Boussinesq Approach (part 2)

Analytical solution of the groundwater flow in an infinitely deep unconfined aquifer

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Environment Protection Program ADAM SZYMAŃSKI Kierzkowo 22A, 84-210 Choczewo

Poland [email protected] [email protected]

Analytical Solution Of The Groundwater Flow In An Infinitely Deep Unconfined Aquifer

Figure 1: Schematic illustration of the model geometry

A model for simulating the groundwater flow in unconfined aquifers is presented. The proposed methodology allows for the closed-form solution of the nonlinear, elliptic, free-surface, boundary value problem. Comparisons with the standard Boussinesq approach based on the Dupuit assumption, the Extended Boussinesq Approach developed by Szymański, and the closed-form solution proposed by Schmitz and Edenhofer are also presented. TECHNICAL REPORT 2009

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Analytical solution of the groundwater flow in an infinitely deep unconfined aquifer

1. Introduction Let us consider a 2-D, steady, groundwater flow in a homogeneous, isotropic, unconfined aquifer Ω = {(x, z): 0 < x < L and 0 < z < h(x)}, where z = h(x) is the unknown freesurface elevation, h(0) = HS, h(L) = HE, HS > HE, and (x, z) represent the spatial coordinates with z the vertical dimension (see Figure 1). In terms of the model proposed the value z = 0 is treated as an artificial boundary and is only used as a reference datum for the definition of the potential energy. Assuming that the free-surface does not interact with the land surface, and the regional gradient m = (HE – HS)/L should be treated as; - ∞ < m ≤ 0, we can consider Ω as a model of an infinitely deep aquifer. 2. A model of the groundwater flow in unconfined aquifers Using the Extended Boussinesq Approach for α = 1 (see Szymański

2009

) one obtains the

following boundary value problem,

vx(x,z) [dh(x)/dx]2 + k dh(x)/dx + vx(x,z) = 0

at z = h(x), 0 ≤ x ≤ L

h(0) = HS, h(L) = HE

HE ≤ z ≤ HS

vx(x,h(x)) = - k a

0≤ x ≤L

(Ia.3)

vz(x,h(x)) = vx(x,h(x)) dh(x)/dx

0 ≤ x ≤ L,

(Ia.4)

(Ia.1) (Ia.2a,b)

where a < 0 is unknown, real constant. We refer to this model as the Local Subsurface Solution (LSS). The solution of the problem (Ia) is presented in Appendix A. In terms of the piezometric head function Φ(x,z) the problem (Ia) can be rewritten as follows, ∂Φ/∂z = ( ∂Φ/∂z)2 + (∂Φ/∂x)2

z = h(x)

(Ib.1)

Φ(x,h(x)) = h(x)

0≤x ≤L

(Ib.2)

∂2Φ/∂x2 + ∂2Φ/∂z2 = 0

in Ω

(Ib.3)

Φ(0,z) = Az + (1 – A) HS

0 ≤ z ≤ HS

(Ib.4)

Φ(L,z) = Az + (1 – A) HE

0 ≤ z ≤ HE

(Ib.5)

∂Φ(x,0)/∂z = A

0≤x ≤L

(Ib.6)

→ vs = - K grad (Φ),

kxz = kzx = 0, kxx = kzz = k > 0,

(Ib.7)

where A = m2(1 + m2)-1. Equations (Ib.1,2) represent the kinematic and dynamic boundary conditions at the free-surface, respectively. Equation (Ib.3) describes the continuity of the flow. Conditions (Ib.4,5) are used to defined the piezometric head on the Dirichlet boundary segments of the domain Ω. Equation (Ib.6) is considered here in terms of the nonhomogeneous Neumann boundary condition. The last relation (Ib.7) is Darcy's law, where K represents the hydraulic

Analytical solution of the groundwater flow in an infinitely deep unconfined aquifer

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conductivity tensor, symmetric with components on the diagonal not equal to 0.

3. The solution of the boundary value problem The solution of the problem (I.b) is given by the following function, Φ(x,z) = A z + (1 – A) h(x)

0 ≤ x ≤ L and 0 ≤ z ≤ h(x),

(2)

where the free-surface elevation h(x) = m x + HS. The velocity components of the flow are defined by relations, vx(x,z) = - k ∂Φ/∂x = - k m (1 + m2)-1 > 0

0 ≤ x ≤ L and 0 ≤ z ≤ h(x)

(3)

vz(x,z) = - k ∂Φ/∂z = - k m2(1 + m2)-1 < 0

0 ≤ x ≤ L and 0 ≤ z ≤ h(x).

(4)

From the relation Φ(x,z) = z + p(x,z) γ-1 one obtains the dynamic pressure in the following form, p(x,z) = γ (1 – A) [h(x) – z]

0 ≤ x ≤ L and 0 ≤ z ≤ h(x),

(5)

where γ denotes the specific weight of the water. Assuming that - ∞ < m ≤ 0 from (4) and (3) one obtains the following inequalities, 0 ≤ │ vz(x,z) │ < k

0 ≤ x ≤ L and 0 ≤ z ≤ h(x),

(6)

0 ≤ vx(x,z) ≤ 0.5k

0 ≤ x ≤ L and 0 ≤ z ≤ h(x).

(7)

vx(x,z) ~ - k m

0 ≤ x ≤ L and 0 ≤ z ≤ h(x)

(8)

vz(x,z) ~ - k m2 ≈ 0

0 ≤ x ≤ L and 0 ≤ z ≤ h(x), and

(9)

p(x,z) ~ γ [h(x) – z]

0 ≤ x ≤ L and 0 ≤ z ≤ h(x)

Furthermore, from │ m │ << 1 follows that;

(10)

which is the so-called Dupuit assumption, i.e., the pressure is nearly hydrostatic.

4. Comparison with existing results 4.1 Comparison of LSS with the Standard Boussinesq Approach (SBA)

Assuming that the regional gradient m should be treated as an arbitrarily small quantity

approaching zero as a limit, the problem mentioned in the previous section can be approximated by the Boussinesq equation (cf., Szymański 2009). Its solution is given by the function hB(x) = (C + D x)1/2, where C = HS2 and D = (HE2 – HS2)/L. Next, putting HS → ∞ one obtains that; hB(x) ~ m x + HS, and vxB(x) = - k dhB(x)/dx ~ - k m. Therefore, if a characteristic depth of an aquifer tends to infinity, the geometrical location of the free-surface elevation and the horizontal velocity component defined by LSS are well described by the closed-form solution of the Boussinesq equation.

Analytical solution of the groundwater flow in an infinitely deep unconfined aquifer

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4.2 Comparison of LSS with the Extended Boussinesq Approach (EBA)

We

present the numerical examples showing the behavior of EBA for α → 1. The parameters used are as follows; α = 0.99, m = - 0.95, and α = 0.999, m = - 0.95.

For α → 1 the LSS is very well approximated by the EBA (cf., Szymański 2009).

Analytical solution of the groundwater flow in an infinitely deep unconfined aquifer

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4.3 Comparison of LSS with Schmitz and Edenhofer 2000(SE)

For

the comparison mentioned the interval [3000, 3050], and the transformations introduced by Szymański 2009 have been applied.

The maximal relative error of the horizontal velocity component is approximately equal to 1.8%.

The maximal relative error of the vertical velocity component is approximately equal to 3.8%.

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Analytical solution of the groundwater flow in an infinitely deep unconfined aquifer

5. Summary and Conclusions A method, called the Local Subsurface Solution (LSS), was introduced for predicting the flow in infinitely deep, unconfined aquifers. The version presented here is an extension of the Extended Boussinesq Approach (EBA) for α = 1. In the method proposed, there is no need to use any linearization procedure with respect to the boundary conditions at the free-surface, and the closed-form solution is obtained. The LSS is characterized by the regional gradient m, where - ∞ < m ≤ 0. Examples are presented to examine the accuracy of LSS against the Standard Boussinesq Approach, the Extended Boussinesq Approach, and the closed-form solution developed by Schmitz and Edenhofer 2000. The LSS can be used as a reasonable approximation to the model suggested by Schmitz and Edenhofer 2000. The solution of LSS does always exist, implying that the velocity components of the regional flow are bounded; 0 ≤ vx(x,h(x)) ≤ 0.5k and 0 ≤ |vz(x,h(x))| < k, where x ∈ [0, L]. 6. References Schmitz, G.H., Edenhofer, J., 2000. Exact closed-form solution of the two-dimensional Laplace equation for steady groundwater flow with nonlinearized free surface boundary condition, Water Resour. Res., 36(7), 1975-1980. Szymański, A., 2009. On a generalization of the Boussinesq approach for modeling the flow in unconfined aquifers, EPP technical report, (http://www.pdfcoke.com/doc/14793154/Boussinesq-approach.)

7. Appendix A “The solution of the problem (Ia)” The solution of the equation (Ia.1) takes the form, x h(x) = 0.5∫ [k/vx(ξ,h(ξ))] {-1 + [1 – 4 (vx(ξ,h(ξ))/k)2]1/2} dξ + HS

0 ≤ x ≤ L,

(A1)

0

where an integration constant has been determined by the boundary condition (Ia.2a) and 0 ≤ vx(x,h(x)) ≤ 0.5k. Inserting (Ia.3) and (Ia.2b) into (A1) one obtains the following equation for the unknown constant a, L

HE = - 0.5∫ a-1 {-1 + [1 – 4 a2]1/2} dξ + HS..

(A2)

0

From (A2) one obtains that, a = m (1 + m2)-1. Next, from (A1) follows that; h(x) = m x + HS..

(A3)