Development of Kriging-approximation simulated annealing optimization algorithm for parameters calibration of porous media flow model

  • Ming-Che Hu
  • Chia-Hui Shen
  • Shao-Yiu HsuEmail author
  • Hwa-Lung Yu
  • Krzysztof Lamorski
  • Cezary Sławiński
Original Paper


This research proposes an innovative Kriging-approximation simulated annealing (KASA) optimization algorithm to increase optimization efficiency and reduce the computation time for the parameter calibration of simulation model. The newly developed KASA optimization algorithm utilizes the simulated annealing algorithm to search the global optimum; meanwhile, Kriging approximation, a statistical estimation method, is incorporated with simulated annealing to interpolate unknown objective values in solution spaces. Furthermore, this research establishes a network-based porous media flow simulation (NET-PFS) model and then KASA is applied in the calibration of the NET-PFS representative pore network. NET-PFS is a pore network based model constructing a representative pore network to approximate soil characteristics and pore geometry with limited access to pore-scale imaging processes. NET-PFS is applied to estimate the permeability of a sand-packing porous media. NET-PFS establishes a framework for simplifying the pore network but remaining the same hydraulic conductivity and the flow status of pore networks with limited information about the pore structure. In the case study, a quartz sand-packing porous media is scanned by X-ray micro computed tomography. The NET-PFS model is applied to estimate the hydraulic conductivity and flow velocity distribution from the original pore network. The results demonstrate the proposed KASA algorithm effectively calibrated the NET-PFS model; in addition, a representative pore network and the determined flow status in the pore network is presented by NET-PFS.


Kriging approximation Simulated annealing Network-based porous media flow Representative pore network 

List of symbols



Unobserved points, \({\text{i}} = 1, 2, 3, \ldots ,{\text{m}}\)


Observed points, \({\text{j}}1,{\text{j}}2,{\text{k}} = 1, 2, 3, \ldots ,{\text{n}}\)


Pores, \({\text{r}}1,{\text{r}}2 = 1, 2, 3, \ldots , {\text{q}}1\)


Inner pores, \({\text{r}}3 = 1, 2, 3, \ldots , {\text{q}}3\)


Boundary pores, \({\text{r}}4 = 1, 2, 3, \ldots , {\text{q}}4\)

Parameters and variables


Length of the arc between pores \({\text{j}}1\) and \({\text{j}}2\)


Diameter of the arc between pores \({\text{j}}1\) and \({\text{j}}2\)


Dynamic viscosity


Distance between points \({\text{j}}1\) and \({\text{j}}2\)


New objective value


Current objective value


Number of simulated annealing iterations


Given node pressure at node \({\text{r}}4\) of the NET-PFS model


Number of arcs in the pore network


Number of pores in the pore network


Number of inner pores in the pore network


Number of boundary pores in the pore network


Temperature of the simulated annealing algorithm


Computational time of the Kriging approximation for the NET-PFS model


Computational time of the NET-PFS model


Kriging estimation of a random variable at unobserved point \({\text{i}}\)


Realization of a random variable at unobserved point \({\text{i}}\)


Realization of a random variable at observed point \({\text{j}}1\)


Kriging weight at point \({\text{j}}1\)


Arc flow from nodes \({\text{r}}1\) to \({\text{r}}2\) of the NET-PFS model


Node pressure at node \({\text{r}}1\) of the NET-PFS model


Pore pressure at the inlet boundary


Pore pressure at the outlet boundary


Random variable with a uniform distribution in the interval of [0,1]

\(\upgamma( {{\text{d}}_{{{\text{j}}1,{\text{j}}2}}^{{}} } )\)

Semivariogram function of distance \({\text{d}}_{{{\text{j}}1,{\text{j}}2}}^{{}}\)


Lagrange multiplier


Expected values of the random variables


Variance of the random variables



Covariance function


Estimation error at unobserved point \({\text{i}}\)


Expected value


Exponential function


Lagrange function at unobserved point \({\text{i}}\)


Total flow rate


Variance function



The authors thank the editors and anonymous referees for thoughtful comments and suggestions. The authors are responsible for the opinions and comments. This research was funded by the Taiwanese Ministry of Science and Technology (MOST-105-2627-M-002-037; 105T612C502; MOST 106-2628-M-002-009-MY3) and National Taiwan University (NTU-CCP-106R891007; NTU-CC-107L892607).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Bioenvironmental Systems EngineeringNational Taiwan UniversityTaipeiTaiwan
  2. 2.Institute of AgrophysicsPolish Academy of SciencesLublinPoland

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