Reservoir Geomechanics Homework 2- Composition, Elasticity, and Ductility Solution

Instructions
This assignment focuses on understanding the relationships between the composition, microstructure, and mechanical properties of unconventional reservoir rocks.
Part 1: Composition, microstructure, and elastic properties

Figure 1: (Left) Ternary composition of two shale samples in wt%. (Right) Optical micrographs of Barnett 25H and Eagle Ford 65H in plane polarized light. Bedding is horizontal.
Table 1: Mineral elastic moduli and densities
Bulk modulus (GPa) Shear modulus (GPa) Density (g/cm3)
Quartz 37 44 2.65
Carbonates 70.2 29 2.71
Clay + TOC 8.5 4.5 1.95
a) Reading ternary diagrams. For each of samples plotted in Figure 1, determine the percentage by weight of each components (clay + TOC, QFP, carbonates).
b) Estimate the density for each sample. Using the component densities provided in Table 1 and the wt% values from (a) to calculate an approximate density for each sample.
c) Calculate elastic moduli. Ultrasonic laboratory measurements indicate that the compressional wave velocities of the Eagle Ford and Barnett sample are 6.0 and 5.0 km/s, and the shear wave velocities are 3.3 and 3.2 km/s. Using the densities determined in (b), calculate the bulk and shear moduli.
d) Calculate effective elastic moduli. The effective modulus can be calculated by summing the contributions from the individual component moduli, where Mi and fi are the modulus and the fraction of the ith component, and Meff is the effective modulus of the composite:
Meff = XfiMi (Iso-strain)
(Iso-stress)
e) Compare your answers for (c) and (d). Do the ultrasonic measurements reflect stiffness components parallel or perpendicular to bedding?
Part 2: Hydraulic fracture propagation in layered media
a) Net pressure required for hydraulic fracture propagation. Use the expression for the critical stress intensity factor for mode I cracks (KIC) to determine the net pressure (P −S3) required to propagate a hydraulic fracture as a function of length (l). Perform the calculation for each layer using the KIC values in the table and plot results on the same axes.
√
KIC = (P − S3)π l
b) Compare the net pressure values in each layer for a fracture of length l = 3 ft. How important is rock tensile strength to hydraulic fracture propagation once fractures begin to propagate?
c) Consider the stress profile in Figure 3. If we apply sufficient pressure to propagate a hydraulic fracture of length l = 1 ft in layer 3, would we expect the fracture to grow vertically into any of the surrounding layers? Which ones and why?

Figure 2: (Left) A hypothetical layered sequence in a normal faulting environment in which Shmin varies as a function of depth. (Right) Values of Shmin and KIC at the center of each layer.
Part 3: Answer the questions in edX