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In this study we refer to a non-steady state, one-dimensional (on the x-axis), unconfined and saturated flow in an aquifer, described by the Boussinesq equation, combined with accretion. In accordance with the above, the moving boundary of the saturated area (toward x → +∝) serves as a horizontal water flux source to the unsaturated area. As time advances, the horizontally saturated zone, lying on the x-axis, becomes wider. A self-similar solution is derived that, after some mathematical manipulation, it is described in terms of Hypergeometric functions. The long-time behaviors of the solution describe the situation at which the water flux, that penetrates horizontally to the non-saturated zone, is equal to the water flux entering into the saturated zone.

In this study, the equation describing unsteady flow in a semi-infinite phreatic aquifer with accretion [

is analyzed. In the above equation,

where

We consider a situation in which the water head distribution in a body of water, lying in the porous medium, at time

where

where

The downstream boundary conditions for the saturated water body on the moving boundary

and the downstream water flux on the moving boundary is given by

where

In general, the problem must be solved for specified initial conditions imposed upon

We will now refer to the circumstances in which the hydraulic head in the aquifer

where

where

In this study we refer to the particular case

Introducing (5a) into (5b), we obtain

Substituting (6b) in (5) and integration we obtain

where

The similarity function

Introducing (8) into (7) combined to yield

Define a new dependent variable

Introducing (10) into (9) we obtain

where

We now define two new functions,

and

Differentiating

We now define a new function

The substitution of (15) in (14) leads to a Riccati equation with respect to

We now define the function

Substituting (17) in (16), we obtain the following linear ODE

where

We now define

which is valid in the domain

The substitution of (19) in (18) then yields the hypergeometric equation

which possesses the general solution

In the above

are expressed via hypergeometric functions [

where the hypergeometric functions

Substituting (19) into (17) using (18a) we obtain

The introduction of (21) and (22) into (23) we obtain the final solution for

Substituting (10) in (13) we obtain

The introduction of (8) and (10) into (12) gives the following expression for

Using the expression for

and

where

The inlet face position, i.e.

It can be observed from (23) that the requirement appearing in (29) can be achieved only if

and in accordance with (19a), the constant

Substituting (29) in (27b) yields the boundary condition parameter defined in (3a)

From the above, it can be observed that

The boundary condition (3c), imposed on the moving front, is determined by equating (27b) to zero by introducing

where

In accordance with the above (i.e.,

which automatically shows that

and it is in accordance with the range for the parameter

The behavior of

where

The flux parameter

The water flux parameter on the moving boundary, that serve as water source for the unsaturated zone where

We will now assume that at the long-time limit, the water flux exchange between the inlet face and the moving boundary (i.e., the water flux to the saturated zone and the water flux to the unsaturated zone) reach some equilibrium. As a result, an additional condition can be formulate as follow

The introduction of (35) into (39), using (38) and (40) we obtain the following equilibrium equation

which is independent on the value of

It can be observed that the downstream branch of the water head profiles is characterized by a steep transition to zero (almost infinite gradient) as can be expected from (38) and (39) (i.e., the water flux on the boundary between the saturated zone and the non-saturated zone possess finite value, as can be observed from (3b) and (3d)).

In general, the solution here developed describes the evolution of the saturated zone, stem from penetration of rainwater and an influent stream from the inlet face. The developed analytical solution can be most useful for verifying numerical solutions involving groundwater transport in an unconfined aquifer.