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A fully coupled numerical model for two-phase flow with contaminant transport and biodegradation kinetics

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A fully coupled numerical model for two-phase flow with contaminant transport and biodegradation kinetics
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  COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Commun. Numer. Meth. Engng  2001;  17 :325–336 A fully coupled numerical model for two-phase owwith contaminant transport and biodegradation kinetics Claudio Gallo 1 and Gianmarco Manzini 2 ; ∗ 1 CRS4 ;  Zona Industriale Macchiareddu ;  Uta ;  Cagliari  ;  Italy 2 IAN - CNR ;  via Ferrata 1 ;  27100 Pavia ;  Italy SUMMARYA fully coupled numerical model is presented which describes biodegradation kinetics and NAPL-aqueous two-phase ow in porous media. The set of governing partial dierential equations is split intwo subsystems, the former one in terms of phase pressure and saturations, and the latter one in termsof contaminant concentration and bacterial population distribution. Non-linear saturation dependence inBrooks–Corey relative permeability functions and capillary pressure eects are incorporated in a mixed-hybrid nite element model. The non-linear degradation kinetics  a la  Monod is taken into account asa source term in the nite-volume discretization of the equation modelling contaminant transport. Theglobal coupling is performed by using a nested block-iteration technique. A set of numerical experimentsdemonstrates the eectiveness of the method. Sensitivity analysis results are also presented. Copyright ?  2001 John Wiley & Sons, Ltd. KEY WORDS : bioremediation; two-phase ow; mixed nite elements; nite volumes 1. INTRODUCTIONAn organic contaminant leaking in groundwater is generally advected as a separate  non-aqueous phase liquid   (NAPL) one, but may also be partially dissolved in the  aqueous phase ,namely water. This dissolved part of NAPL is liable to be degraded by the bacterial pop-ulation naturally present in soil. A major simplication in the mathematical and numericalmodelling of such a process accounts for the contaminant as a solute passively transportedin the aqueous phase [1–3], no taking care of the presence of dierent phases. However, a better understanding of the underlying phenomena needs an explicit description in terms of a multi-phase ow model coupled with a transport equation for the contaminant dissolvedin water [4 ;  5]. The dissolved NAPL contaminant is degraded according with a Monod-typekinetics, see for instance Reference [6]. ∗ Correspondence to: G. Manzini, IAN - CNR, via Ferrata 1, 27100 Pavia, ItalyContract =  grant sponsor: Faculty of Civil Engineering and Geosciences in Delft, The NetherlandsContract =  grant sponsor: Italian C.N.R.Contract =  grant sponsor: Sardinian Regional Authorities Received 6 March 2000 Copyright  ?  2001 John Wiley & Sons, Ltd.  Accepted 4 December 2000  326  G. GALLO AND G. MANZINI It is worth mentioning that transport of contaminants may often take place in rather de-formable porous media. Hence, the multi-phase ow model should also be completed byincluding soil displacement eects, as in References [7–9]. In this work we explore a nu-merical coupling of contaminant transport and biodegradation in the framework of a simpletwo-phase ow model and its preliminary application to 1-D sensitivity analysis. Extensionto more complex scenarios, which may deserve for special numerical treatments, will be thesubject of future work. The discretization relies on a block-iterative splitting algorithm, andseparates the mathematical model in two sets of equations, the former for phase pressures andsaturations and the latter for contaminant transport and biodegradation. Pressure =  saturationequations are re-formulated in terms of total pressure and velocity elds following the ap- proach of Chavent and Jare, see Reference [10]. The main reason for this approach is thatecient numerical methods can be devised to take advantage of the physical properties in-herent in the ow equations. This set of equations is approximated by the discontinuouslowest-order mixed-hybrid Raviart–Thomas nite element method [11]. The saturation equa-tions are then approximated by cell-centre nite volumes and advanced in time by an explicittwo-stage Runge–Kutta scheme. In the second set of equations, a predictor–corrector strategydecouples the advective transport of the dissolved contaminant concentration from the sourcereaction term which describes the consumption by soil bacterial population. The dierentialequation that modellizes growth and decay of bacteria is analytically solved at any iterativestep of the computation after updating the dissolved contaminant concentration eld [3]. Theoutline of the paper is the following. In Section 2 we introduce the mathematical model,in Section 3 we focus on the iterative splitting approach and shortly discuss the numericaltechniques, and in Section 4 we present some preliminary 1-D numerical results.2. MATHEMATICAL MODELThe two-phase ow is governed by the couple of phase conservation equations @@t  (  ‘  s ‘ ) +  ∇ ·  F ‘  = q ‘  +  q ‘ n  (1) @@t  (  n  s n ) +  ∇ ·  F n  = q n  −  q ‘ n  (2)where the subscripts  ‘  and n refer to aqueous and non-aqueous phases,  t   is the time,    isthe porosity,   ‘  and   n  are the constant phase densities,  s ‘  and  s n  are the saturations,  q ‘  and q n  are the mass source =  sink terms and  q ‘ n  is a mass exchange term between the two phases.The advective uxes  F ‘  and  F n  take the usual Darcy’s form F ‘  = −  ‘  ‘ K  · ∇ (  P  ‘  +   ‘ gh )=  ‘ v ‘  (3) F n  = −  n  n K  · ∇ (  P  n  +   n gh )=  n v n  (4)where  P  ‘  and  P  n  are the pressure elds,  K  is the absolute permeability tensor,   ‘  and   n  are the phase mobilities,  v ‘  and  v n  are the phase velocities,  g  is the gravitational constant and  h  is the Copyright  ?  2001 John Wiley & Sons, Ltd.  Commun. Numer. Meth. Engng  2001;  17 :325–336   NUMERICAL MODEL FOR TWO-PHASE FLOW  327head above a given datum. Phase mobilities are dened as  ‘  =  k  rel ;‘  ‘ ;  n  =  k  rel ; n  n (5)where  k  rel ;‘  and  k  rel ; n  are the soil relative permeabilities and   ‘  and   n  are the viscosities.Let us assume that densities and viscosities are constant (isothermal system assumption), andthat, as usual, saturations sum to unity. Following the ideas of Chavent [10], we can re-state(1)–(2) in terms of the aqueous phase pressure  P  ‘  and the total velocity  v T  = v ‘  +  v n  whichsatisfy ∇ ·  v T  =  Q T v T  =  − (  ‘  +   n ) K  · ∇  P  ‘  +  Q S (6)The global source terms  Q S  and  Q T  are given by Q S  = K  ·  q S ; Q T  =  q ‘  ‘ +  q n  n (7)with q S  = − (  ‘  ‘  +   n  n ) g  −   n  P   c ∇  s ‘  (8)where  g = ∇ ( gh ) is the gravity vector. Assuming water as the wetting uid, we introducedin (8) the capillary pressure dened as  P  c  =  P  n  −  P  ‘ .Phase velocities are then given back by v ‘  =   ‘  ‘  +   n ( v T  −  Q S )  −   ‘  ‘ K  ·  g  (9) v n  = v T  −  v ‘  (10)The transport of the dissolved NAPL contaminant concentration  C   by the water bulk motionis described by @ ( s ‘ C  ) @t   +  ∇ ·  ( v ‘ C  )= q ‘ n  −  q  bio  (11)The source term  q ‘ n  takes into account the mass transfer process from NAPL to aqueous phase. This term is linearly proportional to the dierence between the maximum concentrationof NAPL that can be dissolved in water at equilibrium,  C  eq , and its actual concentration,  C  ,i.e. q ‘ n  = s n k  do ( C  eq −  C  ) (12)The mass transfer proportionality coecient is denoted by  k  do .The sink term  q  bio  is responsible for biodegradation and takes the form q  bio  =  N  c m c Y  bio   C  K  1 =  2  +  C    (13) Copyright  ?  2001 John Wiley & Sons, Ltd.  Commun. Numer. Meth. Engng  2001;  17 :325–336  328  G. GALLO AND G. MANZINI where    bio  is the maximum degradation rate,  K  1 =  2  the half-saturation constant,  N  c  the number of bacterial colonies per unit volume,  m c  the mass of a colony and  Y   a yield coecient [6].Finally, the rate of microbial growth =  decay is obtained by balancing bio-mass reproductionand decay1  N  c @N  c @t   =    bio   C  K  1 =  2  +  C   −  k  d   (14)where the parameter   k  d  is the bacterial decay constant [2 ;  6].Finally, some constitutive relations must be specied to get closure of the model. Saturationsdepend on  P  c  via the Brooks–Corey relation [12],  P  c  =  P  d  ∗  s − 1 =‘ e  (15)In Equation (15),  P  d  is the entry pressure,    a tting parameter related to the pore-sizedistribution,  s ‘ e  the eective liquid saturation, dened as  s ‘ e  =(  s ‘  −  s ‘ r  ) =  (1  −  s ‘ r   −  s nr  ), and  s ‘ r   and  s nr  , respectively, the wetting- and non-wetting-phase residual saturations. The relative permeabilities are given by k  rel ;‘  =(  s ‘ e )  ; k  rel ; n  =(1  −  s ‘ e )  (16)the value of the parameter     being tted via regression on experimental data.3. NUMERICAL MODELBasically, our discretization approach consists in an external loop in time, namely  [Global] ,split in two internal loops, namely  [P-S]  and  [C-bio] . The loop  [P-S]  updates the phase pressures  P  ‘  and  P  n  and the saturations  s ‘  and  s n , by solving iteratively system (6) andEquations (1) and (2). The loop  [C-bio]  updates the concentration of the NAPL contaminantdissolved in water and the microbial population distribution.At any time step, the outer loop alternatively iterates on the two inner loops until conver-gence is reached: while  t¡t  max loop  [Global]  until  convergence loop  [P-S]  until  convergence loop  [C-bio]  until  convergence check  global convergence end loop t   = t   +  t  end while Copyright  ?  2001 John Wiley & Sons, Ltd.  Commun. Numer. Meth. Engng  2001;  17 :325–336   NUMERICAL MODEL FOR TWO-PHASE FLOW  329The loop  [P-S]  takes the form loop  [P-S]  until  convergence estimate  ∇ S  t  + t;k  from  S  t  + t;k  solve for  P  t  + t;k  +1 ‘  and v t  + t;k  +1T  [system (6)] estimate v t  + t;k  +1 ‘  and v t  + t;k  +1n  by using  S  t  + t;k  [Equations (9)–(10)] solve for  S  t  + t;k  +1 [Equations (1)–(2)] check   S  t  + t;k  +1 −  S  t  + t;k   6  [P − S] end loop where  S   denotes the vector of unknowns (  s ‘ ; s n ) T ,  k   is the iteration index,   [P − S]  is a user-specied tolerance, and   ·   is the Euclidean norm.At each internal step, system (6) is solved by approximating the normal component of thetotal velocity eld at cell interfaces,  v T · n  say, by the discontinuous lowest-order mixed-hybridRaviart–Thomas nite elements and the aqueous phase pressure  P  ‘  by piecewise constant ele-ments over the computational cells and at cell interfaces [11]. Let us denote these approximateelds by, respectively,  q h ,     h  and   h . The symbol  h  denotes the maximum diameter of all thecells  K   in a given mesh  T h ;  |  K  |  is the cell volume and  @K   the cell boundary. The variationalformulation reads as    K − 1 (  ‘  +   n )  q h  ·  w h  −   K     K     h ∇ ·  w h  +   K    @K   h  w h  ·  n  K   =    Q S  ·  w h   K     K  v h ∇ ·  q h  =    Q T v h  (17)   K    @K  q h  ·  n  K   h  =0where the test functions  w h ; v h  and   h  are taken in the same functional spaces of the corre-sponding unknowns. Standard algebraic manipulations yields a linear problem which is solvedfor the set of Lagrangian multipliers   h  by using the static condensation technique. Backwardsubstitution closes the solution algorithm [11]. This non-conforming nite element methodguarantees that a zero-divergence constraint is well satised by the total velocity eld.This issue is required to fairly simulate the non-linear degradation kinetics, as pointed out inReference [5].The semi-discrete nite volume formulation that approximates Equations (1) and (2) isobtained by integrating separately in a cell-wise fashion each phase saturation equation, ap- plying the divergence theorem, approximating the interface integrals with the midpoint rule,and nally computing the advective uxes by an upwind estimation of the phase velocities v ‘  and  v n . This method takes the form |  K  | d S   K  d t   +   k  ∈  (  K  ) l k   H  ( S  int k   ;S  ext k   ; n k  ) +   k   ∈   (  K  ) l k    H  (bc) k    ( S  int k   ) =  |  K  | Q ( S   K  ) ;  ∀  K   ∈ T h (18) Copyright  ?  2001 John Wiley & Sons, Ltd.  Commun. Numer. Meth. Engng  2001;  17 :325–336
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