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Oil displacement by slug injection: a rigorous justification for the Jouguet principle heuristic

Sergey Matveenko, Nikita Rastegaev

TL;DR

The paper rigorously justifies the Jouguet principle for oil-displacement by slug injection within a one-dimensional, two-phase three-component EOR model. It leverages a Lagrange-coordinate splitting to decouple the chromatographic (concentration) dynamics from the Buckley–Leverett-type flow, enabling a Kruzhkov-type uniqueness framework for W-solutions and a constructive, semi-analytical solution procedure. A second splitting technique simplifies the characteristic system, and the Jouguet condition is shown to be applicable in a principled way, yielding full and partial-Jouguet solution regimes with proven uniqueness. The analysis covers the slug-concentration equation independently, then couples it to the conservation-law dynamics, providing explicit constructions in key regimes (Full, Partial Jouguet) and establishing a rigorous foundation for using the Jouguet principle in practical polymer-slug flooding scenarios. These results enhance the reliability of predictive models for polymer flooding and slug-design in oil reservoirs, with potential implications for optimizing EOR performance.

Abstract

In this paper we discuss a one-dimensional model for two-phase Enhanced Oil Recovery (EOR) floods, primarily for the polymer flood. We improve upon the method for the construction of semi-analytical solutions for the oil displacement by a water slug containing dissolved chemicals given in (Pires, Bedrikovetsky and Shapiro, 2006) and later generalized in (Apolinário, de Paula and Pires, 2020), (Apolinário and Pires, 2021). This method utilizes a transformation into the Lagrange coordinates that splits the equations and allows one to solve the chromatographic one-phase problem separately. The solution is then substituted into a scalar hyperbolic conservation law, which is solved using the method of characteristics. However, there is often a gap in the characteristics near the chemical shock front. It was posited to the authors that the Jouguet principle could be used to close that gap. However, no rigorous justification was given for this approach, and as such it remained a heuristic. We analyze the conditions for the appearance of the gap and its properties, and give a proper argumentation for the Jouguet heuristic and its applicability based on the Kružkov-type uniqueness theorem for the conservation law system. Additionally, a second splitting technique within the Lagrange coordinates is developed that simplifies this analysis and the construction of characteristics. Keywords: Enhanced oil recovery, Polymer flooding, Slug injection, Conservation laws, Hyperbolic systems of partial differential equations

Oil displacement by slug injection: a rigorous justification for the Jouguet principle heuristic

TL;DR

The paper rigorously justifies the Jouguet principle for oil-displacement by slug injection within a one-dimensional, two-phase three-component EOR model. It leverages a Lagrange-coordinate splitting to decouple the chromatographic (concentration) dynamics from the Buckley–Leverett-type flow, enabling a Kruzhkov-type uniqueness framework for W-solutions and a constructive, semi-analytical solution procedure. A second splitting technique simplifies the characteristic system, and the Jouguet condition is shown to be applicable in a principled way, yielding full and partial-Jouguet solution regimes with proven uniqueness. The analysis covers the slug-concentration equation independently, then couples it to the conservation-law dynamics, providing explicit constructions in key regimes (Full, Partial Jouguet) and establishing a rigorous foundation for using the Jouguet principle in practical polymer-slug flooding scenarios. These results enhance the reliability of predictive models for polymer flooding and slug-design in oil reservoirs, with potential implications for optimizing EOR performance.

Abstract

In this paper we discuss a one-dimensional model for two-phase Enhanced Oil Recovery (EOR) floods, primarily for the polymer flood. We improve upon the method for the construction of semi-analytical solutions for the oil displacement by a water slug containing dissolved chemicals given in (Pires, Bedrikovetsky and Shapiro, 2006) and later generalized in (Apolinário, de Paula and Pires, 2020), (Apolinário and Pires, 2021). This method utilizes a transformation into the Lagrange coordinates that splits the equations and allows one to solve the chromatographic one-phase problem separately. The solution is then substituted into a scalar hyperbolic conservation law, which is solved using the method of characteristics. However, there is often a gap in the characteristics near the chemical shock front. It was posited to the authors that the Jouguet principle could be used to close that gap. However, no rigorous justification was given for this approach, and as such it remained a heuristic. We analyze the conditions for the appearance of the gap and its properties, and give a proper argumentation for the Jouguet heuristic and its applicability based on the Kružkov-type uniqueness theorem for the conservation law system. Additionally, a second splitting technique within the Lagrange coordinates is developed that simplifies this analysis and the construction of characteristics. Keywords: Enhanced oil recovery, Polymer flooding, Slug injection, Conservation laws, Hyperbolic systems of partial differential equations

Paper Structure

This paper contains 31 sections, 26 theorems, 130 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Restrictions on problem parameters
  3. Restrictions on the flow function
  4. Restrictions on the adsorption function
  5. Admissible solutions of chemical flood system
  6. Admissible weak solutions
  7. Travelling wave dynamic system
  8. Admissible $s$-shocks
  9. Admissible $c$-shocks
  10. The uniqueness theorem
  11. Lagrange coordinate transformation
  12. Lagrange coordinates
  13. Entropy conditions in Lagrange coordinates
  14. Mapping shocks to Lagrange coordinates
  15. Oleinik, Lax and entropy admissibility for $\mathcal{U}$-shocks
  16. ...and 16 more sections

Key Result

Proposition 1

The following restrictions on admissibility are evident from the properties (F1)--(F4), (A1)--(A3), the Rankine--Hugoniot conditions eq:RH-1 and the analysis of the sign of the right-hand side of eq:dyn_sys_cap_diff:

Figures (6)

  • Figure 1: Examples of (a) flow function $f(s,c)$; (b) adsorption function $a$.
  • Figure 2: Areas $Q_{orig}$ (grey area on the left), and $Q_{lagr}$ (grey area on the right) in the case $s_{init} \neq 0$. The blue line on the left is mapped onto the blue line $\varphi_0(x) = -x s_{init}$ on the right.
  • Figure 4: The function $\mathcal{F}(\mathcal{U},\zeta)$ corresponding to the flow function $f(s, c)$ plotted in Fig \ref{['fig:BL_ads']} (a).
  • Figure 5: Characteristics of the $\zeta$-solution.
  • Figure 6: Trajectories of the $\mathcal{U}$-characteristics in the $(\zeta, \mathcal{U})$ plane. Full Jouguet case.
  • ...and 1 more figures

Theorems & Definitions (48)

  • Definition 1
  • Remark 1
  • Remark 2
  • Proposition 1: Proposition 3.2 MR2024
  • Proposition 2
  • Theorem 1: Theorem 6.1, MR2024
  • Lemma 1: Lemma 4.4, MR2024
  • Corollary 1: Corollary 4.5, MR2024
  • Proposition 3: Proposition 4.6, MR2024
  • Remark 3
  • ...and 38 more