Martha Lien

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...

A 3D computational study of effective medium methods applied to fractured media

This work evaluates and improves upon existing effective medium methods for permeability upscaling in fractured media. Specifically, we are concerned with the asymmetric self-consistent, symmetric self-consistent, and differential methods. In effective medium theory, inhomogeneity is modeled as ellipsoidal inclusions embedded in the rock matrix. Fractured media correspond to the limiting case of flat ellipsoids, for which we derive a novel set of simplified formulas. The new formulas have improved numerical stability properties, and require a smaller number of input parameters. To assess their accuracy, we compare the analytical permeability predictions with three-dimensional finite-element simulations. We also compare the results with a semi-analytical method based on percolation theory and curve-fitting, which represents an alternative upscaling approach. A large number of cases is considered, with varying fracture aperture, density, matrix/fracture permeability contrast, orientation, shape, and number of fracture sets. The differential method is seen to be the best choice for sealed fractures and thin open fractures. For high-permeable, connected fractures, the semi-analytical method provides the best fit to the numerical data, whereas the differential method breaks down. The two self-consistent methods can be used for both unconnected and connected fractures, although the asymmetric method is somewhat unreliable for sealed fractures. For open fractures, the symmetric method is generally the more accurate for moderate fracture densities, but only the asymmetric method is seen to have correct asymptotic behavior. The asymmetric method is also surprisingly accurate at predicting percolation thresholds.

Anisotropic effective conductivity in fractured rocks by explicit effective medium methods

In this work, we assess the use of explicit methods for estimating the effective conductivity of anisotropic fractured media. Explicit methods are faster and simpler to use than implicit methods but may have a more limited range of validity. Five explicit methods are considered: the Maxwell approximation, the T-matrix method, the symmetric and asymmetric weakly self-consistent methods, and the weakly differential method, where the two latter methods are novelly constructed in this paper. For each method, we develop simplified expressions applicable to flat spheroidal “penny-shaped” inclusions. The simplified expressions are accurate to the first order in the ratio of fracture thickness to fracture diameter. Our analysis shows that the conductivity predictions of the methods fall within known upper and lower bounds, except for the T-matrix method at high fracture densities and the symmetric weakly self-consistent method when applied to very thin fractures. Comparisons with numerical results show that all the methods give reliable estimates for small fracture densities. For high fracture densities, the weakly differential method is the most accurate if the fracture geometry is non-percolating or the fracture/matrix conductivity contrast is small. For percolating conductive fracture networks, we have developed a scaling relation that can be applied to the weakly self-consistent methods to give conductivity estimates that are close to the results from numerical simulations.

Identification of subsurface structures using electromagnetic data and shape priors

We consider the inverse problem of identifying large-scale subsurface structures using the controlled source electromagnetic method. To identify structures in the subsurface where the contrast in electric conductivity can be small, regularization is needed to bias the solution towards preserving structural information. We propose to combine two approaches for regularization of the inverse problem. In the first approach we utilize a model-based, reduced, composite representation of the electric conductivity that is highly flexible, even for a moderate number of degrees of freedom. With a low number of parameters, the inverse problem is efficiently solved using a standard, second-order gradient-based optimization algorithm. Further regularization is obtained using structural prior information, available, e.g., from interpreted seismic data. The reduced conductivity representation is suitable for incorporation of structural prior information. Such prior information cannot, however, be accurately modeled with a gaussian distribution. To alleviate this, we incorporate the structural information using shape priors. The shape prior technique requires the choice of kernel function, which is application dependent. We argue for using the conditionally positive definite kernel which is shown to have computational advantages over the commonly applied gaussian kernel for our problem. Numerical experiments on various test cases show that the methodology is able to identify fairly complex subsurface electric conductivity distributions while preserving structural prior information during the inversion.

Robust inversion of controlled source electromagnetic data for production monitoring

Monitoring of the flow pattern during reservoir production is a potential new application for Controlled Source Electromagnetic (CSEM) data. However, it is a challenging problem due to a restricted resolution power in the data and possible high signal to noise ratio. We present a novel 3-D inversion algorithm for estimating changes in the electric conductivity based on time-lapse CSEM data. The inversion is based on a reduced representation of the unknown parameter function, where the degree of freedom in the estimation is determined by the information content in the data. The chosen representation facilitates the estimation of flow patterns with varying structure and also varying degree of smoothness in the transition between the oil and water saturated regions.

Estimation of a Piecewise Constant Function Using Reparameterized Level-Set Functions

In the last decade, the use of level-set functions has gained increasing popularity in solving inverse problems involving the identification of a piecewise constant function. Normally, a fine-scale representation of the level-set functions is used, yielding a high number of degrees of freedom in the estimation. In contrast, we focus on reparameterization of the level-set functions on a coarse scale. The number of coefficients in the discretized function is then reduced, providing necessary regularization for solving ill-posed problems. A coarse representation is also advantageous to reduce the computational work in solving the estimation problem.

Structural joint inversion of AVO and CSEM data using flexible representations

Exploiting information from multiple data sets can improve the quality of the estimated parameters and thereby the reliability of the resulting reservoir predictions. Combination of multiple data types can be done sequentially, where the model parameters are updated based on the information from one data type at the time; or simultaneously, where at each parameter update, the joint information from all available data types is taken into account. The latter approach has the prospect of stabilising the inversion by constraining the inversion with a higher degree of information. In this work, we present a method for structure-coupled joint inversion of controlled source electromagnetic (CSEM) data and seismic AVO data with application to production monitoring. By reformulating the unknown model parameters in terms of common structural relationships, we obtain a common model parameter that shares sensitivity to both data sets. The performance of the structurally coupled inversion is tested on synthetic examples where we compare the results of joint versus separate inversion of the data sets. We also investigate into the robustness of the joint inversion strategy with respect to a specific type of modelling errors.

A multi-level strategy for regularized parameter identification using nonlinear reparameterization with sample application for reservoir characterization

Level-set methods are popular for identifying piecewise constant structures. We propose an approach inspired by the level-set idea to identify coarse scale features of a scalar field, where the transitions between different regions can be both sharp and smooth. The nonlinear and coarse reparameterization structure provides regularization on the inverse problem, which is important if the quality of the available information is poor. In our identification strategy, the resolution of the representation of the parameter function is refined gradually; for several inverse problems, the nonlinearity of the problem is correspondingly carefully increased. Hence, when using a gradient-based optimization routine, the approach may reduce the risk of getting trapped in a local minimum of the objective function. We demonstrate the identification strategy for estimation of fluid conductivity in a porous medium based on sparsely distributed, transient data. The methodology is well suited for determining channels and barriers as well as smooth structures based on limited information.

History matching of dual continuum reservoirs—preserving consistency with the fracture model

Ensemble- and optimization-based parameter estimation is commonly used to calibrate simulation models of fractured reservoirs to measured data. Traditionally, statistical data on small-scale fractures are upscaled to a dual continuum model in a single step, and the subsequent history matching procedure makes adjustments to the upscaled parameters. In this paper, we show that the resulting reservoir models may be inconsistent with the initial fracture description, meaning that the reservoir parameters do not correspond to a physically valid combination of fracture parameters. A number of numerical examples is provided, which illustrate why and when the problem occurs. We utilize an invertible analytical fracture upscaling method, and deviations from the fracture model can thus be quantified in each case. We show that consistency with the fracture model is preserved if fracture parameters are history matched directly, if the relation between inversion variables and fracture parameters is linear, or if an unbiased Bayesian sampling method is used. We also show that preserving consistency is less important if the uncertainty of the fracture upscaling method is large.