Application to the contaminant transport problem

Contaminant transport in underground media is a topic of great interest in the last thirty years due, for instance, to the increasing threat of contamination of groundwater supplies by waste treatments and landfill sites or to the disposal of nuclear radioactive waste [10].

We refer to [5] for a reference regarding modeling issues of contaminant transport. Our model assumes that the computational domainΩ=Ω1∪Ω2∪Ω3∪Ω4, represented in

Fig-20 25 30 35 40 45 50

Figure 2.5: In the top row, we show the number of iterations required to reach conver-gence with a tolerance of 10⁻⁶as function of the optimized parameters for the advection reaction diffusion-diffusion coupling with tangential advection. In the bottom row, we show the dependence onpand the level curves of the objective function in the min-max problem (2.2.10). Physical parameters: ν1=1,ν2=2,η²₁=1,η²₂=2,a2=15, mesh size h=0.01.

ure 2.6, can be partitioned into four layers. In the first one, the contaminant, whose con-centration is described through the unknownu, penetrates mainly thanks to rainfalls and therefore an advection towards the negative y direction is present. The next two layers are formed by porous media so that the contaminant spreads in a diffusive regime described by the Laplace equation. We furthermore suppose that in the second layer, some chem-ical reactions may take place which are synthesized in the reaction term. Finally in the last layer, an underground flow transports the contaminant in thexdirection towards a groundwater supply which is connected to a water well. The problem belongs to the het-erogeneous class, since in different parts of the domain we have different physical phe-nomena, and thus in the last paragraph we use the results discussed in this manuscript to design an efficient domain decomposition method to compute the stationary and time dependent distribution of the contaminant.

y

x Γ1

Γ2

Γ3

Γ4

Γ4

Ω1

Ω2

Ω3

Ω4

−ν1∆u−a2∂yu=0 η2u−ν2∆u=0

−ν3∆u=0

−ν4∆u+a1∂xu=0

Water well

Figure 2.6: Geometry for the contaminant transport problem.

The computational domainΩis set equal to Ω=(0, 8)×(−4, 0), withΩj =(0, 8)×(1− j,−j), j =1...4. On the top boundaryΓ1, we impose a condition on the incoming con-taminant flow, i.e. ^∂_∂^u_y −a2u=1 while on the bottom edgeΓ3we impose a zero Neumann boundary condition ^∂_∂^u_y =0. On the vertical edgesΓ2andΓ4we set absorbing boundary conditions so that

∂u

∂n+pu=0 on {0}×[−3; 0] and {8}×[−3; 0],

∂u

∂n−a1u+pu=0 on {0}×[−4;−3] and {8}×[−4;−3],

wherenis the outgoing normal vector. The parameterp is chosen equal to p=q π^π_h, beingkmin=πandkmax= ^π_h. This choice derives from the observation that imposing

∂u∂n+Su=0, whereS is the Steklov-Poincaré, is an exact transparent boundary condition, see [136, 135]. Thus we replace the expensive exact transparent boundary condition with an approximation of the Steklov-Poincaré operator. We know from [74] thatp=q

π^π_h is indeed a zero order approximation ofS. To solve the system of PDEs, we consider the OSM:

−ν1∆uⁿ₁−a2∂yu₁ⁿ=0 inΩ1, B1(u₁ⁿ)=0 on∂Ω1\eΓ1,

∂n1,2uⁿ₁+p₁₂u₁ⁿ=∂n1,2uⁿ₂⁻¹+p₁₂uⁿ₂⁻¹ oneΓ1,

η²₂u₂ⁿ−ν2∆u₂ⁿ=0 inΩ2, B2(u₂ⁿ)=0 on∂Ω2\ {eΓ1,Γe2},

∂n_1,1uⁿ₂+p21u₂ⁿ=∂n_1,1uⁿ⁻¹₁ +p21uⁿ⁻¹₁ oneΓ1,

∂n2,3uⁿ₂+p₂₃u₂ⁿ=∂n2,3uⁿ₃⁻¹+p₂₃uⁿ₃⁻¹ oneΓ2,

−ν3∆u₃ⁿ=0 inΩ3, B3(u₃ⁿ)=0 on∂Ω3\ {eΓ2,Γe3},

∂n_2,2uⁿ₃+p32u₃ⁿ=∂n_2,2uⁿ⁻¹₂ +p32uⁿ⁻¹₂ oneΓ2,

∂n3,4uⁿ₃+p₃₄u₃ⁿ=∂n3,4uⁿ₄⁻¹+p₃₄uⁿ₄⁻¹ oneΓ3,

−ν4∆uⁿ₄+a1∂xu₄ⁿ=0 inΩ4, B4(u₄ⁿ)=0 on∂Ω4\eΓ3,

∂n_3,3uⁿ₄+p43u₄ⁿ=∂n_3,3uⁿ⁻¹₃ +p43uⁿ⁻¹₃ oneΓ3,

(2.3.1)

whereeΓiare the shared interfacesΓei=∂Ωi∩∂Ωi+1,i=1, 2, 3, the vectorsni,j are the nor-mal vectors on the interfaceΓeipointing towards the interior of the domainΩjand the op-eratorsBi(ui) represent the boundary conditions to impose on the boundary excluding

0

Stationary distribution of the contaminant. Physical parameters:

ν1=0.5,ν2=3,ν3=3,ν4=1,η²₂=0.01,a₂=2,a₁=2.

the shared interfaces. Regarding the Robin parameterspi,j, we choose them according to the two subdomain analysis carried out in this Chapter. Due to the exponential decay of the error away from the interface, see eq. (2.1.3), if the subdomains are not too narrow in they direction, the information transmitted from each subdomain to the neighbouring one does not change significantly and therefore thepi,j from a two subdomain analysis are still a good choice. We remark that this argument does not hold for the Helmholtz equation, for which there are resonant modes for frequenciesk≤ω, whereωis the wave number, which travel along the domains and they do not decay away from the interface.

Figure 2.7 shows the stationary distribution of the contaminant. We observe that due to the advection in theydirection inΩ1, the contaminant accumulates on the interface with Ω2, representing the porous medium, and here we have the highest concentration. Then the contaminant diffuses into the layers below and already in the porous media region it feels the presence of the tangential advection inΩ4. Next we also consider the tran-sient version of equations (2.3.1). We discretize the time derivative with an implicit Euler scheme, so that each equation has a further reaction term equal toη²_{j,t r an}=η²_{j,st at}+_∆t¹ . Figure 2.8 shows the time dependent evolution of the concentrationuover 400 integra-tion steps. The initial condiintegra-tion is set equal to zero on the whole domainΩ.

Table 2.4 shows the number of iterations to reach a tolerance of 10⁻⁶ for the algorithm (2.3.1) both used as iterative method and as a preconditioner for GMRES for the substruc-tured system, see [80] for an introduction to the substrucsubstruc-tured version of (2.3.1). We con-sider both single and double sided optimizations for the parameterspi,jat each interface.

For the time evolution problem, the stopping criterion is max From Figures 2.7 and 2.8, we note that this physical configuration would represent a safe situation since a very small concentration of contaminant manages to get through the

ver-0

Figure 2.8: Evolution of the contaminant concentrationu.

Iterative GMRES

Number of iterations to reach a tolerance of 10⁻⁶for the OSM (2.3.1) used as an iterative method and as a preconditioner. The left side refers to the stationary case while the right side to the transient one where we consider the number of iterations needed to satisfy

the stopping criterion (2.3.2) averaged over 400 time steps.

tical diffusive layers and to reach the right-bottom of the domain, where it could pollute the water well.