Anisotropy Settings
In a 2D model, the anisotropy of the hydraulic conductivity is defined through a set of material properties and not via the Problem Settings dialog. Conductivity [max] (Transmissivity [max] for confined models) defines the maximum conductivity. The direction of the maximum hydraulic conductivity is defined as Anisotropy angle (defining the angle between the x-axis and the maximum hydraulic conductivity). The ratio between minimum and maximum hydraulic conductivity is finally defined as Anisotropy of conductivity.
In a 3D model, the Anisotropy Settings page of the Problem Settings dialog offers different approaches for the definition of anisotropy of hydraulic conductivity. The directions of these main conductivities can be assigned in three different ways:
- Axis-Parallel Anisotropy: Major principal directions of conductivity (Kxx, Kyy and Kzz) are considered to be coincident with the Cartesian coordinates directed along the x,y and z axes.
- General Anisotropy with computed angles: Three main conductivities (K1m, K2m and K3m) are directed along the axes of a transformed coordinate system. The transformation is done automatically using the inclination of each single element. Using this option, anisotropy can be related to the varying inclination of layers.
- General Anisotropy with user-defined angles: Using Eulerian angles, the rotation is always a sequential process. The order of rotation is according to the 'classic' Euler scheme as an intrinsic rotation z-x'-z' which means that there is a rotation around the z axis with angle Φ, a following rotation around the new x axis (x') with angle Θ, and a third rotation around the new z axis (z') with angle Ψ. The positive direction for the angles is counterclockwise.
- General
Anisotropy with user-defined 3D tensor components: Using a Python interpretator, the 9 components of the hydraulic conductivity tensor can be programmed for each element. IFM access is also made possible. This mode gives access to arbitrary definitions of relationship for each tensor entry.
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FEFLOW uses a classic Euler rotation ('Proper Euler angles') on a z-x'-z' basis (rotation around z twice). This should not be confused with Tait-Bryan angles where there is one rotation around each of the axes. The Euler angles used by FEFLOW can therefore NOT be expressed as yaw, pitch and roll (z-y'-x'' rotation scheme). |
The first rotation is done about the vertical (z axis) by angle Φ:
Based on the new axis of the first rotation, a second rotation is added about the new x axis (here called x') by angle Θ:
Finally, the third rotation is executed about the new z axis (called z'' here) by angle Ψ to obtain the new system x''', y''', z''':
Overlaying all three rotations, the following sequence is done:
The tree hydraulic conductivity values are directed along the new coordinate axes x''' (K_1m), y''' (K_2m) and z''' (K_3m).
The option Generate Euler angles from layer slope computes the Eulerian angles from the slope and the calculated angles can be used as an initial distribution for further manual editing.
Single axis rotation
In many cases, a rotation about one axis might be sufficient. Here are the three options as examples:
Single rotation about the z-axis
For a single rotation about the z-axis, the Euler angles Θ and Ψ are given the value zero [°] and the rotation is done with the angle Φ
Single rotation about the x-axis
For a single rotation about the x-axis, the Euler angles Φ and Ψ are given the value zero [°] and the rotation is done with the angle Θ.
Single rotation about the y-axis
For a single rotation about the y-axis, the Euler angles have to be set as follows:
- Φ = 90 [°] (rotates about the z-axis so that the y-axis is located on the original x-axis),
- Θ = Desired value of rotation about the y-axis (indeed rotates about the former x-axis)
- Ψ = -90 [°] (rotates about the new z-axis so that the final y-axis is in its original position again)
Defining Euler angles from classic strike and dip geological information
Formula comparisons between rotation matrices for strike-dip and Euler angles indicate a direct relation between dip β and Euler angle Θ, strike α and Euler angle Φ, while Euler angle Ψ can be kept as zero. Note that back-calculation of Euler angles produces non-unique results (inherent to trigonometry function properties). Given the definitions for strike and dip measures of a geological bed plane:- Strike α [°] (>> Euler angle Φ): degrees counted CW from North
- Dip β [°] (>> Euler angle Θ): degrees counted CW from strike orthogonal direction
Euler angles deduce from:
- Θ = acos(R[2][2]), R[2][2] = cos(β)
- Φ = acos(R[2][1]/sin(Θ)), R[2][1] = -sin(β)cos(α)
- Ψ = asin(R[0][2]/sin(Θ)), R[0][2] = 0