Inclined specimen, total heads, gradient

Total head at a point = elevation head + pressure head, both measured from the datum: $H = z + h_p$. The piezometer reading $h$ at each point is the pressure head.

(i) Total heads

$$H_1 = z_1 + h_1, \qquad H_2 = z_2 + h_2, \qquad H_3 = z_3 + h_3$$

The total head falls steadily along the specimen from point 1 to 3 because of friction loss in the soil; $H_2$ lies between $H_1$ and $H_3$.

(ii) Hydraulic gradient (entry 1 to exit 3)

Head loss between 1 and 3:

$$\Delta H = H_1 - H_3 = (z_1 + h_1) - (z_3 + h_3)$$

The flow path through the soil is the sample length $L$ measured along the incline, so

$$i = \frac{\Delta H}{L} = \frac{(z_1 + h_1) - (z_3 + h_3)}{L}$$

The inclination $\theta$ does not enter explicitly if $L$ is the length along the sample, because the elevation differences are already included in the $z$ terms. (If only the horizontal distance $L_h$ is known, $L = L_h/\cos\theta$.)