{"id":238,"date":"2022-01-13T23:17:25","date_gmt":"2022-01-13T23:17:25","guid":{"rendered":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/chapter\/delayed-compaction-of-aquitards-confining-beds\/"},"modified":"2022-03-30T20:19:41","modified_gmt":"2022-03-30T20:19:41","slug":"delayed-compaction-of-aquitards-confining-beds","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/chapter\/delayed-compaction-of-aquitards-confining-beds\/","title":{"raw":"2.5 Delayed Compaction of Aquitards (Confining Beds)","rendered":"2.5 Delayed Compaction of Aquitards (Confining Beds)"},"content":{"raw":"
\r\n

An aquitard (or confining bed) is a clayey\u2011silty low permeability formation that does not provide an appreciable quantity of groundwater to pumping wells; however, it can transmit appreciable water to adjacent aquifers. While flow in an aquifer is predominantly two\u2011dimensional (2\u2011D) and horizontal, particularly if wellbores are fully penetrating, flow in the aquitards separating the aquifers is mostly 1\u2011D and vertical. In a complex aquifer system (for example, Figure\u00a016) the role played by the intervening aquitards is important as they can represent a significant source of water to the aquifers and can contribute greatly to land subsidence as clay\/silt compressibility c<\/em>b<\/em><\/sub> is usually much larger than that of the sand\/gravel.<\/p>\r\n

\"a)\u00a0Digital<\/p>\r\n

Figure\u00a0<\/strong>16<\/strong>\u00a0<\/strong>\u2011\u00a0<\/strong>a)\u00a0Digital elevation model of the Emilia\u2011Romagna plain, Italy, and b) vertical cross section along the A\u2011A' alignment shown in (a)\u00a0of the complex multi\u2011aquifer system used to supply freshwater to the coastland (modified after Teatini et al., 2006).<\/p>\r\n

Normally aquitard compaction is larger and delayed in time relative to aquifer compaction. The law that governs pore-water decline in the aquitard as a function of time and the factors controlling compaction are explained in this section. Darcy\u2019s law describing the velocity of groundwater flow in an aquitard can be written as shown in Equation\u00a014.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle v_{z}=-K\\frac{\\partial h}{\\partial z}[\/latex]<\/td>\r\n(14)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nwhere:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
v<\/em>z<\/em><\/sub><\/td>\r\n=<\/td>\r\napparent seepage velocity (LT-1<\/sup>)<\/td>\r\n<\/tr>\r\n
K<\/em><\/td>\r\n=<\/td>\r\nhydraulic conductivity (LT-1<\/sup>)<\/td>\r\n<\/tr>\r\n
h<\/em><\/td>\r\n=<\/td>\r\nhydraulic head = z<\/em> + p<\/em>\/\u03b3<\/em> (L)<\/td>\r\n<\/tr>\r\n
z<\/em><\/td>\r\n=<\/td>\r\nvertical coordinate positive downward (L)<\/td>\r\n<\/tr>\r\n
\u2202h<\/em>\/\u2202z<\/em><\/td>\r\n=<\/td>\r\nvertical hydraulic gradient (LL-1<\/sup>)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The hydraulic conductivity is a function of the physical properties of fluid and soil as shown in Equation\u00a015.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle K=k^{*}\\frac{\\gamma }{\\mu }[\/latex]<\/td>\r\n(15)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nwhere:\r\n\r\n\r\n\r\n\r\n\r\n
k*<\/em><\/td>\r\n=<\/td>\r\nintrinsic permeability (L2<\/sup>)<\/td>\r\n<\/tr>\r\n
\u03b3<\/em><\/td>\r\n=<\/td>\r\nspecific weight of water (ML-2<\/sup>T-2<\/sup>)<\/td>\r\n<\/tr>\r\n
\u03bc<\/em><\/td>\r\n=<\/td>\r\ndynamic viscosity of water (ML-1<\/sup>T-1<\/sup>)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Intrinsic permeability is dependent exclusively on the properties of the medium:<\/p>\r\n

k*<\/em> = CD<\/em>2<\/sup><\/p>\r\n

where:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n
D<\/em><\/td>\r\n=<\/td>\r\na representative length of the porous medium (for example, the average grain size) (L)<\/td>\r\n<\/tr>\r\n
C<\/em><\/td>\r\n=<\/td>\r\nappropriate parameter related to the soil type (dimensionless)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Other more complex relationships (depending on porosity, mean pore diameter, and specific surface area) have been developed for intrinsic permeability of reactive clays, especially if salt is dissolved into the pore water (for example, Raffensperger and Ferrell Jr., 1991).<\/p>\r\n

Assume the initial conditions are in equilibrium, and all the hydrologic and geomechanical quantities presented here are incremental with respect to the initial conditions. Let\u2019s balance the weight of water in an elementary soil sample of initial length \u0394z<\/em> and unitary cross\u2011sectional area (shown as 1 in the expressions below) between time t<\/em> and t<\/em>\u00a0+\u00a0\u0394t<\/em>:<\/p>\r\n\r\n