Given:
- Time required for 80% consolidation in a clay layer = 15 years

To find:
- Time required for 80% consolidation in a clay layer that is twice as thick, 15 times more permeable, and 10 times less compressible

Solution:

1. Consolidation time equation:

The time required for consolidation of a soil layer is given by the following equation:

t = (T_v H^2)/(4D_2) x log [(1+e_1)/(1+e_2)]

where,
- t = time for consolidation to be completed
- T_v = coefficient of volume compressibility
- H = thickness of the layer
- D_2 = coefficient of consolidation (also called the coefficient of permeability)
- e_1 and e_2 = void ratios at the start and end of consolidation

2. Comparison of the two clay layers:

Let's compare the two clay layers using the following parameters:

- Thickness: The second clay layer is twice as thick as the first.
- Permeability: The second clay layer is 15 times more permeable than the first.
- Compressibility: The second clay layer is 10 times less compressible than the first.

3. Calculation of parameters for the second clay layer:

- Thickness: If the first clay layer has a thickness of H, then the second clay layer will have a thickness of 2H.
- Permeability: If the coefficient of permeability for the first clay layer is D_1, then the coefficient of permeability for the second clay layer will be 15D_1.
- Compressibility: If the coefficient of volume compressibility for the first clay layer is T_v1, then the coefficient of volume compressibility for the second clay layer will be T_v1/10.

4. Calculation of time required for consolidation of the second clay layer:

Using the equation for consolidation time, we can calculate the time required for the second clay layer as follows:

t_2 = (T_v2 H^2)/(4D_2) x log [(1+e_1)/(1+e_2)]

where,
- t_2 = time required for consolidation of the second clay layer
- T_v2 = coefficient of volume compressibility for the second clay layer
- H = thickness of the second clay layer
- D_2 = coefficient of consolidation (permeability) for the second clay layer
- e_1 and e_2 = void ratios at the start and end of consolidation

Substituting the values from step 3, we get:

t_2 = [(T_v1/10) x (2H)^2]/[4(15D_1)] x log [(1+e_1)/(1+e_2)]

t_2 = (T_v1 H^2)/(120D_1) x log [(1+e_1)/(1+e_2)]

Note: We have used (2H)^2 = 4H^2 in the above equation.

5. Calculation of time ratio:

The time ratio is defined as the ratio of time required for consolidation of the second clay layer to the time required for consolidation of the first clay layer.

time ratio = t_2/t_1