Steady state two dimensional seepage problem

Question

Assignment Problem:

Consider a steady state two dimensional seepage problem as shown above. There are four different kinds of the boundary conditions in this problem. The pressure value is prescribed on Γ₁, and Γ₂; PΓ₁ = 100 kPa, PΓ₂ = 150 kPa. Γ₃ is an impermeable surface, i.e. the flux in the normal direction to the surface is zero. Γ₄ is a partially permeable so that the flux to the surface is given by J = 1 x 10^-4 e m/s

The hydraulic conductivity of the porous medium is k = 4.2 x 10^-6 m/s.

The governing equation of this problem is given by a Laplace equation, i.e.

2P = ∂²P/∂x² + ∂²P/∂y² = 0

Subjected to

P = P₀ on Γ₀ (Dirichlet condition)

-k∂P/∂n = Jn on Γ_J  (Neumann condition)

Q1. Investigate the problem and build a plan for the numerical analysis.

Q2. Build the solution procedure.

Q3. Solve the system and consider to the following questions.

- Provide the pressure field.

- Calculate the amount of water passing throughout Γ* in an hour. (Approximate the thickness = 1m)

- Provide the field of the maximum flux, i.e.

J_x = -k.∂P/∂_x, J_y = -k.∂P/∂y

J = √(Jx² + Jy²)

Q4. Discuss your result.

Use your own linear algebraic solver.

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