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Poromechanics V © ASCE 2013
2423
Poroelasticity of High Porosity Chalk under Depletion
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K.A. Andreassen1 and I.L. Fabricius1
1
Department of Civil Engineering, Technical University of Denmark, Brovej – Building 118, 2800 Kgs. Lyngby, Denmark; PH (+45) 45251781; email: kall@byg.dtu.dk
ABSTRACT
The theory of poroelasticity for the elastic region below pore collapse by means of
three different loading paths gives the possibility to compare the static and dynamically determined Biot coefficient for a set of experimental data with uniaxial loading
on outcrop chalk performed with different levels of pore pressure. The chalk is oilsaturated Lixhe chalk from a quarry near Liège, Belgium, with a general porosity of
45%. Additionally, we compare the theoretical lateral stress to the experimentally
determined lateral stress at the onset of pore collapse.
The static Biot coefficient based on mechanical test results is found to be lower than the pretest dynamic Biot coefficient determined from elastic wave propagation
for the loading path and with less deviation under depletion. The calculated lateral
stress is lower than the experimentally measured lateral stress depending on loading
path. An explanation to this behaviour is pore pressure build up.
INTRODUCTION
Knowledge of the Biot coefficient is important for prediction of basin or reservoir
subsidence especially for depletion under hydrocarbon production. As the depth for
newly discovered reservoirs continually increases so will the pressures involved increase. Consequently, there is a large effect by not applying the correct Biot coefficient due to these higher overburden and pore pressure. There exist several studies of
Biot’s coefficient (Engstrøm, 1992; Fabre and Gustkiewicz, 1997; Ojala and Fjær,
2007; Fabricius et al., 2010; Alam et al., 2012) but most are for hydrostatic conditions
as described by Biot (1941).
In this paper we present a combined poroelastic analysis of laboratory tests on
outcrop chalk from the Pasachalk project (Schroeder et al. 2000; 2003). A number of
tests are under reservoir conditions with elevated pore pressures and have not been
treated by use of poroelastic theory previously.
METHODS
We apply theory of poroelasticity stated with stresses and strains positive for compression (Biot, 1941; Zimmerman, 2000a):
 = 2 + � +  +  � +  ,
Poromechanics V
Poromechanics V © ASCE 2013
2424
 = 2 + � +  +  � +  ,
(1)
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 = 2 + � +  +  � +  ,
 = 2 ,
 = 2 ,
 = 2 ,
where εii denotes the principal strain, εij denotes the shear strain, σii denotes the principal stress, σij denotes the shear stress, G is the shear modulus, λ is Lamé’s parameter, Pp is the pore pressure and α is Biot’s coefficient. For hydrostatic stresses it is
well known that Biot’s coefficient is

 = 1 − dry ,
min
valid for isotropic, monomineralitic materials. On the other hand, uniaxial strain, ∆εxx
= ∆εyy = 0, is generally thought to resemble reservoir conditions, Addis (1997), and
therefore often relevant in petroleum engineering. Thus, following the derivation of
Alam et al. (2012), poroelasticity for uniaxial strain conditions results in the relationship:
(2)
 =  +  ,
with σa the applied axial stress, εa the resulting axial strain, M the compressional
modulus (often termed the oedometer or p-wave modulus) valid for ∆εxx = ∆εyy = 0,
and n the effective stress coefficient for uniaxial strain conditions. The term Mεa is
the effective stress experienced by the soil skeleton. Alam et al. (2012) show that n is
identical to α, which is in accordance with the derivation by Zimmerman (2000a). By
differentiation of eq. (2)

� �

=

1


,
� �


=
1−

,
with σd the difference in stress, σd = σa – Pp. This yields
 =1−

�


�


�


�
(3)
.
Differentiating eq. (2) with respect to σa gives
=−

�


�
�


�
 

(4)
.
Poromechanics V
Poromechanics V © ASCE 2013
2425
From eq. (1) with the conditions of uniaxial strain (∆εxx = ∆εyy = 0) the lateral stress is

� 
1−
 = �
1−2
� 
1−
+�
(5)
,
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as stated by Zimmerman (2000b) and Engelder and Fischer (1994). Finally, differentiating eq. (5) yields

� �


1−2
�
1−
=�
⇒
=�
1−

� � �
1−2
 

(6)
,
used by Addis (1997) and Goulty (2003) among others for reservoir depletion or passive basin response.
Three equations above determine individually the Biot coefficient under uniaxial strain conditions. In a laboratory setting these correspond to the following:
• Eq. (3) with first changing pore pressure with the difference of stress constant
followed by a phase with changing differential stress with constant pore pressure,
• Eq. (4) for changing pore pressure and total axial stress constant followed by a
phase with changing total axial stress with constant pore pressure,
• Eq. (6) for simultaneous change in lateral stress and pore pressure with the total axial stress held constant.
These phases are all relevant for the determination of the static Biot coefficient for
the mechanical test results reported in the Pasachalk project (Schroeder et al., 2000;
2003).
From elastic wave propagation we find the dynamic Biot coefficient by use of
the isoframe model (Fabricius et al., 2007; 2010) as the Pasachalk data set contains
dry and saturated p-wave measurements but not s-wave measurements and thus does
not facilitate direct determination of Biot’s coefficient. The isoframe model is based
on modified upper Hashin-Shtrikman bounds and the isoframe values show how large
part of the solid constitutes the load bearing skeleton,
 =
4
3
4
3
�min + min �fl (min −sat )+�min + min �sat (fl −min )
4
3
4
3
��sat + min �min (fl −min )+�min + min �fl (min −sat )�(1−)
,
(7)
with Kmin the mineral bulk modulus, Gmin the mineral shear modulus, Kfl the fluid
bulk modulus, Ksat the saturated bulk modulus, and φ the porosity. It can be visualised
as a modelled mixture of relatively large hollow spheres and smaller solid spheres.
For a homogenous Reuss model the bulk modulus of a suspension of particles is
 =
+(1−)(1−)
 (1−)
+
fl min
(8)
.
The Modified Upper Hashin–Shtrikman bound for the bulk modulus gives,
Poromechanics V
Poromechanics V © ASCE 2013

HS+
=�
2426

4
3
sus + min
+
(1−)(1−)
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together with the shear modulus


 HS+ = � +
(1−)(1−)

4
3
sus + min
+
+
(1−)
(1−) −1
min +
4
3
min + min
�
�
−1
− ,
=
Additionally, the p-wave modulus is defined as
4
3
(9)
− min ,
min 9min +8min
�  +2 �
6
min
min
.
(10)
4
3
HS+ =  HS+ +  HS+ ,
and by back-calculation of MHS+ to equal the measured p-wave modulus by change of
IF in Equations (7–10) the result is the Biot coefficient from the dry bulk modulus
4  min min (1−)
+4
min
min ) − 3  min (1−)
dry = (
RESULTS
,
dry
=1−
min
.
(11)
The Pasachalk project (Schroeder et al., 2000; 2003) covers aside from a high number
of tests on water- or oil-saturated chalk also a number of tests on oil-saturated chalk
under reservoir conditions. Figure 1 shows an example of the loading path for one of
these. The loading consists of a hydrostatic uploading including increasing pore pressure. This is followed by a deviatoric phase – actually termed triaxial here and in the
Pasachalk project – where the lateral stress is constant while increasing the axial
stress. Finally comes a depletion phase with constant total axial stress while decreasing the pore pressure under uniaxial strain conditions.
The yield point is the onset of acceleration of deformation or the end of the
linear elastic behaviour, which is after the start of the depletion phase. Therefore, we
apply the theory of poroelasticity to obtain Biot’s coefficient and the resulting lateral
stress. Using Eq. (3) for the loading phase and Eq. (4) for the depletion phase determines the Biot coefficient for the example in Figure 1. Using Eq. (5) gives the lateral
stress from the total axial stress with the pore pressure, Poisson’s ratio and the Biot
coefficient. We include the tests numbered 177, 182, 189, 182, and 190 for comparison of the poroelastic behaviour with and without raised pore pressure.
Pretesting measurements of the saturated and dry p-wave exist and Equations
(7–11) form the basis for calculation of the Biot coefficient. These calculations use a
fluid bulk modulus for soltrol of 1.292 GPa valid for 20°C and determined by use of
the procedure listed in Mavko et al. (2009) from the stated density of 0.783 g/cm2 in
the Pasachalk reports.
Poromechanics V
Poromechanics V © ASCE 2013
2427
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(a)
(b)
Figure 1. Sample of testing procedure. Test 198 with (a) time evolution of the
stress and strain and (b) effective stress or pore pressure versus axial strain. The
effective stress is based on the dynamic Biot coefficient of 0.887, see Table 1.
Biot’s coefficient – dynamical versus statical. The above procedures lead to the
results listed in the Table 1. Two tests had the onset of pore collapse before the actual
uniaxial strain loading phase started therefore the lateral stress calculation is not applied.
Table 1. Liège chalk test results with sound velocity measurements. Loading
paths are stated as HxTy: Hydrostatic to x MPa, triaxial (deviatoric) to y MPa.
All are then uniaxially loaded (∆εxx = ∆εyy = 0) to beyond yield.
Initial
Dyn. α,
Dyn. ν,
Load
Pp
before
before
Static α, Static α,
Test path
(MPa) Porosity testing
testing loading
depletion
177
H7
0
0.450
0.888
0.257
181
H7
0
0.453
0.895
0.271
189
H7
0
0.448
0.892
0.271
165
H7T12
38
Yield onset before uniaxial strain loading phase
166
H7T12
38
0.459
0.891
0.267
0.74
0.77
167
H7T12
38
0.454
0.905
0.271
0.78
0.87
172
H7T12
0
0.434
0.887
0.270
173
H7T12
0
0.448
0.882
0.255
170
H7T14.5
38
0.441
0.889
0.273
0.70
0.78
198
H7T17
33
0.436
0.887
0.267
0.74
0.83
141
H7T17
38
0.453
0.903
0.271
–
0.84
142
H7T17
38
0.442
0.890
0.270
–
0.88
174
H7T17
0
Yield onset before uniaxial strain loading phase
182
H12
0
0.438
0.884
0.268
190
H12
0
0.448
0.892
0.270
196
H12T17
33
0.445
0.890
0.263
0.75
0.88
197
H5T18
32
0.453
0.904
0.276
0.71
0.83
Poromechanics V
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Poromechanics V © ASCE 2013
2428
Lateral stress. The loading paths contain hydrostatic and deviatoric loading phases
followed by a uniaxial strain phase. Therefore computation of the part of the total
axial stress contribution to the lateral stress from the uniaxial strain phase is achieved
by subtraction of both of these levels of stress. Following this, the hydrostatic contribution is added to the total stress value while the deviatoric phase do not contain
strain restrictions and consequently do not lead to an increase in lateral stress. Table 2
lists the results. All tests have a loading rate of 4·10–3 MPa/s except tests 189 and
190, which are slower with a loading rate of 1·10–3 MPa/s.
Table 2. Liège chalk test results with lateral stress measurements. Loading paths
are stated as HxTy: Hydrostatic to x MPa, triaxial (deviatoric) to y MPa. All are
then uniaxially loaded (∆εxx = ∆εyy = 0) to beyond yield. Total lateral stress calculated using dynamic properties, α and ν, listed in Table 1.
At yield onset,
At yield onset,
Load
Initial Pp
measured stresses
σr calc. from σa
Test path
(MPa)
Porosity
[σa, σr, Pp] (MPa)
(MPa)
177
H7
0
0.450
15.4
7.5
0
9.9
181
H7
0
0.453
24.0
8.5
0
13.3
189
H7
0
0.448
19.2
7.2
0
11.6
165
H7T12
38
Yield onset before uniaxial strain loading phase
166
H7T12
38
0.459
49.8
42.5
26.8
37.0
167
H7T12
38
0.454
49.8
42.8
28.8
38.5
172
H7T12
0
0.434
20.3
6.5
0
10.1
173
H7T12
0
0.448
19.0
6.7
0
9.4
170
H7T14.5
38
0.441
47.5
39.1
28.2
33.6
198
H7T17
33
0.436
50.0
38.9
25.8
34.6
141
H7T17
38
0.453
50.0
43.7
31.5
38.2
142
H7T17
38
0.442
49.9
41.4
31.0
37.9
174
H7T17
0
Yield onset before uniaxial strain loading phase
182
H12
0
0.438
25.2
12.6
0
16.9
190
H12
0
0.448
18.1
12.2
0
14.3
196
H12T17
33
0.445
50.2
42.8
27.5
40.7
197
H5T18
32
0.453
50.0
35.9
24.8
32.0
DISCUSSION
The static and dynamic Biot coefficients are not expected to be equal due to the strain
amplitude is smaller for the elastic wave propagation than for the mechanical strain
resulting from the loading (Fjær et al., 2008). Others find that there is agreement
within the level of estimated error (Ojala and Fjær, 2007; Alam et al., 2012). The Biot
coefficient is also expected to decrease with applied stress level and as the dynamic
Biot coefficient is determined before the test it should give higher values compared to
Poromechanics V
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Poromechanics V © ASCE 2013
2429
what could be expected during testing (Ojala and Fjær, 2007; Alam et al., 2012) although hysteresis is anticipated, i.e. the loading path influences the Biot coefficient.
The results in Table 1 indicate that hysteresis is present for the Liège Lixhe chalk and
that the dynamic Biot coefficient is higher than the static Biot coefficient. This is
most pronounced for the loading phase and less for the depletion phase.
For the lateral stress calculation the results in Table 2 show higher calculated
lateral stress compared with the measured stress for tests with zero pore pressure
while lower calculated lateral stress for tests at higher pore pressures. To compare the
effect of applying static Poisson’s ratio and static Biot coefficient instead of the dynamic parameters we find the average for the difference between the measured and
calculated total lateral stress. The Poisson’s ratio used is a general value of 0.18 valid
for oil-saturated Liège Lixhe specimens (Schroeder et al., 2000). The average difference goes from 3.5 MPa to 1.4 MPa for the zero pore pressure tests. On the contrary,
for the elevated pore pressure tests by applying static parameters the average difference goes from 3.7 MPa to 6.9 MPa. Therefore, the static parameters seem to yield
the best calculation of the lateral stress based on the zero pore pressure tests while the
dynamic parameters are expressing the higher pore pressure tests well.
An important reason for the discrepancy for the higher pore pressure tests
could be the high rate by which the pore pressure is depleted. This raises concerns
that the pore pressure is not fully dissipated and the pressure build up contributes to a
higher measurement of the lateral stress. The lower onset of yield of tests 189 and
190 compared with the other tests verify this.
Another aspect is time dependent effects. The higher pore pressure tests are in
principle performed with a much slower loading rate for the first part of the test (hydrostatic loading to 2 MPa, with ramp in the pore pressure) and as such this could be
seen as a consolidation phase changing the material properties. Additionally, the
Poisson’s ratio is a mean value and therefore not specific for each test specimen in
Table 2. By obtaining shear wave velocities this could be further investigated.
CONCLUSION
The theory of poroelasticity applied for chalk at reservoir conditions gives consistent
results for the static and dynamic Biot coefficient. Hysteresis exists between loading
and depletion phases. The theoretical lateral stress is compared with the measured
stress giving small differences. Both of these effects could be attributed to the high
depletion rate raising concerns that the pore pressure was not left enough time to dissipate.
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Poromechanics V © ASCE 2013
2430
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