orth,

stigy COs. gar, gndinode

taiTheoretical and numerical simulations of the lm/corner ow are conducted for the corner-shapes. The

l recovecessesalso i07). Grfully i1995).

that the gravity dominates over capillary force. Gravity-dominated et al., 1988) of free gravity drainage suggest that both ow

Journal of Petroleum Science and Engineering 62 (2008) 8086

Contents lists available at ScienceDirect

Journal of Petroleum Sci

.e lprocesses are often modeled experimentally and theoretically(Grattoni et al., 2001; Donato et al., 2006) as free gravity drainage,which implies that no external pressure is applied between the topand bottom of the reservoir. Capillary effects, wettability, interfacialtension may have important impact on gravity-drainage processesand performances (Schechter et al., 1994; Zhou and Blunt, 1998;Shahidzadeh-Bonna et al., 2003). However, in gravity-dominated

mechanisms are inuenced by the presence of dynamic wettinglms left behind the gas/oil interface. The ow behaviors are sensitiveto the geometry of the ow channels. In channels with circular cross-section shape, the liquid in the lm-owing regime exists as acontinuous lm along the wall. In channels with angular cross-sectional area, thewetting uid occupies the corners in addition to thewetting lm. Ransohoff and Radke (1988) numerically solved theconditions, naturally, gravity would be the mSeveral proposed predictive models (Card

Nenniger and Storrow, 1958; Dykstra, 1978; Pbeen found to be unsatisfactory (Schechpredicting the production response of a rese

Tel.: +61 8 9266 4994; fax: +61 8 9266 7063.E-mail address: Q.Xu@exchange.curtin.edu.au.

0920-4105/$ see front matter 2008 Published by Edoi:10.1016/j.petrol.2008.07.006ctice in a gas injectionow rate (pressure), such

two steps bulk liquid (oil) ow below the gas/liquid interface andlm/corner ow above the interface. Micromodel studies (Chatzisgravity-drainage process is to inject gas at ldrainage processes, maintaining gravity stable displacement mode isvital for success. Therefore, the general pra1. Introduction

Gravity drainage is an important oiassistant-gravity-drainage (SAGD) proet al., 2004; Nabipour et al., 2007) and(Jacquin et al., 1989; Nabipour et al., 20processes have been applied success(Bangla et al., 1993; Langeberg et al.,microscopic level with deterministic solutions. The modeling procedure and results are useful for modelingthe performance of gravity dominated improved oil recovery processes, ground water ltration, and watermovement in a CO2 plume in CO2 sequestration processes.

2008 Published by Elsevier B.V.

ry mechanism in steam-(Butler, 1998; Canbolat

n fractured oil reservoirsavity stable gas injectionn elds for many yearsIn gas injection gravity-

deterministic models, e.g., network models, may offer a way toimprove the modeling and predictive capability for gravity-drainageprocesses.

A network is comprised of many individual ow channels. Gravity-dominated uid ow in such channels is assumed for the entirenetwork. In oil reservoir, two-phase oilgas ow is much morecommon and is considered in this work. Oil is the denser and wettingphase and gas is the non-wetting phase in oilgas systems. Thedenser-phase free gravity drainage in a single channel takes place inresults of this study provide basis for more detailed network models, which describe rock-uid systems atangular channels.The uid distribution or theModeling unsteady-state gravity-driven

Josh-Qiang Xu CRC for Greenhouse Gas Technologies, Curtin University of Technology, GPO Box U1987, Pe

a b s t r a c ta r t i c l e i n f o

Article history:Received 1 March 2007Accepted 13 July 2008

Keyword:gravitydrainageuid owcapillary

This paper presents an inveGravity is important in manwater ow in subsurface ingenerally large, e.g., water vthese processes. In particulabulk ow (steady-state) acomplicated ow channelsauthor knows, this work m

j ourna l homepage: wwwost important factor.well and Parsons, 1949;avone et al., 1989) haveter and Guo, 1996) inrvoir. More detailed and

lsevier B.V.w in porous media

WA, 6845, Australia

ation of gravity-driven ow in porous media using angular capillary tubes.uid transport processes, such as ground water ow, oil ow in reservoir, and2 sequestration processes. In these processes, density contrast of the uids iss, or oil vs. gas. Gravity-driven ow of the denser uid largely takes control inravity-driven ow falls into two regimes in regarding to the denser uid: thethe lm/corner ow (unsteady-state) that follows. The geometrically

a porous medium are represented by shaped capillary tubes. As far as thels for the rst time the unsteady-state laminar ow of Newtonian uids in

l of the ow above the uid contact (Fig. 1) is governed by the uid dynamics.

ence and Engineering

sev ie r.com/ locate /pet ro lsteady-state corner owproblem, and dened resistance factors to theow for various corner congurations. Patzek and Kristensen (2001)investigated two-phase steady-state ow in polygon geometries bystudying the corner ow and proposed a universal curve for owconductance in the corners of an arbitrary angular capillary withvariable contact angles. Ransohoff et al. (1987) proposed corner owmodel based on the concept of hydraulic diameter and thin-lm owapproach. Zhou et al. (1997) improved the hydraulic diameter and

thin-lm ow approximation based on the concept that the ow isexplained by thin-lm theory with an effective lm thicknessobtained as the ratio of the ow area to the length of the non-owboundary. They concluded that the new solution is valid for a widerange of geometry and contact angles with error signicantly less than50%. The assumption in the previous works that the liquid distributionis uniform along the ow direction away from the advancing front(interface) may be accurate for capillary dominated displacement(small capillary number, ca=v/) as concluded by Bretherton (1961).

However, in gravity-dominated processes, the lm/corner ow isnot steady-state. In the gas injection gravity-drainage processes,unsaturated oil ow from fractures contributes to the production afterthe early stage. Such unsaturated unsteady-state lm ow has notbeen sufciently addressed previously. Bird et al. (1960) provided a

81J.-Q. Xu / Journal of Petroleum Science and Engineering 62 (2008) 8086solution for thin-lm (2D) ow using analytical approaches. Goodwinand Homsy (1991) presented a boundary integral technical to obtainnumerical solution to the 2D problem in which a contact angleboundary condition is imposed. For the rst time, this work modelsthe gravity-dominated unsteady-state laminar ow of Newtonianuids in angular channels using numerical and theoretical methods.

2. Methodology

As the rst step, the relevant steady-state ow problem is solved.Based on the steady-state solutions, the unsteady-state problem issolved analytically in the three dimensional space for region above theliquid/gas interface for ow in an angular geometry. Mass conserva-tion principle is invoked to obtain the governing partial differentialequation, which is then solved using the method of characteristics.Dimensionless form of the solution provided a convenient method forcalculating the production curve. In addition to the detailed derivationof the unsteady-state corner ow, it has been shown that the problemin other geometries can be solved in a similar manner.

2.1. Steady-state single-phase ow

As shown in Fig. (1), a constant ow potential gradient acts in thevertical direction, z, aligned with the direction of the ow. For single-phase liquid ow, obtaining the velocity prole, u(x,y), is the primarygoal in this part of the study.

The NavierStokes equation can be simplied in the cases underinvestigation. For a steady-state slow motion, the inertial term can beneglected and the ow is in the laminar regime. The uid is assumedto be Newtonian and incompressible. The NavierStokes equationreduces to a simple form,

j2u x; y 1ddz

1

where u(x,y) is the velocity in z direction; d/dz is the potentialgradient. In the case of gravitational force only, d/dz=g; is liquidFig. 1. Gravity drainage in an angular cross-sectional shaped tube.viscosity. For a particular uid system, the right-hand side term ofEq. (1) is a constant, thus Eq. (1) can be transformed into Laplace'sequation (Happel and Brenner, 1973).

Depending on boundary conditions and the geometry, manytechniques may be used to solve Eq. (1). Analytical solutions areavailable for circular, rectangular, and equal-lateral triangle shapes(Happel and Brenner, 1973). However, even for a case that analyticalsolution does exist, e.g., rectangular, the mathematical expression ofthe solution is so complicated that a numerical solution is still neededto evaluate that expression.

In this study, MATLAB pdetool (a nite element method) wasused to solve Eq. (1). For the corner ow problem (Fig. 2(a)), theparameters are: interfacial radius, a=0.001 m; half angle, =/6;density, =1000 kg/m3; gravity constant, g=9.81 m/s2; andviscosity, =0.001 Pa s, respectively. Two types of boundaryconditions are specied: the Dirichlet boundary condition (non-slip condition, zero velocity) at the walls and the Neumannboundary condition, zero normal derivative of velocity, on theinterfacial boundary, where liquid is adjacent to the gas phase.These boundary conditions are appropriate as concluded by Zhouet al. (1997) for corner ows. The solutions are achieved by step-wise mesh renement until the ow rate converges at apredetermined small error (less than 0.01% of the average velocity).Numerical integration of the velocity distribution over the owingarea gives the volumetric ow rate, q (m3/s),

q ZA

Zu x; y dxdy 2

this was done by mapping the nodal solution to a Cartesian gridsystem, because the velocity values are available at nodes of the niteelements. After obtaining q, the average velocity, denoted as, buN isobtained as,

hui q=A 3where A is the cross-sectional area. For a particular geometry shapeand boundary condition, buN is linearly proportional to the drivingforce in a laminar viscous Newtonian ow. For the circular-shape, wehave,

hui r2p8L

r2jp8

r2g8

4

which is the HagenPoiseuille equation (Bird et al., 1960). Eq. (4) statesthat the average velocity is linear to the pressure gradient. For othercross-sectional shapes, the Newtonian uid assumption leads tolinearity of the average velocity and pressure gradient as well,

hui a2pBL

a2jp

a2p

5

where a is a characteristic dimension, and is a constant representinga particular geometric shape. This parameter is similar to theresistance factor dened in Ransohoff and Radke's (1988) work. Weobtain =30.09 for the corner ow problem, which compares toRansohoff and Radke's (1988) result of =31.07.

Eq. (5) suggests that the average velocity is proportional to theinterfacial radius to the second power. Eq. (5) is sufcient in describingthe steady-state ow behaviors in shaped owing channels. However,for the unsteady-state ow, the volumetric ow rate q changes overtime, the equation is not applicable directly. Nevertheless, Eq. (5) isuseful in the unsteady-state modeling.

2.2. Unsteady-state lm ow

Many physical processes, including drainage ow in a duct, involve

dynamic lm drainage along a wall. In drainage in cylindrical

82 J.-Q. Xu / Journal of Petroleum Science and Engineering 62 (2008) 8086capillaries occupied by a non-wetting phase, wetting liquid owoccurs in a lm, too. In a gravity-drainage process, a lm (Fig. 3) isretained on the wall due to the non-slip liquidsolid interface as theliquid drains. The thickness of this liquid lm, (z,t), changes with time

Fig. 3. The lm drainage problem.

Fig. 2. FEA results: velocity prole in the corner (a=0.001 m; =/6, g=9810 Pa/m, (t) and position (z), it becomes thinner over time at any particularposition and it becomes thicker downwards as the uid drains down.

The drainage is a free-surface problem, which can be solved usingnumerical approaches, e.g., nite element method, nite boundaryelement method. Goodwin and Homsy (1991) addressed this problemwith a boundary integral technical. Bird et al. addressed this problemwith a theoretical approach. Excluding evaporation effect, massconservation law applies and can be used to derive a partialdifferential equation that governs the lm ow. Solution to theproblem is (Bird et al., 1960),

= g z=t 1=2 6the detailed derivation for Eq. (6) is omitted since a similar derivationfor corner ow is presented in next section.

2.3. Unsteady-state corner ow

In angular capillaries, the wetting phase occupies the cornerregions as shown in Fig. (4). Volume of liquid lm along the straightwall is much smaller compared to the volume of liquid in the corner.Therefore, the liquid is considered in the corners only (Fig. 4(a)). In theequal-lateral triangle shaped channel, ow in one corner representsthe ow in the entire channel.

=0.001 Pa s). (a) Geometry of the corner, a=0.001 m, =/6. (b) Velocity prole.

With this analytical solution, the thickness (indicated by theradius, a) of the liquid corner is known everywhere in all times.However, the solution is not derived and intended to apply for veryearly time, t0, when Eq. (10) blows up. Furthermore, in order toapply Eq. (10), we need to knowwhere the advancing interface is. Theadvancing velocity and shape of the meniscus is very complex in anon-circular ow channel (Wong et al., 1992). As far as we know, theadvancing velocity in non-circular tube has not been addressedsatisfactorily. It is assumed that the interfacial radius a is a0 (Fig. 4) atthe liquid/gas advancing front, in other words, the liquid occupies theentire corner but without merger with adjacent corners. Thisassumption is consistent with the assumption that the ow onlyoccurs at corners above the interface. With this, the interfaceadvancing velocity is,

v z=t 2ga20= 11

to obtain dimensionless form of Eq. (10), let aD=a/a0, zD=z/L, tD= t/t,where t=L/v, thus, we have,

aD zD=tD 1=2: 12

This dimensionless solution is plotted in Fig. 5(a). A value of zD/tD=1.0 represents the interface advancing front. This characteristiccorner drainage ow curve represents ow in a corner for any uidproperties.

83J.-Q. Xu / Journal of Petroleum Science and Engineering 62 (2008) 8086Initially, the angular capillary is lled completely with the wettingphase. Once the gravity-drainage process starts, the corner owregime is formed above the two-phase interface. We assume twoboundary conditions for this corner ow. One is that at the top of thecapillary tube, the lm thickness is zero instantaneously. The secondcondition is at the bottom of the corner, it is assumed that theinterfacial radius a is a0 (Fig. 4) at the liquid/gas advancing front. Inother words, the liquid occupies the entire corner but without mergerwith adjacent corners. With these assumptions, we start thederivation as following.

First, mass (volume) conservation law is used to derive (Appendix A)the governing partial differential equation Eq. (7) for the liquid columnalong the corner,

2hui a=z a hui=z 2 a=t 0 7using the steady-state solution Eq. (5) obtained from FEM numericalsimulation, which states,

hui ga2= 8in place of buN in Eq. (7), we have,

2ga2= a=z 2ga2= a=z 2=t 0combine the rst two terms,

2ga2= a=z a=t 0: 9

Eq. (9) is a rst order quasi-linear partial differential equation.Method of characteristics is used to solve Eq. (9) (Appendix A),

a = 2g z=t 1=2: 10

Fig. 4. The corner drainage problem: (a) Drainage in one corner. (b) Fluid distribution inthe cross-sectional area of the ow channel.Fig. 5. (a) Characteristic drainage curve for corner ow. (b) Recovery versus dimensionless

time in an equal-lateral triangle shaped tube.

84 J.-Q. Xu / Journal of Petroleum Science and Engineering 62 (2008) 8086From Eq. (12), it is easy to compute the fractional recovery (as afraction of initial uid/oil in place, IOIP) in an equal-lateral triangleshaped channel for tDb1,

Np Z tD0

A3c aD 2

d zD =Z 10

A dzD 13

where A is the dimensionless cross-sectional area of the equal-lateraltriangle (inner circle radius aD=1), equals 331/2; c(aD)2 is the liquidoccupied area in one corner at a particular value of zD; the constant, c,equals (31/2/3) for a half angle of /6. With these values andEq. (13),

Np0:80tD: 14

For tDN1, we have,

Np Z 10

A3c aD 2

d zD =Z 10

A d zD

Np10:2= tD 1=2: 15

Plot Eqs. (14) and (15) in Fig. 5(b) for a full range of tD, thebreakthrough oil production is about 0.80 IOIP, and the productionincreases after breakthrough at a slower rate as described in Eq. (15).Results show a low retention feature of the equal-lateral triangleowing channel, indicating high efciency of gravity-drainageprocess.

3. Discussions

3.1. Apply the analytical approach to other geometries

The above approach can be extended to other geometries. Forexample, in a circular-shaped tube, the uid forms a uniform-thickness lm in a cross-section. The relation Eq. (5) of averagevelocity of the liquid lm and the lm thickness can be obtainedtheoretically or numerically, and it can then be applied in thederivation to obtain the partial differential equation. For othergeometries such as square, since the corner (90) does not restrictthe ows as much as the corner (60) in an equal-lateral triangle, thelm along the wall may be more signicant and need to be taken intoaccount. Thus, the corner ow results would not be applicable directly.A combination of corner and lm ow pattern has to be considered.With lm and corner ow problem solved in this study, thecombination of lm and corner ow can also be solved in principle.The methodology developed in this work may be incorporatedinto network models for more complicated modeling of ow inporous media.

3.2. Capillary force effects

The relative strength of capillary force and gravitational force isoften represented with the dimensionless Bond number, Nb. For owin a capillary tube, a Bond number,

Nb gLr= cos 16

is considered appropriate. In Eq. (16), is the density difference; L isthe length and r is the radius of the tube; is the interfacial tension;and is contact angle. The larger the Bond number is, the less theeffect of the capillary force is. For the case of ow in equal-lateraltriangle shaped tube studied (Fig. 4), assuming g=9810 Pa/m,L=1 m, taking r=0.001 m, =0.025 N/m, and =0 (perfect wetting),then Nb=392.4, indicating the dominance of gravity force.

Based on force balance (Fig. 1) of the liquid in capillary tubes,

the potential gradient is comprised of gravity term and capillaryterm, which can be treated as an upwards force acting at theinterface,

jW G Lz pc= Lz 17jW gpc= Lz : 18

It is clear that the effect of capillary force is relatively small in theearly stage when z is small, thus ow rate is nearly constant asconrmed in experiments (Blunt et al., 1995), until (Lz) becomessufciently small. Previous studies on the interface advances based onforce balances have been conducted and are referred as WRL theory(Van Remoortere and Joos, 1991). From the force balance analyses, aslong as Pc is small or Lz is larger, the effect of capillary force to theow can be neglected.

The effect of capillary force is limited in the dynamics of drainageowin conditions where Bond number is large, for instance, in relatively longor larger channels, such as in a fracture. In addition, in conditions wherethe interfacial tension is low, capillary force is insignicant anyway. Thus,the uid dynamics approach is appropriate for ow in conditions withlarge Bond number and low interfacial tension conditions.

4. Conclusions

With the results and the discussion, we conclude,

4.1 FEA numerical method is used in solving velocity prole forsteady-state ow in a corner geometry.

4.2 Ananalytical approach is developed for theunsteady-state cornerow problem and lm ow problem. The analytical approach forthe triangular geometry can be extended to other geometries.

4.3 Results of uid production (oil recovery) from a single capillarytube can represent the trend of oil recovery process in a porousmedia. The high recovery efciency of gravity-drainage pro-cesses is demonstrated.

NomenclatureConsistent (e.g., SI) units are assumed in all equations in the text.

a interfacial radius in the corner, mc constant, dimensionlessD subscript, dimensionlessg gravity constant, m/s2

L length of the ow channel, mNp dimensionless productionNb Bond number, dimensionlesst time, sbuN average velocity, m/sv interface advancing velocity, m/sx, y, z coordinate, m

Greek symbols half angle of a corner shape factor lm thickness, m viscosity, Pa s interfacial tension, N/m characteristic function density, kg/m3

uid potential, Pa

Appendix A. Derivation of equations for the unsteady-statecorner ow

In angular capillaries, most of the wetting phase ow occurs in

the corner regions as show in Fig. (4). First, use mass (volume)

85J.-Q. Xu / Journal of Petroleum Science and Engineering 62 (2008) 8086conservation to derive the governing partial differential equation forthe liquid column along the corner (Fig. 4(a)). At a particular time andposition, tN0 and zN0, take a control volume with thickness of z asthe shell element and t as time increment. The in-ow to this controlvolume at z is,

huizca2z t

where buNz is the average velocity over the area at z; c=(1/tan()/2+),is a constant only related to the half angle ; az is the interfacialradius; caz2 is the cross-sectional area of the liquid occupied corner; tis a small time interval. Similarly, the out-ow at position z+z, is,

huizzca2zz t

and the accumulation within this shell element is,

ca2ttca2t

z

where cat+t2 and cat2 are the average cross-sectional area at time t+tand t. Using mass (volume) conservation principle, we have,

huizca2z thuizzca2zz t ca2ttca2t

z A:1

the constant c is cancelled out from Eq. (A.1). Then use the relations

azz az a=z z A:2huizz huiz hui=z z; A:3

to substitute buNz+z and az+z in the second term in Eq. (A.1), itbecomes:

huiz hui=z z

az a=z z 2 t

expand the above term, we have,

huiza2z huiz2az a=z z huiz a=z z 2 hui=z za2z hui=z z 2az a=z z hui=z z a=z z 2 t

asz0, neglect the three higher order terms, drop the subscript, z, theleft side of Eq. (A.1) becomes,

2huia a=z a2 hui=z z t:Similarly, the right-hand side,

a2tta2t

z

a2t 2

a=t att a=t t2a2t

z

2a a=t tzthus, Eq. (A.1) becomes,

2huia a=z a2 hui=z 2a a=t 0 A:4using the steady-state laminar lm drainage results Eq. (5) obtainedfrom FEM numerical simulation, which states,

buN=ga2/() in place of buN in Eq. (A.4), we have,

2ga3= a=z 2ga3= a=z 2a =t 0 A:5

combine the rst two terms and cancel-out 2a,

2ga2= a=z a=t 0: A:6

Eq. (A.6) is a rst order quasi-linear partial differential equation.

Method of characteristics is used to solve Eq. (A.6). Suppose for a (z,t)=0, da/d=0 holds, i.e., (z,t) represents characteristic curves on which adoes not change. We have,

da=d a=t dt=d a=z dz=d 0 A:7

comparing Eqs. (A.6) and (A.7) leads to the following two ordinarydifferential equations,

dt=d A:8dz=dt 2ga2= A:9

combine Eqs. (A.8) and (A.9),

dz=dt 2ga2= A:10for a xed a (on a character line), perform integration on Eq. (A.10),

z 2ga2= t z0 A:11apply the boundary condition,

a 0; z 0; t N0 A:12we get, z0=0. Thus, from Eq. (A.11),

a = 2g z=t 1=2: A:13

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86 J.-Q. Xu / Journal of Petroleum Science and Engineering 62 (2008) 8086

Modeling unsteady-state gravity-driven flow in porous mediaIntroductionMethodologySteady-state single-phase flowUnsteady-state film flowUnsteady-state corner flow

DiscussionsApply the analytical approach to other geometriesCapillary force effects

ConclusionsNomenclatureDerivation of equations for the unsteady-state corner flowReferences