Estimation of relative permeability curves using an improved Levenberg-Marquardt method with simultaneous perturbation Jacobian approximation

Abstract

Relative permeability controls the flow of multiphase fluids in porous media. The estimation of relative permeability is generally solved by Levenberg-Marquardt method with finite difference Jacobian approximation (LM-FD). However, the method can hardly be used in large-scale reservoirs because of unbearably huge computational cost. To eliminate this problem, the paper introduces the idea of simultaneous perturbation to simplify the generation of the Jacobian matrix needed in the Levenberg-Marquardt procedure and denotes the improved method as LM-SP. It is verified by numerical experiments and then applied to laboratory experiments and a real commercial oilfield. Numerical experiment indicates that LM-SP uses only 16.1% computational cost to obtain similar estimation of relative permeability and prediction of production performance compared with LM-FD. Laboratory experiment also shows the LM-SP has a 60.4% decrease in simulation cost while a 68.5% increase in estimation accuracy compared with the earlier published results. This is mainly because LM-FD needs 2n (n is the number of controlling knots) simulations to approximate Jacobian in each iteration, while only 2 simulations are enough in basic LM-SP. The convergence rate and estimation accuracy of LM-SP can be improved by averaging several simultaneous perturbation Jacobian approximations but the computational cost of each iteration may be increased. Considering the estimation accuracy and computational cost, averaging two Jacobian approximations is recommended in this paper. As the number of unknown controlling knots increases from 7 to 15, the saved simulation runs by LM-SP than LM-FD increases from 114 to 1164. This indicates LM-SP is more suitable than LM-FD for multivariate problems. Field application further proves the applicability of LM-SP on large real field as well as small laboratory problems.

Multi-objective optimization for oil-gas production process based on compensation model of comprehensive energy consumption using improved evolutionary algorithm

“…FD method is usually expensive in function evaluation. In contrast, the calculation time of Simultaneous Perturbation (SP) method is less (Zhou, 2017), so the SP method is used in the paper to estimate the gradient of the j -th objective function, and the i -th component of gradient is expressed as follows:…”

“…FD method is usually expensive in function evaluation. In contrast, the calculation time of Simultaneous Perturbation (SP) method is less (Zhou, 2017), so the SP method is used in the paper to estimate the gradient of the j -th objective function, and the i -th component of gradient is expressed as follows: where Δ is a n -dimensional vector of random perturbations, and Δi is the i -th component of Δ . For each component of Δ, a simple choice is used to have a Bernoulli distribution ±1 and the probability of each ±1 outcome is 0.5.…”

Interpretation of Gas/Water Relative Permeability of Coal Using the Hybrid Bayesian-Assisted History Matching: New Insights

“…The assisted history matching method in essence solves a mathematical optimization problem with an objective function of minimizing the errors between the simulated and experimental production curves by tuning the relative permeability curves with an optimization algorithm(s). The assisted history matching method has been successfully used to derive water/oil relative permeability curves from core-flooding tests [24][25][26]. Shaw et al [27] used manual history matching to simultaneously derive relative permeability and capillary pressure for gas-water-coal systems.…”

Research on the Recognition Algorithm concerning Geometric Boundary regarding Heat Conduction Based on BEM and CGM

“…When the inversion regarding the geometric boundary of the heat conduction system is conducted, the two-dimension steady-state space should be dispersed firstly; the discrete methods mainly include finite difference method [10] (FDM), finite element method [11] (FEM), and boundary element method [12] (BEM), and they can be used to solve the direct problem of heat transfer theory; and the inversion of the geometric boundary can be realized through various optimization technologies based on solving the direct problem. The inversion method based on optimized technology can be divided into the optimized algorithm based on gradient and the optimized algorithm based on nongradient; the optimized algorithm based on gradient mainly includes conjugate gradient method (CGM), Levenberg-Marquardt [13] (L-MM), and steepest descent method (SDM), while the optimized algorithm based on nongradient mainly includes genetic algorithm [14] (GA), neural network algorithm [15] (NNM), and particle swarm algorithm [16] (PSO).…”