Reservoir Geomechanics Homework 3- Ductility and Stress Magnitudes Solution

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):

a) What do the power law parameters B and n represent? How do they vary with clay + TOC? sample orientation? elastic stiffness?
c) What is the relationship between the amount of creep strain and clay + TOC?
d) 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.
e) 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
a) 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
b) 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.

Part 3: Vertical growth of hydraulic fractures in layered media
a) 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)?
b) Assuming a strike-slip faulting regime, which layer would you stimulate to achieve a wide, confined fracture with limited vertical extent?
c) 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.