Perturbation Solutions for Flow in a Slowly Varying Fracture and the Estimation of Its Transmissivity

Abstract

Flow in fractures or channels is of interest in many environmental and geotechnical applications. Most previously published perturbation analyses for fracture flow assume that the ratio of the flow in the fracture aperture direction to the flow in the fracture length direction is of the same order as the ratio of mean fracture aperture to fracture length, and hence, the dominant flow is in the fracture length direction. This assumption may impose an overly strict requirement for the flow in the fracture length direction to be dominant, which limits the applicability of the solutions. The present study uses the ratio of aperture variation to length as the perturbation parameter to derive perturbation solutions for flow in two-dimensional fractures under both the pressure boundary condition (PBC) and the flow rate boundary condition (FBC). The solutions are cross-validated with direct numerical solutions of the Navier–Stokes equations and with solutions from published perturbation analyses using the geometry of two-dimensional symmetric wedges and fractures with sinusoidally varying walls. The study shows that compared with the PBC solution, the FBC solution is in a closer agreement with simulation results and provides a better estimate of the fracture transmissivity especially when the inertial effects are more than moderate. The improvement is due mainly to the FBC solution providing a more accurate quantification of the inertial effects. The solutions developed in this study provide improved means of analysing the hydraulic properties of fractures/channels and can be applied to complex flow conditions and fracture geometries.

Appendices

Appendix A: Derivation of Perturbation Solutions

In this appendix, we summarize the derivation process of the stream function and perturbation solutions (to the second order) under both the pressure and flow rate boundary conditions.

1.1 A1 Perturbation Solution Under the Pressure Boundary Condition

Equation (14) can be expanded in terms of the perturbation parameter $\epsilon$ using Eq. (17) with terms of different orders. For the 0-order:

$$ \frac{{\partial^{4} \varPsi_{0}}}{{\partial Y^{4}}} = 0 $$

(47)

Integrating Eq. (47) over Y and making use of the 0-order boundary condition:

$$ \left. {\frac{{\partial \varPsi_{0}}}{\partial Y}} \right|_{Y = \pm 1} = 0,\quad \left. {\varPsi_{0}} \right|_{Y = \pm 1} = \pm \frac{{Q_{0}}}{2} $$

(48)

the stream function at the 0-order can be obtained:

$$ \varPsi_{0} = Q_{0} \left({\frac{3}{4}Y - \frac{1}{4}Y^{3}} \right) $$

(49)

Substitute the 0-order stream function into the auxiliary condition [i.e. Eq. (15)], and after integration, the 0-order discharge is derived as:

$$ Q_{0} = \left({\mathop \int \nolimits_{0}^{\omega} \frac{1}{{8\omega h^{3}}}{\text{d}}X} \right)^{- 1}. $$

(50)

Following the same procedure, using the 1-order terms and Eq. (14):

$$ \frac{{\partial^{4} \varPsi_{1}}}{{\partial Y^{4}}} = - 2R_{p} h_{X} \frac{{\partial \varPsi_{0}}}{\partial Y}\frac{{\partial^{2} \varPsi_{0}}}{{\partial Y^{2}}} - R_{p} h_{X} Y\frac{{\partial \varPsi_{0}}}{\partial Y}\frac{{\partial^{3} \varPsi_{0}}}{{\partial Y^{3}}} - R_{p} h\frac{{\partial \varPsi_{0}}}{\partial X}\frac{{\partial^{3} \varPsi_{0}}}{{\partial Y^{3}}} $$

(51)

Integrating Eq. (51) over Y and making use of the 1-order boundary condition:

$$ \left. {\frac{{\partial \varPsi_{1}}}{\partial Y}} \right|_{Y = \pm 1} = 0,\quad \left. {\varPsi_{1}} \right|_{Y = \pm 1} = \pm \frac{{Q_{1}}}{2} $$

(52)

the stream function at the 1-order is obtained:

$$ \varPsi_{1} = \frac{{3R_{p} Q_{0}^{2} h_{X}}}{1120}\left({5Y - 11Y^{3} + 7Y^{5} - Y^{7}} \right) + \frac{1}{4}Q_{1} \left({3Y - Y^{3}} \right) $$

(53)

Substitute the 1-order stream function into the auxiliary condition, and after integration, the 1-order discharge is derived:

$$ Q_{1} = \mathop \int \nolimits_{0}^{\omega} \frac{{9R_{p} Q_{0}^{3} h_{X}}}{{280\omega h^{3}}}{\text{d}}X. $$

(54)

Similarly, for the 2-order:

$$ \begin{aligned} \frac{{\partial^{4} \varPsi_{2}}}{{\partial Y^{4}}} & = R_{p} \left[{h^{3} \frac{{\partial \varPsi_{0}}}{\partial Y}\frac{\partial}{\partial X}\left({\frac{1}{{h^{2}}}\frac{{\partial^{2} \varPsi_{1}}}{{\partial Y^{2}}}} \right) - 2h_{X} \frac{{\partial \varPsi_{1}}}{\partial Y}\frac{{\partial^{2} \varPsi_{0}}}{{\partial Y^{2}}} - h_{X} Y\frac{{\partial \varPsi_{1}}}{\partial Y}\frac{{\partial^{3} \varPsi_{0}}}{{\partial Y^{3}}} - h\frac{{\partial \varPsi_{0}}}{\partial X}\frac{{\partial^{3} \varPsi_{1}}}{{\partial Y^{3}}} - h\frac{{\partial \varPsi_{1}}}{\partial X}\frac{{\partial^{3} \varPsi_{0}}}{{\partial Y^{3}}}} \right] \\ & \quad-\,2\left[{\left({6h_{X}^{2} - 2hh_{XX}} \right)\frac{{\partial^{2} \varPsi_{0}}}{{\partial Y^{2}}} + \left({6h_{X}^{2} - hh_{XX}} \right)Y\frac{{\partial^{3} \varPsi_{0}}}{{\partial Y^{3}}}} \right] \\ \end{aligned} $$

(55)

Using the same approach, the solution of the 2-order stream function is:

$$ \begin{aligned} \varPsi_{2} & = \frac{{R_{p}^{2} Q_{0}}}{{3{,}449{,}600}}\left[h_{X}^{2} \left({2875Y - 8222Y^{3} + 8778Y^{5} - 4488Y^{7} + 1155Y^{9} - 98Y^{11}} \right)\right.\\ &\quad\left. - hh_{XX} \left({1213Y - 3279Y^{3} + 3234Y^{5} - 1518Y^{7} + 385Y^{9} - 35Y^{11}} \right) \right] + \frac{{3Q_{0}}}{40}\left({4h_{X}^{2} - hh_{XX}} \right)\left({Y - 2Y^{3} + Y^{5}} \right) \\ & \quad +\,\frac{{9R_{p} Q_{1} h_{X}}}{560}\left({5Y - 11Y^{3} + 7Y^{5} - Y^{7}} \right) + \frac{1}{4}Q_{2} \left({3Y - Y^{3}} \right) \\ \end{aligned} $$

(56)

Substitute the 2-order stream function into the auxiliary condition to find the solution of the 2-order discharge:

$$ Q_{2} = \mathop \int \nolimits_{0}^{\omega} \frac{{3Q_{0}^{2} h_{X}^{2}}}{{40\omega h^{3}}} - \frac{{Q_{0}^{2} hh_{XX}}}{{20\omega h^{3}}}{\text{d}}X + \mathop \int \nolimits_{0}^{\omega} \frac{{9R_{p} Q_{0}^{2} Q_{1} h_{X}}}{{140\omega h^{3}}}{\text{d}}X + \mathop \int \nolimits_{0}^{\omega} \frac{{13R_{p}^{2} Q_{0}^{4}}}{{26{,}950\omega h^{3}}}\left[{h_{X}^{2} - \frac{3}{4}hh_{XX}} \right]{\text{d}}X $$

(57)

Note that the Reynolds number R_p here is defined as the discharge of a fracture with constant aperture, divided by the kinematic viscosity.

1.2 A2 Perturbation Solution Under the Flow Rate Boundary Condition

The stream function at each order under the flow rate boundary condition can be derived using a similar approach to that discussed in the previous section, but with a different form of boundary condition, i.e. Equation (29). Following the same procedure, the stream functions up to the 2-order are obtained as shown in Eqs. (32)–(34). Expand the auxiliary condition by substituting Eq. (30) into Eq. (28) and retain up to the 2-order:

$$ \begin{aligned}\Delta P & = - \int\nolimits_{0}^{\omega} {\int\nolimits_{0}^{1} {\frac{1}{24\omega}\left[{\epsilon^{2} \frac{{\partial^{2}}}{{\partial X^{2}}}\left({\frac{1}{h}\frac{{\partial {\varPsi}_{0}}}{\partial Y}} \right) + \frac{1}{{h^{3}}}\frac{{\partial^{3} {\varPsi}_{0}}}{{\partial Y^{3}}} + \frac{\epsilon}{{h^{3}}}\frac{{\partial^{3} {\varPsi}_{1}}}{{\partial Y^{3}}} + \frac{{\epsilon^{2}}}{{h^{3}}}\frac{{\partial^{3} {\varPsi}_{2}}}{{\partial Y^{3}}}} \right]}} \\ & \quad + \frac{{R_{q}}}{24\omega}\left[ \frac{\epsilon}{{h^{2}}}\frac{{\partial {\varPsi}_{0}}}{\partial X}\frac{{\partial^{2} {\varPsi}_{0}}}{{\partial Y^{2}}} - \frac{\epsilon}{h}\frac{{\partial {\varPsi}_{0}}}{\partial Y}\frac{\partial}{\partial X}\left({\frac{1}{h}\frac{{\partial {\varPsi}_{0}}}{\partial Y}} \right) + \frac{{\epsilon^{2}}}{{h^{2}}}\frac{{\partial {\varPsi}_{0}}}{\partial X}\frac{{\partial^{2} {\varPsi}_{1}}}{{\partial Y^{2}}}\right.\\&\left.\quad -\, \frac{{\epsilon^{2}}}{h}\frac{{\partial {\varPsi}_{0}}}{\partial Y}\frac{\partial}{\partial X}\left({\frac{1}{h}\frac{{\partial {\varPsi}_{1}}}{\partial Y}} \right) + \frac{{\epsilon^{2}}}{{h^{2}}}\frac{{\partial {\varPsi}_{1}}}{\partial X}\frac{{\partial^{2} {\varPsi}_{0}}}{{\partial Y^{2}}} - \frac{{\epsilon^{2}}}{h}\frac{{\partial {\varPsi}_{1}}}{\partial Y}\frac{\partial}{\partial X}\left({\frac{1}{h}\frac{{\partial {\varPsi}_{0}}}{\partial Y}} \right) \right]{\text{d}}Y{\text{d}}X \\ \end{aligned} $$

(58)

Substituting the stream function solutions [Eqs. (32)–(34)] into Eq. (58), and after integration and rearrangement, the pressure difference up to the 2-order is:

$$ \Delta P = \mathop \int \nolimits_{0}^{\omega} \frac{1}{{8\omega h^{3}}} - \epsilon\frac{{9R_{q} h_{X}}}{{140\omega h^{3}}} - \epsilon^{2} \left[{\frac{{3h_{X}^{2} - 2hh_{XX}}}{{40\omega h^{3}}} + \frac{{13R_{q}^{2}}}{{13{,}475\omega h^{3}}}\left({2h_{X}^{2} - \frac{3}{2}hh_{XX}} \right)} \right]{\rm d}X $$

(59)

Note that the Reynolds number R_q here is defined as half of the given fracture discharge divided by the kinematic viscosity.

Appendix B: Flow in a Two-Dimensional Fracture with Periodic Aperture Variations

The proposed perturbation solutions were tested further for the fracture case with sinusoidal wall profiles, as shown in Fig. 9. The dimensionless half aperture is given by (Basha and El-Asmar 2003; Hasegawa and Izuchi 1983):

$$ h\left(x \right) = h_{\rm m} \left[{1 - a\cos \left({\frac{2\pi x}{l}} \right)} \right] $$

(60)

where h_m is the mean half aperture and α is the magnitude of the wall roughness (Zimmerman et al. 1991), the perturbation parameter here is given by $\epsilon$  = ω/l (ω = 4a). Substituting Eq. (60), together with Eq. (26), into the derived dimensionless PBC perturbation solution, one can obtain the dimensional solution up to 2-order (Gradshteyn and Ryzhik 2014) as

$$ Q = - \frac{{H_{\rm m}^{3}}}{12\mu}\frac{{\Delta p}}{l}Q_{0} \left[{1 - \epsilon^{2} \pi^{2} \frac{{1 - a^{2}}}{{16\left({2 + a^{2}} \right)}}\left({\frac{1}{5} + \frac{{26R_{p}^{2} Q_{0}^{2}}}{{13{,}475}}} \right)} \right] $$

(61)

$$ Q_{0} = \frac{{2\left({1 - a^{2}} \right)^{5/2}}}{{2 + a^{2}}} $$

(62)

where the Reynolds approximation, Eq. (62), can also be found in, e.g. Basha and El-Asmar (2003) and Zimmerman et al. (1991). Using the same procedure for the FBC condition, one can obtain:

$$ \Delta p = - \frac{12\mu lQ}{{H_{\rm m}^{3}}}\frac{{2 + a^{2}}}{{2\left({1 - a^{2}} \right)^{5/2}}}\left[{1 + \epsilon^{2} \pi^{2} \frac{1}{{32\left({1 - a^{2}} \right)^{3/2}}}\left({\frac{1}{5} + \frac{{104R_{q}^{2}}}{{13{,}475}}} \right)} \right] $$

(63)

Fig. 9

An example of a periodically varying fracture with sinusoidal wall profiles

Similar to Sect. 4.2, Eqs. (61) and (63) were compared with simulation results. In general, both solutions agree well with the simulation results as shown in Figs. 10a and 11a, for different $\epsilon$ in the range of 0.12–0.6, where Re and α/h_m are set at 2 and 0.3. The mean effective deviation of PBC and FBC solutions from the numerical solution is generally identical at 1.2%. For different Re in the range of 0.1–30, the results are shown in Figs. 10b and 11b, where $\epsilon$ and α/h_m are set at 0.4 and 0.2, respectively. The mean effective deviation of the PBC solution from the numerical solution is 1.6%, while the FBC solution has more accurate results with a mean effective deviation of 1.4%.

Fig. 10

Comparison of the discharge obtained from the PBC solution and the simulation results. The plots show the effects of the perturbation parameter $\epsilon$ and the Reynolds number Re; a, Q at different $\epsilon$, and b, Q at different Re, where Q is the volumetric flow rate from different solutions normalized by Q_m

Fig. 11

Comparison of the pressure difference obtained from the FBC solution and simulation results showing the effects of the perturbation parameter $\epsilon$ and Reynolds number Re; a, ∆P at different $\epsilon$, and b, ∆P at different Re. ∆P is the pressure difference from different solutions normalized by ∆P_m