FD-DG Method for Heat Front Capture in Thermal Recovery
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Solution Overview
Problem
Conventional numerical methods for simulating hot fluid injection in heavy oil reservoirs suffer from significant numerical dispersion, leading to smearing of sharp temperature fronts and reduced accuracy, requiring excessive grid refinement and increased computational time.
Innovation Solution
Combining the finite difference (FD) method with the discontinuous Galerkin (DG) method to solve the flow and energy equations, allowing for non-constant temperature within grid-blocks and using a slope limiter to stabilize the solution, thereby reducing numerical dispersion and improving accuracy without increasing the number of grid-blocks.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of manufacture
If the finite difference method is used to solve the energy equation, then the solution is simple and easy to implement, but numerical dispersion occurs that smears sharp temperature fronts and reduces accuracy
Solution Approach 1:
The patent segments the numerical method into two parts: using finite difference for the flow equation and discontinuous Galerkin for the energy equation. This segmentation allows each method to be optimized for its specific purpose while avoiding the numerical dispersion problems that affect the finite difference method when applied to the energy equation.
Solution Approach 2:
The patent changes the numerical method parameter from finite difference to discontinuous Galerkin for solving the energy equation. This parameter change transforms the approach from a method that smears sharp fronts to one that captures them accurately, while maintaining computational efficiency through the use of linear basis functions.
2Measurement precision
If grid refinement is increased to improve temperature front accuracy, then measurement precision improves, but computational time and complexity increase excessively
Solution Approach 1:
The patent changes the numerical method parameter from finite difference to discontinuous Galerkin, which fundamentally alters how temperature is computed. This parameter change enables accurate capture of sharp temperature fronts without requiring excessive grid refinement, thereby reducing computational time while maintaining high precision.
Solution Approach 2:
The patent substitutes the traditional finite difference mechanical approach with a discontinuous Galerkin method that uses polynomial basis functions and numerical integration. This substitution replaces a method reliant on fine grid spacing for accuracy with one that achieves accuracy through higher-order approximation, reducing the need for grid refinement and associated computational costs.
3Measurement precision
If the discontinuous Galerkin method is used to solve the energy equation, then temperature accuracy near sharp fronts improves, but device complexity increases
Solution Approach 1:
The patent segments the solution approach by applying discontinuous Galerkin only to the energy equation while using finite difference for the flow equation. This segmentation limits the complexity increase to only where it is needed for accurate temperature calculation, rather than complicating the entire simulation system.
Solution Approach 2:
The patent applies local quality by using discontinuous Galerkin specifically for the energy equation where high accuracy is needed, while maintaining the simpler finite difference method for the flow equation. This localized application of the more complex method minimizes overall system complexity while achieving the required accuracy where it matters most.
4Measurement precision
If fine gridding is used in thermal recovery models, then temperature prediction accuracy improves, but the number of grid blocks and computational resources increase significantly
Solution Approach 1:
The patent changes the numerical method parameter from finite difference to discontinuous Galerkin, which fundamentally alters the relationship between grid resolution and accuracy. This parameter change enables the use of coarser grids while maintaining high temperature prediction accuracy, thereby reducing the number of grid blocks needed in the model.
Data Source
AI summary
A numerical procedure is disclosed to improve the prediction of heat fronts when simulating hot fluid injection in viscous hydrocarbon reservoirs. The mathematical model is composed of the conventional governing equations that describe multiphase fluid flow and energy balance. The reservoir geometry can be partitioned into a regular Cartesian grid or an irregular corner-point geometry grid. The numerical procedure uses the finite different (FD) method to solve the flow equations and the discontinuous Galerkin (DG) method to solve the energy balance equation. The proposed FD-DG method is an alternative to the traditional solution procedure that uses the FD method to solve both the flow and the energy equations. The traditional method has the deficiency that it may require excessive number of grid cells to achieve acceptable resolution of the heat fronts. The proposed FD-DG method significantly reduces numerical dispersion near discontinuities in the solution of the energy equation and therefore provides a better capture of the heat fronts. To obtain a desired accuracy in the energy equation solution, the FD-DG method can be orders of magnitude faster than the traditional method. The superiority of the FD-DG method is that it converges on coarser grids while the traditional method requires much finer grids.