Nonlinear iteration methods for nonequilibrium
multiphase subsurface flow
P.A. Forsyth
a ,*, A.J.A. Unger
b& E.A. Sudicky
c aDepartment of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
b
HydroGeoLogic Inc., Herndon, VA, USA
c
Department of Earth Sciences, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
(Received 4 April 1996; revised 10 October 1996; accepted 14 April 1997)
Fully implicit, fully coupled techniques are developed for simulating multiphase flow with nonequilibrium mass transfer between phases, with application to groundwater contaminant flow and transport. Numerical issues which are addressed include: use of MUSCL or Van Leer flux limiters to reduce numerical dispersion, use of full or approximate Jacobian for flux limiter methods, and variable substitution for increased Newton iteration efficiency. A comparison of the performance of equilibrium and nonequilibrium models is also presented.q1998 Elsevier Science Limited. All rights reserved.
Key words: nonequilibrium multiphase porous media flow, NAPL, Newton iteration,
flux limiter.
1 INTRODUCTION
Many groundwater contamination incidents involve non-aqueous phase (NAPL) liquids. Examples of such contami-nants include gasoline, PCBs, TCE, creosote, and wastes from petrochemical refineries, industrial degreasing opera-tions, and fertilizer manufacturing. Usually, NAPL con-taminants dissolve in groundwater at small concentrations. Consequently, relatively small amounts of NAPL can result in contaminated groundwater plumes many kilometers in extent.
Although many new technologies are being tested in order to remediate NAPL contaminated sites, it is first necessary to determine the extent of the contamination. Also, pump and
treat operations are often initiated to control the extent of the
plume and to increase the rate of dissolution of the non-aqueous contaminant source. Simulation of the original spill and pump and treat operation is therefore of interest.
Most multiphase subsurface flow simulators assume instantaneous equilibrium between phases 1–11. However, there is mounting evidence that the contaminant in the non-aqueous phase cannot be considered to be in instantaneous
equilibrium with the flowing water phase, at least for typical groundwater velocities12.
As a result, there has been interest in simulating the none-quilibrium dissolution of nonaqueous phase liquids (NAPLs) in groundwater contamination problems 12–14. There is some evidence that nonequilibrium mass transfer effects can be an important controlling effect for dissolution of NAPL contaminants12–17. These effects can have impor-tant ramifications for development of the dissolved contami-nant plumes. In addition, nonequilibrium mass transfer between phases can be a controlling factor in the effective-ness of pump and treat methods for site remediation. The dissolution of NAPL contaminants into the aqueous phase is an important process even for air sparging and vacuum extraction10.
The objective of this article is to examine several numer-ical issues related to simulation of multiphase flow with nonequilibrium mass transfer between phases. Briefly, these issues are:
• a comparison of TVD (total variation diminishing) 18–20
and MUSCL21type flux limiters for reduction of numerical dispersion of the dissolved contaminant;
• efficient methods for solution of the discrete non-linear algebraic equations when using a high-order flux limiter;
Advances in Water Resources 21 (1988) 433–449
q1998 Elsevier Science Limited All rights reserved. Printed in Great Britain 0309-1708/98/$19.00 + 0.00 PII: S 0 3 0 9 - 1 7 0 8 ( 9 7 ) 0 0 0 1 0 - 9
• use of variable substitution methods to improve Newton iteration convergence;
• a comparison of the numerical performance of equi-librium and nonequiequi-librium models on the same scenarios, using typical field scale interphase mass transfer data13.
For single-phase reactive transport flows, it is common to use a form of operator splitting22to discretize the equations. The usual rationale for this form of discretization is that, for a large number of species, it is very costly to construct and solve the Jacobian matrix. Also, the transport of each species can be solved independently (if the reactions are split) due to the very weak coupling between the transport equations.
In the multiphase case, the flows in each phase can be decoupled using an IMPES type procedure 23. However, since the flows in different phases are strongly coupled through the nonlinear relative permeability and capillary pressure terms, the timestep restrictions of IMPES methods can be very severe. Consequently, as discussed in19, it is often desirable to use fully implicit methods for simulation of multiphase, multicomponent flows. However, as pointed out in22, use of a low-order method for trace contaminant transport, coupled with an implicit method, may cause poor accuracy. Methods which treat multiphase transport using an IMPES type method also have great difficulty simulating flows near injection/production wells, since the flow velo-cities are very high near these points 19. For this reason, there has recently been some interest in using flux limiter methods for the trace transport terms in implicit subsurface flow models19,20,24.
In this article we will consider a three-phase model with a passive air phase. It will be assumed that there is a single NAPL component, which can partition into the nonaqeuous phase, the aqueous phase, and the solid phase. It will also be assumed that contaminant partitioning and transport in the gas phase is negligible. These assumptions would be suita-ble for simulating NAPL contaminants with low vapor pressures.
Recent work has shown that flux limiters based on TVD (total variation diminishing) ideas 18–20 are effective in reducing numerical smearing for the dissolved contaminant fronts. Alternative high-order flux limiters, which are pop-ular in aerospace applications, are MUSCL methods 21. However, if full Newton iteration is used for the flux limit-ers discretized equations, then this results in a Jacobian with more non-zeros than a simple upstream or centrally weighted discretization. Note that in19, use of the full Jaco-bian is advocated in conjunction with use of a high-order flux limiter. Consequently, this Jacobian will be more expensive to construct and solve compared to an upstream weighted Jacobian. For some applications, it has been found that convergence of the nonlinear iteration is unacceptably slow unless a full Newton iteration is used25. In this work, we will carry out a systematic comparison of nonlinear iterative methods based on: (a) the full Jacobian; (2) an
approximate Jacobian which ignores some of the non-zero entries in the full Jacobian; (3) a Jacobian based on a low order discretization. Comparisons will be carried out using both MUSCL and TVD based limiters.
For unsaturated flow simulations under very dry condi-tions, it has been found that use of variable substitution methods (even though not required by the physics of the problem) can result in an order of magnitude reduction in the number of Newton iterations required for convergence 26. Rather surprisingly, we will demonstrate that a similar variable substitution method can also improve performance by factors of three to five, in the case of multiphase flow. This improvement occurs even for situa-tions where the air–water capillary pressure curves are not very steep, in contrast to the case described in26.
We will also compare the numerical performance of equi-librium and nonequiequi-librium models, in terms of total New-ton iterations, for a range of typical interphase mass transfer parameters.
We emphasize that the scope of this paper is restricted to numerical performance issues. Detailed three-dimensional simulations which study the effects of heterogeneities, dif-ferent interphase mass transfer models, and uncertain con-stitutive relations are described in27.
2 FORMULATION
As mentioned in the introduction, we are considering pro-blems where there are two active phases: water (w) and nonaqueous (n), and a single species NAPL contaminant. The air phase (a) pressure is assumed constant, and the NAPL contaminant is considered to be nonvolatile. The NAPL contaminant can partition into both nonaqeuous and aqueous phases, as well as the solid phase. Conse-quently, these equations can be used to describe the flow of NAPL contaminants having a low vapour pressure in variably saturated media. A similar model, but assuming instantaneous phase equilibrium, was described in11.
The water component conservation equations is:
]
]t[(fSwMwXw)]¼ ¹=·(MwXwVw)þqw, (1)
where Sw ¼ saturation of aqueous phase; Mw ¼ molar density of aqueous phase; Xw ¼mole fraction of water in aqueous phase; Vw ¼aqueous phase velocity; qw ¼water source/sink term.
In eqn (1), we have neglected the dispersive flux of the water component in the aqueous phase, which is reasonable since the mole fraction of water in the aqueous phase is nearly unity, and the mass transport of water will be dominated by water con-vection. (A similar assumption was made in11.)
equation is:
]
]t[f(SnMnþSwMwXnþrbKdMwXn)]
¼ ¹=·(MwXnVw)¹=·(MnVn)
þ=·(fSwDwMw=Xn)þqn, ð2Þ
where Sn¼saturation of nonaqueous phase; Mn¼molar density of nonaqueous phase; Xn¼mole fraction of NAPL contaminant in the aqueous phase; Vn¼nonaqeuous phase velocity; Kd ¼sorption coefficient; rb¼mass density of solid phase; Dw ¼ dispersion/diffusion tensor, aqueous phase; qn¼NAPL contaminant source/sink term.
In the same case of instantaneous phase equilibrium, the phase equilibrium condition is simply:
Xn¼Xpn if Sn.0 (3)
where Xn*¼maximum dissolved NAPL contaminant mole fraction.
In the case of nonequilibrium between phases, eqn (2) becomes two partial differential equations, a conservation law for the NAPL contaminant in the nonaqueous phase, and a conservation law for the NAPL contaminant dissolved in water. In addition, there are two ordinary differential equations which determine the rate of contaminant sorption onto the solid phase, and the rate of contaminant mass trans-fer from the nonaqueous to the aqueous phase:
]
]t[f(SnMn)]¼ ¹R˙¹=·(MnVn)þqnn (4)
]
]t[f(SwMwXn)]¼ þR˙¹A˙=·(MwXnVw)
þ=·(fSwDwMw=Xn)þqnw, ð5Þ
where qnn¼source/sink term, NAPL contaminant in non-aqueous phase; qnw¼source/sink term, NAPL contaminant in aqueous phase;R˙ ¼mass transfer rate, NAPL contami-nant, nonaqueous into aqueous phase; A˙ ¼mass transfer rate, dissolved NAPL into sorbed phase.
The mass transfer rates are given by
˙
A¼lA(rbKdMwXn¹A) (6)
˙
R¼MwlRfSwSbn(Xpn¹Xn), (7)
where lR ¼ time constant for R; A˙ ¼ molar density for sorbed NAPL contaminant;lA¼time constant forA;˙ b¼ exponent in mass transfer rate.
Note that in eqn (7) the rate of mass transfer from the nonaqeuous phase into the water phase uses the model from 13. This model was compared to field-scale experi-ments with good results. Other forms for this mass transfer rate have been suggested, see for example12. More detailed comparison of different forms for the mass transfer rates may be found in27. However, for the purposes of this article, we will use the simple form in eqn (7). A first-order model is used for the mass transfer of dissolved NAPL into the sorbed phase (eqn (6)).
In eqn (1), eqn (2), eqn (4) and eqn (5), the velocity of each phase l¼n, w is given by:
Vl¼K·krl=ml(=Pl¹rlg=D), (8)
where Pl¼pressure in phase l; D¼depth; K¼absolute permeability tensor; rl ¼ mass density of phase l; g ¼ gravitational acceleration; ml¼viscosity of phase l; krl¼ relative permeability of phase l.
There exist the following constraints and constitutive relations among the above variables (a¼air phase):
SnþSaþSw¼1 (9)
Pa¼PnþaPcan(Sa)þ(1¹a)[Pcaw(Sg)¹Pcnw(Sw¼1)]
Pn¼PwþaPcnw(Sw)þ(1¹a)Pcnw(Sw¼1)
where Pclk is the capillary pressure between phase l and phase k (l,k ¼n,w,a), and where
Pa¼air pressure (10)
¼constant(passive air assumption)
Sa¼air saturation,
and where
a¼min(1,Sn=Spn): (11)
Pcan, Pcnw, Pcaw are experimentally determined capillary pressures. Stone’s second model 28 is used for three phase-relative permeabilities. Sn* is a blending parameter used to ensure that the capillary pressure has the correct form as the nonaqueous phase saturation goes to zero5(in which case the capillary pressure reduces to the air–water value Pcaw).
The dispersion/diffusion tensors have the form29
(fSwDw)xx¼aL magnitude of aqueous phase velocity;aTV¼vertical trans-verse dispersity; aTH ¼ horizontal transverse dispersity;
aL ¼ longitudinal dispersity; dw ¼ molecular diffusion;
Note that we have altered the expression for the disper-sion/diffusion tensor (eqn (12)) in29to account for differing horizontal and vertical transverse dispersities30.
3 DISCRETIZATION
Let Nibe the usual Lagrange polynomial C0basis functions. Let Sl, Plbe the saturation and pressure in phase l¼n, w,
and let Xnbe the NAPL mole fraction in water, and let Slj,
Plj, Xnjbe the values of these variables at the jth node. Then,
Sl, Pl, Xncan be expanded in terms of the basis functions in
Mass lumping is used for the time derivative term, back-ward Euler timestepping is used, and a single-point quad-rature method is used for the mobility terms 26,32. This results in evaluation of the mobility terms at a node. We then obtain the following discrete equations:
Water component conservation:
where superscript N is the time level.
The discrete version of the mass conservation of NAPL contaminant, assuming instantaneous phase equilibrium (eqn (2)), is:
NAPL conservation (instantaneous equilibrium):
[f(SnMnþSwMwXnþrbKdMwXw) such thatgijandgij9are non-zero. Further details about the above discretization method can be found in3,5,8,20,32. Note that in eqn (17) the superscript M indicates that gij9 is evaluated partially implicitly 8.
In the case of nonequilibrium between phases, the dis-crete version of eqns (4)–(7) are:
NAPL contaminant in the nonaqueous phase:
[f(SnMn)]
NAPL contaminant in the aqueous phase:
where
Sn9¼max(Sn,0:0): (20)
Equation (20) ensures the correct behaviour for the mass transfer as Sn→0. In eqn (19) AiN
þ1is given by solving the
following equation:
ANi þ1¹ANi
Dt ¼lA[(rbKdMwXn¹A)] Nþ1
i : (21)
In this case (form of mass transfer eqn (6)), AiN
þ1can be
solved simply from eqn (21) and inserted directly into eqn (19). In general, however, it may be necessary to solve eqn (21) numerically.
A TVD type flux limiter weighting in eqn (15), eqn (18), and eqn (19) is given by8,18–20,33:
(Xn)FL(i,j)¼(Xn)ups(i,j)þj(rij)
(Xn)dwn(i,j)¹(Xn)ups(i,j)
2
,
(22)
where (Xn)dwn(i,j) is the downstream point between nodes i and j,
dwn(i,j)¼iþj¹ups(i,j): (23)
The smoothness sensor rijin eqn (22) is
rij¼
(Xn)ups(i,j)¹(Xn)i2up(ups(i,j),dwn(i,j))
kxups(i,j)¹xi2up(ups(i,j),dwn(i,j))k (Xn)dwn(i,j)¹(Xn)ups(i,j)
kxdwn(i,j)¹xups(i,j)k
(24)
A van Leer limiter18,34is used in eqn (22)
j(r)¼0, if r#0 (25)
¼ 2r
1þr, if r.0:
In eqn (24), i2up(i,j) is the second upstream point to node i assuming i is upstream of j and xiis the location vector of node i. There are several methods of selecting i2up(i,j). A geometric construction can be used (i.e. find the neighbour of node i which is most nearly in the direction from node j to node i), or the neighbour node of node i can be selected which has the largest flow into node i (the maximum poten-tial method). Both these techniques are described in detail in8,20. A case where the two methods would differ in the selection of i2up(i,j) is shown in Fig. 1.
An alternative method for flux limiting, which is popular in aerospace computations21, is a MUSCL method:
(Xn)FL(i,j)¼(Xn)ups(i,j)þs
4((1¹sk)D¹ þ(1þsk)Dþ),
D¹¼z((Xn)ups(i,j)¹(Xn)i2up(ups(i,j),dwn(i,j))),
Dþ¼((Xn)dwn(i,j)¹(Xn)ups(i,j)),
z¼ kxdwn(i,j)¹xups(i,j)k2
kxups(i,j)¹xi2up(ups(i,j),dwn(i,j))k2 ,
s¼ 2DþD¹ þe (Dþ)2þ(D¹)2þe
:
(26)
In eqn (26), xi is the location vector of node i and eis a small number which prevents a zero divide. The definition of s in eqn (26) carries out this limiting in a smooth manner. A value ofk¼1/3 was used in eqn (26), which on a regular mesh is an upwind biased third-order scheme. Further details about MUSCL methods can be found in 21.
In the following, we will be using linear triangular or tetrahedral basis functions. Typically, grids will be con-structed by subdividing quadrilaterals or bricks into trian-gles or tetrahedra.
Note that in eqn (15) and eqn (19), the flux limiter is applied only to the dissolved contaminant in the aqueous phase. Flux limiters have also been applied to the relative permeability terms33. However, it is our experience that in practical situations, the numerical dispersion of saturation fronts is small compared to the smearing of dissolved con-centration fronts20. This is because the strongly nonlinear shocks are self-sharpening. Detailed grid refinement studies which demonstrate this effect are shown in20.
4 VARIABLE SUBSTITUTION
The set of discrete equations (eqn (14)) and either eqn (15) (equilibrium) or eqn (18) and eqn (19) (nonequilibrium) represent a system of nonlinear algebraic equations. We will use Newton iteration (or an approximate Newton itera-tion) to solve this system. Consequently, it will be necessary to select a set of primary variables (those variables which are regarded as independent when constructing the Jaco-bian) for various cases. Primary variable switching is often necessary to ensure that the primary variables are physically relevant. Different choices of primary variables Fig. 1. Illustration of determination of the second upstream point
(i2up) for flow between node i and node j, using the geometric method and the maximum potential method. In this example node
i is upstream of node j, and node i2up is the second upstream point
can also have a profound effect on the convergence of the Newton iteration8,9,11,26,35.
4.1 Equilibrium primary variables
The equilibrium equations (eqns (14) and (15)) are two partial differential equations and hence require two primary variables. The state of a node is given by two state indicators {NAPL_STATE, LIQUID_STATE}. The primary variables are given in Table 1. For example, if node i is in the state {napl_on, Sw_primary}, the primary variables are {Sni, Swi}. The transition rules are:
IF(LIQUID_STATE¼Sw_primary AND Swi$tolf)
LIQUID_STATE : ¼Pn_PRIMARY
ELSEIF(LIQUID_STATE¼Pn_primary AND Swi,tolb)
LIQUID_STATE : ¼Sw_primary
ENDIF (27)
IF(NAPL_STATE¼napl_on AND Sni,0)
NAPL_STATE : ¼napl_off
Sni¼0
ELSEIF(NAPL_STATE¼napl_off AND Xni.Xnip)
NAPL_STATE : ¼napl_on
Xni¼Xpni
ENDIF (28)
Note that the LIQUID_STATE variable substitution is not strictly required, since we can always solve the system of eqn (14) and eqn (15) if we use Pnas a primary variable for the LIQUID_STATE. In26it was demonstrated that using the LIQUID_STATE variable switching resulted in a large gain in efficiency for unsaturated flow under dry conditions compared to using Pn as a primary variable. Somewhat surprisingly, we have found using the LIQUID_STATE variable switching to be more efficient when solving multi-phase flow problems. We will demonstrate this effect in some numerical examples. This phenomena has also been observed previously36.
4.2 Nonequilibrium primary variables
In the case of nonequlibrium between phases (eqns (14), (18) and (19)) there are three primary variables required. In this case, there is only one state indicator {LIQUID_ STATE}.
The primary variables are described in Table 2, and the transition rule is given in eqn (27).
5 NONLINEAR ITERATION METHODS
As discussed in Section 1, we will consider fully implicit methods in the following. In our experience, it is essential to solve for the saturation in a fully coupled, fully implicit manner for reliable, robust simulations. It is possible that adaptive implicit methods3may prove to be useful in some situations. As well, we will restrict attention to a single species NAPL contaminant. If there are a large number of reacting chemical components, it may be more attractive to use some form of operator splitting.
5.1 Jacobian selection for Newton iteration
Use of the nonlinear flux limiter (eqns (22)–(26)) results in the flux between two nodes i and j having dependency, in general, on variables at nodes i,j, i2up(i,j), i2up(j,i). Unless otherwise stated, we will determine the second upstream points i2up(i,j), i2up(j,i) (eqns (24)–(26)) using the maxi-mum potential method 20. We will also include some tests using the geometric method for determination of i2up (see Fig. 1).
If full Newton iteration is used to solve the implicit dis-crete equations, then the Jacobian matrix will contain more non-zeros, compared to a discretization which used simple upstream weighting. This is due to the fact that the flux between nodes i and j is a function of variables at i,j,
i2up(i,j), i2up(j,i) (see eqns (22)–(24)). For example, if a
seven-point finite volume stencil is used in three dimensions with the usual upstream weighting, then the full Jacobian with the flux limiter (eqn (22)) will give rise to a 13-point stencil. This will clearly be more expensive then the seven-point molecule, both in terms of Jacobian construction and solution cost. If a numerical method is used to construct the Jacobian, then the efficient technique described in26cannot be used, since the method in26requires that the flux function between nodes i and j be only a function of variables at nodes i and j.
It has been claimed that, in aerospace applications, use of a full Newton iteration with a TVD-based limiter results in slow convergence37. This contrasts with experience with fully implicit MUSCL methods, where the full Newton approach is very efficient25,38. However, these aerospace problems were concerned with steady-state solutions, whereas in this work we are interested in true transient problems. In33, use of a first-order Jacobian is advocated, although no comparisons are made with a full Jacobian iteration.
Table 1. States and primary variables: equilibrium model
NAPL_STATE napl_on napl_off
Primary variable Sn Xn
LIQUID_STATE Pn_primary Sw_primary
Primary variable Pn Sw
Table 2. States and primary variables: nonequlibrium model
LIQUID_STATE Pn_primary Sw_primary
In order to avoid the extra expense of full Newton itera-tion, there are a number of possibilities. Since we are sol-ving a true time-dependent problem, then it is reasonable to suppose that it may not be necessary to use the exact Jacobian. One possibility is to simply ignore the Jacobian entries which lie outside of the sparsity pattern generated using first-order upstream weighting. The simplest and most efficient technique for constructing an approximate Jacobian is simply to use the efficient algorithm described in26, but using the flux-limited form for the flux functions. This method constructs the Jacobian using numerical differ-entiation. The Jacobian is constructed by columns, and, if there are neq unknowns per node, the entire Jacobian is constructed with work equivalent to neq þ 1 residual evaluations. In our case, this means that for the equilibrium case, this method requires the work of three residual evalua-tions, while for the nonequilibrium case, four residual eva-luations are required. If we simply use the algorithm described in26, and use a flux limiter to evaluate dissolved NAPL flux, then this method ignores the derivatives with respect to i2up in eqns (22)–(24)Eq, (25)eqn (26).
Another method is to use the first-order Jacobian, but compute the residual using the high-order method33. This technique is commonly used in aerospace applications25. To be more precise, if xkis the kth iterate for the vector of primary variables at t¼Nþ1, then the nonlinear iteration is
While(not converged) (29)
J(xkþ1¹xk)¼ ¹rk
EndWhile
where J in eqn (29) is the Jacobian constructed using first-order upstream weighting, and the residual vector rk is constructed using a high-order flux-limited method (for the NAPL mole fraction as in eqn (22) and eqn (26)).
To recapitulate, we will investigate three methods for solving the nonlinear algebraic equations:
• Full Newton: the Jacobian and residual are evalu-ated using high-order flux limiter.
• Approximate Jacobian: all Jacobian entries corre-sponding to derivatives with respect to i2up in eqns (22)–(27) are ignored.
• First-order Jacobian: the Jacobian is evaluated using first-order upstream weighting. The residual vector is evaluated using a high-order flux limiter for the NAPL mole fraction.
6 CODE VERIFICATION
The equilibrium model developed in this work was com-pared to the test problems described in9. There was good agreement with the equilibrium, multiphase compositional simulations reported in9.
Several nonequilibrium cases (with various mass transfer
correlations) were compared with the results in 14, with good qualitative agreement. These tests are reported in detail in27.
7 COMPUTATIONAL PARAMETERS
The nonlinear algebraic equations (eqns (14), (15) and (18)) are solved in the following using full or approximate Newton Iteration. A block incomplete LU (ILU) factoriza-tion iterative solver39–42with CGSTAB acceleration43was used to solve the matrix. An ILU (1) (level 1) precondition-ing is used40,41. Reverse Cuthill-McKee (RCM) ordering44 was used to order the unknowns. The tolerances for the Newton iteration are
Pressure tolerance¼0:01 kpa (30)
Saturation tolerance¼0:001
Mole fraction tolerance¼10¹7
with inner iteration tolerances an order of magnitude smal-ler than the Newton iteration tosmal-lerances (eqn (30)). These nonlinear iteration tolerances typically resulted in a cumu-lative (at the end of the run) recumu-lative material balance error of less than 10¹4. Variable timestepping was employed using a method similar to that in 45. This method is based on selecting a desired maximum error in the solution, and predicting a timestep which will result in this error.
8 TEST PROBLEMS
We consider a number of two-dimensional test problems, with different boundary conditions, constitutive data, and LNAPL (less dense than water) and DNAPL (more dense than water) contaminants.
8.1 LNAPL problem
The two-dimensional cross-section for this problem is shown in Fig. 2. The computational domain for these test consist of a 50340 grid, withDx¼1.0 m andDz¼0.1 m. A unit thickness was specified in the y direction (normal to the cross-section). Hydrostatic pressure boundary condi-tions are imposed (on the left and right ends of the region) with the water table elevations as shown in Fig. 2.
The physical properties data are given in Table 3 and Table 4. The longitudinal dispersivity is typical of the Borden sand, based on matching field scale experiments 46. The value of
b in Table 3 (b ¼1.0) is typical of that obtained by matching field-scale experiments13. Laboratory experiments typically haveb . 0.5–0.75 16. Note that in the following we assume that the permeability tensor is diagonal, with diagonal entries in the x and z directions kx,
kz:
K¼
kx 0
0 kz
!
(31)
The Courant number for this problem, based on the max-imum timestep of 10 days was .10. The Peclet number is about .2, for the dispersion parameters in Table 3. The LNAPL data in Table 4 was used for this example. The relative permeability and capillary pressure data are given in Table 5 and Table 6. Hysteresis effects are ignored in all simulations reported in this paper.
This problem is initially fully saturated (Sw ¼1 every-where), and was run to 7 days with no NAPL injection to equilibrate the water table. LNAPL was then injected a rate 1.6 m3/day for 0.5 days at the point shown in Fig. 2. Injec-tion of LNAPL then ceased, and the problem was run to completion. Fig. 3 shows the Log10normalized mole frac-tion (Xn/Xn*) at 11 days for this run, using a nonequilibrium model (eqn (1), eqn (4), eqn (5) and eqn (7)).
Fig. 4 shows the saturation contours for this problem, but using the equilibrium model. Comparing the contours at 23.5 and 365 days, it is clear that a large portion of the LNAPL has been dissolved by the flowing water phase.
In contrast, Fig. 5 shows the LNAPL saturation contours
at 365 days using the nonequilibrium model. Comparing Fig. 5 with Fig. 4 (365 days), we can see that the extent of the NAPL phase is much greater for the nonequilibrium model than for the equilibrium model. Of course, this is to be expected since the nonequilibrium model will tend to dissolve the NAPL phase slower than the equilibrium model.
8.2 DNAPL problem
Fig. 6 shows the computational domain for these tests, which consists of a 503 40 grid, withDx¼1.0 m,Dz¼ 0.1 m. A fine grid (99379) was also used for this problem, with all node spacing halved. A unit thickness was specified in the y direction (normal to the cross-section). Hydrostatic pressure boundary conditions are imposed with the water table elevations as shown in Fig. 6.
All the physical data used for this example was as in Table 3 and Table 4. The Courant number for this problem, Table 3. Physical properties data
kx¼kz 1.0310¹11m2
Xn* 0.00285
Mn 0.74 cp
aL 0.50 m
aTH 0.03 m
aTV 0.001 m
dw 0.0 m2/day
lR 2.0 day¹1
b 1.0
rbKd 0.03
lA 1.0 day¹1
f 0.3
Table 4. Density data for the examples
LNAPL data
Mn 9.03103mol/m3
rn 100.1310¹
3 kg/mol DNAPL data
Mn 11.4310
3 mol/m3
rn 131.1310
¹3 kg/mol Water data
Mw 55.53103mol/m3
rw 18.02310¹3kg/mol
Table 5. NAPL: water data: data set 1
Sw krw krn Pcnw(kpa)
0.20 0.0 0.68 9.0
0.30 0.04 0.55 5.4
0.40 0.10 0.43 3.9
0.50 0.18 0.31 3.3
0.60 0.30 0.20 3.0
0.70 0.44 0.12 2.7
0.80 0.60 0.05 2.4
0.90 0.80 0.0 1.5
1.0 1.0 0.0 0.0
Table 6. Liquid–gas data: data set 1
SwþSn kra krn Pcan(kpa) Pcaw(kpa)
0.20 0.64 0.0 9.0 6.0
0.32 0.46 0.00 3.0 4.5
0.40 0.36 0.0009 2.4 3.9
0.50 0.25 0.045 2.1 3.6
0.60 0.16 0.116 1.8 3.3
0.70 0.09 0.21 1.5 3.0
0.80 0.04 0.34 1.2 2.0
0.90 0.01 0.49 0.9 1.0
0.95 0.00 0.58 0.5 0.5
1.0 0.0 0.68 0.0 0.0
Sn*¼0.2.
based on the maximum timestep of 10 days was .20. The Peclet number is about .2, for the dispersion parameters in Table 3. A few results will also be reported for a case where all dispersion constants are decreased by a factor of 10 compared to the base case in Table 3, which results in convection-dominated flow with a Peclet number of .20. Two variations of this scenario were tested.
• Homogeneous: a constant absolute permeability was used (see Table 3).
• Heterogeneous: the second case used highly hetero-geneous absolute permeability data, with perme-abilities ranging from 10¹10 to 10¹15 m2. The exponent in the mass transfer eqn (7) for the hetero-geneous problem was altered tob¼1.5, while the time constant waslR¼2.0 (days)
¹1
. (This data is typical of field-scale experiments 13.) The Hetero-geneous absolute permeability data was (all perme-ability units m2, and x, z units are m):
kx¼kz¼10
¹15
13#x#16, 2:95#z#3:05
(32)
kx¼kz¼10¹15 10#x#14, 2:75#z#2:85
kx¼kz¼10
¹15
15#x#17, 2:55#z#2:65
kx¼kz¼10¹13 29#x#34, 1:55#z#2:05
kx¼kz¼10
¹14
19#x#29, 2:25#z#2:55 kx¼kz¼10
¹10
9#x#39, 0:65#z#1:15
kx¼kz¼10
¹11
everywhere else
Initially, these problems were fully saturated (Sw ¼ 1 everywhere), and were first run for 7 days to equilibrate the water table (no NAPL injected), and then the DNAPL was injected for 0.1 days at the point shown in Fig. 6 at a rate of 0.8 m3/day. DNAPL injection then stopped, and the problem was run to completion.
Fig. 4. NAPL saturation contours, homogeneous LNAPL problem, equilibrium model.
Fig. 5. NAPL saturation contours, 365 days, homogeneous LNAPL problem, nonequilibrium model. Compare with Fig. 4,
365 days.
Fig. 6. Two-dimensional cross-section computational domain for the DNAPL scenarios.
Fig. 7. Log10normalized mole fraction, 11 days, van Leer flux limiter, homogeneous DNAPL problem, equilibrium model,
coarse grid.
Fig. 8. Log10 normalized mole fraction, 11 days, MUSCL flux limiter 21, homogeneous DNAPL problem, equilibrium model,
coarse grid.
Fig. 9. Log10normalized mole fraction, 11 days, upstream weight-ing, homogeneous DNAPL problem, equilibrium model, coarse
In the following, we will be testing the effect of various nonlinear iteration strategies for use in conjunction with MUSCL and van Leer flux limiters. It is therefore, of inter-est to see what effect use of these different limiters has on the computed solution. To illustrate the effects of these different methods for discretization, we show some results for the homogeneous DNAPL problem, equilibrium model, using the coarse grid (503 40). Figs 7–9 show the log10 normalized mole fraction (Xn/Xn*) contours at 11 days, using van Leer (eqns (23)–(25)), MUSCL (eqn (26)) and upstream methods. Figs 10–12 show the same results using the fine (99379) grid.
Note than Van Leer and MUSCL weighting give similar results at both coarse and fine grids. Upstream weighting
appears to be very slowly converging, and shows considerable numerical dispersion compared to the flux limiter methods. A complete comparison of grid refinement studies showing that the van Leer limiter converges more quickly than upstream weighting as the grid is refined is given in 20, and will not be repeated here.
We will also be using the heterogeneous version of the DNAPL problem in the following. Figs 13 and 14 show the log normalized NAPL contaminant mole fraction contours at 11 days, for both equilibrium and nonequilibrium models. Note that the plume for the nonequilibrium model has not spread as far as the equilibrium model, which is to be expected. As discussed in 20, the use of the maximum potential method for determining the second upstream point would seem to be more physically reasonable compared to the geometric method, especially for heterogeneous problems. Fig. 15 compares the normalized contaminant mole frac-tion contours at 100 days, for the heterogeneous DNAPL problem, nonequilibrium model, using both the geometric and maximum potential methods for location of the second upstream point. Van Leer weighting was used for these examples. Note that there appears to be little difference between using either geometric and maximum potential methods, consistent with the results in 20. However, as will be shown, there is a considerable difference in the numerical performance (in terms of CPU cost) of these two methods.
Fig. 10. Log10normalized mole fraction, 11 days, Van Leer flux limiter, equilibrium model, fine grid.
Fig. 11. Log10normalized mole fraction, 11 days, MUSCL flux limiter, equilibrium model, fine grid.
Fig. 12. Log10 normalized mole fraction, 11 days, upstream weighting, equilibrium model, fine grid.
Fig. 13. Log10normalized mole fraction, 11 days, heterogeneous DNAPL problem, equilibrium model, coarse grid.
Fig. 14. Log10normalized mole fraction, 11 days, heterogeneous DNAPL problem, nonequilibrium.
Fig. 15. Log10normalized mole fraction, 100 days, heterogeneous DNAPL, problem, nonequilibrium model coarse grid. Comparison of maximum potential and geometric second upstream point
8.3 Large unsaturated zone
In order to test the variable substitution strategies, the pro-blem shown in Fig. 16 was used. Note that this propro-blem has a large unsaturated zone, with hydrostatic pressure con-ditions on the left and right hand boundaries (location of water table as indicated in Fig. 16). This is a cross-section, with thickness 0.25 m in the direction normal to the cross-section. The domain was discretized using a 16311 grid,
Dx¼Dz¼0.25 m.
The data in Table 3, Table 5 and Table 6 were used. In addition, another set of relative permeability and capillary pressure data was tested (Table 7 and Table 8). The data in these tables is representative of a Borden sand47. In parti-cular, the capillary pressure data is based on the dimension-less capillary pressure curves in Ref.48.
A number of cases were tested, using both LNAPL and DNAPL data (Table 4), and both equilibrium (eqn (2)) and nonequilibrium (eqn (4), eqn (5), eqn (7)) assumptions. The problem was initially fully saturated, and was run to 7 days with no NAPL injection to equilibrate the unsaturated flow model. From 7.0 to 7.1 days, a NAPL contaminant was injected at a rage of 0.153 m3/day. Injection then ceased, and the simulation was run to 500 days.
9 TEST RESULTS
9.1 LIQUID_STATE variable substitution
As discussed previously, variable substitution for LIQUID_STATE primary variables is not actually required since Pncan be used as a primary variable without switching to Sw. However, in26, it was demonstrated that this type of variable switching speeded up the convergence of the Newton iteration for unsaturated flow problems under very dry conditions. Sometimes an order of magnitude improvement in number of Newton iterations was observed 26. However, this improvement was only seen for very dry conditions, where the capillary pressure– saturation curve had a very large derivative.
We did not anticipate this type of problem for multi-phase (passive air phase) simulations, since we were mainly inter-ested in problems near and below the water table, which have water saturations well above the critical value, due to the capillary fringe. However, rather to our surprise, we found that variable substitution had a large effect for simu-lations involving a NAPL phase.
As a first example, consider the LNAPL Problem (Section 8.1). Two runs were carried out. The first run disabled the LIQUID_STATE variable substitution (eqn (27)) by setting
tolf ¼ tolb ¼ ¹` (see eqn (27)), in which case Pn was always the primary variable. The second run used the vari-able switching parameters tolf¼0.99 and tolb¼0.9. This problem was run to a stopping time of 11 days. Full Newton iteration was used with a van Leer flux limiter. There was virtually no difference between these two methods during the initial part of the run to 7 days, which was simply an unsaturated flow situation (no NAPL). The run using LIQUID_STATE switching used 72 Newton iterations, compared with 80 Newton iterations for the run without LIQUID_STATE switching, to complete the initial 7 days. This indicates that the initial unsaturated flow part of this scenario is not particularly difficult, in contrast to the situa-tion in26. This is to be expected since the capillary pressures do not have the large derivatives with respect to saturation, Fig. 16. Computational domain for test of variable substitution.
Table 7. NAPL: water data: data set 2
Sw krw krn Pcnw(kpa)
0.074 0.0 0.45 9.4
0.1 0.001 0.42 7.0
0.2 0.005 0.34 4.2
0.3 0.025 0.28 3.3
0.4 0.05 0.22 2.8
0.5 0.08 0.17 2.5
0.6 0.14 0.115 2.2
0.7 0.22 0.065 2.0
0.8 0.36 0.025 1.9
0.85 0.46 0.01 1.85
0.9 0.57 0.0 1.8
1.0 1.0 0.0 0.0
Table 8. Liquid–gas data: data set 2
SwþSn kra krn Pcan(kpa) Pcaw(kpa)
0.074 0.65 0.0 9.0 11.8
0.1 0.6 0.0 5.0 6.1
0.2 0.45 0.02 3.1 4.5
0.3 0.34 0.005 2.4 3.8
0.4 0.25 0.09 2.0 3.4
0.5 0.18 0.13 1.75 3.0
0.6 0.12 0.17 1.5 2.8
0.7 0.07 0.22 1.25 2.6
0.8 0.03 0.28 1.0 2.5
0.9 0.0033 0.35 0.9 2.4
0.92 0.0012 0.37 0.88 2.38
0.95 0.0 0.4 0.85 2.35
1.0 0.0 0.45 0.0 0.0
as in26. The run statistics for the complete run (to 11 days) are shown in Table 9. Clearly, the use of LIQUID_STATE variable substitution has a very large effect on performance after the NAPL is present in the system.
At this point, we changed the input data to try to deter-mine the cause of this problem. Problems disappeared, for example, when Pcnw¼0.0, and when the Pcawand Pcanwere changed to straight line curves. Consequently, this problem definitely appears to be due to the non-linearities in the capillary pressure curves. Adjusting the value of Sn
* in eqn (11) occasionally caused improvement, but sometimes made matters worse. It would therefore appear that the switching between Pcan and Pcaw as in eqn (9) was not the primary source of the problem. Note that other models of the two-phase to three-phase capillary pressure transition are similar in spirit. For example in11a Pcawcurve is used when no NAPL is present, and the three-phase Pcnwcurve is used for situations where free NAPL is present. For cases where the NAPL is trapped (i.e. below residual), a weighted average of water- and gas-phase pressures is used for Pn.
An extensive series of tests was carried out to determine the range of liquid saturations where the LIQUID_STATE switching was effective, and where it ceased to be effective. The results are summarized in Table 10. Rather surprisingly, there was little effect on the number of Newton iterations for the parameters (eqn (27)), until an increase was observed at a value of (tolf, tolb)¼(0.4, 0.3). Lowering (tolf, tolb) any further caused the number of Newton iterations to increase to the number observed when the LIQUID_STATE switch-ing was entirely disabled.
Consequently, it would appear that it is necessary to use saturations as primary variables until the liquid saturation is above 0.5. Since the problem in convergence seems to occur near the water table, this suggests that convergence difficul-ties are encountered as the NAPL phase becomes mobile at a given node, but this is not entirely clear. The safest choice
of parameters appears to be use a value of tolfwhich is as large as possible. In the following, we will use tolf¼0.99 and tolb¼0.9.
To study the effect of LIQUID_STATE switching in more detail, a comprehensive series of tests was carried out using the Large Unsaturated Zone problem (see Section 8.3). If the LIQUID_STATE switching mechanism was enabled, we used the following parameters for tolf and
tolbin eqn (27).
tolf¼0:99 tolb¼0:9: (33)
Full Newton iteration was used with a van Leer flux limiter. A number of cases were tested, using both LNAPL and DNAPL data (Table 4), and both equilibrium (eqn (2)) and nonequilibrium (eqn (4), eqn (5), eqn (7)) assumptions. In addition, another set of relative permeability and capil-lary pressure data was tested (Tables 7 and 8). The total Newton iteration counts for these tests are given in Table 11 (CPU time roughly proportional to Newton iteration count). The total iteration counts are given for the complete run to 500 days.
These tests indicate that LIQUID_STATE variable sub-stitution is always faster (in terms of Newton iteration count) than not using LIQUID_STATE switching. The decrease in Newton iteration count varies from more than a factor of 10 to less than a factor of 2. The increase in efficiency obtained using LIQUID_STATE switching is clearly highly dependent on the physical properties data, and on the modelling assumptions (i.e. equilibrium, none-quilibrium). However, in all cases involving LNAPLs, dis-abling the LIQUID_STATE state switching increased the number of Newton iterations by more than a factor of 5.
We have seen the same results in many tests. LIQUID_STATE state variable substitution is never signif-icantly slower than using Pnas a primary variable, and is sometimes an order to magnitude faster. We emphasize that this is a different effect form that in26, since the work in26 considered unsaturated flow in very dry soils. In the problems in this work, the capillary pressure data for two-phase Table 9. Performance of variable switching: LNAPL example,
nonequlibrium model, stopping time 11 days
Variable substitution
Table 10. Performance of variable switching: effect of switch-ing parameters: CPU time proportional to Newton iteration
count
tolf, tolb Total Newton iterations
0.9, 0.8 433
Table 11. Total Newton iteration count for test of LIQUID_STATE switching (Fig. 16): CPU time proportional
to Newton iteration count
Scenario Switching No switching
air–water flow does not give rise to convergence pro-blems, since the number of Newton iterations is virtually the same for both LIQUID_STATE switching and no LIQUID_STATE switching runs, before the NAPL is injected. Convergence difficulties arise only when the NAPL is injected. Although the cure for convergence problems is similar to that used in 26, the cause of these problems appears to be different.
In the following examples, we will always use LIQUID_STATE variable substitution.
9.2 Jacobian selection for the flux limiter: experiments
In this Section, we will carry out some tests to determine the effect of using various approximations to the Jacobian when using a flux limited discretization (eqns (22)–(26))).
Tables 12–14 compare the performance of pure upstream weighting and flux limiters based on van Leer and MUSCL approaches. For the flux limiters, the nonlinear iteration is carried out using the full Jacobian, an approximate Jacobian (derivatives respect to i2up in eqns (22)–(26)) is ignored), and a order Jacobian (Jacobian constructed using first-order upstream weighting). These methods are described in more detail in Section 5.1. All the results in Tables 12–14
were obtained using the maximum potential method (see Fig. 1) for location of the second upstream point.
Examination of these tables reveals that, at least for this interphase mass transfer data, which is similar to that in13, the nonequilibrium models are easier numerical problems than the equilibrium models, for the same scenario. This will be discussed more detail in Section 9.3.
Another interesting result form Tables 12–14, is that there is very little difference between using the full Jaco-bian, the approximate JacoJaco-bian, and the first-order JacoJaco-bian, in terms of total Newton iterations. This holds for MUSCL and van Leer weighting, equilibrium and nonequilibrium models, and all three scenarios. For any given weighting method, model formation, and scenario, the number of Newton iterations varied by at most . 15%. Of course, the CPU times for the full Jacobian methods were about double the CPU cost compared to the approximate and first-order Jacobians, due to the increased size of the Jacobian.
Consequently, on the basis of these results, either the first-order or approximate Jacobian method would appear to be a good choice for solution of the nonlinear equations. There does not seem to be any advantage to using the full Jacobian.
Table 12. Comparison of full and approximate Jacobian, stra-tegies, heterogeneous DNAPL examples, 100 days: CPU times
are for a SUN Sparc-10
Method Total nonlinear
Full Jacobian 250 1540 241
MUSCL weighting
Full Jacobian 435 2261 831
Van Leer weighting
Full Jacobian 427 21398 784
Nonequilibrium model Upstream weighting
Full Jacobian 288 1806 390
MUSCL weighting
Full Jacobian 265 1603 727
Van Leer weighting
Full Jacobian 271 1686 765
Table 13. Comparison of full and approximate Jacobian, stra-tegies, heterogeneous DNAPL examples, 100 days: CPU times
are for a SUN Sparc-10
Method Total nonlinear
Full Jacobian 309 1647 282
MUSCL weighting
Full Jacobian 509 2195 941
Van Leer weighting
Full Jacobian 729 2519 1319
Nonequilibrium model Upstream weighting
Full Jacobian 249 1383 324
MUSCL weighting
Full Jacobian 252 1379 692
Van Leer weighting
Recall from Figs 7 and 8 that MUSCL and van Leer weighting give very similar computed solutions. Tables 12–14 also indicate that, in terms of numerical perfor-mance, there is little difference between MUSCL and van Leer methods. This is somewhat surprising, since the MUSCL method in eqn (26) was specifically designed to use a smooth limiter function. The van Leer limiter function (eqn (25)) has a discontinuity in slope at r¼0.
In order to verify that the above trends are also observed in convection-dominated problems, the homogeneous DNAPL nonequilibrium problem was run again, this time with all dispersion parameters in Table 3 reduced by a factor of 10. This resulted in a grid Peclet number of .20. Table 15 shows that even for this problem, there does not seem to be any advantage of using the full Jacobian, compared to either the first-order Jacobian or the approximate Jacobian. Note that, as shown in Fig. 15, there is very little differ-ence in the computed solution for the Heterogeneous DNAPL problem, using either the maximum potential or geometric method for determining the second upstream point (see Fig. 11) for use in the flux limiter expressions. Again, this is somewhat surprising, since the maximum potential method would appear to be more physically reasonable for heterogeneous problems. This was also reported in20.
In order to determine if there was a difference in comput-ing cost when uscomput-ing maximum potential or geometric methods for determining the second upstream point, the heterogeneous DNAPL scenario was run again, using the geometric method. The run statistics for the computations Table 14. Comparison of full and approximate Jacobian,
stra-tegies, heterogeneous LNAPL examples, 100 days: CPU times are for a SUN Sparc-10
Method Total nonlinear
Full Jacobian 1023 4788 880
MUSCL weighting
Full Jacobian 1319 6365 2246
Van Leer weighting
Full Jacobian 1235 5877 2067
Nonequilibrium model Upstream weighting
Full Jacobian 624 2567 736
MUSCL weighting
Full Jacobian 648 2569 1544
Van Leer weighting
Full Jacobian 599 2427 1439
Table 15. Comparison of full and approximate Jacobian, stra-tegies, homogeneous DNAPL examples, 100 days
Method Total nonlinear
Full Jacobian 268 1829 376
MUSCL weighting
Full Jacobian 295 2056 857
Van Leer weighting
Full Jacobian 277 1877 778
This example used dispersion parameters 10 times smaller com-pared to the base case in Table 3, resulting in a grid Peclet number of .20. CPU times are for a Sun Sparc-10.
Table 16. Comparison of full and approximate Jacobian, stra-tegies, homogeneous DNAPL examples, 100 days
Method Total nonlinear
Full Jacobian 935 2861 2461
Van Leer weighting
Full Jacobian 1020 2954 2676
Nonequilibrium model
Full Jacobian 307 1673 1287
Van Leer weighting
Full Jacobian 345 179 1446
using the geometric method are given in Table 16 and should be compared with Table 12. Note that for the equili-brium models, there is a large increase in the number of Newton iterations required when using the geometric method compared to the maximum potential method, with one exception. The one anomaly occurs when using MUSCL weighting, with the first-order Jacobian (equili-brium model). In this case, the geometric method run has slightly fewer total Newton iterations than the maximum potential computation. However, for the rest of the equili-brium simulations, the number of Newton iterations for the geometric method is about double that of the maximum potential methods. For the nonequilibrium assumptions, the geometric-based runs have slightly more Newton itera-tions compared with the maximum potential runs.
Note that the CPU time requirements for the geometric-based methods, when using the full Jacobian Newton itera-tion, are much greater than the corresponding runs using the maximum potential method. This is because the number of nonlinear iterations has increased for the geometric methods and, as well, the geometric Jacobians have more non-zeros than the maximum potential Jacobians. This results in greater costs for construction and solution of the Jacobian for the geometric Jacobians. The maximum potential Jacobian has fewer non-zeros than the geometric Jacobian since the maximum potential method recognizes that many possible non-zeros in the geometric Jacobian are identically zero in the maximum potential Jacobian. An example of this would be if fluid flows from node j to node i, then i2up(i,j) does not exist (see eqn (24)).
Consequently, even though the solutions for both maxi-mum potential and geometric methods are very similar, the geometric method is generally slower than the maximum potential method.
9.3 Effect of varying mass transfer parameters
In order to determine the effect of varying the mass transfer parameters on the convergence of the nonlinear iteration, several tests were carried out. Table 17 and Table 18 show
the run statistics for the various interphase mass transfer parameters, for the heterogeneous DNAPL and LNAPL sce-narios. The nonlinear iteration was carried out using the first-order Jacobian method, with MUSCL weighting. For comparison, the statistics for the equilibrium runs are also shown. Recall from eqn (7) that the mass transfer rate is proportional tolRSnb. These tests span the range ofbvalues observed in both laboratory and field scale experiments.
For the heterogeneous DNAPL problem (Table 17), the number of Newton iterations is quite insensitive to the changes in the mass transfer parameters. Note that when
b,1, the derivative of the mass transfer rate with respect to Snbecomes infinite as Sn→0. It might be expected that
using a value of b,1 would cause difficulty for the non-linear iteration. However, there is only a slight increase in the number of Newton iterations for theb¼0.5 case. This problem is also fairly insensitive to increasing lR(which corresponds to increasing the mass transfer rate). Note that the total CPU time for the nonequilibrium models is roughly the same as the total CPU time for the equilibrium model, even though the nonequilibrium model solves three conservation equations compared to two for the equilibrium case.
For the LNAPL problem, the Newton iteration counts increase slightly forb¼0.5,lR¼1.0 (Table 18). However, this problem is very sensitive to increasing the mass transfer rate by increasinglR. It is only atb¼2.0,lR¼10.0 that the Newton iteration counts become comparable to thelR¼1.0 case. For problems withb# 1, we can expect some diffi-culty, since the mass transfer rate (as a function of Sn) does not go smoothly to zero as Sn→0 (see eqn (20)). IflRis large, then this problem apparently becomes more severe. Note that the LNAPL problem has more NAPL injected into the domain than the DNAPL problem. As well, the DNAPL rapidly sinks down through the water table and becomes immobile. This results in a fairly small area of NAPL con-tamination. On the other hand, the LNAPL contaminant spreads out over a large area on top of the water table, so that there is a large region of small NAPL saturation. This appears to cause difficulty when the mass transfer rate is large.
Table 17. Effect of interphase mass transfer parameters, (see eqn (7)), heterogeneous DNAPL examples, 100 days
b lR Total nonlinear
iterations
0.5 10.0 319 1788 446
1.5 10.0 280 1753 399
2.0 10.0 285 1809 409
The example used MUSCL weighting with the first-order Jacobian iteration. CPU times are for a SUN Sparc-10.
Table 18. Effect of interphase mass transfer parameters, (see eqn (7)), heterogeneous LNAPL examples, 100 days
b lR Total nonlinear
iterations
0.5 1.0 984 4592 1279
1.0 1.0 706 3046 886
2.0 1.0 699 3037 874
0.5 10.0 4197 16452 5163
1.0 10.0 3675 14630 4465
2.0 10.0 585 2401 723
10 CONCLUSION
Even though the examples used in this study did not have capillary-pressure saturation curves having steep derivatives, we found that use of a liquid state variable substitution method for multiphase flow models typically resulted in a three- to five-fold decrease in the number of Newton iterations. Since the liquid state variable sub-stitution never caused an increase in the number of Newton iterations (compared to always using the pressure as a primary variable), we advocate routine use of liquid state variable substitution for multiphase, passive air-phase formulations.
Nonlinear flux limiters based on TVD (van Leer) or MUSCL methods give very similar solutions for dissolved NAPL plumes. Both these methods showed considerably less numerical dispersion compared to upstream weighting. If the maximum potential method8,20is used for location of the second upstream point, then there does not seem to be any advantage to using a full Jacobian for the nonlinear iteration. Nonlinear iteration methods based on an approx-imate Jacobian (based on ignoring some terms in the full Jacobian), or a first-order Jacobian (based on an upstream weighting) required about the same number of nonlinear iterations as the full Newton iteration, yet resulted in con-siderably less computational cost.
If the geometric method for location of the second upstream point is used, then generally this resulted in more computational cost compared to the maximum poten-tial method. Since the maximum potenpoten-tial method is also easily generalized to an unstructured grid, we recommend that the maximum potential method be used.
For nonequilibrium DNAPL problems, the total Newton iteration counts are relatively insensitive to the interphase mass transfer parameters. In fact, the nonequilibrium model formulation for DNAPL problems results in an easier numerical problem than the equilibrium model. However, for nonequilibrium LNAPL problems, the total number of nonlinear iterations is quite sensitive to the interphase mass transfer rate. As the rate is increased, the problem becomes quite difficult to solve.
In summary, we recommend using a flux limiter, based on either TVD or MUSCL ideas, for simulating the dissolved NAPL plume. If a first-order or approximate Jacobian is used for the nonlinear iteration, then the flux-limited solu-tion can be obtained at little addisolu-tional cost compared to first-order upstream weighting, but a much more accurate solution is obtained. For multiphase flow simulations with a passive air phase, variable substitution should be used to enhance convergence of the nonlinear iteration.
ACKNOWLEDGEMENTS
This work was supported by the National Sciences and Engineering Research Council of Canada, the Waterloo Center for Groundwater Research, and the Information
Technology Research Center, funded by the Province of Ontario, and by Haley and Aldrich Inc. and Garter Lee Ltd.
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