Terzaghi's consolidation problem

The classical Terzaghi's consolidation problem analyzes the one-dimensional deformation delay when compressing soil. A confined soil sample, fully-saturated with water is loaded vertically by a constant stress and its deformation and fluid pressure is observed as a function of time on the vertical axis. This problem is a classical poroelastic problem which can be simplified in one dimension. For more information, please see Chapter 2 in Verruijt (2016).

Setup

A confined soil column of height $var element = document.getElementById("moose-equation-7c61c172-7eae-46c7-8358-9b8ee1ec1121");katex.render("h", element, {displayMode:false,throwOnError:false});$ is loaded vertically at constant stress. The top surface is drained and at $var element = document.getElementById("moose-equation-40e2e8ea-8e9c-4470-92ec-013308bd5d0d");katex.render("t = 0", element, {displayMode:false,throwOnError:false});$ , the vertical force of magnitude $var element = document.getElementById("moose-equation-4df63ae4-1c72-4782-a98b-321655dba303");katex.render("F", element, {displayMode:false,throwOnError:false});$ is applied. This setup is sketched in Figure 1.

Figure 1: Setup for the Terzaghi's problem.

Parameters

Table 1: List of parameters used for the Terzaghi's problem.

Symbol	Value	Unit	Name
$var element = document.getElementById("moose-equation-1e1ca3a2-2705-48f7-89e3-8419c7e7dbd6");katex.render("K", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-26253907-1b65-4b76-88c3-9af1bccb149c");katex.render("4.0", element, {displayMode:false,throwOnError:false});$	Pa	Bulk modulus
$var element = document.getElementById("moose-equation-6233baad-26e4-49e8-ae3a-dea848ebead5");katex.render("G", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-73582293-384e-4550-aa09-3b7371b4e1bd");katex.render("3.0", element, {displayMode:false,throwOnError:false});$	Pa	Shear modulus
$var element = document.getElementById("moose-equation-f13c776e-8798-4d90-b67d-d0c67a2d945d");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-d63f3a57-d91e-4658-98cf-0630478c5479");katex.render("0.6", element, {displayMode:false,throwOnError:false});$	-	Biot's poroelastic coefficient
$var element = document.getElementById("moose-equation-8431819c-276f-4089-bf01-175a9d6e0bad");katex.render("k", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-7236afa0-43d5-419d-af29-cd299132a1d7");katex.render("1.5", element, {displayMode:false,throwOnError:false});$	m $var element = document.getElementById("moose-equation-7bf17822-0c75-4033-94dc-068e9321f75e");katex.render("^{2}", element, {displayMode:false,throwOnError:false});$	Permeability
$var element = document.getElementById("moose-equation-4a9d1008-f7dd-4459-948f-46b8a1ddc383");katex.render("\\mu", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-d69fa51a-1c2a-4a0d-8ade-dfbde0f31362");katex.render("1.0", element, {displayMode:false,throwOnError:false});$	Pa s	Fluid viscosity
$var element = document.getElementById("moose-equation-e26f5b53-b4ab-43d9-a873-e731bd429ab6");katex.render("\\phi", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-7989122e-369d-4c07-8796-6ddea594e781");katex.render("0.1", element, {displayMode:false,throwOnError:false});$	-	Porosity
$var element = document.getElementById("moose-equation-ecd5651f-0cd9-4c80-9635-fbd962c3afe9");katex.render("K_{f}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-8a9bf376-b51a-4880-ac8b-63096e03b4ad");katex.render("8.0", element, {displayMode:false,throwOnError:false});$	Pa	Fluid bulk modulus
$var element = document.getElementById("moose-equation-cf5d1516-b65d-4bed-9e65-ea8aea41735c");katex.render("F", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-af605a52-4d31-4a30-8bcf-b08bbf76667a");katex.render("1.0", element, {displayMode:false,throwOnError:false});$	Pa	Applied force

Here are definitions of some poroelastic parameters:

Solid bulk modulus: $var element = document.getElementById("moose-equation-1f9f53f1-4394-4589-b503-94f91a577d4f");katex.render("K_{s} = \\frac{K}{1 - \\alpha}", element, {displayMode:false,throwOnError:false});$
Storage coefficient: $var element = document.getElementById("moose-equation-b00ef4eb-d042-48b8-8a94-8b185a15cb5e");katex.render("S = \\frac{\\phi}{K_{f}} + \\frac{\\alpha - \\phi}{K_{s}}", element, {displayMode:false,throwOnError:false});$
Confined compressibilty: $var element = document.getElementById("moose-equation-e91ddf91-19e1-49c1-9084-ef8228b74dbf");katex.render("m_{v} = \\frac{1}{K + \\frac{4}{3} G}", element, {displayMode:false,throwOnError:false});$
Consolidation coefficient: $var element = document.getElementById("moose-equation-794a625f-3fb0-4636-b525-d214ada712d5");katex.render("c_{v} = \\frac{k}{\\mu \\left(S + \\alpha^{2} m_{v}\\right)}", element, {displayMode:false,throwOnError:false});$

Solutions

Verruijt (2016) gives the following solution for the fluid pressure:

var element = document.getElementById("moose-equation-a487fa94-7e31-432c-958d-c3a6ebc923db");katex.render("\\frac{p\\left(z, t\\right)}{p_{0}} = \\frac{4}{\\pi} \\sum_{k=1}^{+\\infty} \\frac{{\\left(-1\\right)}^{k-1}}{2k - 1} \\cos \\left[\\left(2k - 1\\right) \\frac{\\pi}{2} \\frac{z}{h}\\right] \\exp\\left[-{\\left(2k - 1\\right)}^{2} \\frac{\\pi^{2}}{4} \\frac{c_{v} t}{h^{2}}\\right].", element, {displayMode:true,throwOnError:false});

For small values of the time parameter $var element = document.getElementById("moose-equation-1e2c76fe-8ed5-415f-98e3-a3cf76a0ba4c");katex.render("\\frac{c_{v} t}{h^{2}} \\ll 1", element, {displayMode:false,throwOnError:false});$ , the pressure solution can be simplified as:

var element = document.getElementById("moose-equation-069f53f5-f2dc-421f-b908-569669cb529b");katex.render("\\frac{p}{p_{0}} \\approx \\text{erfc} \\left(\\frac{h - z}{2 \\sqrt{c_{v}t}}\\right).", element, {displayMode:true,throwOnError:false});

The degree of consolidation $var element = document.getElementById("moose-equation-3d431089-954c-4224-a11d-fe28a07d9fdf");katex.render("U", element, {displayMode:false,throwOnError:false});$ is defined as:

var element = document.getElementById("moose-equation-929f6602-5320-428d-80bd-17e4baf9233a");katex.render("U = \\frac{u - u_{0}}{u_{\\infty} - u_{0}},", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-358e1785-9fcf-4fe3-8f5e-9813d48b903d");katex.render("u", element, {displayMode:false,throwOnError:false});$ is the vertical displacement, $var element = document.getElementById("moose-equation-09bf934c-5459-42d1-a3d5-9327657aab88");katex.render("u_{0}", element, {displayMode:false,throwOnError:false});$ the immediate displacement after application of the vertical load $var element = document.getElementById("moose-equation-5984f064-c98b-464e-919c-ebb68f989db6");katex.render("F", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-4a3066a4-ec8a-42f7-b2f1-bd53462912f5");katex.render("u_{\\infty}", element, {displayMode:false,throwOnError:false});$ the final displacement. The degree of consolidation varies between $var element = document.getElementById("moose-equation-57044662-c47a-4c13-b04f-d026613ff81c");katex.render("0", element, {displayMode:false,throwOnError:false});$ (at the moment of loading) and $var element = document.getElementById("moose-equation-3a1b2435-c63a-4dcb-b78a-4f5896b7f364");katex.render("1", element, {displayMode:false,throwOnError:false});$ (after consolidation has been completed) and evolves over time.

Its expression is given by Verruijt (2016) as:

var element = document.getElementById("moose-equation-9a927f18-77f9-428a-9aa2-388d16576daa");katex.render("U = 1 - \\frac{8}{\\pi^{2}} \\sum_{k = 1}^{+\\infty} \\frac{1}{{\\left(2k - 1\\right)}^{2}} \\exp\\left[-{\\left(2k - 1\\right)}^{2} \\frac{\\pi^{2}}{4} \\frac{c_{v} t}{h^{2}}\\right].", element, {displayMode:true,throwOnError:false});

Figure 2: Fluid pressure solution for the Terzaghi's consolidation problem.

Figure 3: Consolidation solution for the Terzaghi's consolidation problem.

Complete Source Files

terzaghi.i

References

Arnold Verruijt. Theory and problems of poroelasticity. Delft University of Technology, 2016.

@book{Verruijt2016,
    author = "Verruijt, Arnold",
    editor = "Verruijt, Arnold",
    publisher = "Delft University of Technology",
    title = "{Theory and problems of poroelasticity}",
    year = "2016"
}

Mandel s problem

Setup
Parameters
Solutions
Complete Source Files
References