Numerical modeling approaches for multiphase flow and

Transcription

1 Effiient Shemes for Reduing Numerial Dispersion in Modeling Multiphase Transport through Heterogeneous Geologial Media SPECIAL SECTION: TOUGH2 Yu-Shu Wu* and P. A. Forsyth When modeling transport of hemials or solute in realisti large-sale subsurfae systems, numerial issues are a serious onern, even with the ontinual progress made over the past few deades in both simulation algorithms and omputer hardware. The problem beomes even more diffiult when dealing with hemial transport in a vadose zone or multiphase flow system using oarse, multidimensional regular or irregular grids beause of the nown effets of numerial dispersion assoiated with moving plume fronts. We have investigated several total variation diminishing (TVD) or flux-limiter shemes by implementing and testing them in the T2R3D ode, one of the TOUGH2 family of odes. The objetives of this paper are (i) to investigate the possibility of applying these TVD shemes, using multidimensional irregular unstrutured grids, and (ii) to help selet more aurate spatial averaging methods for simulating hemial transport, given a numerial grid or spatial disretization. We present an appliation example to show that suh TVD shemes an effetively redue numerial dispersion. ABBREVIATIONS: REV, representative elementary volume; TVD, total variation diminishing. Numerial modeling approahes for multiphase flow and traer or hemial transport in porous media are generally based on methodologies developed for reservoir simulation and groundwater modeling. They involve solving oupled massonservation equations that govern the transport proesses of all hemial omponents in isothermal and nonisothermal subsurfae systems using finite-differene or finite-element shemes. Sine the 1960s, in parallel with rapid advanes in multiphase flow simulation and groundwater modeling, signifiant progress has been made in understanding and modeling solute transport through porous and fratured media (Sheidegger, 1961; Bear, 1972; Huyaorn et al., 1983; Isto, 1989; Falta et al., 1992; Unger et al., 1996; Forsyth et al., 1998; Wu and Pruess, 2000). Sine the 1970s, transport problems involving solute and ontaminant migration in porous and fratured formations have reeived inreasing attention in the groundwater and soil siene Y.-S. Wu, Lawrene Bereley National Lab., Bereley, CA 94720; P.A. Forsyth, Shool of Computer Siene, Univ. of Waterloo, Waterloo, ON, Canada. Reeived 30 May *Corresponding author (YSWu@lbl.gov). Vadose Zone J. 7: doi: /vzj Soil Siene Soiety of Ameria 677 S. Segoe Rd. Madison, WI USA. All rights reserved. No part of this periodial may be reprodued or transmitted in any form or by any means, eletroni or mehanial, inluding photoopying, reording, or any information storage and retrieval system, without permission in writing from the publisher. literature. As demanded by site haraterization, remediation, and other environmental onerns, many quantitative modeling approahes have been developed and applied (van Genuhten and Alves, 1982; Abriola and Pinder, 1985; Corapioglu and Baehr, 1987; Adenean et al., 1993; Forsyth, 1994). More reently, suitability evaluation of storing high-level radioative wastes in underground, unsaturated fratured ro has generated renewed interest in investigating traer or radionulide transport in a nonisothermal, multiphase fratured geologial system (e.g., Viswanathan et al., 1998; Robinson et al., 2003; Moridis et al., 2003). In addition, the appliation of traer tests, inluding environmental and manmade traers, has beome an important tehnique for haraterizing subsurfae porous-medium systems. Even with the ontinual progress in both omputational algorithms and omputer hardware made in the past few deades, modeling the oupled proesses involved in multiphase fluid flow and hemial migration in porous and fratured media remains a mathematial hallenge. Many unresolved issues and limitations with urrent numerial approahes still exist. One onern is that severe numerial dispersion often ours when using a multidimensional ontrol-volume type numerial grid in field-sale modeling studies. The problem beomes even greater when dealing with traer transport, in whih a general three-dimensional, oarse, irregular grid is used to solve advetion dispersion type governing equations for handling traer transport. To overome these numerial diffiulties, sientists have investigated a number of TVD or flux-limiter shemes and applied them in transport modeling, with varying suess (Sweby, 1984; Liu et al., 1994; Unger et al., 1996; Forsyth et al., 1998; Oldenburg and Pruess, Vol. 7, No. 1, February

2 1997, 2000). Many of these investigations, however, were demonstrated using one- or two-dimensional regular grids. Our wor ontinues the effort to redue numerial dispersion in simulating traer or hemial plumes as they travel spatially through porous or fratured media. In this study, we examine the effetiveness of these TVD shemes as they are used in two- or three-dimensional, irregular, and unstrutured grids. Our objetives are (i) to develop a general sheme for implementing different TVD shemes into multidimensional, irregular, unstrutured grids of porous or fratured media; (ii) to investigate the appliability of these TVD shemes to suh irregular unstrutured grids; and (iii) to help selet more aurate spatial averaging methods for simulating hemial transport, given a numerial grid or spatial disretization. The TVD shemes were implemented using the T2R3D ode (Wu et al., 1996), one of the TOUGH2 family of odes (Pruess, 1991). In the TOUGH2 numerial approah, a subsurfae domain is disretized using an unstrutured integrated finite-differene grid, followed by time disretization using a baward, first-order, finite-differene method. The resulting disrete linear or nonlinear equations are handled fully impliitly, using Newtonian iteration, with the fratured medium handled using a general multiontinuum modeling approah. Also, we present an appliation example to demonstrate that TVD shemes an in general effetively redue numerial dispersion when modeling transport through heterogeneous geologial fratured reservoirs. Model Formulation The physial proesses assoiated with fluid flow and hemial transport in porous media are governed by fundamental onservation laws, represented mathematially (on the marosopi level) by a set of governing equations. In addition, the movement of dissolved mass omponents or hemial speies within a fluid in a multiphase porous-medium system is governed by advetive, diffusive, and dispersive proesses and is further influened by other proesses, suh as radioative deay, adsorption, dissolution and preipitation, mass exhange or partition between phases, and other hemial reations. The formulation disussed in this setion and the next is, for ompleteness, to model a fully oupled fluid and heat flow and traer transport proess. For a simple ase of steady-state flow, the model formulation ould be signifiantly simplified. Governing Equation Consider a multiphase, nonisothermal system onsisting of several fluid phases, suh as gas, water, and oil (nonaqueous phase liquid), with eah fluid phase in turn onsisting of a number of mass omponents. To derive a set of generalized governing equations for multiphase fluid flow, multiomponent transport, and heat transfer, we assume that these proesses an be desribed using a ontinuum approah within a representative elementary volume (REV) in a porous or fratured medium (Bear, 1972). In addition, a ondition of loal thermodynami equilibrium is assumed so that at any time, temperatures, phase pressures, densities, visosities, enthalpies, internal energies, and omponent onentrations (or mass frations) are the same loally within eah REV of the porous medium. Aording to mass- and energy-onservation priniples, a generalized onservation equation of mass omponents and energy in the porous ontinuum an be written as follows: M = G + q + F [1] t where supersript is the index for the omponents, = 1, 2, 3,, N, with N being the total number of mass omponents and with = N + 1 for the energy omponent (thermal energy is treated as a omponent for onveniene); M is the aumulation term of omponent ; G is the deay or internal generation (reation) term of mass or energy omponents; q is an external soure sin term or frature matrix exhange term for mass or energy omponent and energy; and F is the flow term, desribing mass or energy movement, ontributed by multiphase flow, or diffusive and dispersive mass transport, or heat transfer, as disussed and defined below. Under equilibrium adsorption, the aumulation term Eq. [1] for omponent is M = φ ( ρ SX) + ( 1 φ) ρ s ρ w X w K d [2] ( = 1, 2, 3,..., N ) where φ is the porosity of porous media; is an index for fluid phase ( = g for gas, w for aqueous phase); ρ is the density of phase ; S is the saturation of phase ; X κ is the mass fration of omponent in fluid ; ρ s is the density of ro solids; and K d is the distribution oeffiient of omponent between the aqueous phase and ro solids to aount for adsorption effets. In the ase in whih omponents are subjet to a first-order radioative deay, the deay generation term is G = φλ ( S X ) ( 1 ) s w X w K ρ + φ ρ ρ d [3] ( = 1, 2, 3,..., N ) where λ is the radioative deay onstant of omponent. The generation term may also be subjet to other proesses suh as dissolution and preipitation, mass exhange and partition between phases, or hemial reations under equilibrium or nonequilibrium ondition. The aumulation term in Eq. [1] for the heat equation is usually defined as N+ 1 M = ( φρ SU) + ( 1 φ) ρsus [4] where U and U s are the internal energies of fluid and ro solids, respetively. The mass omponent transport is governed in general by proesses of advetion, diffusion, and dispersion. Advetive transport of a omponent or solute is arried by fluid flow, and diffusive and dispersive flux is ontributed by moleular diffusion and mehanial dispersion, or hydrodynami dispersion. These proesses are desribed using a modified Fi s law for the total mass flow term for a omponent, by advetion and dispersion, written as F = ( X ) D ( X ρ v + ρ ) [5] ( = 1, 2, 3,..., N ) Vol. 7, No. 1, February

3 where v is a vetor of the Dary s veloity or volumetri flow, defined by Dary s law to desribe the flow of single or multiple immisible fluids as r v = ( P ρg z) [6] μ where P, μ, and g are pressure, visosity of fluid phase, and gravitational aeleration onstant, respetively; z is the vertial oordinate; is absolute or intrinsi permeability; r is the relative permeability to phase, and g is gravitational onstant (vetor). In Eq. [5], D is the hydrodynami dispersion tensor aounting for both moleular diffusion and mehanial dispersion for omponent in phase, defined by an extended dispersion model (Sheidegger, 1961; Bear, 1972) to inlude multiphase effets as D vv = αt v δ ij + ( αl α T) + φ S τ d δij v [7] ( = 1, 2, 3,..., N ) T L where α and α are transverse and longitudinal dispersivities, respetively, in fluid of porous media; τ is tortuosity of the porous medium; d is the moleular diffusion oeffiient of omponent within fluid ; and δ ij is the Kroneer delta funtion (δ ij = 1 for i = j, and δ ij = 0 for i j), with i and j being oordinate indies. Equation [5] indiates that the mass flow onsists of two parts. The first part, the first term on the right-hand side of Eq. [5], is ontributed by advetion in all phases, and the seond part, the seond term on the right-hand side of Eq. [5], is diffusive flux by hydrodynami dispersion. Heat transfer in porous media is in general a result of both onvetive and ondutive proesses, although in ertain ases, radiation may also be involved. These heat-transfer proesses are ompliated by interations between multiphase fluids, multiomponents, and assoiated hanges in phases, internal energy, and enthalpy. Heat onvetion is ontributed by thermal energy arried mainly by bul flow of all fluids as well as by dispersive mass fluxes. On the other hand, heat ondution or radiation is driven by temperature gradients and may follow Fourier s law or Stefan Boltzmann s law, respetively. Then, the ombined, overall heat flux term, due to onvetion, ondution, and radiation in a multiphase, multiomponent, porous medium system, may be desribed as F N+ 1 = ρ ( h v ) + hd ρ + ( K T) ε σ T T o o ( X ) 4 [8] where h and h are speifi enthalpies of fluid phase and of omponent in fluid, respetively; K T is the overall thermal ondutivity; T is temperature; ε o is a radiation emissivity fator, and σ o (= J m 2 K 4 ) is the Stefan Boltzmann onstant. As shown in Eq. [8], the total heat flow in a multiphase, multiomponent system is determined by heat onvetion of flow and mass dispersion (the first two terms on the right-hand side of Eq. [8]), heat ondution (the third term on the righthand side), and thermal radiation (the last term on the righthand side). Constitutive Relationships To omplete the mathematial desription of multiphase flow, multiomponent transport, and heat transfer in porous media, Eq. [1], a generalized mass- and energy-balane equation, needs to be supplemented with a number of onstitutive equations. These onstitutive orrelations express interrelationships and onstraints of physial proesses, variables, and parameters and allow the evaluation of seondary variables and parameters as funtions of a set of primary unnowns or variables seleted to mae the governing equations solvable. Table 1 lists a ommonly used set of onstitutive relationships for desribing multiphase flow, multiomponent mass transport, and heat transfer through porous media. Many of these orrelations for estimating properties and interrelationships are determined by experimental studies. Numerial Formulation The methodology for using numerial approahes to simulate multiphase subsurfae flow and transport onsists in general of the following three steps: (i) spatial disretization of mass and energy-onservation equations of Eq. [1], (ii) time disretization, and (iii) iterative approahes to solve the resulting nonlinear, disrete algebrai equations. Among various numerial tehniques for simulation studies, a mass- and energy-onserving disretization sheme, based on finite volume or integral finite-differene or finite-element methods, is the most ommonly used approah, and the one disussed here. Disrete Equations The omponent mass and energy balane, as expressed in Eq. [1], is disretized in spae using a ontrol-volume, integrated finite-differene onept (Narasimhan and Witherspoon, 1976; Pruess, 1991). The ontrol-volume approah provides a general spatial disretization sheme that an represent a one-, two-, or three-dimensional domain using a set of disrete meshes. Eah mesh has a ertain ontrol volume for a proper averaging or interpolation of flow and transport properties or thermodynami variables. Time disretization is performed using a baward, first-order, fully impliit finite-differene sheme. The disrete nonlinear equations for omponents in the multiphase system at gridblo or node i an be written in a general form: n, + 1 n, + 1 n, Vi ( Ai + Gi Δ t Ai ) Δt n, + 1 n, + 1 = flowij + Qi [9] j ηi ( = 1, 2, 3,..., N, N + 1) and ( i = 1, 2, 3,..., N) where the supersript also serves as an equation index for all mass omponents with = 1, 2, 3,, N and = N + 1 denotes the heat equation; supersript n denotes the previous time level, with n + 1 the urrent time level to be solved; subsript i refers to the index of gridblo or node i, with N being the total number of nodes in the grid; Δt is time step size; V i is the volume of node i; η i ontains the set of diret neighboring nodes (j) of node i; and i i A and G are the aumulation and deay generation Vol. 7, No. 1, February

4 TABLE 1. Constitutive relationships and funtional dependene. Defi nition Funtion Desription Capillary pressure P = P( S) In a multiphase system, the apillary pressure (P ) relates pressures between the phases and is defi ned as a funtion of fl uid saturation (S ). Equilibrium adsorption Xs = Kdρ X Xs is the mass of omponent sorbed per mass of solids; and the distribution oeffi ient, K d, is treated as a onstant or as a funtion of the onentration or mass fration in a fl uid phase under the loal hemial equilibrium ondition. ρ = density of phase. Equilibrium partitioning First-order deay onstant Fluid density Fluid saturation Fluid visosity Henry s law Mass fration Partitioning oeffi ient Porosity Radioative deay Relative permeability Speifi enthalpies of gas α Kα : ω = ω ln(2) λ = T ρ = ρ ( PT,, X ) S μ = μ P = 1 ( PT,, X ) g = KHω w X = 1 ( ) K : K : P, T, X ωα and ω are the mole fration of omponent in phase α and, respetively; and K α : is the equilibrium partitioning oeffi ient of omponent between phases α and. T is the half-life of the radioative omponent. Density of a fl uid phase is treated as a funtion of pressure (P) and temperature (T), as well as mass ompositions, X ( = 1, 2, 3,, N ). Constraint on summation of total fl uid saturation. The funtional dependene or empirial expressions of visosity of a fl uid is treated as a funtion of pressure (P), temperature (T), and omposition, X. Pg is partial pressure of omponent in the gas phase; K H is Henry s onstant for omponent ; and ω is the mole fration of omponent in the water phase. w Constraint on mass frations X within phase. α = K α α : depends on hemial properties of the omponent and is a funtion of temperature (T), pressure (P ), and omposition, X. ( ) ( ) φ = φ o 1+ Cr P P o CT T T o C C t 0e λ = ( ) = S h r r P g g = Ug + C g φ o is the effetive porosity at a referene pressure, P o, and a referene temperature, T o ; C r and C T are the ompressibility and thermal expansion oeffi ient of the medium, respetively. C is the onentration of omponent in phase and is equal to C 0 at t = 0; λ is the radioative deay onstant. The relative permeability of a fl uid phase in a multiphase system are normally assumed to be funtions of fl uid saturation, S. U g the speifi internal energy of omponent in the gas phase; Pg is partial pressure of omponent in the gas phase; C g onentration of omponent in the gas phase (g/m 3 ). Speifi enthalpy of liquid Thermal ondutivity Internal energy, U P, of liquid phase is a funtion of pressure (P ) and temperature (T). h = U + ρ ( ) KT = KT S The thermal ondutivity of the porous medium is treated as a funtion of fl uid saturation (S ). terms, respetively, at node i. The flow term between nodes i and j, flow ij, and the sin soure term at node i for omponent or thermal energy, Q i, are defined below. Equation [9] has the same form regardless of the dimensionality of the system; that is, it applies to one-, two-, or three-dimensional flow, transport, and heat-transfer analyses. The aumulation and deay generation terms for mass omponents or thermal energy are evaluated using Eq. [2], [3], and [4], respetively, at eah node i. The flow terms in Eq. [9] are generi and inlude mass fluxes by advetive and dispersive proesses, as desribed by Eq. [5], as well as heat transfer, desribed by Eq. [8]. In general, the mass flow term is evaluated as (Wu and Pruess, 2000) flowij = FA, ij + FD, ij [10] ( = 1, 2, 3,..., N ) where F A, ij and F D, ij are the net mass fluxes by advetion and hydrodynami dispersion along the onnetion, respetively, with ( ) A, ij ij, ij F = A X F [11] where A ij is the ommon interfae area between onneted blos or nodes i and j; and the mass flow term, F,ij, of fluid phase is desribed by a disrete version of Dary s law. That is, the mass flux of fluid phase along the onnetion is given by F ρ r, ij = γij ψ j ψi μ [12] where γ ij is transmissivity and is defined differently for finitedifferene or finite-element disretization, and ψ is the flow potential term at node j or i (Pa). If the integral finite-differene sheme (Pruess, 1991) is used, the transmissivity is alulated as A γ ij = D + D ij i j [13] Vol. 7, No. 1, February

5 where D i is the distane from the enter of blo i to the interfae between blos i and j. The flow potential term in Eq. [12] is defined as ψ = P ρ gz [14] i i, i where Z i is the depth to the enter of blo i from a referene datum. For mass omponent transport, the flow term or the net mass flux by advetion and hydrodynami dispersion of a omponent along the onnetion of nodes i and j is determined by D, ij ij ij ( ) F = n A D ρx [15] where n ij is the unit vetor along the onnetion of the two blos i and j. The total heat flux along the onnetion of nodes i and j, inluding advetive, diffusive, ondutive, and radiation terms, may be evaluated, when using a finite-differene sheme, by flow N ij + 1 = ( h) F, ij + () h F D, ij Tj T i + Aij ( KT ) Di D + j A σε T T ( ) ij o o, j i [16] In evaluating the flow terms in Eq. [11 14] and Eq. [16], subsript ij + is used to denote a proper averaging or weighting of fluid flow, omponent transport, or heat-transfer properties at the interfae or along the onnetion between two blos or nodes i and j. The proper weighting of mass fration in Eq. [11] for alulating advetive mass flux is the objetive of this wor, whih is disussed in detail below. The onvention for the signs of flow terms is that flow from node j into node i is defined as + (positive) in alulating the flow terms. Wu and Pruess (2000) presented a general approah to alulating these flow terms assoiated with advetive and dispersive mass transport and heat transfer in a multiphase system, using an irregular and unstrutured multidimensional grid. Spatial and Temporal Weighting and Flux Limiter Shemes As shown in Eq. [11] and [12], in general, two types of spatial weighting shemes are needed in modeling multiphase traer transport. The first one, ( X ), is used in Eq. [11] for estimating the averaged mass fration ) for alulating advetive flux. The other, ( ρr/ μ ), is used in Eq. [12] for mobility weighting of the multiphase flow term. In the literature, fluxlimiter shemes have been used not only for the first type of weighting but also for the seond type (Blunt and Rubin, 1992; Oldenburg and Pruess, 2000). However, it has been observed in pratial simulations (Datta-Gupta et al., 1991) that the numerial smearing aused by saturation fronts is in general muh less severe than that with dissolved onentration fronts. Therefore, in this wor, we fous our attention on the use of the flux limiters for mass fration averaging or modeling onentration plume only, whereas the traditional, full upstream weighting is used in mobility or relative permeability averaging for estimating fluid displaement or saturation fronts. In addition to spatial weighting shemes, temporal weighting also needs to be addressed in numerial formulation. Commonly used temporal weighting shemes inlude fully impliit and Cran Niolson methods. The fully expliit weighting is rarely used beause of its strit limitation in time-step size. Among these shemes, the fully impliit method has proven itself to be most effetive in handling numerial problems assoiated with solving highly nonlinear multiphase flow equations. In partiular, the theoretial analysis of advetive dispersive transport through a one-dimensional finite-volume grid by Unger et al. (1996) indiates that the fully impliit sheme has no limitations in Courant number under various temporal weighting shemes, inluding flux limiters. They demonstrate how fully impliit temporal weighting leads to unonditionally stable solutions for linear advetion dispersion equations. Although fully impliit weighting is only a first-order approximation, with numerial errors of the same size as the time step, it is our experiene (in onduting hundreds of large, field-sale simulations of oupled multiphase flow and hemial transport) that fully impliit temporal shemes always result in stable solutions and that temporal disretization errors, aused by a fully impliit sheme, are of seondary importane in simulation when ompared with many other unnowns. The ey is to have a robust numerial sheme that leads to reliable and stable solutions under different spatial disretizations and various physial onditions. Beause it is impratial to define a Pelet or Courant number for detailed theoretial analyses in most field appliations when using multidimensional, irregular, unstrutured grids, fully impliit temporal weighting should be a first hoie. Seleting proper spatial-weighting shemes beomes ritial when dealing with oupled proesses of multiphase flow, hemial transport, and heat transfer in a fratured medium. This is beause frature and matrix harateristis often greatly differ; for example, there an be a many-orders-of-magnitude ontrast in flow and transport properties, suh as permeability and dispersivity. It is further ompliated in that there are no generally appliable weighting shemes or rules appliable to all problems or proesses (Wu and Pruess, 2000). The following weighting shemes were used for flux alulation in this wor: 1. Upstream weighting for relative permeability and/or mobility 2. Harmoni or upstream weighting for absolute permeabilities in global frature or matrix flow 3. Matrix absolute permeability, thermal ondutivity, and moleular diffusion oeffiients for frature matrix interation 4. Phase saturation based weighting funtions for determining diffusion oeffiients 5. Upstream weighted enthalpies for advetive heat flow 6. Central weighted sheme for thermal ondutivities of global heat ondution Consider the shemati of Fig. 1, representing a multidimensional irregular, unstrutured grid of porous and/or fratured media. To alulate advetive flux between nodes i and j, we also need the information from a seondary upstream node (denoted as i2up), whih is an upstream node to the upstream one, ups(i,j), between nodes i and j (Unger et al., 1996; Forsyth et al., 1998). As shown in Fig. 1, the node i2up is determined by the maximum potential method in terms of maximum fluid Vol. 7, No. 1, February

6 FIG. 1. Shemati for determining the seond upstream blo (i2up) for fl ow between blo i and blo j, using the geometri method and the maximum potential method. influx into the node ups(i,j), whih is implemented at Newtonian iteration level for every onnetion. Various weighting shemes for spatially averaged mass fration or onentration for advetive flux alulation between nodes i and j are summarized as upstream and entral. In the upstream weighting sheme, X = X [17] ( ) ( ) ij + ups( i, j ) where subsript ups(i,j) stands for the upstream node for fluid flow between nodes i and j. For the entral weighting sheme, ( X) + ( X) i j X = [18] ( ) 2 The several flux-limiter or TVD shemes tested inlude the van Leer limiter sheme, the MUSCL method, and the Leonard method. The van Leer flux limiter sheme is expressed as X = ( ) ( ) ( ) ( ) ( X X ) i j X +σ r ij [19] is the mass fration of downstream node of i dwn(, ) ups( i, j) ups( i, j) 2 X where ( ) dwn( i, j ) and j, defined as dwn( i, j) = i+ j ups( i, j) [20] The van Leer weighting fator σ(r ij ) is defined as () rij 0 ( if rij 0) σ = 2rij = 1+ r > ij ( if rij 0) with the smoothness sensor, r ij ( X) ( X) Dups( i, j) + D i2up ups( i, j) i2ups = ( X) ( X) ( Di + Dj) dwn( i, j) ups( i, j) [21] [22] where D ups(i,j) and D i2up are the distanes from the enter of blo ups(i,j) or its upstream blo i2up to their ommon interfae along the onnetion between the blos. The MUSCL method is desribed by X = ( ) ( X ) where s s s + 1 Δ + 1+ Δ ups( i, j) ( ) ( ) [23] 2Δ +Δ + ε s = [24] Δ + Δ + ε Δ = ζ ( X ) ( X ) ups( i, j) Δ = and Di + Dj ζ = D + D ( X ) ( X ) i2up + ups( i, j) dwn ups( i, j) i2up [25] [26] [27] In Eq. [24], ε is a small number, whih prevents a zero divide. The Leonard flux limiter is also desribed by Eq. [19] with the Leonard weighting fator σ(r ij ) defined as σ () r ij = max 0, min(2, 2 r ij,(2 + r ij )/3) [28] with r ij also defined by Eq. [22]. The numerial implementation of these TVD shemes is made in the T2R3D ode (Wu et al., 1996) for simulation of traer transport through an isothermal or nonisothermal system. Numerial Solution Sheme Over the past few deades, a number of numerial solution tehniques have been developed in the literature to solve the nonlinear, disrete equations of flow and transport in porous media. When handling multiphase flow, multiomponent transport, and heat transfer in a multiphase flow system, investigators predominantly use a fully impliit sheme. This is beause of the extremely high nonlinearity inherent in those disrete equations and the many numerial shemes with different levels of expliitness that may fail to onverge in pratie. In this setion, we disuss a general proedure to solve the disrete nonlinear Eq. [9] fully impliitly, using a Newton iteration method. We an write the disrete nonlinear Eq. [9] in a residual form as n, + 1 n, + 1 n, + 1 n, Vi Ri = { Ai + Gi Ai } Δt n, + 1 n, + 1 flowij Qi = 0 [29] j ηi = 1, 2, 3,..., +1; = 1, 2, 3,..., ) N i N Equation [29] defines a set of (N + 1) N oupled nonlinear equations that need to be solved for every balane equation of mass omponents and heat, respetively. In general, (N + 1) pri- Vol. 7, No. 1, February

7 mary variables per node are needed to use the Newton iteration for the assoiated (N + 1) equations per node. The primary variables are usually seleted among fluid pressures, fluid saturations, mass (mole) frations of omponents in fluids, and temperatures. In many appliations, however, primary variables annot be fixed and must be allowed to vary dynamially to aount for phase appearane and disappearane (Forsyth et al., 1998). The rest of the dependent variables are treated as seondary variables, inluding relative permeability, apillary pressures, visosity and densities, partitioning oeffiients, speifi enthalpies, thermal ondutivities, and dispersion tensor (listed in Table 1), as well as nonseleted pressures, saturations, and mass (mole) frations. In terms of the primary variables, the residual Eq. [29] at a node i is regarded as a funtion of the primary variables at not only node i but also at all its diret neighboring nodes j. The Newton iteration sheme gives rise to m ( x, ) ( ), + 1 δ xm, p+ 1 = Ri ( xm, p ) n, + 1 i m p n R [30] x m where x m is the primary variable m, with m = 1, 2, 3,, N + 1, respetively, at node i and all its diret neighbors; p is the iteration level; and i = 1, 2, 3,, N. The primary variables in Eq. [30] need to be updated after eah iteration: x = x +δ x [31] m, p+ 1 m, p m, p+ 1 The Newton iteration proess ontinues until the residuals n, 1 R + n or hanges in the primary variables δx m, p+ 1over an iteration are redued below preset onvergene toleranes. A numerial method is used to onstrut the Jaobian matrix for Eq. [30], as outlined in Forsyth et al. (1995). At eah Newton iteration, Eq. [30] represents a system of (N + 1) N linearized algebrai equations with sparse matries, whih are solved by a linear equation solver. When using the flux limiter shemes, as disussed in the last subsetion, advetive mass flux terms in the disrete Eq. [21] may depend on primary and seondary variables beyond the diret neighboring nodes, suh as at the node i2up. In suh a situation, the Newton iteration disussed here beomes inexat beause the Jaobian matrix does not inlude the ontributions with respet to the primary variables beyond neighboring nodes. Nevertheless, onverged solutions should be orret beause the residuals are exat. This omission in these Jaobian alulations may mae solution onvergene more problemati. Many numerial tests, however, have been made for multiphase traer transport, and no signifiant numerial problems have been observed. Fratured Media The mathematial formulations and flux-limiter shemes disussed above are appliable to both single-ontinuum and multiontinuum media, as long as the physial proesses involved an be desribed in a ontinuum sense within either ontinuum. When handling flow and transport through a fratured ro using the numerial formation of this setion, fratured media (inluding expliit frature, dual, or multiple ontinuum models) an be onsidered speial ases of unstrutured grids (See Fig. 1). Then, a large portion of the wor onsists of generating a mesh that represents both the frature and the matrix system under onsideration. Several frature and matrix subgridding shemes exist for designing different meshes for different frature matrix oneptual models (e.g., Pruess, 1983). One a proper unstrutured grid of a frature matrix system is generated, frature and matrix blos are identified to represent frature and matrix domains, separately. Formally they are treated identially for the solution in the model. However, physially onsistent frature and matrix properties, parameter weighting shemes, and modeling onditions must be appropriately speified for both frature and matrix systems. Appliation One example is presented here to demonstrate the appliation of the TVD shemes disussed above in handling transport through fratured media. The sample problem is based on a two-dimensional site-sale model developed for investigations of the unsaturated zone at Yua Mountain in Nevada. This example shows transport of one onservative (nonadsorbing) traer through unsaturated fratured ro using a two-dimensional unstrutured grid and a dual-permeability oneptualization for handling frature and matrix interation. The two-dimensional west east ross-setional model grid, shown in Fig. 2, has a total of 30,000 frature-matrix gridblos and 74,000 onnetions between them in a dual-permeability mesh. The potential repository is loated in the middle of the model domain, disretized with a loally refined grid (Fig. 2), at an elevation above 1100 m. The two-dimensional model uses the ground surfae as the top model boundary and the water table as the bottom boundary. Both top and bottom boundaries of the model are treated as Dirihlet-type boundaries; that is, onstant (spatially distributed) pressures, liquid saturations, and zero initial traer onentrations are speified along these boundary surfaes. In addition, on the top boundary, a spatially varying, steady-state, present-day infiltration map (as shown in Fig. 3), determined by the sientists of the USGS, is used in this study to desribe the net water reharge, with an average infiltration rate of about 5 mm yr 1 FIG. 2. Two-dimensional west east ross-setional model domain and grid showing lateral and vertial disretization, hydrogeologial layers, repository layout, and several faults inorporated. The rosssetion is loated between two Nevada oordinates of (169, 372 m, 233, 052 m) and (171, 844 m, 233, 858 m). Vol. 7, No. 1, February

8 over the model domain. In addition, an isothermal ondition is assumed in this study. The properties used for ro matrix and fratures in the dual-permeability model, inluding two-phase flow parameters of fratures and matrix as well as faults, were estimated from field tests and model alibration efforts (Wu et al., 2002). We onsider a onservative liquid traer migrating from the repository downward by advetive and dispersive proesses, subjet to the ambient steady-state unsaturated flow ondition. A onstant effetive moleular diffusion oeffiient of (m 2 s 1 ) is used for matrix diffusion of the onservative omponent. Dispersivities used are for matrix: α L, M = 5 m and α T,M = 0.5 m; for fratures: α L,F = 10 m and α T,F = 1 m. Transport starts with a finite amount of the traer initially released into the frature elements of the repository blos. After the simulation starts, no more traer will be introdued into the system, but the steady-state water reharge from the top boundary ontinues. Eventually, all the traer will be flushed out from the two-dimensional system through the bottom, the water table boundary, by advetive and diffusive proesses. Figures 4 and 5 show normalized traer onentration ontours in the frature ontinuum within the two-dimensional model at 10 yr of traer release, simulated using various weighting shemes of spatially averaged mass fration for advetive flux alulation. Comparisons of simulated onentrations between Fig. 4 (entral weighting) and Fig. 5 (TVD MUCSL) show a large differene at the time of 10 yr. Beause all three TVD shemes implemented in this study give similar results for this problem, only the results with MUSCL are shown for the TVD ases in Fig. 5. Figure 6 presents frational umulative mass breathrough urves at the water table, also showing some signifiant differenes between the results using the TVD shemes and the entral weighting. Overall, the simulation results indiate that at early time periods, suh as in the first 10 yr (Fig. 4), the entral weighting sheme underestimates advetive transport, while at later time periods (t > 100 yr) it overestimates advetive transport, beause of the seletion of too high or too low averaged onentration values. In omparison, the TVD shemes provide more aurate, onsistent numerial solutions at all the times. In addition, the TVD shemes are tested and found to have muh better numerial performane than the entral weighting sheme with respet to taing larger time steps or stability. Summary and Conlusions FIG. 3. Net infi ltration rate along the west east ross-setional model as surfae water reharge boundary ondition. FIG. 4. Conentration (C) distributions within the two-dimensional model at 10 yr, simulated using the entral weighting sheme. We have investigated several TVD shemes by implementing them into the TOUGH2 family of odes, using multidimensional irregular unstrutured grids. Our test results show that suh TVD shemes an redue numerial dispersion effetively, if used properly. In addition, FIG. 5. Conentration (C) distributions within the two-dimensional model at 10 yr, simulated using the total variation diminishing (TVD) sheme (MUSCL). Vol. 7, No. 1, February

9 FIG. 6. Breathrough urves of frational umulative traer mass arriving at the water table, sine release from the repository, simulated using the different weighting shemes. numerial performane with TVD shemes is signifiantly better than ommonly used entral weighting and is omparable to fully upstream weighting. It is enouraging that under multiphase onditions using relatively oarse spatial disretization, these TVD shemes provide more aurate simulation results for modeling large-sale field traer transport proesses through heterogeneous, fratured ro than the traditional modeling approahes. Appendix: Nomenlature A ij interfae area between onneted gridblos i and j (m 2 ) A i aumulation term at node i (g or J m 3 ) C onentration of omponent in phase D i, D j distane from enter of first (i) and seond (j) gridblo, respetively, to their ommon interfae (m) d moleular diffusion oeffiient of omponent within fluid (m 2 s 1 ) D hydrodynami dispersion tensor aounting for both moleular diffusion and mehanial dispersion for omponent in phase (m 2 s 1 ) F flow term, desribing mass or energy movement F,ij mass flux of phase, desribed by a disrete version of Dary s law (g s 1 ) F net mass fluxes of omponent by advetion along the Aij, F Dij, onnetion between nodes i and j (g or J s 1 ) net mass fluxes of omponent by hydrodynami dispersion along the onnetion between nodes i and j (g or J s 1 ) flow ij flow term of omponent or thermal energy between nodes i and j (g or J s 1 ) g gravitational aeleration (salar) (m s 2 ) g gravitational aeleration vetor (m s 2 ) i G deay generation terms at node i (g or J m 3 ) h speifi enthalpies of fluid phase (J g 1 ) h speifi enthalpies of omponent in fluid (J g 1 ) absolute or intrinsi permeability (m 2 ) r K T relative permeability to phase overall thermal ondutivity of ro (W m 1 C) distribution oeffiient of omponent between the aqueous phase and ro solids (m 3 g 1 ) M aumulation term of omponent n ij unit vetor along the onnetion of the two blos i and j N total number of nodes in the grid N total number of mass omponents (N + 1 for energy) P pressure, visosity of fluid phase (Pa) q external soure sin term or frature matrix exhange term for mass or energy omponent and energy Q i soure sin terms at node i (g J 1 s 1 ) r ij smoothness tensor for alulating flux limiters n, 1 R i + residual term of mass or energy onservation of omponent (g s 1 ) or (J s 1 ) at node i at time level n + 1. s parameter, defined in Eq. [24]. S saturation of phase. K d T temperature ( C). Δt time step size (s). U internal energy of phase (J/g). U s internal energy of ro solids (J/g). v vetor of the Dary s veloity or volumetri flow (m/s) V i volume of node i (m 3 ) x m generi notation for the mth primary variable (m = 1, 2, 3,, N, and N + 1) x m,p generi notation for the mth primary variable at Newton iteration level p (i = 1, 2, 3, ) X mass fration of omponent in the water phase X w mass fration of omponent in fluid z vertial oordinate (m) Z i depth to the enter of blo i from a referene datum (m) α L Gree Symbols longitudinal dispersivity in fluid of porous media (m) α T transverse dispersivity in fluid of porous media (m) δ ij Kroneer delta funtion (δ ij = 1 for i = j, and δ ij = 0 for i j) ε small, positive number ε o radiation emissivity fator φ effetive porosity of frature or matrix ontinua γ ij transmissivity (m 3 ) η i index set of diret neighboring nodes for node i λ radioative deay onstant of omponent (s 1 ) μ visosity of fluid (Pa s) ρ w density of water (g m 3 ) ρ density of phase (g m 3 ) ρ s density of ro grains (g m 3 ) σ(r ij ) van Leer weighting fator σ o Stefan Boltzmann onstant (= J m 2 K 4 ) τ tortuosity of porous media ψ i flow potential term at node i (Pa) ζ parameter, defined by Eq. [27] Δ parameter, defined by Eq. [25] Δ parameter, defined by Eq. [26] + Vol. 7, No. 1, February

10 Subsript i index of gridblos of neighbors to n; or total number of onneted elements to element n; or matrix. ij between two onneted gridblos i and j, or appropriate averaging between two gridblos ij + ½ appropriate averaging between two gridblos i and j j index of gridblos p Newton iteration level r relative or ro ups(i,j) upstream node for fluid flow between nodes i and j v volume w water index for fluid phase ( = g for gas, = w for aqueous phase) Supersript a air omponent dwn(i,j) upstream node between nodes i and j i2up seondary upstream node index for the omponents, = 1, 2, 3,, N and with = N + 1 for the energy omponent. n previous time step level + 1 urrent time step level t traer w water omponent ups(i,j) upstream node between nodes i and j ACKNOWLEDGMENTS The authors than Lehau Pan for his help with the model grid. Thans are also due to Guoxiang Zhang, Curt Oldenburg, and Dan Hawes for their review of the paper. This wor was supported in part by the U.S. Department of Energy. The support is provided to Bereley Laboratory through the U.S. Department of Energy Contrat No. DE- AC03-76SF Referenes Abriola, L.M., and G.F. Pinder A multiphase approah to the modeling of porous media ontamination by organi ompounds: 1. Equation development. Water Resour. Res. 21(1): Adenean, A.E., T.W. Patze, and K. Pruess Modeling of multiphase transport of multiomponent organi ontaminants and heat in the subsurfae: Numerial model formulation. Water Resour. Res. 29(11): Bear, J Dynamis of fluids in porous media. Amerian Elsevier, New Yor. Blunt, M., and B. Rubin Impliit flux limiting shemes for petroleum reservoir simulation. J. Comp. Phys. 102: Corapioglu, M.Y., and A.L. Baehr A ompositional multiphase model for groundwater ontamination by petroleum produts: 1. Theoretial onsiderations. Water Resour. Res. 23(1): Datta-Gupta, A., L.W. Lae, G.A. Pope, K. Sepehrnoori, and M.J. 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Triay A reative transport model of neptunium migration from the potential repository at Yua Mountain. J. Hydrol. 209: Wu, Y.S., C.F. Ahlers, P. Fraser, A. Simmons, and K. Pruess Software qualifiation of seleted TOUGH2 modules. LBL-39490, UC-800. Lawrene Bereley National Laboratory, Bereley, CA. Wu, Y.S., L. Pan, W. Zhang, and G.S. Bodvarsson Charaterization of flow and transport proesses within the unsaturated zone of Yua Mountain, Nevada. J. Contam. Hydrol. 54: Wu, Y.S., and K. Pruess Numerial simulation of non-isothermal multiphase traer transport in heterogeneous fratured porous media. Adv. Water Resour. 23: Vol. 7, No. 1, February