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Velocity of viscous fingers in miscible displacement: Intermediate concentration

Fedor Bakharev, Aleksandr Enin, Sergey Matveenko, Dmitry Pavlov, Yulia Petrova, Nikita Rastegaev, Sergey Tikhomirov

TL;DR

This work tackles the problem of viscous fingering during miscible displacement in porous media when the injected fluid is more mobile than the resident fluid. The authors refine the transverse flow equilibrium (TFE) model by incorporating an intermediate concentration near finger tips, and they justify these refinements with maximum-principle arguments and two theorems, complemented by CFD simulations. The main contribution is a modified leading-edge and trailing-edge velocity bounds: $v^f ≤ mbar(cstar, cmax)/m(cmax)$ and $v^b ≥ mbar(0, cstar)/m(0)$, which improve accuracy of edge speeds in many cases. The results enhance understanding of mixing-zone growth and have practical implications for optimizing post-flush polymer-flood strategies. The work also highlights the challenge of predicting rear-edge speeds and the need to determine intermediate concentrations from experiments.

Abstract

We investigate one-phase flow in porous medium corresponding to a miscible displacement process in which the viscosity of the injected fluid is smaller than the viscosity in the reservoir fluid, which frequently leads to the formation of a mixing zone characterized by thin fingers. The mixing zone grows in time due to the difference in speed between its leading and trailing edges. The transverse flow equilibrium (TFE) model provides estimates of these speeds. We propose an enhancement for the TFE estimates, and provide its theoretical justification. It is based on the assumption that an intermediate concentration exists near the tip of the finger, which allows to reduce the integration interval in the speed estimate. Numerical simulations were conducted that corroborate the new estimates within the computational fluid dynamics model. The refined estimates offer greater accuracy than those provided by the original TFE model.

Velocity of viscous fingers in miscible displacement: Intermediate concentration

TL;DR

This work tackles the problem of viscous fingering during miscible displacement in porous media when the injected fluid is more mobile than the resident fluid. The authors refine the transverse flow equilibrium (TFE) model by incorporating an intermediate concentration near finger tips, and they justify these refinements with maximum-principle arguments and two theorems, complemented by CFD simulations. The main contribution is a modified leading-edge and trailing-edge velocity bounds: and , which improve accuracy of edge speeds in many cases. The results enhance understanding of mixing-zone growth and have practical implications for optimizing post-flush polymer-flood strategies. The work also highlights the challenge of predicting rear-edge speeds and the need to determine intermediate concentrations from experiments.

Abstract

We investigate one-phase flow in porous medium corresponding to a miscible displacement process in which the viscosity of the injected fluid is smaller than the viscosity in the reservoir fluid, which frequently leads to the formation of a mixing zone characterized by thin fingers. The mixing zone grows in time due to the difference in speed between its leading and trailing edges. The transverse flow equilibrium (TFE) model provides estimates of these speeds. We propose an enhancement for the TFE estimates, and provide its theoretical justification. It is based on the assumption that an intermediate concentration exists near the tip of the finger, which allows to reduce the integration interval in the speed estimate. Numerical simulations were conducted that corroborate the new estimates within the computational fluid dynamics model. The refined estimates offer greater accuracy than those provided by the original TFE model.
Paper Structure (13 sections, 4 theorems, 52 equations, 12 figures, 4 tables)

This paper contains 13 sections, 4 theorems, 52 equations, 12 figures, 4 tables.

Table of Contents

  1. Introduction. Motivation
  2. Theoretical basis for TFE estimate improvement
  3. Description of the numerical model
  4. General considerations
  5. Viscosity functions
  6. Grid
  7. Initial and boundary conditions
  8. Software
  9. Simulations and estimation of velocities of fronts
  10. Analysis of the results
  11. Forward front analysis
  12. Rear front analysis
  13. Conclusions

Key Result

Theorem 1

Let $c(t, x, y)$ be a solution of the TFE model 03-TFE-1, 03-TFE in the strip $(x,y)\in\mathbb{R}\times[0,H]$, with no-flow boundary conditions at $y=\{0,H\}$, that is $q^y(t,x,0)=q^y(t,x,H)=0$, and initial data satisfying Denote by the front tip of the level line for the concentration $r$. Suppose there is a concentration $c^*$, such that for some constant $l_1$ and all $t>0$. Then for any $\w

Figures (12)

  • Figure 1: Profile of polymer concentration along a finger. The example shown was obtained from a simulation with the linear viscosity model, $M=5$ (see Sect. \ref{['sec:viscosity']})
  • Figure 2: Illustration for our expectations for the level line velocities. Green lines correspond to the graph of conjectured theoretical approximation of $v^f$ and $v^b$, defined by formula \ref{['TFE-estimate-mod-1']}, as a function of $c^*$ and $c_*$, respectively. Red lines correspond to expected true values for the front speed $v^f$ and back speed $v^b$ of the level lines for the concentration $c\in[0,c_{\max}]$. The conjecture in terms of the left figure states as follows: after the intersection point of the red and green curves, the red one is constant. Similarly, for the right figure: before the intersection point of the red and green curves, the red one is constant. The blue line corresponds to velocity equal to 1.
  • Figure 3: An example of a viscous finger that is close to the bottom border and moves faster than other fingers. On the right picture an enlarged region is shown. The picture was obtained during a simulation with the quadratic viscosity model, $M = 10$. Several other simulations have thin fingers on horizontal borders.
  • Figure 4: Position of the front for $C/c_{\max} = 0.015, 0.065, 0.115, 0.415, 0.915$, for exponential viscosity with $M = 5$
  • Figure 5: Comparison of the speed of the fastest finger with estimates by \ref{['TFE-estimate-mod-1']}. The green curve corresponds to \ref{['TFE-estimate-mod-1']}, filled dots correspond to \ref{['eq:fronts-regression']}, upper point of the vertical strip corresponds to \ref{['eq:fronts-max']}. The vertical strip shows the difference between \ref{['eq:fronts-regression']} and \ref{['eq:fronts-max']}. The results are shown for linear (left column), quadratic (middle column), and exponential (right column) viscosity with contrasts (from top to bottom) $M = 5, 10, 20, 40$.
  • ...and 7 more figures

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Theorem 2
  • Remark 4
  • Remark 5
  • proof : Proof of Theorem \ref{['Theorem_apost']}
  • Remark 6
  • Lemma 1
  • ...and 3 more