FVM implementation and analysis of borehole Poroelastic problem of mode3

FVM implementation and analysis of borehole Poroelastic problem of mode3

This blog will show the comparison of the numerical solution calculated using FVM code with the analytical solution in Cheng.H.D’s paper, and briefly analyze the results.

1 Display of results

Fig. 1. Variation curve of dimensionless shear stress with dimensionless time and distance

Fig. 2. Curve of dimensionless pore pressure over dimensionless time and distance

Fig. 3. Theoretical solution of dimensionless shear stress curve with dimensionless time and distance

Fig. 4. Theoretical solution of dimensionless pore pressure curve with dimensionless time and distance

2. The analytical solution

Analytical solution of pore pressure value

$$\frac{p}{S_0}=\frac{4}{3}B\left(1+v_u\right)\left[-\sqrt{\frac{a}{r}}\mathrm{erfc}\left(\frac{r-a}{\sqrt{4ct}}\right)+\frac{a^2}{r^2}\right]\cos 2\theta$$
It is not difficult to see from this formula that the pore pressure p is a function of time t and r, and the corresponding parameters can be substituted to draw the analytic solution curve

Analytical solution of tangential stress

$$\frac{{\sigma }_{\theta \theta }}{S_0}=\left[-1+4\frac{v_u-v}{1-v}\sqrt{\frac{a}{r}}\mathrm{erfc}\left(\frac{r-a}{\sqrt{4ct}}\right)-3\frac{a^4}{r^4}\right]\cos 2\theta$$
It is not difficult to see that the tangential stress is a function of time t and r, and the corresponding parameters can be used to draw the analytic solution curve
From the above two formulas, it is also easy to obtain the circumwellbore shear stress and the pore pressure distribution near the borehole at the time t=0+ and the infinite time, which are obtained by numerical solution to accurately calculate. The dark red dotted lines in Fig. 1 and Fig. 2 are analytical solutions. The calculation formula is as follows:
$${\sigma }_{\theta \theta }^{0+}=-\left(1+3\frac{a^4}{r^4}\right)S_0\cos 2\theta$$
$$p^{0^{+}}=\frac{4}{3}{S}_{0}B\left(1+v_u\right)\frac{a^2}{r^2}\cos 2\theta$$

3. Result analysis

Fig. 1 shows the very interesting fact that the compressive stress concentration near the wellbore, which is analytically solved as a quadratic hyperbola, is caused by the far-site stress immediately after the wellbore is drilled. Due to the instantaneous compression of the pore space, the fluid in the pore has no time to drain into the borehole, resulting in a pore pressure peak near the wellbore wall, which gradually decreases over time and moves away from the wellbore wall (see Fig. 2). In this process, the tangential total stress is always positive, and the curve group is shaped like the Nike logo. Since this shape occurs only within 1-1.1 borehole radii, some people call it the stress skin effect. At some point between t=0.001 and t=0.01, the Nike logo disappears. At the borehole wall (r = r0), the tangential stress first decreases and then increases with time (note: according to the rules of material mechanics, the tension is positive, and the compression is negative), and the initial time coincates with the last infinite time.