Haibin Chang

History matching of facies distribution with the EnKF and level set parameterization

In this work, we develop a methodology to combine the Ensemble Kalman filter (EnKF) and the level set parameterization for history matching of facies distribution. With given prior knowledge about the facies of the reservoir geology, initial realizations are generated by commonly used software as the prior guesses of the unknown field. Furthermore, level set functions are used to reparameterize these initial realizations. In the reparameterization process, a representing node system is set up, on which the values of level set functions are assigned using Gaussian random numbers. The mean and the standard deviation of the Gaussian random numbers are designed according to the facies proportion, and the sign of the random numbers depends on the facies type at the representing nodes. The values of the level set functions at the other grid nodes are obtained by linear interpolation. The level set functions on the representing nodes are the model parameters of the EnKF state vector and are updated in the data assimilation process. On the basis of our numerical examples for two-dimensional reservoirs with two or three facies, the proposed method is demonstrated to be able to capture the main features of the reference facies distributions.

Data Assimilation of Coupled Fluid Flow and Geomechanics Using the Ensemble Kalman Filter

Summary In reservoir history matching or data assimilation, dynamic data, such as production rates and pressures, are used to constrain reservoir models and to update model parameters. As such, even if under certain conceptualization the model parameters do not vary with time, the estimate of such parameters may change with the available observations and, thus, with time. In reality, the production process may lead to changes in both the flow and geomechanics fields, which are dynamically coupled. For example, the variations in the stress/strain field lead to changes in porosity and permeability of the reservoir and, hence, in the flow field. In weak formations, such as the Lost Hills oil field, fluid extraction may cause a large compaction to the reservoir rock and a significant subsidence at the land surface, resulting in huge economic losses and detrimental environmental consequences. The strong nonlinear coupling between reservoir flow and geomechanics poses a challenge to constructing a reliable model for predicting oil recovery in such reservoirs. On the other hand, the subsidence and other geomechanics observations can provide additional insight into the nature of the reservoir rock and help constrain the reservoir model if used wisely. In this study, the ensemble-Kalman-filter (EnKF) approach is used to estimate reservoir flow and material properties by jointly assimilating dynamic flow and geomechanics observations. The resulting model can be used for managing and optimizing production operations and for mitigating the land subsidence. The use of surface displacement observations improves the match to both production and displacement data. Localization is used to facilitate the assimilation of a large amount of data and to mitigate the effect of spurious correlations resulting from small ensembles. Because the stress, strain, and displacement fields are updated together with the material properties in the EnKF, the issue of consistency at the analysis step of the EnKF is investigated. A 3D problem with reservoir fluid-flow and mechanical parameters close to those of the Lost Hills oil field is used to test the applicability.

A Probabilistic Collocation-Based Kalman Filter for History Matching

Original SPE manuscript received for review 17 January 2010. Revised manuscript received for review 9 May 2010. Paper (SPE 140737) peer approved 13 July 2010. Summary The ensemble Kalman filter (EnKF) has been used widely for data assimilation. Because the EnKF is a Monte Carlo-based method, a large ensemble size is required to reduce the sampling errors. In this study, a probabilistic collocation-based Kalman filter (PCKF) is developed to adjust the reservoir parameters to honor the production data. It combines the advantages of the EnKF for dynamic data assimilation and the polynomial chaos expansion (PCE) for efficient uncertainty quantification. In this approach, all the system parameters and states and the production data are approximated by the PCE. The PCE coefficients are solved with the probabilistic collocation method (PCM). Collocation realizations are constructed by choosing collocation point sets in the random space. The simulation for each collocation realization is solved forward in time independently by means of an existing deterministic solver, as in the EnKF method. In the analysis step, the needed covariance is approximated by the PCE coefficients. In this study, a square-root filter is employed to update the PCE coefficients. After the analysis, new collocation realizations are constructed. With the parameter collocation realizations as the inputs and the state collocation realizations as initial conditions, respectively, the simulations are forwarded to the next analysis step. Synthetic 2D water/oil examples are used to demonstrate the applicability of the PCKF in history matching. The results are compared with those from the EnKF on the basis of the same analysis. It is shown that the estimations provided by the PCKF are comparable to those obtained from the EnKF. The biggest improvement of the PCKF comes from the leading PCE approximation, with which the computational burden of the PCKF can be greatly reduced by means of a smaller number of simulation runs, and the PCKF outperforms the EnKF for a similar computational effort. When the correlation ratio is much smaller, the PCKF still provides estimations with a better accuracy for a small computational effort.

History matching of statistically anisotropic fields using the Karhunen-Loeve expansion-based global parameterization technique

Traditional ensemble-based history matching method, such as the ensemble Kalman filter and iterative ensemble filters, usually update reservoir parameter fields using numerical grid-based parameterization. Although a parameter constraint term in the objective function for deriving these methods exists, it is difficult to preserve the geological continuity of the parameter field in the updating process of these methods; this is especially the case in the estimation of statistically anisotropic fields (such as a statistically anisotropic Gaussian field and facies field with elongated facies) with uncertainties about the anisotropy direction. In this work, we propose a Karhunen-Loeve expansion-based global parameterization technique that is combined with the ensemble-based history matching method for inverse modeling of statistically anisotropic fields. By using the Karhunen-Loeve expansion, a Gaussian random field can be parameterized by a group of independent Gaussian random variables. For a facies field, we combine the Karhunen-Loeve expansion and the level set technique to perform the parameterization; that is, for each facies, we use a Gaussian random field and a level set algorithm to parameterize it, and the Gaussian random field is further parameterized by the Karhunen-Loeve expansion. We treat the independent Gaussian random variables in the Karhunen-Loeve expansion as the model parameters. When the anisotropy direction of the statistically anisotropic field is uncertain, we also treat it as a model parameter for updating. After model parameterization, we use the ensemble randomized maximum likelihood filter to perform history matching. Because of the nature of the Karhunen-Loeve expansion, the geostatistical characteristics of the parameter field can be preserved in the updating process. Synthetic cases are set up to test the performance of the proposed method. Numerical results show that the proposed method is suitable for estimating statistically anisotropic fields.

Jointly updating the mean size and spatial distribution of facies in reservoir history matching

There exist several methods for history matching of reservoir facies distribution. When using these methods, the facies mean size is usually supposed to be prior information known without uncertainty. However, in reality, it is often difficult to acquire an accurate estimation of the facies mean size due to limited measurement data. Thus, it is more reasonable to treat the facies mean size as an uncertain model parameter for updating. In this work, we propose a methodology to jointly update the mean size and spatial distribution of facies in reservoir history matching. In the parameterization step, we utilize a Gaussian random field and a level set algorithm to parameterize each facies. The range of the Gaussian field controls the facies mean size of the generated facies distribution realizations. To accomplish jointly updating the mean size and spatial distribution of facies, the Gaussian random field is further parameterized by the Karhunen–Loeve (KL) expansion. We choose the range of the Gaussian field and the independent Gaussian random variables in the KL expansion as model parameters. After model parameterization, we use the Levenberg–Marquardt ensemble randomized maximum likelihood filter (LM_EnRML) to perform history matching. Three synthetic cases are set up to test the performance of the proposed method. Numerical results show that for estimating the facies field, the mean size and spatial distribution of facies can be jointly updated to match the reference distribution using the proposed method.

Data Assimilation of Coupled Fluid Flow and Geomechanics via Ensemble Kalman Filter

In reservoir history matching or data assimilation, dynamic data such as production rates and pressures are used to constrain reservoir models and to update model parameters. As such, even if under certain conceptualization the model parameters do not vary with time, the estimate of such parameters may change with the available observations and thus with time. In reality, the production process may lead to changes in both the flow and geomechanics fields, which are dynamically coupled. For example, the variations in the stress/strain field lead to changes in porosity and permeability of the reservoir and hence in the flow field. In weak formations such as the Lost Hills oilfield, fluid extraction may cause a large compaction to the reservoir rock and a significant subsidence at the land surface, resulting in huge economic losses and detrimental environmental consequences. The strong nonlinear coupling between reservoir flow and geomechanics possesses a challenge to construct a reliable model for predicting oil recovery in such reservoirs. On the other hand, the subsidence and other geomechanics observations can provide additional insight into the nature of the reservoir rock and help constrain the reservoir model if used wisely. In this study, the Ensemble Kalman filter (EnKF) approach is used to estimate reservoir flow and material properties by jointly assimilating dynamic flow and geomechanics observations. The resulting model can be used for managing and optimizing production operations and for mitigating the land subsidence. The use of surface displacement observations improves the match to both production and displacement data. Localization is used to facilitate the assimilation of a large amount of data and to mitigate the effect of spurious correlations resulting from small ensembles. Since the stress, strain, and displacement fields are updated together with the material properties in The EnKF, the issue of consistency at the analysis step of the EnKF is investigated. A 3D problem with reservoir fluid-flow and mechanical parameters close to those of the Lost Hills oilfield is used to test the applicability. Introduction The geomechanical behavior of a reservoir is usually only considered through rock compressibility in reservoir simulators. Rock compressibility determines the change of reservoir pore volume with respect to the change of pressure. The stress fields change, however, dramatically during depletion, water injection or different applications of enhanced oil recovery techniques. The changes in the stress field induce various geomechanical phenomena, such as land subsidence, abrupt compaction of the reservoir, induced fracturing, enhancement of natural fractures and fault activation. Among these phenomena, reservoir compaction and surface subsidence are most commonly seen. Well know examples include the sea floor subsidence in the Ekofisk field and Valhall field in the North Sea (Pattillo et al. 1998), subsidence in the Lost Hill field, California (Wallace and Pugh 1993) and in the region of the Boliva Coast and Lagunillas in Venezuela (van der Knapp and van der Vlis 1967). These complicated geomechanical situations require more sophisticated methods to take them into account in order to better predict the production and avoid facility failures. In the past ten years, extensive efforts have been made to couple the fluidflow and geomechanics simulations to model the complex process during hydrocarbon production (Chin et al. 2000; Dean et al. 2006; Samier et al. 2006). Geomechanics module is available in several commercial reservoir simulators to facilitate the coupled modeling of fluid-flow and geomechanical processes. Both multiphase flow problems and geomechanical processes require some parameters to describe the flow or mechanical properties of the reservoir. Typical primary parameters for multiphase phase flow simulation are permeability and porosity. The primary parameters for elastical geomechanical processes are Youngs modulus and Poissons ratio. Both flow and geomechanical parameters exhibit spatial variabilities and none of them can be measured extensively and accurately, thus these parameters are subject to large uncertainty. In contrast, model responses normally can be measured relatively precisely. For example, oil and water production rates are usually measured and recorded regularly. Down hole pressure gauges can

Non-intrusive Stochastic Approaches for Efficient Quantification of Uncertainty Associated with Reservoir Simulations

This paper presents non-intrusive, efficient stochastic approaches for predicting uncertainties associated with petroleum reservoir simulations. The Monte Carlo simulation method, which is the most common and straightforward approach for uncertainty quantification in the industry, requires to perform a large number of reservoir simulations and is thus computationally expensive especially for large-scale problems. We propose an efficient and accurate alternative through the collocation-based stochastic approaches. The reservoirs are considered to exhibit randomly heterogeneous flow properties. The underlying random permeability field can be represented by the Karhunen-Loeve expansion (or principal component analysis), which reduces the dimensionality of random space. Two different collocation-based methods are introduced to propagate uncertainty of the reservoir response. The first one is the probabilistic collocation method that deals with the random reservoir responses by employing the orthogonal polynomial functions as the bases of the random space and utilizing the collocation technique in the random space. The second one is the sparse grid collocation method that is based on the multi-dimensional interpolation and high-dimensional quadrature techniques. They are non-intrusive in that the resulting equations have the exactly the same form as the original equations and can thus be solved with existing reservoir simulators. These methods are efficient since only a small number of simulations are required and the statistical moments and probability density functions of the quantities of interest in the oil reservoirs can be accurately estimated. The proposed approaches are demonstrated with a 3D reservoir model originating from the 9th SPE comparative project. The accuracy, efficiency, and compatibility are compared against Monte Carlo simulations. This study reveals that, compared to traditional Monte Carlo simulations, the collocation-based stochastic approaches can accurately quantify uncertainty in petroleum reservoirs and greatly reduce the computational cost.