Marinoschi, Gabriela; Morega, Alexandru

On the boundary value problem of water infiltration into a nonhomogeneousunsaturated soil

The paper deals with the resolution of an optimization problem related to the nonlinear water infiltration into astratified soil with two layers, for the hydraulic model introduced by Broadbridge and White.

1. Formulation of the problem and determination of the solution

The objectives of the paper are to study the evolution of the volumetric water content in an unsaturated porousnonhomogeneous stratified soil, during a time dependent rainfall infiltration and to solve an optimization problemregarding the jump in water content at the interface between two different soil layers. This model represents anextension of the case studied in [1] for an one-layer soil with an impermeable basement, under a constant rainfallrate.

We consider the water infiltration into a stratified unsaturated soil consisting of two layers characterized by differenthydraulic properties. The flow is assumed to be one dimensional, in the vertical direction, z, positive downward.The flow domain is (0, L) and the layers are separated by the boundary z = zB. Each layer (j = 1, 2) is structurallystable and the corresponding water diffusivity, Dj and hydraulic conductivity, Kj are defined by nonlinear functionsof the volumetric water contents θj , which are functions of time and depth. The initial state of the soil is describedby known depth distributions of volumetric water contents in each layer but different from one layer to the other.Consider infiltration that starts at t = 0 and is driven by a time dependent water flux (rainfall rate) across the soilsurface. This flux is such that hysteresis in flow does not occur. At the lower boundary a general nonlinear fluxcondition is imposed. Moreover the flux is continuous across the interface between the layers, but as it is known,this is no longer true for the volumetric water content that displays a jump controlled by the rainfall rate. In eachlayer the water flow obeys Richards’ equation, written in the diffusive form:

∂θ1

∂t=

∂

∂z(D1(θ1)

∂θ1

∂z) − ∂K1(θ1)

∂z, (z, t) ∈ (0, zB) × (0, T ), (1)

∂θ2

∂t=

∂

∂z(D2(θ2)

∂θ2

∂z) − ∂K2(θ2)

∂z, (z, t) ∈ (zB, L) × (0, T ), (2)

θ1(z, 0) = θ01(z), z ∈ (0, zB), θ2(z, 0) = θ0

2(z), z ∈ (zB, L) (3)

q1(0, t) = u(t), q2(L, t) = αD∗2(θ2(L, t)) + f0(t), t ∈ (0, T ), α > 0, (4)

q1(zB, t) = q2(zB, t), t ∈ (0, T ), (5)

where the water flux is defined by qj(z, t) = Kj(θj) − Dj(θj)∂θj/∂z, j = 1, 2.

The hydraulic functions satisfy the following hypotheses: Dj is positive (Dj ≥ ρj > 0), continuous and monotonicallyincreasing on [θr

j , θsj) and it blows up at the saturation value θs

j ; Kj is bounded, Kj(θ) ∈ [Krj , Ks

j ]. Here θrj represents

the residual value of the moisture and j = 1, 2.

Our analysis focuses on the characterization of the discontinuity of the volumetric water content, which develops atthe interface. Since specific interests may require that certain properties of physical variables should be preservedfor the considered flow, the study refers next to the inverse problem of determining the rainfall rate for which theinterface water content jump is minimum. That will be searched in the class of:

a) all positive functions

b) a prescribed form functions, case in which the control parameter is one of the function characteristics.

PAMM · Proc. Appl. Math. Mech. 3, 466–467 (2003) / DOI 10.1002/pamm.200310503

In the first case we have to minimize the cost functional

Φ(u) =∫ T

0

(u2(t) + (θ2(zB, t) − θ1(zB, t))2)dt, (6)

subject to (1)-(5) and to the constraints u ≥ 0, θj(z, t) ≤ θsj , j = 1, 2. Using optimal control techniques, the optimal

pair (u∗, θ∗1 , θ∗2) is found by solving two coupled systems, the direct one, (1)-(5) and its dual, given by :

∂p1

∂t+ D1(θ∗1)

∂2p1

∂z2+ K

′1(θ

∗1)

∂p1

∂z= 0,

∂p2

∂t+ D1(θ∗2)

∂2p2

∂z2+ K

′2(θ

∗2)

∂p2

∂z= 0, (7)

p1(z, T ) = 0, p2(z, T ) = 0, p1(zB, t) = p2(zB, t), (8)

∂p1

∂z|z=0 = 0, (

∂p2

∂z+ αp2) |z=L = 0, (9)

θ1(zB, t)) = (D∗1)−1(D∗

2(θ2(zB, t))) notation= ζ(θ2(zB , t)) (10)

D2(θ∗2(zB, t))(∂p1

∂z(zB, t) − ∂p2

∂z(zB, t)) = (θ∗2(zB, t) − θ∗1(zB , t))(1 − ζ

′(θ∗2(zB, t)). (11)

The final expression of the controller is found to be

u∗(t) = (−p1(0, t))+, (12)

meaning the positive part of −p1(0, t). We denoted

D∗j (θ) =

∫ θ

0

Dj(ξ)dξ. (13)

In the second case, choosing for instance functions of Gaussian form, u(a, σ, t) = exp[−(x− at)2/(4σt)]/√

4πσt, andconsidering as controller the parameter a, we have to solve the problem

Minimize Ψ(u) = a2 +∫ T

0

(θ∗2(zB, t) − θ∗1(zB, t))2dt.

The system in p remains unchanged and only the relationship between the control parameter and the dual state ismodified, the controller being determined from the relationship

a∗ +∫ T

0

(p1(0, t) + u(a∗, t))∂u

∂a(a∗, t)dt = 0.

These results are applied, as specified in the beginning, for the model developed in [1], in which the dimensionlessform of the hydraulic functions is given by

Dj(Θj) =αj

(cj − Θj)2, Kj(Θj) =

(cj − 1)Θ2j

cj − Θj, Θj =

θj − θrj

θsj − θr

j

,

where Θj is the dimensionless water content. Here αj is known in each layer and cj is a parameter that indicatesthe degree of nonlinearity of the soil (strongly nonlinear for cj → 1 and weakly nonlinear for cj → ∞).

The numerical tests are performed using the FEMLAB software, [2], that implements Galerkin FEM on a nonuniformmesh with Lagrange quadratic elements.

2. References

1 Broadbridge, P.; White, I.: Constant Rate Rainfall Infiltration : A Versatile Nonlinear Model 1. Analytic Solution.Water Resources Research, 24 (1988), 145-154.

2 Femlab v2.3a, Comsol Sweden, 2003.

Dr. Gabriela Marinoschi, Institute of Mathematical Statistics and Applied Mathematics, P.O.Box 1-24,Bucharest, Romania, Prof. Dr. Ing. Alexandru Morega, Dept. of Electrical Engineering, Faculty of Elec-trotechnics, ”Politehnica” University of Bucharest, Bucharest, Romania

Section 17: Mathematical methods of the natural and engineering science 467