Constrained Pressure Residual Preconditioner for Adjoint Equations
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Solution Overview
Problem
Conventional preconditioners are ineffective for solving adjoint equations, especially in large and complex reservoir models, due to their reliance on properties of the Jacobian matrix, making them impractical for efficient gradient calculation in production optimization problems.
Innovation Solution
A constrained pressure residual preconditioner (CPRA) is developed to efficiently solve adjoint equations by constructing a two-stage preconditioner that applies factorization to the transpose of the Jacobian matrix, using an Algebraic Multi Grid (AMG) linear solver and Generalized Minimal Residual acceleration.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional preconditioners are used to solve adjoint equations, then the solution process relies on properties of the Jacobian matrix, but the effectiveness deteriorates for large and complex reservoir models
Solution Approach 1:
The patent applies the inversion principle by constructing a preconditioner for the transpose of the Jacobian matrix (J^T) rather than the Jacobian matrix itself. The conventional CPR preconditioner is designed for J, but the adjoint equation requires solving with J^T. The invention inverts the approach by transposing the preconditioner structure: M_CPRA^-1 = (M_CPR^-1)^T, where M_CPRA is the new preconditioner for J^T and M_CPR is the conventional preconditioner for J. This inversion allows the preconditioner to effectively handle the adjoint equation even for large and complex reservoir models.
2Productivity
If direct coding of adjoint model is performed, then gradient calculation can be achieved, but the code complexity increases and it becomes dependent on the forward model
Solution Approach 1:
The patent uses the copying principle by constructing the adjoint model through systematic transposition of the forward model's Jacobian matrix rather than direct coding. The adjoint equation is formed as J^T * y = d, where J^T is the transpose of the Jacobian matrix from the forward model. This copying approach through transposition allows gradient calculation efficiency while avoiding the complexity of direct adjoint coding and eliminating dependency on forward model updates, as the transposed structure automatically adapts when the forward model changes.
3Ease of manufacture
If conventional preconditioners are applied to adjoint equation, then the existing preconditioning methods can be used, but the convergence acceleration becomes impractical for large reservoir models
Solution Approach 1:
The patent applies parameter changes by transforming the preconditioning approach from working with the Jacobian matrix J to working with its transpose J^T. The key parameter change is in the mathematical structure: instead of M_CPR * J ≈ I, the new preconditioner satisfies M_CPRA * J^T ≈ I, where M_CPRA = (M_CPR)^T. This parameter transformation allows conventional preconditioning techniques to be applied to the adjoint equation, maintaining ease of implementation while achieving practical convergence acceleration for large reservoir models.
Data Source
AI summary
A method, system and computer program product is disclosed for using a constrained pressure residual (CPR) preconditioner to solve adjoint models. A linear system of fluid flow equations comprising a plurality of variables that represent fluid flow properties in a geological formation of a subterranean reservoir is provided. Matrix (Ã)T, which comprises a transpose of a Jacobian matrix associated with the linear system of fluid flow equations, is constructed. A constrained pressure residual preconditioner MCPRA−1 is constructed responsive to the matrix (Ã)T. Matrix equation (Ã)Ty=d is then solved using the constrained pressure residual preconditioner MCPRA−1.