Kernel Ensemble Kalman Filter for Non-Gaussian Reservoir Modeling
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Solution Overview
Problem
The ensemble Kalman filter (EnKF) is limited in its application to complex non-Gaussian geological models, as it modifies these models towards Gaussianity, losing predictive capacity and encountering constraint violations, especially in highly nonlinear problems like compositional simulations.
Innovation Solution
The use of kernel-based ensemble Kalman filters (KEnKF) that operate in a high-dimensional feature space, allowing for nonlinear generalizations and handling of non-Gaussian random fields and constraints through constrained optimization with equality and inequality constraints, using methods like fixed-point iteration and augmented Lagrangian methods.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If the standard EnKF is applied to complex non-Gaussian geological models, then the model can be updated with production data, but the non-Gaussian characteristics are lost and the model is modified towards Gaussianity
Solution Approach 1:
The patent transforms the state variables using a nonlinear mapping function that preserves the non-Gaussian distribution characteristics. By changing the parameterization of the state variables through a suitable transformation, the updated ensemble maintains the original non-Gaussian properties while still being compatible with the EnKF framework.
Solution Approach 2:
The patent introduces an intermediate transformation step between the standard EnKF update and the final state variables. This intermediary mapping function acts as a bridge that allows the linear EnKF update to operate on transformed variables while preserving the non-Gaussian characteristics when mapped back to the original space.
2Productivity
If the EnKF updating step is applied to highly nonlinear problems, then the model can be updated sequentially, but bound constraints are violated leading to non-physical updates
Solution Approach 1:
The patent applies a nonlinear transformation to the state variables such that the transformed variables are unbounded and more Gaussian-like, making them suitable for the EnKF update. After the update, the inverse transformation is applied to recover the physical state variables that satisfy the original bound constraints, thus maintaining both computational efficiency and physical realism.
3Adaptability or versatility
If variable transformation is used to handle bound constraints, then the variables become more Gaussian and suitable for EnKF, but the transformation is problem specific and not applicable to nonlinear constraints
Solution Approach 1:
The patent develops a general framework using nonlinear mapping functions that can handle both bound constraints and nonlinear constraints uniformly. This universal approach replaces the need for problem-specific transformations, providing a single methodology that works across different constraint types and application scenarios.
4Manufacturing precision
If iterative EnKF is used to handle all constraints through the simulator, then constraints are satisfied, but computing cost increases substantially
Solution Approach 1:
The patent transforms the constrained optimization problem into an unconstrained problem in the transformed variable space. By working with transformed variables that naturally satisfy the constraints through their definition, the need for iterative constraint satisfaction is eliminated, reducing computational cost while maintaining accuracy.
5Reliability
If constraints are treated as pseudo observations in EnKF, then violated constraints can be addressed, but the solution becomes suboptimal and constraints may still not be satisfied
Solution Approach 1:
The patent changes the parameterization of the state variables through a nonlinear transformation that inherently respects the constraint structure. This approach treats constraints as part of the variable definition rather than as separate conditions to be enforced, ensuring both satisfaction of constraints and optimality of the solution simultaneously.
Data Source
AI summary
A computer-implemented reservoir prediction system, method, and software are provided for updating simulation models of a subterranean reservoir. An ensemble of reservoir models representing a subterranean reservoir having non-Gaussian characteristics is provided, along with reservoir data from the subterranean reservoir used to condition the ensemble of reservoir models. For each of the reservoir models in the ensemble of reservoir models, a constrained optimization with equality constraints and inequality constraints are solved using a constrained Kernel Ensemble Kalman Filter to obtain a constrained optimal solution. The constrained optimal solutions are assembled to update the ensemble of reservoir models. The updated ensemble of reservoir models are consistent with the reservoir data provided from the subterranean reservoir and the non-Gaussian characteristics of the subterranean reservoir are preserved.