Differential Equations of saturated flow

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Laplace Equation and its potential:

In cartesian coordinates (x, y, z):

div grad =∇2 Ǿ = Ǿxx + Ǿyy  + Ǿzz =0
In two dimensional flow in (x, y) plane:
∇2 Ǿ = Ǿxx + Ǿyy  =0
In spherical coordinates (r, θ, w);
∇2 Ǿ =(1/r2)[( r2 Ǿ r)r +(1/sin θ)(sin θ.Ǿ θ)θ   +(1/sin2 θ) Ǿww] =0

Inhomogeneous Isotropic Media
Under this media, soil and water compressibilities are considered as negligible , which makes the continuity equation as

div (K grad Ǿ) = K ∇2 Ǿ + grad K. grad Ǿ = 0
Anisotropic Media Inhomogeneous (uniform) isotropic media referred to the three principal axis of K (K1, K2 & K3)
q1=K1J1=-K1 Ǿx ; q2=K2J2=-K2 Ǿy ; q3=K3J3=-K3 Ǿz

By continuity in steady flow or for incompressible media , div q = 0 and we obtain,
K1 Ǿxx + K2 Ǿyy + K3 Ǿzz = 0


If the medium is also inhomogeneous :
          (K1 Ǿx)x + (K2 Ǿy)y + (K3 Ǿz)z = 0

Compressible Fluid & Matrix
When compressibility of the solid matrix ( consolidation by rearrangement of the soil particles under external loading or gravity) is taken into account, different equations from different theories are obtained. but a simpler theory was given by Jacob:
“The compressibility of soil solids are neglected, when compared with that of the water and the apparent compressibility by consolidation”



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