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DOI 10.1007/s11242-005-0616-xTransport in Porous Media (2006) 62: 125–138 © Springer 2006
Modeling the Formation of Fluid BanksDuring Counter-Current Flow in Porous Media
G. LI1, Z. T. KARPYN2, P. M. HALLECK2 and A. S. GRADER2,�
1University of Tulsa, Tulsa, Oklahoma, U.S.A.2The Pennsylvania State University, State College, Pennsylvania, U.S.A.
(Received: 22 October 2003; accepted in final form: 4 January 2005)
Abstract. Fluid banks sometimes form during gravity-driven counter-current flow in cer-tain natural reservoir processes. Prediction of flow performance in such systems dependson our understanding of the bank-formation process. Traditional modeling methods usinga single capillary pressure curve based on a final saturation distribution have success-fully simulated counter-current flow without fluid banks. However, it has been difficult tosimulate counter-current flow with fluid banks. In this paper, we describe the successfulsaturation-history-dependent modeling of counter-current flow experiments that result influid banks. The method used to simulate the experiments takes into account hysteresisin capillary pressure and relative permeabilities. Each spatial element in the model fol-lows a distinct trajectory on the capillary pressure versus saturation map, which consistsof the capillary hysteresis loop and the associated capillary pressure scanning curves. Thenew modeling method successfully captured the formation of the fluid banks observed inthe experiments, including their development with time. Results show that bank forma-tion is favored where the pc-versus-saturation slope is low. Experiments documented inthe literature that exhibited formation of fluid banks were also successfully simulated.
Key words: counter-current flow, hysteresis, scanning curves, fluid bank, capillary pressuretrajectory.
1. Introduction
Gravity-driven counter-current flow occurs in many reservoir processes. Forinstance, during gas storage in an aquifer shown in Figure 1, gas per-colates through high permeability layers first and then exchanges placeswith water by counter-current flow. The mechanistic understanding of thiscounter-current flow process will help reservoir engineers to estimate howlong it takes to form gas bubble, which, in turn, help plan efficient gasstorage operations. Briggs and Katz (1966) performed experiments and sim-ulations to study the importance of counter-current flow on bubble forma-tion in this type of gas sntorage. The experiments showed the formation
�Author for correspondence: e-mail: [email protected]
126 G. LI ET AL.
Injection well
(a) EARLY STAGE
(b) LATER STAGE
(c) LATER STAGE WITH GAS BUBBLES
W ATER
GAS
Figure 1. Development of gas bubbles in an aquifer (from Briggs et al., 1966).
of a fluid bank during counter-current flow, which would slow down gasmigration to the top structure of the reservoir. Their simulation with onlya single imbibition capillary pressure curve failed to reproduce the bank.
Barbu et al. (1999) and Karpyn (2001) performed similar counter-currentflow experiments with an X-ray CT scanner to monitor saturation distri-butions. Karpyn’s (2001) experiments showed a fully developed fluid bankand a shock, but Barbu et al. (1999) experiments did not. In this paper, afluid bank is defined as a high fluid saturation region similar to a cloudof accumulated mass. A shock is a sharp change in fluid saturation alongthe direction of flow. Fluid shocks described in this study always form atthe bottom of the sample and move in the vertical direction. Al-Wadahiet al. (2000) successfully simulated a portion of Barbu et al.’s (1999) exper-iments with an S-shaped capillary pressure curve. A similar S-shaped cap-illary pressure curve was applied to Karpyn’s experiments, but it failed toreproduce the fluid bank observed in the experiments. S-shaped capillarypressure curves are not intuitive and fail to work in all situations. Thus,better understanding of counter-current flow is clearly needed.
The purpose of this paper is to provide a mechanistic approach tosimulate the dependence of fluid banks during counter-current flow processes.The proposed approach, called saturation-history-dependent modeling,
MODELING THE FORMATION OF FLUID BANKS 127
takes into account hysteresis in capillary pressure and relative permeabilities.The experiments of Karpyn (2001) and of Briggs and Katz (1966)have been successfully simulated by this method, thus providing physicalinsight into the formation of fluid banks and shocks during gravity-drivencounter-current flow.
2. Hysteresis of Parameters Controlling Counter-Current Flow
In the counter-current flow experiments (Briggs and Katz, 1966; Barbuet al., 1999; Karpyn, 2001), both drainage and imbibition flow regimeswere observed simultaneously. Therefore, two pairs of relative permeabilitycurves should be used in the simulation along with both drainage andimbibition capillary pressure curves. However, if both drainage and imbi-bition capillary pressure curves are used, discontinuities in both saturationprofiles and capillary pressures result at the boundary between the drainageand imbibition flow regimes. The S-shaped capillary pressure curve usedby Al-Wadahi et al. (2000) is a combination of the drainage and imbibi-tion capillary pressure curves with a transition zone. It eliminated the dis-continuities but the representation is not physically sound and can only beused to simulate experiments without fluid banks. In reality, elements in thesample with different local saturation history follow different capillary pres-sure (scanning) curves according to capillary pressure hysteresis that will bediscussed in the following.
Capillary pressure hysteresis is characterized by a closed loop, consist-ing of the bounding drainage and imbibition capillary pressure curves, andtransition scanning curves. When the initial conditions are with irreducibleor maximum saturation, capillary pressure follows one of the boundingcurves. When the direction of saturation change reverses, then the capillarypressure follows along a scanning curve starting from the point at whichthe reversal occurred (Morrow, 1970). Any specific point within a hysteresisloop can thus be reached by many different saturation paths. The path bywhich a saturation is attained at any location has to be specified in orderto obtain a complete description of the subsequent displacement process(Morrow, 1970).
Figure 2 gives the capillary hysteresis loop and some scanning curves,which show the dependence of the future behavior on the saturation his-tory. A drainage process initiated from point O can follow a scanningpath OCA, or ODA to approach the bounding drainage capillary pressurecurve, depending on the saturation history. The path OCA is a continua-tion of a drainage process that started at point E, while the path ODAis a drainage process that started at point F. These different paths willresult in different future flow behaviors. An imbibition process initiated atpoint O may follow various paths depending on the scanning curves it
128 G. LI ET AL.
B
A
C
p c
snw 1.0-swrsnwr0 1.0
OD
Drainage
ImbibitionE
F
Figure 2. Schematic of a capillary hysteresis loop and scanning curves.
followed to arrive at the saturation point. Therefore, to correctly modeloil recovery processes, capillary pressure hysteresis has to be taken intoconsideration with a specified saturation history for each gridblock. Thedevelopment of such a model, called history-dependent modeling, is givenby Li et al. (2005). In this modeling method, different capillary pressure(scanning) curves are assigned to different locations according to the ini-tial saturation and saturation history, and different relative permeabilitycurves are assigned to the different flow regimes. In the experiment stud-ied, the upper part of the sample is a drainage flow regime and is assignedwith drainage-scanning curves and a pair of drainage relative permeabilitycurves. The lower part of the sample is an imbibition flow regime and isassigned with imbibition-scanning curves and a pair of imbibition relativepermeability curves.
3. Application of History-Dependent Modeling to Developmentof Fluid Banks
We have applied history-dependent modeling to Karpyn’s (2001)counter-current flow experiment to reproduce the observed fluid bankand shock. In the history-dependent-modeling method, a distinct capillarypressure curve is assigned to each gridblock in the simulator accordingto its saturation history, and two pairs of relative permeability curves areused. The relative permeability curves and the capillary hysteresis boundingcurves (drainage and imbibition) were acquired for this specific experi-ment using a history-matching procedure. The capillary pressure scanningcurves are generated by a combination of two mathematical representationmethods developed by Killough (1976) and Kleppe et al. (1997).
MODELING THE FORMATION OF FLUID BANKS 129
Karpyn (2001) performed counter-current flow experiments in a100×100×10-mm X-ray-transparent cell filled with glass beads. The threeimmiscible fluids used in the experiment were distilled water, benzylalcohol, and decane, representing water, oil, and gas, respectively. Satura-tion distributions as a function of time and space were monitored using amedical CT scanner. Benzyl alcohol and decane were simultaneously injectedat irreducible water saturation until a homogeneous distribution of phaseswas obtained. Fluid injection took place in the vertical direction. The inletports were placed at the top of the cell, and the outlet ports at the bottom.Then, the cell was closed and two different stages of counter-current flowwere studied: stages 1 and 2.
In stage 1, the counter-current segregation process started from a uni-form initial saturation distribution. At the end of this stage, the heavywetting fluid, benzyl alcohol concentrated at the bottom of the samplewhile the light nonwetting fluid, decane stayed at the top of the sample.In stage 2, the sample was rotated 180 ◦ around the horizontal axis to ini-tiate another counter-current segregation process that started from a non-uniform initial saturation distribution.
History matching was applied to stage 2 of the experiment to extractthe relative permeabilities and capillary hysteresis bounding curves. Stage2 of the experiment is the focus of this study. The parameters obtainedfrom stage 2 were then applied to stage 1 to verify the accuracy of parame-ter calculation. The matches of decane saturation profiles for both stages 1and 2 are shown in Figure 3. In Figure 3, solid lines represent the experi-mental data and dashed lines represent the simulation results. During earlytime of stage 2, there is a region of maximum benzyl alcohol saturation(around 80 mm along the saturation profile), defined in this study as abenzyl alcohol bank. The bank dissipates as gas migrates towards the topof the core. At late time, benzyl alcohol starts to accumulate at the bot-tom of the core forming a decane shock, with a near-stair-step saturationprofile. The benzyl alcohol bank and the decane shock are well captured bysimulation. Figure 4 shows the drainage and imbibition relative permeabil-ity curves. The solid curves are the imbibition relative permeabilities usedfor the lower part of the sample. The dashed curves are the drainage rela-tive permeabilities used for the upper part of the sample.
Figure 5 shows the capillary hysteresis loop together with the scan-ning curves used for modeling. The steady-state injection of benzyl alcoholand decane followed the drainage capillary pressure curve A ◦CODB upto point O, where decane saturation is about 0.4. Then the sample wasisolated, marking the beginning of stage 1, and counter-current segrega-tion occurred. In stages 1 and 2, decane saturation increased at the upperregion of the sample, decreased at the lower region and remained the sameat the middle. In stage 1, the top of the sample continued to follow the
130 G. LI ET AL.
0.1 0.2 0.3 0.4 0.5 0.6 0.70
2
4
6
8
10
Decane saturation
eV
rtic
alop
siti
no(x
0 1m
m)
Finalprofile
Initialprofile
ExperimentSimulation
O
0.1 0.2 0.3 0.4 0.5 0.6 0.70
2
4
6
8
10
Decane saturationV
etric
alop
siit
no(1
x0m
m)
Initialprofile
Finalprofile
Experiment
Simulation
bank
Decane shock
O
(a) stage 1 (b) stage 2
Figure 3. Saturation profile match for Karpyn’s experiment (BA).
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Decane saturation
taleR
ive
Pe
mribae
tily
BA
Decane
DrainageImbibition
Figure 4. Relative permeability curves.
drainage capillary pressure curve ODB, and the lower part followed animbibition-scanning curve OEA. But in stage 2, the sample had differentinitial saturations at different locations, and the local saturation historieswere heterogeneous as well. Thus after rotating, each portion of the samplefollowed a different scanning curve according to the existing saturationvalue and saturation history.
The history-dependent modeling successfully reproduced the spatial andtemporal saturation distribution of Karpyn’s (2001) experiment. The fluid bankand shock are well captured in the simulation. The parameters obtained from
MODELING THE FORMATION OF FLUID BANKS 131
0 0.2 0.4 0.6 0.8 16
7
8
9
10
11
Decane saturation
Cllipa
pyra
rerus se
1 (0-3
tm a)
Drainage
Imbibition
O
B
A
C
D
E
A
Figure 5. Capillary pressure curves.
stage 2 using history-matching method are confirmed to be correct by predict-ing the behavior of stage 1. In the following, application of history-dependentmodeling to literature experiments with fluid banks is discussed.
Briggs and Katz (1966) did similar counter-current flow experiments andtwo of them (Experiments 3 and 4) were chosen for investigation. Thesample was made of glass beads with 762 mm in height and 51 mm indiameter. Water and air were used in both experiments. The saturationswere determined by measuring, at various positions and times, the radio-activity of the water, which contained an I131 tracer. The procedure of theexperiments was as follows: the sample was vacuum-saturated with waterto certain saturation, then the system was closed and rotated by 180 ◦ toinitiate the counter-current flow process.
The saturation-history-dependent modeling was applied to these twoexperiments. Figures 6a and b show the match of saturation profiles forBriggs and Katz’s (1966) experiments. The dotted lines with circle symbolsrepresent the experimental data, and the dashed lines represent history-dependent simulation. The fluid banks are well captured in the simula-tion of both experiments. Briggs and Katz (1966) attempted to simulateexperiment 4 with a single imbibition capillary pressure curve and theirsimulation did not capture the fluid bank (solid line in Figure 6b).
4. Discussion and Analysis
4.1. capillary pressure trajectories and pc front
The reason why history-dependent modeling yields satisfactory matches isthat it includes hysteresis in capillary pressure and relative permeabilities.
132 G. LI ET AL.
Ver
tical
op si
( noitx
01m 0
m)
0 10
8
Sw
0 10
8
Sw
19 MIN
60 MIN
History dependent simulationExperiment
0 10
9
Sw
0 10
9
Sw
60 MIN
150 MIN
History dependent simulation
Ve
tric
laP
ositi
x( n o)
mm 001
ExperimentBriggs and Katz simulation
(a) Experiment 3. (b) Experiment 4.
Figure 6. Saturation profile match in Briggs and Katz’s experiment.
Flow in different regions is controlled by the corresponding relativepermeabilities according to the ongoing process (drainage or imbibition).Different elements in the system follow different capillary pressure curvesaccording to their saturation and history. We define the paths they followas capillary pressure trajectories. The front of these trajectories is called pc
front, which is the locus of instantaneous capillary pressure values alongtheir respective trajectories.
Figures 7–10 show the saturation profiles including the initial and finalconditions with the corresponding modeled capillary pressure trajectories(thin dotted line) and the pc front (thick dashed line) at different timesof stage 2. The lettered points along the saturation profiles in Figures 7a,8a, 9a, and 10a are reflected on the pc front in Figures 7b, 8b, 9b, and10b. Decane saturation at point O stays constant during the process. Abovepoint O it is a drainage process with increasing decane saturation andbelow point O it is an imbibition process with decreasing decane satu-ration. The initial saturation profile between points B and C (Figure 7a)shows an almost constant saturation at the top of the sample. Thisregion of the sample follows the same capillary scanning curve during thedrainage process in stage 2, because they have the same initial saturationand saturation history.
MODELING THE FORMATION OF FLUID BANKS 133
(a) saturation profiles (b) capillary pressure trajectories
0 0.2 0.4 0.6 0.80
2
4
6
8
10
Decane saturation
reV
tplacio
tisi
(no
xm
01m
)
Initialprofile
Finalprofile
AB
E
O
C
D
0 0.2 0.4 0.6 0.8 16
7
8
9
10
11
Decane saturation
sserpyrallipa
Cu
er(1
03-
ta)
m
Drainage
ImbibitionA O
C
E
BD
Figure 7. Vertical saturation profiles with the corresponding capillary trajectoriesand the pc front at t =0.6 h.
(a) saturation profiles (b) capillary pressure trajectories
0 0.2 0.4 0.6 0.80
2
4
6
8
10
Decane saturation
Ver
tciopla
tisi
no(x
10m
m)
Initialprofile
Finalprofile
A
C
O
D
B
E0 0.2 0.4 0.6 0.8 1
6
7
8
9
10
11
Decane saturation
rallipaC
yrp
russee
(10
3-at
m)
Drainage
Imbibition
A
CE
O
D
B
Figure 8. Vertical saturation profiles with the corresponding capillary trajectoriesand the pc front at t =3.0 h.
Right after the segregation process started, decane saturation sharplyincreased in region AB but it remained the same at portion BC (Figure 7a).Figure 7b shows the capillary pressure trajectories (thin dotted line)that segregation in stage 2 followed up to 0.6 h. The capillary pressuretrajectories to the left of point O represent the drainage processes andthe ones to the right of point O represent the imbibition processes.
134 G. LI ET AL.
(a) saturation profiles (b) capillary pressure trajectories
0 0.2 0.4 0.6 0.80
2
4
6
8
10
Decane saturation
reV
ttisopl aci
(no i
xm
01m
)
Initialprofile
Finalprofile
A
E
O
C
D
B
0 0.2 0.4 0.6 0.8 16
7
8
9
10
11
Decane saturation
rallipaC
yrp
r us see
(10
3-)
mc
Drainage
Imbibition
A
E
OC
D
B
Figure 9. Vertical saturation profiles with the corresponding capillary trajectoriesand the pc front at t =6.0 h.
(a) saturation profiles (b) capillary pressure trajectories
0 0.2 0.4 0.6 0.80
2
4
6
8
10
Decane saturation
reV
ttisoplaci
(noi
xm
01m
)
Initialprofile
Finalprofile
O
A
E
D
B
C
0 0.2 0.4 0.6 0.8 16
7
8
9
10
11
Decane saturation
rallipaC
yrp
sseur
e(1
03-
tam
)
Drainage
Imbibition
A
E
O
D
B
C
Figure 10. Vertical saturation profiles with the corresponding capillary trajectoriesand the pc front at t =89.0 h.
Point A followed the first trajectory on the left, because it had the low-est initial saturation. Points B and C followed the same trajectory (thesecond one on the left). The saturation at point E (Figure 7a) did notchange, and the corresponding capillary pressure still stayed on the bound-ing drainage capillary pressure curve (Figure 7b). Points in region OEfollow the imbibition-scanning curves with decreasing decane saturation.The dashed pc front shows the progress made along the trajectories after
MODELING THE FORMATION OF FLUID BANKS 135
0.6 h of segregation. The AB portion of the pc front corresponds to theupper wing of the developing bank also denoted as AB in saturation pro-files (Figure 7a).
At t = 3.0 h (Figure 8), the drainage capillary trajectories continuedto climb towards the bounding drainage capillary pressure curve and theimbibition trajectories moved towards the bounding imbibition capillarypressure curve. The pc front formed an oval shape ABCO with ABC cor-responding to the upper wing of the bank (Figure 8b). At 6.0 h, a decaneshock had formed at the bottom of the sample along DE of the saturationprofile in Figure 9a, which corresponds to the low slope on the pc front inFigure 9b (portion DE).
Since the final saturation profile of the system comes back to the initialcondition, another 180 ◦ rotation will result in the same flow process as inthis stage. The decane shock at the bottom of the saturation profile is fullydeveloped at the end of the process (Figure 10).
The pc front shown in Figures 7b, 8b, 9b, and 10b are displayedtogether in Figure 11, demonstrating how the capillary pressures migratewithin the capillary hysteresis loop. The drainage part (on the left) migratesupward and the imbibition part (on the right) migrates downward. Thefinal and initial conditions overlap.
0 0.2 0.4 0.6 0.8 16
7
8
9
10
11
Decane saturation
Ca
ipll
rap yre
ussre
( 10
3-ta m
)
Drainage
Imbibition
1(6)2
1(6)
3 45
23
45
Figure 11. The pc front at different times: 1. initial condition; 2. t =0.2 h; 3. t =0.6 h;4. t =1.0 h; 5. t =3.0 h; 6. t =89.0 h.
136 G. LI ET AL.
4.2. formation of fluid banks and shocks
Fluid banks were observed in Briggs and Katz’s (1966) and Karpyn’s(2001) experiments. In Karpyn’s experiments, there was also a decane shockformed at the late time at the bottom of the sample. Banks and shocksonly occurred in regions with high wetting phase saturation (Figures 3band 6). The shape of the capillary pressure curve plays an importantrole on the formation of fluid banks and shocks. The positions of thebank correspond to the low slope portions of the pc front (Figure 8).In stage 2 of Karpyn’s experiment, the upper part follows a family ofdrainage-scanning curves. These curves sharply rise and then flatten toasymptotically approach the bounding drainage capillary pressure curve.Therefore, in a short time, these points approach a flat zone of the capillarypressure curve (Figure 9). In the flat zone, the capillary force does not havemuch impact on the flow process. The same reason holds for the formationof decane shock at the late time of the same experiment as the lower partof the sample enters a flat zone on the imbibition capillary pressure curve.This phenomenon is consistent with the observation that without capillaryforce, the gravity force tends to segregate fluids with or without a smalltransition zone (i.e. a sharp front).
The initiation and development of the fluid bank and shock can alsobe shown with the vertical capillary pressure profile along the sample(Figure 12). At 0.6 h and 3.0 h, the upper wing of the bank AB(Figures 7a and 8a) is also shown in the capillary pressure profile inFigure 12. The shock formation at the bottom of the sample at late timeof stage 2 (Figure 10a) corresponds to the almost vertical line portion (lowcapillary pressure gradient in space) in the capillary pressure profile belowpoint O at 6.0 h and 89.0 h in Figure 12.
6 7 8 9 100
2
4
6
8
10
Capillary pressure (10-3 atm)
reV
ticla
sop iti
( nox
01m
m)
0.6 3.0 6.00 89 hrsAB
C
D
E
O
Figure 12. Vertical capillary pressure profiles.
MODELING THE FORMATION OF FLUID BANKS 137
In both Briggs and Katz’s (1966) and Karpyn’s (2001) experiments, glassbead packs were used. The homogeneous nature of the beads creates aporous medium that has portions of the capillary pressure curves that areflat – the small changes in capillary pressure are associated with largesaturation changes. Thus the low capillary pressure gradient is similar tothe behavior of capillary tube models. In a natural rock, the pore space ismuch more heterogeneous than in glass beads. Therefore, the gradient ofthe capillary pressure curves is steeper than in the case of glass beads, min-imizing the tendency to form fluid banks.
Briggs and Katz (1966) pointed out that no ‘bulge’ or fluid bank wouldform if the initial saturation were uniformly distributed, such as at thebeginning of stage 1 in Karpyn’s experiment. This conclusion was madebased on the experimental observation from Templeton et al. (1962), butthis may not be true if the saturations are located in the flat zone on thecapillary pressure curve. In stage 1 of Karpyn’s experiment, fluid migra-tion tended to form a decane shock at the bottom of the sample (Fig-ure 3a) because of the flat capillary pressure curve at low decane saturation(Figure 5).
During counter-current flow, fluid banks form after local fluid exchangeand they do not move but dissipate. During co-current flow, fluid banks(fronts) are due to fluid displacement and move along the displacementdirection. A fluid phase may also accumulate and buildup a bank due todifferent flowing velocity. In this case, the formation of the bank is mainlycaused by external force (injection) and the gravity and capillary forcestend to dissipate the bank. In counter-current flow, fluid banks are causedby the action of gravity and capillarity.
5. Conclusions
The following conclusions can be drawn from this research:
(1) Counter-current flow can be correctly modeled only if local satura-tion history and capillary pressure hysteresis are taken into account.In the history-dependent-modeling method, drainage and imbibitionregions use different relative permeabilities and each gridblock repre-senting the sample is assigned a distinct capillary pressure curve basedon its current saturation and saturation history. It is a general methodthat can be applied to cases with and without fluid banks. However, thesingle pc method can only be applied to cases without fluid banks.
(2) Flat portion of the capillary pressure curve assists in the formation offluid banks and shocks. In the case of counter-current flow, the forma-tion of a bank at the top of the sample depends on the shape of thedrainage capillary pressure curve and the associated scanning curves.
138 G. LI ET AL.
The formation of a shock at the bottom of the sample depends on theimbibition capillary pressure curve and the associated scanning curves.
(3) During the counter-current flow processes, sample elements followdifferent capillary pressure trajectories that depend on the initialsaturation and saturation history. The connection of the end of thetrajectories forms a pc front. The formation of fluid banks and shocksare related to the shape of the pc front.
References
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