ABSTRACT The contribution of the pore fluid pressure in reducing the effective stress during loading of fully saturated high porosity chalk (&gt;40%) has often been assumed to be represented by an effective stress coefficient close to unity. This assumption entails the differential stress; the difference between the total stress and the pore fluid pressure, to equal the stress the rock matrix is exposed to. Laboratory experiments were conducted by simultaneously increasing the total stress and pore pressure. These tests resulted in substantial strains that should not occur if the assumption of an effective stress coefficient close to unity was true. Different explanations for these strains have been discussed, among these consolidation effects, partial saturation effects, micro damage and possible laboratory equipment effects. The strains that were observed during the above mentioned test phase, resulted in a focus on the effective stress coefficient for porous chalk material. The results presented in this study suggest that the effective stress coefficient for high porosity outcrop chalks depends on the applied stress and the pore fluid, and is not a constant nor close to unity as commonly presumed

Bp Norway

Conocophillips Bartlesville

E Omdal

G G Ramos

Hiorth A Iris Madland

Naess K E

R I Korsnes

T G Kristiansen

CiteSeerX

SCA2008-45 1/6
 
EXPERIMENTAL INVESTIGATION OF THE EFFECTIVE 
STRESS COEFFICIENT FOR VARIOUS HIGH POROSITY 
OUTCROP CHALKS 
 
Omdal, E.*, Breivik, H.*, Næss, K.E.*, Ramos, G.G.ConocoPhillips Bartlesville,  
Kristiansen, T.G.BP Norway, Korsnes, R.I.*, Hiorth, A.IRIS and Madland, M.V.* 
*University of Stavanger, 4036 Stavanger, Norway 
edvard.omdal@uis.no 
 
This paper was prepared for presentation at the International Symposium of the 
Society of Core Analysts held in Abu Dhabi, UAE 29 October-2 November, 2008 
 
ABSTRACT  
The contribution of the pore fluid pressure in reducing the effective stress during loading of 
fully saturated high porosity chalk (>40%) has often been assumed to be represented by an 
effective stress coefficient close to unity. This assumption entails the differential stress; the 
difference between the total stress and the pore fluid pressure, to equal the stress the rock 
matrix is exposed to. Laboratory experiments were conducted by simultaneously increasing 
the total stress and pore pressure. These tests resulted in substantial strains that should not 
occur if the assumption of an effective stress coefficient close to unity was true. Different 
explanations for these strains have been discussed, among these consolidation effects, partial 
saturation effects, micro damage and possible laboratory equipment effects. The strains that 
were observed during the above mentioned test phase, resulted in a focus on the effective 
stress coefficient for porous chalk material. The results presented in this study suggest that the 
effective stress coefficient for high porosity outcrop chalks depends on the applied stress and 
the pore fluid, and is not a constant nor close to unity as commonly presumed.  
KEYWORDS. Effective stress coefficient, Compressibility, Outcrop chalk 
 
THEORY 
In solid homogeneous material, the applied stresses will theoretically be evenly distributed 
throughout the substance. Introduction of porosity results in a more complex stress distribution 
at microscopic levels. External load is transmitted at the intergranular contacts. The pore space 
is subjected to a hydrostatic fluid pressure and hence balances the external load. Terzagi5 
proposed the relation in equation 1, called effective stress, σ', as the difference between the 
external load, σtot, and the pore fluid pressure, Pp, however, with a correction factor for the 
pore pressure, the effective stress coefficient, α. This factor obtained theoretical the value of 
the porosity, however experimentally found equal unity. Biot1 listed certain limiting 
assumptions for this theory, among others elasticity. This relation was proposed for soil, yet 
the common interpretation of the effective stress coefficient α present today is given by the 
modulus of the bulk and solid, as described by equation 2.2 
SCA2008-45 2/6
 
p
tot
ptot P
P
'
'
σσ
αασσ
−
=→⋅−=    (1) 
 
s
b
K
K
−= 1α       (2) 
 
Ks and Kb is the hydrostatic strength modulus of solid and bulk respectively. Deformation of 
materials depends primarily on changes in effective stress. If one assumes no time or chemical 
effects, one obtains constant bulk volume if the effective stress is kept constant. 
 
EXPERIMENTAL PROCEDURES AND RESULTS 
Experimental Set-up and Background 
Chalk is a sedimentary rock within the carbonate family. The mineralogy characterization 
classify chalk as a limestone, as it typically contains >90% calcite. High porosity chalk is 
usually related to outcrop, yet deeply buried high porosity formations are found as 
hydrocarbon reservoirs. Under field development and production these formations have during 
primary production and repressurization by water flooding, experienced shifts in its stress 
state. The governing factors for the stress state of in-situ rock matrix are the principal stresses 
in horizontal and vertical direction. However, the total stresses need to be corrected for the 
contribution of any fluid pressure within the pore space, according to the effective stress 
relation described in the theory section. Chalk as material is commonly interpreted as a linear 
elastic weak rock. Further more, high porosity chalk is considered to obtain values of the 
effective stress coefficient, close to unity. Laboratory testing at in-situ stress conditions 
requires both external load and pore pressures and standard laboratory procedure is to 
simultaneously increase the external stresses and the pore pressure. Independent increase of 
external stress and pore pressure could cause the sample to fail, since the external stresses that 
are aimed for are well beyond the elastic limit, as chalk is known as a weak material. With a 
minor difference between the pore pressure and the external stress, this procedure should 
maintain any bonding between the grains in the rock matrix. Several experiments were 
conducted according to this procedure, as presented in Figure 1. The stippled curve in this plot 
is the external stress versus the pore pressure; the total hydrostatic stress is increased to 27 
MPa, while the pore pressure increased to 25 MPa, with a constant differential stress at 2 MPa. 
The external stress has then been increased from 27 to 35 MPa in order to cause the specimen 
to yield, as the typical hydrostatic yield strength of water saturated Stevns Klint chalk is 
around 7 MPa3. The bulk modulus Kb is calculated to 0.48 GPa in Figure 1. If this value is 
used in equation 2, knowing that the solid modulus Ks for calcite is 71±5 GPa, one obtain α 
equal 0.99, i.e., close to 1. This value does not concur with the strains obtained in the build up 
phase. As seen from the total stress versus axial strain curve in Figure 1, this core sample 
experienced around 7‰ of axial deformation during the build-up phase. The curve of the 
SCA2008-45 3/6
 
actual loading, where the differential stress is increased, shows a shift in the deformation 
slope, however, the deformation prior to this last phase is significantly. The interpretation of 
these deformations leads to a further investigation of the effective stress relation. 
 
0
5
10
15
20
25
30
35
40
0 5 10 15 20 25 30
Pore Pressure [MPa] / Axial Strain [‰]
To
ta
l H
yd
ro
st
at
ic
 S
tr
es
s 
[M
Pa
]
∆σ = ∆Pp
Pp = const 
Total Stress vs Pore Pressure
Total Stress vs Axial Strain
7
Kb = 0.48 GPa3Kb
Figure 1. Simultaneously increase of total stress and pore pressure, Stevns Klint core, porosity - 0.489 
 
Investigation of the Effective Stress Coefficient (α) 
The effective stress coefficient has been measured for samples of high porosity outcrop chalk. 
If one increases the fluid pressure in the pores of a specimen, and simultaneously regulates the 
hydrostatic total stresses and hence keeping the volumetric strain constant, (∆εv = 0), there is 
no actual loading taking place, i.e.; the effective stresses are constant2. Internal strain gauges 
have been installed on all cores to be able to control the strain within ±0.1‰. The experiments 
conducted in this study have all been performed by the use of tri-axial loading cells, and all 
samples has been fully saturated under vacuum with its respective pore fluid prior to testing. 
The majority are performed at the rock mechanics facilities at the University in Stavanger, 
(UiS), however some studies are performed at the Rock & Fracture Mechanics Laboratory at 
the ConocoPhillips Research Centre in Bartlesville, (CoP-BTC). As seen in Figure 2, an initial 
regulating margin (differential stress) is established to prevent pressure communication 
between pore and confining space, i.e. if the pore pressure exceeds the confining pressure, 
leakage takes place. The constant bulk volume portion of the test then starts at 80 min. The 
pore pressure is increased at constant load rate, and the total stress is regulated to prevent the 
core from deforming. The effective stress coefficient is then calculated according to equation 
1. At 160 minutes the pore pressure catches up with the confining pressure, and the test is 
finished due to no more regulating margin left. Different test parameters are varied in the 
experiments, such as stress state, core material, pore fluid and loading rate.  
SCA2008-45 4/6
 
0
3
6
9
12
15
18
21
0 20 40 60 80 100 120 140 160 180
Test Progress [min]
Pr
es
su
re
/S
tr
es
s 
[M
Pa
]
-0,2
0
0,2
0,4
0,6
0,8
1
1,2
Ef
fe
ct
iv
 S
tr
es
s 
C
oe
ffi
ci
en
t/V
ol
um
et
ric
 S
tr
ai
n 
[‰
]
Effective Stress Coefficient
Volumetric Strain
Total Stress
Pore Pressure
Regulating 
margin
 
Figure 2. Constant bulk volume test, core 1, pre yield, distilled water as pore fluid 
 
Table 1. Summary of the observations from the constant bulk volume tests 
Core 
no 
Porosity 
[%] Outcrop Lab 
Load rate 
[MPa/min] 
Pore 
Fluid 
Stress 
Regime 
Coefficient 
value α 
Coefficient 
trend 
Elastic 0.60-0.80 Increasing 1 44.29 0.16  Plastic 0.75-0.65 Decreasing 
Elastic 0.60-0.75 Increasing 2 43.23 0.13 Plastic 0.80-0.40 Decreasing 
Elastic 0.55-0.75 Increasing 3 44.29 0.15 Plastic 0.70-0.60 Decreasing 
Elastic 0.70-0.80 Increasing 4 44.40 0.16 
Plastic 0.80-0.68 Decreasing 
5 44.57 0.15 
Distilled 
water 
0.65-0.80 Increasing 
6 44.50 
UiS 
0.15 Ethylene glycol 0.55-0.67 Increasing 
7 47.56 
Stevns 
Klint 
Denmark 
0.03 0.70-0.89 Increasing 
8 36.80 Kansas US 
CoP-
BTC 0.03 
Kerosene
Elastic 
0.89-0.93 Increasing 
 
When measuring the effective stress coefficient, α, a transient effect is observed. There will 
always be some initial consolidation and a small delay before the core starts deforming and the 
mathematical calculation make α start at unity, as the effective stress is assumed equal the 
differential stress. This transient effect is observed as α decline from 1 to 0.6 in Figure 2.  
SCA2008-45 5/6
 
The trend of α is then stabile, yet converging throughout testing. The term elastic and plastic 
phase in Table 1 is used based on knowledge of material strength; test labelled elastic was 
tested with an initial effective stress equal 5 MPa as seen in Figure 2, yet test labelled plastic 
was tested with an initial effective stress equal 15 MPa. The converging trend of the 
coefficient observed show a very concurrent pattern, whereas all values of α in the elastic 
phase show an increasing trend, while the trend is decreasing in the plastic phase, according to 
Table 1. From this it is evident that cohesion influences the effective stress relation. Increasing 
trend in the elastic phase is interpreted as breaking of bonds between the grains which leads to 
increased influence of the pore pressure; the bulk is converging towards failure. In the plastic 
phase, the core is moving towards a more hardened state due to more compact packing of the 
grains. Pore fluid is known to influence the mechanical parameters for chalk3, as is also seen 
for core 5 and 6 in Table 1, which show different values of α, yet very similar in porosity and 
loading rate. The more viscous and inert ethylene glycol results in a lower coefficient for core 
6. 
 
DISCUSSION 
Laboratory Effects 
Tests performed at different laboratories needs in-depth interpretation of laboratory effects. 
For a constant bulk volume test, only a limited number of sources for error exist. Equation 1 
identifies two parameters, total stress and pore pressure, given that the bulk volume is 
constant. The critical source is the measured deformation, hence the bulk volume. UiS-
laboratory used a thicker rubber sleeve surrounding the sample than the stiffer teflon shrinking 
sleeve used at CoP-BTC. Any sleeve will compact during exposing to stresses and the strain is 
measured outside the sleeve. When forcing the sleeved core diameter constant, the diameter of 
the core is exposed to a considerable error. However, core 7 and 8 of different material show a 
significant difference according to Table 1. If the measured α for core 8 is exclusively 
laboratory effects it is a measure of the error in a worst case scenario, i.e., assumed true α is 1. 
If this worst case error is then added to the value of α measured for core 7, the corrected value 
for core 7 is then found to vary from 0.81-0.96. 
 
Impact of Inelasticity 
The material used in this study is limited to very high porosity outcrop chalk, (36-48%). 
Conventional assumptions state that the lower porosity and higher strength, the lower α the 
material obtain. A previous study on chalk6 concluded that α was influenced mainly by 
porosity, and decreasing with decreasing porosity. This study investigated reservoir chalk from 
15-36% porosity. However, the obtained values of α in table 1 contradict these results. The α 
obtained for the lower porosity Kansas core 8 show the highest value. The high porosity 
Stevns Klint material which shows a lower α than expected might be too influenced by in-
elasticity. Further, the study6 also concluded that nonlinearity invalidates most theoretical and 
mechanistic models, which is also observed in this study. The effective stress theory is based 
upon the assumption of elasticity and linearity of stress-strain relations.  
SCA2008-45 6/6
 
Years of experience on chalk reveals that these relations do only to some extent apply for 
chalk4. The procedure when simultaneously increasing the pore pressure and the external 
stress is on infinitesimal time scale really many loading and unloading cycles, which may 
cause irreversible strains to chalk material. These strains have also been discussed as simply 
the cores responding to creep, hence the loading rate has been varied, but no significant effects 
where seen between for example core 4 and 7 in table 1. The majority of the cores have also 
been tested within 2 hours, which minimize any creep behaviour. However, more studies 
should be performed on the subject. 
 
CONCLUSION 
The significance of this work is important for the understanding of the effective stress 
relations of high porosity soft chalk, in particular for core analysis. Chalk testing at in-situ 
conditions requires implementation of high pore pressure. The experimental observations of 
the effective stress relations obtained in this work deviates form the theoretical explanation, 
yet the understanding of the relation is vital. Hence, future work is needed to further discuss 
the relation for high porosity chalk as a very soft non-linear material. This study clearly shows 
a stress and fluid dependence of the effective stress coefficient within the same batch of 
outcrop, as well as a difference between two different outcrops of chalks. 
 
ACKNOWLEDGEMENT  
The authors acknowledge ConocoPhillips and the Ekofisk co-venturers, including TOTAL, 
ENI, StatoilHydro and Petoro, and the Valhall licence, including BP and Shell, for financial 
support and for permission to publish this work. 
 
REFERENCES 
1. Biot, M.A., “General theory of three-dimensional consolidation,” J. Appl. Phys., 1941 
12, 155 
2. Charlez, Ph. A., Rock Mechanics, ISBN 2-7108-0585-5(Vol.1), Éditions Techip, Paris, 
1991 
3. Madland, M.V., Water weakening of chalk, Doctorial Thesis, ISBN 82-7644-257-9, 
2005 
4. Risnes, R and Nygaard, V., “Elasticity in High Porosity Outcrop Chalk,” 2. 
Euroconference on Rock Phys. and Mech. Heriot-Watt Univ., Edinburgh, Scotland, 
1999 
5. Terzaghi, K.Van, „Die Berechnung der Durchassigkeitsziffer des Tones aus dem 
Verlauf der hydrodynamischen Spannungserscheinungen,“ Sitzungsber. Akad. Wise. 
Wien Math Naturwiss. Kl. Abt., 1923, 2A, 132, 105 
6. Teufel, L.W. and Warpinski, N.R., 1990, “Laboratory Determination of the Effective 
Stress Laws for Deformation and Permeability of Chalk,” paper presented at the North 
Sea Chalk Symposium, Copenhagen, June 11-12 


EXPERIMENTAL INVESTIGATION OF THE EFFECTIVE STRESS COEFFICIENT FOR VARIOUS HIGH POROSITY OUTCROP CHALKS

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EXPERIMENTAL INVESTIGATION OF THE EFFECTIVE STRESS COEFFICIENT FOR VARIOUS HIGH POROSITY OUTCROP CHALKS

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