URTeC 2669793 Utilizing A Viscoplastic Stress Relaxation Model to Study Vertical Hydraulic Fracture Propagation in Permian Basin Shaochuan Xu*, Fatemeh S. Rassouli, and Mark D. Zoback, Stanford University Copyright 2017, Unconventional Resources Technology Conference (URTeC) DOI 10.15530/urtec-2017-2669793

This paper was prepared for presentation at the Unconventional Resources Technology Conference held in Austin, Texas, USA, 24-26 July 2017.

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Summary Predicting vertical hydraulic fracture propagation is critically important to optimize production from unconventional reservoirs. In this study, we consider a site in Permian Basin of west Texas, where multiple horizontal wells were drilled to develop a stacked pay. For several years, our research group has been studying use of a viscoplastic stress relaxation model to predict variations of the least principal stress with depth that control vertical hydraulic fracture propagation. Creep measurements have been conducted for clay-rich rocks (Sone & Zoback, 2014) and carbonate-rich rocks (Rassouli & Zoback, 2015), and significant viscoplasticity was observed for both types of rocks. A time dependent constitutive law has been found to accurately describe the viscoplastic behavior of the shales under laboratory conditions. This constitutive law has been used to develop a stress relaxation model for the study area to predict the variations of the least principal stress with depth. The predicted frac gradients are compared with the measured frac gradients from DFITs in several formations as well as the vertical extent of hydraulic fracture propagation indicated by microseismic data. Introduction Viscoplastic stress relaxation in sedimentary rocks leads to a more isotropic state of stress. In normal faulting or normal/strike-slip faulting environments, this causes the least principal stress to increase. Therefore, formations that are more viscoplastic are likely to have higher values of the least principal stress (or frac gradient) and are potential barriers to vertical hydraulic fracture propagation.

Figure 1: Stress relaxation in stacked elastic and viscoplastic layers and the corresponding Mohr diagram

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Figure 1 illustrates the stress relaxation in stacked formations of elastic and viscoplastic layers. In elastic and brittle formations like sandstone, there is very little stress relaxation, and the corresponding Mohr circle is large; in a viscoplastic shale layer with a relatively small degree of ductility, there is a small degree of stress relaxation and a moderate increase in Shmin (as shown in red in the schematic cross section and the corresponding Mohr circle); in a more viscoplastic layer (shown in blue) Shmin increases even more. This phenomenon was observed by Warpinski & Teufel (1989) at the Multiwell Experiment (MWX) in Piceance basin of Colorado, and also by Ma & Zoback (2017) at Woodford shale in Oklahoma. MWX also shows that Shmin in sandstone, instead of being controlled by stress relaxation, is bounded by frictional equilibrium, shown by failure line and corresponding Mohr circle in Figure 1. Theory The time dependent constitutive law, developed from a wide range of creep experiments on unconventional reservoir rocks (Sone and Zoback, 2014), is as follows:

where σ(t) is the differential stress with time; B is a measure of the elastic compliance of the rock; n is stress relaxation index that describes the tendency to exhibit time-dependent deformation; ε0 is the total strain. In practice, the term ε0·t-n is a fitting parameter. Sone and Zoback (2014) showed that B is equal to the reciprocal of Young’s modulus, and Rassouli and Zoback (2015) showed that these constitutive parameters derived from short-term creep tests can be used to predict long-term deformation of shale. Therefore, we apply this constitutive law into the reservoir by writing the differential stress in terms of the overburden stress minus the least principal stress as,

which only holds in normal or normal/strike-slip faulting environments. Case Study We examined the stacked Wolfcamp formations in Permian Basin where one pilot well M and four horizontal production wells A, B, C, and D were drilled. Figure 2 plots the east view and map view of the well trajectories. In wells A, B, and C, DFIT (Diagnostic Fracture Injection Test) measurements were made at three different depths, which are used for estimating in situ least principal stress. Wells A, B, C, and D were hydraulically fractured in sequence to stimulate production; microseismic events were monitored during the frac treatment by the geophones in well M. These data are used to analyze the influence of variations of frac gradients on vertical hydraulic fracture propagation. Geophysical logs as well as core samples are available in well M; core samples are used for laboratory creep experiments.

Figure 2: Trajectories of studied wells with formation tops and the depth coverage of geophones (blue dash line)

σ (t)= ε0

1B(1−n)t

−n (1)

SV − Shmin = ε0

E(1−n)t

−n (2)

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Results

Figure 3: Predicted least principal stress from measured B and n

Figure 4: Least principal stress controlled vertical hydraulic fracture growth (indicated from microseismic events)

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B and n values are measured for two vertical samples at two different depths, 6356ft and 6466ft, respectively. We choose vertical samples to calculate the stress for the reason that studied formations are in normal faulting regime (Xu & Zoback, 2015) and in situ stresses of normal faulting environments on vertical samples are similar to our lab conditions (Rassouli & Zoback, 2015). Three frac gradients are estimated from DFITs, shown by red circles in the right panel of Figure 3. Fortunately, the depth of one DFIT is very close to 6466ft, where we measure B and n values. Therefore, we use measured B, n, and the least principal stress to backward calculate the total strain ε 0, which is about 2×10-3. Then, applying this total strain to the other depth, 6356ft, we predict the frac gradient to be about 0.98 psi/ft, plotted by red dot in the right panel of Figure 3. Converting this frac gradient into least principal stress, we plot both the measured and predicted least principal stresses in the upper right and lower right panels of Figure 4. The least principal stress at 6356ft is large compared with three DFITs, which implies Equation 2 might overestimate the least principal stress. Nevertheless, a frac barrier highly likely exists around this depth indicated in Figure 4, if we assume the depths of microseismic events could represent the vertical growth of hydraulic fractures. Black dash lines are formation tops same as those plotted in Figure 2 and 3, and yellow stars are locations where hydraulic fractures in each well are initiated. There are two features to note in Figure 4: (1) As seen in the upper left panel, when well A was hydraulically fractured in the upper part of the Lower Wolfcamp, the upward propagation of hydraulic fractures appears to have been limited by the predicted high least principal stress in the Upper Wolfcamp. The same behavior is seen in the lower left and lower middle panels, when wells C and D were hydraulically fractured in the Middle Wolfcamp. (2) As seen in the upper middle panel, when well B was hydraulically fractured in the middle of Upper Wolfcamp, where the least principal stress is high, we would expect both upward and downward propagation of hydraulic fractures due to the low least principal stress. However, very few micro-earthquakes are seen below well B, presumably because the micro-earthquakes were already triggered during the fracturing of well A, which is very close (Figure 2). Discussion

Figure 5: Generic plot with frac gradient prediction to illustrate the anisotropy of both Young’s modulus and stress relaxation index, n More viscoplastic constitutive parameters across North America were measured in our research group, which are plotted in Figure 5 labeled by formation names with measurements from Wolfcamp formations labeled by the depths. As almost all these formations are in normal or normal/strike-slip faulting environments, the frac gradient is calculated from Equation 2, assuming the total strain is 0.001 and the overburden gradient is 1 psi/ft. The core samples in the left upper corner of the plot are more viscoplastic since E is low and n is high; the core samples in the right lower corner of the plot are less viscoplastic since E is high and n is low. Interestingly, the Wolfcamp samples have low E and low n, meaning that the samples creep little even though they are compliant. Also, these unconventional reservoir rocks are anisotropic viscoplasically; the vertical samples seem to be more viscoplastic than the horizontal samples as most open dots are located upper left to the open triangles. The third observation is

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that some constitutive parameters are below allowed minimum frac gradient 0.6 psi/ft, due to not considering the mechanism of frictional equilibrium shown in Figure 1. Conclusion Vertical propagation of hydraulic fracture is largely controlled by the variations of least principal stress with depth. In situ least principal stress is possible to be predicted by a viscoplastic stress relaxation model based on power law creep, but further validation is still needed. The fundamental reason for varying amounts of stress relaxation at different depths is the change of mineral composition in different formations. Prediction of the least principal stress can be very helpful when designing hydraulic fracturing operations so that vertical propagation of hydraulic fractures is contained to the production horizon or extending to multiple producing horizons as desired. References Ma, X., & Zoback, M. D. (2017, January 24). Lithology Variations and Cross-Cutting Faults Affect Hydraulic

Fracturing of Woodford Shale: A Case Study. Society of Petroleum Engineers. doi:10.2118/184850-MS Rassouli, F. S., & Zoback, M. D. (2015, November 13). Long-Term Creep Experiments on Haynesville Shale

Rocks. American Rock Mechanics Association. Sone, H., & Zoback, M. D. (2014). Time-dependent deformation of shale gas reservoir rocks and its long-term effect

on the in situ state of stress. International Journal of Rock Mechanics and Mining Sciences, Volume 69, Pages 120-132.

Warpinski, N. R., & Teufel, L. W. (1989, April 1). In-Situ Stresses in Low-Permeability, Nonmarine Rocks. Society

of Petroleum Engineers. doi:10.2118/16402-PA Xu, S., & Zoback, M. D. (2015, November 13). Analysis of Stress Variations with depth in the Permian Basin

Spraberry/Dean/Wolfcamp Shale. American Rock Mechanics Association. Yang, Y., Sone, H., & Zoback, M. D. (2015, September 28). Fracture Gradient Prediction Using the Viscous

Relaxation Model and Its Relation to Out-of-Zone Microseismicity. Society of Petroleum Engineers. doi:10.2118/174782-MS