15.4 Compressibility and Storage Parameters
We have discussed stress; now let’s talk about strain and the relationship between strain and stress. In any given material, a certain amount of stress will result in a certain amount of strain. The ratio of strain to stress is called the compressibility. In terms more specific to what we are discussing here, the compressibility is the amount of shortening or extension—that is, the change in thickness of a layer—that occurs due to a change in stress—in this case, the change in stress associated with a change in pore fluid pressure. We can express this as Equation (114).
| [latex]\alpha = \frac{\frac{\mathrm{\Delta}L}{L}}{\mathrm{\Delta}\sigma'}[/latex] | (114) |
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where (parameter dimensions are dark green font with mass as M, length as L, time as T):
| [latex]\alpha[/latex] | = | compressibility (LT2M-1) |
|---|---|---|
| [latex]\frac{\mathrm{\Delta}L}{L}[/latex] | = | the change in length (thickness) over the total length (:) |
| [latex]\mathrm{\Delta}\sigma'[/latex] | = | the change in effective stress (ML-1T-2) |
Compressibility is just a simple coefficient relating strain to stress, and it can be applied in several ways. If we know the compressibility of a material—for example, a clay confining layer—we can calculate how much the layer will consolidate for a given change in head. To do that, we convert the change in head to a change in pressure, then multiply by the compressibility to calculate the change in length over the total length.
The specific storage is directly related to the compressibility of the aquifer by the following relationship shown in Equation (115).
| [latex]S_{s} = \rho_{w}g\ \phi\ (\alpha + \beta)[/latex] | (115) |
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where (parameter dimensions are dark green font with mass as M, length as L, time as T):
| [latex]S_{s}[/latex] | = | specific storage (L-1) |
|---|---|---|
| [latex]\rho_{w}[/latex] | = | density of water (ML-3) |
| [latex]g[/latex] | = | gravitational acceleration (LT-2) |
| [latex]\alpha[/latex] | = | compressibility of the aquifer framework (M-1LT2) |
| [latex]\phi[/latex] | = | porosity (:) |
| [latex]\beta[/latex] | = | compressibility of water (M-1LT2) |