{"id":289,"date":"2025-02-06T04:56:56","date_gmt":"2025-02-06T04:56:56","guid":{"rendered":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/4-8-evolution-of-primary-consolidation-settlement-with-time\/"},"modified":"2026-03-16T14:01:12","modified_gmt":"2026-03-16T14:01:12","slug":"4-8-evolution-of-primary-consolidation-settlement-with-time","status":"publish","type":"chapter","link":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/4-8-evolution-of-primary-consolidation-settlement-with-time\/","title":{"raw":"4.8 Evolution of primary consolidation settlement with time","rendered":"4.8 Evolution of primary consolidation settlement with time"},"content":{"raw":"The rate of dissipation of excess pore water pressure, and thus the necessary time for the applied loading on the ground surface to be transferred to the soil skeleton resulting in increase of the effective stress, depends on the permeability, <em>k<\/em> of the soil. We can obtain a closed-form analytical expression that provides the rate of dissipation of excess pore pressure, and thus the evolution of settlement with time (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/01\/4.7-HR.png\">Figure 4.7<\/a>), under some simple assumptions:\n<ul>\n \t<li>1-D conditions prevail, with water flowing only vertically, as in the oedometer test (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/01\/4.27-HR.png\">Figure 4.27<\/a>).<\/li>\n \t<li>The soil is fully saturated, homogeneous and isotropic.<\/li>\n \t<li>Darcy\u2019s law is valid, thus flow <em>q<\/em> is equal to the permeability times the hydraulic gradient <em>i <\/em>i.e., <em>q<\/em>=<em>ki<\/em><\/li>\n \t<li>Small strains.<\/li>\n<\/ul>\nUnder the above assumptions, Terzaghi formulated the one-dimensional consolidation equation, describing the variation of excess pore pressure <em>u<\/em> with time <em>t <\/em>and depth <em>z<\/em> as:\n\n<strong>(4.65)<\/strong> [latex]\\dfrac{{\\partial u}}{{\\partial t}} = {c_v}\\dfrac{{{\\partial ^2}u}}{{\\partial {z^2}}}[\/latex]\n\nThe constant <em>c<sub>v<\/sub><\/em>\u00a0in the differential equation 4.65 is called <em>coefficient of consolidation<\/em>, and depends on the coefficient of volume compressibility <em>m<sub>v<\/sub><\/em> and the soil permeability in the vertical direction <em>k<sub>z<\/sub><\/em> as:\n\n<strong>(4.66)<\/strong> [latex]{c_v} = \\dfrac{{{k_z}}}{{{m_v}{\\gamma _w}}}[\/latex]\n\nwhere <em>\u03b3<\/em><sub>w <\/sub>\u2248 10 kN\/m<sup>3<\/sup> is the unit weight of the water. The coefficient of consolidation carries units of length<sup>2<\/sup>\/time.\n\nSolution of the differential equation 4.65 requires knowing the initial distribution of excess pore pressure with depth (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.38-initial-excess-porre-pressure-distribution-with-depth.png\">Figure 4.38<\/a>), and the permeability boundary conditions. For example, assuming uniform excess pore pressure distribution with depth, corresponding to a relatively thin clay layer (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.38-initial-excess-porre-pressure-distribution-with-depth.png\">Figure 4.38<\/a>) and double drainage, both from the top and the bottom of the layer (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.39-boundary-conditions-and-length-of-drainage-path.png\">Figure 4.39a<\/a>):\n<ul>\n \t<li>at <em>t <\/em>= 0, the excess pore pressure will be uniform along the thickness, and equal to the applied pressure at the surface \u0394<em>u<\/em><sub>0<\/sub><em> = <\/em><em>q<sub>ext<\/sub><\/em><\/li>\n \t<li>at the top boundary <em>z<\/em>=0 where drainage is allowed, \u0394<em>u <\/em>= 0<\/li>\n \t<li>at the bottom boundary <em>z=H<\/em> where drainage is allowed too, \u0394<em>u <\/em>= 0<\/li>\n<\/ul>\n[caption id=\"attachment_288\" align=\"aligncenter\" width=\"350\"]<img class=\"wp-image-285 size-full\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2025\/02\/4.38-initial-excess-porre-pressure-distribution-with-depth-e1743559781888.png\" alt=\"Illustration of three initial excess pore pressure \u0394_u distributions: uniform, trapezoidal, and triangular. The uniform pore pressure distribution is applicable to the thinner soil layer, and the triangular pore pressure distribution to the thicker soil layer. \" width=\"350\" height=\"419\"> Figure 4.38. Initial excess pore pressure distribution with depth, for increasing thickness of the compressible layer.[\/caption]\n\nThe drainage boundary conditions suggest that excess pore pressure will immediately fall to zero at the permeable boundaries.\n\nA closed-form solution of Eq. 4.65 exists for the above conditions and provides the variation of the excess pore pressure with depth at different time instances (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.40-HR.png\">Figure 4.40<\/a>):\n\n<strong>(4.67)<\/strong> [latex]\\Delta u\\left( {z,t} \\right) = \\sum\\limits_{m = 0}^\\infty {\\dfrac{{2\\Delta {u_0}}}{M}\\sin \\left( {\\dfrac{{Mz}}{{{H_{dr}}}}} \\right)} {e^{ - {M^2}{T_v}}}[\/latex]\n\nwhere [latex]M = \\dfrac{{\\pi \\left( {2m + 1} \\right)}}{2}[\/latex], and <em>T<sub>v <\/sub><\/em>is the <em>time factor<\/em>, equal to:\n\n[caption id=\"attachment_288\" align=\"aligncenter\" width=\"490\"]<img class=\"wp-image-286 size-full\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2026\/03\/4.39-boundary-conditions-and-length-of-drainage-path.png\" alt=\"The figure on the top presents the flow boundary conditions for a soil specimen where drainage is allowed from both the top and bottom of the sample. The excess pore pressure at the top and bottom of the sample are \u0394u=0 and the length of the drainage path is half the thickness of the sample. The figure on the bottom presents the flow boundary conditions for a soil specimen where drainage is allowed only from the top of the sample. The excess pore pressure at the top is \u0394u=0, and the gradient of the excess pore pressure at the bottom is zero. The length of the drainage path is equal to the thickness of the sample.\" width=\"490\" height=\"292\"> Figure 4.39. Boundary conditions and length of drainage path for (a) double drainage, and (b) drainage only from the top of the soil layer.[\/caption]\n\n<strong>(4.68)<\/strong> [latex]{T_v} = \\dfrac{{{c_v}t}}{{H_{dr}^2}}[\/latex]\n\nThe length of the drainage path, <em>H<sub>dr<\/sub><\/em> depends on the drainage boundary conditions, and is depicted in <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.40-HR.png\">Figure 4.40<\/a>.\n\n[caption id=\"attachment_288\" align=\"aligncenter\" width=\"400\"]<img class=\"wp-image-287 size-full\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2026\/03\/4.40-HR-e1743559804589.png\" alt=\"Graph depicting the distribution of excess pore pressures along a sample of thickness H where drainage is allowed from the top and bottom, at three different time instances defined via the time factor Tv. When Tv = 0 (top figure) the excess pore pressure is uniform across the sample, and equal to the applied stress. When Tv > 0 (mid figure) the excess pore pressure is zero at the top and bottom of the sample, and increases parabolically to its middle, where it attains its maximum value. When Tv is infinite (bottom figure) the excess pore pressure is zero across the entire sample.\" width=\"400\" height=\"433\"> Figure 4.40. Graphical representation of excess pore pressure distribution with depth at different time instances, calculated from Eq. 4.67.[\/caption]\n\nInstead of using the absolute value the of excess pore water pressure, consolidation progress can be described via the <em>degree of consolidation<\/em> (or <em>consolidation ratio<\/em>), <em>U<\/em>(<em>z<\/em>) which defines the amount of consolidation completed at a particular time instance, at a specific depth, <em>z<\/em>:\n\n<strong>(4.69)\u00a0<\/strong>[latex]U\\left( z \\right) = 1 - \\dfrac{{\\Delta {u_z}}}{{\\Delta {u_0}}}[\/latex]\n\nThe degree of consolidation is visualised as the grey shaded area in<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.40-HR.png\"> Figure 4.40<\/a>. From an engineering point of view, we are rather interested in the <em>average<\/em> degree of consolidation, <em>U<\/em> of the whole compressible layer. For the special case where the initial excess pore pressure distribution is assumed constant with depth (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.38-initial-excess-porre-pressure-distribution-with-depth.png\">Figure 4.38<\/a>) the average degree of consolidation is calculated as:\n\n<strong>(4.70)<\/strong> [latex]U = 1 - \\sum\\limits_{m = 0}^\\infty {\\dfrac{2}{{{M^2}}}} {e^{ - {M^2}{T_v}}}[\/latex]\n\nThe variation of the average degree of consolidation with the time factor <em>Tv<\/em> is illustrated in <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.41-variation-of-average-degree-of-consolidation.png\">Figure 4.41<\/a>.\n\nEq. 4.70 and Figure 4.41 are also valid when drainage is not allowed at the bottom of the clay layer. The length of the drainage path in that case will be <em>H<sub>dr <\/sub><\/em>= <em>H<\/em> (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.39-boundary-conditions-and-length-of-drainage-path.png\">Figure 4.39<\/a>). Note that at an impermeable boundary, the gradient of pore pressure will be zero [latex]\\partial u\/\\partial z[\/latex] = 0.\n\n[caption id=\"attachment_288\" align=\"aligncenter\" width=\"700\"]<img class=\"wp-image-288 size-full\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2026\/03\/4.41-variation-of-average-degree-of-consolidation-e1743559821462.png\" alt=\"Graph depicting the variation of the average degree of consolidation U(%) with the time factor T_v,. When the time factor becomes Tv=1 the average degree of consolidation is about 90%.\" width=\"700\" height=\"317\"> Figure 4.41. Variation of the average degree of consolidation <em>U<\/em> with the time factor <em>T<sub>v<\/sub> <\/em>for uniform initial excess pore water pressure distribution.[\/caption]\n\nNote that <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.41-variation-of-average-degree-of-consolidation.png\">Figure 4.41<\/a> can be used to estimate settlement of a 1-D soil column at any given time <em>t<\/em><sub>1<\/sub>. By substituting <em>t<\/em><sub>1<\/sub> into the expression for the time factor <em>T<sub>v<\/sub><\/em> (Eq. 4.68) we can find the average degree of consolidation <em>U<\/em><sub>1<\/sub> at <em>t<\/em><sub>1<\/sub> directly from <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.41-variation-of-average-degree-of-consolidation.png\">Figure 4.41<\/a>. Therefore, settlement at <em>t<\/em><sub>1<\/sub> will be:\n\n<strong>(4.71)<\/strong> [latex]{\\rho _1} = \\dfrac{{{U_1}\\left( \\% \\right)}}{{100}}{\\rho _{pc}}[\/latex]\n\nwhere the primary consolidation settlement <em>\u03c1<\/em><sub>pc <\/sub>is calculated according to Section 4.6.2, assuming linear elastic or non-linear soil behaviour.","rendered":"<p>The rate of dissipation of excess pore water pressure, and thus the necessary time for the applied loading on the ground surface to be transferred to the soil skeleton resulting in increase of the effective stress, depends on the permeability, <em>k<\/em> of the soil. We can obtain a closed-form analytical expression that provides the rate of dissipation of excess pore pressure, and thus the evolution of settlement with time (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/01\/4.7-HR.png\">Figure 4.7<\/a>), under some simple assumptions:<\/p>\n<ul>\n<li>1-D conditions prevail, with water flowing only vertically, as in the oedometer test (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/01\/4.27-HR.png\">Figure 4.27<\/a>).<\/li>\n<li>The soil is fully saturated, homogeneous and isotropic.<\/li>\n<li>Darcy\u2019s law is valid, thus flow <em>q<\/em> is equal to the permeability times the hydraulic gradient <em>i <\/em>i.e., <em>q<\/em>=<em>ki<\/em><\/li>\n<li>Small strains.<\/li>\n<\/ul>\n<p>Under the above assumptions, Terzaghi formulated the one-dimensional consolidation equation, describing the variation of excess pore pressure <em>u<\/em> with time <em>t <\/em>and depth <em>z<\/em> as:<\/p>\n<p><strong>(4.65)<\/strong> [latex]\\dfrac{{\\partial u}}{{\\partial t}} = {c_v}\\dfrac{{{\\partial ^2}u}}{{\\partial {z^2}}}[\/latex]<\/p>\n<p>The constant <em>c<sub>v<\/sub><\/em>\u00a0in the differential equation 4.65 is called <em>coefficient of consolidation<\/em>, and depends on the coefficient of volume compressibility <em>m<sub>v<\/sub><\/em> and the soil permeability in the vertical direction <em>k<sub>z<\/sub><\/em> as:<\/p>\n<p><strong>(4.66)<\/strong> [latex]{c_v} = \\dfrac{{{k_z}}}{{{m_v}{\\gamma _w}}}[\/latex]<\/p>\n<p>where <em>\u03b3<\/em><sub>w <\/sub>\u2248 10 kN\/m<sup>3<\/sup> is the unit weight of the water. The coefficient of consolidation carries units of length<sup>2<\/sup>\/time.<\/p>\n<p>Solution of the differential equation 4.65 requires knowing the initial distribution of excess pore pressure with depth (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.38-initial-excess-porre-pressure-distribution-with-depth.png\">Figure 4.38<\/a>), and the permeability boundary conditions. For example, assuming uniform excess pore pressure distribution with depth, corresponding to a relatively thin clay layer (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.38-initial-excess-porre-pressure-distribution-with-depth.png\">Figure 4.38<\/a>) and double drainage, both from the top and the bottom of the layer (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.39-boundary-conditions-and-length-of-drainage-path.png\">Figure 4.39a<\/a>):<\/p>\n<ul>\n<li>at <em>t <\/em>= 0, the excess pore pressure will be uniform along the thickness, and equal to the applied pressure at the surface \u0394<em>u<\/em><sub>0<\/sub><em> = <\/em><em>q<sub>ext<\/sub><\/em><\/li>\n<li>at the top boundary <em>z<\/em>=0 where drainage is allowed, \u0394<em>u <\/em>= 0<\/li>\n<li>at the bottom boundary <em>z=H<\/em> where drainage is allowed too, \u0394<em>u <\/em>= 0<\/li>\n<\/ul>\n<figure id=\"attachment_288\" aria-describedby=\"caption-attachment-288\" style=\"width: 350px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" class=\"wp-image-285 size-full\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2025\/02\/4.38-initial-excess-porre-pressure-distribution-with-depth-e1743559781888.png\" alt=\"Illustration of three initial excess pore pressure \u0394_u distributions: uniform, trapezoidal, and triangular. The uniform pore pressure distribution is applicable to the thinner soil layer, and the triangular pore pressure distribution to the thicker soil layer.\" width=\"350\" height=\"419\" srcset=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2025\/02\/4.38-initial-excess-porre-pressure-distribution-with-depth-e1743559781888.png 350w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2025\/02\/4.38-initial-excess-porre-pressure-distribution-with-depth-e1743559781888-251x300.png 251w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2025\/02\/4.38-initial-excess-porre-pressure-distribution-with-depth-e1743559781888-65x78.png 65w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2025\/02\/4.38-initial-excess-porre-pressure-distribution-with-depth-e1743559781888-225x269.png 225w\" sizes=\"(max-width: 350px) 100vw, 350px\" \/><figcaption id=\"caption-attachment-288\" class=\"wp-caption-text\">Figure 4.38. Initial excess pore pressure distribution with depth, for increasing thickness of the compressible layer.<\/figcaption><\/figure>\n<p>The drainage boundary conditions suggest that excess pore pressure will immediately fall to zero at the permeable boundaries.<\/p>\n<p>A closed-form solution of Eq. 4.65 exists for the above conditions and provides the variation of the excess pore pressure with depth at different time instances (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.40-HR.png\">Figure 4.40<\/a>):<\/p>\n<p><strong>(4.67)<\/strong> [latex]\\Delta u\\left( {z,t} \\right) = \\sum\\limits_{m = 0}^\\infty {\\dfrac{{2\\Delta {u_0}}}{M}\\sin \\left( {\\dfrac{{Mz}}{{{H_{dr}}}}} \\right)} {e^{ - {M^2}{T_v}}}[\/latex]<\/p>\n<p>where [latex]M = \\dfrac{{\\pi \\left( {2m + 1} \\right)}}{2}[\/latex], and <em>T<sub>v <\/sub><\/em>is the <em>time factor<\/em>, equal to:<\/p>\n<figure id=\"attachment_288\" aria-describedby=\"caption-attachment-288\" style=\"width: 490px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" class=\"wp-image-286 size-full\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2026\/03\/4.39-boundary-conditions-and-length-of-drainage-path.png\" alt=\"The figure on the top presents the flow boundary conditions for a soil specimen where drainage is allowed from both the top and bottom of the sample. The excess pore pressure at the top and bottom of the sample are \u0394u=0 and the length of the drainage path is half the thickness of the sample. The figure on the bottom presents the flow boundary conditions for a soil specimen where drainage is allowed only from the top of the sample. The excess pore pressure at the top is \u0394u=0, and the gradient of the excess pore pressure at the bottom is zero. The length of the drainage path is equal to the thickness of the sample.\" width=\"490\" height=\"292\" srcset=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/4.39-boundary-conditions-and-length-of-drainage-path.png 490w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/4.39-boundary-conditions-and-length-of-drainage-path-300x179.png 300w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/4.39-boundary-conditions-and-length-of-drainage-path-65x39.png 65w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/4.39-boundary-conditions-and-length-of-drainage-path-225x134.png 225w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/4.39-boundary-conditions-and-length-of-drainage-path-350x209.png 350w\" sizes=\"(max-width: 490px) 100vw, 490px\" \/><figcaption id=\"caption-attachment-288\" class=\"wp-caption-text\">Figure 4.39. Boundary conditions and length of drainage path for (a) double drainage, and (b) drainage only from the top of the soil layer.<\/figcaption><\/figure>\n<p><strong>(4.68)<\/strong> [latex]{T_v} = \\dfrac{{{c_v}t}}{{H_{dr}^2}}[\/latex]<\/p>\n<p>The length of the drainage path, <em>H<sub>dr<\/sub><\/em> depends on the drainage boundary conditions, and is depicted in <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.40-HR.png\">Figure 4.40<\/a>.<\/p>\n<figure id=\"attachment_288\" aria-describedby=\"caption-attachment-288\" style=\"width: 400px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" width=\"400\" height=\"433\" class=\"wp-image-287 size-full\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2026\/03\/4.40-HR-e1743559804589.png\" alt=\"image\" srcset=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/4.40-HR-e1743559804589.png 400w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/4.40-HR-e1743559804589-277x300.png 277w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/4.40-HR-e1743559804589-65x70.png 65w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/4.40-HR-e1743559804589-225x244.png 225w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/4.40-HR-e1743559804589-350x379.png 350w\" sizes=\"(max-width: 400px) 100vw, 400px\" \/><figcaption id=\"caption-attachment-288\" class=\"wp-caption-text\">0 (mid figure) the excess pore pressure is zero at the top and bottom of the sample, and increases parabolically to its middle, where it attains its maximum value. When Tv is infinite (bottom figure) the excess pore pressure is zero across the entire sample.&#8221; width=&#8221;400&#8243; height=&#8221;433&#8243;&gt; Figure 4.40. Graphical representation of excess pore pressure distribution with depth at different time instances, calculated from Eq. 4.67.<\/figcaption><\/figure>\n<p>Instead of using the absolute value the of excess pore water pressure, consolidation progress can be described via the <em>degree of consolidation<\/em> (or <em>consolidation ratio<\/em>), <em>U<\/em>(<em>z<\/em>) which defines the amount of consolidation completed at a particular time instance, at a specific depth, <em>z<\/em>:<\/p>\n<p><strong>(4.69)\u00a0<\/strong>[latex]U\\left( z \\right) = 1 - \\dfrac{{\\Delta {u_z}}}{{\\Delta {u_0}}}[\/latex]<\/p>\n<p>The degree of consolidation is visualised as the grey shaded area in<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.40-HR.png\"> Figure 4.40<\/a>. From an engineering point of view, we are rather interested in the <em>average<\/em> degree of consolidation, <em>U<\/em> of the whole compressible layer. For the special case where the initial excess pore pressure distribution is assumed constant with depth (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.38-initial-excess-porre-pressure-distribution-with-depth.png\">Figure 4.38<\/a>) the average degree of consolidation is calculated as:<\/p>\n<p><strong>(4.70)<\/strong> [latex]U = 1 - \\sum\\limits_{m = 0}^\\infty {\\dfrac{2}{{{M^2}}}} {e^{ - {M^2}{T_v}}}[\/latex]<\/p>\n<p>The variation of the average degree of consolidation with the time factor <em>Tv<\/em> is illustrated in <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.41-variation-of-average-degree-of-consolidation.png\">Figure 4.41<\/a>.<\/p>\n<p>Eq. 4.70 and Figure 4.41 are also valid when drainage is not allowed at the bottom of the clay layer. The length of the drainage path in that case will be <em>H<sub>dr <\/sub><\/em>= <em>H<\/em> (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.39-boundary-conditions-and-length-of-drainage-path.png\">Figure 4.39<\/a>). Note that at an impermeable boundary, the gradient of pore pressure will be zero [latex]\\partial u\/\\partial z[\/latex] = 0.<\/p>\n<figure id=\"attachment_288\" aria-describedby=\"caption-attachment-288\" style=\"width: 700px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" class=\"wp-image-288 size-full\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2026\/03\/4.41-variation-of-average-degree-of-consolidation-e1743559821462.png\" alt=\"Graph depicting the variation of the average degree of consolidation U(%) with the time factor T_v,. When the time factor becomes Tv=1 the average degree of consolidation is about 90%.\" width=\"700\" height=\"317\" srcset=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/4.41-variation-of-average-degree-of-consolidation-e1743559821462.png 700w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/4.41-variation-of-average-degree-of-consolidation-e1743559821462-300x136.png 300w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/4.41-variation-of-average-degree-of-consolidation-e1743559821462-65x29.png 65w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/4.41-variation-of-average-degree-of-consolidation-e1743559821462-225x102.png 225w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/4.41-variation-of-average-degree-of-consolidation-e1743559821462-350x159.png 350w\" sizes=\"(max-width: 700px) 100vw, 700px\" \/><figcaption id=\"caption-attachment-288\" class=\"wp-caption-text\">Figure 4.41. Variation of the average degree of consolidation <em>U<\/em> with the time factor <em>T<sub>v<\/sub> <\/em>for uniform initial excess pore water pressure distribution.<\/figcaption><\/figure>\n<p>Note that <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.41-variation-of-average-degree-of-consolidation.png\">Figure 4.41<\/a> can be used to estimate settlement of a 1-D soil column at any given time <em>t<\/em><sub>1<\/sub>. By substituting <em>t<\/em><sub>1<\/sub> into the expression for the time factor <em>T<sub>v<\/sub><\/em> (Eq. 4.68) we can find the average degree of consolidation <em>U<\/em><sub>1<\/sub> at <em>t<\/em><sub>1<\/sub> directly from <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/4.41-variation-of-average-degree-of-consolidation.png\">Figure 4.41<\/a>. Therefore, settlement at <em>t<\/em><sub>1<\/sub> will be:<\/p>\n<p><strong>(4.71)<\/strong> [latex]{\\rho _1} = \\dfrac{{{U_1}\\left( \\% \\right)}}{{100}}{\\rho _{pc}}[\/latex]<\/p>\n<p>where the primary consolidation settlement <em>\u03c1<\/em><sub>pc <\/sub>is calculated according to Section 4.6.2, assuming linear elastic or non-linear soil behaviour.<\/p>\n","protected":false},"author":1,"menu_order":13,"template":"","meta":{"pb_show_title":"","pb_short_title":"4.8 Evolution of primary consolidation settlement with time","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-289","chapter","type-chapter","status-publish","hentry"],"part":215,"_links":{"self":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/289","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":1,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/289\/revisions"}],"predecessor-version":[{"id":290,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/289\/revisions\/290"}],"part":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/parts\/215"}],"metadata":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/289\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/media?parent=289"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapter-type?post=289"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/contributor?post=289"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/license?post=289"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}