

  • Finger width is largest for the maximum mixing case due to homogeneous front of polysolution (displacing phase) which counters water breakthrough.
  • Finger width not the best parameter to assess sweep efficiency due to non-linear effects.
  • In the highly mixed case, shear thinning effect increases the mobility ratio between the displaced phase and displacing phase, thereby causing the homogeneous front to move faster. Front speed increases with omega, the mixing parameter.
  • Due to fast moving front, the efficiency of the recovery is lost and the most mixed polymer results in least COR which is contrary to intuition.
  • New model for shear thinning and polymer mixing is able to capture otherwise elusive anomalous behavior
  • Varying omega spatio-temporally results in capturing the viscosity and subsequently the COR for any given permeability field

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Summary:

  • Competing effects of shear thinning and mobility ratio: Higher shear rates cause a decrease in viscosity ratio (ratio of viscosity of aqueous phase over that of oil phase) due to shear thinning which reduces oil recovery from what it would otherwise be.
  • Finger width not the best parameter to assess sweep efficiency due to non-linear effects.
  • Higher initial polymer concentration (IPC) leads to wider fingers at water breakthrough.
  • Wider fingers observed at higher injection rates (IR) at water breakthrough.
  • Cumulative oil recovered is not a monotonic function of IPC and IR demonstrating competing effects of shear thinning and viscous fingering.

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Main Conclusions:

  • Newtonian model for multi-phase multi-component flow has been extended to include shear-thinning effect of polymer.
  • The shear-thinning effect has been encoded in our MMOC-DFEM based code. This new code has been tested and validated.
  • Numerical results obtained agree with experimental results qualitatively.
  • Simulations with TL mixing model is in progress.
  • In collaboration with Sourav Dutta, we are also developing a model for including dispersion effects and then include this effect in the code.
  • We are also implementing a multiscale model.
  • We are also working on improving our numerical method for higher order accuracy.

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Thanks to Interpore

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Email: daripa@tamu.edu