# Aquifer Storativity (T)

## AQUIFER STORATIVITY

### by Darrel Dunn, Ph.D., PG (Consulting hydrogeologist. Professional Synopsis)

This is a technical page on aquifer storativity. To see a non-technical page on this topic, press here.

Aquifer storativity (also called storage coefficient) of a confined aquifer is defined as S=S_{s}b, where S is storativity (dimensionless), S_{s} is specific storage (L^{-1}) and b is thickness (L) of the aquifer (66). A confined aquifer is an aquifer that is confined between two aquitards (66). Specific storage of a saturated aquifer is defined as the volume of water that a unit volume of aquifer releases from storage under unit decline in hydraulic head (66). Ranges of values for S_{s} for various subsurface materials have been published by Domenico and Mifflin (67). These ranges were calculated from the bulk modulus of compression (E) (ML^{-1}T^{-2}) using S_{s}=γ_{w}α, where γ_{w }is the unit weight of water (MLT^{-2}), and α is the vertical compressibility of the aquifer material (α is the inverse of E)(M^{-1}LT^{2}). This formula for S_{s} assumes that water is incompressible. Table 1 also shows specific storage values corresponding to the values for E calculated using S_{s}=γ_{w}α. A value of 62.4 lb/ft^{3} was used for the unit weight of water. Domenico and Mifflin indicate that these values of specific storage may be high. Their discussion implies that the values were based on S_{s}=k_{v}/c_{v}, where k_{v} is vertical permeability and c_{v} is the coefficient of consolidation. Coefficients of consolidation were apparently obtained from laboratory consolidation tests and k_{v} was from no-load vertical permeability tests. The permeability under no load is likely to be greater than the permeability existing under effective overburden pressure at depth, possibly producing high values for S_{s}. [Coefficient of consolidation (L^{2}T^{-1}) is a parameter used to describe the rate at which saturated clay or other soil undergoes consolidation or compaction when subjected to an increase in pressure. See, for example, *Foundation Engineering* (81).]

Table 1. Range in Values for Bulk Modulus of Compression (E) and Specific Storage

S_{s}=γ_{w}α [After Domenico and Mifflin (67)]

I have expanded the calculations of Table 1 to utilize a formula for S_{s} that includes the compressibility of water. This formula is S_{s}=γ_{w}(α +*n*β), where n is porosity (dimensionless) and β is the compressibility of water (M^{-1}LT^{2}). The results for various porosities are shown in Table 2. The compressibility of water used in this table is 2.1067E-8 ft^{2}/lb from *Groundwater *(66).

Table 2. Range in Values for Compressibility and Specific Storage using S_{s}=γ_{w}(α +*n*β).

The maximum difference in S_{s} calculated with and without the *n*β term is always γ_{w}*n*β; which is 1.32E-7, 2.63E-7 and 3.94E-7 per foot for porosities of 0.10, 0.20, and 0.30, respectively. The ratio of S_{s} calculated with the *n*β term over S_{s} calculated without the *n*β term increases as the porosity increases and the compressibility of the aquifer material decreases. For example, this ratio is 1.13 for fissured and jointed rock with porosity of 0.10, and 1.03 for low compressible sandy gravel. Such differences might be significant in some calculations; for example, an estimate of the amount of water that could be obtained by groundwater mining from a fractured rock aquifer. Groundwater mining is withdrawal of groundwater from an aquifer system faster than it is recharged from sources outside of the aquifer system. When it is as easy to consider the compressibility of water as not, one might as well do it.

Tables 1 and 2 do not include some factors that could affect specific storage. Specific storage may be affected by the presence of gas in the pore space. This may be gas that is present in the aquifer under static conditions, or it may be gas that is exsolved due to decline in head. Decline in head is associated with reduction of pressure in the water, so that the concentration of dissolved gas may exceed the solubility limit, causing exsolution. Gas compressibility is about 100 times aquifer compressibility under pressures commonly observed in shallow confined aquifers (72). Therefore it can be a significant factor in the amount of water yielded due to decline in head.

Water containing dissolved gas may be more than three times more compressible than gas-free water at ordinary aquifer pore-water pressures (less than about 500 psi) (73). Increasing total dissolved solids decreases the compressibility of water (74). Increasing temperature decreases the compressibility of water at temperatures below about 120^{o}F, and increasing ambient pressure reduces the compressibility of water (75). In applications of specific storage where the compressibility of water is a significant factor, these influences on compressibility might also be significant.

Table 2 contains clay materials that are not aquifer materials in the sense that they are not permeable enough to yield economic quantities of water directly to wells. However, clayey materials present as layers and lenses within an aquifer system that yields water to wells from permeable sand and/or gravel layers may affect the storativity of the aquifer system, because the clayey layers yield water by vertical flow to the permeable layers. Indeed, extensometer measurements during long-duration pumping tests in an alluvial aquifer with abundant clay layers in Antelope Valley, California, yielded one specific storage value (3.5E-4) in the plastic clay range of Table 2 (76).

There are two types of specific storage, one related to elastic deformation and the other related to inelastic deformation. For decreases in head to values higher than the previous lowest value (the preconsolidation head), deformation is elastic and the aquifer system recovers, or expands, when head again increases. If head decreases to below the preconsolidation head, effective stress exceeds the historical maximum, and compaction results in permanent deformation of the aquifer system. The specific storage values in Table 1 appear to be ones for inelastic deformation, because specific storage values related to elastic deformation are smaller.

The formula S_{s}=γ_{w}(α +*n*β) is derived assuming that an elemental (smallest representative) volume of an aquifer becomes smaller in its vertical dimension as the aquifer compresses due to reduction in head. See, for example, *Hydraulics of Wells* (69). Davis and DeWiest (70) derived a different formula for Ss by assuming an elemental volume with no change in dimensions as the aquifer compresses. Their result was S_{sf}=γ_{w}[(1-*n*)α +*n*β]. The difference between the two formulas is γ_{w}*n*α. This difference is of academic interest in modeling applications where the model cells do not deform and aquifer thicknesses are held constant.

In both derivations of specific storage, the volume of the solid material in the elemental volume (whether fixed or deforming) is assumed to be constant because "the compressibility of the individual grains is considerably smaller than that of their skeleton ... and is also smaller than the compressibility of water." Note that the compressibility of quartz (for example) is about 1.3E-9 ft^{2}/lb (71). For comparison, the minimum compressibility given for "rock fissured and jointed" is 1.6E-8 ft^{2}/lb; and the compressibility of water is 2.11E-8 ft^{2}/lb.

I searched for published values for specific storage in addition to those given by Domenico and Mifflin. I looked for values derived from (1) laboratory consolidometer tests, (2) in situ extensometer measurements, (3) pumping tests, (4) barometric efficiency, and (5) computer model calibration. I found values in references 82 through 90. These references are not exhaustive, and the tests and measurements are not homogeneous. Consolidometer tests were conducted at various pressures, and some pressures are not given. Some of the consolidometer tests were conducted at consolidation pressures less than the pressure at depths of most aquifers. The hydraulic conductivities necessary to calculate specific storage from consolidometer tests were obtained by various methods. Extensometer data interpretation is somewhat subjective, as is pumping test data. Model solutions are not mathematically unique. Whether or not the consolidation is inelastic or elastic may be subjective in some cases and is not given in many cases. The description of the materials were mostly subjective and not related to any specified classification. Consequently, I used the authors' descriptions of unconsolidated materials to subjectively assign the material to Shepard's classification system (91) to provide a uniform sediment classification for showing the ranges of specific storage and the relation of specific storage to type of subsurface material.

Figure 1 shows all the data I gleaned from the literature plotted regardless of the type of measurement. The inelastic and unspecified specific storage values for clay correspond well with the values for clay given by Domenico and Mifflin, although Domenico and Mifflin subdivide the clay into plastic, stiff, and medium hard. The unspecified values for sand correspond reasonably well with the values for sand given by Domenico and Mifflin. The values given by Domenico and Mifflin seem to agree with the data I found if one assumes that the values of Domenico and Mifflin are for inelastic compression. Figure 1 suggests a sharp decline in specific storage between clay and silty clay and less decline as the grain size becomes coarser. Figure 1 suggests that elastic specific storage is about an order of magnitude less than inelastic specific storage for any type of sediment. The large spread of specific storage for coal suggests that looking at existing data for guidance on the compressibility of coal is not worthwhile and tests should be made for the specific coal of interest.

Figure 1. Specific storage for subsurface materials from consolidometer tests, extensometer data, pumping tests, model calibration, and barometric efficiency plotted together.

Figure 2 breaks down the data displayed in Figure 1 into the type of test used to get the values. Most of the data is from laboratory consolidometer tests. This is a relatively inexpensive test and from this limited set of data it appears that the consolidometer results for inelastic compression are consistent with results from the other types of tests. However, consolidometer tests in the elastic range of consolidation appear to yield higher values for specific storage than extensometer tests and model calibration. Barometric efficiency is not included in Figure 2 because I found only one value. It is for elastic compression of coal and is plotted in Figure 1.

Figure 2. Specific storage values from consolidometer tests, extensometer data, pumping tests and model calibration plotted separately.

The book *Advanced Soil Mechanics* (92) contains a description of how consolidometer testing is used to get the coefficient of consolidation and an estimate of vertical permeability for a sample. Specific storage is calculated by dividing the vertical permeability by the coefficient of consolidation (67). A paper by Polland and others (93) illustrates how extensometer data may be used to estimate elastic and inelastic storage coefficients, and Hantush (69) describes obtaining specific storage from barometric efficiency. Hantush (69) also explains the use of type-curve methods for analyzing pumping test data, and Walton (94) has provided graphs of some type curves. Software is also available for pumping test analysis. Pumping tests may be used for estimating specific storage of aquifers, but they do not yield specific storage of the aquitards so far as I know. A pumping test method that yields the hydraulic diffusivity (ratio of hydraulic conductivity to specific storage) of an aquitard has been described by Neuman and Witherspoon (95), but this ratio would not yield specific storage unless the vertical hydraulic conductivity were known. Step tests may be used to estimate specific storage if data is obtained from a monitoring well located near the pumping well. The concept of storativity has been extended to unconfined aquifers. Water yielded due to aquifer compressibility may be separated from water yielded doe to drainage at the water table in pumping test analysis. See the web page on unconfined aquifer testing using WATEQ.

Given the uncertainties related to specific storage, one might choose to use a value that is conservative for a particular problem. Estimates of specific storage may be improved by consolidometer and permeameter tests of cores in test wells; performing pumping tests, especially pumping tests supplemented with extensometer measurements of compaction; and numerical modeling of pumping test results, especially if piezometers are installed in the aquitards.

**Addendum**

For questions or comments (especially regarding errors and omissions), send an email to ddunn@dunnhydrogeo.com.

If you need a consulting hydrogeologist to perform a hydrologic task involving storativity, send an email.

Posted March 26, 2012