Undrained Cohesion

[jobarr]

Hello,
I am new to this forum, and was hoping to get some help with a soil strength varible that is consfuing me.

I am conducting a hydraulic fracture analysis for a HDD project. To do this, I am using the program MDrill (from the Netherlands).

In their user manual, they refer to "Undrained Cohesion (Cu)" and define it with the following equation:

Cu = c*cos(phi)+p*sin(phi)
where p = (vert effective stress + horiz effective stress)/2

I am having difficulty understanding the logic used in this equation. It resembles Mohr-Coulomb failure criteria, but has a few differences that confuse me. I do not understand the "P" variable, and I do not understand why the cohesion would be multiplied by cos(phi) in the main equation.

I have not been able to find any other references with a similar equation. Any help would be very much appreciated!

Thanks.

 

[mudman54]

it looks like it is "normalized" or something related to the stress ratio; do you have the users manual you can share?

 

[jobarr]

I have uploaded the portion of the manual that presents the Cu value. The equation for Cu is located on the fourth page of the scanned document.

 

[molerat2210]

The P variable is the average of horizontal and vertical effictive stress at the depth. That seems to be a simple calculation. However, I was not aware of any substantial differences between the horizontal and vertical effective stresses. Equation 63 is more interesting to me since it defines cohesion and undrained cohesion. It looks to me that the undrained cohesion can be determined from the unconfined compressive streghth. I guess the manual is saying you could blow out your well if the fluid pressure exceeds that value.

 

[jobarr]

Molerat, thanks for looking at this.

Equation 63 has been confusing me for years. I finally decided to try this forum to see if anyone could provide an explaination.

Equation 63 closely resembles Mohr-Coulomb shear strength, but it has a few differences that really confuse me. I have not been able to find the logic behind the equation. I contacted the program developer in an effort to get them to explain this equation, but they said that they could not answer my question because my company does not have a maintenance contract with them.

Since the program was developed in Europe, I was wondering if they are using the term "undrained cohesion" in place of "shear strength"? Either way, I still don't understand what they are trying to get at.

 

[motorbiketocrank]

This may help

http://www.geoeng.ca/directory/ian pub/2006/Kennedy Skinner Moore 2006 TRR No 1976 pp 172-180.pdf

(see equation 2 which is very similar). Really too tired to sort it all out myself right now but getting the impression it has to do with the fact that friction angle/ratio of vertical to horizontal stress impacts whether the fracture will occur in the top of the hole vs. the sides of the hole (which obviously carry different risks for blowout), etc.

 

[jobarr]

Thanks for the reference. It looks a bit daunting, but I will try to make some sense of it and correlate it to equation 63.

 

[mudman54]

thanks for the attachments jobarr and geobdg; this can be a very useful discussion of cohesion and how to apply laboratory test results in engineering practice. my first step would be to use the example equation for figure 16-1 and see if the proposed equation 63 gives the same answer for Cu. it is too bad that delft software does not derive their equation, but it may be related to at rest pressure, ko = 1-sin(phi)? we know that the undrained shear strength aka cohesion (while defined as 1/2 of an unconfined compressive strength test) is actually dependent on the stress conditions so a correction would be logical. later equations introduce a new cohesion, c, which is unclear but may be related to the cohesion of the drilling mud. calculations for potential frac-out or blow-out are an important topic for our profession and we can't just use a computer program without validation.

 

[mudman54]

the term "cohesion" must be a translation error of the so-called cavity expansion theory.
the Cu term you state in your original question is the numerator of the parameter Q (see geobdg's link).
so, the good news is there's only one cohesion, but it varies up to a maximum value of the lab test i mentioned above.

 

[jobarr]

mudman, thanks for looking into this. It sounds like I need to brush up on my understanding of cavity expansion theory.

One thing that I keep getting hung up on is that in the MDrill manual, right above equation 63, it says "In case no data about the undrained strength of the soil is available, an estimated Cu value can be obtained using the subsequent formula" (see eqn 63).

Ok. So, wouldn't the undrained shear strength of the soil be determined using Mohr-Coulomb failure theory? If you have the information to solve equation 63 (cohesion, phi, unit weight), don't you have the information needed to determine the shear strength of the soil?

 

[dgillette]

I think that equation [Cu = c*cos(phi)+p*sin(phi)] is going to give you drained strength if you apply it with phi' and p' at failure.

Draw a generic Mohr circle (at failure) with its center at p' and MC envelope tangent to it. The failure occurs where the circle is tangent to the envelope (where the failure plane is 45-phi'/2 from the direction of sigma'1, which is oriented 90+phi' away from the horizontal axis on the plot). You can then calculate tau on the failure plane by trig, and I think you will find it matches the above.

Now, to get undrained strength from that seems trickier. Seems like they assume that sigma'1 at failure = sigma'v prior to drilling, and sigma'3 at failure = sigma'h before drilling, which seems like a crude assumption if you don't know whether the material is dilative or contractive.