EFFET DU CONDITIONNEMENT DES TRANSMISSIVITES SUR LES CARACTERISTIQUES SPECTRALES DE LA MATRICE DE L’ECOULEMENT EN MILIEU POREUX
Keywords:
groundwater modeling, condition number, ill-conditioning, modèles numériques, conditionnement, rayon spectral, hydrogéologieAbstract
Usually to determine a measure of the degree of ill-conditioning of an algebraic system, we invoke a condition number for the coefficients matrix. In groundwater modeling these coefficients are depending on the
parameters of the heteregeneous porous medium i.e. trans- missivities T. Therefore an estimate of the condition number of the flow matrix is of a considerable interest to assess the accuracy and the reliability of the solution relevant to the numerical method. Ill-conditioning ofthe system may ascribe to many well-known sources.
Herein we inspect condition numbers of flow matricesarising in groundwater modeling flow through synthetics porous media slowly variable. The flow matrices are build up via the numerical generation of conditional log T fields whose variability is indexed on the corresponding variograms. Numerical experiments of Monte Carlo type allowed to study multiple replications of the flowmatrices incorporating available information into estimating the log T values. Results obtained confirm what it was roughly suspected : condition number and spectral radius of the flow matrix are sensitive to the constraints of the local information only if the distance between these points is greater than the correlation length. In such case, the number of the measurement points increases the influence of the standard deviation σY. Otherwise, the a prioriinformation is unvaluable i.e.redundant
References
Gehlar L.W. – Stochastic subsurface hydrology. Prentice Hall Inc., Englewood Cliffs, New Jersey, 1993.
Delhomme J.P. – Spatial variability and uncertaity in
groundwater flow parameters : a geostatistical approach.
Water Res. Research 15 (2), p. 1979.
Kitanidis, P.K. et Vomvoris, E.G.- A geostatistical approach to the inverse problem in groundwater modeling (steady state)
and one-dimensional simulations. Water Resources Research,
vol. 19, no. 3, p. 677-690, 1983.
Mizell S.A., Gutjahr A.L. et Gehlar L.W. – Stochastic analysis of spatial variability in two-dimensional steady groundwater flow assuming stationary and non-stationary heads. Water Resources Research, vol. 18, no. 4, p. 1053-1067, 1982.
Rubin, Y. et Dagan, G. - Stochastic identification of
transmissivity and effective recharge in steady groundwater
flow : 1. Theory. Water Resources Research, vol. 23, no. 7, p.
-1192, 1987.
Dong A. – Estimation géostatistique des phénomènes régis par des équations aux dérivées partielles, Thèse de Doctorat,
ENMS Paris, 1990.
Zhang, D. Numerical solutions to statistical moment equations of groundwater flow in nonstationary, bounded, heterogeneous media. Water Resources Research, vol. 34, no. 3, p. 529-538, 1998.
McLaughin D. , Wood E.F. – A distributed parameter
approach for evaluating the accuracity of groundwater model prediction, 1. Theory. Water Res. Research 24 (7), p.1034-1047, 1988.
Zimmerman D.A., De Marsily, G., Gotway C.A., Marietta,
M.G., Axness, C.L., R.L., Bras, R.L., Carrera, J., Dagan G.,
Davies, P.B.,Gallegos, D.P., Galli, A., Gómez-Hernandez, J.J., Grindrid, P., Gutjahr,A.L., Kitanidis, P.K, Lavenue, A.M., Mclaughin, D., Neuman, S.P.,Ramarao, B.S., Ravenne, C. et Rubin, Y. (1998). A comparison of seven geostatistically based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow. Water Resources Research, vol. 34, no. 6, p.1373-1413.
Ababou R., McLaughlin D., Gehlar L.W., Tompson A.F.B. –
Numerical simulation of saturated/unsaturated flow fields in
randoly heteregeneous porous media. International symp.
On the stochastic approach to subsurface flow. CNRS, Greco 35, ed. De Marsily, Montvillargenne, p. 294-312, 1985.
Wilkinson J.H. – Rounding errors in algebraic process.
Englewoods Cliffs, N.J., Prentice Hall, 1963.
La Porte et Vignes J. – Algorithmes numériques. Analyse et mise en œuvre, tome 1, Arithmétique des ordinateurs.
Systèmes linéaires, éditions Technip, Paris, 1974.
Larson J.L., Sameh A.H. – Algorithms for roundoff error
analysis – A relative approach. Computing, 24, p. 275-297,
Ehrel J. – Statistical estimation of roundoff errors and
condition numbers. Rapport de recherche n° 1490, INRIA, ,
p 27, septembre 1991.
Golub G.H., Van Loan C.F. – Matrix computations.
University press, 1993.
Ehrel J. – Experiments with data perturbations to study
condition numbers and numerical stability. Computing, 51,
, p. 29-44.
Ebrahimian R., Baldick R. – State estimator condition
number analysis. IEEE Transactions on Power Systems, vol
n°2, may 2001.
El Ghaoui L. – Inversion error, condition number and
approximate inverses of uncertain matrices. Linear Algebra
and its Applications, 342, (1-3), 2002.
Dunagan J.and Teng S.H. – Smoothed analysis of Renegar’s condition number for linear programming in SIAM
Conference on Optimization, 2002.
Mosé R., Siegel P., Ackerer Ph., and Chavent, G. –
Application of the mixed hybrid finite element approximation in a groundwater flow model : Luxury or necessity ? Water resources research, vol. 30, pp. 3001-3012, 1994.
Hoteit H., Erhel J., Mosé R., Phillipe B. and Ackerer P.
Numerical reliability and CPU time for the mixed methods
applied to solve the flow problems in porous media. Rap. De
recherche INRIA n° 4228, p.46, 2001.
Benali A.M. – Hétérogénéités des milieux poreux et
instabilité spectrale de la matrice de l’écoulement, Sciences
& Technologie, n°21, p.75-84, 2004.
Ahmed S., de Marsily G. – Comparison of geostatistical
methods for estimating transmissivity using data on
transmissivity and specific capacity. Water resources
research, vol. 23, n°9, p 1717-1737, 1987.
Deutch C.V. et Journel A.G. - GSLIB : Geostatistical
software Library and User’s Guide. Oxford University
Press, 1992.
Bear J. – Dynamics of fluidsin porous media. Elsevier, New
York, 1972.
Freeze R.A., Marsily de G., Smith L., Massman J. – Some
uncertainties about uncertainty. Proceedings of a DOE/AECL Conf. On Geostatistical sensivity and uncertainty methods for groundwater flow and radionuclides transport, San Francicisco, 1989.
Hoeksema, R.J., Kitanidis, P.K. - Comparison of Gaussian
conditional mean and kriging estimation in the geostatistical
solution of the inverse problem. Water Resources Research,
vol. 21, no. 6, p. 825-836, 1985.
Mantoglu A., Wilson J.L.- Simulation of random fields with
the turnig bands methods. Report n° 264, Ralph Parsons
Lab., Dept of Civil Eng., MIT, 1981.
Carr J.R. – Program Cosim : Co-conditional simulation of
spatial data. A fortran code available from the GMM BBS,
dept of geol. engineering university of Missouri, 1992.
Englund E. , Sparks A. – Geostatistical Environmental
Assesment Software. Env. Monitoring Systems Lab. US
EPA, 1991.
Yates S.R., Yates M.V. – Geostatistic for waste management. Robert Kerr Env. Research Lab., Ada, US EPA, 1990.
Benali A. & de Backer L.W. – Identifiability versus
heterogeneity in groundwater modelling systems. Sciences
& Technologie, n°19, p.75-84, 2003.
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