Time asymptotics of non-darcy flows controlled by total flux on the boundary

Aulisa, Eugenio; Bloshanskaya, Lidia; Ibragimov, Akif

doi:10.1007/s10958-012-0875-3

Cited by 6 publications

(32 citation statements)

References 14 publications

(18 reference statements)

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“…Thus in order to justify the upscaling criteria (23) for general case one should prove convergence of the corresponding time dependent quantity to the time independent one. This property was obtained in [12] and [13] under certain conditions on the boundary data. Namely, let…”

Section: Coarse Scale Equation In Case Of Slightly Compressible Fluidmentioning

confidence: 75%

Upscaling of Forchheimer flows

Aulisa¹,

Bloshanskaya²,

Efendiev³

et al. 2014

Advances in Water Resources

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In this work we propose upscaling method for nonlinear Forchheimer flow in highly heterogeneous porous media. The generalized Forchheimer law is considered for incompressible and slightly-compressible single-phase flows. We use recently developed analytical results [1] and write the resulting system in terms of a degenerate nonlinear flow equation for the pressure with the nonlinearity that depends on the pressure gradient. The coarse scale parameters for the steady state problem are determined so that the volumetric average of velocity of the flow in the domain on fine scale and on coarse scale are close enough. A flow-based coarsening approach is used, where the equivalent permeability tensor is first evaluated following the streamline of the existing linear cases, and successively modified in order to take into account the nonlinear effects. Compared to previous works [2,3], our approach relies on recent analytical results of Aulisa et al.[1] and combines it with rigorous mathematical upscaling theory for monotone operators. The developed upscaling algorithm for nonlinear steady state problems is effectively used for variety of heterogeneities in the domain of computation. Direct numerical computations for average velocity and productivity index justify the usage of the coarse scale parameters obtained for

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Section: Coarse Scale Equation In Case Of Slightly Compressible Fluidmentioning

confidence: 75%

Upscaling of Forchheimer flows

Aulisa¹,

Bloshanskaya²,

Efendiev³

et al. 2014

Advances in Water Resources

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“…In case of slightly compressible fluid we studied different properties of g-Forchheimer equations in [2,4,14]. In this paper we extend the results of our work [4] on asymptotic behavior of the pressure function to the case when the degree of the g-polynomial is arbitrary. In case of ideal gas we discuss both numerical and analytical results for the time dynamics of the solution of the corresponding parabolic equation.…”

mentioning

confidence: 69%

“…While in general porosity depends on pressure, see [21,26,8], we only consider it to be a function of spatial variable x. In case of slightly compressible fluid we studied different properties of g-Forchheimer equations in [2,4,14]. In this paper we extend the results of our work [4] on asymptotic behavior of the pressure function to the case when the degree of the g-polynomial is arbitrary.…”

mentioning

confidence: 99%

Well productivity index for compressible fluids and gases

Aulisa¹,

Bloshanskaya

Ibragimov³

2016

EECT

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We discuss the notion of the well productivity index (PI) for the generalized Forchheimer flow of fluid through porous media. The PI characterizes the well capacity with respect to drainage area of the well and in general is time dependent. In case of the slightly compressible fluid the PI stabilizes in time to the specific value, determined by the so-called pseudo steady state solution, [5, 3, 4]. Here we generalize our results from [4] in case of arbitrary order of the nonlinearity of the flow. In case of the compressible gas flow the mathematical model of the PI is studied for the first time. In contrast to slightly compressible fluid the PI stays "almost" constant for a long period of time, but then it blows up as time approaches the certain critical value. This value depends on the initial data (initial reserves) of the reservoir. The "greater" are the initial reserves, the larger is this critical value. We present numerical and theoretical results for the time asymptotic of the PI and its stability with respect to the initial data.

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“…In general, the functional PI is time dependent. However if the production rate Q(t) stabilizes over time to a constant value Q, then under certain conditions on the boundary data the PI as well stabilizes over time to specific constant value regardless of the initial pressure distribution (see [21,15,20,18,19]). This value is determined by the special pressure distribution, called the pseudo-steady state (PSS) solution such that…”

Section: Background and Problem Descriptionmentioning

confidence: 99%

“…the well-flux becomes constant) the PI becomes constant independent from the production history or even the operating conditions (see references above). The corresponding flow regime is called pseudo-steady state (PSS) and PSS pressure, velocity and PI serve as pseudo-attractor to the transient one, [15,17,18,19]. From reservoir engineering point of view the PSS regime is attained at the time when the perturbation from the well reaches the exterior boundary of the well drainage area.…”

Section: Introductionmentioning

confidence: 99%

Productivity Index for Darcy and Pre-/Post-Darcy Flow (Analytical Approach)

et al. 2017

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We investigate the impact of nonlinearity of high and low velocity flows on the well productivity index (PI). Experimental data shows the departure from the linear Darcy relation for high and low velocities. High-velocity (post-Darcy) flow occurring near wells and fractures is described by Forchheimer equations and is relatively well-studied. While low velocity flow receives much less attention, there is multiple evidence suggesting the existence of pre-Darcy effects for slow flows far away from the well. This flow is modeled via pre-Darcy equation. We combine all three flow regimes, pre-Darcy, Darcy and post-Darcy, under one mathematical formulation dependent on the critical transitional velocities. This allows to use our previously developed framework to obtain the analytical formulas for the PI for the cylindrical reservoir. We study the impact of pre-Darcy effect on the PI of steady-state flow depending on the well-flux and the parameters of the equations.

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Time asymptotics of non-darcy flows controlled by total flux on the boundary

Cited by 6 publications

References 14 publications

Upscaling of Forchheimer flows

Upscaling of Forchheimer flows

Well productivity index for compressible fluids and gases

Productivity Index for Darcy and Pre-/Post-Darcy Flow (Analytical Approach)

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