Alv-Arne Grimstad

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.

Analysis of the ensemble Kalman filter for estimation of permeability and porosity in reservoir models

It has lately been reported several successful applications where the ensemble Kalman filter has been used to estimate reservoir properties such as permeability and porosity. However, a thorough investigation of robustness and performance is still missing for this approach. In this paper we aim at filling this gap by studying the robustness of the methodology. One aspect which is investigated is how the filter depends on the initial ensemble. As the initial ensemble is created in a stochastic way, one can not be certain that the results obtained from one run represent the filter performance. Another aspect of interest is how prior information can be used to obtain best possible initial fields. The influence of geostatistical information on the estimated solutions is studied. In addition, the quality of the estimated fields is investigated by evaluating if the estimated static fields are reasonable when treated as the solution of the history matching problem. The estimation technique has been applied to the widely used PUNQ-S3 reservoir model, which is a small size synthetic 3-D reservoir engineering model. Both permeability and porosity are tuned, and measurements consist of well bottom-hole pressures, water cuts and gas-oil ratios. The initial fields are conditioned on the porosities in the gridblocks where the wells are located. By using a synthetic reservoir model it is possible to calculate the uncertainty of forecasts, and compare this with the true solution.

Nonlinearity, Scale, and Sensitivity for Parameter Estimation Problems

Both sensitivity and nonlinearity are important for the efficiency of an estimation algorithm. Knowledge of a general nature on sensitivity and/or nonlinearity for some class of models can perhaps be utilized to improve the estimation efficiency for this class. For an ODE model, a correlation between high nonlinearity, low sensitivity, and small-scale perturbations, has been reported. Also, it was found that representing the unknown function by a multi-scale basis lead to faster estimation convergence than use of a single-scale local basis. This was explained referring to the above-mentioned correlation. Recently, the existence of such a correlation for a large class of nonlinear models, including the above-mentioned ODE model, was found. Here, we further investigate into utilization of the correlation

Design of Two-Phase Displacement Experiments

This paper presents and demonstrates a systematic approach to the selection of experimental designs leading to accurate estimates of relative permeability and capillary pressure functions for two-phase flow in porous media. The objective is to select the most appropriate experimental designs for determining the flow functions accurately within the saturation range covered by the experimental data. The work is based on a linearized covariance analysis. In this analysis we utilize analytical sensitivity coefficients to calculate confidence intervals for the flow functions. These confidence intervals are estimates of the accuracy with which the flow functions can be determined for a given experimental design. We validate the confidence interval estimates through a Monte Carlo study. A previously reported non-linearity measure seems not to be applicable for determining the utility of the linearized covariance analysis for the porous media fluid flow model.

Simultaneous determination of absolute and relative permeabilities

Data from core analyses, such as residual oil saturation and relative permeabilities, are of great importance for proper exploitation of the petroleum resources. Such quantities are typically determined through interpretation of data acquired during some flooding experiment. In such determinations, the absolute permeabilities are typically represented by a single average value, i.e., the core is assumed homogeneous and isotropic. Recent studies, however, show that the validity of such assumptions can be questioned. When using such assumptions analyzing flooding data, the derived relative permeabilities will depend on the actual core sample heterogeneity, i.e., the variation and distribution of the absolute permeability in the core. A better option would therefore be to determine the absolute and relative permeabilities simultaneously from the data, thereby accounting for heterogeneity effects. In this paper we describe and test a method for such determinations, and discuss some results.

Extension of Methodology for the Determination of Two-phase Porous Media Property Functions

This article addresses estimation of porous media property functions, absolute and relative permeabilities, from experiments on core samples. Typically, the properties are taken to be homogeneous. However, recent studies have shown that such an assumption frequently is erroneous, and may indeed lead to large errors in the estimated properties. The property functions are inaccessible to direct measurements, and have to be estimated through an inverse problem. This article describes an algorithm capable of successfully determining the property functions, taking the heterogeneous nature of the porous medium into account. In this article, this methodology is extended in two ways - the estimation algorithm is made more robust and saturation data have been included. Also, for the first time, experimental data are used for verification purposes.