2-The Effect of Heterogeneityon in-Situ Combustion

The Effect of Heterogeneity on In-SituCombustion: Propagation of Combustion

Fronts in Layered Porous MediaI. Yucel Akkutlu, SPE, U. of Alberta, and Yannis C. Yortsos, SPE, U. of Southern California

Summary

In-situ combustion is a potential method for the recovery of heavyoil. The effect of reservoir heterogeneity, a ubiquitous feature ofoil reservoirs, on in-situ combustion has not been systematicallyaddressed in prior studies, however. In this paper, we presentanalytical models for filtration combustion, namely the combus-tion of a stationary solid fuel, in the specific case where the res-ervoir consists of two layers of different permeability and thick-ness, separated by nearly impermeable shales. We investigate theconditions for the propagation of steady combustion fronts as afunction of some key parameters, including the permeability-thickness contrast R between the layers, the thickness ratio , andthe external heat loss coefficient h.

We find that heterogeneity acts in two distinct ways: It reducesthe temperature of the leading front in the high-permeability layerin all cases, and uncouples the propagation of the fronts in the twolayers if R is smaller than a critical value Rc. The first effect maylead to low-temperature oxidation conditions, and therefore to theeffective extinction of the front in the high-permeability layer. Thesecond leads to a reduced sweep efficiency (and early break-through). However, if R exceeds the critical value, the fronts in thetwo layers travel coherently (with the same speed). This coherenceis identified for the first time. The resulting thermal couplinggreatly retards the front in the more permeable layer, and accel-erates only slightly that in the less permeable one, until the twofronts reach a common velocity.

We study the effects of R, the heat loss rate, and the ratio ofthickness . The coupling is aided by moderate heat losses (smallh), and smaller , which affect the critical value Rc. As in thehomogeneous case, at sufficiently high heat loss rates, steady frontpropagation cannot be sustained and the combustion process be-comes extinct.

The work is useful for the understanding of the viability ofin-situ combustion process in heterogeneous layered reservoirsand the effect of a number of injection, combustion, and reser-voir parameters.

Introduction

The sustained propagation of a front is necessary for the recoveryof oil using in-situ combustion. Compared to other recovery meth-ods, in-situ combustion involves the added complexity of exother-mic chemical reactions and temperature-dependent reaction kinet-ics. Combustion is influenced by a number of processes, includingthe fluid flow of injected and produced gases, the heat transfer inthe porous medium and the surroundings, the kinetics of combus-tion reaction(s), and the heterogeneity of the porous medium. Inthe presence of external heat losses, there exists the possibility ofextinction (i.e., quenching). This paper focuses on the effect ofheterogeneity, which has not been systematically addressed before.

Combustion fronts in porous media have been studied exten-sively in the context of filtration combustion, which is the com-bustion of a stationary solid fuel in a porous medium by an injectedgas oxidant (typically air). An analytical treatment of the front

dynamics is possible using methods similar to the analysis of lami-nar flames in the absence of a porous medium. Britten andKrantz,1,2 for example, provided an asymptotic analysis of in-situcoal gasification using the property that the activation energy ofthe overall (rate-limiting) reaction is large in comparison with thethermal enthalpy.3 In detailed studies, Schult et al.4,5 investigatedfiltration combustion in a homogeneous porous medium, in thedifferent contexts of fire safety and the synthesis of compactedmetal powders (SHS processes). More recently, the microscalemechanisms of forward and reverse filtration combustion in a po-rous medium were studied by Lu and Yortsos6,7 using a pore-network model.

The study of planar forward filtration combustion fronts in ahomogeneous porous medium was undertaken by the present au-thors using a continuum approach.8 They addressed the issue ofsteady-state propagation under both adiabatic and nonadiabaticconditions. External heat losses were modeled by conduction orconvection modes (the former being more appropriate for subsur-face applications). A number of important results were obtained,which for the benefit of the reader will be briefly summarized inthe Preliminaries section. In particular, the dependence of the com-bustion front velocity on injection and combustion parameters wasinvestigated in detail.

In this paper, we extend the asymptotic approach in Ref. 8 tomodel combustion fronts in heterogeneous, and specifically in lay-ered, porous media. Layered systems are prototypical of hetero-geneous reservoirs as they capture the channel-like features ofstreamtubes.9 In typical fluid displacements, the effect of reservoirheterogeneity interacts with the fluid mobility: the displacement ina more permeable layer is accelerated in the case of unfavorablemobility ratio, and retarded in the case of favorable mobility ratio.In such processes and in the absence of crossflow, the couplingbetween the layers only enters through their common inlet andoutlet pressure conditions. This leads to the so-called Dykstra-Parsons regime (e.g., see Yang et al.10). In combustion, however,the propagation of combustion fronts is affected by the local ther-mal coupling between the two layers, not only by common inletand outlet conditions. Typically, the front in the high-permeability(hence high-flow rate) layer will move faster than that in the lowerpermeability layer. Heat transfer preheats the porous medium inthe low-permeability layer, and increases the heat capacity encoun-tered by the high-permeability front. This results in thermal cou-pling, which will retard the high-permeability front and acceleratethe low-permeability front. An important question is whether ornot and under what conditions the two fronts eventually reach acommon velocity and propagate coherently. In addition, of signifi-cant interest is the possible lowering of the temperature in thefast-moving front, as it may essentially extinct the combus-tion process.

In this paper, we provide answers to these questions for the caseof filtration combustion in two layers, thermally coupled by asimple convective heat transfer mode. We assume that the layersdo not otherwise communicate (e.g., they may be sealed from oneanother through an intervening shale, or their fluid mobility mayremain constant, which is the case when the net rate of gas gen-eration because of reaction is small11). Under these conditions, theinjection rate in each layer is constant in time, and proportional toits permeability-thickness product. Both adiabatic and nonadia-batic (external reservoir heat losses) conditions are studied. We

Copyright 2005 Society of Petroleum Engineers

This paper (SPE 75128) was first presented at the 2002 SPE/DOE Symposium on ImprovedOil Recovery, Tulsa, 1317 April. Original manuscript received for review 29 May 2002.Revised manuscript received 2 June 2005. Manuscript peer approved 10 July 2005.

394 December 2005 SPE Journal

focus on how the thermal coupling affects the steady combustionfront propagation in the adjacent layers, on whether or not a stateof frontal coherence develops to maintain the system stable andself-sustaining, and on conditions of extinction. The main tool ofour analysis is the large activation energy asymptotics methodderived in Refs. 8 and 12 for a single layer. Because of the rel-evance of those results to the present case, they are briefly sum-marized below.

PreliminariesSingle-Layer Case

Consider the propagation of a filtration combustion front in a 1Dhomogeneous porous medium. Fuel necessary for the combustionhas already been generated by the processes preceding the com-bustion front and is deposited uniformly in the pore space. There-fore, the reservoir contains ab initio a known amount of fuel. (Thecoupling of fuel generation with the process itself will be discussedin a separate study.) An overall combustion reaction takes placeinside the front and completely consumes the available fuel, leav-ing a fraction of the injected oxidant unreacted (i.e., its consump-tion is incomplete). Because of the heat of combustion, a reaction-leading high-temperature region is established.

The single-layer solution is based on large-activation energyasymptotics, which is a technique based on singular perturbationtheory and used extensively to describe diffusive and premixedflame propagation.3,12 In this approach, the chemical reactions areconfined in a narrow internal region (a traveling shock, acrosswhich there are discontinuities in variables, such as the fluxes ofheat and mass). Appropriate jump conditions relate the change inthese variables across the front. The front structure accounts for thekinetics of the reaction and for internal heat and mass transfer.The problem then reduces to the modeling of the dynamics of afront, on either side of which transport of momentum (fluids), heat,and mass, but not chemical reactions, must be considered. Thismethodology allows us to explicitly incorporate permeability het-erogeneity effects, as will be shown in the following.

Under adiabatic conditions, a sustained front propagation al-ways exists. For a single layer, the (adiabatic) front temperature isgiven by Ref. 8:

f =TfTo

= 1 qVD

av+ VD 1 + q. . . . . . . . . . . . . . . . . . . . . . . . (1)

where subscripts f and o stand for front and initial, respectively,and we defined the nondimensional quantities

q =Qfo

1 cssToand a =

cggi

1 css.

In these definitions, we denoted the heat of reaction by Q, theinitially available fuel density (per total volume) by fo, and thevolumetric heat capacity of the porous medium by (1)css.VDV/vi is the dimensionless front velocity normalized with theinjection velocity vi, and (v)+ refers to the dimensionless gas massflux ahead of the front, which implicitly includes any reaction-generated gas. Eq. 1 shows that for small values of a, usually thecase is that, because of the large volumetric heat capacity of therock with respect to the gas, the front temperature is practicallyindependent of the front velocity VD.

This is not the case under nonadiabatic conditions, however.Then, the front temperature is given by the different expression8

f = 1 +qh

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

where h1+4h/VD2 , ht*(h/H)/[(1)css]. Now, the front tem-perature depends on the front velocity, decreasing with decreasingfront velocities and increasing heat losses. The latter were ac-counted for in Eq. 2 by introducing in the total energy balance avolumetric sink term equal to h(TTo)/H (see also Ref. 8 for thecase of conductive heat losses). Here, T is the absolute temperatureat a given point; To is the temperature of the surroundings, takento be equal to the initial temperature; h is a lumped convective-type coefficient from the reservoir to the surroundings; and H is

the reservoir thickness. Note that the external heat losses increase(hence, the front temperature decreases) as the thickness of theporous media becomes smaller. In addition, t*l*/vi, (l*/vi)denotes a characteristic time (space) and s/(1)css is theeffective thermal diffusion coefficient.

The relation of the front velocity to the front temperature wasderived in Ref. 8:

V D2

= Af exp f1 VD1 + gVD

. . . . . . . . . . . . . . . . . . . . . . . . (3)Importantly, it is valid for both adiabatic and nonadiabatic condi-tions and also applies to the heterogeneous case, as we will seebelow. In Eq. 3, parameter A is defined as

A =asskpYiqEIvi

2 , where I = 0

1f

fdf,

as is the specific surface area per unit volume; k the pre-exponential factor; p the initial total gas pressure; f 1f /fo theextent of fuel conversion depth; and () a dimensionless functionrepresenting dependence of reaction on . We have also intro-duced the Arrhenius number E/RTo; the activation energy E;the universal gas constant R; and the dimensionless stoichiometriccoefficients for oxygen and produced reaction gas, respectively, and ggp, defined in terms of the reaction stoichiometriccoefficients and g by

=vf

o

giYiand =

vgfo

gi.

Eq. 3 contains the dependence of the front velocity on a number ofvariables, such as pressure, the reaction rate constant, and theinjected oxygen concentration Yi. For later use, we note that theterm in parentheses on the right side of Eq. 3 represents the molefraction, normalized with Yi, of the unreacted oxidant leakingahead of the front.

In the adiabatic case, substitution of Eq. 1 into Eq. 3 leadsalways to a unique solution for the front velocity. Typical resultsfor the latter as a function of the injection velocity are shown inFig. 1 for the parameter values of Table 1 and for various injectionmole fractions. The curves in the figure have two limits. At smallinjection velocities (where A is large), the process is at the stoi-chiometric control limit, where the front propagation velocity islinear with the injection velocity and independent of the fronttemperature: VD1/ (or, in dimensional terms, Vvi/). In thislimit, the slope of the curve is proportional to the injected mole

Fig. 1Steady-state adiabatic front propagation velocity vs. in-jection velocity for varying oxygen mole fractions injected intoa single-layer porous medium. The front velocity increases withthe injection rate.

395December 2005 SPE Journal

fraction, as expected from Eq. 3. The second limit, obtained at highrates (where A is small), is the kinetic control limit, where thedimensional front velocity is now independent of the injectionvelocity: VD2 Af exp(/f). We note that in Fig. 1, the kineticcontrol limit is not reached yet. As long as the process is not underkinetic control, either in isolated single layers, or in thermallydecoupled reservoir layers, combustion fronts in higher-permeability layers under adiabatic conditions will travel fasterthan in lower-permeability ones.

Under nonadiabatic conditions, on the other hand, the solutionof Eqs. 2 and 3 can lead to multiple solutions.8 Figs. 2 and 3 showresults of the variation of the front temperature and front velocity

with the injection velocity as a function of the layer thickness (towhich the dimensionless heat loss coefficient is inversely propor-tional). For sufficiently thin layers, the system exhibits threesteady-state solutions, two stable branches (one at a high tempera-ture, where rigorous combustion takes place, and another at a lowtemperature, near the initial temperature), and an intermediate un-stable branch. Extinction and ignition points, Ec and Ic, respec-tively, appear in the form of turning points. As the thickness Hdecreases, the external heat loss rate increases, and the extinctionthreshold rapidly increases, requiring an increasingly larger injec-tion velocity for the reaction to be sustained, as shown in Fig. 2b.Conversely, at larger H, the threshold decreases, and for suffi-ciently large values, multiplicity disappears altogether. The upperbranch is the solution corresponding to a proper combustion front;it approaches and runs parallel to the adiabatic solution at higher rates.

Similar ignition and extinction phenomena have been previ-ously observed in nonadiabatic reactive systems, in the absence ofa porous medium, and reported in the combustion3,12 and reactionengineering13,14 literatures.

Here, we observe an analogous behavior for the problem offiltration combustion in homogeneous porous media. Simlar re-sults are obtained when the heat losses are of the conductive type,but require much more elaborate mathematical manipulations.8,11

Combustion in Layered Porous Media

Consider, now, combustion front propagation in the layered porousmedium shown in the schematic of Fig. 4. The steady-state propa-gation of the combustion fronts in the two layers i and j and theirinteractions is our primary focus. The layers are homogeneous, andin this section are assumed of equal thickness, but their perme-abilities are different, with layer j being the more permeable. Theyare exposed to the same pressure drop; therefore, in the absence ofmobility effects, the injection velocities are proportional to thelayer permeability-thickness product. We assume the presence of athermal coupling across the layers; therefore, the temperaturesdictated by Eqs. 1 or 2, corresponding to thermally isolated layers,do not apply. Indeed, we expect that the faster-traveling front inthe higher-permeability layer will slow down because of the trans-fer of its heat of reaction to the lower-permeability layer, the frontof which should accelerate until perhaps a coherent state is reachedand the front velocities are the same.

Fig. 2Single-layer case: Nonadiabatic combustion front temperature vs. injection velocity for 10 different values of the volumetricheat loss coefficient h/H (kW/m3K). (a) Low injection velocities: (1) 0.004, (2) 0.006, (3) 0.008, (4) 0.01, (5) 0.02, (6) 0.03, and (7) 0.04.The calculated ignition and extinction thresholds are Ic1=(5.7, 144.8), Ec1=(4.5, 193.3), Ic2=(23.5, 141.0), Ec2=(5.4, 211.9), Ec3=(10.8,253.0), Ec4=(15.8, 271.6), and Ec5=(20.8, 286.6). The adiabatic (dashed) line corresponds to f =1+q. (b) High injection velocities. Asthe volumetric heat loss coefficient increases, the ignition point disappears and the extinction thresholds are Ec6=(20.2, 286.2),Ec7=(90.4, 357.5), and Ec8=(322.5, 399.3). The corresponding Ic points are not observed within the range of physically meaningfulinjection velocities.


To proceed with the analysis, we first write dimensionless en-ergy balances for the two layers

uii = i + j i hii 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)ujj = j + i j hjj 1, . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

where we assumed equal thickness and a coherent state. In theseequations, is the dimensionless thermal coupling coefficient be-tween the two layers in the direction transverse to the frontalpropagation, and we also allowed for external heat loss to thesurrounding formations using a convective heat loss coefficient. Inthe remainder of the paper, the latter will be fixed to a constant. InEqs. 4 and 5, prime denotes spatial derivatives with respect to themoving coordinate xVDt with xx/l* and tt/t* referring todimensionless space and time variables, and we have also introduced

ui = avi VD VD, uj = avj VD VD.

As before, parameter a1 represents the ratio of volumetricheat capacity of the gas to the solid matrix.

In the following, we will provide analytical solutions to theproblem, under both adiabatic and nonadiabatic conditions. Al-though fully adiabatic conditions are not likely in a reservoir, theirinvestigation is necessary in order to understand fundamentals ofthe front propagation in a layered system.

Adiabatic Combustion Fronts. Under adiabatic conditions,hihj0 in Eqs. 4 and 5. We will proceed by considering first theproblem when the two fronts are fully separated from each other,namely when their separation distance is infinitely large, .

Fully Separated Adiabatic Fronts. When the fronts are fully-separated, the one in the higher-permeability layer is leading. Itsvelocity is smaller than in the single-layer case, however, becauseof the requirement to preheat the other layer. The front velocity isstill given by Eq. 3 but the temperature is now different. Thesolution of the heat transfer problem for this layer is summarizedin Appendix A. We find that the front temperature is now given by

fj = 1 +q2 +

q2hj

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

where hj1+8/VDj2 . Eq. 6 shows that when the thermal cou-pling is weak (small ), the front temperature reduces to its adia-biatic value for a single layer (Eq. 1). When it is strong (large ),we have the also expected result

fj = 1 +q2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

In either case, the far-upstream temperature for the first layer canbe shown to be

= 1 +

q2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)

The factor of 2 in Eq. 8 reflects the fact that the layer wherecombustion occurs is of a thickness equal to 12 of the total. In general,if the thickness is a fraction x of the total, then Eq. 8 reads instead:

= 1 + xq

(see also the following).To find the front velocity, we first reformulate Eq. 3 to nor-

malize velocities with the mean injection velocity, which is thecontrol variable. Then, it is not difficult to show (e.g., for thedimensionless velocity of the leading front in the higher-permeability layer VDj, but now rescaled with the mean velocity)the expression

VDj2

= Aofj exp fj 21 + R VDj 21 + R + gVDj

, . . . . . . . . . . . . . . . . . (9)

where parameter Ao is now based on the mean velocity, andRkihi/kjhjki/kj is the permeability-thickness ratio, varying inthe range [0,1]. Substitution of Eq. 6 into Eq. 9 gives the solutionfor the front velocity and temperature in layer j.

A similar approach applies for the trailing front in the lower-permeability layer. Under adiabatic conditions, this front sees thetemperature

from Eq. 8 as its initial temperature. The solutionof the heat problem is as described previously, with the exceptionthat the initial (reference) is now different. Expressed in terms ofTo, the trailing front temperature in the lower-permeability layercan be shown to be

fi = 1 + q +q

2hi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)

with hi1+8/VDi2 . The counterpart of Eq. 9 for the velocity ofthe trailing front is now

VDi2

= Aofi exp fi 2R1 + R VDi 2R1 + R + gVDi

. . . . . . . . . . . . . . . . . (11)

Substitution of Eq. 10 into Eq. 11 gives the corresponding frontvelocity and temperature, always under the assumption that thetwo fronts are fully separated.

Fig. 3Single-layer case: Nonadiabatic front propagation ve-locity vs. injection velocity for a single-layer porous medium.The extinction point appears at higher velocities as the volu-metric heat loss coefficient decreases.

Fig. 4Schematic of the notation used for the propagation ofcombustion fronts in a two-layered porous medium.


Typical results for the two propagation velocities are plotted inFig. 5. It is seen that when R is sufficiently small, the trailing frontin the low-permeability front has a smaller velocity than the lead-ing front in the high-permeability layer, suggesting fully separatedfronts. However, as R increases, the difference between the twovelocities decreases and at a critical value, Rc, the fronts reach thesame velocity. This coherent state will be maintained for values ofR above Rc, as will be shown below. The critical value correspondsto the limit when the fronts become fully separated. It is a functionof a number of variables, but in the present paper we will inves-tigate its dependence on the thermal coupling coefficient and thefrontal propagation coefficient Ao.

Shown in Fig. 6 is the critical value surface Rc(, Ao) con-structed by taking VDiVDjVD. It is not difficult to show that Rcis the positive real root of the quadratic:

B 21 + R VD 21 + R + gVD

=

2R1 + R VD 2R1 + R + gVD

, . . . . . . . . . . . . . . . . . . (12)

where

B = fj exp fjfi exp

fi. . . . . . . . . . . . . . . . . . (13)

Fig. 6 shows that the critical value increases (corresponding to alower permeability contrast) as Ao increases (e.g., as the meaninjection rate decreases), and saturates to a constant close to one at

large Ao. The latter signals the onset of the stoichiometric controllimit, where combustion is no longer affected by the heterogeneityand the fronts are decoupled. Coupling, on the other hand, is strongin the kinetic control limit (small Ao), where it persists at evenlarge permeability contrasts. Surprisingly, the effect of the thermalcoupling coefficient is minimal, in the range investigated. Thecritical surface divides two regions, referred to as regions U and Lin Fig. 6.

When the critical value of R is reached or exceeded, the frontsbecome coupled and travel coherently, separated now by a finitedistance . To find their common velocity, the front temperatures,and , the solution of Eqs. 4 and 5 must be sought under theassumption of a fixed separation distance.

Coherent Adiabatic Combustion Fronts. When the frontstravel coherently, we can determine the temperature profiles in thetwo layers by working as before (Appendix B). We find

fj = 1 +q2 1 + exp +

q2hj

1 exp 2 1 + hj. . . . . . . . . . . . . . . . . . . . . . . . . . (14)

for the leading-front temperature and

fi = 1 + q1 + 12hi q

2hiexp 2 hi 1 . . . . . . . . . (15)

for the trailing-front temperature, respectively. These depend onflow rates and the thermal coupling coefficient, as well as on theseparation parameter VD. Compared to the fully separatedcase (infinite ), the coherent fronts are closer to each other, andtheir interaction is also influenced by the separation length, .Substituting Eqs. 14 and 15 into Eqs. 9 and 11 and imposing thecoherent-state condition VDiVDjVD leads to a set of two equa-tions for the determination of the common velocity and the sepa-ration distance, respectively.

Fig. 7 shows the results for the dimensionless separation dis-tance , as a function of the heterogeneity ratio R, for a specificvalue of the heat transfer coupling coefficient and for three differ-ent values of Ao. The separation distance is nil when the perme-abilities are the same (R1, corresponding to the homogeneouscase), but it increases as the permeability ratio decreases. Couplingbetween the two fronts becomes more difficult as the permeabilitycontrast increases, and at the critical value, the fronts becomedecoupled. This behavior is consistent with the findings of Fig. 6.As expected, the coupling persists for a sharper permeability con-trast, provided that Ao is smaller (e.g., larger injection velocity). (Asimilar behavior is also observed for .) Thus, for given parametervalues, frontal coherence is possible when the permeability ratio isnot too small (in the range [Rc, 1.0]). Outside this range, thecoupling does not exist. The range expands as Ao decreases.

Thus, thermal coupling mitigates somewhat the effect of het-erogeneity, as it is associated with better sweep efficiency. Theexistence of the critical value Rc, however, shows that at suffi-

Fig. 5Variation of the two adiabatic front velocities in the low-and high-permeability layers, i and j, respectively, with the per-meability ratio R assuming fully separated fronts. Parameters ofTable 1: Ao=1000 and =0.01.

Fig. 6The critical surface Rc(Ao, ) corresponding to equalpropagation velocities for the fronts in the two layers. Adiabaticfully separated combustion fronts.

Fig. 7The variation of the dimensionless front separation dis-tance as a function of the permeability ratio R for three differ-ent values of Ao and a fixed thermal coupling coefficient(=0.01). Adiabatic conditions for a two-layer system. Thefronts become decoupled below a critical value of the perme-ability ratio.


ciently large permeability contrast, heterogeneity cannot be over-come, with combustion fronts traveling at different velocities, thusresulting in reduced efficiency. While for a moderate permeabilitycontrast, coupling was shown to exist, it is also necessary to showthat it is also stable to small disturbances. This is undertaken in thenext section.

Stability of Thermally Coupled Adiabatic Coherent Fronts.The question we address in this section is whether the coupledstate (e.g., as shown in Fig. 7), is a stable attractor. A simplecriterion for stability is the sign of the derivative of the fronttemperatures in the two layers with respect to the separation dis-tance. From Eqs. 9 or 11, we know that the front velocity is amonotonic function of the front temperature. Hence, a sufficientcondition for the coherent steady states to be stable is that theleading-front temperature decreases, while the trailing-front tem-perature increases, with an increase in the separation distance.Under these conditions, the leading front would slow down, whilethe trailing front would accelerate as the separation distance un-dergoes a slight increase, until the coherent state is reached again.An analogous argument applies for the case in which the separa-tion distance decreases.

Using Eqs. 14 and 15, the calculation of the derivatives isstraightforward. We find

fj

=

q4hj

exp1 + hjexp 2 hj 1 2hj 0.. . . . . . . . . . . . . . . . . . . . . . . . . . (16)

fi

=

q4hi

hi 1exp 2 hi 1 0, . . . . . . . . . . . . (17)where the right-side inequalities arise because hj>1, and(1+hj)exp[/2(hj1)]

where Hi,j1+4h/VDi,j2 and i,j1+(4h+8)/V Di,j2 . Substitu-tion in Eqs. 9 and 11, along with the coherent front requirementVDiVDjVD , leads to two equations that determine the separa-tion distance and the common velocity.

Fig. 10 shows plots of the frontal separation distance as afunction of the permeability ratio R and the heat loss coefficient h.Turning points develop for most heat loss coefficients, yieldingmultiple separation lengths. The upper branch curves upwards, theseparation lengths increasing with R, while the lower branch hasthe behavior of the adiabatic solution. Interestingly, the presenceof heat losses expands the region of coherence, although in anontrivial way. Indeed, the results show that as the heat loss rate isincreased, the fronts tend to propagate closer to each other andmaintain their coherence at values of R as low as 0.25. Ath8.5E6, the first branch yields a peculiar hump at R0.21. Atlarger heat loss rates (h>8.5E6), coherent front propagation ispossible only when the fronts are very close to one another. Theturning point value Ec4 is lower than that observed at low andmoderate heat loss rates. Thus, it appears that once we reach acritical external heat loss rate, coherent front propagation in thesystem bifurcates and goes through two separate propagation re-gimes. The fronts become either apart from each other with alarger separation length (or even become fully separated, as in theadiabatic case), or they propagate closely spaced, thus minimizingthe influence of heat losses. In the latter case, however, extinctionis inevitable and the fronts quench together.

Stability of Nonadiabatic Coherent Fronts. It is interesting tofind the stability of the nonadiabatic fronts under coherent condi-tions. Working as before, the steady coherent fronts will be stableif the following condition is satisfied:

=

VDj VDi 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20)

To analyze Eq. 20, we write VD/(VD/f)(f/) to find

VDj VDi =

VDjfj

fj

VDifi

fi

. . . . . . . . . . . . . . . . . . . . (21)

Expressions for VDi/fi and VDj/fj are obtained from Eqs. 9and 11:

VDjfj 2VD +

2VD2 + g

21 + R VD 21 + R + gVD1 + R=

VD2

fj

VDifi 2VD +

2VD2 + g

2R1 + R VD 2R1 + R + gVD1 + R=

VD2

fi

. . . . . . . . . . . . . . . . . . . . . . . . . . (22)where VD is the common velocity. Then, taking the derivatives ofEqs. 18 and 19 with respect to , we find

fj

=

q4F FH

fi

=

q4G GH, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (23)

where we introduced the auxiliary functions

F = 1 +

exp 2 1 + G = 1

exp 2 1

Now, it is not difficult to show that F() is a monotonically de-creasing function. Thus, given that >H , we will always havefj/

multiple solutions (when R is larger than the critical) or no coher-ent solutions at all. The multiplicity is also reflected in the sepa-ration distance of Fig. 10. The upper branch denotes the stablesolution, and leads to stable separation fronts, as demonstrated inFig. 11 as well. The intermediate branch is unstable (the loweststable branch being the trivial one of the initial temperature). Sur-prisingly, the effect of heat losses on the leading front is negligible,and the resulting curve appears quite similar to the adiabatic case,except for the appearance of multiplicity, with the same qualitativefeatures as in the high-temperature front.

Fig. 13 shows the effect of R on the common front velocity.Interestingly, the curve obtained does not vary significantly withthe heat loss rate, except that higher heat losses extend along therange of the plot. The curves show multiplicity, the upper branchbeing stable and close to the adiabatic front value. The multiplicityis similar in appereance to the homogeneous case, except of coursethat here, the control parameter is R rather than the injection ve-locity. In all cases, the turning points (with infinitely large slopes)signal front extinction. The turning points in the various quantitiesoccur for the same R value.

It is rather interesting that, as the permeability contrast of thelayered system increases, the nonadiabatic combustion fronts be-come fully separated in a manner resembling extinction in thehomogeneous case (see, for example, the similarites between Ec2

and Ec4 in Figs. 12 and 13). Unlike the adiabatic fronts approach-ing their full separation limit asymptotically, however, nonadia-batic fronts develop multiplicity. The latter suggest that one of themultiple solutions is likely to be unstable.

Adiabatic Combustion FrontsUnequal Thickness Layers.The previous sections studied combustion front propagation in alayered porous media, where the two layers have the same thick-ness. In general, this is not the case in the typical application.Layers with different thickness are expected to affect the problemin a nontrivial way. For example, the ratio in the thickness willaffect the temperature of the leading front (e.g., in the fully sepa-rated front case), where the heat capacity of the rock will appeareffectively larger, and therefore lead to a decrease in the fronttemperature if the thickness of the high-permeability layer issmaller. In this section, we will investigate the effect of the thick-ness ratio hj/hi on the propagation of adiabatic fronts. Thenonadiabatic case becomes more complex, and for simplicity willnot be addressed.

When 1, the heat transfer Eqs. 4 and 5 are rewritten as follows:uii = i + j i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (24)

ujj = j +

i j. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (25)

The solution of the heat transfer problem is obtained as previouslydescribed. Omitting the considerable details, we find the follow-ing expressions:

fj = 1 +q

1 + 1 exp 2 1 +

+q

1 + + exp . . . . . . . . . . . . . . . . . . . . . . . . . . . . (26)

fi = 1 + q +q

1 + 1 exp 2 1, . . . . . . . . (27)

where 1+4(1+)/VD2 . Proceeding as before, we can thensolve for the common velocity, the separation distance, and thevarious other properties of the process, as a function of parameterRkihi/(kjhj) and the thickness ratio hj/hi.

Fig. 14 shows the separation distance plotted vs. the heteroge-neity parameter R for different values of . It is clear that theregion of coherence increases as thickness of the more permeablelayer increases. This is quite interesting, as lower values of ,for a fixed R, require increasingly larger permeability contrasts.Fig. 15 shows variation of the front temperatures as a function ofR for different values of for the same conditions as in Fig. 14. As

Fig. 13The variation of the dimensionless coherent front ve-locity with the permeability ratio R for varying heat loss coeffi-cient and Ao=1,000 and =0.01.

Fig. 14The variation of the dimensionless front separationdistance as a function of the permeability-thickness ratio R fordifferent values of the thickness ratio and for Ao=1,000 and=0.01. Adiabatic conditions for a two-layer system. The frontsbecome decoupled below a critical value of the ratio R.

Fig. 12Nonadiabatic front temperatures for the high- and low-permeability layers as a function of the permeability ratio R forvarying heat loss coefficient and Ao=1,000 and =0.01.


in the equal-thickness case, there is a remarkable lack of depen-dence of the curves on parameters other than R. Indeed, the effectof the thickness ratio is only to extend the range of the coherentstate into lower front temperatures (which, however, are probablytoo low to correspond to a proper combustion process, and prob-ably invalidate the large activation energy asymptotics approach).Analogous is the lack of the effect of the thickness ratio on thefront velocity, as shown in Fig. 16.

Conclusions

In this paper we extended the asymptotic approach of Ref. 8 toheterogeneous systems by considering the simpler case of filtrationcombustion in porous media consisting of two layers. Analyticalmodels were developed to delineate the combined effects offluid flow, reaction, and heat transfer on combustion front propa-gation, using as parameters the coefficient of thermal couplingbetween the layers (), the heat transfer to the surroundings (h),the layer permeability-thickness contrast (R), and the layer thick-ness ratio ().

We find that thermal coupling between the layers plays a cru-cial role on the combustion front propagation. In particular, forvalues of the parameter R above a critical value, thermal couplingallows for coherent fronts in the two layers, propagating at thesame velocity. The coupling retards greatly the front in the more-permeable layer, although it accelerates only slightly that in theless-permeable one. In essence, under these conditions, propaga-tion becomes slave to the injection velocity in the lower-permeability layer.

Combustion front propagation is enhanced in the presence ofexternal heat losses and when the more permeable layer is ofsmaller thickness. The front can maintain its uniform structure ina reservoir with a larger permeability contrast. However, its tem-perature drops significantly with the heterogeneity ratio, smallerthickness ratio, and increasing increasing heat loss rates. Below acertain limit, it is questionable that a proper combustion reactioncan be sustained in the high-permeability layer. At high heat lossrates, steady front propagation cannot be sustained and the com-bustion process becomes extinct.

As in the single-layer case, there exists a unique solution underadiabatic conditions, and there exist multiple steady-state solutionsunder nonadiabatic conditions. Importantly, the appearance ofmultiplicity should not always be interpreted as frontal extinction.Depending on the intensity of the external heat losses in the res-ervoir, multiple solutions could appear, indicating the fronts be-coming detached and fully separated.

Nomenclature

a volumetric heat capacity ratio of the gas andsolid

as specific fuel surface area per unit volume, m2/m3A defined in Eq. 3

Ao parameter A of the layered system based on themean injection velocity

B defined in Eq. 13E activation energy of the reaction, kJ/kmoleF auxiliary function defined in Eq. 23G auxiliary function defined in Eq. 23h nondimensional reservoir heat loss coefficienth convective-type external heat loss coefficient

kW/m2KH reservoir thickness, mk pre-exponential factor

l* characteristic length, m nondimensional separation distance between

coherent fronts propagating in the layersp initial total gas pressure in pore space, atmq nondimensional parameter reflecting total heat

content of the reservoirQ heat of reaction, kJ/kg fuelR permeability:thickness ratio between the layerst nondimensional time

t* characteristic time, sTf front temperature, KTo initial temperature, KT absolute temperature, Ku convective term coefficient in the moving

coordinate defined as VDvi injection velocity, m/dayV common propagation velocity of the fronts m/day

VD nondimensional front propagation velocityx nondimensional coordinate

Yi inlet oxygen concentration kg/kg(1 )css heat capacity of solid matrix, kJ/m3K

effective thermal diffusion coefficient, m2/s Arrhenius number fraction of the total thickness thickness ratio between the layersf extent of fuel conversion defined as 1f/f* nondimensional temperaturef front temperature, K effective thermal conductivity, kW/mK nondimensional stoichiometric coefficient for

oxygen

Fig. 16Variation of the dimensionless coherent front velocitywith the permeability-thickness ratio R for different values ofthe thickness ratio and for Ao=1,000 and =0.01. Adiabaticconditions for a two-layer system.

Fig. 15Adiabatic front temperatures for the high- and low-permeability layers (denoted by j and i, respectively) as a func-tion of the permeability-thickness ratio R for different values ofthe thickness ratio and for Ao=1,000 and =0.01.


g nondimensional stoichiometric coefficient for netgas production defined as gp

gp nondimensional stoichiometric coefficient forgaseous-phase combustion products

normalized stoichiometric coefficient for net gasproduction defined as gp

gp normalized stoichiometric coefficient forgaseous-phase combustion products

moving coordinate defined as xVDtof initially available fuel mass density, kg/m3

(v)+ gas mass flux ahead of the front

dimensionless coefficient of thermal coupling

between the layers frontal separation parameter defined as VD porosity dimensionless function representing dependence

of reaction on f stability condition defined in Eq. 20

Subscriptsf index for fronti index for less permeable layerj index for more permeable layero index for initial reservoir state

Acknowledgments

This research was partially funded by the U.S. Dept. of Energy andby the Natural Science and Engineering Research Council ofCanada. We gratefully acknowledge their financial contributions.

References

1. Britten, J.A. and Krantz, W.B.: Linear Stability Analysis of PlanarReverse Combustion in Porous Media, Combustion and Flame (1985)60, 125.

2. Britten, J.A. and Krantz, W.B.: Asymptotic Structure of Planar Nona-diabatic Reverse Combustion in Porous Media, Combustion andFlame (1986) 65, 151.

3. Williams, F.A.: Combustion Theory, Benjamin and Cummings Pub-lishing Co. Inc., Redwood City, California (1985).

4. Schult, D.A. et al.: Forced Forward Smolder Combustion, Combus-tion and Flame (1996) 104, 1.

5. Schult, D.A., Bayliss, A., and Matkowsky, B.J.: Traveling Waves inNatural Counterflow Filtration Combustion and Their Stability, SIAMJ. Applied Math. (1998) 58, 806.

6. Lu, C. and Yortsos, Y.C.: The Dynamics of Forward Filtration Com-bustion at the Pore-Network Level, AIChEJ (2005) 51, No. 4, 1279.

7. Lu, C. and Yortsos, Y.C.: Pattern Formation in Reverse FiltrationCombustion, Physical Review E, submitted for publication (2005).

8. Akkutlu, I.Y. and Yortsos, Y.C.: The Dynamics of In-Situ Combus-tion Fronts in Porous Media, Combustion and Flame (2003) 134, 229.

9. Willhite, G.P.: Waterflooding, SPE Textbook Series, Vol. 3 (1986).10. Yang, Z., Yortsos, Y.C., and Salin, D.: Asymptotic Regimes of Un-

stable Miscible Displacements in Random Porous Media, Adv. WaterRes. (2002) special anniversary issue, 25, 885.

11. Armento, M.E. and Miller, C.A.: Stability of Moving CombustionFronts in Porous Media, SPEJ (December 1977) 423; Trans., AIME,263.

12. Akkutlu, I.Y.: Dynamics of Combustion Fronts in Porous Media,PhD dissertation, U. of Southern California (2003).

13. Nicolis, G. and Prigogine, I.: Exploring Complexity, W.H. Freeman andCo., New York City (1989).

14. Epstein, I.R. and Pojman, J.A.: An Introduction to Nonlinear ChemicalDynamicsOscillations, Waves, Patterns and Chaos, Oxford U. PressInc. (1998).

Appendix A

Consider a combustion front propagating at constant speed in layerj under adiabatic conditions. The fronts are fully separated. Bysolving the corresponding heat transfer equations, we find

i = 1 +q2 expVD

q2hi

exp12 VD + VD2 + 8. . . . . . . . . . . . . . . . . . . . . . . . (A-1)

j = 1 +q2 expVD + exp12 VD + VD2 + 8

. . . . . . . . . . . . . . . . . . . . . . . . (A-2)for the temperature profiles ahead of the front, and

i = q

2hiexp12 VD + VD2 + 8 . . . . . . . . . . (A-3)

j = q

2hjexp12 VD + VD2 + 8 . . . . . . . . . . (A-4)

for the temperatures behind the front.

Appendix B

Consider the case of adiabatic fronts under a coherent state. Bysolving the corresponding heat transfer equations, we find thefollowing (please refer to Fig. 4):

Region I.

i = 1 + q 12h + B q

2hexphexpVD2 1 + h

j = 1 + q + 12h + B q

2hexphexpVD2 1 + h. . . . . . . . . . . . . . . . . . . . . . . . . (B-1)

Region II.

i = 1 +q2 +

C2 expVD

q2h

expVD2 1 h B q2hexp

VD2 1 + h

j = 1 +q2 +

C2 expVD

q2h

expVD2 1 h+ B q2hexp

VD2 1 + h . . . . . . . . . . . . . . . . . . (B-2)

Region III.

i = 1 +A2 expVD BexpVD2 1 + h

i = 1 +A2 expVD BexpVD2 1 + h . . . . . . . . . . (B-3)

where coefficients A, B, and C are given byA = q1 + exp,

B =q

2h1 exp1 + h2,

C = qexp.

Appendix C

Consider the case of nonadiabatic fronts under a coherent state. Bysolving the corresponding heat transfer equations, we find thefollowing (please refer to Fig. 4):

Region I.

i = 1 DexpVD2 1 + C2 expVD2 1 hj = 1 + DexpVD2 1 + C2 expVD2 1 h

. . . . . . . . . . . . . . . . . . . . . . . . . (C-1)


Region II.

i = 1 +E2 expVD1 + h FexpVD2 1 +

q2

expVD2 1 + q2h expVDh 1j = 1 +

E2 expVD1 + h + FexpVD2 1 +

+q

2expVD2 1 + q2h expVDh 1

. . . . . . . . . . . . . . . . . . . . . . . . . (C-2)Region III.

i = 1 +A2 expVD1 + h BexpVD2 1 +

j = 1 +A2 expVD1 + h + BexpVD2 1 +

. . . . . . . . . . . . . . . . . . . . . . . . . (C-3)where the various coefficients are given by

A =qh

1 + exp1 + h2

B =q

21 exp1 + 2

C =q

H1 + exph 12

D =q

21 exp 12

E =qh

exp1 + h2

F = q

2exp1 + 2.

I. Yucel Akkutlu is an assistant professor of petroleum engineer-ing at the U. of Alberta. e-mail: [email protected]. He teachesreservoir engineering courses and conducts research on im-proved oil recovery. Akkutlu holds a BS degree in chemicalengineering from Hacettepe U. as well as MS and PhD degreesin petroleum engineering from the U. of Southern California.Yannis C. Yortsos is the Chester Dolley Professor of PetroleumEngineering in the Dept. of Chemical Engineering at the U.of Southern California. e-mail: [email protected]. Since June2005, he has also served as the Dean of the USC Viterbi Schoolof Engineering and also holds the position of Senior AssociateDean for Academic Affairs. His research interests are in flowand transport processes in porous media. Yortsos holds aPhD degree in chemical engineering from the CaliforniaInst. of Technology. Yortsos served as the Executive Editor ofthe SPE Journal from 20002001, and he is currently on theEditorial Board.


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