Generalized Solution for Double-Porosity Flow Through a Graded Excavation Damaged Zone

Abstract

Prediction of flow to boreholes or excavations in fractured low-permeability rocks is important for resource extraction and disposal or sequestration activities. Analytical solutions for fluid pressure and flowrate, when available, are powerful, insightful, and efficient tools enabling parameter estimation and uncertainty quantification. A flexible porous media flow solution for arbitrary physical dimensions is derived and extended to double porosity for converging radial flow when permeability and porosity decrease radially as a power law away from a borehole or opening. This distribution can arise from damage accumulation due to stress relief associated with drilling or mining. The single-porosity graded conductivity solution was initially found for heat conduction, the arbitrary dimension flow solution comes from hydrology, and the solution with both arbitrary dimension and graded permeability distribution appeared in reservoir engineering. These existing solutions are combined and extended here to two implementations of the double-porosity conceptual model, for both a simpler thin-film mass transfer and more physically realistic diffusion between fracture and matrix. This work presents a new specified-flowrate solution with wellbore storage for the simpler double-porosity model, and a new, more physically realistic solution for any wellbore boundary condition. A new closed-form expression is derived for the matrix diffusion solution (applicable to both homogeneous and graded problems), improving on previous infinite series expressions.

Appendices

Appendix A: Wellbore Storage Boundary Condition

The wellbore storage boundary condition accounts for the storage in the finite borehole arising from the mass balance $Q_{\textrm{in}}-Q_{\textrm{out}}=A_c \frac{\partial h_w}{\partial t}$. $Q_{\textrm{in}}$ $\mathrm {[m^3/s]}$ is the volumetric flow into the borehole from the formation, $Q_{\textrm{out}}$ is possibly time-variable flow out of the well through the pump (Q(t) $\mathrm {[m^3/s]}$), and $\frac{\partial h_w}{\partial t}$ is the change in hydraulic head [m] ($h_w=\frac{p_w}{\rho g} + z$) of water standing in the borehole through time, $p_w$ is change in pressure [Pa] of water in the borehole, $\rho$ is fluid density $\mathrm {[kg/m^3]}$, z is an elevation datum [m], and g is gravitational acceleration $\mathrm {[m/s^2]}$. $A_c$ is the cross-sectional surface area of the pipe, sphere, or box providing storage (it may be a constant or a function of elevation); for a typical pipe, it becomes $A_c=\pi r_c^2$, where $r_c$ is the radius of the casing where the water level is changing. The mass balance is then

$$\begin{aligned} \frac{A_m k_0}{\mu } \left. \frac{\partial p}{\partial r} \right| _{r=r_w} - Q(t) = \frac{A_c}{\rho g} \frac{\partial p_w}{\partial t}, \end{aligned}$$

(35)

where $A_m$ is the area of the borehole communicating with the formation. For the integer m considered here these are $A_0=b^2$, $A_1=2 \pi r_w b$, $A_2=4 \pi r_w^2$ (b is a length independent of the borehole radius).

Assuming the change in water level in the borehole ($h_w=p_w/\left( \rho g\right)$) is equal to the change in formation water level ($h=p/\left( \rho g\right)$), this can be converted into dimensionless form as

$$\begin{aligned} \left. \frac{\partial p_D}{\partial r_D} \right| _{r_D=1} - f_t = \sigma \frac{\partial p_D}{\partial t}, \end{aligned}$$

(36)

where $\sigma =A_c/\left( r_w n_0 c \rho g A_m \right)$ is a dimensionless ratio of formation to wellbore storage; $\sigma \rightarrow 0$ is an infinitesimally small well with only formation response, while $\sigma \rightarrow \infty$ is a well with no formation response (i.e., a bathtub).

Appendix B: Transformation of Modified Bessel Equation

Following the approach of Bowman (1958), alternative forms of the Bessel equation are found; this approach is a simplification of the original approach of Lommel (1868). An analogous approach is applied here to “back into” the desired modified Bessel equation. The equation satisfied by the pair of functions

$$\begin{aligned} y_1&=x^\alpha \textrm{I}_\nu \left( \beta x^\gamma \right) ,&y_2&=x^\alpha \textrm{K}_\nu \left( \beta x^\gamma \right) , \end{aligned}$$

(37)

is sought, where $\alpha$, $\beta$, $\gamma$, and $\nu$ are constants. Using the substitutions $\zeta =yx^{-\alpha }$ and $\xi =\beta x^\gamma$ gives $\zeta _1=\textrm{I}_\nu \left( \xi \right)$ and $\zeta _2=\textrm{K}_\nu \left( \xi \right)$, which are the two solutions to the modified Bessel equation (DLMF 2023, §10.25.1),

$$\begin{aligned} \xi \frac{\textrm{d}}{\textrm{d}\xi } \left( \xi \frac{\textrm{d}\zeta }{\textrm{d}\xi }\right) - (\xi ^2 + \nu ) \zeta = 0. \end{aligned}$$

(38)

Given

$$\begin{aligned} \xi \frac{\textrm{d}}{\textrm{d}\xi } \left( \xi \frac{\textrm{d}\zeta }{\textrm{d}\xi }\right) = \frac{x}{\gamma ^2} \frac{\textrm{d}}{\textrm{d}x} \left( x \frac{\textrm{d}\zeta }{\textrm{d}x} \right) , \end{aligned}$$

(39)

and

$$\begin{aligned} x \frac{\textrm{d}}{\textrm{d}x} \left( x \frac{\textrm{d}\zeta }{\textrm{d}x} \right) = \frac{y''}{x^{\alpha -2}} - \frac{\left( 2 \alpha -1 \right) y'}{x^{\alpha -1}} + \frac{\alpha ^2 y}{x^\alpha }, \end{aligned}$$

(40)

the standard-form equation satisfied by y is

$$\begin{aligned} y'' + \left( 1-2 \alpha \right) y' + \frac{\alpha ^2 y}{x^\alpha } - \left( \beta ^2 \gamma ^2 x^{2\gamma -2} - \frac{\alpha ^2 - \nu ^2 \gamma ^2}{x^2} \right) y = 0. \end{aligned}$$

(41)

This equation can be compared to the Laplace-space ordinary differential Eq. (9), allowing direct use of the product of powers and modified Bessel function (37) as solutions (13).