Reservoir Geomechanics Homework 4- Permeability and Flow Mechanisms Solution

Instructions
This assignment focuses on understanding the relationships between permeability, pore pressure, effective stress, and pore size through analysis of laboratory data.
Part 1: Pressure-dependent permeability

What does the value of χ mean in terms of the relative sensitivity of permeability to changes in pore pressure and confining pressure? Hint: Find the change in log k with the change in Pp for Pc = 4000-6000 psi and take the average. Then, find the change in log k with the change in Pc for Pp = 1000-4000 psi and take the average.

Figure 1: Permeability measurements on a clay-rich, horizontal Eagle Ford sample at various confining pressures and pore pressures. From Heller et al. (2014).
Part 2: The Klinkenberg effect
a) Determine the magnitude of the Klinkenberg effect. First, find the permeability value of each low pressure curve at modified effective stresses 2000, 3000, and 4000 psi. Then, plot the permeability data at each effective stress as a function of inverse pore pressure (1/Pp) (e.g., Unit 10, slide 19).
b) For each effective stress, find the value of the permeability at infinite pore pressure, k∞, and the Klinkenberg correction factor, Kb, using the expression below.

Part 3: Effective pore size
a) Use the expression below to calculate the effective slit pore width for each value of modified effective stress 2000-4000 psi using the following parameters:

Geometric factor, c = 4
Temperature, T = 40 °C
Molar mass, M = 4 g/mol
Pore pressure, Pp = 500 psi
Universal gas constant, R = 8.3145 J/mol°K
Find the viscosity, µ, of helium gas at the given pressure-temperature conditions from the NIST Chemistry WebBook (https://webbook.nist.gov/chemistry/fluid/).
b) Plot the effective slit pore width, w, as a function of effective stress. How does effective pore width vary with effective stress? Provide a physical explanation here. How do these values compare to the pore sizes observed in unconventional reservoir rocks (Unit 9)?
Part 4: Flow mechanisms
a) Under what conditions is slip flow important in gas production? Hint: Calculate the ratio of diffusive to viscous flux at 2000 psi effective stress (the ratio of the Klinkenberg corrected permeability to k∞) as a function of pore pressure.
b) Calculate the mean free path, λ, as function of pore pressure for helium gas using the expression and parameters below.

Temperature, T = 40 °C
Molecular diameter, dm = 2.6e-10 m
Boltzmann’s constant, KB = 1.38064852e-23 m2kg/s2°K
Use your answer to (e) to determine the value of the Knudsen number, Kn, as function of pore pressure at each effective stress. What does your answer signify in terms of the flow mechanisms active during the low pressure permeability measurements (Unit 10, slide 7)?