# On the movement of an LNAPL lens on the water table

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## Abstract

The penetration of light nonaqueous phase liquids (LNAPLs) in quantities that lead to an accumulation in the form of a lens above the water table is considered. First, the three-phase vertical gravity-capillary equilibrium of water, NAPL, and air above the water table is specified. The hypothesis of ‘vertical equilibrium phase distribution’ is used to derive averaged asymptotic equations describing NAPL flow as a thin lens floating above the water table. Some problems of unsteady NAPL lens movement and the development of a NAPL mound, spreading along an inclined or horizontal phreatic surface are discussed and the analytical solutions are obtained.

## Key words

nonaqueous phase liquids unsaturated zone three-phase flow vertical equilibrium LNAPL lens analytical solutions## Nomenclature

*a*constant in Equation (55)

*A*constant in Equation (7)

*C*constant in Equation (25)

- e
_{0}e_{1} constants in Equation (72) (75)

*g*acceleration due to gravity

- F(xyt)
form of the moving boundary of the lens

- i
_{0} vector of the inclination of the water table

*k*absolute permeability of the soil

*k*_{rn},*k*_{rw}relative permeabilities of NAPL an water

*K*water saturated hydraulic conductivity of the soil

*L*initial length of the lens

*m*exponent in the equation for NAPL relative permeability

*P*_{a},*P*_{n},*P*_{w}air, NAPL water pressures

*P*_{caw},*P*_{can},*P*_{cnw}air-water air-NAPL and NAPL-water capillary pressures

*P*_{c}^{*}reduced capillary pressure function

*P*_{c0}characteristic capillary pressure

**Q**_{n}**Q**_{w}NAPL and water flowrates (2-d vectors)

*Q*intensity of the source.

*r*radial coordinate √x

^{2}+ y^{2}*R(u)*function defined by Equation (61)

*S*_{a},*S*_{n},*S*_{w}air and NAPL water saturations

*S*_{nr},*S*_{wr}residual NAPL, water saturations

*S*_{l}total liquid saturation, S

_{n}+ S_{w}- s
effective saturation (Equation (2))

*t*time

*T*integrated NAPL relative permeability

*T*_{w}^{*}integrated water relative permeability in the unsaturated zone

*T*_{0}constant in Equation (45)

*u*NAPL content per unit of the water table surface

*u*_{w}^{*}water content per unit of the water table surface within the lens

*u*dimensionless NAPL content per unit of the water table surface

*u*_{J}the value of

*u*at a jump- u
_{0}, u_{1} constants in Equations (34) and (35)

- U
_{0}, U_{1} total NAPL quantity in a mound

- V
_{0} characteristic velocity

- x, y, z
cartesian coordinates

- z
_{1}, z_{2} upper boundaries of the two-phase (NAPL-water) three-phase zones

- z
_{*} lower boundary of undisturbed unsaturated zone

- z′
elevation of the upper boundary of a point above the moving water table

- α
constant in van Genuchten equation (8)

- β
_{an}, β_{aw} scaling coefficients in Equation (2)

- β
_{0}, β_{1} constants in Equations (34) and (35)

- γ
_{aw}, γ_{ij} surface tensions

- δ
Dirac delta-function

- Δρ
density difference (ϱ

_{w}− ϱ_{n})- ζ
_{1} thickness of the two-phase zone (NAPL water), within the lens

- ζ
_{2} total thickness of the lens

- ζ
^{*} total thickness of the zone, disturbed by the presence of the lens

- η
elevation of the moving water table

- η
_{0} initial elevation of the water table

- η
_{1} deviation of the water table η − η

_{0}- λ
exponent in Equation (66)

- μ
_{n}, μ_{w} NAPL water viscosities

- v
constant in tht van Genuchten equation (8)

- ξ
Boltzmann variable (

*r/√at*)- ρ
_{n} ρ

_{w}NAPL water densities- τ(u)
dimensionless integrated NAPL relative permeability

- φ
soil porosity

- Φ
_{n}, Φ_{w} NAPL water potentials

- Φ(
*U*) function defined by Equation (17)

- ϕ(u)
dimensionless NAPL potential

- χ
characteristic thickness Δϱ/αϱ

_{n}- Ω
_{t} domain occupied by the lens in the

*x, y*-plane- Σ
_{t} moving boundary of the lens

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