A Model for Tracking Fronts of Stress-Induced Permeability Enhancement

Abstract

Using an analogy to the classical Stefan problem, we construct evolution equations for the fluid pore pressure on both sides of a propagating stress-induced damage front. Closed form expressions are derived for the position of the damage front as a function of time for the cases of thermally-induced damage as well as damage induced by over-pressure. We derive expressions for the flow rate during constant pressure fluid injection from the surface corresponding to a spherically shaped subsurface damage front. Finally, our model results suggest an interpretation of field data obtained during constant pressure fluid injection over the course of 16 days at an injection site near Desert Peak, NV.

Appendix: Justification for Neglecting \(\nabla \rho \)

Substituting (2.2), (2.3), and (2.4) into (2.1) yields

$$\begin{aligned} {\widetilde{\beta }}\frac{\partial p}{\partial t} - \frac{k}{\mu }\frac{\rho }{\rho _0}\nabla ^2 p - \frac{2kg\rho \beta }{\mu }\nabla z \cdot \nabla p - \frac{\beta k}{\mu }(\nabla p)^2 = 0, \end{aligned}$$

(6.1)

with

$$\begin{aligned} {\widetilde{\beta }} \equiv \phi \beta + \frac{\rho }{\rho _0}\alpha . \end{aligned}$$

(6.2)

Dividing (6.1) by the term proportional to $\nabla ^2 p$ leads to the dimensionless equation

$$\begin{aligned} \frac{\rho _0}{\rho }\left( \frac{{\widetilde{\beta }} \mu }{k \nabla ^2 p} \frac{\partial p}{\partial t} \right) - 1 - 2\beta \rho _0 g \frac{\nabla z \cdot \nabla p}{\nabla ^2 p} - \frac{\rho _0}{\rho }\beta \frac{(\nabla p)^2}{\nabla ^2 p} = 0. \end{aligned}$$

(6.3)

Now consider a small vertical section of porous material of height $\Delta z$ over which the pressure varies by amount $\Delta p$, and suppose the time variation of $p$ over an interval of time $\Delta t$ is equal to $\xi \Delta p$ for some constant $\xi$. Then the first term on the left hand side is in order of magnitude

$$\begin{aligned} \frac{\xi {\widetilde{\beta }}\mu \Delta z^2}{k\Delta t}, \end{aligned}$$

(6.4)

where we have assumed that $\rho _0/\rho \approx 1$ and that the order of $\nabla ^2 p$ is $\Delta p/\Delta z^2$. Term (6.4) is not in general small compared to unity. The third term has order of magnitude

$$\begin{aligned} 2\beta \rho _0 g \Delta z, \end{aligned}$$

(6.5)

and due to the smallness of $\beta$, only approaches unity for very large values of $\Delta z$. The fourth term varies as

$$\begin{aligned} \beta \Delta p, \end{aligned}$$

(6.6)

and is small compared to unity except for very large values of $\Delta p$. Therefore, for the parameter regime of interest in this study, the dominant balance in Eq. (6.1) is between the first and second terms on the left hand side.