Question

Identify the valid assumptions applied in Laplacian equation:
(A) Water is compressible
(B) The soil is incompressible and porous.
(C) The quantity of water entering the soil in any given time is the same as the quantity flowing out of soil.
(D) Darcy's law is valid for the given soil and hydraulic boundary conditions at the entry and exit are known.
Choose the correct answer from the options given below:

• A. (B), (C) and (D) only.
• B. (A), (B) and (C) only.
• C. (A), (B) and (D) only.
• D. (A), (C) and (D) only.

Hint:

The Laplace equation for two-dimensional, steady-state, isotropic flow is:
\[ \frac{\partial^2 h}{\partial x^2} + \frac{\partial^2 h}{\partial z^2} = 0 \] This equation is solved graphically using flow nets consisting of orthogonal flow lines and equipotential lines.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The Laplacian equation (\( \nabla^2 \phi = 0 \)) is the governing differential equation for steady-state groundwater flow through a porous medium.
It is derived by combining the equation of continuity with Darcy's law, and is used to plot flow nets and calculate seepage discharge under dams and wells.

Step 2: Detailed Explanation:
Let us analyze the assumptions required to derive and apply the Laplace equation:
1.Incompressibility of water: Water is assumed to be completely incompressible.
This means its density remains constant throughout the flow field.
Thus, assumption (A) is invalid because water is assumed to be incompressible, not compressible.
2.Incompressibility of soil skeleton (B): The soil solid skeleton is assumed to be incompressible and stable.
The soil does not compact or expand during flow, meaning the porosity remains constant.
Thus, (B) is a valid assumption.
3.Steady-state flow Continuity (C): The quantity of water entering a given unit volume of soil is equal to the quantity flowing out of it in any given time.
There is no change in water storage over time (\( \frac{\partial \theta}{\partial t} = 0 \)).
Thus, (C) is a valid assumption.
4.Validity of Darcy's Law (D): The flow must be laminar, making Darcy's law (\( v = K \cdot i \)) valid.
In addition, the hydraulic boundary conditions at the boundaries of the flow domain must be known to solve the differential equation.
Thus, (D) is a valid assumption.

Step 3: Final Answer:
The valid assumptions for the Laplace equation are (B), (C), and (D) only.