Abstract
This chapter deals with the basic equation governing groundwater flow. In Sect. 2.1, the equations are presented in their full four-dimensional form (three spatial dimensions + time) with emphasis on the parameters. Section 2.2 introduces Calderón’s approach to determination of the spatially heterogeneous hydraulic conductivity field. To avoid negative hydraulic conductivities in the double constraint methodology (DCM), this approach is based on the square root of the hydraulic conductivity (sqrt-conductivity α). Section 2.3 introduces Stefanescu’s α-center method for parameter estimation, while Sect. 2.4 analyzes Calderón’s approach in more depth using inspiration from Stefanesco’s method. Although this chapter sets the scene for the real subject matter of this book—the double constraint methodology (DCM)—its reading may be skipped by readers who want to go directly to the DCM treated in Chap. 3.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Barber D, Brown B (1984) Applied potential tomography. J Phys E: Sci Instrum 17:723–733
Barber DC, Brown BH (1986) Recent developments in applied potential tomography-APT. In: Bacharach SL (ed) Information processing in medical imaging. Martinus Nijhoff, Amsterdam, pp 106–121
Bear J (1972) Dynamics of fluids in porous materials. American Elsevier Publ. Co., New York
Borcea L (2002) Electrical impedance tomography. Inverse Prob 18:99–136
Borcea L (2003) Addendum to electrical impedance tomography. Inverse Prob 19:9978
Borcea L, Gray GA, Zhang Y (2003) Variationally constrained numerical solution of electrical impedance tomography. Inverse Prob 19:1159–1184. PII: S0266-5611(03)57791-7
Bresciani E, Gleeson T, Goderniaux P, de Dreuzy JR, Werner AD, Wörman A, Zijl W, Batelaan O (2016) Groundwater flow systems theory: research challenges beyond the specified-head top boundary condition. Hydrogeol J 24:1087–1090. https://doi.org/10.1007/s10040-016-1397-8
Brown RM (1996) Global uniqueness in the impedance imaging problem for less regular conductivities. SIAM J Math Anal 27:1049–1056
Brown RM, Uhlmann G (1997) Uniqueness in the inverse conductivity problem for non-smooth conductivities in two dimensions. Commun Part Diff Eqns 22:1009–1027
Bukhgeim AL, Uhlmann G (2002) Recovering a potential from partial Cauchy data. Comm Partial Diff Eqns 27:653–668
Butkov E (1973) Mathematical physics. Addison Wesley Publ. Comp., Reading
Calderón AP (1980) On an inverse boundary value problem. Seminar on numerical analysis and its applications to continuum physics (Rio de Janeiro, 1980). Soc Brasil Mat, 65–73, Rio de Janeiro. Also see the reprint: Calderón AP (2006) On an inverse boundary value problem. Comput Appl Math 25(2–3):133–138
Chavent G, Jaffré J (1986) Mathematical models and finite elements for reservoir simulation. Elsevier, North-Holland, Amsterdam
Chen Z, Zhang Y (2009) Well flow models for various numerical methods. Int J Num Anal Mod 6(3):375–388
Cheney M, Isaacson D, Newell JC (1999) Electrical impedance tomography. SIAM Rev 41:85–101
Duvaut G, Lions JL (1976) Inequalities in mechanics and physics. Springer, Berlin. ISBN 978-3-642-66167-9 (Print) 978-3-642-66165-5 (Online)
El-Rawy M, Batelaan O, Zijl W (2015) Simple hydraulic conductivity estimation by the Kalman filtered double constraint method. Groundwater 53(3):401–413. https://doi.org/10.1111/gwat.12217
El-Rawy M, De Smedt F, Batelaan O, Schneidewind U, Huysmans M, Zijl W (2016) Hydrodynamics of porous formations: Simple indices for calibration and identification of spatio-temporal scales. Mar Petrol Geol 78:690–700. https://doi.org/10.1016/j.marpetgeo.2016.08.018
De Smedt F, Zijl W (2018) Two- and three-dimensional flow of groundwater. CRC Press, Taylor & Francis Group, Boca Raton
Holder DS (2005) Electrical impedance tomography. IOP Publishing, Bristol
Kenig C, Sjöstrand J, Uhlmann G (2007) The Calderón problem with partial data. Ann Math 165:567–591
Morse PM, Feshbach H (1953) Methods of theoretical physics. McGraw-Hill, New York
Nachman AI (1996) Global uniqueness for a two-dimensional inverse boundary problem. Ann Math 143:71–96
Olmstead WE (1968) Force relationships and integral representations for the viscous hydrodynamical equations. Arch Ration Mech Anal 31:380–390
Peaceman DW (1977a) Interpretation of well-block pressures in numerical reservoir simulation. SPE 6893, 52nd annual fall technical conference and exhibition, Denver
Peaceman DW (June 1983) Interpretation of well-block pressures in numerical reservoir simulation with non-square grid blocks and anisotropic permeability. Soc Pet Eng J: 531–543
Peaceman DW (Feb 1991) Presentation of a horizontal well in numerical reservoir simulation. SPE 21217, presented at 11th SPE symposium on reservoir simulation in Ananheim, California, 17–20
Salo M (2008) Calderón problem. Lecture notes, Spring 2008 Mikko Salo, Department of Mathematics and Statistics, University of Helsinki. http://users.jyu.fi/~salomi/lecturenotes/calderon_lectures.pdf
Siltanen S, Mueller JL, Isaacson D (2000) An implementation of the reconstruction algorithm of A. Nachman for the 2-D inverse conductivity problem. Inverse Prob 16:681–699
Stefanescu SS (1950) Theoretical models of heterogeneous media for electrical prospecting methods with direct currents (in French). Comitetul Geologic, Studii Technice si Economice, Seria D, Nr. 2, Studii Technice si Economice, Imprimeria National, Bucuresti: 51–71
Sylvester J, Uhlmann G (1987) A global uniqueness theorem for an inverse boundary value problem. Ann Math 125:153–169
Tóth J (2009) Gravitational systems of groundwater flow. Cambridge University Press, Cambridge
Trykozko A, Zijl W, Bossavit A (2001) Nodal and mixed finite elements for the numerical homogenization of 3D permeability. Comput Geosci 5:61–64
Uhlmann G (2003) Electrical impedance tomography and Calderon’s problem. Med Eng Phys 25:79–90 https://www.math.washington.edu/~gunther/publications/Papers/calderoniprevised.pdf
Uhlmann G (2009) Electrical impedance tomography and Calderón’s problem
Inverse Problems 25:123011 (39 p) doi:https://doi.org/10.1088/0266-5611/25/12/123011
Yeh WWG (1986) Review of parameter identification procedures in groundwater hydrology: The inverse problem. Water Resour Res 22(2):95–108
Zijl W, Nawalany M (1993) Natural groundwater flow. Lewis Publishers, Boca Raton
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2018 The Author(s)
About this chapter
Cite this chapter
Zijl, W., De Smedt, F., El-Rawy, M., Batelaan, O. (2018). Foundations of Forward and Inverse Groundwater Flow Models. In: The Double Constraint Inversion Methodology. SpringerBriefs in Applied Sciences and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-71342-7_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-71342-7_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-71341-0
Online ISBN: 978-3-319-71342-7
eBook Packages: Earth and Environmental ScienceEarth and Environmental Science (R0)