OJFDOpen Journal of Fluid Dynamics2165-3852Scientific Research Publishing10.4236/ojfd.2015.51007OJFD-54351ArticlesPhysics&Mathematics A Short Note on Self-Similar Solution to Unconfined Flow in an Aquifer with Accretion riehPistiner1*Unit for Hydrocarbon Pollution Prevention, Ministry of the Environmental Protection, Haifa, Israel* E-mail:ariehpistiner@gmail.com12022015050151576 February 2015accepted 27 February 2 March 2015© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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.

Boussinesq Equation Self-Similar Solution Hypergeometric Function
1. Introduction

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

is analyzed. In the above equation, is the hydraulic head in the aquifer; and are the normalized position and time coordinates, respectively (i.e.,), and is a time and position dependent function, representing the rain intensity distribution imposed on the aquifer that is given by

where is the rain intensity.

We consider a situation in which the water head distribution in a body of water, lying in the porous medium, at time, is unknown. Initially, at time, the water level on the inlet face of the aquifer suddenly drops, according to the following power law

where is a scaling parameter of the porous medium, and is a negative constant to be determined hereafter. This boundary condition would correspond to an influent stream that supplies water to the aquifer. In addition to this, rainwater begins to penetrate into the aquifer according to (2) and adds rainwater to the saturated water body. As a response to that, water flux at the inlet face is created and possesses the following form

where is a dimensionless inlet flux parameter and is a negative constant to be determined hereafter.

The downstream boundary conditions for the saturated water body on the moving boundary is given by

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

where is the dimensionless flux parameter of the moving boundary, where the area in the domain, is supposed to be a non-saturated zone.

In general, the problem must be solved for specified initial conditions imposed upon. However, as will be shown below, the long-time profile of is independent of the precise form of the initial condition, which governs the hydraulic head at early stages only. However, the long-time profile will be investigated in the next section by the similarity method.

2. Self-Similar Model

We will now refer to the circumstances in which the hydraulic head in the aquifer achieves a certain asymptotic, and is described by a single independent self-similar variable  :

where is a similarity positive function, and are parameters to be determined later. Substituting (2), (4a), (4b) in (1) and after certain mathematical manipulation we obtain

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 is an integration constant.

3. Method of Solution

The similarity function may be defined via a new independent function as follows

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 respectively

and

Differentiating with respect to, using (11) and (12) and selecting a value for the rain intensity, i.e., , we obtain an Abel-type equation of the second kind 

We now define a new function as follows 

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

We now define the function and apply the Riccati transformation  , as follows

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

where

We now define as follows

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  , and and are constants to be determined below. Using the properties of the hypergeometric series, we obtain from (21) and (21a), (21b) the expression for

where the hypergeometric functions and are given by

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 in Equations (15) and (25) and combined with (26), the functions and are given by

and

where can be easily obtained from (19)

The inlet face position, i.e., is obtained from (27a) as follows

It can be observed from (23) that the requirement appearing in (29) can be achieved only if. In accordance with the above, we obtained the value for by equating (21) to zero at, i.e.

and in accordance with (19a), the constant exists in the following range

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

From the above, it can be observed that must be negative

The boundary condition (3c), imposed on the moving front, is determined by equating (27b) to zero by introducing (see (23)). Hence, the downstream parameter (i.e.) is obtained after introducing into (27a). Using the property of the hypergeometric functions (i.e.,) we obtain the downstream parameter

where

In accordance with the above (i.e.,), the denominator of Equation (34) must obey the following inequality

which automatically shows that

and it is in accordance with the range for the parameter in (31).

The behavior of as approaches zero can be obtained from (27b) and is given by

where is a positive constant which is equal to.

The flux parameter for the saturated zone, which appear in (3b), can be obtained by using (8)-(13) as follow

The water flux parameter on the moving boundary, that serve as water source for the unsaturated zone where (i.e., see (37)), can be obtained from (7)

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. Solving (41) implicitly and using (30) and (34), we obtain the value for

4. Short Discussion

Figure 1 illustrates the evolution of the water head in the aquifer for three time intervals.