numerical simulation of thermal and reaction fronts

Chemical Engineering Science 94 (2013) 200–213

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Chemical Engineering Science

0009-25

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n Corr

E-m

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

Numerical simulation of thermal and reaction frontsfor oil shale upgrading

M.S.K. Youtsos n, E. Mastorakos, R.S. Cant

Hopkinson Laboratory, Engineering Department, University of Cambridge, UK

H I G H L I G H T S

c We model shale oil extraction by in situ thermal upgrading using an in house code.c Hot gas injection and conduction heating are found to be viable methods.c Reaction wave progression can be tracked solely by monitoring the thermal wave.c Dimensionless flow rate governs oil recovery by hot gas injection.c Dimensionless depletion region length governs oil recovery by conduction heating.

a r t i c l e i n f o

Article history:

Received 5 September 2012

Received in revised form

14 February 2013

Accepted 15 February 2013Available online 27 February 2013

Keywords:

Petroleum

Shale

Computational fluid dynamics

Multiphase flow

Heat transfer

Pyrolysis

09/$ - see front matter & 2013 Elsevier Ltd. A

x.doi.org/10.1016/j.ces.2013.02.040

esponding author. Tel: þ44 1223 332641; fa

ail address: [email protected] (M.S.K. Youtso

a b s t r a c t

This paper analyses reaction and thermal front development in porous reservoirs with reacting flows,

such as those encountered in oil shale upgrading. A set of dimensionless groups and a 1D code are

developed in order to investigate the important physical and chemical variables of such reservoirs

when heated by in situ methods. Theory necessary for this study is presented, namely shale

decomposition chemical mechanisms, governing equations for multiphase flow in porous media and

necessary closure models. Plotting the ratio of the thermal front speed to the fluid speed allows one to

infer that the reaction front ends where this ratio is at a minimum. The reaction front follows the

thermal front closely, thus allowing assumptions to be made about the extent of decomposition solely

by looking at thermal front progression. Furthermore, this sensitivity analysis showed that a certain

minimum permeability is required in order to ensure the formation of a traveling thermal front.

Compared to varying deposit porosities and kerogen activations energies, varying temperature,

pressure and permeability are more important.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

1.1. Motivation

Enhanced oil recovery methods and unconventional hydro-carbon sources represent the next steps in ensuring today’s evergrowing demands for oil and gas are met.

Shale deposits around the world are estimated to contain ca.3 trillion barrels worth of oil (Fan et al., 2009). There has been arenewed interest in shale in recent years, driven heavily by risingoil prices, resulting in the development of new in situ oil shaleupgrading technologies, for example by Shell (Vinegar, 2006),ExxonMobil (Symington et al., 2010) and Chevron (Looney et al.,2010).

ll rights reserved.

x: þ44 1223 3 32662.

s).

In situ oil shale upgrading (OSU) and heavy oil extractionare both achievable by thermal methods, which involve a mix ofthermal decomposition and combustion reactions (Fan et al.,2009). These methods are analyzed in this paper by the construc-tion of governing equations and their numerical solution andexploration.

1.2. In situ oil shale upgrading

Oil Shale is a solid clay-rich sedimentary deposit containingorganic matter (kerogen) that generates oil upon being heated(Fan et al., 2009). The kerogen weight content in oil shale canrange from 0 to 40% (Lee, 2010). For instance, rich shale in theGreen River deposit in the USA typically will give 0.00035 m3 ofoil per kg of shale; whereas lean shale might be somewherearound 0.00013 m3/kg (Qian and Yin, 2010).

Shale oil has been produced by mining and subsequent surfaceretorting methods for many years, in countries like Australia,

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Table 1Species considered in this simulation, their symbols and their respective phases.

Variable Symbol Phase Molecular

weight

Pcrit

(MPa)

Tcrit

(K)

Carbon dioxide CO2 Gaseous 44.01 7.38 304.18

Heavy oil ho Liquid/

gaseous

339.85 1.40 893.70

Light oil lo Liquid/

gaseous

130.88 2.78 646.30

Hydrocarbon

gas

hc Gaseous 44.10 4.25 370.18

Methane me Gaseous 16.04 4.6 190.18

Kerogen k Solid 647.00 – –

Coke-1 c1 Solid 13.00 – –



M.S.K. Youtsos et al. / Chemical Engineering Science 94 (2013) 200–213 201

China, Estonia and Canada (Qian and Yin, 2010). The presentcontribution however focuses only on in situ heating methods.These methods include: conduction heating, hot gas injectionand in situ combustion. The first method, makes use of simpleconduction heaters drilled into the reservoir and is the preferredmethod of Shell (Vinegar, 2006). The latter two methods involvefluids being injected into the reservoir. In the case of the hot gasinjection, an inert gas is injected with the intent of heating thereservoir solely by its sensible enthalpy. Chevron have outlined amethod whereby CO2 is injected (Looney et al., 2010). In the caseof in situ combustion, hot air is injected in order to combust theshale organic component (kerogen) and subsequently releaseits products (Looney et al., 2010). These methods result in theformation of thermal and reaction fronts. They are travelingstructures with a characteristic set of properties, dependent onthe properties of the injected fluid(s) and the initial reservoirfluids (Woods, 1999).

The interaction between the different fronts is of primeinterest if one wishes to truly understand how a reservoir’sproperties will change in each of these processes. While thereexists an abundant amount of literature on the description ofthermal and reaction fronts for more conventional reservoirprocesses (Akkutlu and Yortsos, 2003; Mailybaev et al., 2010;Phillips, 1991; Woods, 1999) (for instance, in situ combustion)there are no such contributions for shale reservoir processes.

To the knowledge of the authors, there are four other studies,published in the open literature, which deal with the simulationof the in situ extraction of shale oil (Bauman and Deo, 2011;Bauman et al., 2010; Fan et al., 2009; White et al., 2010).Conduction heating and kerogen combustion were investigated.Higher heater temperatures result in faster recoveries, but poten-tially unfavorable oil decompositions (Fan et al., 2009; Whiteet al., 2010). Combining the two methods was found to bebeneficial (Bauman et al., 2010), for example when combustionfollows conduction heating in order to displace the depositedheavy oils. A timespan of 5–10 years is recommended for heating(Bauman and Deo, 2011; Bauman et al., 2010; Fan et al., 2009; Sunet al., 2006; White et al., 2010), while EROI ratios of as high asfour (Bauman et al., 2010) have been reported. EROI is defined asthe energy output (the potential heat of formation when com-busting the fuel) to the energy input (in the form of pressurizedand heated fluid injected into the reservoir) ratio. Numericalstudies employed either in-house (White et al., 2010; Fan et al.,2009) or commercial (Bauman et al., 2010) reservoir simulators,with empirically derived closure models. Shell’s study is the onlylarge scale in situ oil shale upgrading study to date (Bauman et al.,2010). An important conclusion of White et al. (2010) was thatonly considering kerogen decomposition – ignoring oil and cokedecompositions – yields residual liquid oil in the formation,something which is inconsistent with lab and field observations.

Studies have focused extensively on well patterns, extractionrates and efficiencies for conduction heating (Bauman and Deo,2011; Bauman et al., 2010; Fan et al., 2009; Sun et al., 2006;White et al., 2010). There seems to be no analytical work on thesubject of oil shale upgrading, or any contribution which inves-tigates oil shale upgrading by hot gas injection. Finally theevolution of the various thermal and reaction fronts developedthrough heating shale has not been analyzed systematically.

1.3. Objectives

It is thus the objective of this investigation to implement amodel for the thermal and reaction fronts developed when shaleis heated in situ by conduction and hot gas injection. Coupledwith non-dimensional analysis, a deeper physical understandingof these processes is obtained. In order to assess the performance

of each method, a preliminary comparison of oil shale upgradingby thermal conduction and hot inert gas injection is presentedand discussed.

2. Model formulation

2.1. Summary of the model

This study was performed using an in-house code to simulatereacting multiphase flow in porous media. A kinetic modeldeveloped by Burnham and Braun (1990) which is based on amultiple pathway thermal decomposition model, was used. Kero-gen decomposes into components, which at industrially relevanttemperatures and pressures, include gases, liquids and solids.Reservoir properties such as permeability, porosity and depth areinputs to the model and in this paper, values typical of the GreenRiver shale formation in the US are used. The OSU process ismodeled as a Cartesian problem, with heat entering the reservoirvertically through an injection well and extracted at the bottomby a collector well. These wells are represented by appropriateboundary conditions. The phase behavior of the liquids and gaseswas described using the Peng–Robinson equation of state. Oil andgas species densities vary according to reservoir pressure andtemperature. The output of the model is the concentration ofliquid and gaseous species, per unit volume of the reservoir, as afunction of distance and time. In greater detail, the model is givenin the following sections, while the Appendix includes somedetails on the closures used and the non-dimensionalizations.

2.2. Governing equations

Several phases are considered, a solid phase (the depositmatrix) and a gaseous phase made of many species. The spe-cies/lumped species considered are given in Table 1.

In greater detail, there will be three distinct phases, two ofwhich (the gas mixture and the oil) will be present within thepore space of the shale (the void continuum), resulting in theneed for multiphase flow equations. The third phase is the solidmatrix. Since with reservoir flow inertial forces are orders ofmagnitude smaller than viscous forces, the velocity field isderived simply by applying Darcy’s law (Phillips, 1991):

ua ¼�kkra

maðrPa�rgrzÞ ð1Þ

where k is the absolute permeability of the shale matrix, f is theporosity, kra is the relative permeability of phase a, ma is theviscosity of the phase and rPa is the spatial gradient operator and

Fig. 1. Schematic for simulation grid. 1�N refer to the grid nodes for the numerical

solution.

M.S.K. Youtsos et al. / Chemical Engineering Science 94 (2013) 200–213202

z is the depth. The phase symbol a can either denote a gas (g) oroil (o).

The conservation of the liquid (oil) and gas mixture phases isgoverned by

@ðfraSaÞ

@t¼�r � ðrauaÞþqa ð2Þ

where ra is the density of the phase, Sa is the saturation of thephase (ratio of phase volume to total non-solid volume) and qa isthe source term in kg/m3/s.

The transport of components within the phases is governed by

@ðfraicaiÞ

@t¼�r � ðraicaiuaÞþr � ðDiraircaiÞþqai ð3Þ

where cai is the mass fraction of a component i within theformation and Di is the mass diffusivity of component i in theformation. When this equation is summed, Eq. (2) is obtained,since Di ¼D. The pressure field is found by combining massconservation, Darcy’s law for the gaseous phase and the gas law

r¼ PW

ZRTð4Þ

Z is the gas deviation factor. The following pressure equation isobtained:

@ðfSaPaWa=ZaTÞ

@t¼r �

PaWakkrg

mZaRTrPa�

PaWa

ZaTgrz

� �� þqa ð5Þ

where T is the formation temperature, R is the universal gasconstant and W is the mass averaged molecular weight of thegaseous phase. The term q represents the pressure source termdue to chemical reactions. Capillary pressures are accounted forthrough the empirical model provided by Chen et al. (2009).

Finally, the governing equation for the temperature is pre-sented. Only one energy governing equation for the solid matrixand the void continuum is required. The time needed for the flowto traverse a grain tflow (10�4 s) is much greater than the time fora rock grain to absorb the fluid’s heat theat (10�7 s) (Phillips, 1991;Woods, 1999), meaning that all fluid phases will be at the sametemperature as the rock grains. The formation temperatureequation is described as in Chen et al. (2009) and is written as

@

@t

XNa

aðracpaTÞþð1�fÞrscpsT

" #¼�r�

XNa

aðrauacpaTÞþr�ðlrTÞþqe

ð6Þ

where a is the thermal diffusivity of the formation, qe is theenergy source term from chemical reactions. Following Woods(1999) we define Ga as

Ga ¼racpa

fPNa

a ðracpaÞþð1�fÞrscps

ð7Þ

cpa is the specific heat capacity of phase a, Na is the number ofphases (in this case there are two) and the subscript s denotes thesolid phase. This can be physically understood as the dimension-less heat capacity of the injected fluid. Finally, qe is the energysource term from the chemical reactions.

This formulation necessitates the use of two-phase relativepermeability which is obtained from Chen et al. (2009) and detailsare given in the Appendix. With regards to oil shale upgrading,the authors are unaware of any empirical studies to describerelative permeabilities specific to hydrocarbons derives from oilshale upgrading. As such, this study employs empirical dataspecific to typical black oil models, as other studies on oil shaleupgrading have (Bauman and Deo, 2011; Bauman et al., 2010; Fanet al., 2009; White et al., 2010).

Oil shale upgrading is achieved by heating through a boreholeat the top of the formation, with a single collector at the bottom.Boundary conditions at either end of the domain simulate the

injection and collector wells. The injector well boundary condi-tion is that of a constant pressure. As such, the injection rate shallvary depending on the reservoir conditions, prompting a max-imum flow rate to be set at the collector and injector. Vertical1D direct line drive is simulated through the use of cartesiancoordinates. As Liu et al. (2011) note, such a system ‘promotesgravity stable displacement’. One must note the limiting effectsof a 1D system as mainly being constrained to a homogenousreservoir. Inhomogeneities are certainly an important feature ofany reservoir, however as this is the first study of its kind onthermal and reaction fronts for shale oil upgrading, homogeneityis assumed to isolate key physical mechanisms.

There may be some heat loss when thermal OSU methods areemployed. However, Vinsome and Westerveld (1980) have notedthat for large Peclet numbers, longitudinal heat losses are few. Liuet al. (2011) have used heat loss models for 1D simulations.

These models cannot account for how heat losses affect theenergy transport at the boundaries and thus they are unfit for usein this study. However, even without taking into account heatloss, this study can provide much insight into the front dynamicsof thermal OSU.

The schematic shown in Fig. 1 can represent our solution. Heatand mass transfer occur only in the lateral direction.

2.3. Porous medium and transport properties

Lee (2010) has compiled much experimental data on Shale,outlining certain ranges of values that are typical of a Oil Shalesample. The shale properties shall be modeled here as if the shalewere ca. 1 km underground, thus subject to high temperaturesand very high pressures. The shale deposit will be filled withmethane initially. The initial conditions of the shale deposit arepresented in Table 2. Of the several studies that simulated oilextraction from Green River formation shale (Bauman and Deo,2011; Bauman et al., 2010; Fan et al., 2009; White et al., 2010)initial permeability used varied from 0.003 mD to 10 mD. Ahigher value on this range was chosen for three reasons. First,kerogen concentrations used in this study are typical of leandeposits which have higher permeabilities typical of oil shalereservoirs that would be targeted as economically recoverable.Second, hot gas injection would only be applied to oil shale

Table 2Summary of formation initial conditions. P, SK, SC, t and T denote pressure, kerogen

volumetric saturation, coke volumetric saturation, time and temperature respectively.

Property Value Constant? Function of

Permeability 7 mD NO P, SK, SC

(White et al., 2010)

Temperature 313 K NO –

Density kerogen 1200 kg/m3 NO t, T

Density inorganics 2200 kg/m3 YES –

Concentration of

kerogen

240 kg/m3 NO t, T

Concentration of

inorganics

1960 kg/m3 YES –

Kerogen Mass Pct. 10 NO t, T

Thermal conductivity 1.2 W/m K YES T

(Nottenburg et al.,

1978)

Porosity 0.1 NO P, SK, SC

(White et al., 2010)

Specific heat capacity 1335 J/kg K NO T (Qian and Yin, 2010)

Pressure of formation 100 atm NO t

Mass diffusivity 4.8�10�9 m2/s YES –

Table 3Initial inert gas properties for base case at inlet.

Property Value Constant? Function of

Temperature 573 K YES –

Density 141.15 kg/m3 NO P, T

Pressure 230 bar YES –

Specific heat capacity 1110 J/kg K NO T

Viscosity 0.00007942 Pa s NO P, T (Chen et al., 2009)


reservoirs with relatively high initial permeabilities, thus it makessense to assess said method on such reservoirs. Finally, Shell(Vinegar, 2006) and Exxon (Symington et al., 2010) have recentlyreported that initial hot water flooding could dissolve muchinorganic material resulting in ca. an order of magnitude increasein initial permeability. As such, typically higher values of oil shaleinitial permeability are very likely to exist in practice beforeany upgrading takes place. Kerogen decomposition will result inincreased pore space, less the deposited coke residue. Pressureincreases will act to increase the pore radii which result inincreased permeability. These phenomena are captured in theempirical equation for permeability variation, detailed in theappendix.

It was decided to use CO2 as the inert gas for injection, as perChevron’s method. The transport properties of CO2 are mostlytemperature dependent, though the large pressure variations dovary some of the properties significantly (e.g. dynamic viscosity).The initial conditions of the inert gas, within the borehole,are shown below for 573 K and 230 atm, the chosen base casecondition for inert gas injections. The conduction base case issimply a temperature boundary condition at 573 K with no fluidinjection. A minimum bottom hole pressure, as well as a max-imum collection rate, is defined. Fluids are allowed to passthrough the collector boundary at a rate specified by each phasesvelocity, but no more than the maximum allowable collectionrate.

As the fluids travel through the reservoir, they are eventuallycooled. The lighter oil fraction will condense to its liquid phase.Thermodynamic transport properties are detailed in the Appen-dix. Relative permeability equations are also detailed in theAppendix for the oil and gaseous phases.

2.4. Chemistry

Since there is no oxygen in the gas injected, only pyrolysisreactions are considered. The decomposition of kerogen into Oil andother products has been very widely studied but not one singlemodel exists that is used by a majority of these researchers.However, the prevalent model is that of Burnham and Braun(1992) and there is a prevalent tendency to take primary kerogendecomposition as being a first order reaction (Braun and Burnham,1986; Burnham and Braun, 1992; Fan et al., 2009; White et al.,2010). These authors pointed out that all data are consistent with an

alternate pathway mechanism, by which oil is formed from kerogendirectly or from bitumen if such bitumen has been produced by thekerogen. Reactions 1–5, used here, are taken from Burnham andBraun (1992) and detailed in the Appendix (Table 3).

Mineral decomposition is ignored since the maximum tem-perature encountered during simulation is insufficient to activatethese reactions. Water presence is ignored as only small quan-tities may be present in most shales (Fan et al., 2009; Rahm,2011). Table 4 summarizes each reaction’s kinetic parameters.

2.5. Non-dimensionalization

A set of non-dimensional equations was developed for the caseof oil shale upgrading, based upon the methods in Phillips (1991)and Jupp and Woods (2003). By working with the non-dimensionalcoordinate Z, defined as

Z¼ ðGgug=lÞðx�GutÞ ð8Þ

nine dimensionless groups were derived in the Appendix and aregiven below.

P1 ¼lrkrkFk,ho

u2gfCho,maxG

2g

, P2 ¼�lrho

u2gfG

2g

ð9Þ

p1 ¼lðrkrkFk,loþrhorhoFho,loÞ

u2gfClo,maxG

2g

ð10Þ

E1 ¼lHk,rxn

G2g u2

g Tinlet

, E2 ¼lHo,rxn

G2g u2

g Tinlet

ð11Þ

Y ¼lrk

G2g u2

g ð1�fÞ, Le¼

lD

ð12Þ

V ¼1�fGg

fGg, R¼

uo

ugð13Þ

Another dimensionless group was added, in accordance with theobserved phenomena by Jupp and Woods (2003).

b¼ugDx

lð14Þ

The variable ri refers to the rate of pyrolysis of the species i. Fi,j refersto the mass fraction of j produced from the pyrolysis of species i.

It must be noted that the thermal front will travel with thespeed of approximately Ggug (Woods, 1999), i.e. based on theDarcy velocity of the gas, since the Darcy velocity of the oil phaseis significantly (2–3 orders of magnitude) smaller, based ontypical gas and oil viscosities (Chen et al., 2009).

The first two terms can be thought of as ratios of production/depletion rates of heavy oil to thermal front speed. The morepositive the value, the larger the rate of production given a frontspeed. Similarly, p1 is a measure of the rate of production of lightoil to the speed squared of the thermal front. Alternatively theycan be thought of as ratio of the time scale of formation of thethermal front (l=ðGgugÞ

2) to the time scale over which the specificreaction occurs (1=r).

Table 4Kinetic data for decomposition reactions (Burnham and Braun, 1992). Coke 3 and H/C gas activation energies are far too high to be activated at this study’s temperatures of

interest (Burnham and Braun, 1992).

Decomposition reaction Frequency factor (1/s) Activation energy (kJ/mol) DH (J/mol)

Heavy oil (HO) - 0.373 LOþ0.156 Gasþ0.03 Methaneþ0.441 Coke 2 (2) 1.0�1013 213.4 �46 500

Light oil (LO) - 0.595 Gasþ0.115 Methaneþ0.290 Coke 3 (3) 5.0�1011 225.9 �46 500

Kerogen - 0.279 HOþ0.143 LOþ0.018 Gasþ0.005 Methaneþ0.555 Coke 1 (1) 3.0�1013 225.9 �3 35 000

Coke 1 - 0.031 Gasþ0.033 Methaneþ0.936 Coke 2 (4) 1.0�1013 225.9 �46 500

Coke 2 - 0.003 Gasþ0.033 Methaneþ0.964 Coke 2 (5) 5.0�1011 225.9 �46 500

0.5 1 1.5 2 2.5 3 3.5 40

20

40

60

80

100

Distance

Spe

cies

Con

cent

ratio

n (k

g/m

3 )

Light OilHeavy OilCoke 1Coke 2Coke 3Gas mixture

50

100

150

s C

once

ntra

tion

(kg/

m3 )

Light OilHeavy OilCoke 1Coke 2Coke 3Gas mixture


E1 and E2 are simply the non-dimensionalized energy releaseterms. The rate of energy production (power) is non-dimensionalized by the rate of heat traveling past a point perunit second due to the thermal front. Y is a measure of the rate ofdepletion of kerogen to the speed square of the thermal front. Le

is the Lewis number, which is a measure of thermal to massdiffusivity.

The dimensionless number V is a measure of the speed of thefluid front to the speed of the thermal front. R is the ratio of the oilphase to the gaseous phase velocity. Finally, the term b is a ratioof the volumetric flow rate per unit length of fluid to the thermaldiffusivity (effectively a measure of the influence of convection tothat of conduction) and plays an important role in defining thetemperature profile, as will bee seen later in the paper.

2.6. Numerical method

In order to solve the set of differential equations presentedabove the Method of Lines (MOLs) was used. By this method, allthe PDEs were discretized in spatial dimensions but not in thetime dimension. Central differences are used for diffusion andupwind differencing for convection terms. This leaves us with aset of ODEs for which initial value conditions have been provided.The DVODPK (Variable-coefficient Ordinary Differential equationsolver, written in FORTRAN 77) (Brown et al., 1989) was used andrun in fully implicit stiff mode. This solver has been used for awider variety of reacting flow problems (de Paola et al., 2009;Wright et al., 2009) and has been shown to handle stiff sets ofequations well (Byrne, 1992).

0.5 1 1.5 20

Distance

Spe

cie

Fig. 2. (a) Species profiles for Hot gas injection after 30 days and (b) species

profiles for Heat conduction after 90 days.

3. Results and discussion

3.1. In situ gas injection

Fig. 2(a) shows the species concentrations at 30 days for hotgas injection. The oil phases terminate near the reaction front andthere is little heavy oil decomposition (there is very little of thecoke 2 species).

Fig. 3(a) shows the extent of kerogen decomposition withtime, as the rate at which the reaction front penetrates thereservoir is ca. 6�10�6 m/s but decreasing slowly with time.The velocity of the gaseous phases drops as the gases moveoutward into the reservoir, thus it is no surprise that the thermalfront progression also slows down as time passes (Fig. 3(b)).Insignificant thermal decomposition of heavy oil is taking place(there are very small amounts of Coke 2 in the system which is aproduct of heavy oil decomposition). The thermal gradients arerelatively steep when compared to what thermal conductionachieves, as will be discussed later. This means that the injectionrate is large enough in order to ensure that thermal advection isthe dominant heat transfer mechanism.

In Fig. 4, the normalized decomposition rates of heavy oil andkerogen vs. Z are plotted. The reaction rates are normalized with

the rate of decomposition either species would undergo were it’stemperature at the injected temperature. Both reaction frontsretreat to more negative Z values as time passes. Our non-dimensional coordinate Z is based on the location of the thermalfront. Since the thermal front’s plateau is not horizontal and isgrowing in the x-direction, the thermal front (as tracked by Z, lagsbehind the true thermal front. Despite this, the front decomposi-tion rate for kerogen is slowly receding due to slowly decreasingthermal front temperatures. However, heavy oil decompositionrates are growing because of the increasing presence of heavy oil.

Fig. 5(a) yields some very interesting insight into the frontdynamics of the system. The normalized temperature profileshows us where the thermal front is located at 15 days into theheating program. The net heavy oil production of heavy oil ismaximum at the thermal front’s edge. Right ahead of the reaction

0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

Distance (m)

Ker

ogen

(kg

/m3 )

15 days30 days45 days

0 2 4 6 8 10300

350

400

450

500

550

600

distance (m)

Tem

pera

ture

(K

)


Fig. 3. Hot gas injection: (a) profiles of kerogen at 573 K and (b) the evolution of

the thermal front as a function of distance for different times.

−10 −8 −6 −4 −2 00

0.05

0.1

0.15

0.2

0.25

0.3

η

Nor

mal

ised

Ker

ogen

Dec

ompo

sitio

n R

ate


−10 −8 −6 −4 −2 00

0.005

0.01

0.015

0.02

0.025

0.03

η

Nor

mal

ised

Hea

vy O

il D

ecom

posi

tion

Rat

e


Fig. 4. Hot gas injection: (a) profiles of the kerogen pyrolysis rate and (b) profiles

of the heavy oil pyrolysis rate.


front, the ratio of the thermal to the fluid front speed is minimum.To understand why these two observations occur, the followingthree graphs of Fig. 5 are considered.

Fig. 5(b) depicts the variable b vs. the dimensionless coordi-nate Z. As the heating progresses, the b profile shifts upwards,implying thermal convection’s increasing importance. This wouldmean that the velocity of the gas phase is increasing as a result ofpermeability increasing at high pressures and the freeing up ofpore space from the kerogen.

The b profiles are at local minimum around the thermal front,downstream of which they rapidly increase. Downstream of thethermal and reaction fronts, the gas is less hindered by the banksof oil being built up upstream of the thermal front. The localminimum occurs because the heavy oil hinders the flow of thegaseous phase.

Fig. 5(c) shows the plots of the dimensionless depletion/produc-tion terms of Heavy Oil after 15 days. These plots capture severalimportant aspects of the system. They show the decreasing reactionrate, upstream of the thermal front, with time due to kerogendepletion. Also, the abrupt decline just downstream of the thermalfront implies there is very little to no kerogen decompositionhappening. This leads to a very important result: the given reactiontimescales and convection heating timescales have led to the pointwhere the thermal front and reaction front are clearly alignedthroughout the heating process. This was seen in Fig. 5(a).

The non-dimensional production of heavy oil stays positive forthe whole of the heating program. P1þP2 quantifies the extent

of dominance that kerogen decomposition has over heavy oildecomposition and would be negative at a higher temperature.Since a very low temperature will yield a very high dominance,yet low kerogen decomposition rates, this marker can not be thesole point of reference for the extent of kerogen depletion.

Fig. 5(d) depicts the profiles of the dimensionless number 1/V. Thevariable 1/V quantifies the ratio of the thermal to fluid front speed.Perhaps the first thing noticed is that all the radial values are small.This makes sense because the fluid front will always move faster thanthe thermal front. However, two distinct regimes on this graph arenoticed. Upstream of the thermal front, there is an increase in theimportance of the thermal versus the fluid front. This would bedriven by the initial formation of the thermal front followed by theincreasing presence of oil, which slows down the gas front.

What is now observed differently from b is that downstreamof the thermal front, the curves converge. Slightly further down-stream of the thermal front a minimum value for 1/V is reached.This value could be considered a characteristic marker for thisbase case, as it does not seem to be affected by time. Further tothis, downstream of the thermal front, the pressure gradientsdevelop and thus would affect the thermal and fluid front speedsin equal proportions. This would explain why the curves convergedownstream of the thermal front.

If indeed it is the case that the decomposition reactions arewhat lead to the temporal evolution of the 1/V curves upstream ofthe thermal front, then this must mean that the reaction frontends where the thermal front speed is at a minimum.

−3 −2.5 −2 −1.5 −1 −0.5 0 0.50.7

0.8

0.9

1

1.1

η

normalized 1/Vnormalized temperaturenormalized Π1 + Π2

−10 −8 −6 −4 −2 0 20

200

400

600

800

1000

1200

1400

1600

1800

2000

η

β


−3 −2.5 −2 −1.5 −1 −0.5 00

0.02

0.04

0.06

0.08

η

Π 1 +

Π 2


−4 −3 −2 −1 0 1 24

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6x 10−3

η

1 / V

7.5 days10 days15 days

Fig. 5. Hot gas injection: (a) profiles of 1/V, P1þP2 and normalized temperature vs. Z after 15 days. Net heavy oil production peaks at the edge of the thermal front. The

thermal to fluid front speed is at minimum right ahead of the reacting front , (b) profiles of b vs. Z, (c) profiles of P1þP2 vs. Z and (d) profiles of 1/V vs. Z.


When running this base case for 180 days, the non-dimensional production of heavy oil converges ca. 0.036 (at thethermal front). As such, for a given 1/V one can infer characteristicvalue of P, thereby being able to comment on the extent ofdecomposition solely by looking at the thermal front.

The final recovery at 142 days for the gas injection base casewas 0.78 and the energy returned on energy invested (EROI) wasca. 3.0. These recoveries are not achievable in reality due to heatlosses and heterogeneities. However, they serve as a benchmarkfor real world comparison as well as comparison with other idealcases of oil shale upgrading methods. The recovery of hydrocar-bons was defined as the ratio of the total amount of heavy andlight oils extracted at the ‘well node’ to the maximum recoverableamount of oils (assuming no heavy and light oil pyrolysis).Residual oil, which could not be displaced, accounted for mostof the unrecovered oil for the base case run.

3.2. In situ conduction heating

The ‘‘conduction base case’’ (the gas injection boundary con-dition was removed) was run up to 270 days. For the conductionbase case, the thermal front tails are exceedingly long (shown inFig. 6(a)) and they take much longer to develop, compared to thegas injection case. The thickness of the thermal front heats shalevolumes where no reactions are taking place. Due to this, theresidence time is long enough now to permit significant decom-position of heavy oil and coke-1.

Fig. 2(b) shows species concentrations for heat conduction.There is much heavy oil and coke 1 decomposition. The reactionfront takes much longer to progress than the gas injection case,because there is hardly any gas to carry the heat downstream.

Conduction heating does not produce a moving thermal front,but rather an ever stretching thermal profile.

Fig. 6(d) displays two separate regions. The first one is a smallregion between Z¼ 0 and ca. Z¼ 0:25. Here, the 1/V curve evolveswith time. This region, as will be shown, is the region wherekerogen decomposition is being completed, i.e. the depletionregion. This depleted region terminates quite abruptly at ca.Z¼ 0:25, while ahead of the depletion region there is very littlechange in time because no reactions have occurred there yet.

Fig. 6(c) lends credence to the previous reference about thedepletion region terminating at ca. Z¼ 0:25. The P1þP2 curvescollapse onto each other ahead of Z¼ 0:25. The peak representsthe reaction front, which decreases in rate with time.

The volumetric flux per unit length of fluids is constantthroughout time upstream of the reaction front (Fig. 6(b)). Furtherto this, since the reaction front moves in tandem with the 1/Vcurve depletion region, the extent of decomposition based solelyon the kerogen products’ influence on the thermal front’s pro-gression is inferred.

The final recovery at 3012 days for the conduction base casewas 0.75 and the energy returned on energy invested (EROI) wasca. 3.4. Compared with hot gas injection, this method appears tobe more energetically efficient due to the much lower heat inputrates. Compressing gas, heating it and then injecting it is muchmore energy intensive than using resistance heaters.

3.3. Sensitivity analysis

Having described the base case, key parameters in the systemwere varied so as to investigate their respective influence onthermal and reaction front progression. Variation of the initialpermeability, kerogen reaction rates, temperature and pressure ofthe gas injected into the reservoir were investigated. Permeabilityand pressure variations have more effect on gas injection, thusonly this method was considered in those sensitivity analyses. For

−2 −1 0 1 20

0.5

1

1.5

2

2.5

3

3.5x 10−5

η

Ker

ogen

Dec

ompo

sitio

n R

ate

(kg/

m3 s) 270 days

135 days90 days

−2 −1 0 1 2 30

500

1000

1500

2000

η

β


−1.5 −1 −0.5 0 0.5 1 1.50

50

100

150

η

Π1

+ Π

2


−1 −0.5 0 0.5 1 1.5 2 2.5

1

1.5

2

2.5x 10−4

η

1 / V


Fig. 6. Heat conduction: (a) temperature profiles, (b) profiles of b vs. Z, (c) profiles of P1þP2 vs. Z and (d) profiles of 1/V vs. Z.


the conduction heating method, only temperature and kerogenreaction rates were varied and are presented alongside thevariations of temperature and kerogen reaction rates for the hotgas injection simulations.

3.3.1. Permeability variations

Permeability can inhibit fluid progression greatly, leading to adominance of conduction as a heat transfer mechanism, or it canpermit very high flow rates, thus creating a very steep travelingthermal front. It is thus no surprise that permeability enhancingtechniques are becoming more and more advanced (Farcas andWoods, 2007). Any oil shale upgrading process which aims torely on hot gas injection should make sure that reservoir permeabilityis at a favorable level. A high final permeability permits efficientoil shale upgrading relatively quickly. However, below a certainpermeability, the convection as a heat transfer mechanism fails todominate conduction. This leads to very slow overall decompositionrates.

Fig. 7(d) shows the thermal front greatly suffering, with respectto the fluid front, (predominantly upstream of Z¼ 0) from thedecrease in permeability. But this is not only because the thermalfront travels faster (evidence of that provided in Fig. 7(a)), but ratherbecause – as is shown in Fig. 7(b) – the fluid front’s ability to convectenergy declines, as indicated by lower b values. Thus, conductionplays the significant role in the overall shape of the thermal front. Infact for 10 mD, the thermal conduction is the stronger of the twoheating mechanisms, and thus no formation of a traveling thermalfront is observed. As such, it is possible to find a minimumpermeability for which hot gas injection would not induce atraveling thermal front for the given boundary and initial conditions.

In order to give a characteristic b value to each of the simulationruns in Fig. 7(b), the minimum value ahead of the thermal front isselected. This minimum value happens to coincide with the edge of

the thermal and depletion front, as discussed earlier. This isexpected because at this point the hot injected gas is the only fluidflowing through shale that has not seen an increase in permeabilitydue to kerogen decomposition.

It is observed that for bmino1, a traveling thermal front doesnot develop. For 10obmino1000 thermal fronts are forming,however thermal tails are still significant in their influence on theefficiency of heating. Finally, for bmin41000 very small thermaltails and a fast traveling thermal front are expected.

The graph of non-dimensional net production of heavy oil forvarying permeability (Fig. 7(c)), shows a decrease for increasingpermeabilities, because of kerogen depletion.

Recovery rates for high permeability reservoirs are noticeablylarger, with the added benefit that residence times are decreased– thus avoiding unwanted heavy oil decomposition.

3.3.2. Pressure variations

Three inlet pressures for the inert gas were run: 210, 230 and250 bar. Fig. 8(a) and (d) show that higher pressures lead toquicker thermal and reaction fronts. There is no evidence that thethermal front shape benefits from an increase in pressure, onlythat it progresses faster.

Although there is a noticeable increase in volumetric flow rateper unit length with respect to the thermal diffusivity (Fig. 8(b))upstream of the thermal front, this is not the case downstreamof it. Finally, an intermediate pressure leads to a higher non-dimensional net production of oil (Fig. 8(c)) at 15 days.

Recovery rates for highly pressurized injectants are noticeablylarger, with the added benefit that residence times are decreased– thus avoiding unwanted heavy oil decomposition. There is alimit to the increase in efficiency for higher pressure injection,due to the added energy required to compress the fluids.

−6 −4 −2 0 24

5

6

7

8

9

10 x 10−3

η

1 / V

1mD7mD10mD

1 2 3 4 5300

350

400

450

500

550

600

650

Distance (m)

Tem

pera

ture

(K)

1mD7mD10mD

−20 −15 −10 −5 0 50

1000

2000

3000

4000

5000

η

β

1mD7mD10mD

−2 −1.5 −1 −0.5 0 0.5 10

0.02

0.04

0.06

0.08

η

Π1 +

Π2

1mD7mD10mD

Fig. 7. Hot gas injection: (a) temperature profiles for permeability variations, (b) profiles of b vs. Z for permeability variations, (c) profiles of P1þP2 vs. Z for permeability

variations and (d) profiles of 1/V vs. Z for permeability variations.

1 2 3 4 5300

350

400

450

500

550

600

650

Distance (m)

Tem

pera

ture

(K

)

210 MPa230 MPa250 MPa

−10 −5 0 50

1000

2000

3000

4000

5000

η

β


−2 −1.5 −1 −0.5 0 0.5 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

η

Π1

+ Π

2


−6 −4 −2 0 2

4

5

6

7

8

9

10 x 10−3

η

1 / V


Fig. 8. Hot gas injection: (a) temperature profiles for pressure variations, (b) profiles of b vs. Z for pressure variations, (c) profiles of P1þP2 vs. Z for pressure variations

and (d) profiles of 1/V vs. Z for pressure variations.


1 2 3 4 5300

350

400

450

500

550

600

650

Distance (m)

Tem

pera

ture

(K

)

553K573K593K

−10 −8 −6 −4 −2 0 20

500

1000

1500

2000

η

β

553K573K593K

−1.5 −1 −0.5 00

0.02

0.04

0.06

0.08

η

Π1

+ Π

2

553K573K593K

−5 −4 −3 −2 −1 0 1 2

4

4.5

5

5.5

6

6.5 x 10−3

η

1 / V

553K573K593K

Fig. 9. Hot gas injection: (a) temperature profiles for temperature variations, (b) profiles of b vs. Z for temperature variations, (c) profiles of P1þP2 vs. Z for temperature

variations and (d) profiles of 1/V vs. Z for temperature variations.


3.3.3. Temperature variations

Fig. 9(a) shows that increasing the temperature at the inletwill strengthen the thermal front. Increasing the temperatureeven more (to 593 K) increased the thermal front speed, howeverthere is a drop in the 1/V profile at 593 L, upstream of the reactionfront. This is shown in Fig. 9(b) shows a difference in the rate ofkerogen decomposition at higher temperatures. At the highesttemperature, the non-dimensional production of heavy oil hasdecreased because of ensuing decomposition. For a higher tem-perature, while the kerogen decomposes much more readily,heavy oil decomposition is more likely to ensue. Too low atemperature delays the decomposition of kerogen by too muchand leads to low overall decomposition rates (Fig. 9(c,d)). Thismakes sense because the heavy oil has decomposed back into asolid material and thus increased the solid mass that the thermalfront must heat and hinders the subsequent flow of gas.

Even though the thermal tails of the profiles do not divergethat much (Fig. 10(a)) the 1/V profiles in Fig. 10(b) clearly show anincreasingly wide depletion region front for the in situ conductionmethod. The wider, the more efficient because the thermal tailis smaller and does not heat as much kerogen which is notdecomposing.

Recovery rates for high temperature injectants are noticeablylarger. The competing effect of unwanted heavy oil decompositionresults in a presumably ‘optimal’ temperature which is neither toolow nor too high. High temperature conduction recovery rates fortemperature variations display similar behavior.

3.3.4. Kerogen decomposition rate variations

The increase in depletion region length is clearly illustrated,for the heat conduction case, in Fig. 10(c) and (d). A more rapidkerogen decomposition rate results in deeper reaction frontpenetration. The gas travels quicker through the high

decomposition rate case because pore space is freed up fromkerogen much more rapidly. The reaction front is also muchthinner (Fig. 10(d)) due to the increased decomposition rate.

Thus, knowing the kerogen chemistry will allow us to inferdepletion region lengths solely by looking at the progression ofthe thermal front. Hot gas injection cases are now considered.

Fig. 11(a) shows that increasing the kerogen decompositionrate has little effect on the thermal front. This is to be expectedsince the heat release rate is very small compared to the heatinjection rate. In fact, quicker decomposition leads to impededhot gas flow (Fig. 11(b)) and thus a weaker convective front, asobserved.

Fig. 11(c) shows, as expected, quicker decomposition rates forincreased chemistry rates.

Fig. 11(d) shows the 1/V profiles from which much usefulinformation is extracted. The thermal front propagation is hin-dered at high reaction rates due to increased oil presence.Downstream of the front, the profiles converge. This signifies thatno decomposition is taking place, definitive proof the thermal andreaction fronts are aligned.

4. Conclusions

A set of governing equations and closure models for multi-phase reacting flows in Shale reservoirs have been developed andsolved numerically for a one-dimensional formation.

The final recovery at 142 days for the gas injection basecase was 0.78 and the EROI was ca. 3.0. The final recovery at3012 days for the conduction base case was 0.75 and the EROIwas ca. 3.4.

It is observed that for migrating thermal fronts, the reactionfront follows the thermal front closely, thus allowing one toroughly calculate the extent of decomposition solely by looking

−1 0 1 2 3

1

2

x 10−4

η

1 / V

573 K593 K603 K

1 2 3 4 5300

350

400

450

500

550

600

650

Distance (m)

Tem

pera

ture

(K)

573 K593 K603 K

−2 −1 0 1 2 3 4

20

40

60

80

100

120

η

beta

0.5 x BaseBase2 x Base

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

η

Ker

ogen

Dec

ompo

sitio

n R

ate

(kg/

m3 s) 0.5 x Base

Base2 x Base

Fig. 10. Heat conduction: (a) temperature profiles for temperature variations, (b) profiles of 1/V vs. Z for temperature variations, (c) profiles of b vs. Z for kerogen

decomposition rate variations and (d) kerogen decomposition rates vs. Z for kerogen decomposition rate variations.

1 2 3 4 5300

350

400

450

500

550

600

650

Distance (m)

Tem

pera

ture

(K

)


−10 −8 −6 −4 −2 0 20

500

1000

1500

2000

η

β


−1.5 −1 −0.5 0 0.50

0.02

0.04

0.06

0.08

η

Π1

+ Π

2


−5 −4 −3 −2 −1 0 1 2

4

4.5

5

5.5

6

6.5x 10−3

η

1 / V


Fig. 11. Hot gas injection: (a) temperature profiles for kerogen decomposition rate variations, (b) profiles of b vs. Z for kerogen decomposition rate variations, (c) profiles of

P1þP2 vs. Z for kerogen decomposition rate variations and (d) profiles of 1/V vs. Z for kerogen decomposition rate variations.


at thermal front progression. This was true for both the conduc-tion and hot gas injection base cases. There was very little (lessthan 1%) heavy oil decomposition observed for heavy oil in the

hot gas injection base case, however this was not true for theconduction base case where large residence times allowed forsignificant heavy oil decomposition.


Hot gas injection was found to be more effective, in terms ofspeed and heavy oil recovery, for heating the formation. The moredominant thermal conduction is in the formation of the thermalfront, the larger the tail of the temperature profile.

Heavy Oil decomposition becomes very active at high tem-peratures and since it has a high viscosity, it will remain in thereservoir for some time before it is extracted; this being moreprominent in the conduction base case. This will mean that it willalmost certainly decompose completely if the heating tempera-ture is too high, regardless of heating method.

Plotting the ratio of the thermal front speed to the fluid speed(1/V) allows one to infer that the reaction front ends where this ratio isat a minimum, for both heating methods. It was found that for givenboundary conditions and initial conditions, a certain characteristicvalue for each 1/V can be assigned that will characterize the behaviorof the induced thermal front and reaction front. For heat conduction, adepletion region can be defined to characterize each run.

For hot gas injection, the dimensionless variable b must have aminimum value of ca. 1 in order to produce a thermal front whichmigrates through the reservoir.

Compared to varying deposit porosities and kerogen activationsenergies, varying temperature, pressure and permeability, in eithermethod, are more important. Varying the gas injection pressure hadlittle effect on much else than varying the progression of the thermaland reaction fronts. The temperature had a great effect on thedepletion rate, the shape of the thermal front and its progressionand the upstream volumetric flow rate per unit length.

Nomenclature

A
frequency factor (s�1) C concentration (kg/m3) E activation energy (kJ/mol) F product mass fraction from pyrolysis reaction H enthalpy of reaction (J/mol) P pressure (Pa) S saturation ðVpores=VtotalÞ
T
temperature (K) V volume of fluid (m3) W molecular weight Y non-dimensional kerogen pyrolysis rate Z gas deviation factor c species mass fraction f fraction of decomposing species I species subscript k permeability
kr
relative permeability
r
pyrolysis rate (kg/s) G non-dimensional fluid heat capacity P1 non-dimensional rate of production of heavy oil P2 non-dimensional heavy oil pyrolysis rate a phase subscript
b
ratio of influence of convection to conduction heating
Z
non-dimensional coordinate that follows thermal wave
l
thermal conductivity (W/m K)
m
fluid viscosity (Pa s) p1 non-dimensional rate of production of light oil r fluid density (kg/m3)
f
porosity
Acknowledgments

The authors would like to acknowledge the helpful discussionswith Prof. Andrew Woods of the BP Institute, Cambridge. M. Youtsosacknowledges the financial support of National Oil Shale Holdings LLC.

Appendix A. Non-dimensionalization

The injection of a fluid in a 1D pore space at a constant Darcyspeed ug is considered. The kerogen and heavy/light oil decom-pose at a rate given by

Qk ¼�rkCk ðA:1Þ

Qo ¼�roCo ðA:2Þ

where C is the mass concentration of a species per unit reservoirvolume. The oil (non-aqueous liquid phase) flows through thepore space as described by the following equation:

f@ro

@tþr � ðuoroÞ ¼Qo,p�Qo,d ðA:3Þ

Two lumped oil species are considered: light oil and heavy oil.They are both described by Eq. (A.3). For the purposes of thisanalysis, the decomposition of light oil is neglected since it is veryslow compared to the decomposition of heavy (Burnham andBraun, 1992). Thus, the behavior of the heavy oil decompositionreaction will dominate in terms of affecting the fluid dynamicsand chemistry of the system.

Also, the rate at which kerogen is depleted is:

@rk

@t¼�Qk

1�fðA:4Þ

The following procedure is developed based upon the methodsin Phillips (1991) and Jupp and Woods (2003). It is desirable towrite the differential equations in dimensionless form, thus thedimensionless coordinate system is based on that which is ofmost interest: the speed of the thermal front, Gu:

Z¼ ðGgug=lÞðx�GgugtÞ ðA:5Þ

t¼ ½ðGgugÞ2t�=l ðA:6Þ

The dimensionless coordinate Z moves with respect to thedimensional position x, so that x¼0 is given by Z¼ t. Theconcentration variables (which have units kg per total formationvolume) are non-dimensionalized with the maximum possibleconcentrations they can attain. For instance, if there is Ck initiallywithin the reservoir, this is Ck,max, and the maximum possibleconcentration for heavy oil (Cho,max) is 0:279Ck,max according toreaction (1).

O¼ Cho=Cho,max o¼ Clo=Clo,max ðA:7Þ

K ¼ Ck=Ck,max Y¼ T=Tinlet ðA:8Þ

Thus the equations are re-written as

@Y@tþ@Y@Z¼@2Y@Z2þE1þE2 ðA:9Þ

@K

@t¼ YK ðA:10Þ

@O@t þð1þVÞRrdO¼

1

Ler

2OþP1K�P2O ðA:11Þ

@o@tþð1þVÞRrdo¼ 1

Ler2oþP2o ðA:12Þ

with initial conditions and boundary conditions

PðZ,0Þ ¼ 0, pðZ,0Þ ¼ 0, YðZ,0Þ ¼ 1,

Kðz,0Þ ¼ 1, Yð�t,tÞ ¼ 1 ðA:13Þ

A.1. Permeability

Of the several studies that simulated oil extraction from GreenRiver formation shale (Bauman and Deo, 2011; Bauman et al.,


2010; Fan et al., 2009; White et al., 2010) initial permeability usedvaried from 0.003 mD to 10 mD. White et al. (2010) used a modelwhich is a function of porosity and weight composition of thesolid. This makes physical sense because it is expected that thepermeability increases due to the pore pressure expanding thepore radii and due to the decomposition of kerogen which createsnew pore space. The model from White et al. (2010) is employed,namely

k¼ k0

ð1�Ske�ScoÞ3 1

f�1� �

1f�ð1�Ske�ScoÞ

� �2ðA:14Þ

where k0 is the initial permeability at formation pressure and Si isthe formation volume saturation of a component i.

A.2. Porosity

Chen et al. (2009) model the porosity as

f0 ¼f0ð1þCRðp�p0ÞÞ ðA:15Þ

The same rock compressibility factor as White et al. (2010)was used.

A compressibility factor of 4.0�10�11/Pa was used, in linewith typical Shale compressibility factors (Lee, 2010).

f0 was modeled as being purely dependent on f0, the initialporosity (prior to any decomposition) at the formation pressure tobe investigated, and the solid volumetric composition of theformation. Namely

f¼ ð1þf0Þ�SsðCs=rsÞ ðA:16Þ

where f0 is the initial porosity for a given formation pressureand subscript s refers to summing all solid species within theformation.

A.3. Thermal conductivity of Shale

The thermal conductivity model was based on 30 gal/tonne. Assuch, the empirical formulae for lateral and vertical permeabil-ities are based on Nottenburg et al. (1978):

lv ¼ 7� 10�7T2�0:0078Tþ2:9505, R2

¼ 0:8951 ðA:17Þ

lh ¼�5:9172� 10�4Tþ1:2775, R2¼ 0:9873 ðA:18Þ

Units are W/mK

A.4. Specific heat capacity of Shale

Qian and Yin (2010) provide separate relations for specific heatcapacity for kerogen, coke and the inorganic components. Thefollowing are the heat capacity models for each component

cp,ke ¼ 1753:6þ3:3367T J=kg K ðA:19Þ

cp,in ¼ 687:27þ0:92945T J=kg K ðA:20Þ

cp,co ¼ 1008:2þ2:8276T J=kg K ðA:21Þ

A.5. Relativity permeability

The presence of oil in the porous media will hinder the flow ofgas and vice versa. There is little theory for how to describe therelative permeabilities of each phase vs. the saturation of thosephases. Data has been provided by Chen et al. (2009) and curves

have been fitted to the data set. These relations are

kr,g ¼ 93:954S5o�238:07S4

oþ223:42S3o�91:395S2

oþ13:026Soþ0:41153

ðA:22Þ

kr,o ¼ 100ð�154:2S6oþ341:2S5

o�286:3S4oþ120:2S3

o�26:3S2oþ2:85So�0:117Þ

ðA:23Þ

where So is the volume fraction of the oil phase within the porespace. It is interesting to note that the gaseous and oil phasevelocities are likely to drop by almost an order of magnitudebecause of their combined presence. This means it is crucial toensure that oil flows as easily as possible so that the injected gascan also flow uninhibited.

A.6. Oil compressibility

The oleic phase compressibility is calculated by the followingequation from Chen et al. (2009)

co ¼�1433þ17:2T�1180YGþ12:61API

100 000PbðA:24Þ

where YGS is the gas gravity (unit for air), Pb is the bubble pointpressure and API refers to then gravity of the oil.

A.7. Oil and gas viscosity

The American Petroleum Institute gravity (API gravity) mea-sures how heavy a petroleum fraction is compared to water. If theAPI is larger than 10 (1000 kg/m3), it is lighter and floats on water(Chen et al., 2009). The heavy oil is taken to be ca. 15 API(965.0 kg/m3) and light oil is taken as ca. 25 API (903.7 kg/m3).The Beggs–Robinson equation (Beggs and Robinson, 1980) wasused to calculate oil viscosities based on each fraction’s API andtemperature.

For simplicity, no gases were allowed to dissolve into the oil.Thus the ‘dead’ oil viscosity formula from the Beggs–Robinsonequation was used. The relation is

mo ¼ 10C�1 ðA:25Þ

where

C ¼ 10C0T�1:163, C0 ¼ 3:0324�0:02023API ðA:26Þ

Formulas give viscosities in cP therefore must multiply by 1000 toget Pa s (SI unit).

The gas viscosity calculation is a lot more complicated tocalculate. From Chen et al. (2009): ‘‘The gas viscosity mg wasevaluated based on an estimation of the gas density using the realgas law (with a Z-factor correction). The pseudocritical pressureand temperature are corrected for non-hydrocarbon compo-nents.’’ Here, mg was calculated by the Lee–Gonzalez correction(Dempsey, 1965):

mg ¼eFmc

TredðA:27Þ

where F is a function of the reduced temperature (Tred), thereduced pressure (Pred). These reduced properties are simply thenormalization of each individual gas temperature and pressure byits critical temperature and pressure. The corrected gas viscositymc is defined by Carr et al. (1954) as

mc ¼ ð1:709� 10�5�2:062� 10�6YGÞTþYCO2

ð6:24� 10�3

þ8:188� 10�3�6:15� 10�3 logðYCO2

Þþ9:08� 10�3Þ logðYGÞ

ðA:28Þ

where YG is ratio of the gas mixture density to air density and YCO2

is the mass fraction of CO2 within the gas mixture.


A.8. Phase fugacities

The fugacity, f, of a real gas is the effective pressure whichreplaces the true mechanical pressure for chemical equilibriumcalculations (Chen et al., 2009). The fugacity of component i is

f i,a ¼ paxi,aFi,a ðA:29Þ

where xi,a and Fi,a are the mass fraction and fugacity coefficient ofa component i in phase a. For thermodynamic equilibrium, thefugacities of every hydrocarbon component to be distributed bythe relation

poxi,oFi,o ¼ pgxi,gFi,g ðA:30Þ

for i¼ 1,2, . . . ,Nc. The phase mass fractions of each oil and gasfraction are given by xi,o and xi,g . Phase pressures of gas and oil aregiven by pg and po. The fugacity coefficients, Fi,a are functions ofthe gas deviation factor, Za, from the Peng–Robinson equation ofstate, oi is the acentric factor of component i and finally kij, whichis the binary interaction parameter between components i and j.The acentric factor of a component roughly expresses the devia-tion of the shape of the component molecule from a sphere (Fanet al., 2009). Necessary physical properties of the componentsused are detailed in Fan et al. (2009).

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