Combustion and Flame 169 (2016) 51–62

Contents lists available at ScienceDirect

Combustion and Flame

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

Multicomponent effects in liquid–gas filtration combustion

M.A. Endo Kokubun

a , ∗, N. Khoshnevis Gargar b , H. Bruinning

b , A .A . Mailybaev

a

a Instituto Nacional de Matemática Pura e Aplicada – IMPA, Rio de Janeiro, Brazil b TU Delft, Civil Engineering and Geosciences, The Netherlands

a r t i c l e i n f o

Article history:

Received 1 December 2015

Revised 5 April 2016

Accepted 6 April 2016

Keywords:

Filtration combustion

In situ combustion

Medium temperature oxidation

Multicomponent effects

a b s t r a c t

This paper develops the theory of liquid–gas filtration combustion, when an oxidizer (air) is injected into

porous rock containing two-component liquid fuel. We found a qualitatively new combustion mechanism

controlled by the successive vaporization and condensation of the liquid phase sustained by the reaction.

Motivated by the problem of recovery of light oil by air injection, as an enhanced oil recovery method,

we consider a liquid composed of light and medium pseudo-components. The light part is allowed to ox-

idize and vaporize, while the medium part is non-volatile and only oxidizes. The liquid mobility depends

strongly on its composition, with a small viscosity (high mobility) of the purely light component and a

high viscosity for the purely medium (immobile) component. We show that the combined vaporization

and condensation in the combustion wave leads to accumulation of the light component in the upstream

part of the wave, considerably increasing mobility and, therefore, playing a crucial role in the mechanism

of the combustion process. We describe physical implications of this effect, as well as its importance for

applications. The results are confirmed by numerical simulations.

© 2016 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1

p

t

t

s

a

p

o

[

o

c

i

fl

a

fl

c

m

t

t

K

t

t

c

t

l

a

f

i

p

c

a

T

t

r

[

s

d

i

d

o

(

h

0

. Introduction

Filtration combustion, when an oxidizer is injected into a

orous medium containing fuel, has numerous applications in

echnology and nature (enhanced oil recovery by in situ combus-

ion, coal gasification, self-propagating high-temperature synthesis,

molder waves etc.). A large number of theoretical studies in this

rea consider solid (immobile) fuels [1,2] . Then the combustion

rocess can be described using a single-phase flow model for an

xidizer. Examples of such studies are forced forward smoldering

3] , downward buoyant filtration combustion [4] and the advance

f in-situ combustion fronts in porous media [5] . These models

an be extended to study the effects of gas–solid non-equilibrium

n filtration combustion [6] and the transition from smoldering to

aming regimes [7] , for example. When the fuel is a liquid with

relatively low viscosity and boiling temperature, the two-phase

ow model becomes essential. In this case, combustion process

ouples both to the multi-phase flow and to the phase transition

echanism, increasing the complexity of the problem. Recent

heoretical [8,9] , numerical [10] and experimental [11] advances in

his problem showed the fundamental difference of the combus-

∗ Corresponding author.

E-mail addresses: max.akira@gmail.com , maxakira@impa.br (M.A. Endo

okubun), N.khoshnevisgargar@tudelft.nl (N. Khoshnevis Gargar), j.bruining@

udelft.nl (H. Bruinning), alexei@impa.br (A .A . Mailybaev).

h

m

p

t

o

ttp://dx.doi.org/10.1016/j.combustflame.2016.04.008

010-2180/© 2016 The Combustion Institute. Published by Elsevier Inc. All rights reserved

ion process with liquid fuels, compared to the classical filtration

ombustion for solid fuels. In this paper, our goal is to deepen

he understanding of this process by considering multicomponent

iquid fuels. We show that the presence of multiple components in

liquid fuel with different physical properties has a dramatic ef-

ect on the internal structure of the combustion wave, manifesting

tself in a highly counterintuitive way.

The model considered in this work is motivated by the

etroleum engineering applications, where in situ combustion is

onsidered as a technique for enhancing the recovery rate due to

lowering of the oil viscosity thus increasing its mobility [12–14] .

his technique is usually applied for heavy oils. However, due

o thermal expansion and gas drive promoted by the oxidation

eaction and vaporization, it can improve the recovery of light oils

15–17] . When air is injected into the reservoir at medium pres-

ures (10–90 bar), the oxidation mechanism is fundamentally

ifferent for light and heavy oils. The heavy oil undergoes crack-

ng, due to the increase in temperature, which generates coke

eposited on the rock surface. This coke reacts with the injected

xygen, leading to the formation of a high-temperature oxidation

HTO) wave that propagates in the reservoir [5,18] . On the other

and, the oxidation of light oil leads to the scission of liquid

olecules, generating a gaseous product. In this case, the tem-

erature is bounded by the boiling point of the oleic phase, such

hat it does not get large. This is termed the medium-temperature

xidation (MTO) mechanism and recent experiments support the

.

52 M.A. Endo Kokubun et al. / Combustion and Flame 169 (2016) 51–62

s

w

t

s

u

p

o

a

a

c

c

m

o

v

a

c

l

p

h

c

a

t

l

l

c

m

n

t

f

m

e

t

o

s

s

c

t

T

s

e

t

t

s

t

i

h

m

d

a

s

2

g

o

(

m

a

o

r

c

ν

Nomenclature

A rl , A rm

frequency factors for light ( l ) and medium

( m ) component reactions (1/s)

c g heat capacity of gas (J/mol K)

C m

, C o heat capacities of rock and oil (J/m

3 K)

f o , f g oil and gas fractional flow functions

k o , k g oil and gas relative permeabilities (m

2 )

n reaction order

P prevailing gas pressure (Pa)

P atm

atmospheric pressure (Pa)

Q rl , Q rm

enthalpies of light and medium compo-

nent reactions per mole of oxygen (J/mol)

Q v light component vaporization heat (J/mol)

R ideal gas constant (J/mol K)

s o , s g oil and gas saturations

t time (s)

T temperature (K)

T b light component boiling temperature (K)

T bn boiling temperature at P atm

(K)

T ini initial temperature (K)

T ac l

, T ac m

activation temperatures for reactions of

light and medium components (K)

u o , u g , u oil, gas and total Darcy velocities (m/s)

u gl , u gk , u gr Darcy velocities of light ( l ), oxygen ( k ) and

remaining ( r ) gas components (m/s)

u ol , u om

Darcy velocities of light and medium com-

ponent (m/s)

u inj air injection Darcy velocity (m/s)

W v vaporization rate (mol/m

3 s)

W rl , W rm

reaction rates of light and medium com-

ponents (mol/m

3 s)

X l , X m

volumetric fractions in oleic phase

X ini m

medium component fraction in initial oil

x spatial coordinate (m)

Y l , Y k , Y r molar fractions in gas phase (mol/mol)

Y in j

k oxygen fraction in injected gas

λ thermal conductivity of porous rock

(W/m K)

μo , μg viscosities of oil and gas (Pa s)

μl , μm

viscosities of purely light and purely

medium oleic phases (Pa s)

νol , νgl , νom

, νgm

stoichiometric coefficients for light and

medium component reactions

ρo , ρg molar densities of oil and gas (mol/m

3 )

ρ l , ρm

molar densities of purely light and purely

medium oleic phases (mol/m

3 )

ϕ rock porosity

existence of reaction occurring at low temperatures [11] . Pascual

et al. [19] performed a high pressure tube test using light crude

oil to simulate the LTO process, with a stable reaction front at a

temperature of 250 °C. They showed that reservoir oil had excel-

lent burning characteristics, which made the process technically

feasible. When thermal losses through the rock are high, or if

the heat released by the reaction is not sufficient to increase the

temperature significantly, the oxidation reaction occurs not far

from the initial reservoir temperature [20] . In this case, oxidation

is slow and can be incomplete, and oxygen consumption occurs in

a larger reservoir zone [21] .

The MTO process for light oil requires the two-phase (liquid–

gas) flow model. The combustion (MTO) wave was analyzed

theoretically [8,9] for a simplified two-phase model with one

pseudo-component oil (single component liquid fuel) and the

olution to this problem was obtained as a series of traveling

aves. Inside the combustion wave the reaction region separates

he vaporization zone upstream and the condensation zone down-

tream. Note that the position of the vaporization zone in the

pstream part of the combustion wave is opposite to the HTO

rocess, where it travels on the downstream side [18] . The speed

f the wave was shown to be equal to the Buckley–Leverett char-

cteristic speed evaluated at a point separating the vaporization

nd condensation regions (called a resonance condition), and this

ondition allowed determining all macroscopic parameters of the

ombustion wave. The MTO process within a two-component oil

odel was studied in [22] by numerical simulations, where an

leic mixture composed of light (low viscosity) and medium (high

iscosity) fractions was considered. The light component was

llowed to vaporize and oxidize, while the medium non-volatile

omponent only oxidized. Numerical simulations showed that the

ight component has a tendency to accumulate in the upstream

art of the combustion wave despite of vaporization and the much

igher mobility, indicating the importance of multicomponent

onsiderations. However, for mixtures composed predominately of

medium non-volatile fraction, the transient behavior featuring

he HTO process was reported.

In the present work we provide the analytical theory for

iquid–gas filtration combustion with a two-component oil as a

iquid fuel. In our formulation, following the model of [22] , we

onsider a two-component oil mixture, consisting of light and

edium pseudo-components, where the medium component is

on-volatile and immobile. The latter property is represented by

he liquid mobility depending on the composition, which is high

or a purely light oil and vanishes for purely medium oil. Such

odel is the simplest possible that considers multicomponent

ffects and it mimics a wide variety of oil types from very light

o very heavy ones. The result of this paper is the detailed study

f the combustion wave profile providing analytic formulas de-

cribing its macroscopic properties: limiting states and speed. We

how that, due to the phase transition mechanism, the medium

omponent is almost completely expelled from the reaction region

o the downstream side of the wave and, thus, it does not react.

his important property, which is valid for any initial oil compo-

ition cannot be captured with one pseudo-component models,

.g., [8,9] . Furthermore, the described effect is very counterin-

uitive: the medium component decreases the oil mobility and,

hus, would be expected to accumulate on the opposite upstream

ide. The consequences are dramatic: the combustion wave speed

urns out to be independent on the initial oil composition even

f the medium component dominates in the initial oil. For such

eavy initial mixtures, the coupled reaction/flow/phase-transition

echanism in the combustion wave yields a strong increase of gas

rive and provides high downstream oil saturations, both favor-

ble for oil recovery applications. Finally, we perform numerical

imulations supporting our theoretical conclusions.

. Two pseudo-component oil model

We study a combustion front in two-phase flow in which a

aseous oxidizer (air) is injected into a porous rock filled with

il composed of a light and a medium fraction. The light oil

volatile component) can both vaporize and oxidize, whereas the

edium oil (non-volatile component) can only oxidize. In our

pplications we disregard gaseous phase reactions, as annihilation

f free radicals at pore walls reduces drastically the corresponding

eaction rates [23] . We summarize the reaction process of liquid

omponents in the following reaction equations:

νol (light component) + O 2 → νgl (gaseous products) ,

om

(medium component) + O 2 → νgm

(gaseous products) , (2.1)

M.A. Endo Kokubun et al. / Combustion and Flame 169 (2016) 51–62 53

i

d

r

e

m

l

p

l

i

p

r

p

p

w

r

c

w

t

t

t

g

f

i

Y

o

Y

v

t

a

c

p

n

w

α

(

e

t

r

i

w

a

u

S

p

u

u

w

fl

D

t

k

w

i

o

μ

w

i

v

μ

n

s

μ

T

μ

Y

t

S

a

y

e

w

C

Q

T

t

t

u

c

p

ρ

P

p

.e., one mole of oxygen reacts with νoi moles of oleic (liquid) hy-

rocarbons (where i = l, m denotes light and medium components,

espectively) generating νgi moles of gaseous products (H 2 O, CO 2 ,

tc.). This reaction mechanism is used in order to consider the

edium temperature oxidation, when oxygen reacts with oil,

eading to a scission of liquid molecules and generating gaseous

roducts. Such mechanism is fundamentally different from the

ow-temperature oxidation (LTO), when the oxygen molecule is

ncorporated into the liquid oil, generating another liquid com-

ound, and the high-temperature oxidation, where the oxygen

eacts with the coke generated by the cracking due to high tem-

eratures. Water originally present or condensed from the reaction

roducts is neglected. Effect of water in the one-component case

as studied numerically in [10] and it was shown that although

ecovery rates may change, the overall wave structure does not

hange. The changes in our solution due to the presence of water

ill be discussed in future works.

We study two-phase (liquid and gas) one-dimensional flow in

he positive spatial direction x . The oil has saturation s o , describing

he oil occupied fraction of pore volume. The saturation of gas is,

herefore, equal to s g = 1 − s o . In the gaseous phase, we distin-

uish the molar fraction of the light hydrocarbon Y l and the molar

raction of oxygen Y k . The medium oil component does not exist

n the gaseous phase. The remaining components with fraction

r = 1 − Y k − Y l consist of reaction products and inert components

f the injected gas. For the ideal gas, the molar fractions Y l , Y k and

r are equal to the volume fractions.

We assume that the two-component oil has neither heat nor

olume effects due to mixing. Let X m

and X l be the volume frac-

ions of the medium and light components ( X m

+ X l = 1 ), and ρm

nd ρ l be the pure oleic molar densities of the medium and light

omponents, respectively. We remark that the model in a previous

aper [22] was based on molar fractions.

The molar mass balance equations for liquid and gas compo-

ents can be written as [22]

∂

∂t ϕX m

ρm

s o +

∂

∂x ρm

u om

= −νom

W rm

, (2.2)

∂

∂t ϕX l ρl s o +

∂

∂x ρl u ol = −νol W rl − W v , (2.3)

∂

∂t ϕY l ρg s g +

∂

∂x ρg u gl = W v , (2.4)

∂

∂t ϕY k ρg s g +

∂

∂x ρg u gk = −W rl − W rm

, (2.5)

∂

∂t ϕY r ρg s g +

∂

∂x ρg u gr = νgl W rl + νgm

W rm

, (2.6)

here u αj means the Darcy velocity of component j in the phase

. In summary, there are four components, viz., light ( l ), medium

m ), oxygen ( k ), and the rest ( r ). The medium component can only

xist in the oleic phase ( o ), the light component can exist in both

he oleic ( o ) and the gaseous phase ( g ), whereas oxygen and the

emaining gases can only exist in the gaseous phase.

The expressions for velocities of medium and light components

n the oleic phase can be derived as (neglecting mass diffusion,

hich is a small effect in this problem and will be discussed more

head)

om

= X m

u o , u ol = X l u o . (2.7)

imilarly, we approximate the velocities of components in the gas

hase by

gl = Y l u g , u gk = Y k u g , u gr = Y r u g . (2.8)

It is convenient to express the liquid and gas velocities as

o = u f o , u g = u f g , (2.9)

here u is the total Darcy velocity with the oil and gas fractional

ow functions

f o (s o , T , X m

) =

k o /μo

k o /μo + k g /μg , f g = 1 − f o . (2.10)

ependence of the relative permeability functions on their respec-

ive saturations is chosen as

o = k (s o − s or ) 2

(1 − s or ) 2 if s o ≥ s or ; k o = 0 if s o ≤ s or ; k g = ks 2 g ,

(2.11)

here s or is the residual oil saturation and k is the rock permeabil-

ty and has unit of m

2 .

The oleic viscosity μo depends on the pure light and medium

il viscosities, μl and μm

, and we use the expressions

o =

(

X l

μ1 /b

l

+

X m

μ1 /b m

) −b

, μ j = μ j0 exp

(− E j

RT

), (2.12)

ith b = 4 [24] . Here E j is the activation energy for the viscos-

ty of the j -component in the oleic phase and μj 0 is a reference

iscosity. For X l = 1 , one has μo = μl , while for X m

= 1 one has

o = μm

. For simplicity, we assume that the pure medium compo-

ent is very viscid (immobile), i.e, we consider the limit μm

� μl ,

uch that the oleic phase viscosity is modeled as

o = μl X

−b l

. (2.13)

he gas (air) viscosity is given by [25]

g =

7 . 5

T + 120

(T

291

)3 / 2

Pa s . (2.14)

Taking the sum of Eqs. (2.4) –(2.6) and using ( 2.8 ) and (2.9) with

l + Y k + Y r = 1 yields the equality u gl + u gk + u gr = u g as well as

he balance law for the total gas as

∂

∂t ϕρg s g +

∂

∂x ρg f g u = (νgm

− 1) W rm

+ (νgl − 1) W rl + W v . (2.15)

imilarly, taking the sum of Eq. (2.2) (divided by the constant ρm

)

nd (2.3) (divided by the constant ρ l ) and using (2.7) and (2.9) ,

ields the balance law for the total liquid as

∂

∂t ϕs o +

∂

∂x f o u = −νom

ρm

W rm

− νol

ρl

W rl −1

ρl

W v . (2.16)

Assuming that the temperature of solid rock, liquid and gas are

qual, we write the heat balance equation as

∂

∂t

(C eff

m

+ ϕC o s o + ϕc g ρg s g )�T +

∂

∂x ( C o u o + c g ρg u g ) �T

= λ∂ 2 T

∂x 2 + Q rm

W rm

+ Q rl W rl − Q v W v , (2.17)

here �T = T − T ini with the initial reservoir temperature T ini and

effm

≡ (1 − ϕ) C m

is the rock effective heat capacity. In Eq. (2.17) ,

rm

and Q rl are reactive enthalpies and Q v is the vaporization heat.

he heat capacities C m

, C o , c g and the effective thermal conduc-

ivity λ are taken as constants, which is a reasonable approxima-

ion and facilitates the analysis. We neglect heat losses, which are

sually very small in field applications (however, taking into ac-

ount heat losses becomes essential for interpreting laboratory ex-

eriments).

We use the ideal gas law to define

g = P/RT . (2.18)

ressure variations are assumed to be small compared to the

revailing pressure, so we take P ≈ const in (2.18) , as well as in

54 M.A. Endo Kokubun et al. / Combustion and Flame 169 (2016) 51–62

Y

W

W

t

t

d

α

E

(

w

d

S

(

w

θ

w

F

e

Y

w

Y

b

the relationships (2.19) and (2.20) below. In other words, physical

properties of the phases are evaluated at constant pressure.

The partial pressure Y eq

l P of the gaseous hydrocarbon in liquid–

gas equilibrium is derived using the Clausius–Clapeyron relation

with Raoult’s law as (see [25, p. 8.12] )

eq

l P = x l γl P atm

exp

(−Q v

R

(1

T − 1

T bn

)), (2.19)

where T bn is the (normal) boiling point of light hydrocarbon mea-

sured at atmospheric pressure P atm

and x l is the molar fraction of

the light component. We choose the activity coefficient γ l such

that x l γl = X l , representing the volume fraction, which is consid-

ered to take into account part of the non-ideality of Raoult’s law.

The purpose of this is that it simplifies the analysis; with some

effort it is also possible to explicitly take into account the activity

coefficient (see [25, Tables 8–3] ). Relation (2.19) determines the

equilibrium fraction of gaseous hydrocarbon Y eq

l (recalling that

only the light component of the oil vaporizes). The boiling temper-

ature T = T b of light hydrocarbon at pressure P is obtained from

Eq. (2.19) by setting Y eq

l = 1 and X l = 1 .

We express the oxidation (MTO) reaction rates for light and

medium components by the Arrhenius form [26,27]

W rl = ϕA rl X l ρl s o

(P Y k P atm

)n

exp

(−T ac

l

T

),

rm

= ϕA rm

X m

ρm

s o

(P Y k P atm

)n

exp

(−T ac

m

T

), (2.20)

where A rl , A rm

are the frequency factors and T ac l

, T ac m

are the acti-

vation temperatures. We use volume fractions X l and X m

instead of

mole fractions again to take into account part of the non-ideality

of the liquid. The vaporization rate is given by

v = k v l ϕ

(Y eq

l − Y l

)ρg s o X l , (2.21)

which is proportional to the deviation of the light component frac-

tion Y l in the gas phase from its equilibrium value Y eq

l with an

empirical transfer parameter k vl . This formulation can be consid-

ered a consequence of non-equilibrium thermodynamics, see for

instance [23,28] . We assume that k vl is large, describing the situa-

tion close to local thermodynamic equilibrium for the gaseous light

fraction Y l , i.e., very fast vaporization or condensation. The vapor-

ization rate W v vanishes under the conditions

s 0 X l > 0 , Y l = Y eq

l or s o X l = 0 . (2.22)

We note that the physical relations for relative permeability

( 2.11 ), viscosities (2.13), (2.14) , and source terms (2.20), (2.21) are

used as examples, since the developed theory does not rely on

their specific form.

3. Dimensionless equations

In order to render the governing equations dimensionless, we

introduce the ratios

˜ =

t

t ∗, ˜ x =

x

x ∗, θ =

T − T ini

�T ∗, ˜ u =

u

ϕv ∗, ˜ ρg =

ρg

ρ∗g

, (3.1)

where the reference quantities denoted by an asterisk are

t ∗ =

x ∗

v ∗, x ∗ =

λ

C m

v ∗, v ∗ =

Q rl ρ∗g u in j Y

in j

k

C m

�T ∗,

ρ∗g =

P

RT ini

, �T ∗ = T b − T ini . (3.2)

The dimensionless variable θ describes the temperature distribu-

tion with the reservoir condition θ = 0 and the boiling tempera-

ture θ = 1 for purely light oil. The reference quantities t ∗, x ∗ and v ∗

are obtained by considering a wave whose combustion heat raises

he rock temperature to T b . Dimensionless parameters are intro-

uced as

o =

ϕC o

C m

, αg =

ϕc g ρ∗g

C m

, βl =

ρ∗g

ρl

, βm

=

ρ∗g

ρm

, γv =

Q v

Q rl

,

γr =

Q rm

Q rl

, θ0 =

T ini

�T ∗, θh =

Q v

R �T ∗, σ =

ϕv ∗

u in j

. (3.3)

The dimensionless governing equations obtained from

qs. (2.17) , (2.16), (2.15), (2.4), (2.5) and (2.2) are given by

we drop the tildes for simplicity)

∂

∂t ( 1 + αo s o + αg S g ) θ +

∂

∂x ( αo f o + αg F g ) uθ

=

∂ 2 θ

∂x 2 +

σ

Y in j

k

( γr w rm

+ w rl − γv w v ) , (3.4)

∂s o

∂t +

∂u f o

∂x = −νom

βm

w rm

− νol βl w rl − βl w v , (3.5)

∂S g

∂t +

∂uF g

∂x = (νgm

− 1) w rm

+ (νgl − 1) w rl + w v , (3.6)

∂Y l S g

∂t +

∂uF g Y l ∂x

= w v , (3.7)

∂Y k S g

∂t +

∂uF g Y k ∂x

= −w rm

− w rl , (3.8)

∂X m

s o

∂t +

∂X m

u f o

∂x = −νom

βm

w rm

, (3.9)

here the temperature-corrected gas saturation and flux are

efined as

g (s o , θ ) = (1 − s o ) ρg , F g (s o , θ, X m

) = (1 − f o (s o , θ, X m

)) ρg ,

ρg =

1

1 + θ/θ0

. (3.10)

The dimensionless forms of the reaction and vaporization rates

2.20 ) and (2.21) are given by

rm

= a rm

X m

s o Y n

k exp

(− θ ac

m

θ + θ0

), a rm

= t ∗A rm

β−1 m

( P/P atm

) n ,

θ ac m

=

T ac m

�T ∗, (3.11)

w rl = a rl X l s o Y n

k exp

(− θ ac

l

θ + θ0

), a rl = t ∗A rl β

−1 l ( P/P atm

) n ,

ac l =

T ac l

�T ∗, (3.12)

v = κv l (Y eq

l (θ, X l ) − Y l ) ρg X l s o , κv l = k v l t

∗. (3.13)

or the fraction of light hydrocarbon in the gaseous phase in

quilibrium with liquid, we have

eq

l (θ, X m

) = (1 − X m

) Y(θ ) , (3.14)

here we defined the function Y(θ ) as

(θ ) = exp

(θh

θ0 + 1

− θh

θ0 + θ

). (3.15)

The following properties of the dimensionless parameters will

e multiply used in our analysis

M.A. Endo Kokubun et al. / Combustion and Flame 169 (2016) 51–62 55

Fig. 1. Left: schematic of the physical problem under consideration; Right: wave sequence solution with the thermal, combustion, and saturation waves traveling with

different speeds, v T < v < v S . Shown are the schematic profiles of temperature θ, oxygen fraction Y k and oil saturation s o in the rock cylinder.

σ

w

j

o

r

h

i

t

c

p

c

θ

c

Y

c

m

Y

c

t

i

n

4

t

i

t

r

n

a

e

v

T

t

T

g

m

i

w

w

r

s

w

d

t

i

s

i

p

c

5

i

p

p

m

b

(

t

a

i

a

ν

V

s

t

w

n

p

1 , βl 1 , Y(0) 1 , βl

σ=

C m

�T ∗

ϕQ rl ρl Y in j

k

� 1 , (3.16)

here σ is the ratio between the combustion wave and the air in-

ection velocities and β l is the ratio between the gas and the light

il densities. The last approximation is given by realizing that β l / σepresents the ratio between the heat accumulated in the rock and

eat released by the reaction of light oil. Since the heat of vapor-

zation is small, in comparison with the heat released by the reac-

ion, most of the heat is used to raise the rock temperature. The

ondition Y(0) 1 implies a small fraction of gaseous light com-

onent at the reservoir temperature, Y eq

l (0 , X ini

m

) 1 , and it is a

onsequence of large θh in Eq. (3.15) .

The system (3.4) –(3.9) contains six equations for the variables

, s o , u, Y l , Y k and X m

dependent on t and x . The initial (reservoir)

onditions are given by

t = 0 , x ≥ 0 : θ = 0 , s o = s ini o , X m

= X

ini m

,

l = (1 − X

ini m

) Y(0) , Y k = 0 , (3.17)

orresponding to the reservoir containing oil with a specified

edium fraction X ini m

. The injection conditions are given by

x = 0 , t ≥ 0 : αg F g uθ − ∂θ

∂x = 0 , s o = Y l = 0 ,

k = Y in j

k , u = σ−1 , (3.18)

orresponding to the injection of an oxidizer (air) at reservoir

emperature and the dimensionless injection speed σ−1 given by

ts dimensional value u inj in Eq. (3.3) . It is assumed that there is

o gaseous light hydrocarbons in the injected gas, Y l = 0 .

. Wave solutions

For large times, the solution of the problem contains three dis-

inct waves: thermal, combustion and saturation waves, as shown

n Fig. 1 . The thermal wave is slower and it is located upstream of

he combustion wave because of the high thermal capacity of the

ock. In the thermal wave region there is no volatile liquid compo-

ents, and no reaction occurs (then the oxidant fraction is constant

t its injection value). From the analysis of the thermal wave (see

.g., [9] ), one derives the wave speed and the gas flux as

T =

αg

σ, uF g =

u

1 + θ/θ0

≈ σ−1 . (4.1)

he thermal wave speed is small, v T 1, because it is propor-

ional to the ratio of the air to rock heat capacities, see Eq. (3.3) .

he second expression in Eq. (4.1) describes the change of the

as velocity due to thermal expansion and it will be used as a

atching condition for the combustion wave. The saturation wave

s the fastest wave and is located downstream of the combustion

ave and resembles a classical Buckley–Leverett solution. In this

ave, the oil saturation s o is changing, while the other variables

emain constant.

The combustion (MTO) wave, where oxidation and phase tran-

itions occur, is the focus of this work. The gaseous hydrocarbons,

hich are generated by vaporization of the light component, flow

ownstream of the combustion wave towards the region of lower

emperatures, where they condense. Hence, the combustion wave

s composed of the vaporization/condensation front, which is

ustained by the heat released in the oxidation reaction. This wave

s analyzed in detail in the next section. We will show that the

rocesses in the combustion wave are dominated by the light oil

omponent, even when the oil is predominately non-volatile.

. Combustion wave

Our analysis will show that almost no medium oil component

s present in the reaction region of the combustion wave. This sur-

rising result is a consequence of the vaporization/condensation

rocess. Therefore, we facilitate the derivations by neglecting the

edium oil reaction, i.e., setting w rm

= 0 , which will be confirmed

y the final result. For further simplification, we consider νgl = 1

no net gas production by the reaction) and νol 1, which means

hat the amount of light oil consumed by reaction is small. We

lso consider that γ v 1, meaning that the heat of vaporization

s much smaller than the heat of combustion. Thus, as a first

pproximation we set

ol = 0 , γv = 0 . (5.1)

alidity of these simplifications is also confirmed by numerical

imulations described below.

We assume that the combustion wave has a stationary profile

raveling at a constant speed v . In order to analyze its structure,

e change to a moving reference frame with ξ = x − v t . In this

ew coordinate, the governing Eqs. (3.4) –(3.9) for the stationary

rofile take the form

d

dξ( −v + αo ψ o + αg ψ g ) θ =

d 2 θ

dξ 2 +

σ

Y in j

k

w rl , (5.2)

dψ o

dξ= −βl w v (5.3)

dψ g

dξ= w v , (5.4)

56 M.A. Endo Kokubun et al. / Combustion and Flame 169 (2016) 51–62

Fig. 2. Combustion wave profile in the moving reference frame: air injected on the

left side and the wave moves to the right.

w

v

k

u

5

c

M

t

s

W

a

u(

−

w

d

a

d

f

ψ

ψ

T

θ

S

s

s

U

s

s

t

θ

dψ g Y l dξ

= w v , (5.5)

dψ g Y k dξ

= −w rl , (5.6)

dψ o X m

dξ= 0 , (5.7)

where we used Eq. (5.1) and defined

ψ o = u f o − v s o , ψ g = uF g − v S g . (5.8)

The quantities ψ o and ψ g represent, respectively, the oil and gas

fluxes in the moving reference frame. All the dependent variables

are now functions of ξ only. We expect, and it will be confirmed

by the final result, that the wave speed v ∼ 1 due to the choice of

the reference value v ∗ in Eq. (3.2) .

In Fig. 2 we show a schematic profile of the combustion wave.

Upstream of the wave the temperature is θu (unknown), all light

oil is vaporized (which is equivalent to X m

= 1 ) and there is no

gaseous light oil in the injected gas, Y l = 0 . Downstream of the

wave the oil is at its initial temperature θ = 0 and composition

X m

= X ini m

. Since we neglected the reaction of the medium com-

ponent, the existence of a (pure medium) oil saturation s u o at

the upstream side is assumed. As we mentioned, the result will

show that s u o is negligibly small. Therefore, we have the upstream

boundary conditions

ξ → −∞ : θ = θu , Y l = 0 , Y k = Y in j

k , X m

= 1 , s o = s u 0 ,

u =

1 + θu /θ0

σ, (5.9)

where the last equality follows from the match with the thermal

wave conditions (4.1) . The upstream values of the fluxes ψ o and

ψ g given by Eqs. (5.8) and (3.10) are determined by realizing that

f o = 0 and f g = 1 for pure medium oil, see Eqs. (2.10) –(2.13) , such

that

ξ → −∞ : ψ o = −v s u o , ψ g =

1

σ− v (1 − s u o )

1 + θu /θ0

≈ 1

σ, (5.10)

where in the last approximation we used (3.16) and the fact that

v ∼ 1.

We assume complete oxygen consumption in the combustion

wave, leading to the following downstream conditions

ξ → + ∞ : θ = 0 , Y l = (1 − X

ini m

) Y(0) , Y k = 0 , X m

= X

ini m

,

s o = s d o , u = u

d , (5.11)

here the light component fraction Y l is given by the equilibrium

alue (3.14) , while the oil saturation s d o and gas speed u d are un-

nown. Therefore, the unknowns are the limiting states θu , s u o , s d o ,

d and the wave velocity v .

.1. Limiting states

This Section derives the relations for the limiting states of the

ombustion wave that follow from the overall balance conditions.

anipulating Eqs. (5.2) –(5.7) in order to eliminate the source/sink

erms, we obtain the following equations for the conserved

calars

d

dξ

(( −v + αo ψ o + αg ψ g ) θ − dθ

dξ+

σψ g Y k

Y in j

k

)= 0 , (5.12)

d

dξ( ψ o + βl ψ g ) = 0 , (5.13)

d

dξ( ψ o X m

+ βl ψ g (1 − Y l ) ) = 0 , (5.14)

dψ o X m

dξ= 0 . (5.15)

e added Eq. (5.15) into Eq. (5.14) because it will simplify the

nalysis.

Integrating Eqs. (5.12) –(5.15) from upstream to downstream

sing (5.9) –(5.11) , we obtain

−v − αo v s u o +

αg

σ

)θu + 1 = 0 , (5.16)

βl

σ− v s u o = ψ

d o + βl ψ

d g , (5.17)

βl

σ− v s u o = ψ

d o X

ini m

+ βl ψ

d g (1 − (1 − X

ini m

) Y(0)) , (5.18)

v s u o = ψ

d o X

ini m

, (5.19)

here the upstream states are at the left-hand side and the

ownstream states are at the right-hand side in the equations

bove. In Eq. (5.16) we used that d θ/d ξ = 0 for both upstream and

ownstream states.

By manipulating Eqs. (5.17) and (5.18) we can find expressions

or ψ

d o and ψ

d g as

d o = −

(βl

σ− v s u o

)Y(0)

1 − Y(0) , (5.20)

d g =

1

βl

(βl

σ− v s u o

)1

1 − Y(0) . (5.21)

he upstream temperature θu is obtained from Eq. (5.16) as

u =

1

( v + αo v s u o − αg /σ ) . (5.22)

ubstituting Eq. (5.20) in Eq. (5.19) gives the upstream saturation

u o as

u o =

βl

σv

(X

ini m

Y(0)

1 − (1 − X

ini m

) Y(0)

). (5.23)

sing the conditions (3.16) and the consideration v ∼ 1, yields

u o ≈

βl

σv X

ini m

Y(0) 1 , (5.24)

uch that we can neglect the term v s u o in Eqs. (5.20) –(5.22) . Then,

he upstream temperature θu is given by

u =

1

( v − αg /σ ) , (5.25)

M.A. Endo Kokubun et al. / Combustion and Flame 169 (2016) 51–62 57

Fig. 3. Schematic graph for the left- and right-hand sides of Eq. (5.30) as functions

of the oil saturation s o .

a

f

ψ

N

t

b

d

a

ψ

T

a

u

W

u

t

w

w

t

o

s

s

s

a

t

t

T

p

5

r

l

i

r

w

s

c

d

T

m

r

o

f

t

s

(

ψ

ψ

ψ

w

(

(

ψ

ψ

E

s

X

W

a

ψ

ψ

w

e

u

E

g

u

t

N

I

i

t

t

o

t

r

b

nd the downstream mass flux of oil and gas, using Y(0) 1

rom Eq. (3.16) , are given by

d o = −βl Y(0)

σ 1 , ψ

d g =

1

σ� 1 . (5.26)

ote that the term αg / σ in the denominator of Eq. (5.25) is equal

o the thermal wave speed v T , see Eq. (4.1) , which was shown to

e small.

The values of the downstream fluxes ψ

d o and ψ

d g can also be

etermined by Eq. (5.8) in terms of the downstream oil flux f d o

nd saturation s d o as

d o = u

d f d o − v s d o , ψ

d g = u

d (1 − f d o ) − v (1 − s d o ) . (5.27)

he sum of these expressions yields the gas velocity downstream

s

d = v + ψ

d o + ψ

d g . (5.28)

e substitute Eq. (5.26) in Eq. (5.28) and obtain

d = v − βl Y(0)

σ+

1

σ≈ 1

σ. (5.29)

Using the first equation in Eq. (5.27) with the neglected small

erm ψ

d o 1 from Eq. (5.26) and with u d = σ−1 from Eq. (5.29) ,

e obtain the downstream saturation s d o implicitly through

f o (s d o , 0 , X

ini m

) = σv s d o , (5.30)

here we expressed the downstream fractional flow function f d o in

he left-hand side. A typical graph of the left- and right-hand sides

f Eq. (5.30) as functions of s d o is shown in Fig. 3 . The left-hand

ide is the s -shaped fractional flow function, while the right-hand

ide is a linear function of s d o with a small positive slope σ v and

tarting at s o = 0 . The two roots for Eq. (5.30) are given by s d o = 0

nd s d o > 0 , where the relevant root is the positive one [8] .

Eqs. (5.24), (5.25), (5.29) and (5.30) determine the conditions at

he limiting states s u o , θu , u d and s d o , respectively. In order to close

he problem we must obtain an expression for the wave speed v .

his expression is obtained from the analysis of the internal wave

rofile, which will be done in the next Section.

.2. Combustion wave profile

We will show that the wave speed v is determined by the

esonance condition at the point where the vaporization of the

ight component is changed by its condensation. Since vapor-

zation is assumed to be the fastest process, the vaporization

egion (VR) is thin and located in the upstream part of the wave,

here the injected air gets into contact with the oleic phase,

ee Fig. 2 . Downstream of the VR, the liquid and gaseous oil are

lose to thermodynamic equilibrium. As the temperature gradually

ecreases in the downstream direction, slow condensation occurs.

hus, we distinguish a thin vaporization region upstream and a

uch wider condensation region (CR) downstream. Most of the

eaction occurs at highest temperatures, i.e., in the upstream part

f the CR, as we describe below.

Eqs. (5.16) –(5.19) were obtained by integrating Eqs. (5.12) –(5.15)

rom upstream to downstream constant states. In order to obtain

he wave profile, we integrate Eqs. (5.12) and (5.15) from down-

tream up to some point ξ inside the wave, and Eqs. (5.13) and

5.14) from upstream to the same point ξ , which yields

( −v + αo ψ o + αg ψ g ) θ − dθ

dξ+ σψ g

Y k

Y in j

k

= 0 , (5.31)

o + βl ψ g =

βl

σ, (5.32)

o X m

+ βl ψ g (1 − Y l ) =

βl

σ, (5.33)

o X m

= ψ

d o X

ini m

, (5.34)

here the right-hand sides were determined from Eqs. (5.16) –

5.19) , and we neglected the small term v s u o according to Eqs.

3.16) and (5.24) . Eqs. (5.32) and (5.33) yield

o = −βl

σ

(Y l

1 − Y l − X m

), (5.35)

g =

1

σ

(1 − X m

1 − Y l − X m

). (5.36)

q. (5.34) with ψ o from Eq. (5.35) and ψ

d o from (5.26) can be

olved with respect to X m

as

m

=

Y(0) X

ini m

(1 − Y l )

Y(0) X

ini m

+ Y l . (5.37)

e can use (5.37) in Eqs. (5.35) and (5.36) in order to express ψ o

nd ψ g in terms of Y l as

o = −βl (Y(0) X

ini m

+ Y l )

σ (1 − Y l ) ≈ − βl Y l

σ (1 − Y l ) , (5.38)

g =

Y(0) X

ini m

+ 1

σ (1 − Y l ) ≈ 1

σ (1 − Y l ) , (5.39)

here the approximations were made because Y(0) 1 .

From the definitions of ψ o and ψ g , Eqs. (5.8) and (3.10) , we

xpress the gas velocity u as

= v + ψ o + ψ g (1 + θ/θ0 ) . (5.40)

qs. (5.38) and (5.39) with conditions (3.16) allow to express the

as velocity (5.40) in terms of Y l and θ as

= v +

(1 + θ/θ0 ) − βl Y l σ (1 − Y l )

≈ (1 + θ/θ0 )

σ (1 − Y l ) . (5.41)

Substituting (5.38) and (5.39) in (5.31) , after some manipula-

ions and with the use of θu given by Eq. (5.25) , yields

dθ

dξ= −

(1

θu +

(αo βl − αg ) Y l σ (1 − Y l )

)θ +

Y k

Y in j

k (1 − Y l )

. (5.42)

The above expressions are valid at all parts of the wave profile.

ow let us discuss the properties specific for each region, Fig. 2 .

n the VR, d θ / d ξ given by Eq. (5.42) is not large. Since this region

s very thin, we neglect the temperature variation in the VR, such

hat the temperature is constant and given by θu in Eq. (5.25) . For

he same reason, one can neglect reaction in the VR, assuming the

xygen fraction Y k ≈ Y in j

k to be approximately constant. In the VR,

he gaseous light hydrocarbon Y l increases from 0 to its equilib-

ium value Y eq

l (θu , X m

) , while the fraction of medium oil X m

given

y Eq. (5.37) decreases from 1 to some value.

58 M.A. Endo Kokubun et al. / Combustion and Flame 169 (2016) 51–62

Fig. 4. Left-hand side (dashed curve) and right-hand side (dashed straight line) of

Eq. (5.44) as functions of oil saturation s o . The arrows indicate the change of the

graphs with increasing Y l along the wave profile. Open dot represents the upstream

solution, black dot represents the downstream solution and crossed dot represents

solution at the resonance point (tangency of the solid curve and solid line).

g

v

H

t

(

l

t

m

v

s

u

F

a

d

a

c

d

s

t

s

t

c

s

b

t

c

E

m

v

s

fi

F

a

F

o

b

c

i

X

fl

E

T

i

v

S

n

r

W

s

r

o

5

m

In the CR, the gaseous light component is close to thermody-

namic equilibrium, i.e., Y l ≈ Y eq

l (θ, X m

) = (1 − X m

) Y(θ ) . Using this

expression in Eq. (5.37) and solving with respect to X m

yields two

roots. The first (irrelevant) root is X m

= 1 , and the second root is

X m

= X

ini m

Y(0)

Y(θ ) ( 1 − Y(θ ) ) . (5.43)

We assume that the oxidation reaction occurs in the region of high

temperatures with complete oxygen consumption. In fact, the more

detailed analysis [29] shows that most of the reaction is confined

in a small region near the highest temperature θu if the air injec-

tion rate is not large. For large injection rates, the reaction may

spread into a wider part of the CR. Since Y(0) Y(θ ) for high

temperatures θ ≈ θu , one has X m

1 in Eq. (5.43) , i.e., a very

small amount of medium component is available in the reaction

region, irrespective of the initial amount X ini m

. This justifies the ini-

tial hypothesis that the medium non-volatile component plays a

minor role in the reaction.

5.3. Resonance condition

The wave speed v is determined from the so-called resonance

condition , which is satisfied at the point where the VR and CR

meet. For this purpose, we substitute ψ o from Eq. (5.8) and the

gas velocity u given by (5.41) in Eq. (5.38) , and obtain

(1 + θ/θ0 ) f o (s o , θ, X m

) = v σ (1 − Y l ) s o − βY l . (5.44)

Eq. (5.44) determines implicitly the saturation s o in the wave

profile. The left- and right-hand sides of Eq. (5.44) are shown

schematically in Fig. 4 in terms of s o with the other variables fixed.

The left-hand side of Eq. (5.44) is proportional to the fractional

flow function, determined by Eqs. (2.10) and (2.11) , and it is s-

shaped. The right-hand side of Eq. (5.44) is a straight line with a

small inclination, which crosses the vertical axis at the small neg-

ative value −βY l . The solution of Eq. (5.44) is given by the inter-

section between the two dashed curves in Fig. 4 . Upstream of the

VR, the lower intersection point is relevant (open dot) because this

region starts with s o = s u o 1 , according to ( 5.24 ). Downstream of

the CR, the upper intersection point is relevant (black dot) because

in this region s o = s d o > 0 , where s d o is not small, see Fig. 3 .

By continuity, there must exit a tangency represented by the

crossed dot in Fig. 4 at some internal point of the wave profile.

Since the slopes of the left- and right-hand sides of Eq. (5.44) are

equal at this internal point, we use Eq. (5.41) and obtain the tan-

ency condition

=

(u

∂ f o

∂s o

)∣∣∣∣r

. (5.45)

ere the wave speed (left-hand side of Eq. (5.45) ) is equal to

he Buckley–Leverett characteristic speed (right-hand side of Eq.

5.45) ), which we call resonance [9] , denoted by “r ” . In the fol-

owing, we show that resonance corresponds to the point where

he VR and CR meet and the gaseous oil fraction Y l attains a

aximum.

In the VR, the light component fraction Y l increases due to the

aporization, while the temperature remains approximately con-

tant as we described earlier. The increase in Y l yields larger val-

es of f o according to Eqs. (2.10) –(2.13) and (5.37) as shown in

ig. 4 (horizontal arrow), while the straight-line moves downwards

ccording to Eq. (5.44) (vertical arrow in Fig. 4 ). Hence, the coor-

inate s o of the lower intersection in Fig. 4 increases as we move

long the wave profile in the downstream direction. This tendency

annot be extended beyond the tangency configuration (crossed

ot in Fig. 4 ), as this would lead to no solution (no intersection).

In the wide CR, the temperature changes from θu to its down-

tream value θ = 0 according to Eq. (5.42) . Since αo β l > αg , as

he molar heat capacity of the liquid oil is higher than for the gas,

ee Eq. (3.3) , and Y k ≈ 0 (most of oxygen is consumed at highest

emperatures), we see from Eq. (5.42) that d θ / d ξ < 0. Thus, the

hange of the temperature is monotonic, decreasing in the down-

tream direction. The equilibrium light oil fraction Y l = Y eq

l given

y Eqs. (3.14) and (3.15) lowers because of the decrease in the

emperature θ and increase in X m

given by Eq. (5.37) . This de-

rease in Y l and θ yields smaller values of the left-hand side of

q. (5.44) according to Eqs. (2.10) –(2.13) , while the straight-line

oves upwards according to Eq. (5.44) . This corresponds to the di-

ergence away from the tangent configuration in Fig. 4 . The upper

olution s o (bold dot) increases as we move along the wave pro-

le downstream until it reaches the limiting constant state s d o , see

igs. 2 and 3 .

Therefore, our analysis proves that the point, where the VR

nd the CR meet, must correspond to a tangency configuration in

ig. 4 (crossed dot). Because the gaseous oil fraction Y l decreases

n both sides of the resonance point, it attains the maximum given

y Y r l

= Y eq

l (θu , X r m

) . Since ∂ X m

/ ∂ Y l < 0 according to Eq. (5.37) , we

onclude that the medium fraction in oleic phase attains the min-

mum X r m

at the resonance point.

As we showed in Eq. (5.43) with Y(0) 1 , the minimum value

r m

1 . Hence, the medium oil fraction can be neglected in the

ow function f o in Eq. (5.44) at the resonance point. As a result,

q. (5.44) at the resonance point yields

(1 + θu /θ0 ) f o (s r o , θu , 0) = v σ (1 − Y(θu )) s r o − βY(θu ) . (5.46)

he resonance condition (5.45) , with u from Eq. (5.41) , is written

n a similar way as

=

1 + θu /θ0

σ (1 − Y(θu ))

(∂ f o

∂s o

)∣∣∣∣(s r o ,θu , 0)

. (5.47)

ubstituting θu from Eq. (5.25) , the solutions of the system of two

onlinear equations (5.46) and (5.47) for the saturation s r o at the

esonance point and the wave speed v can be found numerically.

ith the wave speed v determined, we proceed to obtain the up-

tream temperature θu from Eq. (5.25) and the downstream satu-

ation s d o from Eq. (5.30) . This determines all unknowns quantities

f the combustion wave limiting states.

.4. Remarks on medium component oxidation

The a priori assumption that the oxidation reaction of the

edium component is negligible was justified by showing that

M.A. Endo Kokubun et al. / Combustion and Flame 169 (2016) 51–62 59

0

2

4

6

8

0 0.2 0.4 0.6 0.8 1

s ou(x

103 )

Xmini

a

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

s od

Xmini

b

0

0.02

0.04

0.06

0.08

0.1

0 0.2 0.4 0.6 0.8 1

Ykm

ax

Xmini

c

Fig. 5. (a) Dependence of the downstream oil saturation s d o in the combustion wave on the initial medium fraction X ini m for the reservoir pressures P = 10 ( ), 40( ) and

70( ) bar. The crosses show the results obtained from numerical simulations at P = 10 bar. (b) Upstream oil saturation s u o (×10 3 ) . (c) Oxygen fraction that reacts with the

residual medium oil.

o

t

m

t

r

w

l

b

w

W

s

c

�

A

ρ

l

o

t

p

i

i

s

d

I

m

w

c

t

Y

e

t

o

T

o

i

v

s

g

w

o

m

t

A

v

t

t

n

i

o

a

m

w

o

r

m

t

i

t

s

6

i

nly a small fraction X m

remains in the region of high tempera-

ures in the CR according to Eq. (5.43) , and a small amount s u o re-

ains upstream according to Eq. (5.24) . In this Section we make

his argument more precise.

The non-volatile and immobile medium oil of saturation s u o may

eact with the oxygen at the upstream side of the combustion

ave, where the temperature is elevated, see Fig. 2 . In dimension-

ess variables, the rate of immobile medium oil deposition is given

y | ψ

u o | = v s u o , which by the stoichiometric relation (2.1) can react

ith the amount v s u o / (νom

βm

) of oxygen, see Eqs. (3.8) and (3.9) .

ith the upstream gas flux ψ

u g = σ−1 , as given by Eq. (5.10) , and

u o from Eq. (5.24) , one finds the maximum oxygen fraction that

an be consumed by the reaction of this residual medium oil as

Y max k =

| ψ

u o |

νom

βm

ψ

u g

=

σv s u o

νom

βm

=

βl X

ini m

νom

βm

Y(0) . (5.48)

ccording to Eq. (3.3) , the ratio βl / (νom

βm

) = ρm

/ (νom

ρl ) , where

m

and ρ l are the pure oleic molar densities of the medium and

ight components, is not expected to be very large. Thus, the value

f �Y max k

is dominated by the small equilibrium fraction Y(0) for

he light oil component at the initial reservoir temperature. In

ractical applications, especially at elevated pressures (see Fig. 5 (c)

n the next Section), the value of �Y max k

is small compared to the

njected fraction Y in j

k , which, once again, confirms our initial as-

umption of a small effect of the medium component reaction.

The actual form of how the residual medium oil is oxidized

epends on the magnitude of the reaction rate w rm

in Eq. (3.11) .

f this rate is not too small at the upstream temperature θu , all

edium oil is consumed in the upstream part of the combustion

ave. This leads to a small temperature elevation and a small de-

rease of the oxygen fraction by the amount of �Y max k

. In order

o take this effect into account, one can use the corrected value

in j

k − �Y max

k instead of Y

in j

k in the analysis of Section 5 .

It is possible, however, that the reaction rate w rm

is very small,

.g., due to a small oil saturation or a higher activation energy. In

his case, we may consider the possibility of the high-temperature

xidation (HTO) wave formation after a certain transient period.

he HTO process is well understood, see e.g., [3,18] . In the case

f low fuel saturation s u o 1 , a considerable temperature increase

s required for HTO. This is possible when the HTO wave speed

H is close to the thermal wave speed v T = αg /σ, resulting in the

uperadiabatic heat accumulation [30] . The thermal wave width

rows as L T ∼√

t [9,30] , while the separation between the two

aves increases linearly as L ∼ (v H − v T ) t . Since we need L > L T ,

ne recovers the time for the HTO wave formation by an order of

agnitude as

H ∼ 1

(v H − v T ) 2 . (5.49)

s v H − v T v T v , expression (5.49) typically yields very large

alues of t H , considerably exceeding the time in which the combus-

ion wave traverses from the injection to the recovery well in prac-

ical applications. This argument shows that the HTO wave does

ot form at relevant times of our problem.

Finally, let us comment on the possible effect of mass diffusion

n the gaseous phase, which was neglected in the model. The size

f gas diffusion layer can be estimated, in dimensional variables,

s l diff ∼ ϕD g / u g , which is much smaller than the size of ther-

al diffusion layer l thermal ∼ λ/ (C effm

v ) because D g ≈ λ/C effm

and the

ave speed v is much smaller than the gas speed u g / ϕ [18] . More-

ver, the diffusion transport only affects the region upstream of the

esonance point, while the combustion parameters are determined

ainly by the processes in the downstream region (dominated by

he equilibrium of gaseous and oleic phases), as we demonstrated

n Section 5 . Thus, the effect of mass diffusion may be expected

o be small. Also, this agrees with the numerical simulations pre-

ented below, where diffusion was taken into account.

. Numerical results and simulations

Here we present some results based on the theory developed

n this work and compare them with numerical simulations. We

60 M.A. Endo Kokubun et al. / Combustion and Flame 169 (2016) 51–62

Table 1

Values of reservoir parameters used in numerical simulations.

A rl = A rm = 4060 1/s Q v = 31.8 kJ/mol λ = 3 W/m K

c g = 29 J/mol K s or = 0.1 μlo = 1 . 32 × 10 −2 Pa s

C effm = 2 MJ/m

3 K s ini o = 0.9 νgm = 1.36

C o = 1.53 MJ/m

3 K T ac l

= 7066 K νom = 0.065

E l = 8364 J/mol T ac m = 15, 550 K νol = 0.090

k vl = 100 1/s T bn = 371 K νgl = 1.36

n = 1 T ini = 300 K ρ l = 6826 mol/m

3

P res = 10 6 Pa u inj = 8 × 10 −7 m/s ρm = 5130 mol/m

3

Q rl = Q rm = 400 kJ/mol O 2 Y in j

k = 0.21 ϕ = 0.3

Table 2

Maximum temperatures T u and speeds v of the combustion

(MTO) wave at different pressures P .

Pressure (bar) Temperature (K) Wave speed (m/s)

10 383 1 . 67 × 10 −7

40 430 4 . 33 × 10 −7

70 454 6 . 43 × 10 −7

H

t

o

F

t

l

v

s

l

t

c

i

o

T

s

c

t

m

m

b

r

a

u

a

t

f

u

o

s

i

fl

s

p

t

c

m

a

o

f

s

c

a

f

g

m

i

f

r

T

s

use the liquid, gas and rock parameters given in Table 1 , taken

from [22] , which correspond to air injection at low rates into a

rock of medium permeability filled with a two pseudo-component

oil. Physical properties of the oil are chosen by considering hep-

tane as a light component and we take hexadecane as a model

medium component, with all the physical properties of hexade-

cane, except for its higher viscosity, thus avoiding the necessity

of using complex mixtures for the model oil. Liquid combustion

can be described in an analogous way to pore diffusion [31] . Then,

the activation energies are chosen by taking half of their values

for gaseous reaction [23,31,32] . Numerical simulations were car-

ried out using the finite element method implemented by COM-

SOL software. For such, we introduce the model equations in their

weak form. We use fifth-order Lagrange elements. The grid size

is 0.025 m with an adaptive time step, which is appropriate for

capturing the multiscale processes in the problem. In the simu-

lations we added the terms describing effective molecular ( D g =10 −6 m

2 / s ) and capillary ( D cap = 5 × 10 −8 m

2 / s ) diffusion into the

governing equations in order to improve numerical convergence.

The MTO temperature and wave speed are obtained as de-

scribed in Section 5.3 using Eqs. (5.46) , (5.47) and (5.25) . Con-

sidering different reservoir pressures, i.e., P = 10 , 40 and 70 bar,

the dimensional results for maximum temperatures and combus-

tion wave speeds are given in Table 2 . Both the temperature and

the speed increase with the pressure. Note that the combustion

wave speed remains comparable to the air injection Darcy veloc-

ity u in j = 8 × 10 −7 m/s. This fact is a consequence of the phase-

transfer mechanism leading to strongly enhanced liquid–gas flow,

a phenomenon that distinguishes the MTO process from HTO. The

theoretical values in Table 2 are independent on the initial oil com-

position, as we explained in Section 5.3 , because the medium com-

ponent is expelled from the reaction region and does not appear in

the resonance condition. This surprising prediction agrees reason-

ably well with the results of numerical simulations: in the case

of a pressure of P = 10 bar, almost identical temperatures T u =396 , 398, 401, 406 K and speeds v = 1 . 08 × 10 −7 , 1 . 06 × 10 −7 ,

1 . 01 × 10 −7 , 9 . 4 × 10 −8 m/s are obtained for different medium

fractions X ini m

= 0 . 2 , 0.4, 0.6, 0.8 in the initial oil, respectively. We

note, however, that our model due to the assumption of absence of

cracking of the heavy component is not applicable for a very heavy

mixture. The discrepancies in the wave parameters obtained theo-

retically and numerically are due to approximations used in the

derivation as well as due to numerical errors. This leads to some

decrease on the wave speed and an increase on the combustion

temperature, when compared to the results given by the theory.

owever, the numerical simulations fully confirm the overall struc-

ure of the combustion wave.

Though not affecting the temperature and speed, the initial

il composition strongly influences the oil saturation profile. In

ig. 5 (a) we show the dependence of the downstream oil satura-

ion s d o on the initial oil composition X ini m

. This saturation remains

arge for all values of X ini m

, increasing when the fraction of non-

olatile component gets larger. The results for three different pres-

ures, shown by solid (10 bar), dashed (40 bar) and dotted (70 bar)

ines, demonstrate a rather weak pressure dependence. The flow at

he downstream side depends on the initial oil composition be-

ause of the change in the oil mobility, quantified by the change

n the oleic viscosity in Eq. (2.13) . As the initial fraction of medium

il in the oleic mixture increases, the mobility of the oil decreases.

hus, in order to keep the wave speed unchanged, the downstream

aturation must increase with X ini m

leading to a considerable in-

rease of gas drive for small gas saturations s d g = 1 − s d o . Comparing

he theoretical results in Fig. 5 (a) with the values obtained by nu-

erical simulations, shown by crosses, we see a reasonable agree-

ent with the theory.

Upstream of the combustion wave, only the medium (immo-

ile) oil is left behind. Figure 5 (b) shows the upstream oil satu-

ation s u o given by Eq. (5.24) . As predicted, this value is small for

ll pressures and initial oil compositions. Still, this small resid-

al oil may react with some fraction of oxygen in the injected

ir, since it is located at the hot upstream part of the combus-

ion wave. Figure 5 (c) shows the estimate (5.48) of this oxygen

raction �Y max k

. We see that the oxygen consumption by the resid-

al medium oil is negligible for larger pressures, even if the initial

leic mixture is predominantly non-volatile. Only for lower pres-

ures this value may become considerable in comparison with the

njected fraction Y in j

k = 0 . 21 .

The large downstream oil saturations s d o in Fig. 5 (a) lead to high

ow rates of oil downstream of the combustion wave, while the

mall upstream oil saturations s u o in Fig. 5 (b) imply almost com-

lete recovery of oil from the reservoir. Also the relatively low

emperature increase in the wave requires less oil to be burned

ompared to the HTO process for heavy oils. All these factors

ake the described combustion (MTO) mechanism attractive for

pplications.

Now we describe the results of numerical simulations focusing

n the case of a pressure P = 10 bar and initial oil with medium

raction X ini m

= 0 . 4 . Figure 6 shows the dependent variables at a

pecific time t = 1 . 2 × 10 9 s when the stationary combustion pro-

ess is developed. Dimensional temperature T , oxygen fraction Y k nd oil saturation s o are shown in Fig. 6 (a), while the oleic medium

raction X m

and gaseous light fraction Y l are shown in Fig. 6 (b) to-

ether with the products X l s o and X m

s o describing the light and

edium oil saturations. One can recognize the combustion wave

n the middle (around x ≈ 110 m) and the slower thermal wave

orming in the upstream (left) part of the domain; the faster satu-

ation wave moved out of the computation region at earlier times.

he combustion wave pushes the oleic mixture forward and is re-

ponsible for displacing the non-volatile oil.

M.A. Endo Kokubun et al. / Combustion and Flame 169 (2016) 51–62 61

Fig. 6. Combustion wave profile for X ini m = 0 . 4 and P = 10 bar. (a) Temperature T , oxygen fraction Y k and oil saturation s o . (b) Medium oil saturation X m s o , light oil saturation

X l s o , gaseous hydrocarbon Y l and oleic medium fraction X m . The arrows indicate the resonance point (RP).

s

i

a

w

c

a

i

a

g

p

b

o

s

b

q

p

o

a

e

a

s

b

o

p

b

b

s

m

c

h

7

f

f

m

m

l

l

t

f

t

s

l

a

m

i

h

n

p

i

n

r

e

r

c

r

t

o

r

t

f

b

b

a

o

p

f

t

t

m

e

p

A

v

3

P

2

R

s

S

(

One can see a good agreement of numerical simulation re-

ults with the combustion wave structure described theoretically

n Section 5 , see also Fig. 2 . The vaporization/condensation front

nd the reaction front travel together, inside the combustion wave,

ith constant speed v . The arrows in Fig. 6 (a,b) indicate the lo-

ation of the resonance point (RP); the inset in Fig. 6 (b) shows

zoom of the figure near this point. In agreement with theoret-

cal results, one can see that the oleic medium fraction X m

attains

minimum at the resonance point, while the fraction Y l of the

aseous light component generated by the vaporization has a sharp

eak at the resonance point, decreasing on the downstream side

ecause of condensation at lower temperatures. A small amount

f medium component X m

s o remains behind the resonance point,

ee the inset in Fig. 6 (b). This residual medium oil does not burn

ecause of the high value of its activation energy.

We note that even if the described traveling wave solution is

ualitatively the same for all initial oil compositions, the transient

rocess of the combustion wave formation is very different for oils

f high or low volatility. A bank of immobile medium oil appears

t the initial stage of the combustion process if X ini m

is large, for

xample, X ini m

= 0 . 8 . Combustion of this oil leads to high temper-

tures reaching typical values for the HTO process. This fact, ob-

erved earlier in numerical simulations [22] , was interpreted as a

ifurcation from the MTO to the HTO combustion with an increase

f the non-volatile component in the initial oil. Our simulations,

erformed for larger time intervals, showed that the MTO com-

ustion wave is formed eventually after (probably long) transient

ehavior; the bifurcation is not observed in the current model, as

tated previously. In view of the results obtained, more detailed

odels are necessary to see the transition to HTO, which must in-

lude the cracking process (pyrolysis) leading to coke deposition at

igh temperatures.

. Conclusions

For filtration combustion of a volatile multicomponent liquid

uel, multiphase flow and phase transitions become as important

or the development of the combustion wave, as the oxidation

echanism. In addition to this fact, we demonstrated that the

ulticomponent aspects of the flow are equally important, if the

iquid components have essentially different physical properties,

eading to qualitative changes in the combustion process. In par-

icular, a component with low mobility and volatility is expelled

rom the reaction region in the downstream direction, opposite

o what would be intuitively expected. As a result, large liquid

aturations are attained downstream of the combustion wave,

eading to considerable increase in the wave speed. These aspects

re demonstrated analytically by studying a two-component liquid

odel originating from the problem of light oil recovery by air

njection, where one pseudo-component is volatile and has a

igh mobility (called light), while the other pseudo-component is

on-volatile and immobile (called medium).

The dominant role of the light component in the combustion

rocess has several practical consequences: the total temperature

ncrease remains small being limited by the light oil boiling point,

o oil is left behind the wave, while the downstream oil flux may

each magnitudes comparable to the injected air flux. These prop-

rties make the described combustion mechanism attractive for oil

ecovery, in comparison with the high-temperature oxidation pro-

ess applied for heavy oils, because of lower oil consumption in the

eaction and applicability to abandoned reservoirs with the poten-

ial of recovering residual oil. Note that the suggested mechanism

f oil recovery is not just due to decrease of oil viscosity, but it is

elated to a complex interplay between the flow and phase transi-

ion mechanism. Our results allow to make quantitative estimates

or recovery rates based on the conditions downstream of the com-

ustion wave. For example, the downstream oil saturation ranges

etween 0.5 and 0.9.

Theoretically, we found a qualitatively new combustion mech-

nism controlled by the successive vaporization and condensation

f the liquid phase sustained by the reaction. Since the multicom-

onent effect is dominant in this process, our analysis is crucial

or understanding filtration combustion of light oils, which are in-

rinsically multicomponent. However, our results show that fur-

her study is necessary in order to fully understand the complex

ulticomponent phenomena in this process, by considering mod-

ls with a large number of components having variable physical

roperties.

cknowledgments

This work was supported by Conselho Nacional de Desen-

olvimento Científico e Tecnológico ( CNPq ) under Grant no.

02351/2015-9 and by Fundação Carlos Chagas Filho de Amparo à

esquisa do Estado do Rio de Janeiro ( FAPERJ ) under Grant nos E-

6/201.210/2014 , E-26/110.658/2012 , E-26/110.114/2013 and Pensa

io E-26/210.874.2014 . The first author acknowledge the financial

upport by Coordenação de Aperfeiçoamento de Pessoal de Nível

uperior (CAPES) and Instituto de Matemática Pura e Aplicada

IMPA).

62 M.A. Endo Kokubun et al. / Combustion and Flame 169 (2016) 51–62

[

[

[

[

[

[

[

References

[1] A. Aldushin , Filtration combustion, in: P. Zarchan (Ed.), Advances in Com-

bustion Science: In Honor of Ya. B. Zel’dovich, AIAA, Reston, USA (1997),

pp. 95–115 . [2] A. Mailybaev , J. Bruining , D. Marchesin , Analytical formulas for in-situ com-

bustion, SPE J. 16 (3) (2011) 513–523 . [3] D. Schult , B. Matkowsky , V. Volpert , A. Fernandez-Pello , Forced forward smol-

der combustion, Combust. Flame 104 (1996) 1–26 . [4] A. Aldushin , B. Matkowsky , D. Schult , Downward buoyant filtration combus-

tion, Combust. Flame 107 (1996) 151–175 .

[5] I. Akkutlu , Y. Yortsos , The dynamics of in-situ combustion fronts in porous me-dia, Combust. Flame 134 (2003) 229–247 .

[6] C. Wahle , B. Matkowsky , A. Aldushin , Effects of gas–solid nonequilibrium infiltration combustion, Combust. Sci. Technol. 175 (2003) 1389–1499 .

[7] A . Aldushin , A . Bayliss , B. Matkowsky , On the transition from smoldering toflaming, Combust. Flame 145 (2006) 579–606 .

[8] A. Mailybaev , D. Marchesin , J. Bruining , Resonance in low-temperature oxida-tion waves for porous media, SIAM J. Math. Anal. 43 (2011) 2230–2252 .

[9] A. Mailybaev , J. Bruining , D. Marchesin , Recovery of light oil by medium tem-

perature oxidation, Transp. Porous Media 97 (2013) 1–27 . [10] N. Gargar , A. Mailybaev , D. Marchesin , H. Bruining , Effects of water on light oil

recovery by air injection, Fuel 137 (2014) 200–210 . [11] N. Gargar , A. Mailybaev , D. Marchesin , H. Bruining , Recovery of light oil by air

injection at medium temperature: experiments, J. Pet. Sci. Technol. 133 (2015)29–39 .

[12] P. Sarathi , In-situ combustion handbook – principles and practices, BDM

Petroleum Technologies, Bartlesville, OK, USA, 1999 . [13] L. Castanier , W. Brigham , Upgrading of crude oil via in situ combustion, J. Pet.

Sci. Technol. 39 (2003) 125–136 . [14] S.M. Farouq Ali , Heavy oil-evermore mobile, J. Pet. Sci. Eng. 37 (2003) 5–9 .

[15] M. Onyekonwu , G. Falade , Recovery of light oil using in situ combustion ther-mal recovery methods, Energy 14 (1989) 153–159 .

[16] S. Ren , M. Greaves , R. Rathbone , Air injection LTO process: an IOR technique

for light-oil reservoirs, SPE J. 7 (1) (2002) 90–99 .

[17] L. Adetunji , R. Teigland , Light-oil air-injection performance: sensitivity to criti-cal parameters, SPE Annual Technical Conference and Exhibition (2005) . Paper

SPE-96844. [18] A. Mailybaev , J. Bruining , D. Marchesin , Analysis of in situ combustion of oil

with pyrolysis and vaporization, Combust. Flame 158 (6) (2011) 1097–1108 . [19] M. Pascual , D. Crosta , P. Lacentre , D. Coombe , Air injection into a mature wa-

terflooded light oil reservoir–laboratory and simulation results for Barrancasfield, Argentina (SPE94092), 67th EAGE Conference & Exhibition (2005) .

[20] C. Clara , M. Durandeau , G. Quenault , T. Nguyen , Laboratory studies for light-oil

air injection projects: potential application in handil field, SPE Reserv. Eval.Eng. 3 (3) (20 0 0) 239–248 .

[21] M. Greaves , S. Ren , R. Rathbone , T. Fishlock , R. Ireland , Improved residual lightoil recovery by air injection (LTO process), J. Can. Pet. Technol. 39 (1) (20 0 0) .

22] N. Gargar , A. Mailybaev , D. Marchesin , H. Bruining , Compositional effects inlight/medium oil recovery by air injection: vaporization vs. combustion, J.

Porous Media 17 (2014) 937–952 .

23] O. Levenspiel , Chemical reaction engineering, Wiley, 1999 . [24] E. Koval , A method for predicting the performance of unstable miscible dis-

placement in heterogeneous media, SPE J. 3 (2) (1963) 145–154 . 25] B. Poling , J. Prausnitz , O. John Paul , R. Reid , The properties of gases and liquids,

McGraw-Hill, New York, 2001 . 26] N. Freitag , B. Verkoczy , Low-temperature oxidation of oils in terms of SARA

fractions: why simple reaction models don’t work, J. Can. Pet. Technol. 44 (3)

(2005) 54–61 . [27] F. Moritis , R. Bell , The low temperature liquid phase oxidation of hydrocarbons:

a literature survey, J. Inst. Pet. 44 (1958) 260–272 . 28] I. Prigogine , Introduction to thermodynamics of irreversible processes, Wiley,

1967 . 29] F. Santos , A. Mailybaev , D. Marchesin , Oxidation wave structure and oxygen

breakthrough for air injection into light oil reservoirs, Comput. Geosci. (2016) .

In Press [30] A. Aldushin , I. Rumanov , B. Matkowsky , Maximal energy accumulation in a su-

peradiabatic filtration combustion wave, Combust. Flame 118 (1999) 76–90 . [31] R. Bird, W. Stewart, E. Lightfoot, Transport phenomena, 2002.

32] P. Harriott, Chemical reactor design, 2003.