Assuming no water enters or leaves the aquifer by virtue of the large areal extent of the loading and inserting the undrained condition, = 0, in Equation 7a gives the undrained pore pressure response to be p = vv/(wgSs). {\displaystyle \sigma _{e}} Jacob (1941) used tidal efficiency to compute indirectly the coefficient of storage because it was more easily determined than the barometric efficiency.
) is typically orders of magnitude less ( V 0000007382 00000 n
or B.E. {\displaystyle S_{s}b\ll \!\ S_{y}} Measuring T.E. p This must be added to the amount induced in the aquifer by the atmosphere loading as in the tidal loading case. The other constitutive equation linearly relates vertical strain to changes in vertical stress and pore pressure. In tidal or barometric loading, the vertical stress is decidedly not constant.
Jacobs goal was to derive the groundwater equivalent of the partial differential equation for time-dependent heat flow, and thereby place the mathematical description of groundwater flow on firmer physical ground than a plausible analogy. This is related to both the compressibility of the aquifer and the compressibility of the water itself.
This ratio is equal to the ratio of p/v. Specific storage is the volume of water released from one unit volume of the aquifer under one unit decline in head. Adding the terms and multiplying by the factor wg gives a specific storage of 2.3 10-4 m-1 and the ratio of sand-to-water compressibility is 3.5. The ratio of aquifer compressibility to water compressibility obtained from tidal efficiency was 1.7 whereas it was 2.8 from pumping test analysis. 0000002167 00000 n Equation 7a is one of two basic constitutive equations of poroelasticity for the special case of areally extensive, vertical loading (Wang, 2000). 0000002318 00000 n As his acknowledgment[8] makes clear, Jacob shared ideas with his colleague, C. V. Theis, at the United States Geological Society (USGS).
e HWn8PC s , is relatively small and usually has an insignificant contribution. ) per change in applied stress (effective stress The international system of units (SI) units of Ss are 1/m as indicated for Equation 2. The other values came from handbooks and were calculated with assumptions. or pore pressure 0000001570 00000 n He compared the relative contributions of aquifer elasticity and water compressibility from tidal responses (e.g., Figure 6) with those from pumping tests at depths between 715 and 800 ft in the Lloyd sand on Long Island of the United States.
Problems related to unsaturated flow are simulated using the numerical solution of Richards Equation, which requires estimation of the specific yield, or the numerical solution of the Soil Moisture Velocity Equation, which does not require estimation of the specific yield. 0000006147 00000 n In terms of measurable physical properties, specific storage can be expressed as. The term nw = 2 10-10 Pa-1 for a porosity of 38%. ) per unit volume. Assuming the aquifer or aquitard is homogeneous: For an unconfined aquifer, storativity is approximately equal to the specific yield ( https://www.un-igrac.org/sites/default/files/resources/files/Groundwater%20book%20-%20English.pdf, https://en.wikipedia.org/w/index.php?title=Specific_storage&oldid=1077048852, All articles with bare URLs for citations, Articles with bare URLs for citations from March 2022, Articles with PDF format bare URLs for citations, Creative Commons Attribution-ShareAlike License 3.0. 0000008614 00000 n In addition, the expansion of the pore water volume in response to a fluid pressure decrease must be included in the water budget for a REV (Equation 5). 87m3E:X/(J(8sIs7+l 0|H"D bD`S> E~p_W)|.>M 6X"m3V)DXYI0[/1l-Qt3!?u:@>K>M8 For a confined aquifer or aquitard, storativity is the vertically integrated specific storage value. It should be emphasized that the storage coefficients S and Ss in Equation 6a or 6b can be measured directly in the field from a pumping test or in the laboratory by adhering to its definition as the ratio of water volume removed from storage due to pore pressure change (Figure 9).
0000001914 00000 n
HWv6vurl[}|:="! With total vertical stress constant, an increase in effective stress is equal but opposite to the decrease in fluid pressure, that is, . = 1 T.E. Therefore, in general, the increment of fluid content must be expressed in terms of changes in both vertical stress and pore pressure.
In soil mechanics, the aquifer is said to be undrained. 32Ve Specific storage in terms of head was computed from its value in terms of pressure using w = 1000 kg/m3 and g = 9.8 m/s2. 0000003625 00000 n s The ratio of the increase in water level in a well to the increase in ocean level, is called tidal efficiency, T.E. 0000001096 00000 n S The first term on the right in Equation 7a is the increment of fluid content associated with a change in vertical stress when there is no change in fluid pressure.
0000004945 00000 n 0000004802 00000 n In hydrogeology, aquifer compressibility is typically more significant than water compressibility. {\displaystyle S_{y}} ="4$SpJP4yB. y The simplest form for an equation is to consider increment of fluid content to be a linear function of both variables. effective vertical stress defined to be the difference between the vertical stress and the pore pressure, The title of Meinzers 1928 paper called attention to the role of compressibility and elasticity of artesian aquifers. As evidence, he presented Terzaghis experiment in which a vertical stress caused porosity loss in sandstone (, Jacob reasoned that extraction of water from an aquifer does not change the total vertical stress. (1966). Jacob then obtained from mass conservation the partial differential equation for radial flow in an aquifer of thickness b that was identical in form to what Theis inferred by analogy with heat conduction. 0000012331 00000 n The writer proposes to derive from scratch the fundamental differential equation governing the flow of water in an elastic artesian aquifer, considering in turn each of the assumptions that are necessary to the derivation of the equation.. 0000001160 00000 n The sign convention is that compressive stress is positive. The assumed large horizontal extent of the loading induces a fluid pressure in the aquifer because the fluid cannot escape as it is being loaded. s Jacob computed the mass balance in a REV over a time step in which there was a change in pressure p. S 330 0 obj << /Linearized 1 /O 333 /H [ 1160 432 ] /L 912911 /E 128139 /N 9 /T 906192 >> endobj xref 330 29 0000000016 00000 n The first term states that the vertical strain is the vertical strain due to a change in vertical stress when there is no change in fluid pressure and the second term is the vertical strain associated with a change in fluid pressure when there is no change in vertical stress. This page was last edited on 14 March 2022, at 08:33. Dividing by aquifer thickness gives specific storage. ! endstream endobj 13 0 obj 1179 endobj 4 0 obj << /Type /Page /Parent 5 0 R /Resources << /Font << /F0 6 0 R /F1 8 0 R /F2 10 0 R >> /ProcSet 2 0 R >> /Contents 12 0 R >> endobj 15 0 obj << /Length 16 0 R /Filter /FlateDecode >> stream In addition to Equation 6a, Jacob derived equations for the water-level response to aquifer loading by water tides or changes in barometric pressure, also in terms of aquifer and water compressibility, which could provide an indirect measurement of coefficient of storage. {\displaystyle S_{s}}
Journal of Geophysical Research, 71(4), 11171122. Water can be added to or removed from storage when vertical stress is applied in addition to when pore pressure changes (compare with Equation 2b). The field produced 500 million barrels (80 million m3) of oil with a pressure drop of 375 psi (2.6 MPa). aO`D#lp]ZRd&3 The amount of water obtained for irrigation from the Dakota aquifer makes clear that large volumes of fluid can be stored in highly compressible earth materials. Nevertheless, the specific storage values in Table 2 are orders of magnitude smaller than specific yield of unconfined aquifers whose values are the aquifer porosity. t The sign convention is that vertical strain and vertical stress are positive in compression. x6_OsIKv+ CvMy= 0000011066 00000 n 0000001592 00000 n
Hb``b````e`AXhmC]'$;O vrw3@%Qi@jxf>~AQSdTMvklpa?NIZ (d `6FQ@{ 6AD l *EL^@) >$90`Hj:&ld0R8Ah@6P@-YC b HWrFTD:[KHH&U43`\x"`1 %`) V~ !xf$dD(xo&7!VpmgF%z-EEX3Vu)kjz 6l%Trj8Pe2MH#bFU{4L"]3NHZ[HY$Dw7s}L^EoF C: jN-~r, 1&{b=r].E@1D`yhDGq n%CO2XH1h2j@)PO'xZez&+4u$S'BKT=%Np1Fw>oLxKDbGS rJ{{yxOdy The in-phase response of water levels in an aquifer to ocean tides (Figure 6) was cited by Meinzer as evidence of aquifer elasticity. 0000011089 00000 n On the storage coefficient and the equations of groundwater flow. The sign convention is that is positive when water is added to the REV and v is positive when the REV is compressed. 0000003648 00000 n 0000009833 00000 n Mass specific storage is the mass of water that an aquifer releases from storage, per mass of aquifer, per unit decline in hydraulic head: Volumetric specific storage (or volume specific storage) is the volume of water that an aquifer releases from storage, per volume of aquifer, per unit decline in hydraulic head (Freeze and Cherry, 1979): In hydrogeology, volumetric specific storage is much more commonly encountered than mass specific storage. With those caveats, Table 2 provides order-of-magnitude values for a small set of geologic materials.
{\displaystyle V_{t}} The values come from a variety of sources. and assuming n and w to be knowns means that v, and hence, Ss can be obtained from Equation 8 or 9 and the ratio of the contribution from water compressibility, nw, to aquifer compressibility, v, can be calculated. Dk~O? 0000009856 00000 n Consequently, the term specific storage generally refers to volumetric specific storage. 4 Jacobs Compressibility Formula for Aquifer Storage. The coefficient of proportionality is v for both terms by the law of effective stress, i.e., Equation 7b is simply a restatement of Equation 3. The vertical compressibility v between points a and b is calculated from the slope to be 7 10-10 Pa 1 after converting from English units. 0000004922 00000 n His derivation was based on three physical principles: (1) A fluid pressure decline equates to an effective increase in vertical stress. C@{> b X endstream endobj 358 0 obj 316 endobj 333 0 obj << /Type /Page /MediaBox [ 0 0 512 710 ] /Parent 327 0 R /Resources << /Font << /F0 335 0 R /F1 334 0 R /F2 336 0 R /F3 341 0 R >> /XObject << /Im1 356 0 R >> /ProcSet 354 0 R >> /Contents [ 338 0 R 340 0 R 343 0 R 345 0 R 347 0 R 349 0 R 351 0 R 353 0 R ] /CropBox [ 0 0 512 710 ] /Rotate 0 >> endobj 334 0 obj << /Type /Font /Subtype /TrueType /Name /F1 /BaseFont /TimesNewRoman /Encoding /WinAnsiEncoding >> endobj 335 0 obj << /Type /Font /Subtype /TrueType /Name /F0 /BaseFont /TimesNewRoman,Italic /Encoding /WinAnsiEncoding >> endobj 336 0 obj << /Type /Font /Subtype /TrueType /Name /F2 /BaseFont /TimesNewRoman,Bold /Encoding /WinAnsiEncoding >> endobj 337 0 obj 1227 endobj 338 0 obj << /Filter /FlateDecode /Length 337 0 R >> stream w Barometric efficiency, B.E., is similarly defined to be the ratio of the increase in water level to the increase in vertical stress (expressed as an equivalent head increase). [1], They are often determined using some combination of field tests (e.g., aquifer tests) and laboratory tests on aquifer material samples.
(2) A fluid pressure decline expels a water volume equal to the loss of porosity associated with aquifer compressibility.
0000012356 00000 n
Storativity is a dimensionless quantity, and is always greater than 0. Some water always remains in the formation, even after drainage; it clings to the grains of sand and clay in the formation. 0000012289 00000 n Terzaghis experiment (Figure 5 and Equation 6b) can be used to estimate the specific storage of loose sand. )NB.Dx7PuSC.Q_lx,V7RE(xrq,T[%;*v0, N3k*I:yh-+,e#r"S|L_c9{jC}}&T*Dg E(^.. 12 0 obj << /Length 13 0 R /Filter /FlateDecode >> stream
1E)E%`f3x,aOf.nn,L|yq `,?O,f(=]C&A3e}{y%_ eL4&?i?[x. The difference was that the coefficient of storage S was expressed in terms of compressibility rather than as a quantity defined by analogy. A difference from the tidal efficiency in terms of well response is that a change p in atmospheric pressure directly changes the water level in the well by -p/wg.
0000002295 00000 n The rock compressibility and specific storage values are for isotropic confining stress, not vertical stress. However, this direct measurement is difficult to perform accurately because fluid storage in tubing connected to the rock samples pore volume must be included in the accounting. S In other words, v is equal to the volume of water per unit volume of a REV that is removed from storage due to aquifer compressibility. Also, the value of specific yield may not be fully realized for a very long time, due to complications caused by unsaturated flow. Specific yield, also known as the drainable porosity, is a ratio, less than or equal to the effective porosity, indicating the volumetric fraction of the bulk aquifer volume that a given aquifer will yield when all the water is allowed to drain out of it under the forces of gravity: It is primarily used for unconfined aquifers, since the elastic storage component, In the field of hydrogeology, storage properties are physical properties that characterize the capacity of an aquifer to release groundwater.
