An inverse problem arising from the displacement of oil by water in porous media

Fernandez-Berdaguer, Elena M.; Savioli, Gabriela B.

doi:10.1016/j.apnum.2009.04.009

Cited by 7 publications

(7 citation statements)

References 9 publications

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“…(For the minimization problem, the more preferable constraints b i ≥ 0 could also have been used, but this leads to different explicit formulas for ⌣ η −1 i depending on whether b i = 0 or b i > 0 holds after the optimization.) Since a i < 0 if we impose (30), all functions ⌣ h ′′ i are decreasing. This fact, together with the continuity constraints (23), implies that all inequalities (29) can be replaced by the single constraint…”

Section: 2mentioning

confidence: 99%

“…Related works. Inverse problems for scalar conservation laws of type (1), where (a portion of) the flux function f b (or −f b ) is identified without any assumed parametric form, can be found in [30,40,41,53]. James and Sepúlveda [40] analyze the minimization of a nonstandard cost function which contains the difference between observed data and the solution of (1) at a fixed time point.…”

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confidence: 99%

“…Kang and Tanuma [41] deal with the case when −f ′′ b (φ) > C > 0 for some constant C and the identification of f b is made from observations of the shock formed for large time for initial data with compact support. Fernández-Berdaguer and Savioli [30,53] consider the displacement of oil by water in petroleum reservoirs where the oil fractional flux function is to be identified. This is done with a relatively advanced method including a least-squares technique in functional spaces.…”

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confidence: 99%

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Convexity-preserving flux identification for scalar conservation laws modelling sedimentation

Bürger

Diehl

2013

Inverse Problems

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The sedimentation of a suspension of small particles dispersed in a viscous fluid can be described by a scalar, nonlinear conservation law, whose flux function usually has one inflection point. The identification of the flux function is a problem of theoretical interest and practical importance for the implementation of plant-scale simulators of continuous sedimentation. For a real suspension, the Kynch test and the Diehl test, which are based on an initially homogenous suspension either filling the whole settling column or being initially located above clear liquid, respectively, provide data points that represent curved (convex or concave, respectively) suspension-supernate interfaces from which it is possible to reconstruct portions of the flux function to either side of the inflection point. Several functional forms can be employed to generate a provably convex or concave, twice differentiable accurate approximation of these data points via the solution of a constrained least-squares minimization problem. The resulting spline-like estimated trajectory can be converted into an explicit formula for the flux function. It is proved that the inverse problem of flux identification solved this way has a unique solution. The problem of gluing together the portions of the flux function from the Kynch and Diehl tests is addressed. Examples involving synthetic data are presented.

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Section: 2mentioning

confidence: 99%

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Convexity-preserving flux identification for scalar conservation laws modelling sedimentation

Bürger

Diehl

2013

Inverse Problems

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“…Effect of the variations of the parameters to identify to the reservoirs pressures' evolution during the pulse test. et al [48] using an inverse method of the prediction-correction type, based on parameter-toinput mapping. Zentar et al [49] used a gradient-based inverse method for the interpretation of pressuremeter tests to identify the modified Cam-Clay parameters, whereas Rangeard et al [50] used the same inverse method to simultaneously identify mechanical parameters and the permeability of a soil.…”

Section: The Pulse Test As a Parameter Identification Problemmentioning

confidence: 99%

Fully coupled poromechanical back analysis of the pulse test by inverse method

Giot¹,

Giraud²,

Auvray³

et al. 2011

Num Anal Meth Geomechanics

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SUMMARYIn this paper, we investigate the pulse test, which is usually considered as efficient for measuring the permeability of weakly permeable porous media. The pulse is first analyzed and we show that it is a fully poromechanical coupled problem. Owing to those couplings, the problem is 2D-axisymmetrical, rather than 1D as is usually considered to be the case. As a consequence, the 1D solutions, for example under constant mean stress hypothesis, although giving good approximates of permeability and storage coefficient, are not rigorous and an enhanced back analysis of the test requires 2D solutions. Therefore, no analytical solution exists, and the problem has to be solved using 2D-axisymmetrical numerical modelings of the pulse test. The finite element method is considered in this paper. We then proceed to formulate the pulse test back analysis as a parameter identification problem, and we focus on intrinsic permeability, Biot coefficient, drained Young's modulus and reservoir compressibility levels. This parameter identification problem is solved by an inverse method consisting of the minimization of a cost-functional, through a gradient-based algorithm. This new method of interpretation of the pulse test is finally applied to laboratory tests on Meuse/Haute-Marne argillite and is shown to give encouraging results.

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“…We often discuss these quantities indirectly through known data, that is, to formulate such problems in the form of inverse problems [14,7]. It can be seen from this that inverse problems are the products of the rapid development of science and engineering, which have been widely used in various fields such as geological engineering [25,10], medicine [4], environment [31], telemetry [28], control [2] and so on.…”

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Accelerating the Bayesian inference of inverse problems by using data-driven compressive sensing method based on proper orthogonal decomposition

Xiong

Chen

Jun

et al. 2021

era

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In Bayesian inverse problems, using the Markov Chain Monte Carlo method to sample from the posterior space of unknown parameters is a formidable challenge due to the requirement of evaluating the forward model a large number of times. For the purpose of accelerating the inference of the Bayesian inverse problems, in this work, we present a proper orthogonal decomposition (POD) based data-driven compressive sensing (DCS) method and construct a low dimensional approximation to the stochastic surrogate model on the prior support. Specifically, we first use POD to generate a reduced order model. Then we construct a compressed polynomial approximation by using a stochastic collocation method based on the generalized polynomial chaos expansion and solving an l 1 -minimization problem. Rigorous error analysis and coefficient estimation was provided. Numerical experiments on stochastic elliptic inverse problem were performed to verify the effectiveness of our POD-DCS method.

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An inverse problem arising from the displacement of oil by water in porous media

Cited by 7 publications

References 9 publications

Convexity-preserving flux identification for scalar conservation laws modelling sedimentation

Convexity-preserving flux identification for scalar conservation laws modelling sedimentation

Fully coupled poromechanical back analysis of the pulse test by inverse method

Accelerating the Bayesian inference of inverse problems by using data-driven compressive sensing method based on proper orthogonal decomposition

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