Seepage in Anisotropic soil - Flow Net, Soil Mechanics | Soil Mechanics Notes- Agricultural Engineering PDF Download

 Seepage in Anisotropic soil

The Laplace equation presented in previous lesson (lesson 19) is valid for isotropic soil. If soil is anisotropic and coefficient of permeability in x andz direction is not same, the Laplace equation is modified as:

\[{k_x}{{{\partial ^2}h} \over {\partial {x^2}}} + {k_z}{{{\partial ^2}h} \over {\partial {z^2}}}=0\]                               (20.8)

where k_x and k_z are the coefficient of permeability in x and z direction, respectively. The Eq. (20.8) can be written as:

\[\frac{{{\partial ^2}h}}{{\frac{{{k_z}}}{{{k_x}}}\partial{x^2}}}+\frac{{{\partial^2}h}}{{\partial {z^2}}}=0\]            (20.9)

          Convert the x a new coordinate system x' such that

\[x'=x\sqrt {{{{k_z}} \over {{k_x}}}}\]                    (20.10)

and  \[\partial {x'^2}=\partial {x^2}{{{k_z}} \over {{k_x}}}\] , Thus, Eq.(20.9) can be written as:

\[{{{\partial ^2}h} \over {\partial {{x'}^2}}} + {{{\partial ^2}h} \over {\partial {z^2}}}=0\]                (20.11)

The Eq.(20.11) is Laplace equation for isotropic soil w.r.t x' and z coordinates. Here x coordinate is transformed to x' coordinate [as per Eq. (20.10)] for converting anisotropic soil medium into a fictitious isotropic medium (by keeping z coordinate unchanged). Thus, during the coordinate transformation horizontal dimension (x dimension) is multiplied by \[\sqrt {{{{k_z}} \over {{k_x}}}}\]  . The value of coefficient of permeability for transformed section is taken as:

\[k'=\sqrt {{k_x}{k_z}}\]                  (20.12)

Thus, in this case the quantity of seepage (Q) is calculated as:

\[Q=\sqrt {{k_x}{k_z}} {h_L}{{{N_f}} \over {{N_d}}}\]               (20.13