Abstract
Determination of parameters characterizing flows in porous media is a complex inverse problem. It is especially difficult to determine confidence intervals of such parameters. In this note, we develop a method based on utilization of bootstrapping in order to find confidence intervals of model parameters, which are determined by minimizing the discrepancy between model predictions and published experimental results. The discrepancy is characterized by the objective function defined as the sum of squared residuals in the points where experimental measurements are taken. A residual is defined as the difference between the experimentally measured value and the model prediction of this value. We utilized bootstrapping to generate surrogate experimental data by randomly resampling residuals and then adding them back to model predictions. The model parameters that give the best fit with a large number of surrogate data were then determined. By analyzing the histograms of best-fit parameter values, we were able to find confidence intervals for these parameters. We utilized the developed method to determine confidence intervals for the effective viscosity and Forchheimer coefficient.
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Abbreviations
- \(b_0, b_1, b_2 \) :
-
Dimensionless parameters defined in Eq. (4)
- \(c_\mathrm{F} \) :
-
Forchheimer coefficient describing the form drag due to solid obstacles
- err:
-
Objective (penalty) function defined in Eq. (7)
- \(K^{{*}}\) :
-
Permeability of the porous medium \((\hbox {m}^{2})\)
- \(p^{{*}}\) :
-
Pressure (Pa)
- r :
-
Dimensionless radial coordinate, \(\frac{r^{{*}}}{R^{{*}}}\)
- \(r^{{*}}\) :
-
Radial coordinate (m)
- \(R^{{*}}\) :
-
Pipe radius (m)
- u :
-
Dimensionless axial velocity, \(\frac{u^{{*}}}{U^{{*}}}\)
- \(u^{*}\) :
-
Axial (z) velocity component \((\hbox {m s}^{-1})\)
- \(\mathbf{u}^{{*}}\) :
-
Fluid filtration velocity vector (m s\(^{-1}\))
- \(\hat{{u}}\) :
-
Surrogate “experimental” value of the axial velocity calculated by using Eq. (8)
- \(U^{{*}}\) :
-
Mean flow velocity \((\hbox {m s}^{-1})\)
- \(\varepsilon \) :
-
Residual, defined as the experimentally determined value minus the model prediction at a given radial location
- \(\mu ^{{*}}\) :
-
Dynamic viscosity of the fluid (Pa s)
- \(\mu _\mathrm{eff}^{*} \) :
-
Effective viscosity of the porous medium (Pa s)
- \(\rho _\mathrm{f}^{*} \) :
-
Density of the fluid \((\hbox {kg m}^{-3})\)
- *:
-
dimensional quantity
- eff:
-
Effective value (for a porous medium)
- f :
-
Fluid
References
Beck, J.V., Arnold, K.J.: Parameter Estimation in Science and Engineering. Wiley, New York (1977)
Chernick, M.R., LaBudde, R.A.: An Introduction to Bootstrap Methods with Applications to R. Wiley, Hoboken (2011)
Davison, A.C., Hinkley, D.V.: Bootstrap Methods and Their Application. Cambridge University Press, Cambridge (1997)
Efron, B., Tibshirani, R.: An Introduction to the Bootstrap. Chapman and Hall, Boca Raton (1993)
Givler, R.C., Altobelli, S.A.: A determination of the effective viscosity for the Brinkman-Forchheimer flow model. J. Fluid Mech. 258, 355–370 (1994)
Hall, P.: Theoretical comparison of bootstrap confidence intervals. Ann. Stat. 16, 927–953 (1988)
Kuznetsov, I.A., Kuznetsov, A.V.: Simulating tubulin-associated unit transport in an axon: using bootstrapping for estimating confidence intervals of best fit parameter values obtained from indirect experimental data. Proc. R Soc. A 473, 20170045 (2017a). doi:10.1098/rspa.2017.0045
Kuznetsov, I.A., Kuznetsov, A.V.: Utilization of the bootstrap method for determining confidence intervals of parameters for a model of MAP1B protein transport in axons. J. Theor. Biol. 419, 350–361 (2017b)
Nield, D.A., Bejan, A.: Convection in Porous Media, 5th edn. Springer, New York (2017)
Singh, K., Xie, M.: Bootstrap: a statistical method. Unpublished Manuscript, Rutgers University, USA. Retrieved from http://www.stat.rutgers.edu/home/mxie/RCPapers/bootstrap.pdf (2008)
Zadeh, K.S.: Parameter estimation in flow through partially saturated porous materials. J. Comput. Phys. 227, 10243–10262 (2008)
Zadeh, K.S., Montas, H.J.: Parametrization of flow processes in porous media by multiobjective inverse modeling. J. Comput. Phys. 259, 390–401 (2014)
Acknowledgements
A. V. K. acknowledges with gratitude the support of the National Science Foundation (Award CBET-1642262) and the Alexander von Humboldt Foundation through the Humboldt Research Award.
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Kuznetsov, I.A., Kuznetsov, A.V. Using Resampling Residuals for Estimating Confidence Intervals of the Effective Viscosity and Forchheimer Coefficient. Transp Porous Med 119, 451–459 (2017). https://doi.org/10.1007/s11242-017-0892-2
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DOI: https://doi.org/10.1007/s11242-017-0892-2