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


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.

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\({\text{i}}\) :

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

\({\text{j}}1,{\text{j}}2,{\text{k}}\) :

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

\({\text{r}}1,{\text{r}}2\) :

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

\({\text{r}}3\) :

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

\({\text{r}}4\) :

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

\({\text{al}}_{{{\text{r}}1,{\text{r}}2}}^{{}}\) :

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

\({\text{ar}}_{{{\text{r}}1,{\text{r}}2}}^{{}}\) :

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

\({\text{dv}}\) :

Dynamic viscosity

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

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

\({\text{obj}}_{\text{new}}^{{}}\) :

New objective value

\({\text{obj}}_{\text{current}}^{{}}\) :

Current objective value

\({\text{p}}\) :

Number of simulated annealing iterations

\({\text{pr}}_{{{\text{r}}4}}^{{}}\) :

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

\({\text{qa}}\) :

Number of arcs in the pore network

\({\text{q}}1\) :

Number of pores in the pore network

\({\text{q}}3\) :

Number of inner pores in the pore network

\({\text{q}}4\) :

Number of boundary pores in the pore network

\({\text{temp}}\) :

Temperature of the simulated annealing algorithm

\({\text{tk}}\) :

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

\({\text{tn}}\) :

Computational time of the NET-PFS model

\({\text{v}}_{\text{i}}^{\text{E}}\) :

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

\({\text{v}}_{\text{i}}^{\text{T}}\) :

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

\({\text{v}}_{{{\text{j}}1}}^{{}}\) :

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

\({\text{w}}_{{{\text{j}}1}}^{{}}\) :

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

\({\text{xf}}_{{{\text{r}}1,{\text{r}}2}}^{{}}\) :

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

\({\text{xp}}_{{{\text{r}}1}}^{{}}\) :

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

\({\text{xp}}_{\text{in}}\) :

Pore pressure at the inlet boundary

\({\text{xp}}_{\text{out}}\) :

Pore pressure at the outlet boundary

\({\text{z}}\) :

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

\(\uplambda\) :

Lagrange multiplier

\(\upmu\) :

Expected values of the random variables

\(\upsigma_{{}}^{2}\) :

Variance of the random variables

\({\text{COV}}\) :

Covariance function

\({\text{ERROR}}_{\text{i}}^{{}}\) :

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

\({\text{EXP}}\) :

Expected value

\({\text{EXPO}}\) :

Exponential function

\({\text{LAGR}}_{\text{i}}^{{}}\) :

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

\({\text{Q}}_{\text{m}}\) :

Total flow rate

\({\text{VAR}}\) :

Variance function


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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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Correspondence to Shao-Yiu Hsu.

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Hu, MC., Shen, CH., Hsu, SY. et al. Development of Kriging-approximation simulated annealing optimization algorithm for parameters calibration of porous media flow model. Stoch Environ Res Risk Assess 33, 395–406 (2019).

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