Effects of Pore-Geometry and Distribution on Rock Strength and Stiffness

1. CE630A
Rock Mechanics
Term Paper Presentation
Pore-Geometry and Distribution on Rock Strength and Stiffness

2. Porosity Models for Determining pore size-
distribution of rocks
• Three new porosity models
1. Fractal model expressing the porosity for pore sizes (>r),
 > 𝑟 = 1 −
𝑟
𝐿
(2−𝐷)
……(1)
Where, L is the total size of the range, D is the fractal dimension(area/vol.) of the
grains
2. If (>rmax ) is not equal to zero, then the pore size region must be
(r-rmax )
 > 𝑟 =
𝑟 𝑚𝑎𝑥
𝐿
3−𝐷
−
𝑟
𝐿
3−𝐷
…..(2)
Where, D is the fractal dimension(area/vol.) of the grains

3. Continued…
3. Modifying previous model, this new model was being proposed
which relates pore-size with porosity,
Then porosity of pore of pore-size (>r),
 > 𝑟 = 1 −
𝑟
𝑟 𝑚𝑎𝑥
3−𝐷
……(3)
where, rmax is the max. pore size
This equation gives the relationship between accumulative porosity
and pore-size, whereas previous relationships gives correlation
between total porosity and grain size.

4. Continued…
Some Previous models for finding relationship between porosity and
other macroscopic parameters of rock and soil
 =
𝐿
𝐿2
3−𝐷
Where, L is the measurement size and L2 length of the rock or soil
under consideration.
For soils, derived by Rieu and Sposito,
 = 1 −
𝑑 𝑚𝑖𝑛
𝑑 𝑚𝑎𝑥
3−𝐷
where, d is aggregate size(max. and min)

5. Experimental Testifying of the Models in Three Dimensional Space
• The pore size distribution of three samples of total volumes (59.2,64.8,60.9)*106
m3 were plotted by comparing the results with experimental data.
(Eq. (6b=1), (8b=2), (11b=3),(Dark point = measured data))(porosity prediction from new models)
(Zhang JiRu , 2010)
Figure shows that predictions from eq. 3 are closest to experimental results and errors corresponding to this
are very small. Predictions from eq. 1 gives max. error and eq. 2 gives lower than 2.

6. Influence of pore geometry and orientation on the
strength of porous rock
• Geometry, orientation and aspect ratio of the pores affect the
strength and stiffness of the porous rock.
• Numerical simulations done by Luke Griffith and others using Rock
Failure Analysis Code on a uniaxially deform 400*200 pixels
rectangular bitmap(1 pixel = ~.1mm) containing elliptical pores show,
1. Fixing aspect ration (0.5), the uniaxial compressive strength and
young’s modulus of porous rock can be reduced by a factor(~2.4 to
~1.3) respectively, as the angle b/w major axis of elliptical pore and
applied stress is rotated from 0 to 90().

7. 2. Another observation (aspect ratio (= minor axis/major axis) =0.5)
Load Applied
 =90,  =0
study shows,
Can sustain minimum stress Can sustain maximum stress
In the above orientation, all other orientations show predictions
between these above results

8. Pore Angle() as a control on mechanical behaviour
• 24 simulations were performed in which we varied (0-90), but kept
porosity(0.1) and pore aspect ratio (.5) const.
• 2 simulations for each sample configuration
• UCS and Y. Modulus decrease from ~225 to ~100MPa and from ~67 to ~50 GPa as (0-90)
(Griffiths, 2017)

9. Pore aspect ratio as control on rock strength
• Varied aspect ratio (0.2-1.0) and pore angle (0-90), but kept the porosity (0.1)
const.
• Plots show that for low angles ( = 0-10) an increase in aspect ration results in reductions to
strength and young’s modulus
(Griffiths, 2017)

10. Continued…
• For higher angles ( = 40-90), strength and young’s modulus
increase with increasing aspect ratio
• For an intermediate angle ( = 20-30), strength and young’s
modulus first increases and then decrease as pore aspect ratio is
increased.

11. Relationship b/w Dry bulk modulus & Porosity
• Two methods to model dry bulk modulus at different porosities at a given
pressure:
1. Pore space stiffness method
2. Critical porosity model
• Pore Space Stiffness Method:
 Basic idea behind this method is that, for a given pressure, 𝐾∅ should stay
constant over a range of porosities, allowing us to compute 𝐾 𝑑𝑟𝑦 at different
porosities using equation:
𝐾 𝑑𝑟𝑦
𝐾 𝑚
=
1
1+
∅
𝐾
, where, 𝐾 =
𝐾∅
𝐾 𝑚

12. Porosity Stiffness Model
A family of dry rock over matrix bulk modulus ratio curves for varying values of K

13. Critical porosity Method:
• where, ∅ 𝑐 is the critical porosity, which is defined as the porosity above which
material starts behaving like suspension.
Observation:
1. At ∅ = 0, 𝐾 𝑑𝑟𝑦 is equal to 𝐾 𝑚 , which is as expected.
2. As ∅ increases, 𝐾 𝑑𝑟𝑦 decreases, which is also as expected.
𝐾 𝑑𝑟𝑦
𝐾 𝑚
= 1 −
∅
∅ 𝑐

14. Analytical Study:
• Now, analytical study was done to check how well they fit a dataset that was
measured by De-Hua Han (1986).
• They used
i. data corresponding to 10 clean sandstones (having different porosities) at
40MPa
ii. data corresponding to single sandstone at varying pressure (from 5 to 40MPa)

15. Continued…
• now to fit both the models to these points, note that both equation can be fit by
simple linear function given by:
𝒚 = 𝒎𝒙
Where, 𝑦 =
𝐾 𝑚
𝐾 𝑑𝑟𝑦
− 1 , 𝑚 =
𝐾 𝑚
𝐾∅
, and 𝑥 = ∅ for Pore space stiffness model
And, 𝑦 =
𝐾 𝑑𝑟𝑦
𝐾 𝑚
− 1 , 𝑚 = −
1
∅ 𝑐
, and 𝑥 = ∅ for critical porosity model
• Using linear regression, we can find the value of m corresponding to both the
equation.
• It comes out that for the pore space stiffness method, the best fit value was
𝐾∅
𝐾 𝑚
=
0.162, and for critical porosity method ∅𝑐 = 0.343.

16. • Apply these best fit values to the points to get the approximate solution 𝑦′, then
we can find the root mean square error (RMSE) from
𝑅𝑀𝑆𝐸 = 𝑖=1
𝑁
(𝑦𝑖 − 𝑦𝑖
′
)2
𝑁
• For the two methods, they found that the RMSE for the pore space stiffness
method was 0.039 and for the critical porosity method was 0.058.
• Thus, pore space stiffness method is a better fit to these points.

17. Pressure vs 𝐾∅ Relationship
• Next, they performed a least square fit to the full set of ten clean samples at each
pressure : 5, 10, 20, 30, 40, and 50 MPa.
P (MPa) ∅ 𝒄 RMSE Kφ /Km RMSE
5 0.289 0.126 0.104 0.094
10 0.311 0.107 0.129 0.076
20 0.329 0.079 0.147 0.055
30 0.338 0.069 0.156 0.044
40 0.343 0.058 0.162 0.039
50 0.348 0.053 0.166 0.038

18. • using the data for the pore space stiffness method, they derived the relationship
between Kφ /Km and pressure (P).
𝐾∅
𝐾 𝑚
= 0.064 + 0.027ln(𝑃)
Using calculus, we can write,
∆𝐾∅ = 0.027𝐾 𝑚
∆𝑃
𝑃

19. Effect of Porosity and crack density on Compressive
Strength
Compressive strength :an important parameter for the analysis
of mechanical response
Early attempt to develop theory for rock failue
• Griffith theory
• Mohr Coloumb theory and failure criteria and so on…….
Each theory consider some parameter to explain observed behaviour.
Here we consider two such models taking porosity and crack density as
two important micromechanical parameter in determining compressive
strength

20. Pore-emanated crack model
• The representative element volume is idealized as a pore
or crack embedded in elastic continuum as shown in fig.1
• Localized stress concentration at pore which propagate
parallel to the maximum compression direction.
• Leads to dilatancy and further propagation leads to
uncontrolled crack propagation and hence fracture is
observed at macro level.
• Analytical approximation equation for unconfined
compressive strength
Fig 1
UCS predicted to decrease with increasing porosity and inverse square root
of pore radius
This model predicts compressive strength s controlled by two microstructural attributes: porosity and pore
radius

21. Discrete element method
• When the bonding at grain contact ruptured and neighbouring grains
move ,this micromechanical model takes into account for this motion.
• The contact point of grain has some tensile and shear strength and if
either thresholds is reached leads to slip at contact points
• Useful to model the micromechanics of failure of both compact and
porous rocks.
• This model can simulate the progressive development of strain
hardening /softening and shear localization as observed in laboratory.
• Better agreement with laboratory data if bond strength in rock is
lower.

22. Sliding wing crack model
• Considers tensile stress concentration at the tip of pre-
existing crack
• Key microstructural attribute is crack density
• Tensile stress concentration at the tip propagate wing crack
in σ1 direction
• These crack coalesce to form a macroscopic shear band .
• The principal stress at the onset of dilatancy is given by
As the density of wing cracks increases , they coalesce to form a macroscopic shear
band in the strain.

23. Test method for pore structure characterization:
• Scanning electron microscope(SEM)
• Nuclear magnetic resonance
• Mercury intrusion porosimetry
• Gas adsorption method
• Micro-CT analysis

24. Scanning electron microscope
• It produces image of a sample by scanning it with a focussed beam of
electrons.It produces high resolution images of mineral and
connectivity of pore throat system.
• However it only gives qualitative pore images and we cannot obtain
quantative data of pore size distribution
• It destroys internal structure and external morphology of the pores.
Nuclear magnetic Resonance:
• It produces response signal of fluid hydrogen nuclei in the rock pore
that being measured using NMR scan.
• Applies to sedimentary rocks such as sandstone and carbonate.
• This method only measures open pores in rock.

25. Mercury intrusion porosimetry
• Mercury a non-wetting liquid , will enter pore space when pressure
exceeds capillary pressure.
• The pore size invaded by mercury is related to the applied pressure by
Washburn Equation:
It is the most commonly used method for pore structure
characterization. Fast determination speed and measuring range of
pore diameter is relatively large.

26. Comparison of pore structure characterize methods

27. References
• Luke Griffiths, Michael J. Heap, Tao Xu, Chong-feng Chen, Patrick Baud; The influence of pore
geometry and orientation on the strength and stiffness of porous rock. Journal of Structural
Geology 96 (2017) 149e160.
• ZHANG JiRu, TAO GaoLiang, HUANG Li & YUAN Lun; Porosity models for determining the pore-size
distribution of rock and soils and their applications. Chinese Science bulletin, December 2010
Vol.55 No.34: 3960–3970, doi: 10.1007/s11434-010-4111-6.
• Patrick Baud, Teng-fong Wong, WeiZhu; Effects of porosity and crack density on the compressive
strength of rocks. International JournalofRockMechanics&MiningSciences67(2014)202–211.
• Zechen Yan, Canshou Chen, Pengxian Fan, Mingyang Wang, Xiang Fang; Pore Structure
Characterization of Ten Typical Rocks in China. EGJE, Vol. 20 [2015], Bund. 2 .
• Brian H. Russell, Tad Smit; The relationship between dry rock bulk modulus and porosity – An
empirical study. CREWES Research Report — Volume 19 (2007)

28. THANK YOU