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Tensiometer reaction related to its filter dimensions

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Abstract

A theoretical model for tensiometers is presented. It is based on a new physical considerations: the tensiometer filter is a quasi-saturated porous medium and the transmission fluid in the cavity is in hydrostatic equilibrium and is incompressible. The evolution equations form a complete system which could be used and coupled in a wide number of situations once filter dimensions and geometry have been correctly defined. The model is applied to tensiometer design and leads to new design recommendations. It predicts the existence of two distinct evolution modes for tensiometers. The time constant of the first varies linearly with the ratio of filter thickness to contact area and that of the second varies according to the square of the filter thickness and is independent on the contract area. The model leads to the formulation of an equation for fine-filter tensiometers. This extends Richards and Neal's equation by taking fine-filter geometry and gravity into account.

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Abbreviations

A :

area of surfaceS i

A n ,B n :

coefficients defined in Appendix B

C :

filter capacity

da :

boundary integration element

g :

constant gravity vector field

K :

permeability

L :

filter thickness

M f :

mass of transmission fluid exchanged for a unit variation of the potential

M mn :

components of a matrix defined in Appendix B

n :

porosity

n :

outward unit vector to filter rim (boundary)

N :

number of terms (Appendix B)

p :

pressure

P i ,p e :

internal, external pressure

q :

volume flux

r :

variable defined in Appendix B

r n :

coefficients defined in Appendix B

S :

global sensitivity and gauge sensitivity

S i ,S e ,S r :

filter rim

S f :

saturation of filter

t :

time

U :

velocity of the transmission fluid in the cavity

V :

volume of filter

V c :

volume of cavity

V p :

volume of parasitic fluid

x :

positional vector

z :

spatial coordinate

Β p :

compressibility of the parasitic fluid

Φ :

potential

Φ e :

potential outside of tensiometer

Φ i :

potential inside of tensiometer

Φ 0 :

initial potential

Φ p :

potential of parasitic fluid

η :

adimensional parameter defined in (5.8)

κ :

conductance

Μ :

dynamic viscosity

π :

pi

ρ :

density of transmission fluid

ρ p :

density of parasitic fluid

Τ :

temporal parameter and time constant

Ω :

adimensional temporal coordinate

ζ :

adimensional spatial coordinate

∇:

gradient operator

a·b :

scalar product ofa andb

a×b :

vector product ofa andb

Φ′:

partial derivative ofΦ with respect toζ

\(\dot \phi\) :

partial derivative ofΦ with respect toΩ

\(\bar \phi _e (t)\) :

mean geometrical value ofΦ e(t,x) defined in (4.7)

x ε V :

x belongs toV

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Authors and Affiliations

  1. Lombardi Engineering Ltd., Via Rinaldo Simen 19, CH-6648, Minusio-presso-Locarno, Switzerland

    Mohamed El Tani

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  1. Mohamed El Tani
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El Tani, M. Tensiometer reaction related to its filter dimensions. Transp Porous Med 20, 235–250 (1995). https://doi.org/10.1007/BF01073174

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  • DOI: https://doi.org/10.1007/BF01073174

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