Trond Mannseth

Adaptive Multiscale Permeability Estimation

With multiscale permeability estimation one does not select parameterization prior to the estimation. Instead, one performs a hierarchical search for the right parameterization while solving a sequence of estimation problems with an increasing parameterization dimension. In some previous works on the subject, the same refinement is applied all over the porous medium. This may lead to over-parameterization, and subsequently, to unrealistic permeability estimates and excessive computational work. With adaptive multiscale permeability estimation, the new parameterization at an arbitrary stage in the estimation sequence is such that new degrees of freedom are not necessarily introduced all over the porous medium. The aim is to introduce new degrees of freedom only where it is warranted by the data. In this paper, we introduce a novel adaptive multiscale estimation. The approach is used to estimate absolute permeability from two-phase pressure data in several numerical examples.

Sensitivity study of marine CSEM data for reservoir production monitoring

Using numerical and analytical modeling, we assess the feasibility of marine controlled-source electromagnetic (CSEM) data for monitoring the flooding front during water flooding of an oil reservoir. We discuss the ability of time-lapse CSEM data to resolve changes in the electric conductivity small enough to be of interest for monitoring purposes. Measurement and modeling errors are discussed briefly, and analytical calculations comparing time-lapse signals with a certain type of time-lapse modeling errors are performed. Numerical calculations, performed with a volume integral-equation method, are used to study the effect on the electromagnetic (EM) fields of relevant conductivity changes. The numerical investigation includes robustness with respect to survey parameter values, discrimination between different front shapes, and ability to overcome time-lapse error. Simulated signals are found to be strong enough to overtake typical measurement errors and are fairly robust toward perturbations of survey parameters. It is found analytically and numerically that certain modeling errors experience a high degree of time-lapse cancellation.

Combined Adaptive Multiscale and Level-Set Parameter Estimation

We propose a solution strategy for parameter estimation, where we combine adaptive multiscale estimation (AME) and level-set estimation (LSE). The approach is applied to the nonlinear inverse probl...

Parameter sampling capabilities of sequential and simultaneous data assimilation: II. Statistical analysis of numerical results

We assess and compare parameter sampling capabilities of one sequential and one simultaneous Bayesian, ensemble-based, joint state-parameter (JS) estimation method. In the companion paper, part I (Fossum and Mannseth 2014 Inverse Problems 30 114002), analytical investigations lead us to propose three claims, essentially stating that the sequential method can be expected to outperform the simultaneous method for weakly nonlinear forward models. Here, we assess the reliability and robustness of these claims through statistical analysis of results from a range of numerical experiments. Samples generated by the two approximate JS methods are compared to samples from the posterior distribution generated by a Markov chain Monte Carlo method, using four approximate measures of distance between probability distributions. Forward-model nonlinearity is assessed from a stochastic nonlinearity measure allowing for sufficiently large model dimensions. Both toy models (with low computational complexity, and where the nonlinearity is fairly easy to control) and two-phase porous-media flow models (corresponding to down-scaled versions of problems to which the JS methods have been frequently applied recently) are considered in the numerical experiments. Results from the statistical analysis show strong support of all three claims stated in part I.

Permeability estimation with the augmented Lagrangian method for a nonlinear diffusion equation

We consider numerical identification of the piecewise constant permeability function in a nonlinear parabolic equation, with the augmented Lagrangian method. By studying this problem, we aim at also gaining some insight into the potential ability of the augmented Lagrangian method to handle permeability estimation within the full two-phase porous-media flow setting. The identification is formulated as a constrained minimization problem. The parameter estimation problem is reduced to a coupled nonlinear algebraic system, which can be solved efficiently with the conjugate gradient method. The methodology is developed and numerical experiments with the proposed method are presented.

Relation Between Level Set and Truncated Pluri-Gaussian Methodologies for Facies Representation

Truncated Gaussian/pluri-Gaussian representations and level set representations are two methodologies for implicit representation that can be applied to large-scale subsurface structures. Identification of facies in a petroleum reservoir from dynamic well data is a common field of application for these representation methodologies. No comparison of the methodologies has appeared in the literature to date. The paper seeks to improve on this situation by comparing selected level set and truncated Gaussian/pluri-Gaussian representations in detail. Strong similarities are found, in the sense that every truncated Gaussian/pluri-Gaussian representation considered has a level set counterpart. Furthermore, the transition from a truncated Gaussian/pluri-Gaussian representation to the corresponding level set representation is easily accessible. In addition to the comparison aspect, this paper also introduces a novel level set representation—the hierarchical level set representation—that removes a difficulty present in existing level set representations in a shape estimation setting. It is shown that the hierarchical level set representation corresponds to a well-known truncated pluri-Gaussian representation.

Parameter sampling capabilities of sequential and simultaneous data assimilation: I. Analytical comparison

We assess the parameter sampling capabilities of some Bayesian, ensemble-based, joint state-parameter (JS) estimation methods. The forward model is assumed to be non-chaotic and have nonlinear components, and the emphasis is on results obtained for the parameters in the state-parameter vector. A variety of approximate sampling methods exist, and a number of numerical comparisons between such methods have been performed. Often, more than one of the defining characteristics vary from one method to another, so it can be difficult to point out which characteristic of the more successful method in such a comparison was decisive. In this study, we single out one defining characteristic for comparison; whether or not data are assimilated sequentially or simultaneously. The current paper is concerned with analytical investigations into this issue. We carefully select one sequential and one simultaneous JS method for the comparison. We also design a corresponding pair of pure parameter estimation methods, and we show how the JS methods and the parameter estimation methods are pairwise related. It is shown that the sequential and the simultaneous parameter estimation methods are equivalent for one particular combination of observations with different degrees of nonlinearity. Strong indications are presented for why one may expect the sequential parameter estimation method to outperform the simultaneous parameter estimation method for all other combinations of observations. Finally, the conditions for when similar relations can be expected to hold between the corresponding JS methods are discussed. A companion paper, part II (Fossum and Mannseth 2014 Inverse Problems 30 114003), is concerned with statistical analysis of results from a range of numerical experiments involving sequential and simultaneous JS estimation, where the design of the numerical investigation is motivated by our findings in the current paper.

Multiscale estimation with spline wavelets, with application to two-phase porous-media flow

We consider the inverse problem of recovery of unknown coefficient functions in differential equations. The set of PDEs constituting the current forward model describes a special case of two-phase porous-media flow. The focus of the paper is on the influence of different length scales on parameter estimation efficiency. The investigation into these issues is facilitated by applying a multiscale spline wavelet parametrization of the unknown function. Earlier investigations with an ODE forward model found that use of the multiscale Haar parametrization had a positive effect on the estimation efficiency of a quasi-Newton algorithm. Recently, a way to systematically enhance these effects has been suggested. In this paper, we further this approach with the Levenberg-Marquardt algorithm. This results in three variants of the Levenberg-Marquardt algorithm, each incorporating a possibility to enhance multiscale effects. Through numerical experiments with the PDE forward model, we assess the estimation efficiency of the variants when varying the enhancement of multiscale effects.

Domain decomposition Fourier finite element method for the simulation of 3D marine CSEM measurements

We present a novel numerical method based on domain decomposition for the simulation of 3D geophysical marine controlled source electromagnetic (CSEM) measurements. Parts of the computational domain where it is reasonable to represent geoelectric properties in 2D, are discretized combining 2D mixed finite elements (FE) and Fourier expansion. The remaining part is discretized utilizing standard 3D FE methods. The method delivers high-accuracy simulations of marine CSEM problems with arbitrary 3D geometries while it considerably reduces the computational complexity of full 3D FE simulations for typical marine CSEM problems. For the particular scenarios considered in this work, the total CPU time required by the novel method is reduced approximately by a factor of five with respect to that needed by full 3D FE formulations.

Permeability Identification from Pressure Observations: Some Foundations for Multiscale Regularization

The interrelation (denoted SNS) between sensitivity, nonlinearity and scale, associated with the inverse problem of permeability (fluid conductivity) identification from fluid pressure observations in porous-media flow, is considered. The family of models considered includes both single-phase and two-phase flows, with applications to groundwater flow/primary recovery in petroleum reservoirs and to water flooding of petroleum reservoirs, respectively. SNS is important for regularization of both of these ill-posed inverse problems, but so far, SNS has been shown to exist only for single-phase flow. Several multiscale/multiresolution estimation techniques, explicitly or implicitly based on SNS, have, however, been developed and applied to practical permeability estimation both for single- and two-phase flows. In this paper, SNS is shown to exist for one-dimensional, two-phase flow. Moreover, very similar approaches are applied to show the existence of SNS both for single- and two-phase flows. To convey some ...