## Deriving and Solving the Equations of Consolidation

In an earlier post we discussed consolidation settlement. For situations where soil a) decreases its volume due to rearrangement of the particles and b) does so over a relatively long period of time due to difficulties in expelling pore water pressures, we need to know how long it takes to reach maximum settlement, in addition to know what that settlement is and what deflections we might achieve along the way.

This is generally a two part process: a) determining the dissipation of pore water pressure and b) determining the amount of settlement at a given time. This post will focus on the first part of the process; we will deal with the second in a later post. This is a well-worn path in geotechnical engineering but hopefully this derivation is a little simpler than others.

## Developing the Differential Equation

### Relating Porosity to Strain

Let us begin by considering the system above (from Tsytovich (1976)), a uniformly loaded soil layer which is saturated (always) and clay (usually.) The first thing we need to note is that, while the diagram uses z as the variable of length in the vertical direction, from this post we will use the variable x (sorry for the confusion.) The second thing is that we assume the water and the solids to be incompressible. The third thing is that all the changes that take place do so because of changes in the voids where the water is resident. We first note the definition of porosity as

$n={\frac {V_{{v}}}{V_{{t}}}}$ (1)

where

• n = porosity
• Vv = volume of voids
• Vt = total volume

That being the case, the relationship between the porosity of the soil (due to changes in the volume of the voids) and the flow rate of the water can be expressed as

${\frac {\partial }{\partial x}}q(x,t)=-{\frac {\partial }{\partial t}} {\it n}(x,t)$ (2)

We can envision a differential volume having a height x and an area A. Since the problem is one-dimensional, the areas cancel out and the ratio of the void height to the total height is

$n={\frac {x_{{v}}}{x_{{t}}}}$ (3)

where

• xv = height of the voids
• xt = height of the solids

We want to determine the change in porosity from some state 0 to some state 1, just as we did with void ratio in this post. That change can be expressed as follows:

${\frac {x_{{{\it v0}}}}{x_{{{\it t0}}}}}-{\frac {x_{{{\it v1}}}}{x_{{{\it t0}}}-x_{{{\it v0}}}+x_{{{\it v1}}}}}$ (4)

We can assume that the change in void volume xv0 – xv1 << xt0, in which case Equation (4) can be simplified to

$\Delta{{{\it n}}}={\frac {x_{{{\it v0}}}-x_{{{\it v1}}}}{x_{{{\it t0}}}}} = \frac{\Delta x}{x_t}$ (5)

Now we can say, for the small increments we are dealing with here, that the change in strain is equal to the change in porosity,

$\Delta n = \Delta \epsilon$ (6)

From this and our previous post, we can thus use the change in strain to come to the following:

$\Delta n = m_v \Delta \sigma_x$ (7)

Stating this differentially,

${\frac {\partial }{\partial t}}{\it n}(x,t)=m_{{v}}{\frac {\partial }{\partial t}}\sigma_{{x}}(x,t)$ (8)

In this way we emphasise that both porosity and uniaxial stress are functions of depth and time. We now combine Equations (2) and (8) to yield

${\frac {\partial }{\partial x}}q(x,t)=-m_{{v}}{\frac {\partial }{\partial t}}\sigma_{{x}}(x,t)$ (9)

### Including Permeability

Darcy’s Law (or more properly d’Arcy’s Law) states that

$q(x,t) = -k{\frac {\partial }{\partial x}}H(x,t)$ (10)

where k is the coefficient of permeability and H is the hydraulic head. We then combine Equations (9) and (10) to obtain

$k{\frac {\partial ^{2}}{\partial {x}^{2}}}H(x,t)=m_{{v}}{\frac {\partial }{\partial t}}\sigma_{{x}}(x,t)$ (11)

Noting that

$H(x,t) = \frac{u(x,t)}{\gamma_w}$ (12)

where $\gamma_w$ is the unit weight of water and $u(x,t)$ is the excess pore water pressure generated by the decrease in porosity, we substitute this with the result of

${\frac {k{\frac {\partial ^{2}}{\partial {x}^{2}}}u(x,t)}{{\it \gamma_w }}}=m_{{v}}{\frac {\partial }{\partial t}}\sigma_{{x}}(x,t)$ (13)

### Putting It All Together

We now define the parameter

$c_v = \frac{k}{m_v \gamma_w}$ (14)

At this point we need to stop and make an observation: this whole process wouldn’t be worth too much if $c_v$ wasn’t constant (or reasonably so.) The reason it is is that the coefficient of permeability $k$ and the coefficient of volume compressibility $m_v$ both decrease as the void ratio/porosity decrease, and do so at roughly the same rate. That being the case, we substitute Equation (14) into Equation (13) to have at last

$c_{{v}}{\frac {\partial ^{2}}{\partial {x}^{2}}}u(x,t)={\frac {\partial }{\partial t}}\sigma_{{x}}(x,t)$ (15)

## Tidying up the Physics, Governing Equation, Boundary and Initial Conditions, and the Solution

It should be evident that getting to Equation (15) was a major triumph for geotechnical theory. But it’s also evident that there’s one glaring problem: the dependent variable on the left hand side is not the same as the one on the right. It is here that we need to clarify some assumptions behind our equations.

A basic assumption in consolidation theory is that, when the load at the surface is applied, all of this additional load is initially borne by the pore water pressure. Because of the aforementioned permeability, the water will want to “head for the exits,” i.e., the permeable boundaries of the layer being compressed. When the particles have rearranged themselves and the excess pore water has been squeezed out, the settlement should stop (until secondary compression kicks in.) During this process the load on the pore water is being progressively transferred to the soil particles until consolidation has stopped (which, in theory, it never does, as we will see) and the load is completely handed off to the particles.

That being the case, the consolidation equation (15) should be rewritten as

$c_{{v}}{\frac {\partial ^{2}}{\partial {x}^{2}}}u(x,t)={\frac {\partial }{\partial t}}u(x,t)$ (16)

At this point we should invoke the effective stress equation

$\sigma_x(x,t) = p - u(x,t)$ (17)

where p is the applied pressure, and do the following to determine the time and distance history of the vertical stress $\sigma_x(x,t)$:

• Determine the solution of Equation (16), which will be our governing equation.
• Determine the initial and boundary conditions for the problem.
• Solve the problem for $\sigma_x(x,t)$ using Equation (17).

There are several ways to solve the governing equation for this problem. Verruijt employed Laplace Transforms to accomplish this; these are very useful, as was demonstrated by Warrington (1997) for the wave equation. For this analysis, we will use separation of variables, a method which is not as fundamental as one might like (this is a demonstration of a method that is) but which is easier to follow than most of the others.

The separation of variables begins by assuming the solution is as follows:

$u(x,t) = X(x)T(t)$ (18)

This means that the solution is a product of two functions, one of time and the other of distance, and that those functions are separate one from another. Substituting this into Equation (16) and rearranging a bit yields

${\frac {{\frac {d^{2}}{d{x}^{2}}}X(x)}{X(x)}}={\frac {{\frac {d}{dt}}T(t)}{{\it c_v}\,T(t)}}$ (19)

Since the left and right hand sides are equal to each other, they are equal to a third variable, which we will designate as $\beta^2$. In reality they are equivalent to the eigenvalue $\lambda = \beta^2$, but the utility of the squared term will become evident. In any case,

${\frac {{\frac {d^{2}}{d{x}^{2}}}X(x)}{X(x)}}=-{\beta}^{2}$ (20a)

${\frac {{\frac {d}{dt}}T(t)}{{\it c_v}\,T(t)}}=-{\beta}^{2}$ (20b)

The solutions to these equations are, respectively,

$X(x)={\it C_1}\,\cos(\beta\,x)+{\it C_2}\,\sin(\beta\,x)$ (21a)

$T(t)={e^{-{\it c_v}\,{\beta}^{2}t}}{\it C_3}$ (21b)

We need to pause and consider the boundary conditions. As the problem is shown at the top of the article, the boundary conditions are as follows:

$X(0) = 0$ (22a)

$X'(h) = 0$ (22b)

Equation (22a) represents a Dirichelet boundary condition and Equation (22b) represents a Neumann boundary condition. While the equation for $X(x)$ can certainly be solved for this boundary condition, a simpler way would be to do the following:

• Mirror the problem shown above at x = h so that you have two permeable boundaries.
• You now have a layer of 2h thickness with Dirichelet boundary conditions on both sides. At the centre of this new layer, there is no flow; the water above it flows upward, and the water below flows downward.
• Problems in the field can have either one or two permeable boundaries. The distance h is NOT the thickness of the layer but the longest distance the trapped pore water must travel to escape. Confusing h with the thickness of the layer is a common mistake and should be avoided at all costs.

That said, the second boundary condition is now

$X(2h) = 0$ (22c)

Returning to Equation (21a), because of Equation (22a) $C_1 = 0$ as the cosine is by definition unity at this point. If we then set x = 2h and X(2h) = 0, for real, non-zero values of $C_2$ the boundary condition can be satisfied if and only if

$\beta = 1/2\,{\frac {n\pi }{h}}$ (23)

This is the square root of the eigenvalues. Any integer value of n > 0 is valid for this, and we will have recourse to them all to produce a complete orthogonal set (Fourier Series) to solve the problem.

This change will affect both Equations (21a) and (21b). Combining constants into the coefficient $B_n$, substituting the results into Equation (18) and making it an infinite sum for the Fourier series yields

$u(x,t) = {\it B_{n}}\,\sin(1/2\,{\frac{n\pi\,x}{h}}){e^{-1/4\,{\frac{{\it c_v}\,{n}^{2}{\pi}^{2}t}{{h}^{2}}}}}$ (24)

At this point we need to consider our initial conditions $f(x)$. Although many different distributions of initial pore pressure are possible, the simplest one–and the one most commonly used–is a uniform pressure p, which is the same as the surface pressure compressing the layer. For a complete orthogonal set, the coefficients $B_n$ can be computed by (Kreyszig (1988)) as

${\it B_n}=\int _{0}^{2\,h}f(x)\sin(1/2\,{\frac {n\pi \,x}{h}}){dx}{h}^{-1}$ (25)

Substituting $f(x) = p$ and performing the integration,

$B_n = 2\,{\frac{p\left(1-\cos(n\pi)\right)}{n\pi}}$ (26)

Substituting this into Equation (24) and taking the complete sum, we have at last

$u(x,t)=\sum_{n=1}^{\infty}2\,p\left(1-\cos(n\pi)\right)\sin(1/2\,{\frac{n\pi\,x}{h}}){e^{-1/4\,{\frac{{\it c_v}\,{n}^{2}{\pi}^{2}t}{{h}^{2}}}}}{n}^{-1}{\pi}^{-1}$ (27)

More simply we can say that

$u(x,t)=\frac{4p}{\pi}\left(\sin(1/2\,{\frac{\pi\,x}{h}}){e^{-1/4\,{\frac{{\it c_v}\,{\pi}^{2}t}{{h}^{2}}}}}+1/3\,\sin(3/2\,{\frac{\pi\,x}{h}}){e^{-9/4\,{\frac{{\it c_v}\,{\pi}^{2}t}{{h}^{2}}}}}+1/5\,\sin(5/2\,{\frac{\pi\,x}{h}}){e^{-{\frac{25}{4}}\,{\frac{{\it c_v}\,{\pi}^{2}t}{{h}^{2}}}}}\cdots\right)$ (28)

The vertical pressure on the soil skeleton is determined by combining Equations (17) and (28) to yield

$\sigma_{x}(x,t)=p\left(1-\frac{4}{\pi}\left(\sin(1/2\,{\frac{\pi\,x}{h}}){e^{-1/4\,{\frac{{\it c_v}\,{\pi}^{2}t}{{h}^{2}}}}}+1/3\,\sin(3/2\,{\frac{\pi\,x}{h}}){e^{-9/4\,{\frac{{\it c_v}\,{\pi}^{2}t}{{h}^{2}}}}}+1/5\,\sin(5/2\,{\frac{\pi\,x}{h}}){e^{-{\frac{25}{4}}\,{\frac{{\it c_v}\,{\pi}^{2}t}{{h}^{2}}}}}\cdots\right)\right)$ (29)

It is interesting to note that, although Equation (27) includes all positive non-zero values of n, only the odd ones end up in Equations (28) and (29). For even values of n, $B_n = 0$.

## Conclusion

At this point we have derived the equations of pressure dissapation for consolidation settlement. In a future post we will deal with the second part of the problem, namely how much of the total anticipated settlement has taken place at any given time from initial loading.

## Reference

Kreyszig, E. (1988) Advanced Engineering Mathematics. Sixth Edition. New York: John Wiley and Sons

Posted in Geotechnical Engineering, Soil Mechanics

## The “New” NAVFAC DM 7.1 (Soil Mechanics) is Now in Print

Recently, NAVFAC revised their half-century old NAVFAC DM 7.1 and have released it to the public. Now we offer this classic (formally designated as UFC 3-220-10) in print format. Click here or on the photo at the right to order.

The chapters are as follows:

1. IDENTIFICATION AND CLASSIFICATION OF SOIL AND ROCK
2. FIELD EXPLORATION, TESTING, AND INSTRUMENTATION
3. LABORATORY TESTING
4. DISTRIBUTION OF STRESSES
5. ANALYSIS OF SETTLEMENT AND VOLUME EXPANSION
6. SEEPAGE AND DRAINAGE
7. SLOPE STABILITY
8. CORRELATIONS FOR SOIL AND ROCK

As was the case before, there is a wealth of information here, updated and expanded. Now you can order your copy in print today!

Posted in Uncategorized

## NAVFAC DM 7.1 Gets 2022 Refresh — GeoPrac.net

Nearly every geotechnical engineer has referred to the venerable 1986 NAVFAC DM 7.1 Manual on Soil Mechanics. There are numerous useful charts, correlations, tables, and equations for a variety of geotechnical challenges, which is probably […]

NAVFAC DM 7.1 Gets 2022 Refresh — GeoPrac.net

That didn’t take long…after our post The “Before and After” of NAVFAC DM 7, I was made aware of this. This has been posted to our Soil Mechanics page. As far as a print version is concerned, we are considering this, the old version continues to be available.

Posted in Geotechnical Engineering

## The “Before and After” of NAVFAC DM 7

It’s not an overstatement to say that NAVFAC DM 7.01 (Soil Mechanics) and DM 7.02 (Foundations and Earth Structures) are the two most consequential documents on geotechnical engineering to be produced in the last century, and remain that to this day. This is in spite of the fact that it is now a half century old. Sales of hard copies of DM 7.01 and DM 7.02 have supported this site for many years.

So what is the history of this document? And has NAVFAC (the Naval Facilities Engineering Command) updated it? This brief article seeks to answer that question by looking at the documents themselves. We don’t have “inside information” or history, but hopefully we can shed some light and give you an opportunity to look at the history of this important document.

## What is a “DM” and the Original DM 7

The acronym (the military loves these) “DM” stands for “Design Manual. NAVFAC put out a series of documents to give guidance to designers and builders of facilities for the United States Navy. “DM-7” refers to the design manual for geotechnical engineering, entitled Soil Mechanics, Foundations and Earth Structures. The “original” DM-7 was issued in March 1971; the first change version dated September of that year can be downloaded here. Even it, however, was a successor to the “NAVDOCKS DM-7” dated February 1962 (you’ll see that reference in many older texts.)

## From One to Three Documents

In 1982-3 the Navy opted to divide DM-7 into three parts:

The DM-7.01 and DM-7.02 documents we offer in print and download are minor revisions of those documents issued in 1986.

As J. Ledlie Klosky of the United States Military Academy noted in the preface to our version of DM-7.01:

This extraordinary document, published in 1982, is now considerably out‐of‐date and, except as UFC 30220‐10N, is no longer a sanctioned publication of the US Government. It is provided here as a reference because of the incredible density of highly practical geotechnical design guidance it contains. It is also of significant historical interest, and when combined with DM 7.2, it represents perhaps THE principle compendium of geotechnical knowledge used by designers between 1982 and around the turn of the century. It is a testament to the strength of the document that some of the design methods presented are still in use today. The importance of the Federal labs (particularly FHWA, Bureau of Reclamation, Army and Navy labs) in pushing the practice of geotechnical engineering forward between 1930 and around the time of the publication of this manual cannot be overstated, and this manual is a testament to that heritage. Thus, you are holding in your hands (or in your computer memory) a great reference for preliminary design guidance and a knowledge artifact that will be recognized by nearly every senior practicing geotechnical engineer.

It’s unlikely that the Navy had any idea that this document would be such a “hit” in the industry. That’s for a variety of reasons. One of them, as Klosky notes, is that it came at the end of a very fruitful era of research and advancement in geotechnical engineering, starting with pioneers such as Terzaghi, Casagrande, Taylor, Tschebotarioff, Spangler and many others, and supported both by military and civilian agencies of the United State government. Another is that, especially with the 1982/6 versions, the figures and text were largely free of copyright, which guaranteed that they ended up in just about every geotechnical text and reference book (including this one.)

In the late 1990’s the coming of the internet made widespread dissemination of information simple and economical. DM-7.01 and DM-7.02 became a part of that when the Navy put out an online version of this, which this site disseminated widely even through events such as 9/11 (when many military sites went “dark” even with information such as this) and the revisions we will discuss below. The scanning and OCR technology of the time were not up to what we have now, and so the version has a “long in the tooth” air about it, but in those years it helped to disseminate the valuable information contained in DM-7.

## The Coming of UFC 3-220

With a new millennium the needs of the military for construction guidance did not stand still, especially with the conflicts in Iraq and Afghanistan. Additionally the military worked to homogenize its guidance across the service branches, thus the Navy’s Design Manuals because part of the UFC (Unified Facilities Criteria) program. For geotechnical engineering this meant the UFC 3-220 series. Relating to the DM-7 series there were two rounds of revision, the first in 2004-5, and the second in 2012 with more recent revisions.

### The 2004-5 Round:

Although Soil Mechanics was pretty much left alone, Foundations and Earth Structures was broken up into specific topics. This reflects the fact that, of the two major volumes, the second has aged more poorly than the first.

### The 2012 Round:

With this round the Navy decided to make their chief document a “front end” for the International Building Code. The document is UFC 3-220-01, Geotechnical Engineering, in this case Change 1 dated 3 November 2021.

## Some Thoughts

It is unfortunate that, in adopting the IBC, the Navy has decided to leave behind a generation and more of reference materials which have enriched the geotechnical industry. But we have this legacy still available, and trust that is is helpful to you.

Update: they didn’t leave them behind after all. Right after this was posted, we learned that they had updated DM 7.01 and a direct update for DM 7.02 is in the works. You can read about this here and watch the video below for more information.

Posted in Geotechnical Engineering

## Geotechnical Site Characterization Back in Print

We’re in the process of revising our bookstore, putting some titles into broader distribution and retiring others that haven’t sold as well. One of those which we started to retire was Geotechnical Site Characterization, the FHWA’s latest publication on laboratory and field testing. We’ve had technical issues with this book from the start and first decided to drop it from our offereings.

Some looks at our traffic, however, tell us that you’re interested in this book. So we’ve put it back in our lineup, and you can find it here.