hysteresis in three-phase flow: experiments, modeling and reservoir...

Copyright 2000, Society of Petroleum Engineers Inc.

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AbstractA proper modeling of tertiary recovery processes such as gasinjection or WAG (Water Alternating Gas) requires anadequate three-phase flow model. This allows to better predictthe recovery efficiency, gas storage reservoir performance aswell as the well injectivity.

For gas drainage, a previous paper [25] presented a new three-phase flow model based on a theoretical analysis and validatedthrough experimental approach. For WAG injection, there isan additional complexity due to the need to model theimbibition that occurs when gas saturation decreases. Totackle the modeling of hysteresis problem, a comprehensiveapproach was followed. First, successive drainage andimbibition experiments were conducted under variousconditions of initial saturations. A new three-phase modeltaking into account the hysteresis is presented and validated onthe experiments.

Indeed, as shown in previous experimental studies, hysteresiswas found to depend not only on the drainage/imbibitionprocess (saturation history) but also on the cycle considered(displacement history) where cycle names the association oftwo consecutive displacements (drainage and imbibition). Inthis study, a relevant analytical expression of the hysteresis isproposed avoiding any negative effect of numericalinstabilities. The new formulation was implemented in areservoir simulator and WAG experiments have beensuccessfully simulated. The impact on breakthrough time,overall recovery efficiency was tested through large scalereservoir simulations.

IntroductionWhatever the nature of a field, hydrocarbon or undergroundgas storage, exploitation often leads to large sweeping of thereservoir by fluids. For hydrocarbon reservoir, it can happennaturally when aquifer support is strong or when gas capexpands downwards. Most of the time, artificial pressuremaintenance is needed and gas or (and) water are injecteddepending on reservoir properties and fluid availability. Foralmost two decades, WAG (Water Alternating Gas) injectionstrategies have been developed to improve sweep efficiency atboth macroscopic and microscopic scales [1,2]. A kind ofWAG also occurs when old waterflooded reservoirs areconverted into gas storage. Alternated sweeping of gas andwater results from annual cycle of pressurization anddepressurization [3]. These examples illustrate that large partsof a reservoir can be subject to successive drainage andimbibition. This generates hysteresis on relative permeabilitiesthat must be considered to make numerical simulations fullyrepresentative.

Hysteresis on relative permeabilities has been experimentallyevidenced with various measurement methods in two-phaseflow [4,5,6,7,8,9] and more seldom in three-phase flow[10,11,12]. These studies show that relative permeabilities arenot only functions of saturations but also depend on thesaturation history and especially their sense of variation. Thiskind of hysteresis is related to a strong decrease of the non-wetting phase mobility (referred from now as gas) duringimbibition. Complexity is added in three-phase flow sincerelative permeabilities are found to depend also on cyclehistory if cycle denotes the association of two successivedisplacements (drainage and imbibition). This kind ofhysteresis is mainly related to significant gas relativepermeability reduction along the cycles [11,13].

The Land's formula is largely used whatever the modelingapproach considered [14]. It consists of linking initial gassaturation with residual gas saturation after imbibition andassuming that the relation remains valid along a scanningcurve between the trapped gas saturation (Sgt) and the free gassaturation (Sgf). Knowing that Sg is equal to the sum of Sgt andSgf, it enables to derive Sgf at any saturation. Distinct

SPE 65127

Hysteresis in Three-Phase Flow: Experiments, Modeling and Reservoir SimulationsP. Egermann*, O. Vizika*, L. Dallet**, C. Requin** and F. Sonier***

* IFP: Institut Français du Pétrole, ** GDF: Gaz De France ,*** SMC: Simulation & Modelling Consultancy

2 P. EGERMANN, O. VIZIKA, L. DALLET, C. REQUIN AND F. SONIER SPE 65127

approaches have been considered to include hysteresis effectin relative permeabilities:

Analytical expressions of relative permeabilities derivedfrom porous medium description are modified thanks toLand trapping formula in order to calculate hystereticexpressions where only mobile fluids are considered [14].

Killough method relies on interpolation formula betweeninput bounding curves to calculate scanning curves.Weighting coefficients are function of Sgt and curvature ofscanning curves can be adjusted by a parameter [15].

Carlson method uses only drainage curve with Landformula to deduce bounding and scanning imbibitioncurves. He stated that imbibition gas relative permeabilityat Sg is equal to drainage gas relative permeability at Sgf.Then, all imbibition curves are found paralell [16].

In numerical simulation with three-phase flow, hysteresis ismost of the time treated by combining two-phase model ofKillough or Carlson with one version of Stone's model (1 or 2)[17,18]. This approach has been used to simulate WAGexperiments [19,20,2]. If this is efficient to take into accounttrapping impact with three phases, it fails reproducing cyclehysteresis, as scanning curves are reversible. A moresuccessful three-phase hysteretic model is proposed by Skaugeet al [13]. In drainage, empirical expression includingreduction factor function of Sw is used and cycle associatedimbibition is calculated with Carlson like method. Waterrelative permeability is found by interpolation between twoinput bounding curves. Oil relative permeability results fromStone 1 method where residual oil saturation is implementedas a function of Sgt, which enables to reach values beyond Sorg.

This paper proposes another methodology to tackle theproblem of hysteresis with three phases. Specific experiments,consisting of successive gas and water injections at ambientconditions, are first presented. Corresponding relativepermeabilities, derived by history matching, are discussed interms of hysteresis. This enables to propose a newmethodology in the modeling part. Cycle hysteresis on relativepermeabilities is seen as a consequence of hysteresis betweentrapping and untrapping phenomena. Hence, Land's formula isused with distinct constants depending on the displacementconsidered and the history. The methodology is applied withfractal relative permeability expression to derive hystereticexpressions. Implementation of the new model in a reservoircode is presented in the last section. Validation examples onlaboratory experiments are provided and sensitivity study on across section is also added to evaluate impact at a large scale.

ExperimentsRock/fluids systemAll the experiments were conducted on a Estaillades

limestone. Petrophysical properties of the core are gathered inTable 1. Although none specific wettability test has beenperformed, the core is supposed to be strongly water-wet:

Two-phase oil/water relative permeability curves showthat water permeability is low at residual oil saturation

and the crossing point of the two curves is higher than 0.5(Figure 1).

Naturally, this rock is water-wet and oil never stayed incontact with rock for a long time, which excludessignificant wettability alteration.

Main specificity of Estaillades limestone is to exhibitbimodal porosity:

Vugs sized between 100 and 200 µm in the lowestcemented parts.

Smaller pores ranged between 1 and 50 µm.These confer to the pore size distribution a typical shape withtwo slopes in a Log-Log diagram (Figure 2).

The liquids used are Soltrol 170 as oleic phase, brine 30 g/lNaCl as aqueous phase and Nitrogen as gas phase. Fluidproperties are detailed in Table 2. From the correspondingvalues of IFT, the spreading coefficient S,

S wg wo og= − +γ γ γ( ) ,

which denotes the ability of oil to spread on water in presenceof gas, is found positive, equal to 4.8 mN/m. It shows that filmflow of oil is favored by the experimental context.

Experimental set-upThe apparatus is represented on

Sketch 1. At the inlet, it is possible either to inject gas at afixed pressure through a regulator or to inject brine at a fixedrate through a pump. Pressure drop along the core is recordedby a differential sensor. At the outlet, fluids are collected intoa separator. Oil and water productions are measured directlyby reading evolution of the menisci positions. Two devices areused for the gas phase:

When gas just breaks through or when water is injected,production is low and a sensitive device is needed. Hence,the gas line is connected directly to a second upside downseparator initially filled with water. Raw measurementsare first corrected because pressure is lower in the upperpart of the second separator, which gives the totalproduction. Gas production is derived by subtractingliquid productions recorded independently in the firstseparator. Further in the experiment, this device was alsoused to perform gas rate measurement during gasinjections. At the outlet, gas line was opened to thesecond separator for a given time and correspondingproduction was measured.

When gas rates were higher direct measurement was donewith a Labflow. The working range of this equipmentgoes from one to 500 cc/mn.

Experimental procedureThree-phase hysteresis experiments

Injections of gas and water were successively performed intothe core. Each flooding was pursued after the BT of theinjecting phase to insure that the displacement mode (drainageor imbibition) is the same in the whole core. This radicallydiffers from WAG injection experiment where several slugs of

SPE 65127 HYSTERESIS IN THREE-PHASE FLOW: EXPERIMENTS, MODELING AND RESERVOIR SIMULATIONS 3

each phase coexist into the porous medium and makeinterpretation difficult in terms of hysteresis.

Several initial saturation conditions were considered fromirreducible water saturation to residual oil saturation.Establishment of these initial conditions was achieved:

By displacement of one phase until production ceased, toreach end points saturations.

By a steady state injection with adjustment of flow ratesof each phase, to reach the expected intermediatesaturation along the core.

Whatever the initial state, sequence of injections was thesame:

Gaz injection at 500 mb (D1) (except TRI2 at 700 mb bysteps)

Brine injection at 10 cc/h (I1) Gaz injection at 500 mb (D2) (except TRI2 at 700 mb by

steps) Brine injection at 10 cc/h (I2)

D and I respectively denote Drainage and Imbibition whereasthe number indicates the chronology according to thedisplacement mode.

Land's constant experimentReview of the literature shows that the Land's formula is acorner stone in practically all the published modelingapproaches of relative permeability taking into accounthysteresis. Then, complementary experiments were performedwith only gas and water to collect information about Land'sconstant in Estaillades limestone.

In those experiments, core is first fully saturated with brine.Then, gas is injected at a fixed pressure until productionstabilizes. Finally, imbibition is performed with brine toresidual gas saturation (Sgr). Experiments were realized withvarious gas injection pressures so that a large range of gassaturation (Sgi) was reached before imbibition.

ResultsExperimental curves

Four three-phase experiments were conducted as detailed inTable 3. Figure 3 and 4 show evolution of experimental curvesfor the tertiary case TRI2 during the two first injections:

D1: initially, only water is produced but production of anoil bank was observed when gas breaks through atroughly 2000 seconds. It suggests a reconnection ofresidual oil by gas phase. Gas rate progressively increasesafter the BT as liquid saturations decrease (the jumpobserved at 9000 seconds corresponds to an increase ofinjection pressure from 450 to 700 mb, see Table 3).

I1: only gas is produced initially but production of an oilbank was also observed just before the water breaksthrough suggesting reconnection process due to waterinvasion. Pressure drop curve is very typical with amaximum value. This phenomenon is related to the oilbank mobilization. When injection starts, pressureincreases because water replaces gas. As oil mobilizes,

pressure drop due to the bank becomes higher and higher.Although oil bank extension is limited, contribution onthe pressure signal is significant because oil viscosity ishigh. When oil production begins, gas production stopsand oil is integrally replaced by water of lower viscosity,which makes the pressure drop decrease.

Saturation pathwaysFigure 7 represents evolution of saturation pathways. It showsthat the four pathways converge to the same area during D1injection. Oil production was observed whatever the initialsaturation. Oil also was produced during I1 in all theexperiments. For D2 and I2 injection phases, saturationpathways remain parallel to the gas/water axis showing that oilsaturation remains stable.

Relative permeability determinationMethodology is already described elsewhere [21]. It relies onhistory matching of experimental curves with results ofnumerical simulations. A reservoir code adapted to laboratoryconditions was used (ATHOS). Relative permeabilities areintroduced as tables functions of two saturations (Sg, Sw).Values are adjusted by trial and error method until a goodagreement is reached. Examples of fitting are provided onFigure 3, 4, 5 and 6.

KrwFrom Figure 8, no hysteresis can be detected for the mostwetting phase. This result is in contradiction with publishedworks on three-phase hysteresis [11]. Main difference betweenthe two experimental approaches is operational conditions. Inreference [11], experiments were conducted at 100 barswhereas our apparatus works at ambient conditions. It issuspected that pressure increase due to water injectioncompresses the gas located in the upstream part of the core,which tends to hide hysteresis influence. However, it impliesthat the wetting phase is not strongly affected by saturationhistory as it is observed in two-phase flow.

KrgHysteresis effect is very important on the non-wetting phase asshown on Figure 9, 10 and 11. During the two first injectionsD1 and I1, a classical cycle of hysteresis is observed with anincrease of Krg during drainage and a strong decrease duringimbibition. No reversibility is observed during the secondhysteresis cycle (D2 and I2). Krg follows the same generaltrend with an increase associated to the drainage and adecrease to the imbibition, but the whole curves are shifted onthe left (high gas saturation). This shift suggests a kind ofoffset related to the trapped gas saturation. At equal gassaturation, Krg is lower with cycle chronology making gasmobility decrease. This behavior is also observed in reference[13]. It was observed in all the cases whatever the initialsaturations and shows that hysteresis in three-phase isparticular because it is sensitive to:

The nature of the displacement considered (drainage orimbibition), which means that Krg depends on saturationhistory (as in two-phase flow).


The chronology of the cycle considered if a cyclerepresents the association of two consecutivedisplacements (drainage and imbibition). This hysteresisis proper to three-phase and means that the processdepends on cycle history.

KroAs shown on saturation pathways, oil saturation variations arenot significant enough along the cycles to put on lighthysteresis effect on the oil relative permeability. Nevertheless,impact of hysteresis is important on residual oil saturationevolution. In tertiary conditions (TRI2), oil production wasobserved during each injection step of the first hysteresis cyclemaking residual oil saturation decrease. Hence, hysteresiseffect contributes to lower oil saturation in the porousmedium.

LandHysteresis constant of Land is derived as demonstrated inFigure 12, where Sgr is plotted as a function of Sgi. A manualfitting with analytical expression [14],

Lgigr

CS

1

S

1 =− ,

gives the best value CL equal to 0.8. This relatively low valuesuggests that trapping effect is important in Estaillades. This isqualitatively in good agreement with general trend (hightrapping in carbonate rocks [22]).

ModelingBackground on the fractal modelThis approach is fully described elsewhere [23,24,25] and wassuccessfully used to model three-phase relative permeabilitiesobtained during gas drainage experiments on water-wet[26,25] and intermediate-wet samples [27]. Principle relies ona fractal pore picture derived from capillary pressure curve.Fluids are assumed to flow in concentric layers within thesame fractal pore and phases are distributed according to theirwettability (water on the pore wall, gas in the bulk of the poreand oil sandwiched). Relative permeability expressions areobtained by adding contribution of each capillary occupied bya phase with Poiseuille's law. From its nature, the modelenables to take into account a realistic fluid distribution withinthe pore structure as imposed by the solid-liquid interactions(wettability) and the liquid-liquid interactions (spreadingconditions).

Fundamental expressions are reminded hereafter: From the correlation :

P Sc w

1

D 2L∝ −

where Sw is the saturation of the wetting phase, a fractallinear dimension DL can be deduced from the slope of thecapillary pressure curve obtained by mercury intrusion.

ββwiwrw SSK −= where

L

L

D2

D4

−−

=β

+−= ββ )S(SSKK orwL(2Ph.)roro

In these expressions, the irreducible water saturation, Swi isassumed to be immobile. Sor is a part of the residual oil Sorw

that corresponds to a given water saturation Sw, where Sorw isthe maximum residual oil saturation left in place by awaterflooding. In the following, Sor will be denoted by(Sor)Sgt=0 to avoid confusion with general definition of residualoil with hysteresis (notation from reference [13]). Because gas is the non-wetting phase, it occupies the porousspace starting from the bulk of the pores.

( )K K Srg rg L= −max. 14α

LD−

=2

1α , where DL is the linear fractal dimension of the

porous medium and SL is the total liquid saturation.

Hysteresis formulationMain features of three-phase hysteresis are related to the cycledependence. It confers to Krg curves a particular evolutionalong the cycles. Gas mobility is progressively reduced atequal gas saturation because curves are shifted towards thehigh gas saturation level. If two successive cycles could bededuced by simple translation, it would mean that the trappedgas after imbibition is not remobilized and does not participatein gas flow during the following drainage. Actually, this is notexactly the case, but it indicates that the hysteresis of Krg

between I1 and D2 can be attributed to hysteresis in terms oftrapped gas saturation. Hence, hysteresis on Krg can be entirelyexplained by non-reversibility between trapping anduntrapping phenomena.In the modeling approaches, Land's formula is used tocalculate free gas saturation during a displacement on ascanning curve to estimate effective permeability of gas. AsSgf decreases during imbibition, it leads to permeabilityreduction. Land's constant is set unique and only dependent onthe porous medium. This implies that Sgf calculation is thesame under imbibition and secondary drainage and thenconducts to a reversibility of the scanning permeability curves.If this concept fits requirements of two-phase flow, it fails indescribing cycle hysteresis. Hence, it is proposed to conserveLand's formulation but to use it differently. Distinct constantsare used depending on the displacement mode and also thehistory. This enables to restitute directly hysteresis betweentrapping and untrapping of the non-wetting phase.

The model proposed hereafter provides Land's constant in anykind of situations whatever the displacement nature andchronology. In the following, CL, CT and CU denotesrespectively the Land, trapping and untrapping constants. Themodel must fulfil several conditions to be in accordance withexperimental behavior:

CT is equal to CL in all the trapping phase (imbibition)because it corresponds exactly to the conditions whereLand's constant can be used straightforward.


When Sgr is low, it is expected that CU is close to CL

because little gas is trapped in the biggest pores andmakes untrapping easiest.

When Sgr increases, untrapping is harder making CU

becoming higher than CP. When a large fraction of gas has been already trapped

(high Sgt), it is supposed that a kind of reversibility isreached. This is attributed to the remaining presence ofinaccessible trapped gas saturation in the smallest poresoccupied by gas.

An empirical expression of CU was built to fit the aboverequirements:

LLUM2gr

gt

1Drg

minrgrgU C)CC(

S

S

K

KKC +−

−=

λ

where: CUM denotes the maximum value of the untrapping

constant. Krgmin is low mobility curve. KD1

rg is the high mobility curve (first drainage). Sgr2 is the maximum trapped gas saturation. λ is an empirical coefficient to play on CU evolution.

First term in CU expression is calculated at gas saturationcorresponding to the beginning of the previous imbibitionwhereas the second term is calculated at the beginning of thedrainage followed. Figure 13 shows general view of elements,which are included in the model. From its expression CU iscompletely displacement dependent. CU passes through amaximum as suggested by requirements. Position andamplitude of this maximum can be easily adjusted with λ andCUM.

Residual oil saturationPrevious works [28,13] showed that low residual oil saturationcan be obtained in presence of trapped gas. It was proved thatgeneral evolution of Sor can be reasonably estimated with:

gtsgtoror aSSS −= =0)(

In this expression, reduction of oil saturation is directly linkedto the trapped gas through the coefficient a. This behavior isphysically explained in terms of apparent saturation. One partof the trapped gas saturation is seen as an oil phase and makesproduction possible even beyond (Sor)Sgt=0. This explanationalso suggests that the other part of the trapped gas rathercontributes to increase water saturation.

If we consider the experiment TRI1, it comes a constant aequal to 0.45 which is in good agreement with default rangegiven in [13]: between 0.3 and 0.5.

Relative permeability expressionsIn this part, water-wet relative permeability expressions aremodified to take into account trapped gas saturation.

KrgThis is the easiest expression to modify as free gas flows in thecenter of the fractal pore. Apparent saturation of liquid phaseis SL+Sgt instead of SL. It comes the general expression,

4max ))(1( α

gtLrghystrg SSKK +−=

As )S(K ghystrg =Krg(Sgf), we obtain a Carlson like result but

hysteresis is fully taken into account through calculation of thetrapped gas saturation. Synthetic evolution of Krg along WAGinjection cycles is given in Figure 14.

KroOil relative permeability is also deduced by apparentsaturation consideration. As seen above, SL becomes SL+Sgt

and Sw becomes Sw+(1-a)Sgt. Then,]))()1(()[(2 0

ββ=+−+−+= sgtorgtwgtLroro SSaSSSPhKK

This expression insures that Kro is equal to zero when Sor isreached.

KrwIf one part of trapped gas contributes to make water apparentsaturation higher, its presence impacts flow behavior of thewetting phase [11]. In the following expression, it is supposedthat flowing Sw is deprived from a quantity proportional to thetrapped gas saturation related to water.

ββwigtwrw SSaRSK −−−= ))1((

R is a permeability reduction coefficient.

Numerical simulationsModel implantationThe three-phase relative permeability fractal model coupledwith hysteresis model has been implanted in the GENESYS©reservoir simulator developed by SMC consulting company[29,30]. This implantation has been achieved in a way thatallows the new fractal and hysteresis modules to be easilyconnectable to other commercial reservoir softwares.Two numerical schemes are available allowing solvingequations in pressure and saturation with explicit schemes ormixed implicit and explicit schemes. The choice of the schemeis depending of the complexity of the case to solve and itsresulting stability.

Validation on experimental dataSimulation of TRI2

Estaillades core has been meshed using a regular gridding of100 meshes in the main axis. All the petrophysicalcharacteristics of rock have been implemented and keptconstant in each cell. The two first injection periods of theexperiments held in initial tertiary conditions (TRI2) havebeen simulated with the new methodology. Comparison withexperimental data is provided on Figure 15.

Very good agreement is reached for liquid production curvesin both injection sequences. Amplitude and production time ofoil bank mobilization, observed during D1 and I1, isparticularly well simulated. It suggests that the code succeedsin mobilizing oil at low saturation so that small residual value


can be reached as observed experimentally. Water productionis also correctly predicted even if roughly 1 cc difference isobserved with experimental data at the end of D1. Gas rate isless efficiently matched especially when pressure wasincreased up to 700 mb. Nevertheless, first part of the curve isin good accordance and breakthrough time is found close toexperimental observation.

Comparison with Stone I modelThe same simulation has been performed using Stone I model(Figure 16). A matching of the same quality of the experimentwas impossible to be reproduced with that model. Thisexperiment starts with tertiary condition where oil saturation isclose to Sorw. Stone I model can make residual oil saturationdecreasing down to Sorg but not further down.

Fontainebleau Sandstone core simulationAn experiment of WAG made on a sandstone core [20] hasbeen simulated. The core is a composite sample made withtwo samples whose two permeabilities are 100mD and 163mD. The porosity is respectively 10% and 11.16%. Thisexperiment was performed in reservoir conditions (200 bars80°C) with a core in tertiary condition. 14 slugs of water andgas were injected within a 83000 seconds experiment. Slugsize is equal to 3% of the pore volume.

Water, oil and gas productions are fairly well matched asshown on Figure 17. Gas and oil breakthroughs are also ingood agreement with experimental observations. A morepronounced scatter appears between experimental oil recoveryand the simulated one but it has to be mentioned that total oilrecovery is very low (around 4 cm3). The difference betweenresults of the simulation and the experiment reaches amaximal value of less than 1 cm3, which is the same order ofthe experimental uncertainty. As a consequence, it is notpossible to make a fair judgement on the quality of thematching of the experiment dealing with oil phase.Nevertheless, the simulation of this WAG experiment givespromising results dealing with water and gas recoveries.Hysteresis on gas phase is a prevalent phenomenon in a coreWAG experiment. The good agreement on productionrecoveries tends to prove that the trapping and untrappingphenomena are fairly well modeled.

Large scale simulationWAG on a cross section

A cross section has been extracted from the depleted oil fieldSaint Martin de Bossenay bought by Gaz De France a fewyears ago in order to convert it into a gas storage. A WAGphenomenological study has been performed on that crosssection.This study aims at testing the stability of the numerical modelwith up to 50 alternate cycles but also at making varioussensitivity studies on petrophysical parameters, in order tomeasure their influence on the extent of the three-phase zonein the reservoir. Another interest is to check out impact of thenew model in comparison with classical approaches (Stone,…)

in order to observe if the differences observed at the core scaleare still present at the reservoir scale.

One injector well has been placed in the center of the structureand two producers are located at each side of the cross section(Figure 18). The injector alternatively injects gas at the bottomof the perforations and water at the top of them in order todelay partition of the phases due to gravity.The previous model composed of 5 horizontal layers has beenrefined into 9 layers. The total cross section has 720 meshes.A horizontal barrier exists modeled by layer 5 on the right sideof the cross section between the injector and the producer 2.The average porosity is close to 10% and average absolutepermeability is about 100 mD. All the main characteristics aresummarized in Table 4.

Oil in the reservoir is undersaturated and pressure is keptabove the bubble point during whole the production. The firstpart of the production is performed through natural depletionduring only 100 days in order to prevent reservoir pressuredecreasing below bubble point. Then, a waterflooding isapplied during 2400 days, followed by an enhanced oilrecovery with water alternate gas injection to 8100 days.

Stability of the modelThe model presents a very good stability and a very short CPUtime decreased down to 15 minutes for the whole cross sectionsimulation. The IMPES scheme allows long time step duringthe simulation without any instability.

Sensibility studyKv/Kh sensibility

In the simulations shown on Figure 19, different values ofKv/Kh from 0.01 to 1 are considered. Evolution ofpermeability ratio is obtained by progressively decreasingvalue of vertical permeability. As seen on Figure 19,significant differences are observed according to thepermeability ratio value. The best recovery is obtained withthe lower ratio. In this case, segregation between the phases isreduced which makes three-phase area, just near the WAGinjector well, become larger. Extension of this three-phasearea improves recovery, as low residual oil saturation can bereached inside. Moreover, three-phase area tends to moderategas segregation through cycle hysteresis. It has been observedthat the lower the Kv/Kh ratio, the sooner is the waterbreakthrough during waterflooding stage and the latter is thegas breakthrough during WAG flooding.

Comparison with different recovery schemesDifferent injection schemes have been simulated andcompared to the WAG base case (slug of 3% PV) on Figure20. Water flooding gives recovery results very close to WAGrecoveries. This result is not surprising due to St Martin deBossenay field petrophysical properties. First, Sorg estimatedfrom laboratory experiments is very close to Sorw and as aconsequence gas displacement can only provide a very lowincremental recovery. The second reason is the ratio Kv/Kh


equal to 0.1. As a consequence, the three-phase zone has notan extension important enough close to WAG injector. WAGinjection is also simulated with higher rates (5% PV) andlarger slugs (7% PV). It gives incremental oil production butno significant influence of the slug size has been observed.This certainly results from the restricted extension of thethree-phase area, which hides effect at the reservoir scale.

Comparison between different modelsIn this last part, influence of the three-phase permeabilitymodel on the oil recovery is explored. Hysteresis is introducedprogressively with only saturation history dependence andthen with complete model. Run made with no hysteresis at allpredicts the lowest recovery. This is similar to results found inreference [13] and is explained by absence of gas trapping,which does not enhance oil recovery. Complete hysteresismodel gives lower recovery than basic model with no cycledependence. This may result from overall cross sectionmorphology. Anyway, it shows that differences exist betweenthe two cases and that three-phase hysteresis particularity hasto be fully taken into account to run more representativesimulations.

ConclusionsA new analytical model for three-phase relative permeabilityhysteresis has been presented. It is based on an existing three-phase model for drainage conditions and on experimentalobservations revealing the particular behavior of hysteresis inthree-phase flow context. Indeed the new model takes intoaccount drainage/imbibition hysteresis and cycle hysteresiswhich is characteristic of three-phase flow.

The main results of the present work can be summarized asfollows:

In three-phase flow, strong hysteresis has been observedfor the most non-wetting phase (gas). Hysteresis of thewater relative permeability is by far less important.

Two hysteresis types have been observed: a mechanism(drainage/imbibition) and a cycle hysteresis (history).Cycle hysteresis is attributed to the differences betweengas trapping and untrapping phenomena.

The fractal pore model, used successfully in prediction ofthree-phase relative permeabilities for gas injectionprocesses, has been extended here to take into accountthree-phase hysteresis. The Land’s approach has beenused for drainage/imbibition hysteresis. For the cyclehysteresis, different Land’s constants were introducedwhich depend on the history.

The model implemented in a reservoir simulator has beenvalidated on a WAG injection experiment for which allthree production curves are reproduced rathersuccessfully. Compared to the standard models it turns tobe much better, by efficiently describing the incrementaloil recovery at successive water and gas injections.

Numerical simulations in a cross section showedsignificant differences compared to the classical approach.

They demonstrated that it is of prime importance to takeinto account complete three-phase hysteresis in order topredict correctly WAG efficiency. They also confirmedthat in large scale Kv/Kh is an important factor on theextent of the three-phase zone, which in turn influencesthe WAG scheme overall efficiency.

AcknowledgementsThis work was partially funded by FSH (Fonds de Soutien desHydrocarbures)The authors thank M. Renard who performed the experiments.

Nomenclature a = residual oil saturation reduction factorBT = breakthroughCL = Land's constantCT = Trapping constant (related to imbibition)CU = Untrapping constant (related to drainage)CUM = Maximum value of untrapping constantD1 = primary drainageD2 = secondary drainageDL = fractal dimensionK = permeability (m2)Kri = relative permeability of fluid i

hystriK = hysteretic relative permeability of fluid i

Krg max = maximum gas relative permeabilityKro(2Ph) = oil relative permeability from water/oil

imbibition testIrgK = value of Krg at the beginning of the

previous imbibition1D

rgK = primary drainage value of Krg at the gas

saturation corresponding to IrgK

minrgK = low mobility curve

Pc = capillary pressure (N/m2)PV = pore volume (m3)S = spreading coefficient of oil on waterSo = oil saturationSg = gas saturationSw = water saturationSL = liquid saturation (So+Sw)Swi = irreducible water saturationSgr = residual gas saturationSgi = initial gas saturation before imbibitionSgt = trapped gas saturationSgf = free gas saturationSorg = residual oil saturation after gasfloodingSorw = residual oil saturation after waterflooding(Sor)Sgt=0 = residual oil saturation before gas trappingSor = residual oil saturationα = 1/(2-DL)β = (4-DL)/(2-DL) exponentγij = interfacial tension (mN.m-1)ij = gw, go, ow


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4 Braun E.M., Holland R.F. :’’ Relative permeabilityhysteresis: laboratory measurements and a conceptualmodel’’, SPERE Aug 1995, 222.

5 Geffen T.M. et al :’’Experimental investigation of factorsaffecting laboratory relative permeability measurements’’,Trans AIME 1951, Vol 192, 99.

6 Osaba J.S. et al :’’ Laboratory measurements of relativepermeability’’, Trans AIME 1951, Vol 192, 47.

7 Land C.S. :’’ Comparison of calculated with experimentalimbibition relative permeability’’, Trans AIME, Dec1971, Vol 251, 419.

8 Delaplace P., Lenormand R. :"Etude de l'hysteresis de Pc-Kr en diphasique gaz/huile: expériences et modélisation",IFP report n°43198, Sept 1996.

9 Colonna J. , Brissaud F. , Millet J.L. :’’ Evolution ofcapillary and relative permeability hysteresis’’, TransAIME 1972, Vol 253, 28.

10 Eleri O. O., Graue A., Skauge A., Larsen J. A. :’’Calculation of three-phase relative permeabilities fromdisplacement experiments with measurements of in-situsaturation’’, SCA 9509, San Francisco, 12-14 Sept 1995.

11 Skauge A., Larsen J.A. :’’Three-phase relativepermeabilities and trapped gas measurements related toWAG processes’’, SCA 9421, Stavanger, 12-14 Sept1994.

12 Skauge A., Aarra M. :"Effect of wettability on the oilrecovery by WAG", 7th IOR symp, Moscow, 26-28 Oct1993.

13 Larsen J.A. , Skauge A. :’’ Methodology for numericalsimulation with cycle-dependent relative permeability’’,SPEJ, June 1998.

14 Land C.S. :’’ Calculation of imbibition relativepermeability for two and three phase flow from rockproperties’’, Trans AIME 1968, Vol 243, 149.

15 Killough J.E. :’’ Reservoir simulation with hystory-dependent saturation functions’’, SPEJ Feb 1976, TransAIME 261.

16 Carlson F.M. : ’’ Simulation of relative permeabilityhysteresis to the non wetting phase’’, SPE 10157, ATCE,San Antonio Texas, 4-7 Oct 1981.

17 Stone H.L. :"Probability model for estimating three-phaserelative permeability", JPT, vol 22, pp214-218, 1970.

18 Stone H.L. :"Estimation of three-phase relativepermeability and residual oil data", JCPT, vol 12, 4, pp53-61, 1973.

19 Morel D. , Latil M. :’’ Basic study of sweep efficiencyimprovement by water alternate gas injection’’, Hamburg,27-29 Oct 1987.

20 Minssieux L. , Duquerroix J-P. :’’ WAG flow mechanismin presence of residual oil’’, SPE 28623, 69th ATCE, NewOrleans, 25-28 Sept 1994.

21 Moulu, J-C., Kalaydjian, F. and Martin, J-M. : "Performance and Numerical Interpretation of GasDrainage Core Tests under Secondary and TertiaryConditions", Paper SCA 9508, presented at the SCASymposium, San Francisco, Sept. 12-14, 1995.

22 Irwin D.D., Batycky J.P. :"The successive displacementprocess: oil recovery during blowdown", SPERE, Nov1997.

23 Lenormand, R. “Gravity-assisted inert gas injection:micromodel experiments and model based on fractalroughness”, The European Oil and Gas Conference,Altavilla Milica, Palermo, Sicily, October 9-12 (1990).

24 Vizika, O. “Effect of the Spreading Coefficient on theEfficiency of Oil Recovery with Gravity Drainage”,Symposium on Enhanced Oil Recovery, 205th NationalMeeting of ACS, Denver CO, March 28 - April 2, 1993.

25 Moulu J-C., Vizika O., Kalaydjian F., Duquerroix J-P :’’A new model for three phase relative permeabilities basedon a fractal representation of the porous medium’’, SPE38891, Annual Technical Conference and Exhibition, Oct5-8, 1997, San Antonio.

26 Kalaydjian, F.J-M., Moulu, J-C., Vizika, O. andMunkerud, P-K.: "Three-phase flow in water-wet porousmedia : determination of gas/oil relative permeabilitiesunder various spreading conditions", Paper SPE 26671,presented at the 1993 SPE Annual Technical Conferenceand Exhibition, Houston,Texas, Oct. 3-6.

27 Moulu J-C., Vizika O., Egermann P., Kalaydjian F. :"Anew three-phase permeability model for variouswettability conditions", ATCE, Houston, 3-6 Oct 1999.


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29 Eymard,R. and Sonier,F.: " Mathematical and NumericalProperties of Control-Volume Finite-Element Scheme forReservoir Simulation". SPE 25267, Presented at the12th SPE Symposium on Reservoir Simulation, March1993. Publication: SPERE ( Nov. 1994 ) 263-289.

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Kw (mD) φ Diameter (cm) Length (cm) PV (cm3)Estaillades core 215 0.23 4 26.15 76.6

Table 1: Estaillades carbonate core properties

Density (kg/m3) Viscosity (cp) @ std IFT (mN/m)Soltrol 70 770 2.76 γ ow = 40 3.Brine 30 g/l 1019 0.93 γ wg = 72 1.

Nitrogen 1.16 0.018 γ og = 27

Table 2: Fluids properties

Nom Nature Injection conditions Initial stateSw Sg

Final stateSw Sg

TRI1Secondary

D1I1D2I2

500 mb10 cc/h500 mb10 cc/h

0.49 00.46 0.260.61 0.180.56 0.25

0.46 0.260.61 0.180.56 0.250.59 0.22

TRI2Tertiary

D1I1D2I2

700 mb by steps10 cc/h700 mb by steps10 cc/h

0.69 00.47 0.290.58 0.240.48 0.34

0.47 0.290.58 0.240.48 0.340.58 0.24

TRI3Intermediate

D1I1D2I2

500 mb10 cc/h500 mb10 cc/h

0.61 00.49 0.260.61 0.200.53 0.30

0.49 0.260.61 0.200.53 0.300.61 0.20

TRI4Intermediate

D1I1D2I2

500 mb10 cc/h500 mb10 cc/h

0.58 00.48 0.250.57 0.210.51 0.29

0.48 0.250.57 0.210.51 0.290.57 0.24

Table 3: Three-phase hysteresis experiments

Reservoir Pressure 146Average Porosity 10%Average absolute Permeability 100 mDSorw 28-30%Swi 40-45%

Table 4: Cross section characteristics

Labflow

Gas separator

Liquidseparator

Pressureregulator

Nitrogen

Core

Pump

Manometer

Sketch 1: experimental apparatus


Figure 1: oil/water relative permeability

Figure 2: Estaillades pore size distribution

Figure 3: TRI2 experimental results (1)




0

5

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30

0 5000 10000 15000 20000Time (s)

Prod

uctio

n cc

0

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sure

mb

Water experiment

Oil experimentWater simulated

Oil simulatedGas experiment

Gas simulatedDP experiment

DP simulated

D1

I1

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uctio

n cc

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Water simulated

Oil simulated

Gas simulated

DP experiment

Dp simulated

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0 2000 4000 6000 8000 10000 12000

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cc/

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Gas catcher

Labflow

Gas rate simulated

BT exp

0

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0 2000 4000 6000 8000Time (s)

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rate

cc/

mn

Gas rate experiment

Gas simulated

0

0.2

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1

0.3 0.4 0.5 0.6 0.7 0.8Sw

Kr

Oil

Water

100

1000

10000

100000

0.1 1

Swetting

1/r

(cm

-1)


Figure 7: Saturation pathways

Figure 8: TRI2 Krw evolution

Figure 9: TRI1 Krg evolution



Figure 12: Land constant determination

TRI2

TRI1

TRI4

TRI3

OilWater

Gas

0,2

0,2

0,2

0,4

0,4

0,4

0,6

0,6

0,6

0,8

0,8

0,8

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.4 0.5 0.6 0.7 0.8

Sw

Krw

D1

I1

D2

I2

0

0.02

0.04

0.06

0.08

0.1

0 0.1 0.2 0.3Sg

Krg

D1

I1

D2

I2

0

0.02

0.04

0.06

0.08

0 0.05 0.1 0.15 0.2 0.25 0.3Sg

Krg

D1

I1

D2

I2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.1 0.2 0.3 0.4

Sg

Krg

D1

I1

D2

I2

0.2

0.24

0.28

0.32

0.36

0.4

0.25 0.35 0.45 0.55 0.65Sgi

Sgr

Experiment

Land: C=0.8


Krg

Sgr1 Sgr2 SgM1 SgM2

Extra trapped gas

High mobilityCurveKD1

rg

KMrgmin

Low mobilityCurveKrg min

Locus of maxima

I1

Figure 13: elements of the three-phase hysteresis model

Figure 14: Evolution of Krg along a WAG injection

Figure 15:Validation on TRI2

Figure 16: Comparison with Stone 1 method

Figure 17: Simulation of WAG experiment

Producer 1WAG

injector

Producer 2

Figure 18: Example of gas saturation distribution

0

0 .0 5

0 .1

0 .1 5

0 .2

0 .2 5

0 .3

0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6Sg

Krg

H ig h m o bility

L o w m o bility0

10

20

30

40

50

60

70

0 20000 40000 60000 80000

Time seconds

oil a

nd w

ater

rec

over

y cc

0

1

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Gas

rec

over

y cc

Simulated oilSimulated waterExperimental oilExperimental waterSimulated gasExperimental gas

0

5

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30

0 5000 10000 15000 20000

Time seconds

Liq

uid

prod

ucti

on c

c

0.0

0.1

0.2

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0.4

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0.6

Gas

rat

e cc

/s

Experimental waterExperimental oilSimulated waterSimulated oilExperimental gas rateSimulated gas

0

5

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0 5000 10000 15000 20000 25000

Time seconds

Liq

uid

prod

ucti

on c

c

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Gas

rat

e cc

/s

wateroilSimulated waterSimulated oil

gas rateSimulated gas rate


Figure 19: Influence of Kv/Kh ratio

Figure 20: Influence of recovery scheme

Figure 21: Influence of the model

0

100

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800

0 2000 4000 6000 8000 10000

Time days

Oil

Pro

duct

ion

Mm

3

kv/kh=1

kv/kh=0,1

kv/kh=0,01

0

100

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0 2000 4000 6000 8000 10000

Time days

Oil

prod

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m3

Waterflooding

Water + gas injection

Reference WAG

WAG high rates

WAG high rates + large slugs

0

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0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Time days

Oil

pro

duct

ion

Mm

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Stone I model

No hysteresis

hysteresis between drainage and imbibition

Complete hysteresis

hysteresis in three-phase flow: experiments, modeling and reservoir...

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