Anisotropy is a phenomenon for which the permeability *k* is not equal along the x- and y Cartesian axis, *k\({}_{x}\)\({}_{x}\)* and *k\({}_{y}\)\({}_{y}\)*, respectively. It can be notated that for isotropic conditions *k\({}_{x}\)\({}_{x}\)* = *k\({}_{y}\)\({}_{y}\)* (see Figure 12.6a), and for anisotropic conditions *k\({}_{x}\)\({}_{x}\)* \(\neq \) *k*\({}_{y}\)\({}_{y}\) (see Figure 12.6b).

To express the amount of flow along the x- and y-axes of a Cartesian coordinate system, the following equations are valid to compute the flow along these direction; *q\({}_{x}\)* and *q\({}_{y}\)*, respectively (Strack ODL (1989), *Groundwater Mechanics*, Princeton Hall, Inc., Englewood Cliffs, New-Jersey):

\( \seteqsection {12} \)

\( \seteqnumber {2} \)

\begin{equation} \label {GrindEQ.1} \left [\begin {array}{c} {q_{x} } \\ {q_{y} } \end {array}\right ]=\left [\begin {array}{cc} {-k_{xx} } & {-k_{xy} } \\ {-k_{yx} } & {-k_{yy} } \end {array}\right ]\left [\begin {array}{c} {\frac {\partial h_{x} }{\partial x} } \\ {\frac {\partial h_{y} }{\partial y} } \end {array}\right ] \end{equation}

From equation *(12.2)*, it can be seen that in anisotropic conditions (*k\({}_{x}\)\({}_{x}\)* \(\neq \) *k*\({}_{y}\)\({}_{y}\)), flow along the x-direction is not influenced solely by the hydraulic gradient along this x-axis, but also by a hydraulic gradient along the y-axis. The permeability’s *k\({}_{xy}\)* and *k\({}_{xy}\)* are equal to each other and depend on the angle *\(\varphi \)* of the principal axis to the x-axis:

\( \seteqsection {12} \)

\( \seteqnumber {3} \)

\begin{equation} \label {GrindEQ.2} \begin {array}{l} {k_{xx} =f\times T\times \cos (\varphi )^{2} +T\times \sin (\varphi )^{2} } \\ {k_{xy} =k_{yx} =((f\times T)-T)\times \cos (\varphi )\times \sin (\varphi )} \\ {k_{yy} =f\times T\times \sin (\varphi )^{2} +T\times \cos (\varphi )^{2} } \end {array} \end{equation}

For values *\(\varphi \)=0.0; \(\varphi \)=90.0; \(\varphi \)=180.0; \(\varphi \)=270.0, k\({}_{xy}\)* and *k\({}_{xy}\)\({}_{ }\)*become 0.0.

Anisotropy is expressed by an angle *\(\varphi \)* and anisotropic factor *f*. The angle *\(\varphi \)* denotes the angle along the main principal axis (highest permeability *k*) measured in degrees from north (0\({}^{\circ }\)), east (90\({}^{\circ }\)), south (180\({}^{\circ }\)) and west (270\({}^{\circ }\)). The anisotropic factor *f* is **perpendicular** to the main principal axis. The factor is between 0.0 (full anisotropic) and 1.0 (full isotropic), see Figure 12.7.

Most optimally, the model discretisation should follow the configuration of the anisotropy, see Figure 12.8a. However, anisotropy could be folded in many different directions (principal directions), which probably yield for anisotropy in many angles throughout the modeling domain. With the chosen mathematical method (finite-differences) in iMODFLOW, it is impossible to fold the model network according to the anisotropy, see Figure 12.8b.

Since the principal direction of the permeability is not aligned to the axes of the modeling network, it is necessary to add extra flow terms to the finite difference equation to take into account the diagonal flow, caused by the anisotropy, see Figure 12.9.

For more detailed explanation on the computation of these extra flow terms, see Vermeulen PTM (2006) *et al. Limitation to Upscaling of Groundwater Flow Models dominated by Surface Water Interaction,* Water Resources Research 42, W10406, doi:10.1029/2005WR004620*.*

For each cell in the model network, anisotropic angles *\(\varphi \)* and factors *f* can be specified. For those situations where a single model cell contains more than one of these anisotropic parameters, they will be up-scaled to the model cell. For the anisotropic angle, the most frequent occurrence will be used, as for the anisotropic factor, a mean value will be computed. This seems to be the most robust and fair trade-off between a coarsened model network and loss in detail.

The ANI horizontal anisotropy corresponds with the TRPY variable specified in the MODFLOW BCF package and the HANI variable specified in the MODFLOW LPF package.