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There is a correspondence between flow in a reservoir and large scale permeability trends. This correspondence can be derived by constraining reservoir models using observed production data. One of the challenges in deriving the permeability distribution of a field using production data involves determination of the scale of resolution of the permeability. The Adaptive Multiscale Estimation (AME) seeks to overcome the problems related to choosing the resolution of the permeability field by a dynamic parameterisation selection. The standard AME uses a gradient algorithm in solving several optimisation problems with increasing permeability resolution. This paper presents a hybrid algorithm which combines a gradient search and a stochastic algorithm to improve the robustness of the dynamic parameterisation selection. At low dimension, we use the stochastic algorithm to generate several optimised models. We use information from all these produced models to find new optimal refinements, and start out new optimisations with several unequally suggested parameterisations. At higher dimensions we change to a gradient-type optimiser, where the initial solution is chosen from the ensemble of models suggested by the stochastic algorithm. The selection is based on a predefined criterion. We demonstrate the robustness of the hybrid algorithm on sample synthetic cases, which most of them were considered insolvable using the standard AME algorithm.There is a correspondence between flow in a reservoir and large scale permeability trends. This correspondence can be derived by constraining reservoir models using observed production data. One of the challenges in deriving the permeability distribution of a field using production data involves determination of the scale of resolution of the permeability. The Adaptive Multiscale Estimation (AME) seeks to overcome the problems related to choosing the resolution of the permeability field by a dynamic parameterisation selection. The standard AME uses a gradient algorithm in solving several optimisation problems with increasing permeability resolution. This paper presents a hybrid algorithm which combines a gradient search and a stochastic algorithm to improve the robustness of the dynamic parameterisation selection. At low dimension, we use the stochastic algorithm to generate several optimised models. We use information from all these produced models to find new optimal refinements, and start out new optimisations with several unequally suggested parameterisations. At higher dimensions we change to a gradient-type optimiser, where the initial solution is chosen from the ensemble of models suggested by the stochastic algorithm. The selection is based on a predefined criterion. We demonstrate the robustness of the hybrid algorithm on sample synthetic cases, which most of them were considered insolvable using the standard AME algorithm.

We consider two-phase fluid flow through heterogeneous cores with given injection rates. In these problems, measured pressure drop over the core is an important information source for recovering permeability. This gives rise to the inverse problem of inverting the pressure data, p(t), into knowledge of the permeability distribution, K(x), in the core. Given a global pressure model, a description of the resolution power of pressure data with respect to permeability identification is possible. We examine how the flow pattern – width and front structure of the two-phase zone – affects the resolution power. We further briefly discuss the value of additional data, e.g., lateral saturation data, for the ability to recover permeability.We consider two-phase fluid flow through heterogeneous cores with given injection rates. In these problems, measured pressure drop over the core is an important information source for recovering permeability. This gives rise to the inverse problem of inverting the pressure data, p(t), into knowledge of the permeability distribution, K(x), in the core. Given a global pressure model, a description of the resolution power of pressure data with respect to permeability identification is possible. We examine how the flow pattern – width and front structure of the two-phase zone – affects the resolution power. We further briefly discuss the value of additional data, e.g., lateral saturation data, for the ability to recover permeability.

Adaptive multiscale estimation of a spatially dependent diffusion function is considered. When utilizing pressure data, the method has been successful in identifying coarse-scale structures of the diffusion function. In this article the method is used with pressure and fluid rate data as information sources. The estimate is expected to be improved when additional data are included. However, this expectation is not always met, and in some cases the simultaneous use of these data types in the estimation might even cause problems. The reason for the lack of improvement is investigated together with methods for successful estimation utilizing pressure and fluid rates.

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.

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.

We consider accuracy assessment for the inverse problem of recovery of unknown coefficient functions in differential equations from data containing random errors. The set of PDEs constituting the current forward model describes a special case of two-phase porous-media flow. We are concerned mainly with two issues. (1) When is it valid to calculate parameter accuracies for the current nonlinear estimation problem by a linearized method, linearized covariance analysis (LCA)? (2) Can the validity of LCA be assessed without performing an accurate, but computationally very expensive, Monte Carlo analysis (MCA)? For both issues, special emphasis is put on parameter subsets for which LCA predicts high accuracy. The curvature measures of nonlinearity (CMNs) are a potential alternative to MCA. CMNs are approximate, but considerably less expensive to compute. In this paper, we apply LCA, CMNs and MCA to several instances of the current model. We address issue 1 by comparing LCA and MCA results, and issue 2 by including also CMN results in the analysis. It is found that CMN and MCA results lead to identical and negative conclusions concerning the validity of LCA. However, if the real concern is parameter subsets where LCA predicts high accuracy, these conclusions, based on calculations involving all of the parameters, were often misleading. Use of specially designed subset CMNs is essential to avoid this. A potential explanation, which may have implications also for other parameter estimation problems, is presented.

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.