SPE 152218

Three-Phase Relative Permeability and Hystresis Model for Simulation of Water Alternating Gas (WAG) Injection H. Shahverdi, M. Sohrabi, Centre for Enhanced Oil Recovery and CO2 Solutions, Heriot-Watt University

Copyright 2012, Society of Petroleum Engineers This paper was prepared for presentation at the Eighteenth SPE Improved Oil Recovery Symposium held in Tulsa, Oklahoma, USA, 14–18 April 2012. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract

Water- Alternating- Gas (WAG) injection in waterflooded reservoirs can increase oil recovery and extend the life of these reservoirs. Reliable reservoir simulations are needed to predict the performance of WAG injection before field implementation. This requires accurate sets of relative permeability (kr) and capillary pressure (Pc) functions for each fluid phase, in a three-phase flow regime. The WAG process also involves another major complication, hysteresis, which is caused by flow reversal happening during WAG injection. Hysteresis is one of the most phenomena manipulating the performance of WAG injection and hence, it has to be carefully accounted for.

In this study we have benefited from the results of a series of coreflood experiments that we have been running since 1997 as a part of the Characterization of Three-Phase Flow and WAG Injection JIP (joint industry project) at Heriot-Watt University. In particular we focus on a WAG experiment carried out on a water wet core to obtain three-phase relative permeability values for oil, water and gas. The relative permeabilities exhibit significant and irreversible hysteresis for, oil, water and gas. The observed hysteresis, which is due to the cyclic injection of water and gas during the experiment, is not predicted by the existing hysteresis models.

We present a new three-phase relative permeability model coupled with hysteresis effect for modelling of the observed cycle-dependent relative permeabilities taking place during WAG injection. The approach was successfully tested and verified using the measured three-phase relative permeability values obtained from WAG experiment. In line with our laboratory observations, the new model predicts the reduction of the gas mobility during consecutive water and gas injection cycles as well as the increase in oil relative permeability happening in consecutive water injection cycles.

Introduction

Estimation of three-phase relative permeability is needed for a variety of oil recovery techniques, such as water drive of reservoirs at pressure below the bubble point, water alternating gas injection, hot gas/oil/water systems in thermal recovery, and low pressure gas recycling in condensate fields with aquifers. Considerable efforts have been directed towards gaining a better understanding of three phase flow in porous media and in particular determination of three-phase relative permeability values. However, an accurate estimation of three-phase relative permeabilities still remains a challenging task for the petroleum industry. While for two-phase relative permeability (oil/water, gas/oil, and gas/water) there are only two principal displacement paths, i.e. the saturation of one phase may either increase or decrease, in contrast, in the case of three-phase relative permeability there are an infinite number of different displacement paths. This is because any three-phase displacement involves the variation of two independent saturations. It is therefore impractical to measure relative permeability for all possible three-phase displacements that may occur in a reservoir including, for instance immiscible WAG injection.

The current standard approach used in industry for determination of three-phase relative permeability is to employ two-phase kr and use one of the existing correlations (e.g. Stone, Baker) to calculate relative permeability at three-phase flow

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circumstance. Stone (1970) presented a probability method which uses two sets of two-phase data to predict the relative permeability of the intermediate wet phase in a three-phase system. This model is such that it will yield the correct two-phase data when only two phases are flowing, and will provide interpolated data for three-phase flow that are consistent and continuous functions of the phase saturations. He modified his models (1973) by incorporating gas and water relative permeability in calculation of three-phase kr of the oil in order to get better agreement with experimental data, especially in the region of low oil saturations. Both Stone models were originally proposed for preferentially water wet systems in which water and gas relative permeabilities only depend on the water and gas saturations, respectively. Baker (1988) proposed a simple three-phase relative permeability for oil, water and gas based on saturation-weighted interpolation between two-phase relative permeability data in which three-phase kr of each phase is assumed to be function of two saturations. He showed that interpolation model provide a better fit to the experimental data than the other models that were available at the time. Hustad and Hansen (1995), proposed an empirical correlation (IKU) for three-phase relative permeabilities and phase pressure for reservoir simulators. The formulation is based on three sets of two-phase data and properly accounts for six, two-phase, residual or critical saturations. The model uses only two-phase data and an interpolation technique to obtain three-phase properties by a systematic weighting procedure based on the saturations and end point values. Hustad and Holt (1992) carried out a hydrocarbon gas injection test into a water-flooded vertical core under near miscible conditions. He modified Stone’s first model by imposing an exponent parameter on the saturation term of the oil relative permeability to match the production and pressure data.

Characteristic parameters describing multiphase flow in porous media are process dependent. In particular, relative permeabilities are considered to be dependent on saturation and saturation history. This latter dependency is described in the literature as relative permeability hysteresis. Hysteresis on relative permeability has been experimentally observed in two-phase (Osoba et al., 1951; Land, 1971; Braun and Holland, 1995) and in three-phase flow (Skauge and Aarra, 1993; Eleri et al., 1995b) in porous media.

A few three-phase relative permeability models have been developed incorporating both hysteresis and interfacial tension effects. Jerauld (1997) developed a three-phase relative permeability correlation based on Prudhoe Bay field data. The model incorporates hysteresis in gas, oil, and water relative permeability as well as the dependence of relative permeability on composition and gas/oil interfacial tension (IFT). The functional forms chosen to correlate the relative permeability data were based on interpretation of the pore-level mechanisms that determine fluid flow. Blunt (2000) presented an empirical model for three-phase relative permeability which allows for changes in hydrocarbon composition, hysteresis, and the trapping of oil, water, and gas. The model is based on saturation-weighted interpolation between the two-phase relative permeabilities. Layer drainage of oil flow is also accounted for in the oil relative permeability for low oil saturations. Hustad and Browning (2010) proposed a coupled formulation (ODD3P) for three-phase relative permeability for implicit compositional reservoir simulation. The formulation incorporates primary, secondary, and tertiary saturation functions. Hysteresis and miscibility were applied simultaneously to both capillary pressure and relative permeability function.

Oil recovery by immiscible WAG is dependent on the saturation cycles that occur in a core-flood or in the reservoir. In order to predict WAG behaviour in the reservoir from experimental results, numerical models with an effective cycle-dependent hysteresis description of the three-phase oil, water and gas relative permeabilities should be considered. Skauge and Larsen (1994) conducted some WAG coreflood experiments under different wettability condition and then three-phase relative permeability was obtained from different cycle of gas and water. The results indicated that the residual oil saturation can be significant lower by three-phase flow compared to two-phase waterflood or gas injection. Every phase relative permeability depicted irreversible hysteresis effect during various water and gas injection. A relative permeability model was developed based on cycle-dependent hysteresis effects occurring during WAG injection (Larsen and Skauge, 1998). The model accounted for reduced mobility and irreversible hysteresis loops during three-phase flow. This model uses experimental wetting and non-wetting relative permeabilities as input data as well as the knowledge of relations between maximum non-wetting saturation and trapped non-wetting saturation.

Our previous study (Shahverdi et al., 2011) on the evaluation of the most widely used three-phase kr models showed that existing methods available in the reservoir simulators could not adequately predict the results of the near-miscible WAG experiments. In this paper based on the experimental observations, a new kr model coupled with hysteresis effect is proposed to obtain three-phase kr values involved in a WAG process. Three predominant features have been incorporated in this approach to predict the relative permeability for different injection scenario taking place in the reservoir e.g. water injection, gas or oil injection. First, it gives three-phase kr values for each individual phase i.e., oil, water, gas as a function of two independent fluid saturations. Second, two-phase relative permeabilities of the three sets of oil/water, oil/gas and gas/water are taken into account and used to calculate three-phase relative permeability (unlike all existing models that only account for two sets of two-phase kr for calculating three-phase relative permeability). Third, an appropriate saturation function has been introduced to the kr model in order to capture the cyclic hysteresis impact occurring during alternating injection of water and gas.

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Three-Phase Relative Permeability modelling

Theory

One of the key factors controlling multiphase flow in porous media is the fluid distribution pattern which is a function of pore geometry, wettability and interfacial tension (IFT) between fluids (or spreading coefficient). A general configuration of the fluid distribution applicable to all wettability conditions while three immiscible fluids coexist in the porous medium is that each individual fluid is in contact with two other fluids as shown schematically in Figure 1.

Figure 1: A schematic figure of three-phase fluid distribution in porous media

As implied in Figure 1, the flow of the oil under three-phase conditions is influenced by two other phases relative permeabilities i.e. rogk , rowk . Furthermore, it can be assumed that these two phase relative permeabilities independently affect the three phase oil relative permeability. This assumption can mathematically be described by the following equation:

( )3 2 2Ph Ph Phro row rogk k k∝ + (1)

The superscripts “3Ph” and “2Ph” refer to three-phase and two-phase, respectively. This assumption is also made in some of the existing kr models which predict three-phase relative permeability based on two phase data, e.g., Baker (1988), IKU (Hustad and Hansen, 1995). As presented in Figure 1, gas(1) is in contact with water(1) and water(2) is in contact with gas(2). In other words, rogk is influenced by rgwk and rowk is influenced by rwgk , hence 3Ph

rok can be expressed as:

( )3 2 2 2 2Ph Ph Ph Ph Phro row rwg rog rgwk k k k k∝ + (2)

A saturation function term is imposed on the above equation to convert the proportional sign to the equality:

( )3 2 2 2 2Ph Ph Ph Ph Phro row rwg rog rgwk f k k k k= × + (3)

( , , )(1 )(1 )

oo w g

w g

Sf S S SS S

=− −

(4)

Where iS denotes the normalized saturation of i-phase defined based on the direction of the flow i.e. imbibitions or drainage and the initial status of the grid block. This function is introduced in such a way to be able to capture the additional hysteresis effect which is raised by switching the flow from two phase to three phase. Substituting equation 4 in equation 3, the equation for three-phase oil relative permeability using two-phase data is:

3 ( , )(1 )(1 )

Ph oro w g row rwg rog rgw

g w

Sk S S k k k kS S

⎡ ⎤= +⎣ ⎦− − (5)

Similarly to the kro, the following equations are proposed for water and gas relative permeability under three-phase circumstance:

3 ( , )(1 )(1 )

Ph wrw o g rwo rog rwg rgo

g o

Sk S S k k k kS S

⎡ ⎤= +⎣ ⎦− − (6)

Oil Water(2 Gas(2Gas Water(1)

rogk rowk rwgkrgwk

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3 ( , )(1 )(1 )

gPhrg w o rgo row rgw rwo

o w

Sk S S k k k k

S S⎡ ⎤= +⎣ ⎦− −

(7)

Where 3Phrwk and 3Ph

rgk are three-phase water and gas relative permeability respectively and rijk is two-phase relative permeability of i-phase in presence of j-phase and must be computed at Si = 1- Sj which is different i-phase saturation than that in the three-phase situation. Moreover, two-phase kr in Equation 5 to 9 should be selected from an appropriate scanning curve based on the direction of flow in which the three-phase flow is occurring.

As clearly demonstrated in equations (5) to (7), all three-phase relative permeabilities are given as a function of two independent saturations rather than only their own saturation. Figure 3, shows a series of three-phase oil relative permeability curves as a function of water and gas saturations generated by applying equation (5) and using a synthetic two-phase data. The physical constraint of the relative permeability curves is that kri of each phase must monotonically increase by increasing its own saturation ( 0ri

i

dkdS

> ) which is met by equations (5) to (7). Furthermore these equations ensure continuity while

switching from three-phase state to two-phase state. In other words, the three-phase relative permeability in equations (5) to (7) becomes two-phase relative permeability once one the phases’ saturation, other than its own saturation, becomes zero (if 1i jS S+ = then 3 2Ph Ph

ri rijk k= ). In the case of presence of irreducible water saturation in the system, neither two-phase oil nor gas relative permeability is unity at the maximum corresponding saturation (=1-Swc). Hence, all two-phase oil and gas relative permeabilities in equations (5) to (7) must be divided by the maximum relative permeability reached at 1-Swc.

The normalized phase saturations in equations (5) to (7) are defined as below:

***

*

1 gow

ggg SSS

SSS

−−−

−= ,

***

*

1 gow

ooo SSS

SSS−−−

−= ,

***

*

1 gow

www SSS

SSS−−−

−= (8)

where *iS is defined based on the direction of the flow i.e. gas injection, oil injection, water injection which is different for

each relative permeability formula given in equations (5) to (7). For three-phase gas relative permeability (7), *iS are defined

as:

1- gas injection : startgg SS =* , wcw SS =* , 0* =oS

2- oil injection : gtg SS =* , wcw SS =* , startoo SS =*

3- water injection : gtg SS =* , startww SS =* , *

o otS S=

where , ,start start startg o wS S S are saturation of gas, oil and water at the beginning of gas injection, oil injection and water injection,

respectively. The gtS and otS are trapped gas and oil saturation attained during either water injection or oil injection.

During simultaneous flow of three phases in porous media, two types of kr-hysteresis effect can be happen; one belongs to the two-phase hysteresis and the other one is attributed to the three-phase hysteresis. Two-phase kr-hysteresis effect is incorporated in three-phase kr model by using a suitable two-phase kr scanning curve. The second kind of hysteresis relevant to the three-phase flow is accounted for by introducing a normalized saturation term in equations (5) to (7) that can be different for different process i.e. gas injection, oil injection, and water injection. While a gas injection process is taking place in which the gas saturation is increasing in a grid block then *

gS in equation (8) is equal to the startgS . Hence, the higher the

initial gas saturation is the lower the value of the normalized gas saturation ( gS ) will be and consequently a lower value for gas relative permeability will be obtained by equation (7). This fact is consistent with the fact that krg reduces by increasing the number of gas cycles during WAG injection.

Another kind of flow which might happen in a grid block is the reduction of the gas saturation by advancement of a wetting phase i.e. water or oil that causes trapping of some gas inside the porous medium. The amount of this trapped gas saturation strongly affects the gas relative permeability during the WAG or any other sort of three-phase flow. This impact was incorporated in the saturation function (8) of the krg model by imposing initial wetting phase saturation ( start

oS or startwS ) at the

beginning of the imbibition process and the trapped gas saturation (gtg SS =* ) obtained at the end of the imbibition.

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The *iS in equation (8) applicable to the three-phase oil relative permeability (equation (5)) is defined for different process as

below:

1- gas injection : * 0gS = , wcw SS =* , *o orgS S=

2- oil injection : gtg SS =* , wcw SS =* , startoo SS =*

3- water injection : gtg SS =* , startww SS =* , *

o otS S=

where orgS is the residual oil saturation reached by advancement of the gas into the oil. Hysteresis effect of the oil relative permeability is pronounced while water saturation is increasing in a grid block to displace the oil phase. This fact is accounted for in the kro model by imposing the initial water saturation and trapped oil saturation attained by the water injection process.

The *iS in equation (8) applicable to three-phase water relative permeability (equation (6)) is defined as below for all kind of

process:

* 0gS = , *w wcS S= , * 0oS =

Model verification

Coreflood experiments

Prior to performing coreflood experiments, a preparation procedure was followed to determine properties of the core and also check the degree of its homogeneity. Initially, the core was weighed and the length/area measurements taken to estimate the bulk volume. Then the core was loaded into a core holder and an X-ray scan was run to examine homogeneity of the core prior to performing the coreflood tests. The helium porosity and nitrogen permeability of the core were then measured, followed by evacuation of the core, after which it was saturated with a 1% brine solution. The amount of brine imbibed into the core was measured and gave a second measurement of pore volume. A 1% sodium chloride/calcium chloride brine solution was used in order to desensitise any clay minerals and prevent them from swelling and restricting flow. The water permeability was measured, prior to core characterization. Tracer analysis of the core was then conducted to ensure that there are no major heterogeneities, such as fractures or permeability layers, in the core, which, if not detected, might influence the results of multiphase flow experiments. This procedure also provides an accurate measurement of the pore volume of the core. The physical properties of the core is given in Table 1. Core was chosen to be long enough to minimize the capillary end effect while performing flooding tests (Fatemi et al., 2011).

The hydrocarbon fluid system used in the coreflood experiments was prepared from a binary mixture of methane (C1) and n-butane (n-C4). To eliminate mass transfer during the displacement experiments, all the fluids (oil, gas and brine) were pre-equilibrated at the test pressure and temperature of 1840 psia and 100°F and were kept under these conditions in high-pressure transfer vessels and in a temperature-controlled oven. Table 2 shows the measured properties of the fluid system at the conditions of the experiments. The critical point pressure of the hydrocarbon mixture (mixture of C1 and n-C4) was about 1865 psia and hence, at the pressure of the experiments (1840 psi), the system was very close to its critical point and hence, nearly miscible (Sohrabi et al., 2007).

In calculating three-phase relative permeability using Equation 5 to 7, two-phase kr of different fluid systems including oil/water, oil/gas and gas/water are required. Hence a comprehensive set of unsteady-state two-phase experiments were conducted in order to obtain accurate two-phase relative permeability data on the cores for which three-phase relative permeability values would be calculated. A coreflood simulator (Weatherford, 2005) was employed to obtain two-phase relative permeabilities of the unsteady-state experiments by matching the production and differential pressure data of the simulation with those resulting from the experiments.

Then, a three-cycle WAG injection experiment was carried out in order to obtain three-phase kr values. The WAG experiments began with a water flood period followed by a gas injection cycle. These cycles of water and gas injection were repeated three times. Figure 4 demonstrates the saturation path for WAG experiment through 65mD water-wet core as ternary diagrams which is determined by volumetric material balance calculation, knowing the amount of injected and produced fluid and initial saturation of the core. The notation Gi and Wi (i=1, 2,..) in these ternary diagrams refers to the saturation paths of ith gas injection and water injection, respectively. Three-phase kr of different cycles of WAG experiment were determined using in-house three-phase coreflood simulator (Shahverdi et al., 2011) developed at Heriot-Watt University.

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The methodology proposed in this study was applied to calculate three-phase kr values of various cycles of WAG injection by employing measured two-phase kr. In order to assess the accuracy of this model, the calculated three-phase kr by model were compared with the corresponding three-phase kr which is directly obtained from WAG experiment.

Results and discussion

Figure 5 compares krg curves versus gas saturation obtained from various gas injection cycles of the WAG experiment and simulated by our three-phase kr model. As can be seen, there is a good agreement between the experimentally derived values and those predicted by the model. The reduction of the gas relative permeability in WAG cycles which is one of the well-known features of WAG injection is clearly captured and predicted by the proposed model. The physical reason for the reduction of gas relative permeability by increasing the number of WAG cycles is that any water injection performed prior to the gas injection traps some amount of the mobile gas, rendering it immobile in the core. Such trapped (non-wetting) gas will not contribute towards the mobility of the new total injected gas. In other words, every new gas injection cycle, following a water injection cycle, commences with higher amount of immobile gas, which reduces the mobility of the total in-situ gas.

Figure 6 shows gas relative permeability obtained from the experiment and the model for the second and the third water injection cycles of the WAG test. Since the third water injection period started with higher initial gas saturation, the trapped gas saturation in which the gas relative permeability becomes zero for the third water is higher than that for the second water injection. Consequently, the gas relative permeability during the third water injection is shifted to the right by the amount of trapped gas saturation such that both curves are almost parallel. The reason is that during the water flood the amount of total gas saturation inside the core is composed of two elements, flowing gas (Sgf) and trapped gas (Sgt), in which only the flowing gas saturation contributes to the gas mobility. The following relationship exists between flowing gas and trapped gas:

g gf gtS S S= + (9)

Therefore, the same value for krg can occur at different (total) gas saturations for the 2nd and 3rd water injections because their flowing gas saturation could be the same.

Water relative permeability versus water saturation obtained from the experiment and predicted by the model for different gas injection cycles are presented in Figure 7. As can be seen, there is a close match between the krw of the experiment and the model prediction during all gas and water injection cycles. As the gas advances forward during gas injection, it displaces the water phase in the porous medium which consequently reduces the water mobility. The results also reveal that krw is reduced by performing successive gas injection cycles (cycle hysteresis), which is attributed to the increase of trapped gas saturation. This point is demonstrated in Figure 2 which schematically shows gas injection process into a pore throat occupied by mobile water and some trapped gas resulted from a previous water injection. As can be seen, as the amount of the trapped gas in the pore increases, it further restricts the flow of water inside the pore and therefore decreases the relative permeability of water.

Figure 2: Schematic picture of a pore filled by water and trapped gas which is displaced by gas flooding.

The water relative permeabilities of 2nd and 3rd water injection obtained from experiment and model are plotted versus water saturation, in Figure 8. The predicted values of krw by model show good agreement with the corresponding measure relative permeability data. Furthermore, this Figure highlights that krw of the water-wet core does not exhibit a considerable hysteresis effect during successive water injections.

Figure 9 shows oil relative permeabilities obtained from the experiment and the model for various water and gas injection cycles. As the gas advances in the porous medium, it displaces the oil and henceforth reduces the oil relative permeability. As shown in this Figure, cyclic hysteresis effects observed in the oil relative permeability (kro) during different gas injection

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periods are not significant whereas kro obtained for different water injection periods exhibit substantial cyclic hysteresis. Hysteresis in the oil relative permeability during water injection is attributed to the trapped oil saturation caused by increasing water saturation in the porous medium. The fluid saturation values obtained in coreflood experiments on both water-wet and mixed-wet cores demonstrate that trapped hydrocarbon (oil + gas) saturation remains constant during various water injection periods. Since the trapped gas saturation continually increases by cyclic water and gas injection, the immobile oil saturation reduces successively. Therefore, the residual oil saturation reached by the 3rd water injection is less than that by the 2nd water which influences kro positively as shown in Figure 9. This mechanism is similar to what was described for the gas relative permeability (Equation 9) where increasing trapped gas saturation in the subsequent water injection shifts the krg curve horizontally to a higher gas saturation for each water cycle compared to the previous one. For the oil phase, because of decreasing immobile (residual) oil, the kro of the 3rd water injection moves to lower oil saturation (left hand side) compared to the 2nd water injection.

The relative permeabilities predicted by our model and those obtained by some widely used three-phase kr models including Stone1 (Stone, 1970), Stone2 (Stone, 1973), Stone-Exponent (Hustad and Holt, 1992), Larsen-Skauge (1998), Baker (Baker, 1988) and Saturation-Weighted-Interpolation (SWI) (Baker, 1988) were used in Eclipse software (Schlumberger, 2011) to simulate the WAG performance in our coreflood experiment. Figure 10 shows the comparison of the oil recovery as a fraction of Sorw (the remaining oil after the first water flood) obtained from the experiment and predicted by the various models. As can be seen the step-wise oil recovery profile due to the cyclic injection of water and gas is well predicted by our method whereas using other models oil recovery is significantly overestimated or underestimated. The predictions made by the existing models can be very variable and various models can predict vastly different three-phase kr values from the same 2-phase data. While some models perform better than others, all of the three-phase kr models examined in this study fail to predict the continued production of oil after the breakthrough of the gas, which is one of the features of gas and WAG injection experiments at low gas-oil interfacial tension (Sohrabi et al., 2005; Sohrabi et al., 2008).

Conclusions

1- A new three-phase relative permeability model coupled with hysteresis effects has been developed based on using two-phase relative permeability data. The model calculates three-phase relative permeabilities of oil, water and gas as a function of two independent saturations. The model ensures continuity when switching from three-phase flow regime to two-phase state. The approach was successfully tested and verified using the results of a WAG coreflood experiment.

2- Cyclic hysteresis effects which occurs in some oil recovery techniques like WAG injection is accounted for in the new three-phase kr model by introducing an appropriate saturation function relevant to the type of injection process e.g. imbibitions, drainage.

3- All three sets of two-phase relative permeability data i.e., oil/water, oil/gas and gas/water system are incorporated in our methodology for calculating three-phase kr values whereas all the other existing kr models only account for two sets of relative permeability (oil/water and oil/gas) for determining three-phase relative permeability.

ACNOWLEDGMENT This work was carried out as part of the ongoing three-phase flow and Water-Alternating-Gas (WAG) Injection joint industry project (JIP) in the Institute of Petroleum Engineering of Heriot-Watt University. The project is equally sponsored by, Statoil, BHP Billiton, Chevron, Dong Energy, Petrobras, the UK Department of Energy and Climate Change (DECC) and Total E & P Uk, which is gratefully acknowledged.

The Authors would also like to thank Mr. Mobeen Fatemi for carrying out the coreflood experiments.

REFERENCES

Blunt, M.J., 2000, An Empirical Model for Three-Phase Relative Permeability: SPE Journal, paper SPE 67950, (12).

Braun, E.M., and Holland, R.F., 1995, Relative Permeability Hysteresis: Laboratory Measurements and a Conceptual Model: SPE Reservoir Engineering, paper SPE 28615, (08).

Eleri, O., A.Graue, A.Skauge, and J.A.Larsen, 1995, CALCULATION OF THREE-PHASE RELATIVE PERMEABILITIES FROM DISPLACEMENT EXPERIMENTS WITH MEASUREMENTS OF IN-SITU SATURATION presented at the SCA,

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Fatemi, S.M., Sohrabi, M., Jamiolahmady, M., Ireland, S., and Robertson, G., 2011, Experimental Investigation of Near-Miscible Water-Alternating-Gas (WAG) Injection Performance in Water-wet and Mixed-wet Systems, paper SPE 145191, presented at the Offshore Europe, Aberdeen, UK.

Hustad, O.S., and Browning, D.J., 2010, A Fully Coupled Three-Phase Model for Capillary Pressure and Relative Permeability for Implicit Compositional Reservoir Simulation: SPE Journal, paper SPE 125429-PA, (12).

Hustad, O.S., and Hansen, A.G., 1995, A Consistent Correlation For Three-Phase Relative Permeabilities and Phase Pressure Based on Three Sets of Two Phase Data, presented at the 8th European IOR symposium, Vienna, Austria.

Hustad, O.S., and Holt, T., 1992, Gravity Stable Displacement of Oil by Hydrocarbon Gas After Waterflooding, paper SPE 24116, presented at the SPE/DOE Enhanced Oil Recovery Symposium, Tulsa, Oklahoma.

Jamiolahmady, M., Danesh, A., Sohrabi, M., and Ataei, R., 2007, Gas-Condensate Flow in Perforated Regions: SPE Journal, paper SPE 94072-PA, (03).

Jerauld, G.R., 1997, General Three-Phase Relative Permeability Model for Prudhoe Bay: SPE Reservoir Engineering, paper SPE 36178, (11).

Land, C.S., 1968, Calculation of Imbibition Relative Permeability for Two- and Three-Phase Flow From Rock Properties, paper SPE 1942, (06).

Larsen, J.A., and Skauge, A., 1998, Methodology for Numerical Simulation With Cycle-Dependent Relative Permeabilities: SPE Journal, paper SPE 38456, (06).

Osoba, J.S., Richardson, J.G., Kerver, J.K., Hafford, J.A., and Blair, P.M., 1951, Laboratory Measurements of Relative Permeability, paper SPE 951047-G,

Schlumberger, 2011, Eclipse, Reservoir Simulator.

shahverdi, h., Jamiolahmady, M., Sohrabi, M., Ireland, S., Fatemi, S.M., and Rabertson, G., 2011a, Evaluation of Three-Phase Relative Permeability Models for WAG Injection Using Water-Wet and Mixed-Wet Core Flood Experiments, paper SPE 143030, presented at the SPE EUROPEC/EAGE Annual Conference and Exhibition, Vienna, Austria.

Shahverdi, H., Sohrabi, M., and Jamiolahmady, M., 2011b, A New Algorithm for Estimating Three-Phase Relative Permeability from Unsteady-State Core Experiments: Transport in Porous Media, 90(3), p. 911-926.

Skauge, A., and Aarra, M., 1993, Effect of wettability on the oil recovery by WAG, presented at the 7th IOR symp, Moscow.

Skauge, A., and Larsen, J.A., 1994, Three-phase relative permeabilities and trapped gas measurements related to WAG processes, presented at the SCA, Stavanger.

Sohrabi M., Tehrani D.H. and Al-Abri M.; (2007): Performance of Near-Miscible Gas and SWAG Injection in a Mixed-Wet Core, SCA 2007-26, presented at International Symposium of the Society of Core Analysts held in Calgary, Canada, 10-12 September.

Sohrabi, M., Danesh, A., Tehrani, D., and Jamiolahmady, M., 2008, Microscopic mechanisms of oil recovery bynear-miscible gas injection: Transp. Porous Media, 72, p. 351-367.

sohrabi, m., Danesh, A., and Tehrani, D.H., 2005, Oil Recovery by Near-Miscible SWAG Injection, paper SPE 94073, presented at the SPE Europec/EAGE Annual Conference, Madrid, Spain.

Stone, H.L., 1970, Probability Model for Estimating Three-Phase Relative Permeability: SPE Journal of Petroleum Technology, paper SPE 2116, (02).

Stone, H.L., 1973, Estimation of Three-Phase Relative Permeability And Residual Oil Data, paper SPE 73-04-06, (10-12).

Weatherford, 2005, Sendra, Coreflood Simulator.

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Table 1: Rock properties

Table 2: Properties of hydrocarbon mixture and brine at 1840psia and 37.8 °C

Figure 3: A typical three-phase relative permeability generated by model

Figure 4: Saturation path of WAG experiment into core.

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Figure 5: Gas relative permeability versus gas saturation for various cycles of gas injections obtained from experiment and predicted by our kr model.

Figure 6: Gas relative permeability versus gas saturation for 2nd and 3rd waterinjections obtained from experiment and predicted by our kr model.

Figure 7: Water relative permeability versus water saturation for various cycles of gas injections obtained from experiment and predicted by our kr model.

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Figure 8: water relative permeability versus water saturation for 2nd and 3rd water injections obtained from experiment and predicted by our kr model.

Figure 9: Oil relative permeability versus oil saturation for various cycles of water and gas injections obtained from experiment and predicted by our kr model.

Figure 10: Oil production as fraction of Sorw (the remaining oil after first water flood) versus the injected pore volume obtained from WAG experiment and predicted by various three-phase models including our kr correlation. G1 stands for the first gas injection period, G2 denotes the second gas injection period, and so forth.