An implicit numerical model for multicomponent … Pub...aReservoir Engineering Research Institute, Palo Alto, CA, USA bYale University, New Haven, USA article info Article history:

Advances in Water Resources 85 (2015) 64–78

Contents lists available at ScienceDirect

Advances in Water Resources

journal homepage: www.elsevier.com/locate/advwatres

An implicit numerical model for multicomponent compressible

two-phase flow in porous media

Ali Zidane a, Abbas Firoozabadi a,b,∗

a Reservoir Engineering Research Institute, Palo Alto, CA, USAb Yale University, New Haven, USA

a r t i c l e i n f o

Article history:

Received 17 April 2015

Revised 1 August 2015

Accepted 8 September 2015

Available online 16 September 2015

Keywords:

Implicit scheme

MFE

FV

Two-phase

Peng–Robinson

Newton–Raphson

a b s t r a c t

We introduce a new implicit approach to model multicomponent compressible two-phase flow in porous

media with species transfer between the phases. In the implicit discretization of the species transport equa-

tion in our formulation we calculate for the first time the derivative of the molar concentration of component

i in phase α (cα, i) with respect to the total molar concentration (ci) under the conditions of a constant volume

V and temperature T. The species transport equation is discretized by the finite volume (FV) method. The

fluxes are calculated based on powerful features of the mixed finite element (MFE) method which provides

the pressure at grid-cell interfaces in addition to the pressure at the grid-cell center. The efficiency of the pro-

posed model is demonstrated by comparing our results with three existing implicit compositional models.

Our algorithm has low numerical dispersion despite the fact it is based on first-order space discretization.

The proposed algorithm is very robust.

© 2015 Published by Elsevier Ltd.

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1. Introduction

In two-phase compositional flow, most of the numerical models

in the literature are based on explicit approximation of the species

balance equation [1]. The nonlinearity between the phase molar

and/or mass concentration and the total concentration in the implicit

method requires some complicated derivatives that do not appear in

an explicit scheme. In the explicit scheme, one of the major draw-

backs is time step selection based on the Courant–Freidrichs–Levy

condition, known as the CFL condition. The CFL condition requires the

time step to be less than the necessary time for flow to pass through

one mesh block. The impact of this condition becomes severe when

one grid-block has a relatively small size compared to the size of

the simulation domain. Numerical modeling of fractured reservoirs

is a clear example of small and large grid-cells. Most of the numer-

ical simulators that deal with fractures require either to have small

mesh elements to explicitly model the fractures (e.g. single-porosity

models) [2–4], or to have very small elements near the fractures (e.g.

cross-flow-equilibrium approach) [5–8]. In this case the numerical

simulation becomes – computational wise – expensive, if an explicit

scheme is used in the numerical model. In compositional multiphase

flow, the use of an explicit method is preferred for the following rea-

sons: (i) significant numerical dispersion usually consorted with the

∗ Corresponding author.

E-mail address: [email protected] (A. Firoozabadi).

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http://dx.doi.org/10.1016/j.advwatres.2015.09.006

0309-1708/© 2015 Published by Elsevier Ltd.

mplicit approximation, and (ii) complexity of applying a straightfor-

ard Newton method to solve the nonlinear system of equations in

he compositional multiphase flow. The importance of using implicit

cheme arises when the use of an explicit scheme is – CPU wise –

xpensive as mentioned above. This will be demonstrated in the core

f this manuscript when we compare computational cost of both im-

licit and explicit schemes using the same gridding and then reduc-

ng the size of one or multiple grid-blocks to study how the CFL con-

ition would affect the CPU time in the explicit scheme. Evidently,

he CPU cost of the implicit scheme will not be affected by the size of

he grid-block as much as by the number of the grid-blocks instead.

Despite the fact the implicit approximations are incorporated in

ll of the commercial models, very few publications could be found

hen an implicit scheme are discussed in compressible multiphase

ow. Most of the literature goes back to early publications. Coats

9] presented a fully implicit scheme for compositional multiphase

ow. In his model the set of unknowns consist of pressure, and phase

ole fractions. These variables are referred to as natural-type vari-

bles. Fussel and Fussel [10] presented a formulation that uses a dif-

erent set of variables based on phase compositions. These variables

re referred to as the mass variables. Later, Chien et al. [11] proposed

sing the equilibrium ratios as a set of unknowns beside the pres-

ure and the overall concentrations rather than saturations and phase

ompositions.

In Fig. 1 we show a comparison of our model to two different com-

ercial models that we denote by CM-1 and CM-2. The comparison

s based on a modified 3-component mixture example presented in

A. Zidane, A. Firoozabadi / Advances in Water Resources 85 (2015) 64–78 65

CM

-1C

M-2

OU

RM

OD

EL

Fig. 1. Comparison of our model to CM-1 and CM-2 with different mesh refinements;

modified coats example.

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9]. The modified example is in two-phase flow in 1-D. Results show

igh numerical dispersion from CM-1, even with very fine mesh com-

ared to CM-2. Only CM-2, after removing the over and under-shoots

ives comparable results to our model. We should note that we also

ompared the results to a third commercial model (CM-3). We do not

how the results of CM-3 since it does not converge when a more

efined mesh than 120 grid-blocks is used in the modified Coats ex-

mple. All of the three commercial models (CM-1, 2 and 3) are based

n first order finite difference discretization in space and time. To

emonstrate the low numerical dispersion in our model, we com-

are results of 1-D and 2-D examples (with our model) to an explicit

igher-order method (discontinuous Galerkin method) in Section 4.

The most recent work on compositional multiphase implicit

cheme is reported in [12]. These authors chose the set of unknowns

ased on the overall molar density of species and the pressure. The

hase compositions are updated in a post-processing step using con-

tant volume and temperature flash routines.

In this paper we present a new model that solves implicitly

he species balance equation. The species transport equation is dis-

retized by using a finite volume (FV) approximation. The Newton–

aphson (NR) method is used to solve implicitly the species trans-

ort equation. The calculation of the derivatives in our formulation

ill be discussed later in detail. The total flux is calculated by the hy-

ridized mixed finite element method (MFE). The latter provides ac-

urate calculation of the velocity field even in highly heterogeneous

edia when compared to the traditional finite element and finite vol-

me methods [1,5,7,13–20]. The strength of the MFE method is from

he calculation of the pressure inside a finite element and the traces

f the pressures at the interfaces of each finite element in the compu-

ational domain. The flux of each phase is deduced from the total flux

y using the phase mobility coefficient as will be discussed later. In

his paper the effect of gravity is taken into account. In two-phase

ow, the calculation of the phase fluxes is not trivial with gravity

ffect. The difficulty arises from up-streaming the mobility of each

hase. Without gravity, both phases flow in the same direction as the

otal flux. With gravity, however, counter-current flow may develop

t the finite element interfaces due to the density contrast. Updat-

ng the phase mobilities based on the values at the previous time

tep is not consistent. A phase could appear/disappear from one time

tep to another. To resolve this complexity, we have developed an ef-

cient method to upstream the phase derivatives based on the up-

ated phase mobilities and phase fluxes at the current time step. In

his paper we use the same upstreaming technique of the phase mo-

ilities as in [1,7]. Once evaluated, the phase fluxes are then coupled

ith our upstreaming technique of the phase derivatives discussed in

etails in Section 3.

The compressible behavior of each phase is described by the

eng–Robinson equation of state [21]. The computation of compo-

ition of each phase is provided by the equality of fugacities of each

omponent in both (vapor and liquid) phases. This calculation is com-

only known in the literature by flash [22]. For a given pressure P

nd temperature T the flash calculation is performed at each finite

lement of the computational domain and the calculation is known

s PT-flash calculation. However, when an implicit scheme is used to

olve the species transport equation, the derivatives of concentration

f each component of each phase is computed with respect to the

otal concentration at constant volume V and temperature T. In this

ork we will show how the derivatives can be calculated without the

T-flash.

In addition to the new derivatives that appear in the species trans-

ort equation in the implicit scheme, we couple the volumetric fluxes

o the species equation differently from the work of [9,12] and oth-

rs. The pressure in our formulation is calculated in a preprocessing

tep in order to update the fluxes based on the converged values of

he molar densities. We believe that this approach reduces the num-

er of iterations per time step when compared to the implicit update

f the pressures with the molar densities and compositions. The nu-

erical examples in this work will be compared to a higher-order ex-

licit method, that is, the discontinuous Galerkin (DG) method. The

G method is based on a linear approximation of the molar density

nside each grid cell. As a result, we have three degrees of freedom

n each grid cell for the transport variable. A finite difference time

iscretization is used in both models. The aim of this work is to show

hat our implicit model can produce accurate results even when com-

ared to a higher-order method such as the DG. We believe that the

mplicit scheme can be more efficient than the explicit scheme in

roblems that the CFL condition has a severe constraint on the time

tep as we mentioned above.

The rest of the paper is organized as follows: in the next section

e provide the differential equations describing the multicomponent

ompressible two-phase flow in porous media. Then we present the

iscretization of the pressure and the species balance equations. We

resent seven numerical examples to demonstrate the efficiency and

ccuracy of the proposed algorithm.

. Mathematical model

.1. Species balance equation

The mass balance of component i in compressible two-phase (gas

nd oil) flow of nc-component mixture is given by the following

quations:

66 A. Zidane, A. Firoozabadi / Advances in Water Resources 85 (2015) 64–78

3

t

t

t

n

P

3

t

G

w

s

i

i

n

e

∂u

w

d

t

3

t

c

w

∂f

c

g

t

T

w(

w

i

e

∑

∅∂czi

∂t+ ∇ .

(∑α

cαxi,αvα

)= Fi, i = 1 . . . nc in � × (0, τ ) (1)

The above equation is subject to the following constraints:

nc∑i=1

zi =nc∑

i=1

xi,α = 1 ∀α (2)

In the above equations, ∅ denotes the porosity, vα the velocity of

phase α, c the overall molar density of the mixture; zi and Fi are the

overall mole fraction and the sink/source term of component i in the

mixture, respectively. cα is the molar density of phase α and xi, α is

the mole fraction of component i in phase α. � is the computational

domain and τ denotes the simulation time and nc is the number of

components. We neglect diffusion in Eq. (1).

The velocity for each phase α is given by the Darcy law:

vα = −Kkrα

μα(∇p − ραg) = −λαK(∇p − ραg), α = o, g (3)

where K is the absolute permeability, krα , μα and ρα are the rela-

tive permeability, dynamic viscosity and mass density of phase α re-

spectively, with λα = krα/μα; p is the pressure and g is the gravita-

tional acceleration. Capillary pressure is neglected, in a forthcoming

work both capillary and diffusion effects will be included. The rela-

tive permeabilities are calculated as a function of the saturation using

a quadratic relationship [23,24], and to find the phase viscosities we

follow Lohrenz et al. [25].

2.2. Pressure equation

The pressure equation is calculated using the concept of total vol-

ume balance [26,27] given by:

∅Ct∂ p

∂t+

nc∑i=1

V̄i∇(∑

α

cαxi,αvα

)=

nc∑i=1

V̄iFi (4)

where Ct is the total compressibility and V̄i is the total partial molar

volume of component i (see [22]).

The local thermodynamic equilibrium implies the equality of the

fugacities of each component in the two phases:

fo,i

(T, p, x j,o

)= fg,i

(T, p, x j,g

), i = 1, . . . nc; j = 1, nc − 1 (5)

The phase and volumetric behavior are modeled by the Peng–

Robinson equation of state [21]:

cα = Nα

Vα= p

ZαRT, ρα = cα

nc∑i=1

xi,αMi

Zα3 − (1 − Bα)Zα

2 +(Aα − 3Bα

2 − 2Bα

)Zα

−(AαBα − Bα

2 − Bα3)

= 0 (6)

where Nα , Vα , Zα , cα , ρα , are the number of moles, volume, compress-

ibility factor, molar and mass density of phase α. Mi is the molar

weight of component i. R is the universal gas constant and T is the

temperature. Aα and Bα are the parameters of the PR-EOS which de-

pend on pressure, temperature and composition of each phase [22].

The saturation of each phase is calculated from

Sα = c

cαωα (7)

ωα = Nα/∑β

Nβ , where β represents different phase indices. Using

Eq. (7), the saturation constraint could be then written in the follow-

ing form:

1 − c

(ωo

co+ ωg

cg

)= 0 (8)

The above equation is used as a criterion for the selection of a time

step.

. Numerical discretization

In the numerical model we calculate the traces of the pressure at

he element interfaces followed by the calculation of the pressure at

he center of each element with a simple back substitution. In the

ransport equation we update the molar densities of all the compo-

ents followed by the phase compositions and saturations within the

T-flash package at each NR iteration.

.1. Discretization of the transport equation

In the following we show how we solve implicitly the species

ransport equation. Using a FV integration in Eq. (1) we get:

K,i = ∅|K| czn+1i,K

− czni,K

t+

∑E∈∂K

∑α

[c̃αxi,α

n+1qα,K,E

]− |K|Fi = 0 (9)

here |K| is the surface area of the finite element K, c̃αxi,α is the up-

tream value of cαxi, α , and qα, K, E is the normal flux of phase α at the

nterface E of element K.

In the above equation, GK, i is a function of czi, K and the surround-

ng elements of the element K. In the Newton–Raphson method one

eeds to evaluate the term ∂GK,i/∂czn+1i

, that is:

∂GK,i

∂czn+1i

= ∅|K|t

+∑

E∈∂K

∑α

[∂ c̃αxi,α

∂czn+1i

n+1

qα,K,E

](10)

The two major complexities in the above equation are: (i) the

valuation of the term ∂cαxi, α/∂czi, and (ii) upstream the derivative

˜cαxi,α/∂czi if one or more of the surrounding elements are in the

pstream direction with respect to element K. In the next section we

ill show how to calculate these derivatives and how to upstream a

erivative if the surrounding elements are in different phases from

he element K.

.1.1. Evaluation of the derivative ∂cαxi, α /∂czi

We represent czi by ci and cαxi, α by cα, i. From the definition of

he molar density of component i in phase α, cα, i, the derivative of

α,i(= nα,i/Vα) with respect to the total molar density ci could be

ritten in the following form:

∂cα,i

∂ci

= Vα∂nα,i/∂ci − nα,i∂Vα/∂ci

Vα2

(11)

In order to calculate ∂cα, i/∂ci one needs to evaluate ∂nα, i/∂ci and

Vα/∂ci. To do so we use: (i) thermodynamic equilibrium based on

ugacities, and (ii) constant volume at each grid-cell element in the

omputational domain.

At constant volume and temperature (the conditions inside each

rid-cell) the fugacity of each component i in phase α, fα, i is a func-

ion of the number of moles of all the components in the same phase.

he variation of the fugacity fα, i with respect to each component l is

ritten as follows:

∂ fα,i

∂nl

)nl

=∑

j

(∂ fα,i

∂nα, j

)p,nα, j

(∂nα, j

∂nl

)nl

+(

∂ fα,i

∂ p

)nα

(∂ p

∂nl

)nl

(12)

here the vector nl = (n1, n2, . . . nl−1, nl+1, . . . nc). The total volume

ndex is dropped for brevity.Using Eq. (12) for both phases with the

quality of fugacities in Eq. (5) we obtain

j

(∂ fg,i

∂ng, j

+ ∂ fo,i

∂no, j

)∂no, j

∂nl

+(

∂ fo,i

∂ p− ∂ fg,i

∂ p

)∂ p

∂nl

= ∂ fg,i

∂ng,l

(13)

A. Zidane, A. Firoozabadi / Advances in Water Resources 85 (2015) 64–78 67

Fig. 2. Overall mole fractions of the 5 components in our algorithm and the DG method at 60% PVI with 200 elements; Example 1.

a

g

V

c(

nd gas phases; it is equal to the pore volume of the finite element

rid-cell:

t = Vo + Vg (14)

The variation of the volumes with respect to variation of moles of

omponent i could be then written as:

∂Vt

∂ni

)ni

=(

∂Vo

∂ni

)ni

+(

∂Vg

∂ni

)ni

= 0 (15)

68 A. Zidane, A. Firoozabadi / Advances in Water Resources 85 (2015) 64–78

Table 1

Relevant data of the domain and initial conditions (C1/C2/C3/C4/C5 in mole fraction);

Example 1.

Porosity 0.2

Permeability 10 md

Injection rate 10 PV/year

Temperature 311 K

Pressure 69 bar

Injected fluid C1

Initial fluid (mole fraction) 0.25 C2/0.25 C2/0.25 C3/0.25C4

Relative permeability (oil and gas) coefficients Endpoint = 1, power = 2

Residual saturation 0

Table 2

CPU time (s) for the implicit and the DG explicit scheme;

Example 1.

Number of elements DG Implicit Ratio

50 1.2 5.6 4.6

100 3.3 12.4 3.8

200 5.7 20.5 3.6

Cut-2 8.12 20.7 2.5

Cut-10 14.7 20.9 1.4

Cut-50 23.4 21.2 0.9

Table 3

Relevant data of the domain and initial conditions (C1/C2 are in mole fraction);

Example 2.

Porosity 0.2

Permeability 10 md

Injection rate 31.025 PV/year

Temperature 311 K

Pressure 69 bar

Injected fluid C1

Initial fluid C2

Relative permeability (oil and gas) coefficients Endpoint = 1, power = 2

Residual saturations 0

Table 4

CPU time (s) for the implicit and the DG explicit

scheme; Example 2.

Number of elements DG Implicit

36 (6 × 6) 0.5 1

100 (10 × 10) 1.5 4

400 (20 × 20) 7 22

1600 (40 × 40) 32 51

3600 (60 × 60) 304 228

10,000 (100 × 100) 3214 712

40,000 (200 × 200) 13,256 2671

Table 5

Relevant data of the domain; Example 3.

Porosity 0.2

Permeability 10 md

Injection rate (at atmospheric conditions) 97.33 PV/year

Temperature 403.15 K

Pressure 276 bar

Relative permeability (oil and gas) coefficients Endpoint = 1, power = 2

Residual saturations 0

Table 6

Initial and injected compositions (mole fractions);

Example 3.

Component Initial in the domain Injected

CO2 0.0086 1.

N2 0.0028 0.

C1 0.4451 0.

C2–C3 0.1207 0.

C4–C5 0.0505 0.

C6–C10 0.1328 0.

C11–C24 0.1660 0.

C25+ 0.0735 0.

o

s

i

t

3

T

i

M

o

v

From the equation of state for each phase:(∂Vα

∂ni

)ni

= RT

[−NαZα

p2

∂ p

∂ni

+ 1

p

(Nα

∂Zα

∂ni

+ Zα∂Nα

∂ni

)](

∂Zα

∂ni

)ni

=nc∑

j=1

(∂Zα

∂nα, j

)p,nα, j

(∂nα, j

∂ni

)ni

+(

∂Zα

∂ p

)nα

(∂ p

∂ni

)ni

(16)

Using Eq. (16) in Eq. (15) and after arrangements:(−NgZg

p2− NoZo

p2+ Ng

p

∂Zg

∂ p+ No

p

∂Zo

∂ p

)∂ p

∂ni

+ 1

pNg

∂Zg

∂ng,i

+ 1

pZg

+ 1

p

(nc∑

j=1

(No

∂Zo

∂no, j

+ Zo − Ng∂Zg

∂ng, j

− Zg

)∂no, j

∂ni

)= 0 (17)

Eqs. (13) and (17) form a system of equations for each component

i in the mixture which are used to calculate ∂nα, i/∂ni and ∂p/∂ni.

Once these derivatives are evaluated we use them in Eq. (11) to find

∂cα, i/∂ci knowing that for every quantity X:

∂X

∂ci

= ∂X

∂ni

Vt (18)

Once ∂cα, i/∂ci are evaluated, the phase fluxes qα, K, E are used to

upstream the derivative ˜∂cαxi,α/∂czi at each interface of element K.

However, the upstream depends on the number of phases inside the

adjacent element of K that we denote by K′ as follows (this applies for

all the adjacent elements of K):

(i) If qα, K, E ≥ 0 then ˜∂cαxi,α/∂czi = (∂cαxi,α/∂czi)K ; the subscript

K denotes that the derivative is evaluated in element K.

(ii) If qα, K, E < 0 then four different cases should be considered:

(1) When elements K and K′ are both in two-phase then we

simply upstream each phase derivative from K′ to K. Hence˜∂cαxi,α/∂czi = (∂cαxi,α/∂czi)K′

(2) If the element K is in two-phase and K′ is in single phase

then the derivative of the phase in K′ is set to one and the

derivative of the absent phase is set to zero. This is readily

deduced from the fact that if there is only oil in K′ then the

molar concentration of the oil phase is the total molar con-

centration of component i and hence its derivative is unity.

The zero value of the absent phase derivative follows the

same logic.

(3) If the element K is in single phase and K′ is in two-phase

then the opposite procedure in (2) is followed.

(4) If both elements are in single phase, then it is a combination

of (2) and (3) together.

We have examined the derivative (∂cαxi, α/∂czi)V, T for a number

f conditions; it varies in the range of −1.2 to 1.2. A detailed plot that

hows the variation of the derivatives as a function of pore volume

njection (PVI) is presented in Appendix A. It is the small variation

hat may have a profound effect on convergence in our formulation.

.2. Discretization of the total and phase fluxes

The total flux is approximated by the lowest order Raviart–

homas space (RT0). The resulting velocity in the MFE has a marked

mprovement compared to traditional FD methods [28,29]. In the

FE method, the total velocity in each grid-cell K is written in terms

f the normal fluxes across each interface E of K as follows:

=∑α

vα =∑

E∈∂K

qK,EwK,E (19)

A. Zidane, A. Firoozabadi / Advances in Water Resources 85 (2015) 64–78 69

a) 6×6 b)10×10

c)20×20 d)40×40

e)60×60 f)100×100

g)200×200

Fig. 3. Mesh refinements used; Example 2.

Table 7

Relevant parameters for Peng–Robinson EOS; Example 3.

Component Acentric factor Tc (K) Pc (bar) Mol. weight (g/mol) Vc (m3/kg)

CO2 0.239 304.14 7.375E+01 44 0.00214

N2 0.039 126.21 3.39E+01 28 0.00321

C1 0.011 190.56 4.599E+01 16 0.00615

C2–C3 0.11783 327.81 4.654E+01 34.96 0.00474

C4–C5 0.21032 435.62 3.609E+01 62.98 0.00437

C6–C10 0.41752 574.42 2.504E+01 110.21 0.00425

C11–C24 0.66317 708.95 1.502E+01 211.91 0.00443

C25+ 1.7276 891.47 0.76E+01 462.79 0.00417

70 A. Zidane, A. Firoozabadi / Advances in Water Resources 85 (2015) 64–78

a)20×20, Implicit b)40×40, Implicit

c) 60×60, Implicit d)60×60, DG

Fig. 4. Methane overall mole fraction in different mesh refinements in the implicit scheme (a, b, c) and in the DG explicit scheme (d) at 70% PVI. Distances in meter; Example 2.

Table 8

The symmetric binary interaction parameter matrix; Example 3.

CO2 N2 C1 C2–C3 C4–C5 C6–C10 C11–C24 C25+

CO2 0.0

N2 0. 0.0

C1 0.15 0.1 0.0

C2–C3 0.15 0.1 0.0346 0.0

C4–C5 0.15 0.1 0.0392 0.0 0.0

C6–C10 0.15 0.1 0.0469 0.0 0.0 0.0

C11–C24 0.15 0.1 0.0635 0.0 0.0 0.0 0.0

C25+ 0.08 0.1 0.1052 0.0 0.0 0.0 0.0 0.0

Table 9

Performance of our model and DG with

3600 grid blocks; Example 3.

CPU (min) Newt/t

DG 14 –

Our model 16 2.88

3

K

∅

w

where wK, E is the RT0 basis function across edge E of element K and

qK, E is the normal flux at interface E and is calculated through the

average cell pressure of K and the traces of pressure at the interfaces

of K as follows:

qK,E = αK,E pK −∑

E′∈∂K

βK,E,E′t pK,E′ − γK,E (20)

The coefficients αK, E, βK,E,E′ and γ K, E depend on the geometri-

cal shape of the element and the mobility. For more details about

these coefficients and the MFE formulation the reader may refer to

[15,16,30–33].

Once the total velocity is evaluated we can calculate the velocity

of each phase independently by using Eqs. (3) and (19):

vα = fα(v − Gα) (21)

with

fα = λα∑β λβ

and Gα ={λo(ρo − ρg)g, if α = gλg(ρg − ρo)g, if α = o

(22)

u

.3. Discretization of the pressure equation

Inserting Eq. (21) into Eq. (4) and integrating over a finite element

we obtain after using Gauss’s theorem on the divergence term:

|K|Ctp

t+

nc∑i=1

∑α

∑E∈∂K

V̄i,K

∫E

cαxi,α fα(v − Gα).nE =nc∑

i=1

V̄i,KFi,K

(23)

The coefficient cαxi, α , fα and V̄i are evaluated at the element center

ithout considering higher-order spatial approximation. The same

pwind technique that is used in species transport equation (based

A. Zidane, A. Firoozabadi / Advances in Water Resources 85 (2015) 64–78 71

Fig. 5. Overall mole fraction of component 1, and the gas saturation profiles of our model and DG and the gas saturation of CM-2 at 65% PVI with 3600 elements; Example 3.

Table 10

Performance of our model and the CM-1 and CM-2 commercial simulators; Example 3.

Elements CM-1 CM-2 Our model

#T #Newt Newt/t #T #Newt Newt/t #T #Newt Newt/t

400 68 192 2.82 26 67 2.56 71 186 2.62

1600 133 374 2.82 102 270 2.64 121 329 2.72

3600 149 433 2.91 102 291 2.85 162 466 2.88

o

p

4

n

t

1

D

e

t

fi

t

f

3

4

l

t

T

E

t

t

c

T

2

P

p

b

i

n the direction of the phase flux) cannot be used here since the

hase fluxes are not known yet.

. Numerical examples

In the following we present seven numerical examples with the

umber of species varying from 2 to 10 in structured grids to inves-

igate the efficiency of our model. In the first example the domain is

-D with 5 components. In the rest of the examples the domain is 2-

and the number of components varies between 2, 8 and 10. In the

xamples, the time step size in the implicit model is on average 2–50

imes larger than in the DG model. Within the NR method, the very

rst initial guess is from the initial compositions of oil and/or gas in

he domain. During the simulation, the initial guess at time step n is

rom the solution at the previous time step n−1. An Intel Core-i5 PC,

GHZ CPU, 4 GBRAM is used in all the runs.

.1. Example 1: 5-component mixture in 1-D

In this 1-D example, methane is injected at one side of a 50 m

ong domain. Production is at the opposite side. The domain is ini-

ially saturated with a mixture of equal mole fractions of C2/C3/C4/C5.

he high number of components implies large matrix inversions (see

q. (10)) at each time step. At each iteration the number of ma-

rix inversions is equal to the number of components. The size of

he matrix to be inverted equals the number of grid-blocks in the

omputational domain. The relevant data of the domain are given in

able 1. Three different mesh refinements are used with 50, 100 and

00 grid-blocks. The overall compositions of the 5 components at 60%

VI are plotted in Fig. 2 with the 200 grid-blocks for both the im-

licit and DG models. There is agreement for all of the components

etween our implicit results and the DG results. The CPU cost of the

mplicit scheme and the higher-order explicit scheme with the same

72 A. Zidane, A. Firoozabadi / Advances in Water Resources 85 (2015) 64–78

Fig. 6. Gas saturation and overall mole fraction of CO2 at two PVI with 7680 elements; Example 4.

o

e

c

(

e

w

c

t

T

t

e

r

p

c

gridding is shown in Table 2. Obviously, the explicit scheme is more

efficient when the CFL condition does not put a serious limitation

on the time step. However, in Table 2 we show the CPU time ratio

of the implicit scheme over the explicit. Even with these coarse

meshes, the CPU ratio reduces from 4.6 to 3.6 when the number of

elements increases to 200. Despite the fact that the matrix inversion

is more expensive when the number of elements increases, the CFL

condition with a more refined mesh affects the CPU time. For further

examination, we select one grid-block (first block on the left side) of

the 200-element mesh in both models (DG and implicit) and divide

its size by 2, then divide it by 10 (i.e. one grid-block has the size of half

of the rest of the elements in the first case, and in the second case one

grid-block has one tenth the size of the rest of the elements). We re-

fer to these cases in Table 2 by cut-2 and cut-10. When the size of

ne grid-block reduces to half and to one tenth of the original 200

lements mesh, the CPU time ratio reduces from 2.5 down to 1.4 with

ut-10. Furthermore, if we reduce the size of one grid-block 50 times

cut-50, Table 2) the implicit scheme is now more efficient than the

xplicit scheme with almost the same level of accuracy (Fig. 2). Here

e note that: (1) the results from the original and the 3 cuts (cut-2,

ut-10, and cut-50) are exactly the same as shown in Fig. 2, and 2)

he CPU time for the implicit scheme is the same in the three cuts.

he 0.2 and 0.3 s are just numerical fluctuations. Having one element

hat is smaller by a factor of 50 and even more than the rest of the

lements in the computational domain is very common in fractured

eservoir simulation. If one describes flow in small grids by the im-

licit method and larger grids by the explicit method, the method is

alled the adaptive implicit method.

A. Zidane, A. Firoozabadi / Advances in Water Resources 85 (2015) 64–78 73

Fig. 7. Gas saturation (left) and composition of methane (right) at 10% and 60% PVI with 1600 elements; Example 5.

Table 11

Performance of our model and DG with 7680 grid

blocks; Example 4.

CPU (min) Newt/t

DG-mesh-1 41 –

Our model-mesh-1 33 2.82

DG-mesh-2 78 –

Our model-mesh-2 33 2.83

4

o

t

t

fi

1

p

d

g

m

b

p

fi

s

s

m

a

4

w

t

g

a

g

m

f

m

n

fi

w

t

m

d

e

m

f

r

i

o

t

w

n

a

p

T

.2. Example 2: 2-component mixture in 2-D

In this example the domain is 50 × 50 m2. Methane is injected at

ne corner to displace the initially saturated propane to the produc-

ion well at the opposite corner of the domain. The relevant data of

his example are given in Table 3. We examine 7 different mesh re-

nements from a very coarse mesh of 36 to 100, 400, 1600, 3600,

0,000 and 40,000 elements (Fig. 3). In Fig. 4 we show the com-

osition (overall mole fraction) profile of methane at 70% PVI with

ifferent mesh refinements in our model and DG model; there is

ood agreement (Fig. 4c, d). The CPU time for different mesh refine-

ents is shown in Table 4. Results show that as the size of the grid-

lock decreases (more refined mesh) the difference between the ex-

licit scheme and the implicit model reduces. With the 3600-element

ne mesh, the implicit scheme becomes faster than the DG explicit

cheme. This is due to the effect of the CFL condition in the explicit

cheme and to the efficiency of our scheme. At 40,000 elements our

odel is 5 times faster than DG. The average number of Newton iter-

tions per time step in this example is 2.75.

.3. Example 3: 8-component mixture in 2-D

The domain in this example is 50 × 50 m2; it is initially saturated

ith an 8-component oil. CO2 is injected at one corner to displace

he oil to the opposite corner. The relevant data for this example are

iven in Table 5 and the compositions of the initial and injected fluids

re given in Table 6. The PR-EOS parameters for each component are

iven in Table 7 and the symmetrical binary interaction parameters

atrix is given in Table 8. Similar to the previous example, we use dif-

erent mesh refinements, and show the results for the more refined

esh of 3600 elements. The overall compositions of the first compo-

ent and the gas saturation are shown in Fig. 5. The composition pro-

les and the gas saturation demonstrate the accuracy of our scheme

hen compared to the higher-order DG method. The CPU time with

he 8-component mixture is more expensive than the 2-component

ixture since the number of matrix inversions at each iteration is

irectly related to the number of components and the number of

lements (grid-blocks). In our implicit model, from a 2-component

ixture to an 8-component mixture, the CPU time increases by a

actor of 4.2. With the 8-component mixture, the implicit method

equires around 16 min (Table 9) to converge to the results shown

n Fig. 5 compared to 3.8 min with a 2-component mixture. On the

ther hand, the CPU time of the explicit DG scheme is 14 min due to

he CFL constraint on a refined mesh. In case of a more refined mesh

ith the explicit scheme, the use of our implicit model becomes a

atural choice. The gas saturation of our model and the DG model

re compared to CM-2 (Fig. 5). As Fig. 5 shows, CM-2 produces non-

hysical oscillations in the gas saturation near the production well.

here are no oscillations in our model and in the higher-order DG

74 A. Zidane, A. Firoozabadi / Advances in Water Resources 85 (2015) 64–78

Fig. 8. Gas saturation (a) and compositions of the first (b), fourth (c) and seventh (d) components at 60% PVI with 3600 elements; Example 6.

Fig. 9. Permeability distribution; Example 7.

m

i

p

w

g

1

o

b

F

e

r

model. The efficiency of our model is compared with CM-2. CM-2 in

turn is more accurate than CM-1 (see Fig. 1). Since our model and the

commercial code are run in different machines, the CPU time would

not give clear demonstration of the efficiency. Therefore, we choose

to compare three parameters; the total number of time steps, the to-

tal number of Newton iterations and the average number of Newton

iterations per time step during the whole simulation. A total number

of 102 time steps and 291 Newton iterations are required by CM-2 in

the refined mesh of 3600 elements, compared to 162 time steps and

466 Newton iterations in our model. As a result the average number

of iterations per time step is 2.85 in CM-2 compared to 2.88 in our

model (Table 10). However as Fig. 5 shows the numerical dispersion

in CM-2 in 2-D is much higher than in our model. To demonstrate the

low numerical dispersion in our model we compare the results to the

explicit higher-order DG method (Fig. 5); results show the profiles in

our model and in the explicit DG are about the same.

In Appendix B, we provide more detailed study of CPU time and

gas saturation profile of this example by comparing the results from

our model to the DG model at different mesh refinements. We also

provide gas saturation plots over the diagonal for both models to

quantify the numerical dispersion in the implicit model compared to

the DG.

4.4. Example 4: 10-component mixture in 2-D domain

We consider a 500 × 150 m2 reservoir initially saturated with

10-component oil. The relevant data of the domain, the initial and

injected fluid compositions are the same as in Example 3. The 8-

component mixture of the last example is adapted to 10-component

ixture by dividing the C1 component and the C2–C3 component

nto C1-a, C1-b and C2–C3-a, C2–C3-b, respectively. In this case the

hase behavior will not be affected but the performance of the model

ill be influenced by the additional two components and the fine

ridding used in this larger domain. The domain is discretized by

60 grids in the x-direction and 48 grids in the y-direction, a total

f 7680 structured grid elements (mesh-1). With this discretization,

oth the horizontal and vertical elements have a length of 3.125 m.

ig. 6 shows the composition of CO2 and the gas saturation at differ-

nt PVI. For brevity we do not show the overall compositions for the

est of the components. The efficiency of our model in this example is

A. Zidane, A. Firoozabadi / Advances in Water Resources 85 (2015) 64–78 75

Fig. 10. Gas saturation along the diagonal at 15% PVI (a) and 30% PVI (b); Example 7.

c

i

a

(

u

n

s

t

t

4

d

d

Fig. 12. Relative L2 norm error as a function of PVI; Example 7.

t

C

t

4

i

W

r

e

a

5

T

f

i

p

t

t

(

t

i

S

g

a

ompared to the higher-order DG method. The CPU time of our model

s about 33 min; it is in the order of 41 min in the DG and the aver-

ge number of Newton iteration per time step is 2.82 in our model

Table 11). To examine the effect of CFL condition in this example, we

sed different refinements on x and y-directions than mesh-1. The

umber of grids on x and y directions in mesh-1 and mesh-2 are the

ame. As a result, the total number of elements in both meshes is

he same. In mesh-2 we set the size of one element at the center of

he domain (x = 250 m) to 0.625 m and its adjacent two elements to

.375 m, and the rest of the elements are set to 3.125 m. With similar

iscretization on y-axis, we have one grid-block at the center of the

omain with surface area of 1/25th of elements in mesh-1. The CPU

Fig. 11. Gas saturation distribution at 50% PVI in the implicit mod

ime in our model remains almost the same (33 min). In the DG the

PU time increases to 78 min due to the CFL condition resulting from

he small grid-block in the domain.

.5. Example 5: 2-component mixture in 2-D with gravity

In this example we consider a vertical domain (that is, with grav-

ty). The input data of this example are the same as in Example 2.

ith gravity, counter-current flow may develop; this could add more

estrictions on the time step. In this example we demonstrate the

fficiency of our model when the gravitational effect is taken into

ccount. In this 2-component mixture, the CPU time is 125 s. It is

1 s without gravity with the same 1600-element mesh refinement.

he average number of Newton iterations per time step increases

rom 2.7 without gravity to 2.92 with gravity. However, this increase

s expected due to the fact that without gravity, when a phase ap-

ears/disappears, the phase flux is always in the same direction as

he total flux (whether it is influx or out-flux). In an implicit update,

his means, more elements contribute to the molar density update

in fact to the derivative of the molar density) at the interface where

he flux is evaluated. With gravity, the two adjacent elements of one

nterface could then contribute to the calculation (as discussed in

ection 3) and hence more iterations are required to reach conver-

ence. The gas saturation and the composition of methane are shown

t different PVI in Fig. 7. In a more refined mesh of 3600 elements

el (a) and explicit DG (b) using 3600 elements; Example 7.

76 A. Zidane, A. Firoozabadi / Advances in Water Resources 85 (2015) 64–78

a

o

s

i

m

o

t

u

(

5

v

n

e

t

t

t

n

s

1

m

s

5

i

t

p

b

(results not shown) the CPU time becomes 6 min for a simulation

time of 2 years.

4.6. Example 6: 8-component mixture in 2-D with gravity

In this example we consider injection of CO2 into a vertical do-

main saturated with 8-component oil. Dimensions and properties of

the domain are the same as Example 3. The high number of compo-

nents affects CPU time. With gravity the number of iterations at each

time step increases, because the complexity of the flow increases as

discussed earlier. The high number of components and the gravity ef-

fect combined should have a significant effect on the CPU time. The

average number of Newton iterations per time step is 3.21 compared

to 2.72 without gravity. In [12], the authors used this example and

report that the CPU time is 96.7 h for a simulation time of 1.36 years

with 3200 elements. In our model, the CPU time is 32 min for a sim-

ulation time of 2 years, and a mesh refinement of 3600 elements. The

results from our code are the same as in [12] (Fig. 8).

We should note that the total number of iterations per time step

could be reduced if we decrease the maximum allowed time to 10−3

years; the CPU time in this case increases to 52 min and the average

number of iterations per time step reduces to 2.65. We show in Fig. 8

the gas saturation and the compositions for different components at

60% PVI.

4.7. Example 7: 8-component mixture in 2-D heterogeneous media

In the last example we simulate injection of CO2 in heterogeneous

media. The domain is divided into four zones with 3 different perme-

Fig. A1. Variation of the derivatives (a–d) and

bilities such that the permeability distribution is symmetric along

ne of the diagonals in the horizontal domain (see Fig. 9). We demon-

trate low numerical dispersion in our implicit model, by compar-

ng the saturation distribution along the diagonal to the explicit DG

odel. Fig. 10 demonstrates that as in the 1-D example, in 2-D as well

ur model produces low numerical dispersion even when compared

o the higher-order explicit scheme. A good agreement of the gas sat-

ration in the implicit and DG models is observed at 15% and 30% PVI

Fig. 10). For reference we show in Fig. 11 the gas saturation profile at

0% PVI in both models (implicit and DG) with 3600 elements. To in-

estigate the convergence of both models, we calculate the relative L2

orm error of the gas saturation compared to a fine mesh with 8100

lements at different PVI. Fig. 12 illustrates the variation of the rela-

ive error for both models as a function of PVI. As indicated in Fig. 12,

he highest error for both models is less than 3% and decreases to less

han 0.5% at higher PVI. The oscillations at 10 and 25% PVI are due to

umerical fluctuations within a range of 0.4%, therefore they are in-

ignificant in the overall error. We performed the same example with

0 times lower permeabilities in all zones (i.e. 10 md instead of 100

d and 25 md instead of 2.5 md) and the relative error stays in the

ame range (0.5–3%).

. Conclusions

In this work we have introduced an efficient numerical model for

mplicit treatment of the species transport equation in compositional

wo-phase flow in porous media. The efficiency of our model is com-

arable and in general superior to existing commercial codes that are

ased on implicit methods. The numerical dispersion in our model is

gas saturation (e) as a function of PVI.

A. Zidane, A. Firoozabadi / Advances in Water Resources 85 (2015) 64–78 77

l

o s

g

t

A

2

i

o

(

w

a

t

ess than in the commercial models we have examined. The efficiency

f our model is due to two factors:

(i) The calculation of the derivatives of the molar concentration

of each component at constant volume in each phase with re-

spect to the total molar concentration. The derivatives in all of

our examples vary in the range [−1.2, 1.2]. The small variation

of the nonlinear coefficients in the Newton method result in a

fast convergence at each time step. The number of Newton it-

erations per time step is less as well as the CPU time. The small

range of variation of the calculated derivatives may explain the

low numerical dispersion in our model compared to existing

implicit codes.

(ii) We update the phase fluxes based on the converged solution

of the molar concentrations and mole fractions. This coupling

between the species transport equation and the phase fluxes,

reduces the number of iterations at each time step.

Fig. B1. Gas saturation distributions of the implicit and DG m

In all the examples, the maximum number of iterations per time

tep did not exceed 5. The average number of iteration without

ravity is generally less than 3 and with gravity it is generally less

han 4.

ppendix A

To show the variation of the derivatives with PVI we use Example

and replace the number of components by 2 instead of 5 for simplic-

ty. In Fig. A1 we show the variation of the derivatives as a function

f PVI for a C1 (species 1) injection into a domain saturated with C3

species 2).

When a component is in liquid phase (e.g. species 1) its derivative

ith respect to the total molar density is 1. However, after injection,

second (vapor) phase may appear. The derivative ∂cliquid, 1/∂c1 falls

o 0.4 since species 1 is now divided into two phases. At higher PVI

odels at 75% PVI; for domain properties see Example 3.

78 A. Zidane, A. Firoozabadi / Advances in Water Resources 85 (2015) 64–78

[

[

[

[

[

[

[

the mixture goes back to single phase, so the derivative goes back

to 1.

Appendix B

To study the numerical dispersion of our model compared to DG,

we show the gas saturation distribution in the domain and along the

diagonal (Fig. B1). Comparison is made with different mesh refine-

ments from a coarse mesh of 100 elements to a more refined mesh of

1600 elements. With each mesh refinement we provide the CPU time

in seconds for the implicit and explicit DG models.

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