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# Terzaghi's One-Dimensional Consolidation Theory

## Assumptions

1. Loading is one-dimensional. Settlement and flow of water is vertical.
2. Compressibility is constant
3. Permeability is constant
4. Flow is controlled by Darcy's law
5. Secondary compression does not occur
6. Deformations are small, such that strains may be computed from undeformed geometry
7. Soil is saturated and uniform

## Volume Change Based on Flow

Darcy's law:

$$\qquad\frac{dQ}{dt} = k i A = k i dx dy$$;

Flow out of element:

$$\qquad dQ_{out} = k i_z dx dy dt$$

Hydraulic gradient at depth z:

$$\qquad i_z = \frac{1}{\gamma_w}\frac{\partial u_e}{\partial z}$$

Flow into element:

$$\qquad dQ_{in} = k i_{z+dz} dx dy dt$$

Hydraulic gradient at depth z + dz:

$$\qquad i_{z+dz} = i_z + \frac{\partial i_z}{\partial z}dz$$
$$\qquad i_{z+dz} = \frac{1}{\gamma_w}\frac{\partial u_e}{\partial z} + \frac{1}{\gamma_w}\frac{\partial^2 u_e}{\partial z^2}dz$$

Change in flow:

$$\qquad dQ = dQ_{out} - dQ_{in}$$ (Compression positive)

$$\qquad dQ = -\frac{k}{\gamma_w}\frac{\partial^2 u_e}{\partial z^2}dx dy dz dt$$

Assuming volume change is due entirely to flow of water out of the element:

$$\qquad dV = dQ$$

$$\qquad dV = -\frac{k}{\gamma_w}\frac{\partial^2 u_e}{\partial z^2}dx dy dz dt$$

### Volume Change Based on Phase Relations

Volumetric strain:

$$\qquad\epsilon_v = -\frac{de}{1+e}$$ (Compression positive)

Compressibility:

$$\qquad a_v = -\frac{de}{d\sigma_v}$$; (Compression positive)

$$\qquad de = -a_v d\sigma_v$$

Change in effective stress is equal to negative of change in pore pressure:

$$\qquad d\sigma_v = -du_e$$;
$$\qquad de = a_v du_e$$

Partial derivative expression for $$du_e$$:

$$\qquad du_e = \frac{\partial u_e}{\partial t}dt$$

Substituting into volumetric strain:

$$\qquad\epsilon_v = -\frac{a_v}{1+e}\frac{\partial u_e}{\partial t}dt$$

Volume change:

$$\qquad dV = \epsilon_v dx dy dz$$

$$\qquad dV = -\frac{a_v}{1+e}\frac{\partial u_e}{\partial t}dt dx dy dz$$

### Equate Volume Change Expressions

$$\qquad -\frac{k}{\gamma_w}\frac{\partial^2 u_e}{\partial z^2}dx dy dz dt = -\frac{a_v}{1+e}\frac{\partial u_e}{\partial t}dt dx dy dz$$

Collect terms:

$$\qquad\frac{k}{\gamma_w}\frac{1+e}{a_v}\frac{\partial^2 u_e}{\partial z^2} = \frac{\partial u_e}{\partial t}$$

Define coefficient of consolidation, $$c_v$$:

$$\qquad c_v = \frac{k}{\gamma_w}\frac{1+e}{a_v}$$

Terzaghi's one-dimensional consolidation equation:

$$\qquad c_v\frac{\partial^2 u_e}{\partial z^2} = \frac{\partial u_e}{\partial t}$$

Commonly the subscript "e" is dropped from the "u" term, based on the implicit assumption we are talking about excess pore pressures here:

$$\qquad c_v\frac{\partial^2 u}{\partial z^2} = \frac{\partial u}{\partial t}$$

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