On reconstruction of dynamic permeability and tortuosity from data at distinct frequencies

Ou, Miao‐jung Yvonne

doi:10.1088/0266-5611/30/9/095002

Cited by 12 publications

(2 citation statements)

References 67 publications

(135 reference statements)

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“…For example, if one wants to recover a signal corrupted by a low-pass convolution filter, then one needs to recover an entire function from its measured values on an interval [2,11]. Another large class of inverse problems can be termed 'Dehomogenization' [7,26], where one wants to reconstruct some details of microgeometry from measurements of effective properties of the composite. The idea of reconstruction is based on the analytic properties of effective moduli [3,17,24] of composites.…”

Section: Introductionmentioning

confidence: 99%

Explicit power laws in analytic continuation problems via reproducing kernel Hilbert spaces

Grabovsky¹,

Hovsepyan²

2020

Inverse Problems

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The need for analytic continuation arises frequently in the context of inverse problems. Notwithstanding the uniqueness theorems, such problems are notoriously illposed without additional regularizing constraints. We consider several analytic continuation problems with typical global boundedness constraints that restore well-posedness. We show that all such problems exhibit a power law precision deterioration as one moves away from the source of data. In this paper we demonstrate the effectiveness of our general Hilbert space-based approach for determining these exponents. The method identifies the "worst case" function as a solution of a linear integral equation of Fredholm type. In special geometries, such as the circular annulus or upper half-plane this equation can be solved explicitly. The obtained solution in the annulus is then used to determine the exact power law exponent for the analytic continuation from an interval between the foci of an ellipse to an arbitrary point inside the ellipse. Our formulas are consistent with results obtained in prior work in those special cases when such exponents have been determined.

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Section: Introductionmentioning

confidence: 99%

Explicit power laws in analytic continuation problems via reproducing kernel Hilbert spaces

Grabovsky¹,

Hovsepyan²

2020

Inverse Problems

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“…In the case of the fluid flow through a porous medium subject to a time-harmonic pressure gradient, the permeability depends on the frequency and is referred to as the dynamic permeability. The theory of dynamic permeability is established [5, 8, 12, 22] and further developed by Ou [35].…”

Section: Introductionmentioning

confidence: 99%

Integral representation of hydraulic permeability

Bi¹,

Ou²,

Zhang³

2022

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this paper, we show that the permeability of a porous material (Tartar (1980)) and that of a bubbly fluid (Lipton and Avellaneda. Proc. R. Soc. Edinburgh Sect. A: Math. 114 (1990), 71–79) are limiting cases of the complexified version of the two-fluid models posed in Lipton and Avellaneda (Proc. R. Soc. Edinburgh Sect. A: Math. 114 (1990), 71–79). We assume the viscosity of the inclusion fluid is $z\mu _1$ and the viscosity of the hosting fluid is $\mu _1\in \mathbb {R}^{+}$ , $z\in \mathbb {C}$ . The proof is carried out by the construction of solutions for large $|z|$ and small $|z|$ with an iteration process similar to the one used in Bruno and Leo (Arch. Ration. Mech. Anal. 121 (1993), 303–338) and Golden and Papanicolaou (Commun. Math. Phys. 90 (1983), 473–491) and the analytic continuation. Moreover, we also show that for a fixed microstructure, the permeabilities of these three cases share the same integral representation formula (3.17) with different values of contrast parameter $s:=1/(z-1)$ , as long as $s$ is outside the interval $\left [-\frac {2E_2^{2}}{1+2E_2^{2}},-\frac {1}{1+2E_1^{2}}\right ]$ , where the positive constants $E_1$ and $E_2$ are the extension constants that depend only on the geometry of the periodic pore space of the material.

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Time domain numerical modeling of wave propagation in poroelastic media with Chebyshev rational approximation of the fractional attenuation

Xie

2022

Math Methods in App Sciences

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In this work, we investigate the poroelastic waves by solving the time‐domain Biot‐JKD equation with an efficient numerical method. The viscous dissipation occurring in the pores depends on the square root of the frequency and is described by the Johnson‐Koplik‐Dashen (JKD) dynamic tortuosity/permeability model. The temporal convolutions of order 1/2 shifted fractional derivatives are involved in the time‐domain Biot‐JKD model, causing the problem to be stiff and challenging to be implemented numerically. Based on the best relative Chebyshev approximation of the square‐root function, we design an efficient algorithm to approximate and localize the convolution kernel by introducing a finite number of auxiliary variables that satisfy a local system of ordinary differential equations. The imperfect hydraulic contact condition is used to describe the interface boundary conditions and the Runge‐Kutta discontinuous Galerkin (RKDG) method together with the splitting method is applied to compute the numerical solutions. Several numerical examples are presented to show the accuracy and efficiency of our approach.

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On reconstruction of dynamic permeability and tortuosity from data at distinct frequencies

Cited by 12 publications

References 67 publications

Explicit power laws in analytic continuation problems via reproducing kernel Hilbert spaces

Explicit power laws in analytic continuation problems via reproducing kernel Hilbert spaces

Integral representation of hydraulic permeability

Time domain numerical modeling of wave propagation in poroelastic media with Chebyshev rational approximation of the fractional attenuation

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