Boussinesq Approach(Part 1)

7/30/2019 Boussinesq Approach(Part 1)

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An extention of the Standard Boussnesq Approach

Adam Szymanski

adi [email protected]

Environment Protection Program

Technical Report

2014

Abstract

We consider a closed-form solution to the nonlinear two-point boundary value problem,

vx(x, z )(dh

dx)2 + k

dh

dx + vx(x, z ) = 0, (1)

h(0) = H S, (2)

h(L) = H E, (3)

α =

V c(0, H S)

V c(0, H E),

(4)

where: x ∈ [0, L], α ∈ [0, 1], H E ≤ z ≤ H S , z = h(x), V c(x, z ) = [vx(x, z )2 + vz(x, z )2]1/2. We define the functionsvx(x, z ) and vz(x, z ) as follows,

vx(x, h(x)) = −0.5kb(a + bx)−1/2, (5)

0 ≤ vx(x, h(x)) ≤ 0.5k, (6)

vz(x, h(x)) = 0.5k[−1 + [1 − 4(vx(x, h(x))

k )2]1/2], (7)

where: a > 0, b < 0 are unknown real constants, and k , L, H S, H E are known real constants greater than zero.The boundary value problem (1 − 7) presents an extension to the Standard Boussinesq Approach (SBA) and

describes the groundwater flow on the free-surface of an unconfined aquifer. We call it the Extended BoussinesqApproach (EBA).

1 Introduction

In the report we consider the steady state flow in a homogeneous, isotropic, unconfined aquifer represented by a 2-vertical slab of length L, bounded by two-constant-head boundary conditions H S and H E . Relation (1) describes texact nonlinear differential equation governing the horizontal Darcy velocity component vx(x, h(x)) of the groundwatflow at the free-surface elevation, where the function vx(x, h(x))) is defined by (5), and the unknown function h(presents the geometrical location of the free-surface elevation. The coefficient k denotes the spatially constant saturathydraulic conductivity. Relation (6) shows that the horizontal Darcy velocity component should be bounded. Tphysical explanation of (6) is presented by Szymanski (2011). The existence of the bottom of an aquifer is described the dimensionless coefficient α. In terms of (1−7) the SBA based on the Dupuit assumption is treated as an approximati

for the regional gradient m = (H E−H S)L approaching zero as the limit. The detailed description of SBA is presented b

Bear (1979). However, it is not always clear when the SBA will faithfully reproduce the physical phenomena that o

attempts to model. Thus, in this report we follow the line presented by Baiocchi and Capelo (1984) and develop suan extension of SBA that allows the regional gradient to be considered as a small, finite quantity. Formally, it meathat the vertical Darcy velocity component vz(x, h(x)) does always exist in contrary to the SBA, where vz(x, h(x)) ≡Thus, the EBA is physically consistent. The present report is organized as follows: in section 2 we rediscussed some basissues related to the SBA. In section 3 we demonstrate the closed-form solution of the boundary value problem proposeand in section 4 an application of the Weierstrass Approximation Theorem in hydrology is shown. Finally, in sectionconclusions are drawn.

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2 Standard Boussinesq Approach

Let us consider the linearized version of the two-point boundary value problem (1 − 7) in the following form,

kdh

dx + vx(x, z) = 0, (

h(0) = H S , (

h(L) = H E , (1

α =

H E H S , (1

where: x ∈ [0, L], H E ≤ H S , z = h(x), and the function vx(x, z) and vz(x, z) are given by,

vx(x, h(x)) = −0.5kb(a + bx)−1/2, (1

vz(x, h(x)) ≡ 0, (1

where: a > 0, b < 0 are unknown real constants. The solution of the problem (8 − 13) can be find in a closed form,

h(x) = (a + bx)1/2, (1

where: a = H 2S , and b = (H 2

E−H 2

S)

L . Note in passing that (14) is the positive solution of (8 − 13). By the positive solutiof (8 − 13) we understand a function h(x) which is positive on 0 ≤ x ≤ L and satisfies the differential equation (8) an

boundary conditions (9, 10, 11). Additionally, d2

h(x)dx2 < 0 for x ∈ [0, L] and the graph of (14) is convex upwards. We refer

the problem (8 − 13) as the SBA. The solution (14) is well known in groundwater hydrology. For the detailed descriptioof the practical importance of the SBA the reader is refereed to Troch et al. (2013). From the heuristic point of view, tSBA is assumed to be valid in mildly sloping aquifers characterized by nearly horizontal flow conditions. This assertionbased on the flow being slowly-varying in the x-direction, and the pressure being nearly hydrostatic. However, it is clethat the regional gradient m should be treated as an arbitrarily small quantity approaching zero as a limit to make thSBA consistent. Thus, the relations (14), and (12) should be considered as an approximation for describing the flow othe free-surface of groundwater. Unfortunately, (14) has recently been used for checking MODFLOW-NWT model (sNiswonger et al.(2011)).

3 Extended Boussinesq Approach

Let us now describe the basic difference between the SBA and the EBA. It is easy to observe that the solution of SBformally exists for an arbitrary value of the regional gradient m ≤ 0. In the case of EBA there exists an admissible set values for the regional gradient that depends on the geometrical characteristics of an aquifer. It means that if a regiongradient considered for predicting the flow in an aquifer does not belong to the above-mentioned set, the solution of EBdoes not exist. From the physical point of view the following assumptions are required:1. the regional gradient m is considered to be finite and sufficiently small i.e, |m| 1,2. the value z = 0, which is treated as a horizontal impervious bottom of an aquifer, is only used as a reference datum fothe definition of potential energy, and3. the components of the infiltration/exfiltration velocity at the free-surface elevation have been assumed to be equal zero.The assumption |m| 1 is related to the relation (5), because the considered boundary value problem (1 − 7) should seen as a correction of the SBA. The physical meaning of the second assumption can be explained in the following waWe are interested to develop the model using parameters related to flow characteristics at the free-surface elevation.

this way we only need an information that the bottom of an aquifer does exist, but we are not obliged to consider texact value for the parameter H S (a characteristic depth of an aquifer). For the description of the bottom of an aquifthe idea of a kinematic deepness is introduced. We call the coefficient α the kinematic deepness of an aquifer (cf., (11From the formal point of view α can also be considered as a parameter describing the kinematic characteristics at thfree-surface. Using the Fundamental Theorem of Calculus the solution of (1) takes the form

h(x) = 1

2

x0

[k/vx (ξ, h(ξ ))]

−1 +

1 − 4(vx (ξ, h(ξ )) /k)2

dξ + H S , (1

where an integration constant has been determined by the boundary condition (2). From (5) follows that vx(x,h(x))dx >

(i.e., vx(x, h(x)) is a strictly increasing function). Furthermore, we can show that d2h(x)dx2 < 0 for x ∈ [0, L]. Thus, t

graph of the free-surface elevation (15) is convex upwards.

2



3.1 Solution of EBA

Inserting (5) into (15) we obtain,

h(x) = 2

3b−2[(a + bx)3/2 − a3/2 − (a − b2 + bx)3/2 + (a − b2)3/2] + H S . (1

Next, using (16) and (3) we obtain the first equation needed for determining the unknown coefficients a and b. The seconone is obtained by means of (4). Finally, we have

F 1(c1, f 1) = 0, (1

F 2(c1, f 1) = 0, (1

where

F 1(c1, f 1) = (c1 + f 1)3/2 − c3/21 − (c1 + f 1 − f 21 )3/2 + (c1 − f 21 )3/2 − (3/2)mf 21 , (1

F 2(c1, f 1) = α2{1 − [1 − f 21 /(c1 + f 1)]1/2} − 1 + [1 − f 21 /c1]1/2, (2

where: c1 = aL2 , and f 1 = b

L .The parameters of the nonlinear system of equations (17, 18) are m and α. The unknowns are c1 and f 1. Of course, ocan use the Newton-Raphson method to find the solution of (17, 18) with appropriate accuracy. Unfortunately, it is nthe most efficient method. Thus, we want to present here the new technique which is more effective.

3.2 1-D bracketing search method

The basic idea of our method is to reduce the dimension of the system (17 , 18), so that the 1-D bracketing search methcan be used. From the relation (18) follows that,

D = f 1/c1 = (C − 1 + M 2)/(1 − C ), (2

whereM 2 = f 21 /c1, (2

C = [(α2 − 1 + (1 − M 2)1/2)/α2]2. (2

Using the relation (17) one obtains,

1 = (1.5m)2D2M 2/[(1 + D)3/2 − 1 − (1 + D − M 2)3/2 + (1 − M 2)3/2]2, (2

where0 < M 2 < (1 − H ), (2

H = (1 − α2)2. (2

The equation (24) can be considered as the 1-D form of the system (17, 18). Indeed, having M 2 as a solution of tequation (24) we obtain the equivalent form of the system (17, 18) as follows,

f 1 = D c1, (2

f 21 = M 2 c1. (2

Of course, for solving (24) the 1-D bracketing search method should be used.

3.3 Solution of EBA for α = 1

The solution of the boundary value problem (1 - 7) for α = 1 can be find in a closed form. The free-surface elevationgiven by

h(x) = mx + H S , (2

and the Darcy velocity components are calculated as follows,

vx(x, h(x)) = −km

1 + m2, (3

vz(x, h(x)) = −km2

1 + m2. (3

3



4 Weierstrass Approximation

Let us now assume that the appropriate approximation of the free-surface elevation was obtained using GPR methoIn this case, combining the relations (29), (30) and (31) with the well-known Weierstrass Approximation Theorem oobtains the numerical method for calculating the velocity field on the free-surface of groundwater.

5 Summary and Conclusions

The Standard Boussnesq Approach has been extended to incorporate the vertical flow conditions induced by the existen

of the free-surface elevation in unconfined aquifers. The proposed model is suitable for predicting the geometrical locatiof the free-surface elevation, and both velocity components at the free-surface. It is a nonlinear model that is based othe flow characteristics at the free-surface elevation. In our method there is no need to use any linearization proceduwith respect to the boundary conditions at the free-surface, and the closed-form solution is obtained. The modelcharacterized by two parameters, namely; regional gradient m and the so-called kinematic deepness of an aquifer denotby α. Recently, we have used the method presented above for checking the FEFLOW model (see Szymanski (2014)).

6 References

Baiocchi, C., Capelo, A., 1984. Variational and Quasivariational Inequalities Application to Free Boundary ProblemJohn Wiley, New York.

Bear, J., 1979. Hydraulics of Groundwater, McGraw-Hill, New York.

Niswonger, R.G., Panday, Sorab, and Ibaraki, Motomu, 2011. MODFLOW-NWT, A Newton formulation for MODFLOW2005: U.S. Geological Survey Techniques and Methods 6A37, 44 p.

Szymanski, A., 2011. Darcy velocity on the free surface, EPP technical report.

Szymanski, A., 2014. Our Problems Connected With FEFLOW, EPP technical report.

Troch, P. A., et al. (2013), The importance of hydraulic groundwater theory in catchment hydrology: The legacy Wilfried Brutsaert and Jean-Yves Parlange, Water Resour. Res., 49, 50995116, doi: 10.1002/wrcr.20407.

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Boussinesq Approach(Part 1)