Description
Instructions
This assignment focuses the relationship between the viscoplastic properties (ductility) of reservoir rocks and the state of stress in different lithofacies.
Part 1: Creep and stress relaxation
Sone & Zoback (2013b) describe the time-dependent deformation (creep) in terms of a viscoelastic power law with the form (Unit 7, slide 10):
- What do the power law parameters B and n represent? How do they vary with clay + TOC? sample orientation? elastic stiffness?
- Consider the following samples: Barnett-2, Haynesville-1 and Eagle Ford-1 (vertical samples). Using the lower limits of B and n values provided in Table 1, calculate the amount of creep strain, , that would occur due to the application of 30 MPa differential stress over times of 1 yr, 100 kyr, and 100 Myr. Construct a table of the results and/or a scatter plot of the creep strain as a function of time with the points colored based on the clay + TOC.
- What is the relationship between the amount of creep strain and clay + TOC?
- The creep compliance function J(t) of the viscoelastic power law model is given by:
J(t) = Btn
Plot log J(t) versus log t for each sample and show how the values of B and n are obtained.
- For each sample, calculate the accumulated differential stress, σ(t), for a constant strain rate of over 150 My using the following expression:
Figure 1: Ternary composition of relatively high (-1) and low (-2) clay + TOC sample groups from different shale basins. From Sone & Zoback (2013a).
Table 1: Power law constitutive parameters for each sample group (Sone & Zoback, 2013b).
| Vertical samples | Horizontal samples | |||
| B (10−5 MPa−1) | n | B (10−5 MPa−1) | n | |
| Barnett-1 | 3.5-4.2 | 0.015-0.024 | 2.0-2.6 | 0.012-0.021 |
| Barnett-2 | 1.2-1.8 | 0.011-0.027 | 1.6-1.6 | 0.009-0.010 |
| Eagle Ford-1 | 2.6-8.5 | 0.028-0.095 | 1.7-2.3 | 0.024-0.053 |
| Eagle Ford-2 | 2.2-7.1 | 0.019-0.085 | 1.7-1.8 | 0.023-0.049 |
| Haynesville-1 | 3.7-8.9 | 0.023-0.081 | 1.8-2.7 | 0.027-0.062 |
| Haynesville-2 | 1.6-3.1 | 0.025-0.060 | 1.5-1.8 | 0.011-0.049 |
Part 2: Effects of viscoplastic creep on stress magnitudes
- For a normal faulting environment, calculate the lower bound on the least principal stress using the following parameters:
Depth, d = 9000 ft
Coefficient of friction, µ = 0.6
Pore pressure gradient = 0.5 psi/ft
Vertical stress, Sv = 1.1 psi/ft
- Viscoplastic stress relaxation. The variation in differential stress with time is given by the expression below (Unit 7, slide 17):
where 0, the total strain, is a fitting parameter. What is = 100 Myr for E = 40 GPa? Use the plot below to find the value of n from the linear fit line.
Figure 2: Correlation between Young’s modulus and power law parameter n for several shale reservoir samples. From Ma & Zoback (2018).
- How much would you expect the value of Shmin to evolve due to viscoplastic stress relaxation? Use t = 100 Myr and the upper/lower bound values of n to determine the range of Shmin
Part 3: Vertical growth of hydraulic fractures in layered media
- Figure 3 shows Shmin magnitudes as a function of depth for a layered sequence. Based on this stress profile, which formation is the least ductile (most brittle)?
- Assuming a strike-slip faulting regime, which layer would you stimulate to achieve a wide, confined fracture with limited vertical extent?
- Suppose that stimulating layer E results in horizontal hydraulic fractures. What does this tell you about the relative stress magnitudes in layer E?
Figure 3: Variation of the minimum horizontal stress with depth in a layered sequence.
