| Peer-Reviewed

A Parameter Estimation Technique for a Groundwater Flow Model

Received: 31 August 2020     Accepted: 12 October 2020     Published: 16 December 2020
Views:       Downloads:
Abstract

In this paper, the problem of ill-posedness of solution in identifying multiple groundwater flow parameters from hydraulic head data and other ancillary data was assessed. The solution approach to the parameter identification problem is sought by applying the Least Squares, the Adjoint, the Conjugate Gradient Method and a proposed Parameter Transformation Method. Numerical test for a 1D and 2D flow models governed by PDEs were used to assess the accuracy and stability of the proposed method. The proposed method gave an appreciable solution estimates with minimal error-norm compared with the o ptimisation techniques explored in the study as a measure to the PTM The results revealed that when the adapted methods and the PTM were simulated numerically on a 1D and 2D test problems, the PTM gave a more stable solution estimates with a residual norm-error value of 2.23500 for the 1D test problem compared with that of the Adjoint method which prove to be the comparing solution with a norm-error value of 2.66500. For the 2D test case, the results also revealed that the PTM was stable with a residual norm-error value of 10.98310 compared with that of the Conjugate Gradient method with value of 86.562. Thus in conclusion, the study revealed that the PTM is capable of yielding realistic solution estimates compared with the studied optimisation methods.

Published in Mathematical Modelling and Applications (Volume 5, Issue 4)
DOI 10.11648/j.mma.20200504.11
Page(s) 202-213
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Ill-Posed Problem, Parameter Transformation Method, Optimisation Techniques

References
[1] Beven, K. J. (2006), A manifesto for the equifinality thesis, J. Hydrol., 320 (1–2), 18–36.
[2] Bruckner G, Handrock-Meyer S, and Langmach, H (1998), An inverse problem from 2D ground-water modelling, Inverse Problems, 14 (4): 835-851.
[3] Cacuci, D. G., (1981), Sensitivity Theory for Nonlinear Systems: Nonlinear Functional Analysis Approach, J. Math. Phys., 22, 2794.
[4] Carter RD, Kemp LF, Pierce AC, and Williams DL (1974) Performance Matching with constraints, Society of Petroleum Engineering Journal, 14 (02), 187-196.
[5] Chavent G, Dupuy M, and Lemmonier P (1975) History Matching by Use of Optimal Theory, Society of Petroleum Engineers Journal, 15 (01), 74-86.
[6] Dietrich C. R, and Newsam G. N (1990), Sufficient Conditions for Identifying Transmissivity in a Confined Aquifer, Inverse Problems, 6 (3): L21–L28.
[7] Irsa, J, and Zhang Y (2012), A Direct Method of Parameter Estimation for Steady State Flow in Heterogeneous Aquifers with Unknown Boundary Conditions, Water Resources Research, 48: W09526.
[8] Jacquard, P., and C. Jain, (1965), Permeability Distribution from field Pressure Data, Trans Society of Petroleum Engineers, Vol. (54), 281-294.
[9] Knowles I, and Yan A (2007) The reconstruction of groundwater parameters from head data in an unconfined aquifer, Journal of Computational and Applied Mathematics, 208 (1): 72-81.
[10] Knowles I, Le T, and Yan A (2004) On the recovery of multiple flow parameters from transient head data, Journal of Computational and Applied Mathematics, 169 (1): 1-15.
[11] LeVeque RJ. (2007) Finite difference methods for ordinary and partial differential equations, Steady state and time dependent problems, Society for Industrial and Applied Mathematics, Philadelphia, USA, 341p.
[12] Neuman, S. P, Fogg, G. E and Jacobson E. A (1980), A statistical approach to the inverse problem of aquifer hydrology, Case study, Water Resources Research, Vol. 16, No. 1, pp. 33-58.
[13] Neuman, SP. and S. Yakowitz. (1979), “A statistical approach to the inverse problem of aquifer hydrology 1Theory”, Water Resour. Res., Vol. 15, No. 4, pp. 845-860.
[14] Nguyen VT, Nguyen HT, Tran TB, Vo AK (2015) On an inverse problem in the parabolic equation arising from groundwater pollution problem, Boundary Value Problems, 67, DOI 10.1186/s13661-015-0319-3.
[15] Oblow EM (1978) Sensitivity Theory for Reactor Thermal-Hydraulics Problems, Nuclear Science and Engineering, 68 (3), 322-337.
[16] Sun NZ (1999), Inverse Problems in Groundwater Modeling, 1st edn., Kluwer Academic Publishers, Boston, USA, 337p.
[17] Sun, N. Z., and Yeh, WWG (1990a), Coupled inverse problems in groundwater modeling: 1. Sensitivity analysis and parameter identification, Water Resour. Res., 26 (10), 2507–2525.
[18] Sun, NZ. and Yeh, WG. (1990), “Coupled Inverse Problems in Groundwater Modelling: Sensitivity Analysis and Parameter Identification”, Water Resources Research, Vol. 26, No. 10, pp. 2507-2525.
[19] Sun, NZ., and Yeh, WWG (1990b), Coupled inverse problems in groundwater modeling: 2. Identifiability and experimental design, Water Resour. Res., 26 (10), 2527–2540.
[20] Sykes, JF., Wilson, JL and Andrews, RW (1985), Sensitivity analysis for steady state groundwater flow using adjoint operators, Water Resour. Res., 21 (3), 359–371.
[21] Tarantola, A., (1987), Inverse Problems Theory: Methods for Data Fitting and Model Parameters Estimation, Elsevier, New York.
[22] Townley, LR and Wilson, JL (1985), Computationally Efficient Algorithms for Parameter Estimation and Uncertainty Propagation in numerical Models of groundwater flow. Water Resources Research, 21 (12), 1851-1860.
[23] Vainikko G, and Kunisch K (1993) Identifiability of the transmissivity coefficient in an elliptic boundary value problem, Zeitschrift für Analysis und ihre Anwendungen, 12 (2): 327-341.
[24] Vemuri, V., and W. J. Karplus (1969), Identification of nonlinear parameters of ground water basins by hybrid computation, Water Resour. Res., 5 (1), 172–185.
[25] Vogel CR (1999) Sparse matrix computations arising in distributed parameter identification, SIAM J. Matrix Anal. Appl. 20 (4): 1027-1037.
[26] Vogel CR (2002) Computational Methods for Inverse Problems (Frontiers in Applied Mathematics), 1st edn., Society for Industrial and Applied Mathematics, Philadelphia, USA, 183p.
[27] Wilson, J. L., and D. E. Metcalfe (1985), Illustration and verification of adjoint sensitivity for steady state groundwater flow, Water Resour. Res., 21 (11), 1602–1610.
[28] Yeh WWG (2015) Review: Optimisation methods for groundwater modelling and management, Hydrogeology Journal, 23 (6): 1051-1065.
Cite This Article
  • APA Style

    Joseph Acquah, Francis Benyah, Jerry Samuel Yao-Kuma. (2020). A Parameter Estimation Technique for a Groundwater Flow Model. Mathematical Modelling and Applications, 5(4), 202-213. https://doi.org/10.11648/j.mma.20200504.11

    Copy | Download

    ACS Style

    Joseph Acquah; Francis Benyah; Jerry Samuel Yao-Kuma. A Parameter Estimation Technique for a Groundwater Flow Model. Math. Model. Appl. 2020, 5(4), 202-213. doi: 10.11648/j.mma.20200504.11

    Copy | Download

    AMA Style

    Joseph Acquah, Francis Benyah, Jerry Samuel Yao-Kuma. A Parameter Estimation Technique for a Groundwater Flow Model. Math Model Appl. 2020;5(4):202-213. doi: 10.11648/j.mma.20200504.11

    Copy | Download

  • @article{10.11648/j.mma.20200504.11,
      author = {Joseph Acquah and Francis Benyah and Jerry Samuel Yao-Kuma},
      title = {A Parameter Estimation Technique for a Groundwater Flow Model},
      journal = {Mathematical Modelling and Applications},
      volume = {5},
      number = {4},
      pages = {202-213},
      doi = {10.11648/j.mma.20200504.11},
      url = {https://doi.org/10.11648/j.mma.20200504.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20200504.11},
      abstract = {In this paper, the problem of ill-posedness of solution in identifying multiple groundwater flow parameters from hydraulic head data and other ancillary data was assessed. The solution approach to the parameter identification problem is sought by applying the Least Squares, the Adjoint, the Conjugate Gradient Method and a proposed Parameter Transformation Method. Numerical test for a 1D and 2D flow models governed by PDEs were used to assess the accuracy and stability of the proposed method. The proposed method gave an appreciable solution estimates with minimal error-norm compared with the o ptimisation techniques explored in the study as a measure to the PTM The results revealed that when the adapted methods and the PTM were simulated numerically on a 1D and 2D test problems, the PTM gave a more stable solution estimates with a residual norm-error value of 2.23500 for the 1D test problem compared with that of the Adjoint method which prove to be the comparing solution with a norm-error value of 2.66500. For the 2D test case, the results also revealed that the PTM was stable with a residual norm-error value of 10.98310 compared with that of the Conjugate Gradient method with value of 86.562. Thus in conclusion, the study revealed that the PTM is capable of yielding realistic solution estimates compared with the studied optimisation methods.},
     year = {2020}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - A Parameter Estimation Technique for a Groundwater Flow Model
    AU  - Joseph Acquah
    AU  - Francis Benyah
    AU  - Jerry Samuel Yao-Kuma
    Y1  - 2020/12/16
    PY  - 2020
    N1  - https://doi.org/10.11648/j.mma.20200504.11
    DO  - 10.11648/j.mma.20200504.11
    T2  - Mathematical Modelling and Applications
    JF  - Mathematical Modelling and Applications
    JO  - Mathematical Modelling and Applications
    SP  - 202
    EP  - 213
    PB  - Science Publishing Group
    SN  - 2575-1794
    UR  - https://doi.org/10.11648/j.mma.20200504.11
    AB  - In this paper, the problem of ill-posedness of solution in identifying multiple groundwater flow parameters from hydraulic head data and other ancillary data was assessed. The solution approach to the parameter identification problem is sought by applying the Least Squares, the Adjoint, the Conjugate Gradient Method and a proposed Parameter Transformation Method. Numerical test for a 1D and 2D flow models governed by PDEs were used to assess the accuracy and stability of the proposed method. The proposed method gave an appreciable solution estimates with minimal error-norm compared with the o ptimisation techniques explored in the study as a measure to the PTM The results revealed that when the adapted methods and the PTM were simulated numerically on a 1D and 2D test problems, the PTM gave a more stable solution estimates with a residual norm-error value of 2.23500 for the 1D test problem compared with that of the Adjoint method which prove to be the comparing solution with a norm-error value of 2.66500. For the 2D test case, the results also revealed that the PTM was stable with a residual norm-error value of 10.98310 compared with that of the Conjugate Gradient method with value of 86.562. Thus in conclusion, the study revealed that the PTM is capable of yielding realistic solution estimates compared with the studied optimisation methods.
    VL  - 5
    IS  - 4
    ER  - 

    Copy | Download

Author Information
  • Mathematical Sciences Department, University of Mines and Technology (UMaT), Tarkwa, Ghana

  • Mathematics and Statistics Department, University of Cape Coast (UCC), Cape Coast, Ghana

  • Geological Engineering Department, University of Mines and Technology (UMaT), Tarkwa, Ghana

  • Sections