3.6 WP5: Poroelasticity

3.6.1 Exercise 1: Biot coefficient determination

The file BiotCoeffExperiment.xlsx has data from a laboratory experiment on a reservoir sandstone that shows axial and radial deformations caused by alternating variations of confining stress $P_c$ and pore pressure $P_p$.

a.

Plot pressure and stresses as a function of time.

b.

What are the dPc/dt and dPp/dt loading rates when either confining or pore pressure are increased??

c.

Fit a straight line to the data to obtain a unique relationship between $\varepsilon_{vol}$ and $\sigma_{mean}$ (effective), and calculate the bulk Biot coefficient $\alpha$. Assuming isotropic elasticity and that the Poisson Ratio is 0.25, what is the Young's modulus?

d.

Plot together the volumetric strain with Terzaghi’s and Biot’s effective stresses.

e.

If permeability is k = 100 mD, the fluid is water, porosity is 0.28, and the sample length is 0.05 m with top and bottom drainage, what is the characteristic time for pore pressure diffusion $T_{ch}$? How does it compare to the pressure/stress loading time? Would it be drained or undrained loading? Note: you need the diffusivity parameter for which you have to look up for properties of water.

f.

EXTRA: Use the theory of transverse isotropic poro-elasticity to figure out the stress paths needed to measure directly $\alpha_h$ and $\alpha_v$.

Figure 3.1: Variation of volumetric strain with Terzaghi's effective stress.

3.6.2 Exercise 2: Depletion stress path

For this problem you have to use the geomechanical module of reservoir simulator CMG https://www.cmgl.ca/. The software is available to UT Austin students here: http://pge.utexas.edu/LRC/

a.

Review the files CMG_Geomechanics_Tutorial.pdf and CMG_Running_InputFile.pdf.

b.

Change the vertical stress and well schedule as shown in the figure below (example files: Injection1.dat and Production1.dat.

Figure 3.2: Schematic cross section of reservoir model for depletion.

c.

What is initial boundary condition in each direction? (i.e. constant stress or zero displacement).

d.

Plot 1 - Plot minimum principal total stress (Total stress I), vertical total stress (Total stress K), and pore pressure (Pressure) vs time. (**Note: Please remove initial data (time = 0) when you plot).

e.

Plot 2 - Plot minimum principal stress (y-axis) vs pore pressure (x-axis), and verify the slope of the curve is similar with $\alpha \frac{\left(1-2\nu\right)}{(1-\nu)}$ ($\alpha$ is the Biot coefficient and $\nu$ is Poisson’s ratio - **Note: Please remove initial data (time = 0) when you plot pressure and stresses).

f.

Run the simulation again using Biot coefficient from the previous laboratory problem, repeat the question “d” using the new simulation result and plot on the same figure.

g.

Plot the stress path with Mohr circles for the initial (0.1 days) and final time (100 days).

h.

Plot the stress path in the $(p',q)$ space for the same period of time.

i.

What is the absolute minimum pressure to create a hydraulic fracture (i.e. minimum principal total stress) at the end of the simulation when bottom-hole pressure is BHP = 240 psi? Compare with the analytical solution.