Exercise 2: Finite difference modelling

Consider the groundwater reservoir outlined in fig. 1. Groundwater is flowing from well no. 1 to well no. 2. It is assumed that the flow is 1-dimentional and steady state. In the region between well no.1 and well no.2 four different soil types are present – each with different hydraulic conductivity, K_a, K_b, K_c, K_d.

K_d

K_c

K_b

K_a

Figure 1: Vertical section of groundwater reservoir.

The flow domain is discretizied info 5 cells with nodal points located in the box centre, see figure 2.

Figure 2: Finite difference discretization.

Please answer the following questions:

Question 1: Calculate the effective hydraulic conductivity between node 2 and node 3 (K₂₃) and between node 3 and node 4 (K₃₄) when the numerical discretization is as shown on figure 2. Formulate the equations symbolic – i.e. length of box 1 =dx₁, length of box 2 =dx₂, etc.

Question 2: Formulate finite differences equations (symbolic) for cell 2, 3 and 4 and let cell 1 and 5 constitute the boundary conditions. Rewrite the equations on the form Ah_i + Bh_i-1 + Ch_i+1=D.

Question 3: The hydraulic conductivity for each of the four different soil types are determined to

K_a= 0.01 m/s

K_b= 0.005 m/s

K_c= 0.1 m/s

K_d= 0.002 m/s

Solve the three finite difference equations and determine the hydraulic heads, h₂, h₃ and h₄