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WATER RESOURCES RESEARCH, VOL. 29, NO. 11, PAGES 3727-3740, NOVEMBER 1993

Modeling of Multiphase Transport of Multicomponent Organic Contaminants and Heat in the Subsurface: Numerical Model Formulation

A. E. ADENEKAN l AND T. W. PATZEK

Department of Materials Science and Mineral Engineering, University of California, Berkeley

K. PRUESS

Earth Sciences Division, Lawrence Berkeley Laboratory, University of California, Berkeley

A numerical compositional simulator (Multiphase Multicomponent Nonisothermal Organics Trans- port Simulator (M2NOTS)) has been developed for modeling transient, three-dimensional, nonisothermal, and multiphase transport of multicomponent organic contaminants in the subsurface. The governing equations include (1) advection of all three phases in response to pressure, capillary, and gravity forces; (2) interphase mass transfer that allows every component to partition into each phase present; (3) diffusion; and (4) transport of sensible and latent heat energy. Two other features distinguish M2NOTS from other simulators reported in the groundwater literature: (1) the simulator allows for any number of chemical components and every component is allowed to partition into all fluid phases present, and (2) each phase is allowed to completely disappear from, or appear in, any region of the domain during a simulation. These features are required to model realistic field problems involving transport of mixtures of nonaqueous phase liquid contaminants, and to quan.tify performance of existing and emerging remediation methods such as vacuum extraction and steam injection.

1. INTRODUCTION

The contaminants most commonly disposed off into the ground, but also encountered in spills and leaks, are petroleum hydrocarbons and halogenated aliphatic compounds that are widely used as industrial solvents. These organic liquids are commonly referred to as nonaqueous phase liquids (NAPLs) in the groundwater literature.

The potential for groundwater contamination by NAPLs is significant because of their physical and chemical properties. Although these organic liquids are designated as "nonaqueous," i.e., immiscible with water, their solubilities in water are, in fact, sufficient to render large quantities of groundwater unfit for human use. Trichloroethylene (TCE), for example, has a solubility of 1100 mg/L of water at 20øC, whereas the established allowable level of TCE in drinking water is 5 txg/L. Therefore groundwater contacting a separate phase TCE will remain contaminated above the allowable drinking water level for a long time and over a large distance downgradient, despite dispersion. In addition, many NAPLs have high vapor pressures at ambient temperature, partition strongly into the surrounding gas phase, and form a gas-phase contaminant plume that spreads due to molecular diffusion and, in some cases, advection. The contaminants transported in the gas phase may partition into groundwater beyond the extent of the groundwater plume associated with NAPL dissolution.

A NAPL released into the subsurface may migrate as a separate nonaqueous phase. The extent of this migration is governed by the density and viscosity of the NAPL, its

l Now at Exxon Production Research Company, Houston, Texas.

Copyright 1993 by the American Geophysical Union.

Paper number 93WR01957. 0043-1397/93/93WR-01957505.00

quantity, and the rate of release. When introduced into the subsurface, the NAPL migrates downward through the unsaturated zone; lateral spreading caused by capillarity and soil heterogeneity may also accompany this vertical migration. As the NAPL moves through the unsaturated zone, it leaves residual liquid trapped in the pore space by capillary forces. If the NAPL is released in a sufficient quantity and/or at a sufficient rate, some of it will reach the water table. At this point further movement of the NAPL is determined largely by its density. NAPLs which are less dense than water (e.g., gasoline and other petroleum distillates) will remain above the water table and continue to flow laterally in a direction determined by the dip of the water table. NAPLs which are denser than water will continue their downward

migration through the saturated zone, until they encounter a low permeability layer. These NAPLs will then accumulate and migrate downdip on top of such a layer. One should remember, however, that the downward migration will occur only if the NAPL reaches a displacement pressure sufficient to overcome the capillary pressure at the liquid-liquid interfaces.

Two conclusions can be drawn immediately from the foregoing discussion.

1. Once released into the subsurface, a volatile organic compound may be transported as a solute in groundwater, vapor, or constituent of a NAPL, whose chemical composition varies with time and distance from the spill. Therefore assessment of the potential hazard to groundwater and evaluation of remediation alternatives may require a multiphase model which accounts for interphase mass transfer of the chemical components. We refer to such a system as a compositional multiphase system.

2. The threat posed to groundwater may persist even if the NAPL is immobile. The primary challenge in groundwater cleanup is to remove the NAPL that serves as a subsurface source and causes the groundwater and gas phase

3727

3728 ADENEKAN ET AL.: MODELING OF MULTIPHASE TRANSPORT OF ORGANIC CONTAMINANTS

plumes to grow and persist; it is not enough to remove the contaminants dissolved in the groundwater.

In situ NAPL recovery techniques commonly used today include free product recovery, pump and treat, and vacuum extraction [e.g., American Petroleum Institute, 1989; Con- ner, 1988; Johnson et al., 1990]. For denser than water NAPLs below the water table, free product recovery can be extremely ditficult unless the location of the NAPL pool is known. However, even in situations where free product is "successfully" recovered, a substantial volume of NAPL remains trapped in soil by capillary forces. Presently, almost all remediation schemes to remove contamination trapped below the water table rely on pumping the affected zone to purge it of contaminants. These schemes are ineffective because even a small quantity of an almost insoluble NAPL in the subsurface will provide a continuous source of contamination and greatly prolong the pumping time relative to those cases where only dissolved contaminants are present. Indeed, experience has shown that remediation by pump and treat has been very long and expensive at many sites [Mackay and Cherry, 1989].

Even though at present no proven technology exists to recover NAPLs trapped below the water table, steam flooding looks promising [Hunt et al., 1988a, b; Udell and Stewart, 1989; Yuan, 1991; Aines et al., 1992; Adenekan and Patzek, 1992; Adenekan, 1992]. Steam flooding is governed by several complex and coupled processes, such as multiphase fluid flow; interphase mass transfer, evaporation, condensation, and dissolution; highly nonlinear composition-dependent phase properties; and strongly nonlinear heat transport. These complexities make numerical methods the only realistic choice in the solution of practically relevant problems.

During the last decade, several numerical simulators for NAPL transport in multiphase subsurface systems have been reported in the groundwater literature. The majority of these simulators do not consider interphase mass transfer of the NAPL components and either do not include a gas phase or assume a stagnant gas phase with uniform atmospheric pressure [e.g., Faust, 1985; Faust et al., 1989; Kuppusamy et al., 1987; Kaluarachchi and Parker, 1989]. A few of the reported simulators consider interphase mass transfer of one or more NAPL components to the gas and water phases but assume that the NAPL is immobilized in the unsaturated

zone [e.g., Baehr and Corapcioglu, 1987; Sleep and Sykes, 1989]. The simulators presented by Abriola and Pinder [1985a, b] and Forsyth [1988] allow for interphase mass transfer and flow of the NAPL and water phases, but the gas phase is assumed to be stagnant. The simulator developed by Falta [ 1990] and Falta et al. [ 1992a, b] is the only one in the groundwater literature that allows for interphase mass transfer and movement of all three fluid phases. This code, STMVOC, is also the only NAPL transport code that includes thermal energy transport. However, STMVOC considers only a single-component NAPL.

On the other hand, much of our current knowledge of the physics of multiphase flow and the relevant numerical solution methods come from the petroleum reservoir engineering literature. In the last 20 years, sophisticated thermal and compositional simulators have been developed to model oil reservoirs [e.g., Coats, 1974, 1978, 1980; Young and Stephenson, 1983] and it appears that some of these codes could also be used to study NAPL contamination problems.

However, usually this is not the case because oil recovery and NAPL transport/recovery are dominated by different mechanisms. For example, reservoir engineers are generally not interested in the relatively insignificant quantity of oil dissolved in the water phase; neither is diffusion considered important. On the other hand, dissolution and transport of organic compounds in the water phase and diffusion of organic vapors in the gas phase may be quite important in NAPL contamination/remediation studies. Also, because oil reservoirs are generally deep and confined, oil industry simulators usually assume no-flux boundaries for a modeled domain. In subsurface contamination problems, we are generally dealing with shallow systems and are usually interested in evaluating the exchange of contaminants between the subsurface and the atmosphere. Yet another example of how the differences in emphasis enter into code formulation is the treatment of appearance and disappearance of phases. Most oil reservoir codes assume that the oil phase cannot completely disappear from a grid block. This is justifiable because most crude oils contain heavy, nonvola- tile components. In contrast, the goal of many NAPL contamination cleanup efforts is to completely remove the NAPL. Therefore codes used to study transport of NAPLs need to be more flexible in dealing with the appearance and disappearance of phases.

The primary objective of this work is to develop a numerical simulator which is more general than those listed above in its applicability to contaminant transport and remediation problems. The numerical simulator developed in this study, Multiphase Multicomponent Nonisothermal Organics Trans- port Simulator (M2NOTS), accounts for flow of all three fluid phases in response to viscous, gravity, as well as capillary forces, and can be used to model transport in one, two, or three space dimensions and arbitrary geometry. M2NOTS is fully compositional. Therefore the NAPL phase may consist of any number of user-specified chemical components, and each component is allowed to partition into all other phases present. The partitioning of a component among the phases is calculated from the assumption of local equilibrium. Mechanisms of interphase mass transfer include evaporation and condensation of NAPL components and wa•ter, dissolution of NAPL into the water phase, and Henry's law partitioning of chemical components between the water and gas phases. Adsorption of NAPL components on the solid grains is also included. M2NOTS is nonisothermal; therefore heat transport may occur by advection of the fluid phases and conduction. Heat exchange due to multicomponent diffusion is also accounted for.

The issue of phase appearance and disappearance is important for the types of systems being considered. A typical application will generally have regions in which only one or two phases are present. For example, three phases will be present in the unsaturated zone below a contaminant source. At the same time, a NAPL might be present, along with water below the water table near the source (a two- phase condition). Away from the source, the NAPL might not be present above or below the water table, implying a two-phase condition above the water table and a one-phase condition below it in this part of the system. In addition, while it is true that capillary forces (residual saturation) will prevent the complete disappearance of a phase due solely to mechanical forces, interphase mass transfer mechanisms such as evaporation and dissolution make the complete

ADENEKAN ET AL.: MODELING OF MULTIPHASE TRANSPORT OF ORGANIC CONTAMINANTS 3729

disappearance of a phase possible. The present approach to the appearance and disappearance of phases is more rigor- ous than previous attempts [Abriola and Pinder, 1985b; Falta et al., 1992a, b l, which were inherently susceptible to poor convergence [Forsyth and Sammon, 1984]. The algorithm used in this work allows for any grid block to contain any single phase, any combination of two phases, or all three phases at any time during the simulation, i.e., any of the phases is allowed to completely disappear.

The starting point for the present work is the TOUGH2 simulator developed by Pruess [1987, 1991] at the Lawrence Berkeley Laboratory. TOUGH2 is a three-dimensional code for simulating coupled heat and multiphase fluid transport in porous and fractured media. TOUGH2, and its earlier version, TOUGH, have been verified by comparison with several geothermal reservoir and unsaturated flow solutions [Pruess and Wang, 1984; Pruess, 1987b; Moridis and Pruess, 1992]. The coding of TOUGH2 is quite general and its modular architecture makes it suitable for extensions and

enhancements. The flow module of TOUGH2, that is, the routines in which the fluxes and accumulation terms are

calculated, is written in a general fashion for the mass balances of an arbitrary number of components distributed among any number of phases. To calculate the fluxes and accumulation terms, this module needs only the thermophysical properties of the phases, e.g., density, viscosity, enthalpy, and mole fractions. The specific nature of the fluid system does not affect the flow module. This information is needed only in the equation of state (EOS) module that calculates the thermophysical properties of the system. This way, different fluid mixtures can be simulated with the same flow module, the thermophysical properties of the specific fluid mixture of interest being provided by an appropriate EOS module. The modular architecture of the TOUGH2

code allows for flexible interfacing of the flow module with EOS modules representing different kinds of fluid systems. The EOS modules released with version 1.0 of the TOUGH2

code were all limited to (1) two-component systems (water and water with tracer; water and carbon dioxide; water and air; and water and hydrogen) and (2) flow systems with at most two phases.

The new EOS module in M2NOTS handles a three-phase system (gas, water, and NAPL) with water, air, and any number of user-specified hydrocarbons as components. In addition to providing values of all thermophysical parameters to the flow module, the EOS module must (1) recognize the phases present in each grid block and (2) diagnose the appearance or disappearance of these phases in each grid block and take an appropriate action after each Newton- Raphson iteration.

The well model in TOUGH2 has also been enhanced in the

development of M2NOTS. In particular, injection of fluids into multilayer wells under pressure constraints can now be handled. This is necessary to simulate steam injection. For a production well on deliverability, the pump can be located in any layer to which the well is open. M2NOTS also allows injection of hydrocarbon fluids of specified composition.

The governing equations are introduced in section 2. The M2NOTS model and its assumptions are described in sec- tions 2.1 and 2.2. In section 2.3, we discuss the specific approach and correlations used to calculate all thermophysical properties of the system required by the model. The calculations of phase equilibria are presented in section 2.4.

The treatment of heat losses into the confining beds is described in section 2.5.

The numerical algorithm used to solve the governing equations is described in section 3. The space and time discretization schemes are described in section 3.1. Selec-

tion of primary variables and substitution of these variables are discussed in section 3.2. Our handling of two- and three-phase capillary pressures is described in section 3.3. The direct and iterative method of solving the discretized nonlinear algebraic equations is presented in section 3.4. The boundary and initial conditions handled by M2NOTS are described in section 3.5, and the algorithm used to track the appearance and disappearance of phases in section 3.6. Finally, section 3.7 describes the injection and production well model in M2NOTS.

2. MODEL FORMULATION

2.1. Main Assumptions

The following assumptions have been made in the development of the M2NOTS model.

1. The Darcy equation adequately describes multiphase fluid flow in porous media.

2. The phases are in local chemical and thermal equilibrium. Local thermal equilibrium implies that the fluids and the rock minerals in any small volume element are at the same temperature. There is general consensus [e.g., Coats, 1974] that this is a good working assumption in most cases of practical interest. On the other hand, there is ongoing controversy over the local chemical equilibrium assumption [e.g., Baehr et al., 1989; Miller et al., 1990; Zalidis et al., 1991].

3. Molecular diffusion in the water phase and the NAPL is described by constant effective diffusion coefficients. The molecular diffusion coetficients of gas components are assumed to be equal to their respective binary diffusion coefficients in air. This is done only for expediency; calculation of multicomponent diffusion coefficients is complicated and it is not clear at all that such calculations will improve the accuracy of simulations. However, there is nothing intrinsic in the current development that requires these simplifica- tions. The general theory of diffusion in multicomponent gases is covered by Cussler [1984] and Hirschfelder et al. [1954]. Diffusion in liquid mixtures is discussed by Ghai et al. [1973, 1974].

4. Energy changes (as reflected by temperature in the energy equation) caused by acceleration and viscous dissi- pation are negligible.

5. Adsorption of organic compounds on the rock obeys a linear isotherm.

6. No chemical reactions take place (to be addressed in the future).

2.2. Mathematical Model

The equations describing nonisothermal multiphase flow of multicomponent fluids in porous media are based on conservation of mass for each chemical component, Darcy' s law, and conservation of thermal energy. The conservation of mass for each component i is expressed as

• =w,o,g

(1)

3730 ADENEKAN ET AL.' MODELING OF MULTIPHASE TRANSPORT OF ORGANIC CONTAMINANTS

where the subscripts w, o, and g denote water, oil (or NAPL; henceforth, oil and NAPL are used interchange- ably); and gas phases respectively; F i is the molar flux of component i = 1, ß ß ß , N; (b is the porosity; St• is the phase saturation; Pt3 is the phase molar density; Or is the mass density of rock grains; wit is the adsorbed mass of component i per unit mass of rock grains; xit3 is mole fraction of component i in phase/3; q i is the generation rate of i per unit volume of porous medium; and d[/[i is the molar weight of component i.

The total molar flux of component i, F i, is a sum of advection in each of the three phases due to bulk flow of these phases as described by the Darcy equation, and diffusion:

13 = w ,o ,g

kkrt• x •,sP ,s ß (VP,s

+ ckSl3r 13P 13Dil3 ' •7Xil3 (2)

Here k is the diagonal intrinsic permeability tensor; krt• is the relative permeability of the phases; /x• is the phase viscosity; •t3 is the mass density of the phases; Pt3 is the pressure in phase/3; Di/3 = Dil31 is the diffusion coefficient tensor of component i in phase/3, with Di/3 being scalars; rt3 is the tortuosity of the flow of phase /3; and g is the acceleration of gravity.

Using (2) in (1) gives

O [ •) r TMir at (1 -- 4)) q- 4) • S l3 P l3 X i l3 '/[/[i 13 = w ,o ,g

kkrl 3 • V ß Xil3Pl3 t3 = w , o , g I.t t3

•- (VPt• - e t3g)

+ ckSI3rl3pl3Dil3' •7Xil3] + qi Similarly, the thermal energy balance is given by

at (1 - qb)•rCp,T + qb Z St3pl3UI3 13 = w ,o,g

13 = w,o,g

kkrfi

--' (VPt• - •t•g) +h' V T 1 + qheat

(3)

(4)

where T is the temperature; C pr is the rock grain specific heat capacity; h is the effective thermal conductivity tensor of the porous medium; U t3 is the molar internal energy of the phases; H t3 is the molar enthalpy of the phases; and q heat is the heat generation rate per unit volume of porous medium.

For a nonisothermal system with N chemical components, there is one equation (3) per component plus equation (4). However, the dependent variables in these equations total 4N + 25; hence additional 3N + 24 independent relationships are needed.

1. Phase saturations sum to unity:

.

unity'

• s• = 1 (5) 13 = w ,o,g

In each phase, component mole fractions sum to

N

Z Xil3=l i=1

fl = w, o, g (6)

3. Relative permeabilities are functions of saturations:

k•t3 = k•t3(Sw, So, Sg) 13 = w, o, g (7)

such as Corey's [1954, 1956] expressions for two-phase flow and the Stone II model [Stone, 1973] for k•o in three-phase flow.

4. Capillary pressures (Pcow = Po - Pw, Pcgw = Pg - Pw, and Pcgo = Pg - Po) are functions of saturations:

Pcgw = Pcgw(Sw, So, Sg), Pcgo = Pcgo(Sw, So, Sg),

Pcow = Pcgw- Pcgo (8)

such as Parker et al. [1987] expressions. Henceforth the subscript on P g will be dropped and P will denote the gas phase pressure which by convention is used as reference pressure.

5. Phase densities, internal energies, enthalphies, and viscosities are functions of temperature, pressure, and phase compositions (12 relationships):

pj = pj(P, T, x lj, ''' , X Nj), ''', j = 1, 2, 3 (9)

6. At equilibrium all chemical components partition among all the phases (2N relationships)

Xiw Xiw • = Kiwo(P, T, Xiw, Xio), X io X ig

• = Kiwg(P, T, Xiw, Xig)

Kiwg Kiog= , i- 1, 2,''', N (10)

Kiwo

Kiw o, Kiwg, and Kiog are the equilibrium constants (or K factors) for water-oil, water-gas, and oil-gas partitioning, respectively.

7. Chemical components dissolved in water adsorb on the rock grains'

Wir/Wiw = Ko,(T) i= 1, 2,... N (11)

where K o, are the constant adsorption coefficients calculated as functions of the amount of organic carbon present in the soil and the octanol-water partitioning coefficients of the compounds [Karickhoff et al., 1979].

8. Porosity is a function of pressure and temperature:

ck (P, T) = ck (Pinit, Tinit)[ 1 + e e(P - Pinit)

+ e t(T - Tinit)] (12)

where the subscript init represents the initial conditions, and the hysteresis effects in compacting and expanding the pore space are assumed to be negligible.

9. Effective thermal conductivity tensor is a function of phase saturations:


X(Sw, So, Se) = (1 - •b)X + cbS•lAw (13)

estimated from the simplified parallel model [Somerton, 1958], where I is the identity tensor; •r is the thermal conductivity tensor of the rock minerals; Aw is the thermal conductivity of the water; and S• = S w + So is the total liquid saturation. Equation (13) follows from an observation [e.g., Prats, 1982; Reid et al., 1987] that thermal conductivity of a fluid-saturated rock is dominated by that of the rock matrix, followed by those of liquids, and gas contribution to the conductivity is usually negligible (thermal conductivity of a liquid is 10-100 times larger than that of a gas). Equation (13) may be easily modified to permit a more general relationship.

10. Tortuosity of gas flow is a function of porosity and gas saturation:

rg(qb, Sg)= • 1/3S7/3 g (14)

estimated from the Millington and Quirk [1961] model. Both liquid phase to•uosities are currently assumed to be adjust- able parameters to be specified by the user.

2.3. Thermophysical Properties

A multiphase, compositional, and nonisothermal simulator cannot function without a robust PVT and transport property package. In the current implementation, we use a cubic equation of state to handle the thermodynamic properties of the gas phase, International Steam Tables [Interna- tional Formulation Committee, 1967] to handle the properties of the aqueous phase, and the method of corresponding states to describe the properties of the oil phase. We have tried to make as few simplifying assumptions as possible, and therefore we should be able to model a broad range of pressure and temperature conditions of a system at hand.

2.3.1. Gas phase. Thermodynamic behavior of the gas phase is described by the cubic $oave [1972] and Redlich and Kwong [ 1949] Equation of State (SRK EOS), commonly used for petroleum reservoir fluids and other nonpolar organic substances. Stated in terms of the gas phase compressibility factor Z, the SRK EOS for a pure component is

Z 3 - Z 2 + (A* - B* - (B*)2)Z- A'B* = 0 (15)

where A* = aP/R2T 2, B* = bP/RT; a, b, etc. are given by Reid et al. [1987]. For a gas phase that is a mixture of two or more chemical components, the mixing rules recommended by Reid et al. [1987] are used to calculate a and b for the mixture (hereafter referred to as a m and bin). With all the coefficients known, (15) can be solved and the largest root is the gas phase compressibility factor Ze. The molar density of the gas phase is then given by

P a = P/R TZ a (16)

Gas phase enthalpy is calculated as

H e = H(P, T)- Hø(T) + Cp dr ref

ß • x/a + (xIq20)ahIq20 (17) i•H20

where Tre f is the reference temperature; H(P, T) - Hø(T) is the real gas departure enthalpy; f rrref Cf dT is the enthalpy of an ideal gas; C• is the ideal gas heat capacity; and hH2o is the molar enthalpy of water vapor.

As (17) indicates, the enthalpy of a gas mixture is treated as consisting of two parts: the enthalpy of water vapor on one hand and the enthalpy of the gaseous mixture of air and hydrocarbon vapors on the other hand. A weighted average based on the mole fractions of these two parts is used to calculate the enthalpy of the entire gas mixture. Based on the SRK EOS, the departure enthalpy of the gas minus water vapor is calculated as

a m

H(P - PH2o, T) - Hø(T) = •mm In Z+B* rn

T Oam Z ln•+RT(Z- 1) (18)

b m OT Z + B*m

where PH20 is the partial pressure of water vapor in the gas mixture and

O• -= 2 T Z Zxi!7xj!7 i•H20 j•H20

where

1/2

ajTci) 1/2] -t- t• i Pci (19)

t• i •- 0.480 + 1.574ro i -- 0.176r0/2 (20)

The subscripts i andj denote pure component parameters, •o is acentric factor, and the subscript rn denotes mixture values of previously defined parameters. All properties of the chemically pure water are calculated from appropriate equations in the International Steam Tables [International Formulation Committee, 1967]. The above scheme for calculating gas phase enthalpy was adopted here because in numerical testing the SRK EOS gave poor approximations of the water vapor enthalpy.

The choice of Tre f is arbitrary as long as it is used consistently; here it is taken to be 0øC. In a multiphase system however, the temperature alone cannot specify a unique reference state; the distinction between a gaseous state and a liquid state is important. The reference state used here is a liquid state at Tref; that is, a liquid substance has a zero enthalpy at 0øC. The term multiplying the summation term in (17) is the enthalpy of the mixture (minus water vapor) relative to an ideal gas state at Tre f. It has to be corrected for the enthalpy of an ideal gas at Tre f relative to the adopted reference state which is a liquid state at Tre f. This involves calculating the enthalpy of vaporization which is discussed later together with oil phase properties.

The ideal gas heat capacity C• for a pure component i is estimated using the method of Joback [Reid et al., 1987]:

cpOi = a i + bit + ci T2 + di T3 (21) where the parameters ai, b i, ci, and d i are experimentally determined constants whose values are tabulated by Reid et al. [1977] for over 500 organic compounds. With Cpøi determined for each component present in the gas phase, a mole


fraction-weighted average is then used to calculate the ideal gas heat capacity of the mixture.

The gas phase viscosity is computed as a function of pressure, temperature, and composition using the method of Wilke [1950]. This method is a simplification of the Chap- man-Enskog kinetic theory and it gives the gas mixture viscosity as

N

XigtX i

•g=E N i=1

E XJgC•)iJ j=l

(22a)

[1 + (•i/•j)1/2(,/Otj/d•i)1/412 •ij = [8(1 + ,/Oti/•j)] 1/2 (22b)

where /z i and /z s are pure vapor viscosities at the system pressure and temperature and Ati is the molecular mass of the ith component.

In the actual implementation of these equations in the simulator, the water vapor and air components were com- bined and treated as a single pseudocomponent with the viscosity calculated from the method of Hirschfelder et al. [1954], but with the water vapor viscosity input from the International Steam Tables [International Formulation Committee, 1967]. Pruess [1987] used this method and reported that it agrees with experimental data up to 150øC and over the entire air-water composition range to within 4%. The hydrocarbon component viscosities are calculated using a corresponding states method [Lucas, 1980].

2.3.2. Oil phase. The molar density (or molar volume) of the oil phase is calculated as a function of temperature and composition. It is assumed that this density is independent of pressure, a reasonable assumption in light of the low compressibility of most liquids. It is also assumed that the hydrocarbon components form ideal mixtures; i.e., the partial molar volume of each NAPL component is equal to its molar volume at the same temperature. This assumption should hold for NAPLs whose constituent molecules are

similar in size, shape, and intermolecular forces. Mixtures of hydrocarbons usually fit this description. The oil mixture molar volume V o is then

Vø = E XiøVi (23) i

where v i is the molar volume of the ith component (as a liquid) at the system temperature. Given a component molar volume v/• at a reference temperature T R, the modified Rackett equation [Reid et al., 1987] is used to estimate that component's molar volume at temperature T as

V i = v/•(0.29056 - 0.08775•o i) qoi

qo i - (1 - T ri) 2/7 _ (1 -- TrR•) 2/7 (24)

in which T r = T/Tci and T R = TR/Tci - Tci being the i ri

critical temperature of component i. The calculation of the enthalpy of a pure liquid hydrocar-

bon component is done in several steps:

œ = (H œ Hsœ) (HSœ Hsv) (HSV o) H L- HTres -- + -- + -- H _ SV_ L '4-(n O- nOTre) '4-(nOTref nTSrVe) '4-(nTref nTre) (25)

r is the liquid where H r is the liquid enthalpy at T and P; HrreS enthalpy at Tr_ef and the corresponding vapor pressure Pvat•(Tref); H Sœ is the saturated liquid enthalpy at T and Pvat•(T); H Sv is the saturated vapor enthalpy at T and

ß SV

Pray(T), Hrreg is the saturated vapor enthalpy at Tre f and Pvap(Tref); H" is the ideal gas enthalpy at T; and H•r q is the ideal gas enthalpy at Tre. f.

•o H sv •r The contributions H sv - H ø and ,, rre• -- rre• e gas phase departure enthalpies discussed in section 2.3.. The term (H ø H o - rr,•) is an ideal gas enthalpy and it is calculated from frTr•f C• dT. The two terms H sœ - H sv and HTS v -

L _ HTr• are enthalpies of vaporization. Finally, H L •gSL represents the effect of pressure on liquid enthalpy; it is ignored here because it is usually small relative to the other terms.

The enthalpy of vaporization of a pure liquid AHv is estimated in two steps. First, the enthalpy of vaporization AHvb at the normal boiling point T b is computed by the Chen method [Reid et al., 1987]:

3.978Tbr- 3.958 + 1.555 In Pc AHvb = RTcTb (26)

r 1o07 - Tbr

Tb• --= Tb/Tc

Then the enthalpy of vaporization at the temperature of interest is calculated from the Watson correlation [Reid et al., 1987]:

( • _ Tr ) 0.375 AHv = AHab T r = T/T c (27) Tb r

Once the liquid enthalpy of each component has been estimated in the manner enumerated above, the enthalpy of the oil phase is calculated as

Ho = • Xio(H L- HTLre)i (28) i

where the summation is over the hydrocarbon components; the contributions of water and air dissolved in the oil are

neglected. Note that (28) neglects the heat of mixing in the oil phaseß

To use (25), saturated vapor pressures of pure components are required. The vapor pressure of a hydrocarbon component is a function of temperature and it is calculated from the Wagner equation [Reid et al., 1987]:

Pvat• = Pc exp [[a(1 - Tr) + b(1 - Tr)

+ c(1 -- Tr) 3 + d(1 - Tr)6]/Tr] (29)

where T r = T/Tc and Pc and Tc are the critical pressure and temperature of the component, respectively. Values of the constants a, b, c, and d for several organic compounds are given in Appendix A of Reid et al. [1987]. Where these parameters are not available for a compound of interest, provision has been made to use an alternative correlation to calculate vapor pressure. The Antoine correlation is generally less accurate than the Wagner method but its accuracy


should be sufficient in most cases. The Antoine correlation

gives the vapor pressure as

In Pvap = A T + C (30)

where A, B, and C are empirical constants tabulated by Reid et al. [1977] for over 500 compounds.

The temperature dependence of the viscosity of an oil component in the liquid state is estimated according to

t In p, = A' + •+ C'T + D'T 2 (31)

T

where A', B', C', and D' are constants;/a is in centipoise and T in Kelvins. This is the equation that Yaws et al. [1976] used to fit experimental viscosity data for several organic compounds. Van Velzen et al. [1972] used a similar correlation with C' and D' set to zero. Values of the constants

published by both groups are summarized in chapter 9 of Reid et al. [ 1987]. Given the pure component viscosities/a i, the oil mixture viscosity/a o is calculated from

H xi ø /'t'o -- /J'i (32) i

This is one of the many methods proposed in the literature to correlate liquid mixture viscosities; there is no agreement on which is the best one.

2.3.3. Water phase. It is assumed that the dissolved contaminants do not affect the thermodynamic properties of the water phase. This is a reasonable assumption considering that the organic compounds of interest are only slightly soluble in water. Therefore density, enthalpy, vapor pressure, and viscosity of the water phase are taken to depend only on pressure and temperature. These properties are calculated from Steam Table equations given by the Inter- national Formulation Committee [1967].

2.4. Phase Equilibria

Expressions for the equilibrium constants Kiwo, Kiw, and Kiog that appear in (10) are developed in this section. Chemical equilibrium requires that the chemical potential of each component be equal in all the phases. For a real gas, the chemical potential can be expressed in terms of fugacity f, which is an effective pressure, and a fugacity coefficient % Equality of chemical potentials is then equivalent to equality of fugacities:

f iw --f io f iw =fie f io =fie (33)

where f iw is the fugacity of component i in the water phase, and so on. The fugacity of component i in a liquid phase is related to the mole fraction of i in that phase through the fugacity equation

f iw = Xiw'riwf i R f io = Xio'riof i R (34)

where 'riw and 'rio are the fugacity coefficients of i in the water and oil phases, respectively, and f/• is the reference fugacity of the pure liquid i at the temperature of the system. The assumption is made here that gas phase fugacity coeflS- cients are all unity, makingf/• equal to the vapor pressure of

pure liquid i, Pvap,i. The following assumptions have been made in calculating phase equilibria described by (33).

1. The NAPL mixture is ideal in the sense that the

fugacity coefficient of each component is equal to unity. This is a good assumption if the oil components are chemically similar. For the commonly encountered NAPL mixtures, the error incurred by assuming ideality should be small. Other mixtures such as azeotropes may exhibit large departures from the ideality assumption. Future enhancements of the simulator should address this issue.

2. The gas phase behaves like an ideal mixture, that is, the partial pressure of a component is equal to the fugacity of that component.

2.4.1. Water-oil equilibrium. Using (34) in (33) and assuming 'rio = 1 leads to

Xio = Xiw'r iw (35)

The water phase fugacity coefficient is estimated by applying (35) to the pure chemical i, in which case Xio and 'rio are both unity, thus - sol "/iw is equal to 1/x søl Because the organic iw ß

chemicals of interest have low water solubilities (which means that solute-solute interactions are insignificant), it is a good assumption that the fugacity coefficient is constant over the possible concentration range (zero to (usually) less than one part per thousand). Therefore given the solubility of an organic component, one can calculate its fugacity coefficient (taken to be constant) as inverse of the mole fraction in water at the solubility limit. So, the equilibrium constant for water-oil partitioning is

_ sol Kiw o = Xiw/Xio = Xiw (36)

The solubility of a chemical is an increasing or decreasing function of temperature. Generally, the solubility of an organic compound tends to increase with temperature, while the solubility of a noncondensible gas tends to decrease with increasing temperature. Unfortunately, solubility data for most organic liquids are available only for a narrow temperature range and the various correlations that seek to predict solubility as a function of temperature do not always give accurate results. In the present work, if the required data are available, solubility of a compound is calculated as

Xiw=a + bT + cT 2 + dT 3 (37)

where a, b, c, and d are determined by fitting the data. Otherwise, if solubility is known at only one temperature, it is assumed to be constant. Kerfoot [1991] treats this topic in more detail.

2.4.2. Water-gas equilibrium. With the assumption of gas phase fugacity coefficients being unity, (33) leads to

Pi = (Pvap ' sol• ,ilXiw )Xiw (38)

where P i is the partial pressure of component i. This is a statement of Henry's law with the Henry's constant equal to Pvap,i/x søl This relationship assumes that the gas mixture iw ß

obeys Dalton's law of partial pressures. The water-gas equilibrium constant is then given by

sol Xiw

=•P (39) Kiwg Pvap,i 2.4.3. Oil-gas equilibrium. Once Kiw o and Kiwg are

calculated as shown above, Kiog = Kiwg/Kiw o .

3734 ADENEKAN ET AL.' MODELING OF MULTIPHASE TRANSPORT OF ORGANIC CONTAMINANTS

2.5. Heat Loss to Confining Beds

The semianalytical and simple method of Vinsome and Westerveld [ 1980] is used to calculate conductive heat loss to the confining beds. This method requires no grid blocks outside of the fluid flow domain, and it has been found to yield good accuracy for both short- and long-term heat exchange. Vinsome and Westerveld [1980] proposed to rep- resent the temperature profile in a semi-infinite conductive rock layer by a trial function of the form:

T(x, t) - Tinit-- (Tf- Tinit q- px + qx 2)

ß exp (-2x/(Kt)•/2) (40)

where x is the distance from the boundary; Tinit is the initial temperature in cap or base rock (assumed uniform); Tf is the time-varying temperature at cap or base rock boundary; p and q are time-varying fitting parameters; t is the time measured from the instant at which the interface temperature first begins to change; K = A/PC is the thermal diffusivity of the rock; and A, •, and C are the thermal conductivity, density, and specific heat capacity of the rock, respectively. Each grid block in the top and bottom layers of the flow domain will have an associated temperature profile in the adjacent confining layer as given by (40). The coefficients p and q will be different for each grid block; they are determined from the conditions that (1) there is continuity of thermal energy flux at the interface and (2) thermal energy in the confining layer is conserved.

3. NUMERICAL ALGORITHM

3.1. Space and Time Discretization

The integral finite difference method (IFDM) is used to discretize the flow domain into arbitrarily shaped polyhe- drons, constructed by drawing perpendicular bisectors to lines connecting the nodal points. The ability to accomodate arbitrarily shaped grid blocks derives from the fact that the IFDM does not make reference to any global coordinate system, so that there is no predetermined limit on the number of neighbors that a grid block can have. In this respect, the IFDM has the flexibility of the finite element method. In fact, it may be shown [Dalen, 1979] that the basic Galerkin finite element method with lumping and some restfictions on the node placement is equivalent to the IFDM. At the same time, the IFDM handles spatial gradients in a conceptually simpler manner of the classic cell-centered finite difference method. Detailed description of the IFDM are given by Edwards [ 1972], Sorey [ 1975], and Narasimhan and Witherspoon [1976]. Recently, a hybrid control volume finite element method has been proposed [Fung et al., 1993] to improve the representation of wells in reservoir simulation.

The discretized equations of the IFDM are obtained by integrating (3) and (4) over an arbitrary but finite flow domain I with volume V l, and applying the divergence theorem. The mass conservation equation for component i then becomes

q- •b • S fl p [3Xi[3 d V ! • d•i 13=w,o,9

=fr n'FidA+fv qidV (41) ! !

I

A/m

/77

Fig. 1. Center-to-center distance and interface between grid block I and its neighbor m.

where F i is given by (2), F l is the bounding surface of l, and n is the outward unit normal vector. The first term on the

fight-hand side represents the net rate (moles per unit time) at which component i leaves volume element I. This term is approximated as a sum of averages over surfaces shared by I and its neighbors m (see Figure 1). Therefore the spatial gradients which appear in this term are calculated at the respective interfaces and are approximated by first-order finite differences between block-center values. With the

assumption that V l is independent of time (i.e., there is no compaction), the integral and the derivative on the left-hand side can be interchanged. The time derivative is then approximated by a first order finite difference. With a fully implicit treatment of the interblock flow terms, (41) becomes

vl I[ I l • (1 qbl) •OrWir I e I e I l -- + •bl(PwSwxiw + PoSoxio d•i

In+l[ l I I lol l + p•lS•xo) - (1- •1) •OfXir '•i q- *l(pwOwXiw

n} {( llm I I I l,•l l, lm 5 rw + PoSoXio + pgogXig ) -- • (kA) PwXiw I'•w/ m

[ m l (pm _ pl) q- (Pcgw - P cgw) ß dl m -- •l(• w COS a) lm

( / [ m l + PoXio k.ro lm (pm_ pt) + (Pcao- Pcao) P'o/ dim

( I lm[(Pm--Pl) --g(0o COS a•)lm + pgXig krg L --•/(•09 COS o•)lm }n+l

{ rn l -- • (A) lm (4Sw'rwPwDiw) lm Xiw -- Xiw rn d l rn

rn Xio -- X + (4• S o roP o D io) lm d lm

I }n+l + (qbSg•.gpgDig)l m X•g -- Xig ' _ Vl{qi}n+ 1 dl m -- 0 (42)

ADENEKAN ET AL.' MODELING OF MULTIPHASE TRANSPORT OF ORGANIC CONTAMINANTS 3735

where a is the angle between the unit normal n and the direction of gravity, A is the area of the interface between l and m, and d lm is the distance between the nodal points of I and m. In this equation, the superscripts n and (n + 1) refer to the nth and (n + 1)th time levels respectively, and At - t n+l - tn; for example, all quantities within the braces { ... }n+• are evaluated at the (n + 1)th time level. Quan- tities with superscript I are the grid block averages for Vt. The superscript lm denotes those quantities that are evaluated at the interface between volume elements I and m. Note

that in the present formulation, (42) enforces conservation of mass of every component at each time step. Therefore it is impossible to have negative mole fractions in the converged solution. If during the Newton-Raphson iteration a particular mole fraction becomes negative it is reset to zero and the iteration process continues. If the iteration process still does not converge, we then and only then cut the time step.

The simulator employs upstream weighting to calculate these interface terms except for intrinsic permeability which is based on harmonic weighting, and the mass density in the gravity term which is calculated from an arithmetic average. Experience has shown that upstream weighting is required for multiphase flows in order to avoid convergence to nonphysical solutions [e.g., Aziz and Settari, 1979]. It has also been proven [Sammon, 1988] that upstream weighting techniques are capable of calculating a convergent solution in one-dimensional reservoir problems with gravity- segregation effects. Thus upstream weighting provides a perfectly correct approach to this type of problem, thereby justifying its use for multidimensional reservoir simulation. Using the water phase as an example, upstream weighting is defined by

= (krw/law) [twl

if flow is from I to m

krw lm = (krw/• w) rn if flow is from m to I

(43)

lm X iw -- X[w if flOW is from I to rn (44)

lrn rn X iw = X iw if flow is from rn to l

lm I if flOW is from I to m Pw = Pw (45)

lm m if flow is from m to l Pw = Pw

The direction of flow is given by the sign of

[(pm pl) + (Pcgw l •}lm lm] _ rn -- Pcow) - g cos a d (46)

Flow is from I to rn if this quantity is less than zero and vice versa. Upstream weighting is also used to calculate interface fluid enthalpies in the energy balance equation. The effective diffusion coefficients ( qbSw ?wDw) Ira, ( qb?oS oD o) tin, and (•bSa tad i) lrn are defined as harmonic averages.

As upstream weighting may cause excessive front smearing, we have used M2NOTS to solve several one- dimensional Buckley-Leverett problems and made compar- isons with the published analytical and numerical results [e.g., Faust, 1985]. The degree of front smearing was small and comparable to those in the literature.

The mass balance of component i was used in the forego-

ing discussion to illustrate the discretization procedure; a similar procedure is followed for the energy equation. For isothermal problems, however, the energy balance is omit- ted, thus saving memory and computer time. For a flow region discretized into NB volume elements, the result is a system of NB(N + 1) coupled, nonlinear algebraic equations.

3.2. Primary Variables and Variable Substitution

In section 2 it was shown that there are as many governing equations and constitutive relations as unknowns. In principle therefore the system is closed and can be solved. Let us consider the number of independent variables that are needed to completely specify the thermodynamic state of a flow system consisting of N components which are distributed among NPH phases. From Gibbs' phase rule, the number of thermodynamic degrees of freedom in such a system is F - N + 2 - NPH. This equation applies only to intensive properties of the system, but not to the relative amounts of phases present in the system. Information about the latter is contained in the phase saturations, and since only (NPH - 1) of these saturations are independent, the total number of degrees of freedom is F' - N + 2 - NPH + NPH- 1 - N + 1. Note that F' is independent of the number of phases present and is fully determined by the number of chemical components in the system. This implies that from all the unknowns one can choose a set of (N + 1) variables for every grid block, use the constitutive relations to express the remaining unknowns in terms of this set, and then solve the algebraic equations given above for the NB such sets. The (N + 1) variables chosen for each grid block are called the primary variables there. The choice of primary variables for a given grid block is not arbitrary; it depends on the phases present. In a multiphase system where phases are allowed to appear and disappear, some grid blocks may contain only one phase, some two phases, and others three. A choice of primary variables that is appropriate for one of these cases may be inappropriate for the others.

Consider the following example of a four-component system: water, air, and two hydrocarbon components that will be labeled HC1 and HC2. From the above discussion, there are five primary variables for this system. To fully specify the condition of a three-phase grid block, two saturations are required, say, S w and S a. Pressure and temperature are also chosen as primary variables. If the mole fraction of HC1 in the oil phase is chosen as the fifth primary variable, it is possible to solve for the mole fraction of all components in all phases by (1) using the equilibrium constants to express the mole fraction of the components in the water and gas phases in terms of their mole fractions in the oil phase; (2) imposing the three constraint equations •'i Xiw = •'i Xio = •'i Xifl = 1; and (3) solving the resulting set of three simultaneous equations. In the case of a grid block that contains only two phases, say, water and gas, only one saturation is independent. Again, pressure and temperature are chosen as primary variables. Here, two mole fractions in either of the phases may be chosen as primary variables. The two remaining mole fractions can then be calculated by using the equilibrium constants and the two constraint equations, 5'.i Xiw = Y'.i xia = 1.

For a system in which up to three phases may be present,


there are seven possible phase combinations: (1) water only, (2) oil only, (3) gas only, (4) water and oil, (5) water and gas, (6) oil and gas, and (7) water, oil and gas. As is shown above, different sets of primary variables apply to each of these cases.

3.3. Two- and Three-Phase

Capillary Pressures

In a two-phase oil-water system, when oil saturation is low, capillary pressure is also low and can be neglected. In the special case of drying a two- or three-phase gas-liquid (air, water, and/or oil) system (such as in pressure cycling of an electrically heated contaminated aquifer), we continue to use the capillary pressure at the irreducible liquid saturation even as the actual water or oil saturation falls below its

irreducible value. This is acceptable because regardless of the pressure gradient, one can not flow a liquid below its ireducible saturation; however, the liquid droplets can still evaporate.

3.4. Solution of Discretized Equations: Newton-Raphson Iteration

In general, the system of NB(N + 1) equations represented by (42) must be solved iteratively. A residual-based Newton-Raphson technique is used here. The Newton- Raphson technique converges quadratically and has been widely used to solve systems of nonlinear algebraic equations. Here, residual refers to the amount by which an approximate solution to the system represented by (42) fails to conserve mass and energy. A residual is defined for each chemical component and thermal energy in each grid block. For example, the left-hand side of (42) (multiplied through At/V l) is equal to the residual of component i in grid block I by definition. An exact solution to the system of equations will make each residual identically equal to zero. The system represented by (42) can be written as

R(X) = 0 (47)

where R and X are vectors of residuals, and primary variable unknowns, respectively. The Newton-Raphson procedure involves approximating (47) with a Taylor series expansion about an assumed solution X k. This leads to the linearized matrix equation

ox ] = (48) where k is the iteration level, and higher-order terms have been neglected. This equation represents a linear system of NB(N + 1) equations that has to be solved at each iteration. The matrix of partial derivatives in (48) is called the Jacobian matrix. Each row of the Jacobian matrix represents the partial derivatives of the residual of a component in a particular grid block with respect to all the unknown primary variables in all grid blocks. Since the residuals in a grid block are affected only by the values of primary variables in that grid block and those in its neighbors, the Jacobian matrix has a sparse structure with many zeros.

All the partial derivatives are calculated by numerical differencing. At the completion of a Newton iteration, the primary variables are updated. Based on these values, the

residual of each component in each grid block is calculated by successively evaluating the left-hand side of (42). The following steps are then carried out for each primary variable: (1) the primary variable is incremented by a small user-specified amount (we are using 64-bit arithmetic and have found a normalized increment between 10 -7 and 10 -8 to be satisfactory); (2) those residuals whose values can potentially be affected by changing the value of the primary variable in step 1 are recalculated; (3) the corresponding partial derivative is calculated as the change in the residual divided by the increment used in step 1.

At each iteration, the linear system (48) is solved using either a direct or an iterative solver. Direct solution is

accomplished with the program package MA28 from the United Kingdom Atomic Energy Authority, Harwell [Duff, 1977]. MA28 performs a sparse version of lower-upper (LU) decomposition with partial pivoting and back substitution on matrices with random sparsity structure. This direct solver has been found to be competitive (in terms of CPU requirements) with the iterative solvers for problems in which the total number of equations is less than 2000. It is widely known that for larger problems, memory requirements make the use of direct solvers impractical. Two iterative solvers, both of which are based on the preconditioned conjugate- gradient method, are currently available with the simulator: SOLVEN, written by Zyvoloski [1990] of the Los Alamos National Laboratory and DSMGCG [IBM, 1990], which is part of IBM's Engineering and Scientific Subroutine Li- brary. The interface of SOLVEN with the TOUGH2 code, developed by Bullivant and O'Sullivan [Bullivant, 1990] at the Auckland University in New Zealand was adapted for use with M2NOTS.

Iterations are continued until the normalized residual of

every component is smaller than a user-specified convergence tolerance. The normalized residual is defined as the residual of a component divided by the amount of that component in the corresponding grid block at the beginning of the time step. The converged solution at the end of a completed time step is used as the assumed solution for the first iteration of the next time step.

The initial time step size is specified by the user. For every time step for which convergence is achieved in fewer than a user-specified number of iterations, the size of the next time step is doubled. If, for a time step, convergence is not achieved within a specified number of iterations (say, 8), the time step is reduced by a user-specified factor. Any failure in solving the linear equations will also result in automatic reduction in time step size.

3.5. Boundary and Initial Conditions

Boundary conditions are of two basic types. Dirichlet conditions prescribe values of primary variables such as pressure, temperature, saturations and mole fractions on the boundary. Neumann conditions prescribe fluxes of mass and energy crossing boundary surfaces. A common case of the Neumann boundary condition is the "no flux" condition, which in the integral finite difference framework is handled by not specifying any flow connections across the boundary.

Dirichlet boundary conditions are handled by using additional inactive grid blocks. These grid blocks are called inactive because no mass and energy balance equations are set up for them and their primary thermodynamic variables

ADENEKAN ET AL.' MODELING OF MULTIPHASE TRANSPORT OF ORGANIC CONTAMINANTS 3737

are not included in the list of unknowns. However, the time-invariant values of primary variables in the inactive grid blocks are used in setting up flux terms in the balance equations for their neighboring active grid blocks.

Initial conditions for a simulation are given by specifying values of the (N + 1) primary variables in each grid block at the start of the simulation.

3.6. Appearance and Disappearance of Phases

As was indicated earlier, every fluid phase is allowed to completely disappear from any grid block during the simulation. Similarly, each of the phases may appear in a grid block during the simulation. The treatment of phase appearance and disappearance in M2NOTS is described here.

At the conclusion of every Newton-Raphson iteration, the phases present in each grid block are checked for thermodynamic compatibility with the new values of the primary variables. Where it is recognized that a phase has appeared or disappeared, the primary variables in that grid block are changed accordingly.

It is easier to recognize the disappearance of a phase than it is to recognize the appearance of one. This is because those phase saturations which are primary variables contain the information necessary to identify when a phase has disappeared from a grid block. For example, consider a grid block that contains water and gas at the conclusion of the kth iteration. For this grid block, water saturation is one of the primary variables. After the Newton-Raphson changes for the (k + 1)th iteration, AX[, are calculated, the water saturation will be greater than unity if the gas phase is to disappear or it will be less than zero if the water phase is to disappear.

The appearance of a phase is more complicated and generally involves checking if a thermodynamic criterion is satisfied. For example, a gas phase will evolve in a hitherto single-phase water grid block if the sum of partial pressures (at the grid block temperature) of the components in the water phase exceed the grid block pressure. A similar criterion involving the sum of mole fractions in a "would be" NAPL phase is used to identify the evolution of a NAPL phase in a grid block that previously contained only a wa•ter phase. At the conclusion of a Newton iteration, the mole fractions of all components in the water phase in such a grid block are known. The equilibrium constants K iw o are then used to calculate the mole fraction of each component i in the would be NAPL. The sum of component mole fractions in this NAPL is then compared to unity. NAPL evolves if the sum is greater than unity; otherwise, the grid block remains single-phase water. That is, the criterion

9

• Xiw/Kiwo •1 (49) i

is used to check for NAPL evolution in grid blocks that had contained only the water phase. The examples given above naturally extend to transitions between two- and three-phase conditions.

Note that none of the equilibrium constants are set to zero in the current model formulation. In case of air dissolving in water, the value of Ki•vg, i = air, is extremely small. This in principle may cause problems with the Newton-Raphson

iteration. If, in addition to air, other more soluble components are present, the Jacobian matrix becomes even stiffer, and more iterations for the Newton-Raphson method may be required to obtain convergence.

There is another aspect that makes handling the appearance of a phase more complicated than handling the disappearance of a phase. This involves the reinitialization, on the appearance of a phase, of the values of the primary variables in a grid block, especially those of mole fractions. Reinitial- ization here does not refer to switching to a different set of primary variables as described in section 3.2; rather it refers to how values are assigned to the primary variables after a phase evolves. It is necessary to reinitialize those primary variables that are mole fractions because their values prior to, and after the evolution of the new phase may be very different. For example, when a NAPL evolves from a water phase, it is because the water phase had been "too rich" in the NAPL constituents. Due to different solubilities of NAPL constituents, the composition of a water phase that will be in equilibrium with the evolved NAPL may be drastically different from the composition of the water prior to extracting the NAPL. In the course of this study, it was found that the rate of convergence of the Newton iterations subsequent to the evolution of•a phase is very sensitive to the reinitialization of primary variables.

When a phase evolves' in a grid block, component mole fractions in those phases that were originally present must necessarily change. The problem is one of equilibrating a fluid mixture of known overall composition at the temperature and pressure of interest;it is essentially what is referred to as a flash calculation in the chemical engineering literature [e.g., Prausnitz et al., 1980]. In the context of the present work, a grid block in which a phase is evolving corresponds to a separation stage, the fluid mixture in that grid block along With its known composition prior to evolution of the new phase constitutes the feed stream, and the phases present in the grid block after evolution of the new phase constitute the product streams.

A flash calculation involves simultaneous solution of the

material balan ce as well as the phase equilibrium relations. As a result, the composition of each thermodynamically stable phase and the overall mole fractions are calculated with the Newton-Raphson method.

3.7. Treatment of Wells

M2NOTS provides various options for specifying th.• injection or withdrawal of heat and fluids. Multiblock and single block injection/producing wells are handled. Experi- ence has shown that for the best convergence rates 0f:i•'•½ Newton iteration, the source terms in (42) should be treat•'d implicitly; i.e., the source terms should be written at the

,

latest iteration level. Each conservation equation (42) for each grid block I that contains a well will have the source term, qi.

3.7.1. Specified injection rates. The simplest case of fluid injection is to specify a constant rate or a table of time-dependent rates of injection of chemical components into a grid block. This is equivalent to specifying a constant rate or a table of time-varying rates of injection for a fluid phase whose composition is also specified. For example, a NAPL can be injected into a grid block at a specified mass rate but the mole fractions of all chemical components in the

3738 ADENEKAN ET AL..' MODELING OF MULTIPHASE TRANSPORT OF ORGANIC CONTAMINANTS

NAPL must also be given. Injection of water that contains a small amount of air is handled in a similar fashion. The

temperature of the injected fluid is also specified by the user. In these cases, the value of the source term q i in each of the conservation equations (42) for grid block I is known a priori and that value is simply used when the residual of i in I is being calculated.

3.7.2. Bottomhole pressure-constrained wells. A more general and perhaps more realistic way of specifying fluid injection or production is based on the injectivity index or productivity index model [Coats, 1978; Peaceman, 1978]. Many petroleum reservoir simulators [e.g., Coats, 1980; Modine et al., 1992] make provision for a well to be either on a rate constraint (Neumann boundary condition) or on a bottomhole pressure constraint (Dirichlet boundary condition). These codes also allow for the possibility to switch from one constraint to another in different time periods. For example, a rate or Neumann condition may exist for a certain period of time, after which it becomes a Dirichlet condition. This may occur in a pressure-constrained well which is produced at a constant volumetric rate, provided that the bottomhole pressure does not drop below a limiting value. When this value is reached, the simulator automati- cally maintains that pressure and the well produces at the maximum (but time dependent) rate possible under that condition. The well is then said to be on deliverability.

In the present version of M2NOTS, the bottomhole pressures specified by the user are always enforced. The specified bottomhole pressures can be time dependent; i.e., wells can be injected into, produced from, or shut in according to arbitrary user-specified schedules. There is no provision for a well to switch from pressure constraint to rate constraint. Thus a multilayer injection well or production well is always on deliverability. Future work will enhance this aspect of the code, making it more flexible.

4. CONCLUSIONS

The primary objective of this study was to develop a numerical simulator that can be used to (1) examine the migration of nonaqueous phase contaminants in the subsurface and (2) predict the response of a NAPL-contaminated subsurface system to various cleanup techniques, especially steamflooding.

An integral finite difference numerical simulator M2NOTS, presented in this paper, is capable of simulating multiphase transport of multicomponent organic compounds and heat in porous media. The starting point for the present work was the TOUGH2 simulator, developed by Pruess [1987, 1991]. TOUGH2 is a three-dimensional code for simulating coupled heat and multiphase fluid transport in porous and fractured media. The main extension to TOUGH2 in the present study is the addition of a general and efficient EOS module that handles a three-phase system (gas, water, and NAPL) in which the components are water, air, and an arbitrary number of user-specified hydrocarbon components.

The formulation of M2NOTS is quite general. The governing equations include (1) advection of all three phases in response to pressure, capillary, and gravity forces; (2) interphase mass transfer that allows every component to partition into each phase present; (3) diffusion; and (4) transport of sensible and latent heat energy. No artificial constraints are

imposed on the absence or presence of a phase in any part of the domain; a robust algorithm is used to check for phase appearance and disappearance and each phase is allowed to appear or disappear from any grid block during a simulation. M2NOTS has been shown to perform very well in simulating transient, strongly nonisothermal, and highly coupled problems [see Adenekan and Patzek, 1993].

Acknowledgments. This work was funded by the University of California at Berkeley. Partial funding was provided by the Office of Technology Development of the U.S. Department of Energy (DOE) contract ADS 1504-02 to Lawrence Livermore National Laboratory, under subcontract award 5808-54: task 1 and task 2 to the University of California, Berkeley. Additional funding was provided by Lawrence Berkeley Laboratory under contract DE-AC03- 76SF00098 with the U.S. DOE. Some computer time was provided by Cray Research, Inc.

REFERENCES

Abriola, L. M., and G. F. Pinder, A multiphase approach to the modeling of porous media contamination by organic compounds, 1, Equation development, Water Resour. Res., 21(1), 11-18, 1985a.

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A. E. Adenekan, Exxon Production Research Company, P.O. Box 2189, Houston, TX 77252.

T. W. Patzek, Department of Materials Science and Mineral Engineering, 324 Hearst Mining Building, University of California, Berkeley, CA 94720.

K. Pruess, Earth Science Division, Lawrence Berkeley Labora-

tory, University of California, Berkeley, CA 94720.

(Received October 1, 1992; revised July 12, 1993;

accepted July 19, 1993.)

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