Exercise 3 Solution

The cumulative land subsidence is due to the compaction of the three layers (Equation 13):

η_tot = η_aquifer1 + η_aquitard + η_aquifer2

The compaction of each aquifer can be obtained by means of Equation 12:

η_aquifer1 = t_aquifer1 c_b,aquifer1 ∆σ_z,aquifer1

η_aquifer2 = t_aquifer2 c_b,aquifer2 ∆σ_z,aquifer2

The ultimate aquitard compaction amounts to (Equation 28):

η_aquitard = t_aquitard c_b,aquitard 0.5 (∆σ_z,aquifer1 + ∆σ_z,aquifer2)

The sum becomes:

η_tot = (t_aquifer1 c_b,aquifer1 ∆σ_z,aquifer1) + (t_aquitard c_b,aquitard 0.5 (∆σ_z,aquifer1 + ∆σ_z,aquifer2)) + (t_aquifer2 c_b,aquifer2 ∆σ_z,aquifer2)

Taking into account the relationships between the compressibility of the three layers, c_b,aquifer1 = 2 c_b,aquifer2 and c_b,aquifer1 = 0.1 c_{b, aquitard} we can write the expression in terms of the compressibility of aquifer 1:

η_tot = (t_aquifer1 c_b,aquifer1 Δσ_z,aquifer1) + (t_aquitard 10 c_b,aquifer1 0.5(∆σ_z,aquifer1 + ∆σ_z,aquifer2)) + (t_aquifer2 0.5 c_b,aquifer1 ∆σ_z,aquifer2)

Substituting the known subsidence (0.1 m), the decreased pressure in aquifers 1 and 2 (20 m and 15 m, and converting those pressures from height of water to kg/m³, so 20,000 kg/m³and 15,000 kg/m³), and their thicknesses (20 m and 40 m), as well as the 15 m aquitard thickness, results in:

There are 9.806710^‑5 bars per 1 kg/m² , so:

0.1 m = (20 m c_b,aquifer1 1.96 bar) + (15 m 10 c_b,aquifer1 0.5 (1.96bar + 1.47 bar)) + (40 m 0.5 c_b,aquifer1 1.47bar)

carrying out the multiplications produces:

0.1 m = (39.2 m bar c_b,aquifer1) + (257.4 m bar c_b,aquifer1) + (29.4 m bar c_b,aquifer1)

0.1 m = 326.1 m bar c_b,aquifer1

and,

c_b,aquifer2 = 0.5 c_b,aquifer1 = 1.5 × 10^-4 bar^-1

and

c_b,aquitard = 10 c_b,aquifer1 = 3.1 × 10^-3 bar^-1.

Return to Exercise 3