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 $var element = document.getElementById("moose-equation-ee82a246-32b3-4941-a6a7-d65d820ed864");katex.render("h", element, {displayMode:false,throwOnError:false});$ is loaded vertically at constant stress. The top surface is drained and at $var element = document.getElementById("moose-equation-3fd4aa5d-0f3a-4672-bc53-33b0ce75b277");katex.render("t = 0", element, {displayMode:false,throwOnError:false});$ , the vertical force of magnitude $var element = document.getElementById("moose-equation-7280797a-f0ca-4b3e-9e52-963ae8ea2d53");katex.render("F", element, {displayMode:false,throwOnError:false});$ 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
$var element = document.getElementById("moose-equation-c4eebd4a-e768-4830-af07-b4b43152f76c");katex.render("K", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-f6b98d98-98e9-42f8-8600-765532fa6bdb");katex.render("4.0", element, {displayMode:false,throwOnError:false});$	Pa	Bulk modulus
$var element = document.getElementById("moose-equation-564acb09-80d5-48f2-ae88-21d12227ad75");katex.render("G", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-35a0e0f3-e18b-4766-8879-713588775480");katex.render("3.0", element, {displayMode:false,throwOnError:false});$	Pa	Shear modulus
$var element = document.getElementById("moose-equation-ffdcfe6f-5cfd-40f1-8c47-53025b199306");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-32e6dd87-3fa3-4176-b637-a8cc1b837be5");katex.render("0.6", element, {displayMode:false,throwOnError:false});$	-	Biot's poroelastic coefficient
$var element = document.getElementById("moose-equation-9d7550be-33f3-42d3-959c-b7aa27f5b651");katex.render("k", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-fa729567-8add-4f7d-8fd9-09134bdb3e9c");katex.render("1.5", element, {displayMode:false,throwOnError:false});$	m $var element = document.getElementById("moose-equation-b5875c95-81f1-4fbc-b6fc-a6c9b4fcd124");katex.render("^{2}", element, {displayMode:false,throwOnError:false});$	Permeability
$var element = document.getElementById("moose-equation-9c7a2862-4b60-4229-9781-2a8a8c15a7c2");katex.render("\\mu", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-7cdd67d5-2f46-43bd-9f26-0b0af59f571c");katex.render("1.0", element, {displayMode:false,throwOnError:false});$	Pa s	Fluid viscosity
$var element = document.getElementById("moose-equation-092aa69e-c65b-4b96-aca1-14841ab15641");katex.render("\\phi", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-a047f0de-5c80-438f-ba4a-9227a5307a3b");katex.render("0.1", element, {displayMode:false,throwOnError:false});$	-	Porosity
$var element = document.getElementById("moose-equation-b4932d84-e5ff-4c5c-a5c0-f83eb6fafefc");katex.render("K_{f}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-6a4c14dc-eef3-4cd0-9849-a993978126d0");katex.render("8.0", element, {displayMode:false,throwOnError:false});$	Pa	Fluid bulk modulus
$var element = document.getElementById("moose-equation-bb233abb-35fa-419d-b3cf-3829916d0c5a");katex.render("F", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-1ade188c-2621-4313-9aee-a0e044a3dc70");katex.render("1.0", element, {displayMode:false,throwOnError:false});$	Pa	Applied force

Here are definitions of some poroelastic parameters:

Solid bulk modulus: $var element = document.getElementById("moose-equation-a7e0c37c-9cda-4851-8c77-c12aae2aedc1");katex.render("K_{s} = \\frac{K}{1 - \\alpha}", element, {displayMode:false,throwOnError:false});$
Storage coefficient: $var element = document.getElementById("moose-equation-45996f46-16e7-4217-b79c-704344717137");katex.render("S = \\frac{\\phi}{K_{f}} + \\frac{\\alpha - \\phi}{K_{s}}", element, {displayMode:false,throwOnError:false});$
Confined compressibilty: $var element = document.getElementById("moose-equation-4d99c588-39e9-4950-bdae-e952bbd7db23");katex.render("m_{v} = \\frac{1}{K + \\frac{4}{3} G}", element, {displayMode:false,throwOnError:false});$
Consolidation coefficient: $var element = document.getElementById("moose-equation-7b876a7e-fd74-4c48-af49-08a9e485f37f");katex.render("c_{v} = \\frac{k}{\\mu \\left(S + \\alpha^{2} m_{v}\\right)}", element, {displayMode:false,throwOnError:false});$

Solutions

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

var element = document.getElementById("moose-equation-a95b7301-593f-4a2d-8635-04646b0c4053");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 $var element = document.getElementById("moose-equation-121a8f18-eb89-45a8-95f8-4222a522378c");katex.render("\\frac{c_{v} t}{h^{2}} \\ll 1", element, {displayMode:false,throwOnError:false});$ , the pressure solution can be simplified as:

var element = document.getElementById("moose-equation-ef7ce10d-d3b6-4bec-8bc0-cf96dc94542f");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 $var element = document.getElementById("moose-equation-6344eb6b-d26c-4495-926a-0fe255eb46cd");katex.render("U", element, {displayMode:false,throwOnError:false});$ is defined as:

var element = document.getElementById("moose-equation-17dde01b-b436-4f31-9056-97b10792d7d4");katex.render("U = \\frac{u - u_{0}}{u_{\\infty} - u_{0}},", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-314d7115-d66a-424d-9815-56f4620612a2");katex.render("u", element, {displayMode:false,throwOnError:false});$ is the vertical displacement, $var element = document.getElementById("moose-equation-5c1cf75a-aad2-45aa-ab01-fa134e7af042");katex.render("u_{0}", element, {displayMode:false,throwOnError:false});$ the immediate displacement after application of the vertical load $var element = document.getElementById("moose-equation-2b41493b-e72e-4a0d-98e7-dc1118d1dec2");katex.render("F", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-43ddb3be-7a6c-4c01-a157-958aa8882ede");katex.render("u_{\\infty}", element, {displayMode:false,throwOnError:false});$ the final displacement. The degree of consolidation varies between $var element = document.getElementById("moose-equation-9f2aceef-2d20-411a-a5cc-b56106c82df8");katex.render("0", element, {displayMode:false,throwOnError:false});$ (at the moment of loading) and $var element = document.getElementById("moose-equation-7da1854b-b140-4be4-8876-30600bee727e");katex.render("1", element, {displayMode:false,throwOnError:false});$ (after consolidation has been completed) and evolves over time.

Its expression is given by Verruijt (2016) as:

var element = document.getElementById("moose-equation-0fba0233-f3c2-4863-ba5e-b66be096995e");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