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Transient flow towards a well in an aquifer including the effect of fluid inertia

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Abstract

Transient propagation of weak pressure perturbations in a homogeneous, isotropic, fluid saturated aquifer has been studied. A damped wave equation for the pressure in the aquifer is derived using the macroscopic, volume averaged, mass conservation and momentum equations. The equation is applied to the case of a well in a closed aquifer and analytical solutions are obtained to two different flow cases. It is shown that the radius of influence propagates with a finite velocity. The results show that the effect of fluid inertia could be of importance where transient flow in porous media is studied.

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Abbreviations

b :

Thickness of the aquifer, m

c 0 :

Wave velocity, m/s

k :

Permeability of the porous medium, m2

n :

Porosity of the porous medium

p(\(\bar x\),t):

Pressure, N/m2

Q :

Volume flux, m3/s

r :

Radial coordinate, m

r w :

Radius of the well, m

s :

Transform variable

S :

Storativity of the aquifer

S d(r, t):

Drawdown, m

t :

Time, s

T :

Transmissivity of the aquifer, m2/s

\(\bar v\)(\(\bar x\),t):

Velocity of the fluid, m/s

\(\bar x\) :

Coordinate vector, m

z :

Vertical coordinate, m

α :

Coefficient of compressibility, m2/N

β :

Coefficient of fluid compressibility, m2/N

ε :

Relaxation time, s

φ(r, t):

Hydraulic potential, m

μ :

Dynamic viscosity of the fluid, Ns/m2

ξ :

Dimensionless radius

ρ :

Density of the fluid, Ns2/m4

σ(ξ, τ):

Dimensionless drawdown

τ :

Dimensionless time

ζ, x :

Dummy variables

δ 0,δ 1 :

Auxilary functions

References

  1. Bear, J.,Hydraulics of Ground Water. John Wiley and Sons, New York (1979).

    Google Scholar 

  2. Bear, J. and Bachmat, Y., Macroscopic modelling of transport phenomena in porous media. 2: Application to mass, momentum and energy transportTransp.Porous Media, 1 (1986) 241–270.

    Google Scholar 

  3. Bear, J. and Bachmat, Y.,Introduction to Modelling of Transport Phenomena in Porous Media. Kluwer Academic Publishers, Dordrecht (1990) pp. 177, 268–286, 300–301.

    Google Scholar 

  4. Bear, J. and Verruijt, A.,Modelling Groundwater Flow and Pollution. Kluwer Academic Publishers, Dordrecht (1987) pp. 56–65.

    Google Scholar 

  5. Carslaw, H.S. and Jaeger, J.C.,Conduction of Heat in Solids. Oxford University Press, Oxford (1980) pp. 334–339.

    Google Scholar 

  6. Erdelyi, A. (ed.),Higher Transcendental Functions. Vol. 2, McGraw-Hill Book Company, New York (1953) p. 62.

    Google Scholar 

  7. Foster, W.R., McMillen, J.M. and Odeh, A.S., The equations of motion of fluids in porous media: 1. Propagation velocity of pressure pulses.Soc. Pet. Eng. J. 7 (1967) pp. 333–341.

    Google Scholar 

  8. Morin, R.H. and Olsen, H.W., Theoretical analysis of the pressure response from a constant flow rate hydraulic conductivity test.Water Resour. Res. 23 (1987) 1461–1470.

    Google Scholar 

  9. Muskat, M.,The Flow of Homogeneous Fluids Through Porous Media, McGraw-Hill Book Company, New York (1937) pp. 130–131.

    Google Scholar 

  10. Oroveanu, T. and Pascal, H., On the propagation of pressure waves in a liquid flowing through a porous medium.Rev. Mec. Appl., Acad. R.P.R. 4 (1959) pp. 445–448.

    Google Scholar 

  11. Pascal, H., Pressure wave propagation in a fluid flowing through a porous medium.Int. J. Eng. Sci. 24,No. 9 (1986) 1553–1570.

    Google Scholar 

  12. Polubarinova-Kochina, P. Ya.,Theory of Ground Water Movement. Princeton University Press, Princeton (1962) pp. 22–23, 481–484.

    Google Scholar 

  13. Rehbinder, G., Darcyan flow with relaxation effect.Appl. Sci. Res. 46 (1989) 45–72.

    Google Scholar 

  14. Shapiro, A.M., Interpretation of oscillatory water levels in observation wells during aquifer tests in fractured rock.Water Resour. Res. 25 (1989) 2129–2137.

    Google Scholar 

  15. van der Kamp, G., Determining aquifer transmissivity by means of well response tests: the under damped case.Water Resour. Res. 12 (1976) 71–77.

    Google Scholar 

  16. Whitaker, S., Flow in porous media 1: A theoretical derivation of Darcy's law.Transp. Porous Media 1 (1986) 3–25.

    Google Scholar 

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Author notes

  1. Torbjörn Löfqvist

    Present address: Division of Physics, Luleå University of Technology, S-951 87, Luleå, Sweden

  2. Göran Rehbinder

    Present address: Department of Hydraulics Engineering, The Royal Institute of Technology, S-100 44, Stockholm, Sweden

Authors and Affiliations

  1. Division of Fluid Mechanics, Luleå University of Technology, S-951 87, Luleå, Sweden

    Torbjörn Löfqvist & Göran Rehbinder

Authors
  1. Torbjörn Löfqvist
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  2. Göran Rehbinder
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Löfqvist, T., Rehbinder, G. Transient flow towards a well in an aquifer including the effect of fluid inertia. Appl. Sci. Res. 51, 611–623 (1993). https://doi.org/10.1007/BF00868003

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  • DOI: https://doi.org/10.1007/BF00868003

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