Specific recharge along the top boundary is 1 mm/day. Write equation for head along the boundary as function of conductivity and recharge variable. Is it h(head)=R(recharge)/K(conductivity)?
To determine the equation for the head along the top boundary, we can start by considering Darcy's law for groundwater flow. Darcy's law relates the groundwater flow velocity (q) to the hydraulic conductivity (K) and the hydraulic gradient (∇h):
q = -K ∇h
Where:
- q represents the specific discharge (volume of water flowing per unit area per unit time)
- K is the hydraulic conductivity (a measure of how easily water can flow through a porous medium)
- ∇h is the hydraulic gradient (the change in head over a given distance)
Next, we can assume that the flow is one-dimensional, meaning it only occurs in the vertical direction. Therefore, the hydraulic gradient (∇h) simplifies to the change in head (∆h) divided by the distance (∆x):
∇h = ∆h / ∆x
Now, let's consider the specific recharge (R) along the top boundary. Specific recharge represents the rate at which water is added to the groundwater system per unit area. We can express it as:
R = q / A
Where:
- R is the specific recharge (volume of water per unit area per unit time)
- q is the specific discharge (volume of water flowing per unit area per unit time)
- A is the area (horizontal cross-sectional area perpendicular to the flow direction)
Since we assumed one-dimensional flow, the area (A) can be represented as the width (w) multiplied by the distance (∆x):
A = w * ∆x
Combining the previous equations, we can express specific recharge as:
R = (-K ∆h / ∆x) / (w * ∆x)
Simplifying further, we have:
R = -K ∆h / (w * ∆x^2)
Finally, rearranging the equation to solve for ∆h, we obtain:
∆h = -R * (w * ∆x^2) / K
This equation represents the change in head (∆h) along the top boundary as a function of the specific recharge (R), width (w), distance (∆x), and hydraulic conductivity (K).