The Riemann problem for three-phase foam flow in porous media

This study presents a methodology to solve the Riemann problem for three-phase foam flow in porous media under local equilibrium conditions, overcoming the challenges of an umbilic point to classify wave structures and analyze oil bank formation for applications in enhanced oil recovery and carbon storage.

The Gist

Imagine you are trying to push a thick, sticky blob of oil out of a sponge using a stream of gas. This is a common challenge in oil recovery, but there's a problem: gas is like a slippery, fast-moving ghost. It tends to "finger" through the oil, creating little tunnels that bypass the oil entirely, leaving most of it stuck in the sponge.

To fix this, engineers use foam. Think of foam as a traffic jam for the gas. The bubbles in the foam act like speed bumps, slowing the gas down and forcing it to push the oil out more evenly.

This paper is a mathematical study of exactly how that "traffic jam" moves through the sponge (porous rock) when you mix gas, water, and oil together. The authors, Luis Fernando Lozano, Grigori Chapiro, and Dan Marchesin, created a detailed map of how these fluids interact.

Here is a breakdown of their work using simple analogies:

1. The "Traffic Map" (The Riemann Problem)

In math, a "Riemann problem" is like asking: "If I suddenly switch the traffic from a slow line to a fast line, what happens?"

• The Setup: Imagine a long hallway. On the left side, you are injecting a mix of foamed gas and water. On the right side, the hallway is filled with oil and water.
• The Question: When the injection starts, how do the waves of gas, water, and oil move? Do they crash into each other? Do they smooth out? Do they form a specific pattern?

The authors mapped out every possible way these fluids can arrange themselves as they move through the rock.

2. The "Speed Trap" (The Umbilic Point)

Usually, in fluid dynamics, waves travel at different speeds, like cars on a highway with different speed limits. But in this specific three-phase foam mix, there is a special spot called an umbilic point.

• The Analogy: Imagine a roundabout where all the lanes merge into one, and suddenly, the speed limit for a slow car and a fast car becomes exactly the same.
• The Challenge: At this point, the usual rules for predicting traffic flow break down. It's like a traffic light that turns green for everyone at once, causing confusion. The authors had to develop a special "traffic control" method to figure out what happens when the fluids hit this confusing spot.

3. The "Oil Bank" (The Treasure Chest)

One of the most exciting findings in the paper is the oil bank.

• The Analogy: Imagine you are pushing a crowd of people (oil) through a door. Sometimes, instead of everyone spreading out evenly, the people bunch up into a tight, dense group right in front of the door before moving through.
• The Result: The authors found that under certain conditions (specifically when injecting a mix of foamed gas and water), the oil doesn't just trickle out; it forms a concentrated "bank" or a thick wave of oil that moves ahead of the gas.
• Why it matters: This is great news for oil recovery. A concentrated oil bank means you can collect more oil at once, rather than having it scattered and hard to find. The paper provides a mathematical formula to predict exactly when and where this "oil bank" will form.

4. The "Traffic Rules" (Wave Types)

The authors classified the movement of fluids into different types of "waves," similar to how traffic moves:

• Rarefaction Waves: Like a crowd spreading out smoothly when a door opens. The fluids spread out gradually.
• Shock Waves: Like a sudden traffic jam forming instantly. The fluids crash together into a sharp boundary.
• Composite Waves: A mix of both, where the crowd spreads out a bit and then suddenly jams.
• Non-Classical Waves: These are the tricky ones that happen near the "speed trap" (umbilic point). They don't follow the standard rules of traffic flow and require special math to understand.

5. The "Proof" (Validation)

The authors didn't just draw pretty pictures; they proved their math works.

• The Test: They took their mathematical predictions and ran them through a computer simulation (a digital version of the sponge).
• The Result: The computer simulation matched their math perfectly. They also compared their results to other studies and found that their "traffic map" agreed with real-world observations of how foam moves oil.

Summary

In short, this paper is a user manual for the physics of foam in oil wells.

• It explains how to predict the movement of gas, water, and oil when foam is used.
• It solves a tricky mathematical puzzle where the usual rules don't apply (the umbilic point).
• It identifies the specific conditions needed to create an oil bank, a phenomenon that helps engineers get more oil out of the ground efficiently.

The authors emphasize that their work helps improve the computer programs engineers use to design oil recovery projects, making those projects more accurate and reliable. They did not claim to invent a new chemical or a new drilling technique; rather, they provided the mathematical "blueprint" to understand how existing foam techniques behave in complex situations.

Technical Summary

Technical Summary: The Riemann Problem for Three-Phase Foam Flow in Porous Media

Problem Statement
Gas injection in porous media is a critical technique for Enhanced Oil Recovery (EOR), aquifer remediation, and Carbon Capture, Utilization, and Storage (CCUS). However, these applications often suffer from excessive gas mobility, leading to fingering and reduced displacement efficiency. Foam injection is a promising solution to control gas mobility by forming stable lamellae. Despite its industrial relevance, the mathematical modeling of three-phase foam flow (water, oil, and foamed gas) is highly complex due to non-linear interactions and intricate wave structures. Numerical simulations of such systems often lack rigorous convergence analysis. Consequently, there is a need for analytical solutions to the Riemann problem (displacement of one state by another) to validate numerical simulators, improve physical understanding, and calibrate models. A specific challenge in this domain is the presence of an "umbilic point" where characteristic wave velocities coincide, complicating the classification of stable wave structures, particularly when gas viscosity exceeds that of oil and water.

Methodology
The authors formulate a mathematical model for one-dimensional, horizontal, incompressible three-phase foam flow. Key assumptions include:

• Local Equilibrium: Foam texture is assumed to be at its maximum level, resulting in a constant Mobility Reduction Factor (MRF) for the gas phase.
• Relative Permeability: The system utilizes the Corey model with quadratic exponents ($n_w = n_o = n_g = 2$) to describe relative permeabilities, a choice that simplifies analysis while capturing non-linear dynamics.
• Governing Equations: The conservation of mass for water and oil leads to a system of hyperbolic partial differential equations. The system is non-strictly hyperbolic due to the existence of an umbilic point within the saturation triangle.

To solve the Riemann problem, the authors employ Non-Classical Conservation Law Theory. The methodology involves:

1. Wave Curve Construction: Using a global continuation algorithm (implemented via the ELI software package) to construct forward and backward wave curves (rarefactions, shocks, and composite waves). This approach is necessary because traditional local continuation methods fail in the presence of non-local branches of the Hugoniot locus.
2. Admissibility Criteria: The study adopts the viscous profile criterion to select physically meaningful shock solutions. This allows for the identification of non-classical waves, specifically undercompressive (u-shocks) and transitional rarefactions, which occur along invariant segments of the saturation triangle.
3. Classification: The saturation triangle is partitioned into specific regions ($\Gamma_1$ through $\Gamma_8$) based on bifurcation loci and inflection points. For each region, the authors systematically construct the solution structure for various left (injection) and right (reservoir) states, identifying sequences of waves that satisfy velocity compatibility.

Key Contributions

• Complete Classification: The paper provides a comprehensive classification of Riemann problem solutions for the injection of foamed gas and water mixtures under a wide range of initial conditions, specifically where gas viscosity is high (due to foam).
• Non-Classical Wave Analysis: The authors explicitly characterize the role of non-classical waves (undercompressive shocks) in connecting states in the presence of an umbilic point near the gas vertex. They define the conditions under which these waves are admissible and how they alter the wave sequence compared to classical solutions.
• Oil Bank Formation Criteria: A significant contribution is the analytical derivation of conditions leading to oil bank formation (a temporary increase in oil saturation between two shock fronts). The authors prove that oil banks form when the right state $R$ lies below the f-inflection locus and the left state $L$ is within a specific interval $[G, L_R)$.
• Validation: The analytical estimates are validated through direct numerical simulations using an implicit finite-difference scheme, showing qualitative agreement with the constructed wave structures.

Results
The study analyzes four specific industrial scenarios to demonstrate the applicability of the derived solutions:

1. Foamed Wet Injection (High Water Saturation): The solution involves an s-composite wave followed by an f-shock, resulting in the formation of an oil bank. This matches qualitative behaviors observed in more complex models in the literature.
2. Foamed Wet Injection (High Oil Saturation): In this scenario, the solution consists of composite waves involving rarefactions and shocks but does not result in an oil bank, as the oil saturation profile increases monotonically.
3. Dry Foam Injection (CO2/Nitrogen): The solution features a u-composite wave (rarefaction followed by an undercompressive shock) and an f-shock. An oil bank forms, and the gas wavefront is observed to move faster than in wet injection scenarios.
4. FAWAG (Foam-Assisted Water-Alternating-Gas): This scenario involves a nearly pure gas injection into an oil-rich reservoir. The solution includes a complex sequence of s-composite, u-composite, and f-shock waves. No oil bank is formed in this specific configuration.

The results confirm that injecting a two-phase fluid mixture (foamed gas and water) can enhance oil displacement efficiency through oil bank formation, a phenomenon dependent on the specific location of the initial and injection states within the saturation triangle.

Significance
The paper claims that its analytical solutions serve as a robust tool for:

• Calibrating Numerical Simulators: Providing exact solutions to test the accuracy and convergence of complex foam flow simulators used in the oil industry.
• Uncertainty Quantification: Offering a framework to analyze the impact of variations in physical parameters (such as viscosity and MRF) on displacement efficiency.
• Physical Insight: Improving the general understanding of wave interactions in three-phase foam flow, particularly the role of non-classical waves and the conditions for oil bank formation.

The authors emphasize that their analysis encompasses realistic viscosity parameters found in industrial applications (e.g., Pre-salt reservoirs) and that the solution structures remain stable under variations in these parameters, provided the umbilic point remains within a specific region defined by the viscosity ratios. The work complements previous studies by extending the classification to cases where the umbilic point is near the gas vertex, a scenario relevant to high-viscosity foam applications.