Numerical simulation of contaminant biodegradation by higher order methods and adaptive time...

Digital Object Identifier (DOI) 10.1007/s00791-004-0139-yComput Visual Sci 7: 61–78 (2004) Computing and

Visualization in Science

Regular article

Numerical simulation of contaminant biodegradation by higher order methodsand adaptive time stepping

Markus Bause, Peter Knabner

Institut für Angewandte Mathematik, Universität Erlangen-Nürnberg, 91058 Erlangen, Germany(e-mail: [email protected], [email protected]; http://www.am.uni-erlangen.de/am1/am1.html)

Received: 26 August 2003 / Accepted: 26 November 2003Published online: 17 August 2004 – Springer-Verlag 2004

Communicated by: G. Wittum

Abstract. In this work we present and analyze a reliableand robust approximation scheme for biochemically react-ing transport in the subsurface following Monod type kinet-ics. Water flow is modeled by the Richards equation. Theproposed scheme is based on higher order finite elementmethods for the spatial discretization and the two step back-ward differentiation formula for the temporal one. The re-sulting nonlinear algebraic systems of equations are solvedby a damped version of Newton’s method. For the linearproblems of the Newton iteration Krylov space methods areused. In computational experiments conducted for realisticsubsurface (groundwater) contamination scenarios we showthat the higher order approximation scheme significantly re-duces the amount of inherent numerical diffusion comparedto lower order ones. Thereby an artificial transverse mix-ing of the species leading to a strong overestimation of thebiodegradation process is avoided. Finally, we present a ro-bust adaptive time stepping technique for the coupled flowand transport problem which allows efficient long-term pre-dictions of biodegradation processes.

1 Introduction

Groundwater contamination by biodegrading organic com-pounds has become a serious and widespread environmentalproblem in industrialized countries. Major organic contam-inants include petroleum fuels (gasoline, diesel), petroleumbyproducts (coal tar, coal-tar creosote), and chlorinated sol-vents. In many cases, groundwater contains a mixture of or-ganic contaminants, either due to the complex mixture inmany non-aqueous phase liquids (NAPLs; e.g., gasoline) ordue to co-disposal/co-spillage (e.g., landfill leachates). Thedegradation of these contaminants is controlled to a largeextent by the biological and geochemical conditions in thegroundwater. Fortunately, biodegradation tends to attenuate atleast some organics during groundwater transport.

The question of whether active remediation is required,or whether natural processes of attenuation (passive remedia-

tion) will be sufficient is a critical issue in “real world” situa-tions. Passive or intrinsic remediation is generally preferred,if feasible, due to the potential to, firstly, eliminate perma-nently contaminants through biogeochemical transformationor mineralization and, secondly, avoid expensive biological,chemical and physical treatments. However, the possible at-tenuation of organic compounds and the impact of that con-tamination on a groundwater resource is difficult to predictsince field sampling limitations make it difficult to developan accurate mass balance. Numerical models can be used tohelp answer these questions, predict the long-term evalua-tion of the contaminant plume, and evaluate factors limitingbiodegradation. However, a predictive capability for decision-making can only be found in advanced contaminant transportmodels which include the full range of the controlling pro-cesses and efficient, accurate and reliable numerical methodsfor solving the equations. Although the understanding of con-servative transport and the effect of medium heterogeneityon transport are now well-advanced, methods for modellingbioreactive processes, in particular at the field scale, are lesswell understood; cf. [10]. Also, the accurcay of numericaltechniques in the context of bioreactive transport has been lit-tle explored so far. This work provides a contribution to thedevelopment of reliable numerical methods for the simulationof biochemically reacting contaminant transport.

Our performed computer experiments have borne out thatthe efficiency and accuracy of the numerical discretizationof the bioreactive transport model can be ensured by usinghigher order approximation schemes. This is of particularimportance for the spatial discretization. Using lower orderapproximation schemes may lead to an overrepresentationof the transverse mixing of the substances and, thereby, toan overprediction of the biodegradation process; cf. [10, 30].Completely wrong solutions are obtained, even if the spa-tial grid is locally refined and adapted to the solution. Higherorder methods help to overcome these difficulties due to theirless inherent numerical diffusion. This will be shown in com-putational experiments of realistic biodegradation processes.For the governing equations of the involved species (electrondonor, electron acceptor and microbial population; cf. Sect. 2)we propose a discretization by conforming quadratic finite

62 M. Bause, P. Knabner

elements in space and the two step backward differentiationformula in time which are of third and second order accu-ray, respectively. This will be compared to a lower orderapproximation based on linear finite elements. Throughout,advection-dominated transport of the mobile species (electrondonor and acceptor) introducing local numerical instabilitiesin the solution is handeled by the streamline upwind Petrov–Galerkin method (SUPG); cf. [23, 24]. We carefully compareour numerical scheme to a recently published adaptive finitevolume approach (cf. [26, 30]) by recomputing some compu-tational experiments of biodegradation processes presentedin [26, 30].

In our simulations, the underlying flow field (volumetricflux) in the transport equations for the species is either pre-scribed analytically or computed by solving numerically theRichards equation (cf. Sect. 2) of unsaturated-saturated flowwhich is defined by coupling a statement of flow continuitywith the Darcy law; cf., e.g., [14]. In the saturated regime theparabolic-elliptic degenerate Richards equation simplifies tothe elliptic single phase Darcy flow problem; cf. Sect. 2. Sincethis paper is primarily devoted to the accurate and reliablesimulation of the biodegradation process, variably saturatedflow is only considered in a final illustrative simulation. Allother studies are restricted to saturated flow.

Generally, we favor mixed finite element methods (cf.,e.g., [8]) for solving the flow problem; cf. [4]. Appreciableadvantages of mixed finite element methods are their inher-ent conservation properties and the fact that they providea flux approximation as part of the formulation itself. A dis-advantage of the approach is that it leads to a system ofequations with an indefinite matrix. This is a considerablesource of trouble for solving the linear system. Usually, ei-ther the Lagrange multiplier technique or the mixed Schurcomplement algorithm is applied; cf. [31]. Further, the lowestorder Raviart–Thomas element RT 0, which is typically usedto solve the Richards equation (cf., e.g., [4, 22, 37]), yieldsa flow field approximation of first order accuracy only, meas-ured in the L2-norm. For the lowest order Brezzi–Douglas–Marini element BDM1 quadratic convergence can at most beexpected; cf. [8, 31]. However, in order to obtain the optimumconvergence rate for the quadratic finite element discretiza-tion of the transport equations for the species (cf. Sect. 2),a third order accurate approximation of the flow field isneeded; cf. Sect. 6. This is thus not provided by the lowestorder mixed finite element methods RT 0 and BDM1. Here,for the sake of simplicity, we use an approximation of theflow problem based on conforming cubic finite elements.This gives us the required third order accurate approxima-tion of the flow field if the coefficient functions (e.g., hy-draulic permeability) are sufficiently smooth. However, forrough parametrizations the BDM1 discretization seems moreadvantageous; cf. Sect. 6. Alternatively, the BDM2 elementmay be used which we plan for the future but is not done heredue to the greater complexity of solving the resulting indef-inite system of equations and our focus in this study on thebiodegradation process. This element combines mass conser-vation properties with a third order accurate approximation ofthe flow field.

The plan for the paper is as follows. In Sect. 2 we in-troduce the mathematical model describing the transport andMonod type biodegradation of organic contaminants in thesubsurface and the Richards equation of variably saturated

water flow. In Sect. 3 the numerical discretization and stabi-lization techniques are described and the discrete problemsto be solved are formulated. Further, the treatment and solu-tion of the resulting nonlinear algebraic systems of equationsis briefly explained. In Sect. 4 we first compare our numer-ical methods and computations to some recently publishedresults [26, 30]. In these computational studies the flow fieldis explicitly given. Section 5 is devoted to a comparison of thequadratic finite element approach to the linear one. In Sect. 6the discretization of the saturated flow problem is reviewed.The quality of the mixed BDM1 and, in particular, the con-forming cubic finite element approximation of the flow fieldis carefully analyzed. This is done numerically and theoretic-ally. In Sect. 7 we first introduce a robust adaptive time step-ping procedure for long-term simulations. To illustrate its ef-fectiveness, we then compute, using a numerically calculatedsaturated flow field, the advancement of a m-xylene plume.This is based on realistic laboratory-derived or field-measuredinput parameters. In our final simulation (cf. Sect. 8) theRichards equation of variably saturated flow is coupled to thetransport and biodegradation model. In a complex scenarioinvolving unsaturated and saturated zones subsurface contam-ination by m-xylene is simulated and studied. The paper endswith some concluding remarks.

2 Governing equations

Microbial activity in the subsurface is dependent on thebioavailability of all substrates utilized by the microorgan-isms. The main substrates are the electron donor, the electronacceptor, and the primary carbon source. In the standard caseof metabolic aerobic degradation, oxygen is the electron ac-ceptor and the contaminant to be degraded acts both as theelectron donor and the primary carbon source; cf. [7]. Sincewe focus in this paper on an accurate and reliable approxima-tion of the underlying model equations, only the basic processof aerobic degradation of a single substrate will be consid-ered. The principles hold equally for multiple electron donorsor acceptors.

For this simple scenario, biomass growth is assumed tofollow the double Monod kinetics, also refered to as the dou-ble Michaelis–Menten law; cf. [7, 10, 34]. Then the governingequations for the electron donor cD [M L−3], electron accep-tor cA [M L−3] and immobile biomass cX [M L−3] are respec-tively given by

∂t(ΘcD)−∇ · (DD∇cD −qcD) = −µ ,

∂t(ΘcA)−∇ · (DA∇cA −qcA) = −αA/Dµ ,

∂t cX + kdcX = YΘ

(1 − cX

cXmax

)µ , (1)

where the Monod term µ [M L−3 T−1] is defined by

µ = Θµmax cXcD

KD + cD

KID

KID + cD

× cA

K A + cA

KIA

KIA + cA. (2)

Thus, we have to consider a coupled system of partial and or-dinary differential equations. In this paper we refer to (1), (2)

Numerical simulation of contaminant biodegradation by higher order methods and adaptive time stepping 63

along with initial and boundary conditions as the Monodmodel. In (1), Θ [–] denotes the volumetric water content, q[L T−1] the Darcy velocity vector (volumetric flux) and Di ,i = D, A, [L2 T−1] with

Di = (Θdi +βt |q|) I + (βl −βt)q ⊗q

|q| (3)

the diffusion-dispersion tensor following the Scheideggerparametrization (cf. [33]) in which di , i = D, A, [L2 T−1] isthe molecular diffusion, I [–] the identity matrix and βl [L]and βt [L] are the longitudinal and transverse dispersivities,respectively. The constant αA/D [–] denotes the electron ac-ceptor to donor mass ratio, kd [T−1] is the first order decayrate for the biomass, Y [–] is the microbial yield coeffi-cient per unit electron donor consumed (mg biomass per mgelectron donor) and cXmax [M L−3] is the maximum biomassconcentration.

In (2), µmax [T−1] denotes the maximum growth rate,Ki , i = D, A, [M L−3] is the half-utilization constant of theelectron donor and acceptor, respectively, and KIi , i = D, A,[M L−3] is the Haldane inhibition concentration of the elec-tron donor and acceptor, respectively. The inhibition termKIi/(KIi + ci), i = D, A, as proposed by Haldane [19] andAndrews [3], yields a slower microbial growth and, therefore,a slower effective electron donor utilization rate at higherconcentrations; cf. [34].

In the numerical model unrealistically high biomass con-centrations are avoided by introducing the term 1 − cX/cXmax

in the third equation of (1). If in the model microbial growthis not restricted, simulated microbial concentrations may be-come very large, especially in source areas with continuouselectron donor and acceptor supply. In real aquifers, the sizeof the biomass is limited, for example, due to lack of space,production of inhibitory metabolites, lack of nutrients, pro-tozoal grazing, sloughing of microbial mass and viral attack.The constant cXmax represents the maximum microbial con-centration at which the biomass reaches a quasisteady state.

We consider solving the equations (1), (2) over (0, T )×Ωwhere Ω ⊂ Rd , d = 2, 3, is a two- or three-dimensionalbounded domain and the system (1), (2) is supplied with ini-tial conditions

cD(0, ·) = cD,0 , cA(0, ·) = cA,0 , cX(0, ·) = cX,0 (4)

in Ω at t = 0 and nonhomogeneous Dirichlet and Robinboundary conditions for ci , i = D, A,

ci = gi on (0, T )×ΓD ,

(qci − Di∇ci) · ν = hi on (0, T )×ΓR . (5)

Here, ν denotes the outer unit normal to the boundary ∂Ω =ΓD ∪ΓR of the domain Ω.

For the two-dimensional case d = 2 the existence ofa global unique nonnegative solution

cD, cA ∈ W1,2p ((0, T )×Ω) , p > 2 ,

cX ∈ C1 ([0, T ]; C(Ω

))(6)

to the equations (1)–(5) for any given T ∈ (0,∞) was re-cently proved; cf. [28]. We observe that even the nonnegative-ness of the concentrations cD, cA and cX is ensured. The proof

can be carried over to the three-dimensional case. For thedefinition of the spaces Wl/2,l

p ((0, T )×Ω) we refer to [27].In our computational experiments, the Darcy velocity vec-

tor q [L T−1] in (1), (3) and (5) is either prescribed analyti-cally or computed numerically by, first, solving the Richardsequation of unsaturated-saturated flow in soil which in itspressure formulation reads as

∂tΘ(ψ)−∇ · (Kskrw(ψ)∇(ψ + z)) = 0 in (0, T )×Ω ,

ψ = g on (0, T )×ΓD ,

− Kskrw∇(ψ + z) · ν = h on (0, T )×ΓF ,

ψ(0, ·) = ψ0 in Ω (7)

with ∂Ω = ΓD ∪ΓF and, then, putting

q = −Kskrw(ψ)∇(ψ + z) . (8)

The pore spaces may contain both water and air. The “sat-urated zone” is the portion of the medium that is watersaturated while the “unsaturated zone” contains both, wa-ter and air. These zones may vary with time and space.In (7) and (8), ψ [L] denotes the pressure head and z [L] isthe elevation head at any point which is simply the heightof that point (against the gravitational direction). The hy-draulic head p = ψ + z represents the height of the watercolumn above some reference elevation (z = 0). The constantKs = kag/µw [L T−1] denotes the hydraulic permeability inthe saturated zone where ka [L2] is the absolute permeabil-ity, [M L−3] is the water density, g [L T−2] is the accel-eration constant and µw [M L−1 T−1] is the water viscosity.Finally, krw [–] denotes the relative permeability of water toair. Here, for the sake of simplicity, we tacitly assume thatthe porous medium is isotropic. In the anisotropic case Ks be-comes a second-order symmetric tensor; cf. [12]. Functionalforms for Θ(ψ) and krw(ψ), both of which are bounded, havebeen derived in the literature. In this work we use the relationsproposed by van Genuchten and Mualem [16, 29]. Precisely,in the unsaturated zone for ψ < 0 they read as

Θ(ψ) = Θr + (Θs −Θr)θ , θ = 1

(1 + (−αψ)n)m (9)

and

krw(ψ) = √θ

(1 − (

1 − θ1/m)m)2

(10)

where m = 1 −1/n. Here, Θr [–] denotes the residual watercontent, Θs [–] is the saturated water content and α, n [–],n > 1, are some positive real numbers. For other parametriza-tions of Θ and krw we refer to the literature; cf. [20, 35].

Clearly, (7) is parabolic in the unsaturated regime and re-duces to the elliptic Darcy problem

−∇ · (Ks∇(ψ + z)) = 0 in Ω ,

ψ = g on ΓD , −Ks∇(ψ + z) · ν = h on ΓF (11)

in the saturated zone where Θ equals the porosity of the soil.Alt and Luckhaus [2] state the following existence and regu-larity result for the Richards equation (7):


Θ(ψ) ∈ L∞ (0, T ; L1(Ω)

),

∂tΘ(ψ) ∈ L2 (0, T ; W−1,2(Ω)

),

q ∈ L2 (0, T ; L2(Ω)

). (12)

For the definition of the spaces we refer to [1]. Equations (11)define a standard elliptic problem for that numerous existenceand regularity results are known; cf., e.g., [17].

The established regularity of the solutions to (1)–(5) and,in particular, to (7) is, by far, too weak to justify the appli-cation of higher order approximation schemes. However, onemay expect that a higher regularity of the solutions still holdsin some sense locally. This might be sufficient to get a sig-nificant advantage of higher order approximation schemesover lower order ones. Such superiority of the higher ordermethods is confirmed by our performed computations as weshall see in Sects. 4 and 5. Further, higher order regularity re-sults for the solution to (1)–(5) will be established in a forth-coming paper; cf. [6].

3 Discretization and solution techniques

We shall now describe our numerical methods and solu-tion techniques for solving the model equations (1)–(11).For the spatial discretization we use conforming finite elem-ent methods. In our computational studies, either the wholeboundary of Ω or at least some part of it may be curved. Insuch a case we adapt the mesh to the boundary by using theisoparametric counterparts of the finite elements introducedbelow. Since this is a standard technique to deal with curveddomains and in order to simplify the notation, the bound-ary approximation is tacitly ignored in the following, and Ωis supposed to be a polyhedral domain. In our computationswe will consider non-vanishing Dirichlet boundary values gi ,i = D, A in (5) and g in (7) and (11), respectively. How-ever, for the sake of simpilcity, the variational formulationof (1)–(5), (7) and (11) is given for homogeneous Dirich-let boundary conditions only. Nonhomogeneous boundaryvalues are incorporated by standard techniques; cf. [9].

Let Th = K be a finite decomposition of mesh size hof the domain Ω in to closed subsets, triangles in two di-mensions and tetrahedrons in the three dimensional case. Thedecompositions are assumed to be regular, i.e., “face to face”.We use standard conforming P1 and P2 elements for theMonod model (1)–(5) and conforming P3 elements for theRichards equation (7) and the Darcy problem (11), respec-tively. The approximation space V l

h , with l ∈ 1, 2, for theelectron donor and acceptor Ci , i = D, A, Xl

h , with l ∈ 1, 2,for the biomass CX and Qh for the pressure head Ψ are thusdefined as follows:

V lh =

Ci ∈ C(Ω

) ∣∣Ci |K ∈ Pl(K) for K ∈ Th∩ W1,2

0,ΓD,

Xlh =

CX ∈ C(Ω

) ∣∣CX |K ∈ Pl(K) for K ∈ Th

.

By Pj(K), j ∈ N, we denote the space of continuous polyno-mials of maximum degree j . Further, W1,2

0,ΓD= c ∈ W1,2(Ω) |

c = 0 on ΓD, where c = 0 on ΓD has to be understood in thesense of traces. Hence, the spatial discretization of cD, cA and

cX is formally of (l +1)th order accuracy with respect to theL2(Ω)-norm.

For the temporal discretization of problems (1)–(5)and (7) we consider a mesh tn, n = 0, . . . ,N with t0 = 0and tN = T , for the time variable t and define τn = tn+1 − tn.Due to the generally high stiffness of semidiscretizations toflow and transport problems, implicit schemes should be pre-ferred in the choice of time-stepping methods for solvingthese problems. The backward Euler method is robust and hasexcellent stability properties (cf. [18]), but it is inaccurate dueto its first convergence order only and also strongly damp-ing. So, it should only be used for nonstationary calculationswhich aim to iterate towards the steady limit. A scheme hav-ing similar stability properties as the backward Euler methodbut being of second order accuracy is the two step backwarddifferentiation formula BDF2 (cf. [18]) which we use in ourcomputations.

Firstly, we suppose that sequences q(tn),Θ(tn) ∈W1,∞(Ω)× L∞(Ω) are explicitly prescribed. The modifi-cations that are to be made if only finite element approxi-mations Qn,Θn ∈ L∞(Ω)× L∞(Ω) of q(tn),Θ(tn) aregiven which are obtained by solving numerically the Richardsequation (7) or the Darcy problem (11), respectively, aredescribed later. Let l ∈ 1, 2 be fixed and PZh denote the L2-projection onto the finite element space Zh . The discretizationof the Monod model (1)–(5) by the Galerkin method and thetwo step backward differentiation formula BDF2 now readsas follows:

Set C0i = PVl

hci,0 and C0

X = PXlhcX,0. For all time steps

n = 0, . . . ,N −2 compute approximations Cn+2i ∈ V l

h,i = D, A, and Cn+2

X ∈ Xlh by solving the equations

γn+2⟨Θ(tn+2)C

n+2i , Vi

⟩−γn+1⟨Θ(tn+1)C

n+1i , Vi

⟩+γn

⟨Θ(tn)C

ni , Vi

⟩+ τn+1⟨q(tn+2) ·∇Cn+2

i , Vi⟩

+ τn+1⟨Di(tn+2)∇Cn+2

i ,∇Vi⟩

− τn+1⟨(q · ν)Cn+2

i , Vi⟩ΓR

+ τn+1⟨∇ ·q(tn+1)C

n+2i , Vi

⟩= −τn+1

⟨αA/D U

n+2, Vi⟩− τn+1 〈hi , Vi〉ΓR

, (13)

for all Vi ∈ V lh, i = D, A, and

γn+2 Cn+2X −γn+1 Cn+1

X +γn CnX + τn+1kd Cn+2

X

= τn+1Y

Θ(tn+2)

(1 − Cn+2

X

cXmax

)Un+2 , (14)

for all nodes (xj)j=1,... ,M associated with degrees of freedomof Cn+2

X , where Un+2 is defined by

Un+2 = Θ(tn+2) µmax Cn+2X

Cn+2D

KD +Cn+2D

KID

KID +Cn+2D

× Cn+2A

K A +Cn+2A

KIA

KIA +Cn+2A

.


In (13), we have set αA/D = 1 for i = D and αA/D = αA/D fori = A. By 〈·, ·〉 and 〈·, ·〉ΓR we denote the standard L2-innerproduct in L2(Ω) and L2(ΓR), respectively. Further, in (13)and (14) we use the abbreviations

γn+2 = 1 + τn+1

τn+1 + τn, γn+1 = 1 + τn+1

τn,

γn = τ2n+1

(τn+1 + τn)τn.

Clearly, (14) formulates a pointwise condition for all nodesassociated with degrees of freedom. Using instead a varia-tional equation, analogously to (13), leads to stability prob-lems and severe oscillations.

As usual, for the test functions Vi we choose the ba-sis functions of V l

h . Further, let CiX ∈ Xl

h , i = n, n +1, n +2,be represented in terms of the finite element basis func-tions WjM

j=1 of Xlh , i.e., Xl

h = spanWj | 1 ≤ j ≤ M and

CiX = ∑M

j=1 ξ ij Wj , with i = n, n + 1, n + 2, where the vec-

tor ξ i = (ξ i1, . . . , ξ i

M) denotes the degrees of freedom of CiX .

Then, equation (14) amounts to solving the system of equa-tions in the unknown vector ξn+2,

γn+2 Θ(tn+2)ξn+2 −γn+1 ξn+1

+γn ξn + τn+1kd ξn+2

= τn+1Y

Θ(tn+2)

(1 − ξn+2

cXmax

)Un+2 , (15)

where Un+2 = (Un+2(x1), . . . ,Un+2(xM)) and the xj are thenodes associated with the degrees of freedom ξn+2

j .Since the BDF2 is a two step method, we need a start-

ing procedure to compute appropriate approximations C1D,

C1A and C1

X of cD(t1, x), cA(t1, x) and cX(t1, x), respectively.Here, the first time step is done by performing M substepsof the backward Euler method with step size τ0/M. In ourcomputations we use M = 4.

If the transport problem (1) is advection-dominated, thenthe proposed finite element discretization (13) can introduceinstabilities in the solutions which can typically be detectedas oscillations in the studied fields, primarily where steepgradients are present. These oscillations can even be largeenough to impede the convergence of the solution. In order toaddress these problems several stabilization techniques havebeen developed. However, the search of the optimal stabi-lization method for advection-dominated problems is still anactive research area; cf. [38]. Here, we use the streamlineupwind Petrov–Galerkin method (SUPG); cf. [23, 24]. Thisamounts to introducing artificial diffusion in the longitudinaldirection only. No artificial transverse diffusion is added sincethis would cause a fictitious mixing of the species in the re-active transport model and lead to an overestimation of thebiodegradation rates; cf. Sect. 4. Stabilizing equations (13) bythe SUPG method yields the following modified variationalproblems:

Set C0i = PVl

hci,0 and C0

X = PXlhcX,0. For all time steps

n = 0, . . . ,N −2 compute approximations Cn+2i ∈ V l

h,i = D, A, and Cn+2

X ∈ Xlh by solving the equations

γn+2⟨Θ(tn+2)C

n+2i , Vi + δ q(tn+2) ·∇Vi

⟩−γn+1

⟨Θ(tn+1)C

n+1i , Vi + δ q(tn+2) ·∇Vi

⟩+γn

⟨Θ(tn)C

ni , Vi + δ q(tn+2) ·∇Vi

⟩+ τn+1

⟨q(tn+2) ·∇Cn+2

i , Vi + δ q(tn+2) ·∇Vi⟩

+ τn+1⟨Di(tn+2)∇Cn+2

i ,∇Vi⟩

− τn+1⟨(q · ν)Cn+2

i , Vi⟩ΓR

− τn+1⟨∇ · (Di(tn+2)∇Cn+2

i

), δ q(tn+2) ·∇Vi

⟩+ τn+1

⟨∇ ·q(tn+1)Cn+2i , Vi + δ q(tn+2) ·∇Vi

⟩= −τn+1

⟨αA/D U

n+2, Vi + δ q(tn+2) ·∇Vi⟩

− τn+1 〈hi, Vi〉ΓR(16)

for all Vi ∈ V lh, i = D, A.

In (16), the operator “∇” denotes the gradient understoodin a piecewise sense with respect to the decomposition Th ,rather than in the distributional sense with respect to Ω. Theintegrals over Ω are computed elementwise, i.e., by summingup the corresponding integrals over the elements K ∈ Th . Thestabilization parameter δ = δ(K) is chosen locally as

δ = δ0 max

δ1hK

‖q(tn+2)‖L2K (K)

− Θdi

‖q(tn+2)‖2L2

K (K)

, 0

where δ0, δ1 are tuning parameter, hK is the local mesh size ofelement K and ‖q‖L2

K (K) = (|K |−1∫

K |q|2 dx)1/2 is the meanvelocity on K . Throughout, we use δ1 = 1.

It remains to describe the approximation of the flow prob-lems (7) and (11). Due to the specific structure of our soft-ware, we found it more convenient to solve (7) and (11) forthe hydraulic head p = ψ + z instead of the pressure head ψ.We use standard conforming P3 elements. The discrete spacefor the hydraulic head P is

Lh = P ∈ C

(Ω

) | P|K ∈ P3(K) for K ∈ Th∩ W1,2

0,ΓD,

and the approximation of p and the flux q is formally offourth and third order accuracy, respectively. The discretiza-tion of (7) by the Galerkin method and BDF2 now reads asfollows:

Set P0 = PLh (ψ0 + z). For all time steps n = 0, . . . ,

N −2 compute approximations Pn+2 ∈ Lh by solving thevariational equation

γn+2⟨Θ

(Pn+2 − z

),Φ

⟩−γn+1⟨Θ

(Pn+1 − z

),Φ

⟩+γn

⟨Θ

(Pn − z

),Φ

⟩+ τn+1

⟨Kskrw

(Pn+2 − z

)∇ Pn+2,∇Φ⟩

= −τn+1 〈h,Φ〉ΓF(17)

for all Φ ∈ Lh .


Again, the first time step is done by performing M substeps ofthe backward Euler method with stepsize τ0/M. If only satu-rated flow is considered in terms of problem (11), we obtaininstead of (17):

Compute an approximation P ∈ Lh by solving the varia-tional equation

〈Ks∇ P,∇Φ〉 = 〈h,Φ〉ΓF(18)

for all Φ ∈ Lh .Finally, let us describe the modifications to be made in the

equation (13) and (16), respectively, if the flow field q(tn) andwater content Θ(tn) are no longer prescribed analytically butcomputed by solving the problem (17) or (18), respectively.In such a case we simply replace q(tn) by its “finite elementapproximation”

Qn = −Kskrw(Pn − z)∇ Pn (19)

and Θ(tn) by computing Θ(Pn − z) in terms of (9). However,the term 〈∇ ·q(tn+2)C

n+2i , Vi〉 in (13) needs some further con-

sideration. Since Pn ∈ W1,∞(Ω) does not admit second orderdistributional derivatives, the divergence of Qn can only bedefined in a local rather than in a distributional sense. On theother hand, setting

q = −Kskrw∇(ψ + z) (20)

and noting that krw = 1 in the saturated regime, the first equa-tion in (11) implies that ∇ ·q ≡ 0, hence ∇ ·qci = 0 in (1).Therefore, if the flow field q is assumed to satisfy the Darcyflow problem (11), the term 〈∇ ·q(tn+1)C

n+2i , Vi〉, i = D, A,

is omitted in (13). Mixed finite element methods which weplan to use in the future (cf. Sect. 1) avoid the difficulty ofhandling the term 〈∇ · q(tn+2)C

n+2i , Vi〉 since they provide

a flow field approximation Qn of q(tn) with ∇ · Qn ∈ L2(Ω)as part of the formulation itself. Let us now consider the caseof variably saturated subsurface flow. Semidiscretizing theRichards equation (7) in time by applying the BDF2 and re-calling definition (20) we get

γn+2 Θ(ψn+2

)−γn+1 Θ(ψn+1

)+γn Θ (ψn)+ τn+1∇ ·qn+2 = 0 . (21)

If an approximation of q(tn) is computed in terms of(17) and (19), we replace τn+1〈∇ · q(tn+2)C

n+2i , Vi〉 by

〈−(γn+2 Θ(Pn+2 − z) − γn+1 Θ(Pn+1 − z) + γn Θ(Pn − z))Cn+2

i , Vi〉, as proposed by (21).In our considered model (1)–(11) there is no recoupling of

the transport and biodegradation problem (1)–(5) to the flowproblem (7) and (11), respectively, i.e., the species concentra-tions cD, cA and cX do not affect the water flow. Therefore,in each time step we first solve the flow problem (17), in-dependently from the transport problem (13)–(15). For theresulting nonlinear systems of equations a damped version ofNewton’s method is used. The linear problems of the Newtoniteration are solved by standard Krylov space methods like,for instance, GMRES with SSOR preconditioning; cf. [25].For the future we plan to use multigrid methods as linearsolver which is motivated by our former experiences withcomputing variably saturated subsurface flow; cf. [4]. In the

simple case of a single electon donor and acceptor and a sin-gle biomass an alternative treatment of the transport andbiodegradation problem (1)–(5) seems possible. After a tem-poral discretization of the equations (1), one may resolvethe time-discrete version of the third equation in (1) for thebiomass concentration cX and substitute the resulting iden-tity into the time-discrete version of the first two equationsin (1). Thus, the biomass concentration is eliminated from thenonlinear system of equations and can be computed in a post-processing procedure which leads to smaller systems of linearequations to be solved. However, a generalization of such ap-proach to the case of multiple microbial populations and, inparticular, its implementational realization seems more com-plex than an explicit treatment of the ordinary differentialequations for the biomass species. Therefore, the approach isnot considered here.

4 Quadratic finite element versus adaptive finite volumeapproximation

Our first numerical tests are devoted to a comparison of theproposed quadratic finite element approximation, i.e., l = 2in the definition of the finite element space V l

h , to an adap-tive implicit upwind monotone vertex centered finite volumeapproach which was recently published by Ohlberger andRohde [30] and Klöfkorn, Kröner and Ohlberger [26]. We re-compute the model problems considered in these publicationsby our methods.

In [30], the following simpler model for the biodegrada-tion of a contaminant is considered

∂t(ΘcD)−∇ · (Θd ∇cD −qcD) = −αDµ ,

∂t(ΘcA)−∇ · (Θd ∇cA −qcA) = −αAµ , (22)

where the Monod term µ is defined by

µ = cXcD

KD + cD

cA

K A + cA. (23)

The concentration of the biomass cX which acts as a catalystfor the reaction is kept constant and dispersion is neglected.Only the molecular diffusion Θd is incorporated in the equa-tions. The following data are chosen:

Ω : (0, 0.5)× (0, 1) , Θ : 0.2 , q : (0,−1)T , d : 0.0001 ,

αD : 5.0 , αA : 0.5 , KD, K A : 0.1 , cX : 1.0 .

As initial conditions, we choose cD(0, x) = 0.0 andcA(0, x) = 0.1 in Ω. The contaminant or electron donor, re-spectively, is injected at the middle part of the inflow bound-ary. Precisely, we prescribe cD(t, x) = 1.0 and cA(t, x) = 0.0for (t, x) ∈ (0, T ]× (0.225, 0.275)×1 and cD(t, x) = 0.0and cA(t, x) = 0.1 for (t, x) ∈ (0, T ] × (0, 0.5)\(0.225,0.275)×1. On the left and right boundary we use zero fluxboundary conditions, i.e., (Θd∇ci −qci) ·ν = 0 for i = D, A,and on the lower boundary an outflow boundary condition isassumed, i.e., Θd∇ci ·ν = 0 for i = D, A. We use a fixed timestep size τn = 5.0 ×10−3. The computations are done, firstly,on a fixed finite element grid with 6446 elements that is pre-adapted to the profile of the solution at the final time T = 5


and, secondly, on an almost uniform finite element mesh with1860 elements. The grids are visualized in Fig. 1.

Hence, at the initial time we have a small amount ofelectron acceptor (e.g., oxygen) everywhere in the porousmedium. The electron donor (e.g., oil) or contaminant, re-spectively, is injected at the upper boundary for t > 0. Now,the electron donor is transported by the flow field with vel-ocity q to the lower boundary of the domain Ω. Simultan-eously, the contaminant is degraded by a reaction betweenelectron donor, acceptor and (constant) biomass. However,the reaction is restricted to those regions where the concen-trations of the species are sufficiently large. Basically, it is theinterface between the electron donor and the surrounding re-gion where still enough electron acceptor is available. Now, ifa numerical method with much artifical diffusion is used, theinterface between the electron donor and acceptor smears outand the reaction takes places in the larger region. The contam-inant is then degraded too fast; cf. [10, 26, 30].

Figure 2 illustrates the results of our simulation at timeT = 5 when a steady state has already been reached. In theeither cases, on the pre-adapted grid shown in the left col-umn and on the almost uniform grid shown in the rightcolumn, SUPG stabilization was used with δ0 = 1.0 andδ0 = 1.0 ×102, respectively, to avoid spurious oscillations.Our results are consistent with the adaptive finite volumecomputations published in [30]. On the pre-adapted grid, theprofile of the acceptor (cf. Fig. 2) looks even less diffusivethan in [30] and, consequently, a higher contaminant con-centration can be found at the outflow boundary. In [30], themaximum contaminant concentration at the outflow bound-ary is between 40% to 50% of the concentration injected atthe inflow boundary. Apparently, the quadratic finite elem-ent approximation introduces less artificial diffusion than thefinite volume method considered in [30]. On the almost uni-form grid, our results are completely different from thoseobtained by Ohlberger and Rohde [30] on a very fine uni-form mesh. In [30], the biodegradation of the contaminantis dramatically overpredicted and a completely wrong solu-

Fig. 1. Locally pre-adapted grid (6446 elements) and almost uniform grid(1860 elements)

Fig. 2. Comparison of computed concentrations of electron donator (top)and acceptor (bottom) at T = 5 on pre-adapted (left) and on almost uniform(right) grid

tion is obtained. The observed contaminant plume stretchesonly across two third of the domain, approximately, anddoes not expand to the outflow boundary. Too much nu-merical diffusion is introduced by the approximation schemewhich leads to an artificial transverse mixing and, therefore,an overestimation of the biodegradation rate. Our numeri-cal method does not show this shortcoming. Even on thecoarse uniform mesh with only 1860 elements instead of6446 elements in the case of the pre-adapted grid, the con-taminant plume covers the whole length of the domain. Theconcentrations nearly match the adaptively computed resultsin [30].

Consequently, the proposed quadratic finite element ap-proach along with SUPG stabilization seems to be less diffu-sive than the implicit upwind monotone finite volume methodpresented in [30] and is capable of predicting the correctcontaminant biodegradation even on relatively coarse uni-form meshes. However, the SUPG stabilization does gener-ally not produce a monotone scheme which might be a sourceof trouble for the construction of efficient iterative solvers,for instance, multigrid methods; cf. [38] and the discussiontherein. Further, we remark that, very likely, the same toodiffusive results would be obtained by the finite element ap-proach if a monotone upwind discretization of the convectiveterm would be used. In the next section we will reconsiderthis model problem and compare the quadratic finite elementapproximation with a linear one.


Next, we recompute a more complex example consideredin [26]. Originally, this test problem was introduced and ana-lyzed in [10]. The model equations now read as

∂t(ΘcD)−∇ · (D∇cD −qcD) = −µ ,

∂t(ΘcA)−∇ · (D∇cA −qcA) = −αA/Dµ ,

∂tcX + kdcX = Y

Θ

(1 − cX

cXmax

)µ , (24)

where µ is defined by (2). Here, the Haldane inhibition termsare neglected which amounts to putting KID, KIA to infinityin (2). Restricting here microbial growth to cXmax does not al-ter the problem significantly as our computed results will con-firm. For the diffusion-dispersion tensor D the parametriza-tion (3) is used. The following data are chosen (cf. Fig. 3):

Ω : (0, 20)× (0, 80) , Ω2 :x ∈ Ω

∣∣∣∣x − (10, 75)T∣∣ ≤ 1

,

q : (8.0 ×10−6 · sin ((80 − x2)/80 ·24) ,−2.0 ×10−5)T

,

Θ : 0.3 , d : 1.0 ×10−9 , βt : 0.002 , βl : 0.01 ,

αA/D : 2.81 , kd : 5.79 ×10−7 , Y : 0.09 , cXmax : 1.0 ×10−6 ,

µmax : 6.43 ×10−4 , KD : 2.0 ×10−6 , K A : 2.0 ×10−7 .

Here, time is measured in seconds, length in meter and con-centration in mg/l. In our simulation we use the constant timestep size τ = 0.036 (days) and an almost uniform grid with12998 elements which is solely stronger refined in the stripe[8, 12]× [0, 80], similarly to the pre-adapted grid in the pre-vious example (cf. Fig. 1). We consider the initial conditions

cD(0, x) = 0.0 , cA(0, x) = 5.0 ×10−6 in Ω1 ,

cD(0, x) = 2.0 ×10−6 , cA(0, x) = 0 in Ω2 ,

cX(0, x) = 1.0 ×10−9 in Ω = Ω1 ∩Ω2 .

Hence, at the initial time we have a small amount ofbiomass and, except for the well, electron acceptor (e.g.,oxygen) everywhere in the porous medium, and an elec-tron donor or contaminant (e.g., oil) is injected into thewell. As boundary conditions we use cD(t, x) = 0.0 andcA(t, x) = 2.0 ×10−6 on the upper, left and right bound-ary for t ∈ (0, T ] and the outflow condition D∇ci · ν = 0,i = D, A, on the lower boundary for t ∈ (0, T ]. In [26], thetest problem is not specified carefully, in particular, with re-spect to the injection of the contaminant. Moreover, an incor-rect velocity field is given and, compared to [10], a multipli-cation by Θ is missing in the monod term (2) of the electrondonator’s and acceptor’s transport equation and additionallydone in the dispersive part of the diffusion-dispersion ten-sor D. Concerning this factor Θ, we use the specificationsof the original work [10]. Altogether, we tried to recover thesetting of [26]. Nevertheless, there might still be minor differ-ences in the parameters and computational realization of theinjection well. We model the injection well by resetting afterevery time step the concentrations of electron donator and ac-ceptor in Ω2 to the initial values. Objectively, this seems to

Fig. 3. Problem setting and steady state biomass concentration at T = 180(days)

Fig. 4. Steady state plumes of electron donor (left) and acceptor (right) atT = 180 (days)

be (at least) close to the approach in [26] and amounts to in-troducing appropriate additional source and sink terms on theright side of the first two equations in (24).

Again, the contaminant is transported by the flow fieldand degraded by a reaction between electron donor, accep-tor and biomass, thereby, increasing the biomass concentra-tion. Figures 3 and 4 show the results of our simulation attime T = 180 (days) when a steady state has nearly been


reached. The plume of the electron donor expands to the out-flow boundary of the domain. This is consistent with [10, 26].However, in contrast to [26], our computation was done ona “coarse” almost uniform grid whereas in [26] a highlyadapted mesh was used. This shows that the quadratic fi-nite element approximation along with SUPG stabilizationleads to a reliable method to simulate and predict contaminantbiodegradation based on Monod type models.

5 Quadratic versus linear finite element approximation

In this section we compare the quadratic finite elementapproximation, i.e., l = 2 in the definition of the finiteelement space V l

h , to a discretization based on linear fi-nite elements, i.e., l = 1. We recompute the first exampleof Sect. 4. All parameters and boundary and initial condi-tions remain unchanged except for cA(t, x) = 1.0 now, insteadof cA(t, x) = 0.1 there, if (t, x) ∈ (0, T ]× (0, 0.5)\(0.225,0.275)×1. Then, a stronger biodegradation of the electrondonor can be observed and the differences between linear andquadratic finite elements become more dramatic.

Figure 5 shows the results of our simulation at time T = 5.The computations based on linear finite elements were doneon the pre-adapted grid shown in Fig. 1. For the computations

Fig. 5. Steady state plumes of electron donor (top) and acceptor (bottom)at T = 5 based on linear finite element approximation on pre-adapted grid(left) and quadratic finite element approximation on almost uniform grid(right)

with quadratic finite elements we used an almost uniform gridwith 4318 elements. In the either cases SUPG stabilizationwas applied with δ0 = 1.0 ×101. In the second case, a con-taminant concentration of CND = 0.22 is found for T = 5 inthe midpoint (0.25, 0)T of the outflow boundary leaving morethan 20% of the inflowing contaminant undegraded. In thefirst case, a complete degradation within the computationaldomain is predicted which might be a fatal error in prac-tice. This allows us to draw the conclusion that, similarly tothe finite volume approach considered before, the linear finiteelement approximation also introduces too much numericaldiffusion. This shortcoming can hardly be compensated by anadaptive grid refinement strategy. Hence, higher order finiteelement approximations are to be prefered when simulatingMonod type biodegradation processes.

6 Cubic versus BDM1 finite element approximation

In this section we shall analyze more precisely the spatialdiscretization of the flow problems (7) and (11). For sim-plicity, this is only done for the fully saturated regime de-scribed by (11). For two linear elliptic problems we shallnow compare the lowest order mixed BDM1 finite elementapproximation (cf. [8]) to a conforming one based on cubicfinite elements as proposed in Sect. 3. Here, we recall thatthe BDM1 approach yields a second order accurate approxi-mation of the flux variable as part of the formulation and, inaddition, locally preserves mass whereas the cubic approxi-mation converges of third order with respect to the norm ofW1,2(Ω). The BDM1 element might therefore exhibit differ-ent but smaller error constants such that over a computation-ally realistic range of step sizes the errors of both approachesmight be nearly of equal size. For smooth coefficient func-tions this conjecture can not be confirmed by numerical ex-periments. However, if the coefficient functions are rough,then the mixed approach may in fact prove to be superior tothe higher order conforming one. In this study we do not con-sider the lowest order Raviart–Thomas element since it givesfirst order convergence only; cf. [31]. For the future we planto implement and test the BDM2 element which combinesthird order convergence of the flux variable with local massconservation. In this section we shall further prove for a sim-ple transport problem that the third order accurate flow fieldapproximation yields the desired optimum third order conver-gence rate for the quadratic finite element discretization of thespecie’s concentration.

To compare the mixed BDM1 finite element method withthe conforming cubic one, we consider the elliptic modelproblem

−∇ · (k∇u) = f in Ω = [0, 1]2 ,

u = 0 on ∂Ω . (25)

In our first numerical experiment, let k ≡ 1 and the solution uof (25) be given by

u(x1, x2) = x1(1 − x1)x2(1 − x2) .

Table 1 shows the discretization errors q − Q for the fluxq = −k∇u and its finite element approximation Q in theenergy norm ‖σ‖k = ‖k−1/2σ‖L2(Ω). The mixed BDM1 com-


Table 1. Comparison of ‖q − Q‖k for mixed BDM1 (cf. [13]) and conform-ing cubic (P3) finite element discretization

elements first experiment second experimentBDM1 P3 BDM1 P3

32 8.61×10−3 4.27×10−4 8.25×10−2 4.00×10−2

128 2.27×10−3 5.07×10−5 2.36×10−3 2.24×10−2

512 5.80×10−4 7.20×10−6 5.96×10−4 1.73×10−2

2048 1.46×10−4 9.12×10−7 1.46×10−4 1.20×10−2

8192 3.66×10−5 1.11×10−7 3.72×10−5 8.60×10−3

32 768 9.22×10−6 1.43×10−8 9.41×10−6 6.07×10−3

putations were presented in [13]. The cubic finite elementapproach leads to smaller errors. Here, we only focus on theaccuracy of the approximations and not on the efficiency ofthe calculations since in our software the flow and transportproblem are currently solved, for simplicity, on the same fi-nite element mesh. We are aware of the fact that the mixedBDM1 method might be faster and less memory consuming(cf. [13]) such that the BDM1 based computations could havebeen done on a finer mesh within equal computing time.

In our second numerical experiment we consider the het-erogeneous case of a discontinuous diffusion coefficient kdefined by

k(x1, x2) =

1 for u0(x1, x2) ≤ 0.03125 ,

10 else ,

where u0(x1, x2) = x1(1− x1)x2(1− x2). The model problemis not artificial. In subsurface flow, a layering of the soil typ-ically leads to discontinuous hydraulic permeabilities. Thesolution of (25) is now given by

u(x1, x2) =

u0(x1, x2) for u0 ≤ 0.03125 ,

0.03125 + (u0−0.03125) ·0.1 else .(26)

The computed errors ‖q − Q‖k are summarized in Table 1.Again, the results for the BDM1 element were extractedfrom [13] where an adaptive quadrature rule was used withinthe triangles covering the discontinuity of k to improve theconvergence rate of ‖q − qh‖k. The computations for thecubic finite element approach are based on a standard quadra-ture rule. Thus, we conclude that for discontinuous coefficientfunctions the lower order mixed BDM1 finite element ap-proach may be superior to the conforming cubic one. Thisbehavior can also be understood by the analysis of the twoapproaches as we will briefly show in the following twolemmas.

Lemma 1. Let the family Th of triangulations of Ω be regu-lar and quasi-uniform and the coefficient function k be con-tinuous on each triangle K of Th. Let u, q with q = −k∇udenote the solution of the model problem (25). Considera dual mixed approximation of (25) based on the BDM1element with solution UBDM1 , QBDM1 ; cf. [8, 31] for details. Ifq ∈ W2,2(Ω), then there holds∥∥q − QBDM1

∥∥L2(Ω) ≤ ch2

∥∥ q∥∥

W2,2(Ω). (27)

Proof. The assertion follows by standard arguments of fi-nite element error estimation. Let Jh denote the standard

interpolation operator of the flux q ∈ H(div; Ω) in the fi-nite element space Vh = ∏

K∈T hP1(K)∩ H(div; Ω); cf. [8]

for the exact definition of Jh . Here, H(div; Ω) = σ ∈ L2(Ω) |∇ ·σ ∈ L2(Ω). Actually, q has to be slightly smoother thanonly requiring q ∈ H(div; Ω) in order to let the interpolationoperator Jh be well-defined; cf. [8]. For instance, supposingthat q ∈ Ls(Ω)∩ H(div; Ω) with s > 2 is sufficient.

Now, the Galerkin orthogonality of the dual mixed formu-lation along with the coercivity and continuity of the corres-ponding bilinearform a(σ, ϕ) = 〈k−1σ, ϕ〉, σ, ϕ ∈ H(div; Ω),yields after some manipulations∥∥Jhq − QBDM1

∥∥L2(Ω)

≤ ‖a‖α0

‖q − Jhq‖L2(Ω) ,

where ‖a‖ is the continuity constant of the bilinearform a(·, ·)and α0 denotes its coercivity constant. Since Jh : Ls(Ω)∩H(div; Ω) → Vh satisfies (cf. [8])

‖q − Jhq‖L2(Ω) ≤ ch2‖q‖W2,2(Ω)

for q ∈ W2,2(Ω), the triangle inequality directly implies

‖q − QBDM1‖L2(Ω) ≤ ch2‖q‖W2,2(Ω) .

Thus, the error estimate (27) is proved.

Lemma 2. Consider now the conforming approximationof (25) based on cubic finite elements with solution UP3;cf. Sect. 3. If u ∈ Wi,2(Ω)∩ W1,2

0 (Ω), with i ∈ 1, . . . , 4,then there holds the estimate (cf. [9, 31])∥∥k1/2∇(u −UP3)

∥∥L2(Ω)

≤ chi−1‖u‖Wi,2 (Ω) .

Now, for the solution (26) of problem (25) we find

q = −k∇u =(

(1 −2x1)x2(1 − x2)

x1(1 − x1)(1 −2x2)

)∈ W2,2(Ω) ,

whereas ∇u admits a discontinuity at u0 = 0.03125. We ob-serve that the discontinuities of k and ∇u cancel out eachother. Hence, u lacks the smoothness which is needed, dueto Lemma 2, to get an improvement by the cubic finite elem-ent approach. On the other hand, q satisfies the regularityassumption of Lemma 1. Therefore, the BDM1 discretizationyields a higher convergence rate and smaller errors. Althoughthe solution lacks regularity, the cubic finite element approxi-mation is still superior to a linear one considered in [13].

In our final numerical experiment we consider the modelproblem (25) with the heterogeneous diffusion coefficientk(x1, x2) = 1+10i · x2, solution u(x1, x2) = x1(1− x1)x2

2 andboundary condition u(x1, 1) = x1(1 − x1) and u(x1, x2) = 0else. Table 2 contains the errors ‖q − Q‖k for the cubic finiteelement approach and i = 1, 2, 3. The computed numbers aremore than one order of magnitude smaller than the BDM1based results of the first example for a constant diffusion coef-ficient, although the error seems to be bounded above (mono-tonically) in terms of the diffusion coefficient. Therefore, weconclude that for smooth coefficient functions the cubic finiteelement approach leads to more accurate approximations ofthe flux than the lower order but locally mass conservative


Table 2. Cubic finite element discretization error ‖q − Q‖k for continuouscoefficient function k with large gradient

elements i = 1 i = 2 i = 3

32 1.07×10−3 3.14×10−3 9.84×10−3

128 1.24×10−4 3.61×10−4 1.13×10−3

512 1.77×10−5 5.17×10−5 1.62×10−4

2048 2.27×10−6 6.63×10−6 2.08×10−5

8192 2.73×10−7 7.96×10−7 2.49×10−6

32 768 3.50×10−8 1.02×10−7 3.20×10−7

BDM1 approach. This justifies our proposed discretizationof the coupled flow and transport problem (cf. Sect. 3) forsoils with continuous hydraulic properties. However, in thediscontinuous case convergence order is lost at jumps of theparameters.

Finally, we shall show for a simple model problem thata third order accurate approximation of the flow field infact yields the optimum third order convergence rate for thequadratic finite element approximation of the specie’s con-centration in the transport equation. We consider the steadylinear problem

−∇ · (D∇u)+q ·∇u + ru = f in Ω ,

u = 0 on ∂Ω , (28)

where D, q, r are given sufficiently smooth functions on Ω.The matrix D is supposed to be symmetric and positive def-inite, i.e., ξT D(x)ξ ≥ α|ξ|2, with α > 0, for all ξ ∈ Rd andx ∈ Ω. Moreover, r ≥ 0 and ‖q‖L∞(Ω) < α/CP where CP de-notes the Poincare constant, i.e., ‖v‖L2(Ω) ≤ CP‖∇v‖L2(Ω) forall v ∈ W1,2

0 (Ω). Further, we assume that ‖r‖L∞(Ω) < ∞ and‖dij‖L∞(Ω) < ∞, with i, j = 1, . . . , d, where D = (dij)

di, j=1.

Hence, we have

〈D∇u,∇u〉+ 〈q ·∇u, u〉+ 〈ru, u〉 ≥ ε‖∇u‖2L2(Ω)

with ε = (α−‖q‖L∞(Ω)CP) > 0. Thus, the lemma of Lax–Milgram ensures the existence of a unique weak solutionu ∈ W1,2

0 (Ω) to (28). Finally, we further suppose that for anyprescribed f ∈ W1,2(Ω) the boundary value problem (28) ad-mits a solution u ∈ W3,2(Ω) satisfying

‖u‖W3,2(Ω) ≤ c‖ f ‖W1,2(Ω) . (29)

This estimate only holds under restrictions on the coefficientsand regularity of ∂Ω; cf., e.g., [17]. We tacitly assume thatthey are fulfilled. The notation of the function spaces is stan-dard; cf. [1].

Let now a finite element approximation Q of the velocityvector q be given, as for instance obtained by solving theDarcy flow problem (18) or the Richards equation (17), thatsatisfies

‖q − Q‖L2(Ω) ≤ Cqh3 (30)

with some constant Cq independent of the finite element meshsize parameter h. Then we consider solving the finite dimen-sional variational problem

〈D∇U,∇V 〉+ 〈Q ·∇U, V 〉+ 〈rU, V 〉 = 〈 f, V 〉 (31)

for all V ∈ V 2h ; cf. Sect. 3 for the definition of V 2

h . Supposing‖Q‖L∞(Ω) < α/CP , the existence of a unique solution U ∈ V 2

hto (31) is again ensured by the lemma of Lax–Milgram.

First, we have the following auxiliary lemma.

Lemma 3. Under the above-made assumptions the solutionsu and U of (28) and (31), respectively, satisfy for any givenf ∈ W1,2(Ω) the estimate

‖∇(u −U)‖L2(Ω) ≤ ch2 , (32)

where c depends on the data and is independent of h < 1.

Proof. Combining the weak form of (28) with (31) gives

〈D∇(u −U),∇V 〉+ 〈r(u −U), V 〉= −〈q ·∇u, V 〉+ 〈Q ·∇U, V 〉= −〈(q − Q) ·∇u, V 〉− 〈Q ·∇(u −U), V 〉 (33)

for all V ∈ V 2h . Now we choose V = Ihu − U where Ih

denotes the standard interpolation operator in the finiteelement space V 2

h ; cf. [9, 31] for details. Since by assump-tion u ∈ W3,2(Ω), a Sobolev imbedding inequality impliesu ∈ C(Ω) such that Ihu is well-defined and, moreover, satis-fies (cf. [9, 31])∥∥∇(u − Ihu)

∥∥L2(Ω) ≤ ch2

∥∥ u∥∥

W3,2(Ω). (34)

Recalling the positive definiteness of D and non-negativity ofr, relation (33) then implies (‖ · ‖ = ‖ · ‖L2(Ω))

α‖∇(Ihu −U)‖2

≤‖D‖L∞(Ω)‖∇(Ihu −u)‖ ‖∇(Ihu −U)‖+‖r‖L∞(Ω)‖Ihu −u‖ ‖Ihu −U‖+‖∇u‖L∞(Ω)‖q − Q‖ ‖Ihu −U‖+‖Q‖L∞(Ω)‖∇(u − Ihu)‖ ‖Ihu −U‖+‖Q‖L∞(Ω)‖∇(Ihu −U)‖ ‖Ihu −U‖ ,

where ‖D‖L∞(Ω) =(∑d

i, j=1 ‖dij‖2L∞(Ω)

)1/2. Using the Poin-

care inequality, the assumption ‖Q‖L∞(Ω) < α/CP and theSobolev imbedding ‖∇u‖L∞(Ω) ≤ c‖u‖W3,2(Ω), it follows that

‖∇(Ihu −U)‖ ≤ c (‖∇(u − Ihu)‖+‖q − Q‖) . (35)

Together, inequalities (35), (34), (30) and (29) prove the as-sertion (32).

Lemma 3 now enables us to establish the desired optimumthird order convergence of U to u in L2(Ω).

Theorem 1. Under the above-made assumptions the solu-tions u and U of (28) and (31), respectively, satisfy for anygiven f ∈ W1,2(Ω) the estimate

‖u −U‖L2(Ω) ≤ ch3 , (36)

where c depends on the data and is independent of h < 1.


Proof. To prove (36), we use the well-known trick of Aubin–Nitsche. For any prescribed χ ∈ L2(Ω), let φ ∈ W2,2(Ω)∩W1,2

0 (Ω) denote the unique strong solution of the variationalproblem

〈D∇v,∇φ〉+ 〈q ·∇v, φ〉+ 〈rv, φ〉 = 〈χ, v〉 (37)

for all v ∈ W1,20 (Ω). Its existence is tacitly assumed. We will

return to this later. With right-hand side χ = u −U and testfunction v = u −U , equation (37) yields (‖ · ‖ = ‖ · ‖L2(Ω))

‖u −U‖2 = 〈D∇(u −U),∇φ〉+ 〈r(u −U), φ〉+ 〈q ·∇(u −U), φ〉 . (38)

Choosing V = PV 2hφ in (33) where PV 2

hdenotes the L2-

projection onto PV 2h

and combining the resulting equationwith (38) gives

‖u −U‖2 =⟨D∇(u −U),∇

(φ− PV 2

hφ)⟩

+⟨r(u −U), φ− PV 2

hφ⟩+

⟨q ·∇(u −U), φ− PV 2

hφ⟩

−⟨(q − Q) ·∇U, PV 2

hφ⟩

.

By the Cauchy–Schwarz and Poincare inequality we find

‖u −U‖2 ≤ (‖D‖L∞(Ω) +‖r‖L∞(Ω)C2P

+‖q‖L∞(Ω)CP)‖∇(u −U)‖∥∥∥∇

(φ− PV 2

hφ)∥∥∥

+‖q − Q‖ (‖∇u‖+‖∇(u −U)‖) ∥∥∥PV 2hφ

∥∥∥L∞(Ω)

. (39)

There holds (cf. [9, 31])∥∥∥∇(φ− PV 2

hφ)∥∥∥ ≤ ch‖φ‖W2,2(Ω) . (40)

Further we have (cf. [21, Lemma 4.4 and 4.3])∥∥∥PV 2hφ

∥∥∥L∞(Ω)

≤ c‖φ‖W2,2(Ω) . (41)

From (39) along with (32), (40), (30), (29) and (41) we thenconclude

‖u −U‖2 ≤ ch3‖φ‖W2,2(Ω) . (42)

It remains to estimate ‖φ‖W2,2(Ω). First, putting v = φ in(37), recalling χ = u −U and using the assumptions aboutD, q, r and the Poincare inequality, we find

ε‖∇φ‖2 ≤ CP‖u −U‖ ‖∇φ‖with ε = (α−‖q‖L∞(Ω)CP) > 0. Therefore, it follows

‖φ‖W1,2(Ω) ≤ c‖u −U‖ . (43)

The variational problem (37) is equivalent to the weak formof the boundary value problem

−∇ · (D∇φ)+ rφ = χ in Ω ,

φ = 0 on ∂Ω (44)

with right-hand side χ = χ +q ·∇φ+ (∇ ·q)φ and χ = u −U.By the Hölder inequality along with Sobolev imbedding re-sults and (43) we deduce

‖χ‖ ≤ c(‖u −U‖+‖q‖W2,2(Ω)‖φ‖W1,2(Ω)

) ≤ c‖u −U‖ ,

if q ∈ W2,2(Ω) is assumed. Hence, χ ∈ L2(Ω). We sup-pose that the boundary value problem (44) admits, for anyχ ∈ L2(Ω), a strong solution φ ∈ W2,2(Ω) with appropriatea priori estimate which typically requires regularity assump-tions about the coefficient functions and ∂Ω; cf. [17]. Thenwe conclude

‖φ‖W2,2(Ω) ≤ c‖χ‖ ≤ c‖u −U‖ . (45)

Finally, (42) together with (45) prove (36).

7 Adaptive time stepping and m-xylene degradationin saturated flow regime

Assuming that a priori no information for the complex flow,transport and degradation processes is available, then “controlby hand” is often useful for testing accuracy and robustnessbut not for practical applications and long-term simulations.Therefore, in this section we shall first develop a stable, ro-bust and good working adaptive time stepping procedure forthe transport and biodegradation model (1)–(5). This is re-lated to the question of how to perform an adaptive timestep control for highly implict approaches to highly stiffproblems. We emphasize the key words implicit and stiffsince this restriction eliminates most “simple” error controlmechanisms of predictor-corrector type from the list of pos-sible approaches. For backward differentiation formulae thepredictor-corrector technique is commonly considered to beuseful for estimating the local truncation error even in thecase of stiff problems; cf., e.g., [11, 36]. However, it is notapplied here.

To explain this, we consider for simplicity the ordinarydifferential equation

y′ = F(t, y) (46)

with given initial value y(0) = y0. The m-step BDFm appliedto (46) reads as

m∑k=0

γk yj+k = τj+m−1 F(tj+m, yj+m)

with τk = tk+1 − tk and some coefficients γk = γk( j + m),k = 0, . . . , m; cf. [18] for details. The leading term d[l]

j+m ofthe local BDFm truncation error,

d[l]j+m = Cm+1τ

mj+m−1y(m+1)(tj+m) (47)

with known error coefficient Cm+1, can be estimated by

τj+m−1d[l]j+m ≈ E := yj+m − y[0]

j+m

tj+m − tj(48)

where y[0]j+m = p(tj+m) is the predictor and p interpolates the

points (tj+k−1, yj+k−1) for k = 0, . . . , m. The relation (48) is


derived from (47) by approximating y(m+1)(tj+m) with thecorresponding backward difference and, then, rewriting thebackward difference in terms of yj+m − y[0]

j+m; cf. [11, 36] fordetails. Since higher order terms are omitted in the represen-tation (48) of the truncation error, the notation “≈” instead of“=” is used. The estimate E in (48) now allows a step sizecontrol. Although the predictor y[0]

j+m is of explicit type, it re-mains usable for stiff problems without severe restrictions onthe time step sizes. To show this, we consider an equidistantgrid and the case m = 1. Then the first BDF1 step equals theimplicit Euler scheme

y1 = y0 + τF(t1, y1) . (49)

The predictor y[0]2 for y2 now reads as

y[0]2 = y0 +2τF(t1, y1) . (50)

By combining (49) with (50) we observe that y[0]2 can alterna-

tively be defined in terms of the implicit midpoint rule withstepsize 2τ ,

y[0]2 = y0 +2τF

(t1,

y[0]2 + y0

2

),

which is A-stable and, thus, allows for step sizes that are notrestricted at all due to any stability constraint. However, theimplicit midpoint rule is not L-stable which is well-known tobe a considerable source of trouble. Our performed computa-tions of reactive multicomponent systems have also borne outthat the L-stability of the temporal discretization is an import-ant property to be ensured. Otherwise, the calculations mayexhibit spurious oscillations and negative concentrations.

Consequently, the BDFm predictor can be expected tohave less excellent stability properties (for instance, noL-stability as in the case m = 1 considered above) than theBDFm step itself. Due to this lack of stability, which mightlead to a bad estimation of the truncation error, we do notuse the BDFm predictor-corrector technique. Our approachrelies on calculating a lower order discretization with bet-ter stability properties than the BDFm predictor y[0]

m andestimating the error in terms of this approximation. How-ever, this requires solving an additional nonlinear problem ineach time step. Due to its lower numerical cost, the BDFmpredictor-corrector technique might be preferable for weaklyto moderately stiff problems. This and further approaches toadaptive time stepping for the BDF2 approximation of thecoupled flow and transport problem (1)–(7) will be investi-gated more precisely in a forthcoming work.

A practicable approach to time stepping is the followingheuristic technique which is based on the estimation of thelocal truncation error and is close to that presented in [38].Even if the mathematical motivation looks like being crude,the result is in fact a working error indicator. For that, we con-sider saturated flow described by (11). In Sect. 8 the conceptis carried over to variably saturated flow modeled by (7). Wesuppose that the former time level is tn+1 and we want to cal-culate a solution at tn+2 = tn+1 + τ . Now, our aim is to find anappropriate value for τ such that the following relation holdsfor the functional J(·) applied to the exact solution and thediscrete solution at the new time level tn+2,∣∣∣∣∣∣J(ci(tn+2))− J

(Cn+2

i

) ∣∣∣∣∣∣ ∼ TOLi , i = D, A, X , (51)

where TOLi are prescribed tolerances. The functional J(·)may represent directly the concentration values, measured ina properly chosen norm, ||| · ||| = ‖ · ‖L2(Ω), for instance, orJ(·) denotes local user-defined quantities as fluxes, for in-stance. Now, we implicitly make the (heuristic) assumptionthat the error at levels tn and tn+1 is zero. Further we assumethe asymptotic error expansion

J(ci(tn+2))− J(Cn+2

i

) ∼ (τ(i)

)2e2(ci)+O

((τ(i)

)3)

(52)

with an error term e2(ci) independent of the time step.We perform two calculations with step size τ , the first

one with the BDF2 and the second one with the first orderL-stable BDF1 (backward Euler scheme). Then we comparethe computed approximations. Thus, only numerical schemeswith excellent stability properties are involved. For the BDF1we assume an expansion


i

)∼ τ(i)e1(ci)+ (

τ(i))2e2(ci)+O

((τ(i))3

)(53)

with error terms e1(ci), e2(ci) independent of the time stepand (nearly) the same function e2(ci) as in (52). Our approachcannot be verified by rigorous mathematical arguments, butall performed calculations have led to convincing results.Moreover, it should not be forgotten that all these techniqueslead to error indicators in contrast to rigorous error estima-tors such that somewhat heuristic arguments may be allowed.

So we calculate local differences RELi,τ ,

RELi,τ = J(Cn+2

i

)− J(Cn+2

i

). (54)

By a combination of relations (52) and (53) we can obtain fore1(ci) that

e1(ci) ∼ J(Cn+2

i

)− J(Cn+2

i

)τ

,

and hence∣∣∣∣∣∣J(ci(tn+2))− J(Cn+2

i

) ∣∣∣∣∣∣∼ τ(i)

τ

∣∣∣∣∣∣J(Cn+2

i

)− J(Cn+2

i

) ∣∣∣∣∣∣ .This gives us an error indicator for the lower order BDF1 ap-proximation. It is realistic to suppose that the BDF2 yieldsa better approximation than the BDF1. This allows us to con-clude that∣∣∣∣∣∣J(ci(tn+2))− J

(Cn+2

i

) ∣∣∣∣∣∣ τ(i)

τ

∣∣∣∣∣∣J(Cn+2

i

)− J(Cn+2

i

) ∣∣∣∣∣∣ .This last relation leads to the following estimate for τ if therelative error shall be related to the given tolerance TOLi asdemanded in (51),

τ(i) TOLiτ

|||RELi,τ ||| (55)


for i = D, A, X. Finally we put τ = minτ(i) | i = D, A, Xwith τ(i) = TOLi τ/|||RELi,τ |||. Of course, this result is pes-simistic since the BDF2 error is estimated from above bythe BDF1 error which seems to be rather crude. Suppos-ing e1 ∼ e2 which may be expected for nearly steady prob-lems, then we get by repeating the arguments from above that(τ ≤ 1 without loss of generality)


i

) ∼(τ(i)

)2

τ − τ2J

(Cn+2

i

)− J(Cn+2

i

)and, therefore,

(τ(i)

)2 ∼ TOLiτ − τ2

|||RELi,τ ||| . (56)

Neglecting τ2 in the nominator of (56) and supposing that theright-hand side is bounded by one, a larger time step τ(i) isthus proposed by (56). In this work, we use the pessimisticvariant (55).

The corresponding time step control based on (55) or (56),respectively, is rather easy to implement. For a given step sizeτ we perform the time step twice, with BDF2 and BDF1.Then we calcute the local differences RELi,τ and τ as de-scribed above. If the resulting value τ is much smaller thanthe actually used time step τ for its prediction we repeat thetime step with τ = τ . If the value for τ is larger than theused τ , or only slightly smaller (say less than 50%) we acceptthe result and perform the next time step with τ . However,the additional numerical cost of the time stepping proced-ure may be quite large since our predictor step consists ofsolving a complete nonlinear problem. Additionally, the so-lution of this subproblem is even (almost) worthless since itis less accurate (first order accuracy only) and only neededfor the control of the time step. However, approaches of theproposed type have proved to work efficiently and particu-larly robust for fully implicit schemes with correspondingtime steps which are not at all restricted due to any stabilityconstraint; cf. [38].

Now we simulate groundwater contamination by m-xylene in a rectangular domain (cf. Fig. 6) with two im-permeable regions (ellipses) inside denoted by E1 and E2,respectively. The saturated flow field is computed numerically

Fig. 6. Computational domain and calculated pressure head ψ (profile andisolines)

by solving the Darcy problem (11). Concentrations are ap-proximated by quadratic finite elements, i.e., in V 2

h and X2h ,

respectively; cf. Sect. 3. We use reliable field-measured andlaboratory-derived input parameters given in [34]. They haveproven to describe adequately field scale degradation pro-vided that all controlling factors are incorporated in the fieldscale model. The advancement of the m-xylene plume is com-puted by the proposed adaptive time stepping procedure withfunctional J(c) = c and ||| · ||| = ‖·‖L2(Ω). The following dataare chosen in the model equations (1)–(3) and (11):

Ω : (0, 6)× (0, 10)\(E1 ∪ E2) ,

Θ : 0.33 , d : 7.4 ×10−5 , βt : 0.03 , βl : 0.36 ,

αA/D : 2.16 , kd : 0.025 , Y : 0.52 , cXmax : 1.0 ,

µmax : 1.13 , KD : 7.9 ×10−1 , K A : 1.0 ×10−1 ,

KID : 9.17 ×101 , KIA : ∞ , Ks : 4.5 ×10−2 . (57)

Here, E1 and E2 are ellipses with center in (2, 6.5)T and(4, 8.5)T, respectively, both with semi-axes equal to 1.5 and0.5 and rotated π/8 and −π/6, respectively, in counter-clockwise direction. Time is measured in days, length in me-ter and concentration in mg/l. In contrast to [34], we choosea small but non-vanishing decay rate kd for the biomass inorder to incorporate all components of the model (1)–(5).This does not lead to any significant changes in the computedresults. Further, we use slightly modified values for αA/D ,cXmax and µmax. This is only done to stress the characteristiceffects of the biodegradation process.

The boundary conditions for the hydraulic head p = ψ + zwith z ≡ x2 are

p = 0 for x ∈ [0, 6]× 0 ,

p = 20 for x ∈ [0, 6]× 10 ,

Ks∇ p · ν = 0 else.

The initial conditions for the species are

cD(0, x) = 0.0 , cA(0, x) = 2.0 , cX(0, x) = 3.0 ×10−2

in Ω. The electron donor m-xylene is injected at the upperinflow boundary. More precisely, for t > 0 we prescribe theboundary conditions

cD(t, x) = 2.24 and cA(t, x) = 2.55 (58)

for x ∈ [1, 5]× 10,cD(t, x) = 0.0 and cA(t, x) = 2.0

for x ∈ ([0, 1)×10)∪ ((5, 6]× 10) and

Di∇ci · ν = 0 , i = D, A ,

for x ∈ ∂Ω\([0, 6]× 10).The computations were done on an almost uniform grid

with 8586 elements. The calculated pressure head ψ is visu-alized in Fig. 6. The concentration profiles of the contaminant


m-xylene and the electron acceptor at time T = 10, 44 and300 days are shown in Fig. 7. At T = 300 a quasi-steady statehas been reached. The corresponding biomass concentrationscan be found in Fig. 8. For visualization purposes we have re-stricted the color range of the biomass to [0, 0.5]. Of course,the maximum concentrations is higher, namely, cXmax = 1.0(cf. (57)). We observe the following behavior of the m-xyleneplume which seems to be quite typical for scenarios as con-sidered here. First, the m-xylene plume starts infiltrating intothe domain. Due to the relatively large utilization rates, thelarge microbial yield and the presence of high donor and ac-ceptor concentrations, this leads first to a rapid growth of thebiomass and then to a strong consumption of the electronacceptor in a thin layer at the inflow boundary. A biomass bar-rier is generated which bars the m-xylene front from expand-ing. The contaminant is nearly completely degraded close tothe surface. After some period, when the electron acceptor

Fig. 7. Concentration of electron donor m-xylene (left) and electron accep-tor (right) at T = 10 (top), T = 44 (middle) and T = 300 (bottom) days

Fig. 8. Concentration of biomass with color range restricted to [0, 0.5]at T = 44 and T = 300 days (top) and adaptively generated time stepsizes (bottom) with accepted step size τ (∗) and corresponding predictionsτ(D) (+), τ(A) () and τ(X) () for each time step n = 0, . . . ,N −2

becomes depleted outside the layer, the m-xylene plume be-gins on a larger time scale to separate and infiltrate into thewhole domain. In the concentration profile of the biomass atT = 300 we again observe behind the ellipses in downstreamdirection a sharp interface between electron donor and accep-tor where the contaminant is degraded.

The behavior of the plumes is also reflected by the com-puted step sizes (cf. Fig. 8) of our adaptive time steppingalgorithm which thus works good and saves computing time.As initial guess for the first time step size τ0 we used τ0 =5.0 ×10−2 and as tolerances TOL D = TOL A = 5.0 ×10−4

and TOL X = 5.0 ×10−5. We made the experince that in orderto equilibrate the predicted time step sizes τ(D), τ(A) and τ(X)

the tolerance TOL X for the ordinary differential equation ofthe biomass has to be chosen much smaller than the toler-ances TOL D and TOL A for the electron donor and acceptor.A factor between 5 and 10 seems to be a good choice for thedifference in the tolerances. In our calculation we restrictedthe maximum time step size to τmax = 1.0 and the minimumtime step size to 1.0 ×10−3. At first, the time step size iscontrolled by the incompatibility between the boundary andinitial data and the increase of the electron donor and acceptorconcentration which is basically due to their transport. Then,the rapid growth of the biomass governs the time steppingprocess before the depletion of the acceptor inside the do-main becomes the limiting factor. The larger step sizes at theend of the simulation indicate that an almost steady state has


been reached. The last time step is chosen such that the fi-nal time T = 300 is hit exactly. Without any adaptive strategythe different time scales of the involved processes can not besimulated efficiently and with high accuracy.

8 Degradation of m-xylene in variably saturated flowregime

In our final numerical experiment we analyze the bioreactivetransport of m-xylene in a variably saturated soil. The flowfield is now computed by the Richards equation (7) whichamounts to solving in each time step the variational prob-lem (17). Concentrations are approximated by quadratic finiteelements, i.e., in V 2

h and X2h , respectively; cf. Sect. 3. Our

adaptive time stepping has thus to be augmented with a timestep control for the flow problem. This is simply done by ap-plying the (heuristic) arguments of the previous section to theRichards equation (7). Hence, in each time step problem (7)is solved twice, firstly, with BDF2 and, secondly, with BDF1(backward Euler method). Then a local difference REL P,τ

and time step size τ(P) is calculated as proposed by (54) and(55). Finally, we set τ = minτ(i) | i = D, A, X, P. Our per-formed computations have confirmed that this is a practicableand robust approach to handle the coupled flow and bioreac-tive transport problem.

In our numerical experiment we consider the domainshown in Fig. 9 which is filled with Touchet Silt Loam(cf. [16]) and the parametrization (9), (10) of van Genuchtenand Mualem with the constants

Θr : 0.131 , Θs : 0.396 , α: 0.423 ,

n = 2.06 , Ks = 4.96 ×10−2 .

Time is measured in days and length in meter. The initial con-dition (cf. Fig. 9) for the hydraulic head p = ψ + z with z ≡ x2is

p(0, x) = 1.0 for x ∈ Ω = (0, 2)× (0, 3)\(R1 ∪ R2)

with R1 = [0, 1.2] × [2.25, 2.50] and R2 = [0.8, 2.0]×[1.5, 1.75]. We choose the boundary conditions

p =

3.20 for t ∈ (7 · i, 7 · (i +1)]1.00 for t ∈ (7 · (i +1), 7 · (i +2)] , x ∈ Γ1 ,

p = 1.0 for t > 0 , x ∈ Γ2 ,

Ks∇ p · ν = 0 for t > 0 , x ∈ ∂Ω\(Γ1 ∪Γ2)

for i = 0, 2, 4, . . . , where Γ1 = [0, 1] × 3 and Γ2 =2×[0, 1]. Prescribing p = 3.2 on Γ1 means that a water col-umn of height 0.2 (meter) rises above Γ1. Thus we wet thedomain Ω periodically, i.e., if t ∈ (7 · i, 7 · (i +1)]. The inter-vals (7 · (i +1), 7 · (i +2)] represent drying periods. Actually,we do not change the boundary condition for p on Γ1 instan-taneously but linearly over a small period of length 1/2 (day).This seems physically reasonable and avoids numerical oscil-lations due to the discontinuity of the data. The computationswere done on an almost uniform grid with 7324 elements. Theproposed adaptive time stepping was used. For the first wet-ting period t ∈ (0, 7] the calculated pressure head ψ is shownin Fig. 10.

Fig. 9. Computational domain and pressure head ψ (profile and isolines) atinitial time T = 0

The initial conditions for the species are

cD(0, x) = 0.0 , cA(0, x) = 2.0 , cX(0, x) = 3.0 ×10−3

in Ω. The contaminant m-xylene is injected at Γ1. More pre-cisely, for t > 0 the boundary conditions read as

cD(t, x) = 3.36 and cA(t, x) = 2.55 for x ∈ Γ1 ,

Di∇ci · ν = 0 , i = D, A , for x ∈ ∂Ω\Γ1 .

We use the degradation parameters given in (57).The calculated concentration profiles of the contaminant

m-xylene and biomass at time T = 7, 14, 90 and 200 daysare visualized in Fig. 11. For visualization purposes the color

Fig. 10. Pressure head (profile and isolines) at T = 0.4 and T = 0.8 (top)and T = 1.2 and T = 7.0 (bottom) days


range of the biomass is restricted to the interval [0, 0.25].Similarly to the example of Sect. 7, the contaminant plumefirst expands, then retreats and finally, when the acceptor be-

Fig. 11. Concentration of electron donor m−xylene (left) and biomass(right) with color range restricted to [0, 0.25] at T = 7 (top), T = 14,T = 90 and T = 200 (bottom) days

comes depleted, continues infiltrating into the domain. How-ever, during the drying periods t ∈ (7 · (i +1), 7 · (i +2)] thecontaminant expansion slows down strongly or even stopswhich prolongs the spread of m-xylene. The alternate wet-ting and drying phases also lead to the pattern in the biomassconcentration profile at T = 90 (cf. Fig. 11). We note that in-side the region close to Γ1 with high biomass concentrationand strong contaminant degradation the flow becomes alter-nately unsaturated and saturated due to the specific choice ofthe boundary condition for p at Γ1. Hence, our test problemin fact couples variably saturated subsurface flow with biore-active contaminant transport.

All calculations were done on a single processor PentiumIV PC with 1024 RAM. This limited the size of the com-putable problems and the number of usable finite elements.However, for problems which are similar to those consideredin Sects. 7 and 8 (cf. [5, 6, 32]) we compared the computedhigher order finite element solutions to the results which weobtained on a Linux cluster with 36 processors by using lowerorder conforming and mixed finite element discretizations butnow on very fine meshes. Thereby we checked the numeri-cal convergence of our calculations. For the future we plana parallel implementation of the proposed higher order dis-cretization which then will allow us a more rigorous studyand verification of the numerical convergence.

9 Conclusions

In this work we presented and analyzed carefully a higherorder approximation scheme for biochemically reacting con-taminant transport in the subsurface based on Monod kinet-ics. The approach reliably predicted the expected degrada-tion rates and prevented an overestimation due to its lessamount of inherent numerical diffusion. In realistic simu-lations on complex geometries and with an additional nu-merical computation of the flow field the robustness of themethod was shown. For the coupled flow and transport prob-lem an adaptive time stepping technique was developed antested. However, we are aware of the fact that an adap-tive spatial grid refinement is still necessary to further im-prove the accuracy and efficiency of the method. This isour work for the future. The paper strikingly underlines thepower of higher order discretizations. Therefore, the dis-continuous Galerkin method with a local adaptation of thepolynomial degree might be an interesting and promisingapproach to be applied to the considered contaminant trans-port and degradation model. This is also a project for thefuture.

Acknowledgements. The authors thank the referees for their comments forimproving the presentation of the paper.

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