Jan-Erik Nordtvedt

A new reconstruction algorithm for process tomography

In this work a new reconstruction algorithm for use with oil/gas pipe flow imaging has been developed. Accurate images of representative oil/gas distributions (that is, flow regimes) occurring in flow pipes has been obtained. A capacitance sensor system has been used. In the algorithm the oil/gas distribution has been represented by a set of parameters describing the oil/gas interface. A finite-element-based mathematical simulator of the multi-electrode capacitance sensor system has been developed. The simulator is capable of calculating the capacitances for a set of input parameters (namely for a given oil/gas distribution). The reconstruction algorithm calculates the image by finding the parameters that give fewest discrepancies between calculated and measured capacitances, using an optimization routine. All regimes tested in this work have been successfully reconstructed with the new algorithm, and the reconstructions are compared with images produced by the well-known linear back projection algorithm. The new algorithm has been tested using data from a capacitance imaging system; however, it can in principle be used with other imaging techniques.

Estimation of porous media flow functions

Properties important for describing the flow of multiple fluid phases through porous media are represented as functions of state variables (fluid saturations). A generalized procedure is presented to obtain the most accurate estimates of the multiphase flow functions from the available experimental data. The procedure is demonstrated for several different experimental designs, including a novel experiment in which fluid saturations are measured using nuclear magnetic resonance imaging. A method to evaluate the accuracy of the estimates is presented, and its use for assessing experimental design is demonstrated.

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.

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.

Estimation in the Presence of Outliers: The Capillary Pressure Case

The inversion of laboratory centrifuge data to obtain capillary pressure functions in petroleum science leads to a Volterra integral equation of the first kind with a right-hand side defined by a set of discrete data. The problem is ill-posed in the sense of Hadamard [4]. The discrete data lead to a discretized equation of the form

Calculating the Relative Permeability and the Capillary Pressure Functions from In Situ and Effluent Measurements: An Error Analysis

A\overrightarrow c = \overrightarrow b + \overrightarrow \varepsilon ,

Assessing the validity of a linearized accuracy measure for a nonlinear parameter estimation problem

, where b→ represents the observation vector, A is an ill-conditioned matrix derived from the forward problem, c→ is the coefficients in a representation of the inverse capillary function, i.e., parameters to be determined, and ∈→ is the error vector associated with b→. If ∈→∼N(0,σ2), and satisfies the Gauss-Markov (G-M) conditions, then an estimate, c→ λ, of c→ is BLUE [9]. In the presence of outliers, the G-M conditions and/or the normality assumption can be violated.