T&P formula for capillary rise

[fattdad]

Dear All,

Equation 20.4 of my version of Terzaghi & Peck (second edition, 1967) shows the height of capillary rise as follows:

hc=c/(e*D10), where:

hc = height of capillarity rise (cm)
c = empirical coefficient ranging from 0.1 to 0.5 (cm^2 - typical value of 0.3 for granular soils)
e = void ratio
D10 = grain size diameter with 10 percent finer (cm)

I'd like to know, just who devised this equation? When was it published? What data set was it based on? I've had occasion to use this equation over the years and would like to do some further research. Any help out there?

f-d


¡papá gordo ain’t no madre flaca!

 

[dgillette]

My old fluids book (Li and Lam 1964) shows the pressure differential across the water surface in a cylindrical capillary tube as

4 sigma * cos(alpha) / D

where sigma is the surface tension (0.005 lb/ft), alpha is the contact angle between the meniscus and the tube wall, and D is the tube diameter. [The cylindrical tube is a special case; the more general form covers a non-cylindrical tube or between parallel plates.]

The dimensions seem to check out OK between empirical and theoretical eqns. Looks like D10 is analogous to D*e, as a measure of equivalent "tube" diameter, and c is empirically analogous to [4 sigma cos(alpha)]/gamma-w. (The rise would be equal to pressure differential/gamma-w.)

There is no reference given for that equation, but another book relates its derivation to LaPlace! Maybe Li and Lam considered it unnecessary because, to them, it seemed like citing a reference for f=ma. Perhaps T&P themselves began with the theoretical equation for a tube, and adjusted it to fit soil mechanics concepts (like D10) and an empirical factor to cover all of the aspects that aren't so easily included analytically, like the coarser fraction of the grain-size distribution, particle shape, etc.

So, no, I don't actually know the answer to your question.

 

[fattdad]

I like your approach, however. . .

f-d

¡papá gordo ain’t no madre flaca!

 

[BigH]

fattdad - T&P do not give any references as you note. I checked Taylor - he talks about it but with tube equation - but again no direct references except for trying to figure out which "overall" reference from the back of book applies - many in German or Austrian . . . I have a book on soil physics and am attaching a few pages from it. No direct acknowledgment to T&P - but you might find something in it - they give a graph of rise vs cap size. I'll keep looking . . .

 

[fattdad]

Thanks BigH - I'll do my homework and report back.

f-d

¡papá gordo ain’t no madre flaca!

 

[vulcanhammer]

FWIW, I had to research this topic a little when writing the revision for Soils in Construction.

The bottom line of my research can be found towards the end of this:

http://www.vulcanhammer.net/utc/ence361/s2005/361-sl8.pdf

The derivation is based, in part, on the original T&P treatment of the subject.

http://www.pz27.net

 

[fattdad]

Vulcanhammer:

"Soils in Construction" was my introduction to geotechnical engineering. I took an introduction class when I was studying geology and that was the textbook (back in 1976 or so). I consulted that book prior to my post, just to see what it referenced - small world.

If I take your example with a D10 at the No. 200 seive and use "C" values of 0.1 cm2 and 0.5 cm2 (well and a void ratio of 0.3), I'd get an hc of 0.5 to 2.3 meters. So, the equation you cite must have some tacit assumption of a void ratio of 0.3 and a c-value of 0.5 cm2.

I'd like to know just where this "c" value came from and to what extent the equation was derived using void ratio.

I just have a desire to know how this equation was thought up!

Thanks for the reply!

f-d

¡papá gordo ain’t no madre flaca!

 

[vulcanhammer]

Looks like one of those things where everyone just copied off of everyone else.

Holtz & Kovacs (p. 174) notes that "In soils, it is common to assume the effective pore diameter is about 20% of the effective grain size (D10)." That's where the D10/5 comes from, and their formula is the same as T&P and Soils in Construction.

H & K also note that "Using this assumption (the one above) we can estimate a theoretical height of capillary rise and the corresponding capillary pressure for a fine-grained soil. This assumption points up the importance of pore size, not grain size, as the controlling factor in capillarity. Research at Purdue University (Garcia-Bengochea et.al., 1979) has shown that, depending upon the type of compaction and the molding water content, you can get very different pore size distributions in the same soil. The geologic history, soil struture and fabric of natural soils also varies tremendously, and it is possible to have very different pore size distributions in soils with the same D10."

http://www.pz27.net

 

[BigH]

Don - I think that this is what the Indian soil physics book was saying in their graph of rise vs capillary size - I'd really be interested in seeing this to some real conclusion . . .

 

[fattdad]

So, I sent Ning Lu (Colorado State University) an email and he recommended:

“Unsaturated Soil Mechanics” of Lu and Likos published by John Wiley and Sons (

http://inside.mines.edu/~ninglu/Lu sub pages/Lu book.html),

where there are two sections on capillary rise; Section 4.1 Height of Capillary Rise and Section 4.3 Rate of Capillary Rise (p133-160). He says there is theory and experimental validations of various models in this book.

I just have to go to the interlibrary loan to review it. . .

For those that enjoy calculus and complicated math, here is something else to review:

http://inside.mines.edu/~ninglu/Lu publications/Lu paper 33(1)-2004.pdf

Enjoy!

f-d

¡papá gordo ain’t no madre flaca!