Kernel Ensemble Kalman Filter for Non-Gaussian Reservoir Prediction
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Solution Overview
Problem
The ensemble Kalman filter (EnKF) is limited in its application to non-Gaussian random fields, as it modifies models towards Gaussianity, leading to questionable predictive capacity for complex geological models like channel systems, as it only preserves two-point statistics and not multi-point geostatistics.
Innovation Solution
The use of kernel methods to create a nonlinear generalization of the EnKF, allowing for the representation of non-Gaussian random fields by mapping data into a high-dimensional feature space, thereby nonlinearizing the Kalman gain and update equations, and using high-order polynomial kernels to preserve multi-point statistics and geological realism.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If the EnKF is applied to non-Gaussian random fields, then the model can be updated to match production data, but the model is modified towards Gaussianity and loses predictive capacity for complex geological models
Solution Approach 1:
The patent transforms the probability density function from a scalar field to a function that maps spatial coordinates to probability densities. This dimensional transformation allows the EnKF to operate in a space where non-Gaussian characteristics are preserved while still achieving good match to production data, resolving the contradiction between data fitting and predictive capacity.
Solution Approach 2:
The patent changes the parameter representation by using a transformation function that maps Gaussian variables to non-Gaussian variables. This allows the EnKF to work with non-Gaussian random fields without modifying the fundamental Gaussian nature of the filter algorithm, thereby preserving both data match quality and predictive capacity for complex geological models.
2Measurement precision
If the EnKF is applied to complex non-Gaussian geological models, then production data can be matched, but multi-point geostatistics are lost and only two-point statistics are preserved
Solution Approach 1:
The patent introduces a transformation function as an intermediary that maps Gaussian random fields to non-Gaussian random fields. This intermediary allows the EnKF to preserve multi-point geostatistics by operating in the transformed space where the complex geological patterns and their spatial relationships are maintained, while still achieving good match to production data.
3Productivity
If the EnKF is used for continuous model updating, then real-time monitoring and optimization are enabled, but the method is limited to Gaussian random fields
Solution Approach 1:
The patent makes the EnKF universal by introducing a transformation function that can handle both Gaussian and non-Gaussian random fields. This allows the same continuous model updating framework to be applied to a wide range of geological models with different statistical characteristics, thereby enhancing both productivity and adaptability.
Data Source
AI summary
A reservoir prediction system is disclosed that uses a kernel-based ensemble Kalman filter (EnKF) capable of representing non-Gaussian random fields characterized by multi-point geostatistics. The EnKF uses only the covariance and cross-covariance between the random fields (to be updated) and observations, thereby only preserving two-point statistics. The kernel-based EnKF allows the creation of nonlinear generalizations of linear algorithms that can be exclusively written in terms of dot products. By deriving the EnKF in a high-dimensional feature space implicitly defined using kernels, both the Kalman gain and update equations are nonlinearized, thus providing a completely general nonlinear set of EnKF equations, the nonlinearity being controlled by the kernel. By choosing high order polynomial kernels, multi-point statistics and therefore geological realism of the updated random fields can be preserved.