Ismael Herrera andMultilayered Aquifer Theory

By Alex Cheng, University of MississippiSimposio Ismael Herrera Avances en Modelación Matemática en Ingeniería y Geosistemas

UNAM, Mexico, Miércoles 28 de septiembre

EARLY PIONEERS OF GROUNDWATER FLOW

Henry Philibert Gaspard Darcy(1803-1858)

Darcy’s Law (1856)

Arsene Jules Emile Juvenal Dupuit (1804-1866)

Steady state flow toward pumping well in unconfined and confined aquifers (1863)

Dupuit Approximation

Philip Forchheimer (1852-1933)

Laplace equation (1886)

Charles V. Theis(1900-1987)

Theis, C. V. (1935), The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using ground water storage, Transactions-American Geophysical Union, 16, 519-524.

LEAKY AQUIFER THEORY

Charles E. Jacob (?-1970)

Jacob, C. E. (1946), Radial flow in a leaky artesian aquifer, Transactions, American Geophysical Union, 27(2), 198-205.

Mahdi S. Hantush (1921–1984)

Hantush, M. S., and C. E. Jacob (1955), Non-steady radial flow in an infinite leaky aquifer, Transactions, American Geophysical Union, 36(1), 95-100.

MULTILAYERED AQUIFER SYSTEM

S.P. Neuman & P.A. Witherspoon (1969)

I. Herrera (1969, 1970)

• Herrera, I., and G. E. Figueroa (1969), A correspondence principle for theory of leaky aquifers, Water Resources Research, 5(4), 900-904.

• Herrera, I. (1970), Theory of multiple leaky aquifers, Water Resources Research, 6(1), 185-193.• Herrera, I., and L. Rodarte (1972), Computations using a simplified theory of multiple leaky aquifers,

Geofisica International, 12(2), 71-87.• Herrera, I., and L. Rodarte (1973), Integrodifferential equations for systems of leaky aquifers and

applications .1. Nature of approximate theories, Water Resources Research, 9(4), 995-1004.• Herrera, I. (1974), Integrodifferential equations for systems of leaky aquifers and applications .2.

Error analysis of approximate theories, Water Resources Research, 10(4), 811-820.• Herrera, I. (1976), A review of the integrodifferential equations appraoch to leaky aquifer

mechanics, Advances in Groundwater Hydrology, September, 29-47.• Herrera, I., and R. Yates (1977), Integrodifferential equations for systems of leaky aquifers and

applications .3. Numerical-methods of unlimited applicability, Water Resources Research, 13(4), 725-732.

• Herrera, I., A. Minzoni, and E. Z. Flores (1978), Theory of flow in unconfined aquifers by integrodifferential equations, Water Resources Research, 14(2), 291-297.

• Herrera, I., J. P. Hennart, and R. YATE (1980), A critical discussion of numerical models for muItiaquifer systems, Advances in Water Resources, 3, 159-163.

• Hennart, J. P., R. Yates, and I. Herrera (1981), Extension of the integrodifferential approach to inhomogeneous multi-aquifer systems, Water Resources Research, 17(4), 1044-1050.

• Chen, B., and I. Herrera (1982), Numerical treatment of leaky aquifers in the short-time range, Water Resources Research, 18(3), 557-562.

MATHEMATICAL FORMULATION AND NUMERICAL SOLUTION

Solution Mesh for General Groundwater Problem (3 Spatial + 1 Temporal = 4D)

Neuman-Witherspoon Formulation

(3 spatial + 1 temporal) = 4D

Herrera Integro-Differential Formulation

(2 Spatial + 1 Temporal) = 3D

MY ACQUAINTANCE WITH PROF. HERRERA

Cheng & Ou (1989) Laplace Transform + FDM

2 Spatial Dimension = 2D

Cheng & Morohunfola (1993) Laplace Transform + BEM (1 Spatial Dimension)

1 Spatial Dimension = 1D

Green’s Function (Pumping Well Solution)

Two Aquifer One Aquitard System

OTHER COLLABORATIONS AND COMMON RESEARCH AREAS

Trefftz Method

Walter Ritz (1878–1909)

Erich Trefftz (1888-1937)

Trefftz, E. (1926), Ein Gegenstück zum Ritz’schen verfahren (A counterpart to Ritz method), in Verh d.2. Intern Kongr f Techn Mech (Proc. 2nd Int. Congress Applied Mechanics), edited, pp. 131-137, Zurich.

“Ritz’s idea was to use variational method and trial functions to minimize a functional, in order to find approximate solutions of boundary value problems. Typically, trial functions are polynomials or elementary functions. Trefftz’s contribution was to use the general solutions of the partial differential equation as trial functions.” Cheng & Cheng (2005), History of BEM

International Workshop on the Trefftz MethodFirst Workshop: Cracow, May 30-June 1, 1996.Second Workshop: Sintra, Portugal, September 1999.Third Workshop: University of Exeter, UK, 16-18 September 2002.Fourth Workshop: University of Zilina, Slovakia, 23-26 August 2005.Fifth Workshop: Katholieke Universiteit Leuven, Belgium, 2008.Trefftz/MFS 2011: National Sun Yat-sen University, Taiwan, 2011.