Global weak solutions for a compressible gas-liquid model with well-formation interaction

Evje, Steinar

doi:10.1016/j.jde.2011.07.013

Cited by 34 publications

(25 citation statements)

References 28 publications

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“…In a recent paper [5] we studied a similar model, but where the well-formation interaction was characterized by a rate function A(x, t) assumed to possess certain properties like L ∞ ([0, 1]) boundedness and H 1 ([0, 1]) regularity. More precisely, the model took the following form:…”

Section: Introductionmentioning

confidence: 99%

“…In this sense the model (1) relies on new arguments compared to the model without well-formation interaction terms [9,10,20,21]. It is also quite different from the arguments used in [5], where we take advantage of the fact that we know that the term A(x, t) in (7) is pointwise bounded.…”

Section: Introductionmentioning

confidence: 99%

“…As this gas ascends in the well it will typically experience a lower pressure. This leads to decompression of the gas, which in turn potentially can provoke blow-out-like scenarios; see [1,5,6] and references therein for more details.…”

Section: Introductionmentioning

confidence: 99%

“…Since the momentum is given only for the mixture, we need an additional closure law which connects the two phase fluid velocities. For more general information concerning two-phase flow dynamics we refer the reader to [3,5] and references therein. See also [22,23,2] for other results related to the drift-flux model.…”

Section: Introductionmentioning

confidence: 99%

“…The price to pay is that the resulting model takes a more complicated form, as we will see below. In the work [5] we had to impose a constraint of the form…”

Section: Introductionmentioning

confidence: 99%

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A Compressible Two-Phase Model with Pressure-Dependent Well-Reservoir Interaction

Evje¹

2013

SIAM J. Math. Anal.

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Abstract. This paper deals with a two-phase compressible gas-liquid model relevant for modeling of gas-kick flow scenarios in oil wells. To make the model more realistic we include a natural pressure-dependent well-formation interaction term allowing for modeling of dynamic gas influx/efflux. More precisely, the interaction between well and surrounding formation is controlled by a term of the form A = qw(Pw − P ) which appears in the gas continuity equation where qw is a rate constant, and Pw is a critical pressure, whereas P is pressure in the well. Consequently, an additional coupling mechanism is added to the mass and momentum equations. We obtain a global existence result for the new model. One consequence of the existence result is that as long as the well initially is filled with a mixture of gas and liquid, the system will regulate itself (in finite time) in such a way that there does not exist any point along the well where all the gas vanishes, e.g., by escaping into the formation. Similarly, the result guarantees that neither will any pure gas region appear in finite time, despite that gas is free to enter the well from the formation as long as the well pressure P is lower than the critical pressure Pw.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The price to pay is that the resulting model takes a more complicated form, as we will see below. In the work [5] we had to impose a constraint of the form…”

Section: Introductionmentioning

confidence: 99%

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A Compressible Two-Phase Model with Pressure-Dependent Well-Reservoir Interaction

Evje¹

2013

SIAM J. Math. Anal.

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Large time behavior of solutions to the inviscid liquid‐gas two‐phase flow model

Wang

2022

Math Methods in App Sciences

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The purpose of this paper is to gain some insight into the existence theory and large time behavior of the inviscid liquid-gas two-phase flow model in R 3 . Main focus is on the influences of the damping on the qualitative behavior of solutions of the corresponding Cauchy problem under only the smallness assumption on H 3 -norm of the initial perturbation. The energy method combined with the low frequency and high frequency decomposition is used to obtain the desired decay of the solution and hence global existence. As a byproduct, L 2 -L 2 decay rates are obtained for the solution and its derivatives. This result implies that the friction effect of the damping contributes to enhance the decay rate of the velocity, even if there is no the additional L p -norm boundedness condition of the initial perturbation.

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Optimal decay rates of a nonconservative compressible two‐phase fluid model

Wang

et al. 2023

Z Angew Math Mech

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We are concerned with the time decay rates of strong solutions to a nonconservative compressible viscous two‐phase fluid model in the whole space . Compared to the previous related works, the main novelty of this paper lies in the fact that it provides a general framework that can be used to extract the optimal decay rates of the solution as well as its all‐order spatial derivatives from one‐order to the highest‐order, which are the same as those of the heat equation. Furthermore, for well‐chosen initial data, we also show the lower bounds on the decay rates. Our methods mainly consist of Hodge decomposition, low‐ and high‐frequency decomposition, delicate spectral analysis, and energy method based on finite induction.

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Global weak solutions for a compressible gas-liquid model with well-formation interaction

Cited by 34 publications

References 28 publications

A Compressible Two-Phase Model with Pressure-Dependent Well-Reservoir Interaction

A Compressible Two-Phase Model with Pressure-Dependent Well-Reservoir Interaction

Large time behavior of solutions to the inviscid liquid‐gas two‐phase flow model

Optimal decay rates of a nonconservative compressible two‐phase fluid model

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