Anisotropic Darcy Velocity, Three-Well Hydraulic Gradient, and Soil Phase Relations: 18-Geol-A2 Hydrogeology, December 2019, Q1

Question 1: Anisotropic Darcy Velocity, Three-Well Hydraulic Gradient, and Soil Phase Relations (equal value)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. (a) Conductivity tensor $K_{xx}=10^{-3}$, $K_{xy}=5.0\times10^{-4}$, $K_{yy}=10^{-4}$ cm/sec; gradients $dh/dx=0.015$, $dh/dy=-0.008$. (b) Well X at the origin, head 180 m.a.s.l.; Well Y 300 m due north, head 200 m.a.s.l.; Well Z 300 m due east, head 190 m.a.s.l. (c) moist mass $M_t=1020$ g, total volume $V_t=650\ \text{cm}^3$, oven-dry mass $M_s=950$ g, grain density $\rho_s=2.62\ \text{g/cm}^3$.

Find. (a) Darcy velocities $q_x$, $q_y$ and the magnitude/direction of the resultant. (b) the magnitude and direction (bearing) of the hydraulic head gradient. (c) dry bulk density, porosity, and saturation of the soil.

Approach. Part (a) applies the tensor form of Darcy's law component by component, then combines the two velocity components into a magnitude and angle. Part (b) is the classical three-point (plane-fit) problem: with Well X as the origin, the two vectors to Y and Z each fix one linear equation in the unknown gradient components $\partial h/\partial x$, $\partial h/\partial y$, solved simultaneously. Part (c) is a standard four-measurement soil phase-relation reduction from the given masses, volume, and grain density.

Part (a) — anisotropic Darcy velocity components. With $q_x=-(K_{xx}\,dh/dx+K_{xy}\,dh/dy)$ and $q_y=-(K_{xy}\,dh/dx+K_{yy}\,dh/dy)$: $$q_x=-\big[(10^{-3})(0.015)+(5.0\times10^{-4})(-0.008)\big]=-1.1\times10^{-5}\ \text{cm/sec},$$ $$q_y=-\big[(5.0\times10^{-4})(0.015)+(10^{-4})(-0.008)\big]=-6.7\times10^{-6}\ \text{cm/sec}.$$ Converting to SI: $q_x=\boxed{-1.10\times10^{-7}\ \text{m/s}}$, $q_y=\boxed{-6.70\times10^{-8}\ \text{m/s}}$ (both components negative, so flow is toward $-x,-y$).
Magnitude and direction. $$|q|=\sqrt{q_x^2+q_y^2}=\sqrt{(1.10\times10^{-7})^2+(6.70\times10^{-8})^2}=\boxed{1.29\times10^{-7}\ \text{m/s}}.$$ The angle from the $+x$ axis is $\theta=\operatorname{atan2}(q_y,q_x)=\boxed{-148.7^{\circ}}$, i.e. $148.7^{\circ}$ measured into the third quadrant — equivalent to a compass bearing of about S58.7°W if $+x$=east and $+y$=north.

Darcy velocity vector plotted on the local $x$-$y$ axes: both components negative, so the resultant points into the third quadrant, about S59°W.

Part (b) — setting up the three-point plane fit. Taking Well X as the origin $(0,0)$ with head $h_0=180$ m, the vector to Y is $(0,300)$ with $\Delta h_{XY}=200-180=20$ m, and the vector to Z is $(300,0)$ with $\Delta h_{XZ}=190-180=10$ m. Fitting a planar head surface $h=h_0+a\,x+b\,y$ gives two independent equations: $$300\,b=20\ \Rightarrow\ b=\frac{\partial h}{\partial y}=0.0667,\qquad 300\,a=10\ \Rightarrow\ a=\frac{\partial h}{\partial x}=0.0333.$$
Gradient magnitude and direction. $$|\nabla h|=\sqrt{a^2+b^2}=\sqrt{0.0333^2+0.0667^2}=\boxed{0.0745\ \text{(m head per m)}}.$$ Both components are positive (head increases toward $+x$/east and $+y$/north), so the gradient points into the first quadrant at $\theta=\operatorname{atan2}(b,a)=\boxed{63.4^{\circ}}$ from east, i.e. bearing N26.6°E — head rises fastest toward the northeast, roughly on a line between Y and Z.

Three-well layout (X at the origin, Y 300 m north, Z 300 m east) with the fitted head-gradient vector.

Part (c) — solid and void volume. The oven-dried solids occupy $V_s=M_s/\rho_s=950/2.62=362.6\ \text{cm}^3$, leaving void volume $V_v=V_t-V_s=650-362.6=\boxed{287.4\ \text{cm}^3}$.
Dry bulk density and porosity. $$\rho_{d}=\frac{M_s}{V_t}=\frac{950}{650}=\boxed{1.46\ \text{g/cm}^3},\qquad n=\frac{V_v}{V_t}=\frac{287.4}{650}=\boxed{0.442\ (44.2\%)}.$$
Saturation. Pore-water mass $M_w=M_t-M_s=1020-950=70$ g, so with water density 1 g/cm³, $V_w=70\ \text{cm}^3$: $$S_r=\frac{V_w}{V_v}=\frac{70}{287.4}=\boxed{0.244\ (24.4\%)}.$$

Quantity	Result
(a) $q_x$, $q_y$	−1.10×10⁻&sup7; m/s, −6.70×10⁻⁸ m/s
(a) $\|q\|$, direction	1.29×10⁻&sup7; m/s at 148.7° from +x (≈S58.7°W)
(b) Gradient $\|\nabla h\|$, direction	0.0745 (m/m) at 63.4° from east (≈N26.6°E)
(c) Dry bulk density $\rho_d$	1.46 g/cm³
(c) Porosity $n$	44.2%
(c) Saturation $S_r$	24.4%

Concept & Theory

Topic. The anisotropic (tensor) form of Darcy's law, the three-point (plane-fit) method for estimating a regional head gradient from three wells, and soil phase relations.

Key relations. $q_i=-K_{ij}\,\partial h/\partial x_j$ for a symmetric 2-D conductivity tensor; planar head fit $h=h_0+a\,x+b\,y$ solved from two well-pair vectors; $n=V_v/V_t$, $\rho_d=M_s/V_t$, $S_r=V_w/V_v$.

Why this works. When hydraulic conductivity varies with direction, a single scalar $K$ cannot describe flow that is not aligned with the head gradient — the off-diagonal term $K_{xy}=K_{yx}$ couples the $x$- and $y$-gradients into each velocity component. Three non-collinear head measurements are the minimum needed to fix a planar head surface in two dimensions (two independent slopes), exactly analogous to the structural-geology three-point problem for a dipping bed. Soil phase relations follow directly once the solid volume is separated from the total volume using the grain density.

Common pitfall. Dropping the cross term $K_{xy}$ (treating the tensor as if only $K_{xx}$, $K_{yy}$ mattered) silently rotates the computed Darcy velocity away from its true direction; using only two wells for the gradient leaves one degree of freedom undetermined (a third, non-collinear point is required); and reporting bulk density as the MOIST mass over volume when "dry bulk density" is asked for uses the oven-dry mass instead.

Source. Freeze & Cherry, Groundwater, Ch. 2.6 (anisotropic conductivity tensors, three-point gradient estimation) and Ch. 2.11 (soil phase relations).

Question 1 of 5: Anisotropic Darcy Velocity, Three-Well Hydraulic Gradient, and Soil Phase Relations

Question 1: Anisotropic Darcy Velocity, Three-Well Hydraulic Gradient, and Soil Phase Relations (equal value)