The minimum structure solution to the inverse problem

Kitanidis, Peter K.

doi:10.1029/97wr01619

Cited by 35 publications

(24 citation statements)

References 21 publications

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“…Furthermore, it originally required that the flow equations could be linearized (small variance assumption). Only in recent years was this approach extended to quasilinear and even nonlinear problems (e.g., see Kitanidis, 1995Kitanidis, , 1997Kitanidis, , 1998Kitanidis, , 1999.…”

Section: Cokrigingmentioning

confidence: 99%

“…One can examine two limiting cases: in the case where the range of the a priori covariance is very wide, a differential second-order operator comparable to equation 15 intervenes in the regularization term; contrariwise, if the range is so small that the a priori errors over two neighboring meshes are not correlated, the effect, in the absence of log-transmissivity data, is equivalent to a regularization that minimizes the quadratic mean of the log-transmissivities. In hydrogeology, Kitanidis (1997) explored the relation between the geostatistical approach and the so-called Backus-Gilbert smallest solution, and generalized the flatness criterion for one, two, or three dimensions (Kitanidis, 1999).…”

Section: Robustness Regularization and The Cauchy Problemmentioning

confidence: 99%

See 1 more Smart Citation

Four decades of inverse problems in hydrogeology

Marsily¹,

Delhomme²,

Coudrain³

et al. 2000

Theory, Modeling, and Field Investigation in Hydrogeology: A Special Volume in Honor of Shlomo P. Neumans 60th Birthday

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We review the main stages of the evolution of ideas and methods for solving the inverse problem in hydrogeology; i.e., the identification of the transmissivity field in single-phase flow from piezometric data, in mainly steady-state and, occasionally, transient flow conditions. We first define the data needed to solve an inverse problem in hydrogeology, then describe the numerous approaches that have been developed over the past 40 years to solve it, emphasizing the major contributions made by Shlomo P. Neuman. Finally, we briefly discuss fitting processes that start by defining the unknown field as geological images (generated by Boolean or geostatistical methods).The early attempts at solving the inverse problem were direct, i.e., the transmissivity field was directly determined by using stream lines of the flow and inverting the flow equation along these lines. Faced with the poor results obtained in this manner, hydrogeologists have tried many different ways of minimizing the balance error representing an integral of the mass-balance error for each mesh for a given transmissivity field. These attempts were accompanied by constraints imposed on the transmissivity field in order to avoid instabilities.The idea then emerged that the unknown field should reproduce the local observations of the pressure at the measurement points instead of minimizing a balance error. Second, it should also satisfy a condition of plausibility, which means that the transmissivity field obtained through the inverse solution should not deviate too far from an a priori estimate of the real transmissivity field. This a priori notion led to the inclusion of a Bayesian approach resulting in the search for an optimal solution by maximum likelihood, as expounded later.Simultaneously, the existence of locally measured values in the transmissivity field (obtained by pumping tests) allowed geostatistical methods to be used in the formulation of the problem; the result of this innovation was that three major approaches came into being: (1) the definition of the a priori transmissivity field by kriging; (2) the method of cokriging;(3) the pilot point method. Furthermore, geostatistics made it possible to pose the inverse problem in a stochastic framework and to solve an ensemble of possible and equally probable fields, each of them equally acceptable as a solution.

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Section: Cokrigingmentioning

confidence: 99%

Section: Robustness Regularization and The Cauchy Problemmentioning

confidence: 99%

Four decades of inverse problems in hydrogeology

Marsily¹,

Delhomme²,

Coudrain³

et al. 2000

Theory, Modeling, and Field Investigation in Hydrogeology: A Special Volume in Honor of Shlomo P. Neumans 60th Birthday

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show abstract

“…In subsurface hydrology, however, the geostatistical framework is more common. Kitanidis (1997) and independently Maurer et al (1998) showed that the two approaches are mathematically equivalent to each other.…”

Section: Erdal and O A Cirpka: Joint Inference Of Recharge And Cmentioning

confidence: 99%

Joint inference of groundwater–recharge and hydraulic–conductivity fields from head data using the ensemble Kalman filter

Erdal¹,

Cirpka²

2016

Hydrol. Earth Syst. Sci.

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Abstract.Regional groundwater flow strongly depends on groundwater recharge and hydraulic conductivity. Both are spatially variable fields, and their estimation is an ongoing topic in groundwater research and practice. In this study, we use the ensemble Kalman filter as an inversion method to jointly estimate spatially variable recharge and conductivity fields from head observations. The success of the approach strongly depends on the assumed prior knowledge. If the structural assumptions underlying the initial ensemble of the parameter fields are correct, both estimated fields resemble the true ones. However, erroneous prior knowledge may not be corrected by the head data. In the worst case, the estimated recharge field resembles the true conductivity field, resulting in a model that meets the observations but has very poor predictive power. The study exemplifies the importance of prior knowledge in the joint estimation of parameters from ambiguous measurements.

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“…These methods, usually based on gradient minimization, have been described in the literature. 2 Automated history matching has seen good success in single-phase problems and in matching pressure. However, because of the large number of equations involved in a field-scale problem, the non-linear properties of flow parameters, and the subjective nature of 'matching,' automatic techniques have found limited practical application.…”

Section: Introductionmentioning

confidence: 99%

History Matching Finite Difference Models With 3d Streamlines

Emanuel¹,

Milliken²

1998

Proceedings of SPE Annual Technical Conference and Exhibition

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TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractIn the past several years, 3D streamline simulation (3DSL) software has become available to the industry through commercial vendors, university coop projects, and DOE sponsored research. This paper describes the use of 3D streamlines to assist in history matching conventional finite difference reservoir simulation models.One of the most difficult tasks in history matching individual well performance in a finite difference model is the identification of the grid blocks that affect the performance of a well. The task is exacerbated by the progression to large, complex, geostatistical models. This paper describes a technique that integrates, in a simple computer program, streamline and finite difference simulation to identify the grid blocks that affect each well and to effect changes to the model for history matching. Two case studies are presented: case 1 is a thick, stratified 60 well eolian sandstone producing under pattern waterflood and case 2 is a 70 well carbonate anticline producing under partial aquifer drive. The two cases illustrate the efficiency of the streamline-based code and the potential of the process to greatly simplify history matching.

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The minimum structure solution to the inverse problem

Cited by 35 publications

References 21 publications

Four decades of inverse problems in hydrogeology

Four decades of inverse problems in hydrogeology

Joint inference of groundwater–recharge and hydraulic–conductivity fields from head data using the ensemble Kalman filter

History Matching Finite Difference Models With 3d Streamlines

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