Refraction of flow lines at soil interfaces — a flow line meets the interface between soils of permeability k1 and k2 making angles theta1 and theta2 with the normal before and after crossing. According to the law of deflection at a boundary between dissimilar isotropic soils, which relation holds?

Introduction / Context:
When seepage transitions from one soil layer to another with a different coefficient of permeability, the flow lines “refract” at the interface, similar in spirit to optical refraction. The correct relationship is essential for sketching accurate flownets across stratified deposits and for estimating exit gradients and seepage quantities.

Given Data / Assumptions:

• Steady-state, 2D, Darcian flow in isotropic layers.
• Angles theta1 and theta2 are measured from the normal to the interface.
• Continuity of head and discharge holds across the boundary.

Concept / Approach:

Equate tangential components of specific discharge across the boundary using Darcy’s law: q_t = k * i_t. For a given head drop across an elemental strip, the tangential hydraulic gradient i_t is proportional to tan(theta), leading to k1 * tan(theta1) = k2 * tan(theta2). This is sometimes rearranged as tan(theta1) / tan(theta2) = k2 / k1. The result preserves orthogonality between flow lines and equipotentials while changing the “mesh aspect ratio” as k varies.

Step-by-Step Solution:

Verification / Alternative check:

Construct a flownet with equal potential drops; the rectangles distort in proportion to permeability contrast, and checking the ratio of tangents against k values confirms the law numerically.

Why Other Options Are Wrong:

sin/cos forms do not arise from tangential Darcy continuity with angles referenced to the normal; equality of angles ignores permeability contrast; the inverted ratio in option D is simply a rearrangement but is not the standard compact expression.

Common Pitfalls:

Measuring angles from the tangent rather than the normal; applying the relation to anisotropic media without coordinate transformation.

Final Answer:

k1 * tan(theta1) = k2 * tan(theta2)