1. Stresses in the Subsurface

1.1 Scalar, vector, and tensor variables in the subsurface

Figure 1.1: Scalar, vector and tensor quantities.

1.2 Cauchy's equilibrium equation and vertical stress

Figure 1.2: Cauchy's equilibrium equations.

Figure 1.3: Simple application of Cauchy's equilibrium equations for vertical stress.

1.3 Principal stresses and subsurface stress regimes

Figure 1.4: Principal stresses and principal directions.

Figure 1.5: The total and Terzaghi's effective stress tensors.

Figure 1.6: Stress regime classification.

1.4 Stress invariants and stress path

Figure 1.7: 3D Mohr circle.

Figure 1.8: Isotropic and deviatoric stress tensors.

Figure 1.9: Stress tensor invariants and stress paths.

Figure 1.10: Stress tensor invariants $p'$ and $q$.

1.5 WP1: Subsurface Stresses, Stress Tensor and Invariants

1.5.1 Exercise 1: Reading a stress log

The figure below displays principal stresses in a well from the Vaca Muerta Formation

Figure 1.11: Calibrated stress log for an unconventional formation(Image credit: SPE-180965-MS).

The figure does not show absolute depth values. This is a way oil companies “sanitize” their data, so that they do not reveal too much information that competitors could use to their advantage. We do know that depth grid spacing is 25 m. You will have to digitize this image to get numerical values. I recommend using https://apps.automeris.io/wpd/.

Tasks:

1. Calculate the average pore pressure (PPRS_3) gradient between depth A and depth B in [MPa/km] and [psi/ft]
2. Calculate the average vertical stress (SigV) gradient between depth A and depth B
3. Calculate a reasonable guess for depth A
4. Write out the principal stress tensors (as matrices 3x3) at depths A, B, C, D and E assuming vertical stress is a principal stress
5. Classify A, B, C, D and E according to stress regime (Normal, Strike Slip, Reverse Faulting)
6. Plot 3D Mohr circles of effective stresses for A, B, C, D and E
7. Plot $p$-$q$ points for A, B, C, D and E
8. Plot $I_1$-$J_2$ points for A, B, C, D and E
9. In which direction would a hydraulic fracture open-up in the interval under study in this formation? Justify.

1.5.2 Exercise 2: Computing total vertical stress

Go to https://github.com/dnicolasespinoza/GeomechanicsJupyter and download the files 1_14-1_Composite.las and 1_14-1 deviation_mod.dev. The first one is a well logging file (.LAS). You will find here measured depth (DEPTH - Track 1) and bulk mass density (RHOB - Track 8). Track 3 also shows bulk density correction (DRHO). Add RHOB to DRHO to obtain the corrected bulk mass density. The second file has the deviation survey of the well. Use this file to calculate true vertical depth subsea (TVDSS) as a function of measured depth (MD) in the well logging file. Water depth at this location is 104 m TVDSS. You may assume an average bulk mass density of 2 g/cc between the seafloor and the beginning of the density data. Summations with discrete data sets can be easily done through a ‘for’ loop or with a spreadsheet.

1. Plot all available tracks with depth in the y-axis.
2. Calculate and plot vertical stress using the density log.
3. Calculate and plot hypothetical ‘hydrostatic’ pore pressure.

Note for Python users: you may want to use https://lasio.readthedocs.io/en/latest/index.html library to read LAS files.

1.6 Stress projection on a plane: normal and shear stress

Figure 1.12: Stress tensor projection on a plane: 2D case.

Figure 1.13: Stress tensor projection on a plane: 3D case.

1.7 The geographical coordinate system

Figure 1.14: The geographical coordinate system.

1.8 WP2: 3D Stress Projection on Fractures and Faults

1.8.1 Exercise 1: Verification of critically stressed fractures intersecting a wellbore

Borehole imaging http://petrowiki.org/Borehole_imaging in the well from the Vaca Muerta Formation (Project 1) shows the presence of several fractures below Depth E. We would like to know the shear stresses $\tau$ and effective normal stresses $\sigma_n$ acting on these fractures.

Figure 1.15: Fracture mapping from wellbore images (Image credit: [Zoback, 2013]).

Additional borehole images at the depth of leak-off tests show that the azimuth of $S_{Hmax}$ is 90$^{\circ}$ (i.e., in East-West direction).

Tasks to be implemented in Matlab, Python, or similar software:

1. Input the principal stress tensor at Depth E in the principal directions coordinate system and calculate the tensor in the geographical coordinate system.
2. Generate 100 randomly distributed fracture orientations (strike and dip) and compute their effective normal stress and shear stress. Plot all in a $\sigma_n$ versus $\tau$ diagram together with the 3D Mohr circle(s).
   - EXTRA 1 (not required): color symbols for each point $(\sigma_n,\tau)$ from blue to red (e.g., JET colormap) according to the value of $\tau / \sigma_n$.
   - EXTRA 2 (not required): plot all fractures in a stereonet projection with symbols colored by the value of $\tau / \sigma_n$. About stereonets: Section 5.22 in https://dnicolasespinoza.github.io
3. Are there any fractures prone to shear slip (assume friction coefficient $\mu = 0.5 \pm 0.1$)? What fractures are likely to be hydraulically conductive based on the $\tau / \sigma_n$ criterion and which others are not?
4. Wellbore images actually show that there are two major fracture sets:
   - Set 1: strike = $60 ^{\circ} \pm 10^{\circ}$, dip = $80^{\circ} \pm 5^{\circ}$;
   - Set 2: strike = $10^{\circ} \pm 10^{\circ}$, dip = $60^{\circ} \pm 5^{\circ}$;
   Plot results for 20 fractures for each set (errors represent one standard deviation of a normal distribution). Which fracture set is more likely to slip in shear? Why?

Help: check Prof. Foster’s widget for the convention on rotation angles for the principal stress orientations.

1.8.2 Exercise 2: Verification of fault reactivation and available injection pressure window

The following figure and table summarize the geometry of faults at the High Island Block 24-L near the coast of the Gulf of Mexico and the Texas-Louisiana border.

Figure 1.16: (a)Structure map of HC Sand and (b) reservoir model by Jianping Xu.

Table 1.1: Summary of faults simplified to planar surfaces.

Fault ID	Segment origin	Segment end	Average Depth	Dip
	(Easting, Northing) [ft]	(Easting, Northing) [ft]	[ft]
F1a	(308,16772)	(3338,16106)	7250	53$^{\circ}$SW
F1b	(3338,16106)	(8116,16556)	7150	53$^{\circ}$S
F1c	(8116,16556)	(12481,17589)	7050	53$^{\circ}$SE
F1d	(12481,17589)	(17478,21367)	7300	53$^{\circ}$SE
F2	(5535,14938)	(7824,16209)	7500	52$^{\circ}$SE
F3	(15162,12008)	(17994,14322)	7800	55$^{\circ}$SE
F4a	(713,13675)	(3623,14480)	7400	56$^{\circ}$SE
F4b	(3623,14480)	(6489,14394)	7400	56$^{\circ}$S
F4c	(6489,14394)	(10095,12178)	7500	56$^{\circ}$SW
F4d	(10095,12178)	(14704,11738)	7700	56$^{\circ}$S
F4e	(14704,11738)	(18768,9753)	8300	56$^{\circ}$SW
F4f	(18768,9753)	(22832,7690)	8350	56$^{\circ}$SW
F5a	(462, 13250)	(2738,10767)	8000	56$^{\circ}$SW
F5b	(2738,10767)	(4596,7317)	8200	56$^{\circ}$SW
F6a	(2408,11310)	(5771,11029)	8000	30$^{\circ}$SW
F6b	(5771,11029)	(8638,11059)	8100	30$^{\circ}$S
F6c	(8638,11059)	(11090,11633)	8150	30$^{\circ}$SE
F7a	(2988,10960)	(6842,8356)	8200	50$^{\circ}$SW
F7b	(6842,8356)	(8538,6029)	8150	50$^{\circ}$SW
F7c	(8538,6029)	(11813,4162)	8200	50$^{\circ}$SW
F8a	(9939,2580)	(12936,4584)	8500	56$^{\circ}$SE
F8b	(12936,4584)	(15436,7092)	8400	56$^{\circ}$SE
F8c	(15436,7092)	(22831,7497)	8400	56$^{\circ}$S
F9	(13586,93)	(15185,6435)	8450	37$^{\circ}$NW

Tasks:

1. Calculate the stress tensor for each fault. The direction of $S_{hmin}$ is N-S.
2. Calculate the effective normal stress $\sigma_n$ and shear stress $\tau$ on each fault and plot the results together in a normalized 3D Mohr circle with x-axis ( $\sigma_n / \sigma_v$) and y-axis ( $\tau / \sigma_v$). Assume hydrostatic pore pressure d$P_p/$   d$z = 0.44$ psi/ft.
3. Are there any faults prone to reactivation (assume friction coefficient $\mu = 0.6$)? If yes, what faults are those?
4. Assuming that fluid injection would shift Mohr circles to lower $\sigma_n$ values without change of $\tau$ values (circle moves to the left without change of diameter - conservative assumption), what is the maximum allowable pressure increase without causing fault reactivation?
5. Plot the faults as straight lines in a 2D top view map and identify faults closer to reactivation. Suggestion: color segments according to the value of $\tau / \sigma_n$ where red is high likelihood for reactivation and blue is low likelihood for reactivation.