{"id":239,"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\/time-factor-and-compaction-profile\/"},"modified":"2022-03-30T20:22:19","modified_gmt":"2022-03-30T20:22:19","slug":"time-factor-and-compaction-profile","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/chapter\/time-factor-and-compaction-profile\/","title":{"raw":"2.6 Time Factor and Compaction Profile","rendered":"2.6 Time Factor and Compaction Profile"},"content":{"raw":"
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The (positive) compaction \u03b7<\/em>(t<\/em>) <\/span>of the half aquitard at time t <\/span><\/em>is:<\/p>\r\n

[latex]\\displaystyle \\eta (t)=\\int_{0}^{b\/2}c_{b}(p_{0}-p)dz[\/latex]<\/p>\r\n

Substituting Equation\u00a020<\/a>, and integrating term by term leads to Equation\u00a022.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\eta (t)=\\frac{4}{\\pi ^{2}}c_{b}\\Delta p_{0}b\\sum_{n=0}^{\\infty }\\frac{1}{(2n+1)^{2}}\\left\\{1-\\frac{1}{\\textup{exp}\\left ( [(2n+1)\\pi \/b]^{2}c_{v}t \\right )} \\right\\}[\/latex]<\/td>\r\n(22)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

For t<\/em>\u00a0=\u00a00 we get \u03b7<\/em>(0) = 0, while for t<\/em>\u00a0=\u00a0\u221e<\/em> Equation\u00a022 becomes Equation\u00a023.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\eta (\\infty )=\\frac{4}{\\pi ^{2}}c_{b}\\Delta p_{0}b\\left ( 1+\\frac{1}{3^{2}}+\\frac{1}{5^{2}}+\\frac{1}{7^{2}}+... \\right )[\/latex]<\/td>\r\n(23)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The above series converges to \u03c0<\/em>2<\/sup>\/8 and therefore \u03b7<\/em>(\u221e<\/em>) is the ultimate compaction of the half aquitard, as was already pointed out for the various aquifers represented by Equations\u00a04 and 5 and is expressed here as Equation\u00a024.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\eta (\\infty )=c_{b}\\Delta p_{0}\\frac{b}{2}[\/latex]<\/td>\r\n(24)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

It may be interesting to compute the time needed by the aquitard to achieve a given percentage of full compaction, \u03b7<\/em>(\u221e<\/em>), provided by Equation\u00a024. To this aim we introduce the dimensionless time factor T<\/em>v<\/em><\/sub> (Equation\u00a025) as defined by Terzaghi (1923).<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle T_{v}=4\\frac{c_{v}t}{b^{2}}[\/latex]<\/td>\r\n(25)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Denoting the percentage compaction as w<\/em>, Equation\u00a026 shows that it is a function of only T<\/em>v<\/em><\/sub>.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle w(T_{v})=\\frac{8}{\\pi ^{2}}\\sum_{n=0}^{\\infty }\\frac{1}{(2n+1)^{2}}\\left\\{1-\\frac{1}{\\textup{exp}[\\pi ^{2}(2n+1)^{2}T_{v}\/4]} \\right\\}[\/latex]<\/td>\r\n(26)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The behavior of w<\/em>(T<\/em>v<\/em><\/sub>) is shown in Figure\u00a017. There are basically two ways to use Figure\u00a017:<\/p>\r\n\r\n