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with θ the volumetric water content:
θ = nS r
[8.23]
The variable n designates the porosity defined as the ratio between the voids and
the total volume (voids plus solid particles). The right member of this equation [8.22]
is called the storage term. Assuming that the solid particles are not deformable, the
following relation:
∂n
∂t =(1−n) ∂ε v
[8.24]
∂t
can be obtained where ε v is the relative total volume change. Hence the storage term
can be formulated as
∂θ
∂t =(1−n)S r ∂ε v
∂t + n ∂S r
∂p c
∂t
[8.25]
∂p c
Then, by substituting the expressions of v and ∂θ/∂t in [8.22], we obtain the
equation of the water transfer in unsaturated soils:
=(1−n)S r ∂ε v
∂t + n ∂S r
∂p c
∂t
div
k(S r ) ∇(h)
[8.26]
∂p c
This equation involves two characteristics of the water behavior in soil: the
permeability as a function of the degree of saturation and the water retention curve. This
curve gives the degree of saturation according to the capillary pressure p c = u a − u w ,
with u a the air pressure, and u w the water pressure.
As far as the permeability is concerned, we assume simply that it varies in a linear
manner with the degree of saturation and the water isotropic permeability k 0 :
k = S r k 0
[8.27]
As for the water retention curve, we use the Van Genuchten model [VAN 80b]:
1 − S r0
S r = S r0 +
[8.28]
1+(αp c ) β 1−1/β
with S r0 the residual saturation degree, and α and β the calibration parameters. Even
if this model is commonly used, it does not take into account the hysteresis effect
experimentally observed between drying and wetting phases.
Finally, the hydromechanical coupling is done with the help of Bishop's effective
stress concept [BIS 63]:
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