Horizontal Stress Calculations and Effective Medium Theory in the Bakken
Jesse Havens
Department of Geophysics
Abstract
Horizontal Stress:
The first subject I will discuss is the horizontal stress. The horizontal stress is also known as the
closure stress in engineering applications. If the pore pressure exceeds the minimum horizontal
stress and the tensile strength of the formation, then a tensile fracture will occur. Therefore,
the minimum horizontal stress is the key formation parameter when we design hydrofracture
stimulations.
The problem that we encounter in the Bakken is that the upper and lower shales are highly
anisotropic, and we do not have enough information from the well logs to calculate the true
horizontal stress. This causes people to make approximations that may or may not be accurate.
I have tested two approximations and deemed them to be insufficient in characterizing the
horizontal stress of the Bakken.
First, allow me to introduce the transversely isotropic (TI) stress coupling coefficient:
Ko =
C13
C 33
=
Eh  v zx
Ev  1  v xy




(1)
,where C13 and C33 are stiffness coefficients (Thomsen, 1986), Eh and Ev are the horizontal and
vertical Young’s Moduli, and vzx and vxy are Poisson’s ratios. The minimum horizontal stress is
equal to the effective vertical stress multiplied by Ko.
One might guess that Eh and Ev can be calculated using the associated horizontal and vertical
velocities, but this is not true. We do not have enough information in the wellbore to calculate
any of the values in the third term of Eq. (1). The value we can calculate from the wellbore is
C33 (C33=Vp2(0)*ρ). Shown in Figure 1a is the approximated Ko versus the actual Ko for the
dataset measured by Vernik and Nur, 1992.
The tendency with this approximation is to overestimate the stress coupling coefficient.
Another popular approximation is to set C12=C13. C12 can be calculated from the horizontal
velocities (fast p and s). In Figure 1b I am also showing the result of C12 versus C13 and clearly
C12 is consistently larger than C13 leading to an overestimation of Ko.
In both cases if we apply these techniques we will over-predict the horizontal stress. For this
reason, I have decided to take an alternative route to calculate Ko. In the Vernik and Nur
dataset there are strong correlations with the total organic carbon (TOC) plotted against the
stiffness coefficients. At the wellbore we have the ability to estimate the TOC content with the
Schmoker and Hester, 1983 density correlation:
TOC=154.4/ρ – 57.261. We can take C33 from the velocity recorded at the wellbore, then
calculate C13 from the TOC. Figure 2 shows a comparison of a given Ko and a TOC calculated Ko.
Since the Schmoker and Hester correlation is only valid in the shales, there tend to be spikes on
the edges. Future work will include a more subtle transition from the shales into the
surrounding formations.
(b)
1
25
0.8
20
C13 (Gpa)
Approximated Ko (unitless)
(a)
0.6
0.4
15
10
5
0.2
0
0
0
0.2
0.4
0.6
Ko (unitless)
0.8
1
0
5
10
15
20
25
C12 (Gpa)
Figure 1: (a) Approximated Ko versus the true Ko calculated with the Vernik and Nur, 1992
dataset. (b) C12 versus C13.
The most striking difference is in the upper shale. Many calculations tend to always predict an
increase in horizontal stress in the upper and lower shales, which is accentuated with increasing
anisotropy, but in this case the upper shale has a lower horizontal stress than the lower shale.
At this point the absolute magnitudes of the calculation are not trustworthy, but the trend of
less horizontal stress in the upper shale (6,695-7,000ft.) compared to the lower shale (6,7556,780ft.) is of great importance.
The key term that we are missing in our horizontal stress calculations is the energy dissipation,
which can only be acquired with an off-angle p-wave velocity measurement. Figure 2 is
suggesting that extreme amounts of TOC (~20 wt. %) will cause a large amount of energy
dissipation, meaning energy will be lost compressing these extremely weak layers, and
therefore will not be transferred into horizontal stress. In many shale formations the
approximations stated above may do a fair job of estimating horizontal stress, but we are
speculating that the organic content causes these estimations to over-predict the horizontal
stress even though they increase the anisotropy.
Moving forward, we will switch from the direct density-TOC correlations to an actual physical
fine-layering model that can include the mineralogy determined at the wellbore (Backus, 1962).
This will alleviate the unrealistic spikiness of Ko calculated in Figure 2 and incorporate more
information from the wellbore.
Ko (unitless)
0
0.2
0.4
0.6
0.8
1
6680
6700
Depth (ft.)
6720
6740
Given
Calculated
6760
6780
6800
Figure 2: Ko estimated from the horizontal stress gradient provided by Schlumberger (Given)
versus Ko calculated from the Schmoker and Hester TOC correlation and the Vernik and Nur
dataset. The well log is from the Freda Lake Field in Saskatchewan.
Effective Medium Theory:
The Bakken is known for abrupt changes in mineralogy. Each facies has distinct depositional
characteristics, but at any given well the components that make up that facies can change
dramatically, and ultimately affect production. That is why an effective medium should account
for the changes in geometrical relations from facies to facies as well as the mineralogy.
I have recently received X-ray Diffraction data (XRD) for 13 samples from the Freda Lake well in
Figure 2, and QEMScan data will be completed shortly. I am applying the Self-Consistent
Approximation to the Bakken. Eqs. 1 and 2 are the formulations from Berryman, 1995. The P
and Q values are the geometric coefficients, which are generally solved for specific values
(spheres, cracks, needles, etc.). Since obtaining exact solutions causes the equations to be
nonlinear, and dependent on specific measured samples (P1 depends on K1 and Km1), we will
estimate the geometric coefficients by simply solving for P and Q.
N
 v (K
i 1
i
i
 K m ) Pi  0
(2)
i
  m )Qi  0
(3)
N
 v (
i 1
i
As a first pass I have taken the vertical velocities from the well log in Figure 2 and solved for P
and Q values for the entire middle Bakken section. These results are reported in Table 1. The
associated calculated moduli versus measured moduli are plotted in Figure 3.
Mineral
P
Q
Dolomite
-0.2
-0.1
Quartz
0.5
0.05
Feldspar
1.5
1
Clay
-1
-0.2
Other
0.4
-0.2
Porosity
1
1.5
Table 1: Initial P and Q values for Eq. 1 and 2
50
35
45
30
35
G Measured (GPa)
K Measured (GPa)
40
30
25
20
15
10
25
20
15
10
5
5
0
0
0
10
20
30
K Calculated (GPa)
40
50
0
5
10
15
20
G Calculated (GPa)
25
30
35
Figure 3: Measured bulk modulus (K) and shear modulus (G) values versus the calculated values
using Table 1 and Eq. 2 and 3
Future Work:
In the following weeks we will forward model the mineralogy with PowerBench and that will
allow us to apply the Backus model to the shales and the SCA model to individual facies in the
middle Bakken.
Laboratory static and dynamic measurements are still underway, and these can easily be
implemented into the above models. The final result will be a mineral-based model of the
elastic properties that can be used in hydrofracture models and seismic exploration.
References:
Vernik, L. and Nur, A.: 1992, Ultrasonic Velocity and Anisotropy of Hydrocarbon Source Rocks,
Geophysics 57, 727-735.
Schmoker, J. W., and T. C. Hester, 1983, Organic carbon in the Bakken Formation, United States
portion of Williston basin: AAPG Bulletin, v. 67, no. 12, p. 2165–2174.
Backus, G. E., 1962, Long-wave elastic anisotropy produced by horizontal layering: J.
Geophys. Res., 67 , 4427-4440.
Berryman, J.G., 1995, Mixture theories for rock properties, American Geophysical Union
Handbook of Physical Constants, edited by T. J. Ahrens (AGU, New York, 1995), pp. 205-228.