Understanding Water Table Profile in Unconfined Aquifer Flow

The question asks about the shape of the steady water table profile in a specific scenario: one-dimensional flow, without recharge, in an unconfined aquifer situated between two water bodies (like rivers or lakes) that maintain constant water levels.

To determine the shape of the water table profile, we typically use Darcy's Law and apply certain assumptions, especially when dealing with unconfined aquifers. A common approach for this type of problem is based on the Dupuit-Forchheimer assumptions.

Applying Dupuit-Forchheimer Assumptions

The Dupuit-Forchheimer assumptions simplify the analysis of flow in unconfined aquifers. The key assumptions relevant to this problem are:

• The hydraulic gradient is equal to the slope of the water table: $\frac{dh}{dx}$, where $h$ is the height of the water table above the impermeable base and $x$ is the horizontal distance.
• Flow lines are horizontal, and equipotential lines are vertical. This implies that the hydraulic head is constant along a vertical line.

While these assumptions are approximations, they work well for many practical problems where the slope of the water table is small.

Deriving the Water Table Profile Equation

Consider a one-dimensional flow system along the x-axis in an unconfined aquifer with an impermeable base at $z=0$. The water table is at height $h(x)$ above the base. The flow is steady (does not change with time) and there is no recharge (no water added from the surface).

According to Darcy's Law, the discharge per unit width, $q$, is given by:

$$q = -K A \frac{dh}{dx}$$

where $K$ is the hydraulic conductivity and $A$ is the cross-sectional area of flow. In a one-dimensional flow system of unit width, the flow area is the saturated thickness, which is $h(x)$. So, $A = h$.

Applying Darcy's Law with the Dupuit assumptions:

$$q = -K h \frac{dh}{dx}$$

Since the flow is steady and there is no recharge, the discharge $q$ must be constant along the flow path (conservation of mass). We can rearrange the equation:

$$q \, dx = -K h \, dh$$

Now, integrate both sides. Let's assume the flow is from $x_1$ to $x_2$ with water table heights $h_1$ and $h_2$ respectively.

$$\int_{x_1}^{x_2} q \, dx = \int_{h_1}^{h_2} -K h \, dh$$

$$q (x_2 - x_1) = -K \left[ \frac{h^2}{2} \right]_{h_1}^{h_2}$$

$$q (x_2 - x_1) = -K \left( \frac{h_2^2}{2} - \frac{h_1^2}{2} \right)$$

$$q (x_2 - x_1) = \frac{K}{2} (h_1^2 - h_2^2)$$

For a general point $(x, h)$ between the two boundaries, let's consider the flow from $x_1$ with head $h_1$ to a point $x$ with head $h$.

$$q (x - x_1) = \frac{K}{2} (h_1^2 - h^2)$$

Rearranging this equation to express $h^2$ as a function of $x$:

$$h_1^2 - h^2 = \frac{2q}{K} (x - x_1)$$

$$h^2 = h_1^2 - \frac{2q}{K} (x - x_1)$$

$$h^2 = -\frac{2q}{K} x + \left( h_1^2 + \frac{2q}{K} x_1 \right)$$

This equation is of the form $h^2 = Ax + B$, where $A = -\frac{2q}{K}$ and $B = h_1^2 + \frac{2q}{K} x_1$. This is the equation of a parabola with the axis horizontal (along the x-axis) and opening towards the direction of decreasing $h^2$. Since $q$ is the discharge rate and $K$ is the hydraulic conductivity, both positive, $A$ is negative, indicating the parabola opens towards decreasing $x$ (if flow is in the positive x direction). This describes a parabolic shape for the water table profile ($h$ as a function of $x$).

Summary of the Water Table Profile

Under the specified conditions (one-dimensional flow, unconfined aquifer, no recharge, steady state, between two water bodies with fixed heads), the water table profile follows a parabolic shape. The height of the water table squared ($h^2$) varies linearly with the horizontal distance ($x$).

Therefore, the steady water table profile in this specific unconfined aquifer scenario is a parabola.

Revision Table: Unconfined Aquifer Flow Concepts

Additional Information: Groundwater Flow Profiles

The shape of the water table or piezometric surface depends heavily on the type of aquifer (confined or unconfined), the presence of recharge or discharge, and the boundary conditions.

• Confined Aquifer: In a confined aquifer under similar steady, one-dimensional flow conditions between two fixed heads, the piezometric surface profile would be a straight line. This is because the cross-sectional area of flow ($A$) is constant (the aquifer thickness $b$), so Darcy's Law becomes $q = -K b \frac{dh}{dx}$. Integrating this with constant $q$ results in a linear relationship between $h$ and $x$.
• Unconfined Aquifer with Recharge: If there is uniform recharge across the unconfined aquifer, the steady water table profile will deviate from a simple parabola. The governing equation will include a recharge term, leading to a different parabolic-like shape or a more complex curve depending on the specific recharge distribution and boundary conditions.
• Specific Boundary Conditions: The shape can also be influenced by different boundary conditions, such as a constant flux boundary instead of a constant head boundary.

Understanding the assumptions and conditions is crucial for correctly predicting the groundwater surface profile.