The role of sonic shocks between two- and three-phase states in porous media

Castañeda, Pablo; Furtado, Frederico

doi:10.1007/s00574-016-0134-1

Cited by 4 publications

(11 citation statements)

References 6 publications

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“…Such models comprise what we call the second class. Preliminary related work of mathematical nature appeared in a preprint [12] and in conference proceedings [13,14]. The present paper contributes the solution for convex Corey models, the second class.…”

Section: Introductionmentioning

confidence: 99%

On a universal structure for immiscible three-phase flow in virgin reservoirs

et al. 2016

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We discuss the solution for commonly used models of the flow resulting from the injection of any proportion of three immiscible fluids such as water, oil, and gas in a reservoir initially containing oil and residual water. The solutions supported in the universal structure generically belong to two classes, characterized by the location of the injection state in the saturation triangle. Each class of solutions occurs for injection states in one of the two regions, separated by a curve of states for most of which the interstitial speeds of water and gas are equal. This is a separatrix curve because on one side water appears at breakthrough, while gas appears for injection states on the other side. In other words, the behavior near breakthrough is flow of oil and of the dominant phase, either water or gas; the nondominant phase is left behind. Our arguments are rigorous for the class of Corey models with convex relative permeability functions. They also hold for Stone's interpolation I model [5]. This description of the universal structure of solutions for the injection problems is valid for any values of phase viscosities. The inevitable presence of an umbilic point (or of an elliptic region for the Stone model) seems to be the cause of this universal solution structure. This universal structure was perceived recently in the particular case of quadratic Corey relative permeability models and with the injected state consisting of a mixture of water and gas but no oil [5]. However, the results of the present paper are more general in two ways. First, they are valid for a set of permeability functions that is stable under perturbations, the set of convex permeabilities. Second, they are valid for the injection of any proportion of three rather than only two phases that were the scope of [5].

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Section: Introductionmentioning

confidence: 99%

On a universal structure for immiscible three-phase flow in virgin reservoirs

et al. 2016

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“…at the value ℓ * for the Welge point it is actually satisfied λ s (Γ (ℓ * )) = f 2 (Γ (ℓ * ))/ℓ * , as proven in [9]. (The related Bethe-Wendroff point S * = Γ (ℓ * ) belongs to the extension boundary.)…”

Section: Example 2: Choosing a Parametrization Coordinatementioning

confidence: 73%

“…The continuity of the EFF itself holds because of the adding of f 1 (R) in both liftings (7) and (9): from Lemma 1 we have that f(ℓ) = f 1 Γ (ℓ) holds in particular at U * , and since from (9) we have that f(u * 1 ) = f 1 (U * ) holds, also the EFF continuity. From rarefaction to shock curve.…”

Section: The Complete Eff Constructionmentioning

confidence: 92%

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Explicit Construction of Effective Flux Functions for Riemann Solutions

Castañeda

2018

Springer Proceedings in Mathematics &Amp; Statistics

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“…The path from J 2 is connected to I by an admissible 2‐Lax shock. Given that J 1 , J 2 , and J 3 all fit the same injected fractional flows, the path from J 1 or J 3 would jump first to J 2 with zero velocity and then follow the same track as the path from J 2 to I (Castañeda & Furtado, ). However, any perturbation that takes I off the boundary curve would always result in negative velocities along any sort of path from J 2 , which is physically unacceptable.…”

Section: Displacing State Among Multiple Steady Statesmentioning

confidence: 99%

Three‐Phase Fractional‐Flow Theory of Foam‐Oil Displacement in Porous Media With Multiple Steady States

Tang

Castañeda

Marchesin

et al. 2019

Water Resources Research

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Understanding the interplay of foam and nonaqueous phases in porous media is key to improving the design of foam for enhanced oil recovery and remediation of aquifers and soils. A widely used implicit-texture foam model predicts phenomena analogous to cusp catastrophe theory: The surface describing foam apparent viscosity as a function of fractional flows folds backwards on itself. Thus, there are multiple steady states fitting the same injection condition J defined by the injected fractional flows. Numerical simulations suggest the stable injection state among multiple possible states but do not explain the reason. We address the issue of multiple steady states from the perspective of wave propagation, using three-phase fractional-flow theory. The wave-curve method is applied to solve the two conservation equations for composition paths and wave speeds in 1-D foam-oil flow. There is a composition path from each possible injection state J to the initial state I satisfying the conservation equations. The stable displacement is the one with wave speeds (characteristic velocities) all positive along the path from J to I. In all cases presented, two of the paths feature negative wave velocity at J; such a solution does not correspond to the physical injection conditions. A stable displacement is achieved by either the upper, strong-foam state, or lower, collapsed-foam state but never the intermediate, unstable state. Which state makes the displacement depends on the initial state of a reservoir. The dependence of the choice of the displacing state on initial state is captured by a boundary curve. Plain Language SummaryFoam has unique microstructure and reduces gas mobility significantly. Foam injection into geological formations has broad engineering applications: removal of nonaqueous phase liquid contaminants in aquifers and soils, oil displacement in reservoirs, and carbon storage. Key to the success of foam is foam stability in the presence of oil or nonaqueous phase liquid. An experimentally validated foam model describes foam properties as a function of water, oil, and gas saturations. This model predicts that some injected fractional flows of phases correspond to multiple possible injection states with different saturations: strong-foam state with low mobility, intermediate state, and collapsed-foam state with high mobility. We show how to determine the unique displacing state, using three-phase fractional-flow theory and the wave-curve method. A physically acceptable displacing state is the one that gives only positive wave velocities. The choice of the displacing state depends on the initial state; the nature of the dependence is captured by a boundary curve. If the collapsed-foam state makes a displacement, that means ineffective gas-mobility control and, even in the absence of viscous instability, very slow oil displacement. Our findings and approach presented can help to predict the displacing state for a given initial state in geological formations.

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The role of sonic shocks between two- and three-phase states in porous media

Cited by 4 publications

References 6 publications

On a universal structure for immiscible three-phase flow in virgin reservoirs

On a universal structure for immiscible three-phase flow in virgin reservoirs

Explicit Construction of Effective Flux Functions for Riemann Solutions

Three‐Phase Fractional‐Flow Theory of Foam‐Oil Displacement in Porous Media With Multiple Steady States

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