# Dynamical Effects of Variations of Conduit Area in a Karst Conduit

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## Abstract

Variations of the cross-sectional area of a conduit embedded within a porous medium have two effects on flow and transport. First, the hydraulic gradient required to convey a specified flux of water is increased. Second, local exchanges of water and pollutants between the conduit and surrounding porous medium are induced, forming a hyporheic zone. Both these effects are quantified in the case that the variations are periodic and the fractional change of conduit flux induced by the variations is small. The resistance to flow increases dramatically as the variations increase in amplitude. The volume of the hyporheic zone is proportional to the square of the wavelength of the variations. If the wavelength is large, the volume of the hyporheic zone can be far larger than that of the conduit, permitting the sequestration of contaminants for long periods of time.

## Keywords

Karst Conduit Hyporheic zone Flow Transport## List of symbols

*a*Conduit radius; a function of axial position (L)

- \(a_\mathrm{c}\)
Mean conduit radius

- \(a_\mathrm{eff}\)
The effective conduit radius, given by (27)

*f*Friction factor (–)

*g*Local acceleration of gravity (L/T\(^{2}\))

*h*Conduit head (L)

- \(h_\mathrm{m}\)
Matrix head

*i*- \(K_{0}, K_{1}\)
Kelvin functions

- \(i_\mathrm{c}\)
Regional hydraulic gradient for a smooth conduit; see (19)

*j*A summation index

*K*Hydraulic conductivity (L/T)

*n**Q*Conduit flux (L\(^{3}\)/T)

- \(Q_\mathrm{c}\)
Mean conduit flux

*q*Specific flux from matrix to conduit; see (13) (L\(^{2}\)/T)

- \(\bar{{q}}\)
Mean specific flux; see (59)

*r*Dimensional cylindrical radial coordinate (L)

- \({\hat{\mathbf{r}}}\)
Unit vector in the cylindrical radial direction

**u**Velocity of flow within the matrix (L/T)

*z*Dimensional axial position (L)

- \({ \hat{\mathbf{z}}}\)
Unit vector in the axial direction

*Z*Length of a single oscillation of the conduit; see (2) (L)

- \(Z_{Q}\)
Axial distance over which the exchange flux equals the conduit flux, quantified by (61)

- \(\alpha \)
Dimensionless conduit radius; see (6). This function is even in \(\zeta \) (–)

- \(\alpha _{1}, \alpha _{2}\)
- \(\alpha _\mathrm{eff}\)
The effective dimensionless conduit radius, given by (27)

- \(\alpha _{\mathrm{min}}\)
Minimum value of dimensionless conduit radius

- \(\alpha _{\mathrm{max}}\)
Maximum value of dimensionless conduit radius, given by (9)

- \(\beta \)
A function of \(\zeta \) given by (39) (–)

- \(\gamma \)
Dimensionless regional hydraulic gradient; see (23) (–)

- \(\zeta \)
Dimensionless axial position; see (3) (–)

- \(\eta \)
Dimensionless conduit head; see (24). This function is odd in \(\zeta \) (–)

- \(\eta _\mathrm{i}\)
Dimensionless matrix head in the inner domain, given by (37)

- \(\eta _\mathrm{m}\)
Dimensionless matrix head; see (29)

- \(\eta _\mathrm{mp}\)
Periodic portion of the dimensionless matrix head; see (30)

- \(\eta _\mathrm{o}\)
Periodic portion of the matrix head in the outer domain, given by (36)

- \(\eta _\mathrm{p}\)
Periodic part of the dimensionless conduit head; see (24)

- \(\eta _{1}, \eta _{2}\)
- \(\eta _{j}\)
- \(\kappa \)
Dimensionless hydraulic conductivity, given by (54) (–)

- \(\lambda \)
A prescribed (dimensionless) wavenumber (–)

- \(\rho \)
Dimensionless cylindrical radius; see (16) (–)

- \(\rho _\mathrm{c}\)
\( \rho - \rho _{0}\)

- \(\rho _\mathrm{h}\)
The mean radius of the hyporheic zone; see (51)

- \(\rho _{\mathrm{max}}\)
Maximum radius of the hyporheic zone; see (52)

- \(\rho _{0}\)
Maximum radius of the conduit boundary; see (34)

- \(\psi \)
Dimensionless stream function for matrix flow, given by (48); this function is even in \(\zeta \) (–)

- \(\bar{{\psi }}\)
Mean value of the stream function on the domain boundary \(\rho =\lambda \alpha _{\mathrm{\max }} \)

- \(\psi _{\mathrm{\max }} \)
Maximum value of \(\psi \), given by (49)

- \(\Delta \psi \)
Deviation of the stream function on the conduit boundary from the mean; see (56)

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