SPE 165324-MS

Novel Three-Phase Compositional Relative Permeability and Three-Phase Hysteresis Models Mohammad R. Beygi, Mojdeh Delshad, Venkateswaran S. Pudugramam, Gary A. Pope, and Mary F. Wheeler, SPE, The University of Texas at Austin

Copyright 2013, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Western Regional & AAPG Pacific Section Meeting, 2013 Joint Technical Con ference held in Monterey, California, USA, 19−25 April 2013. 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 Mobility control methods have the potential to improve coupled enhanced oil recovery and CO2 storage technique

(CO2-EOR). There is a need for improved three-phase relative permeability models with hysteresis including especially the

effects of cycle dependency so that more accurate predictions of these methods can be made. We propose new three-phase

relative permeability and three-phase hysteresis models applicable to different fluid configurations in porous medium under

different wettability conditions. The relative permeability model also includes compositional effects. Three-phase parameters

are estimated based on saturation-weighted interpolation of two-phase parameters. The hysteresis model is an extension of

the Land trapping model but with a dynamic Land coefficient introduced. The trapping model estimates a constantly

increasing trapped saturation for intermediate- and non-wetting phases. The model overcomes some of the limitations of

existing three-phase hysteresis models for non-water wet rocks and mitigates the complexity associated with commonly

applied models in numerical simulators. The model is validated using multi-cyclic three-phase water-alternating-gas

experimental data for non-water wet rocks. Numerical simulations of a carbonate reservoir with and without hysteresis were

used to assess the impact of the saturation direction and saturation path on gas entrapment and oil recovery.

Background and Literature Review Unlike water-wet rocks, mixed-wet reservoirs show a continuous oil phase in three-phase flow resulting in small, yet non-

zero, values of the oil relative permeability (Salathiel, 1973). The dependence of phase isoperms on one or two phase

saturation in the saturation space is complex and difficult to predict a priori by empirical models (van Dijke et al., 2001). The

contact angle plays a crucial role in fluid configurations in porous media. Moreover, by studying three-phase displacement

processes and pore-scale mechanisms in etched micromodels, it was revealed that fluid configurations can be categorized

based on the phase spreading coefficient (Cs) defined as a balance of interfacial tensions (Adamson, 1960):

1 32 1 2 2 3SC (1)

The necessary condition for a spreading oil layer (phase 2) between water and gas phases (phase 1 and phase 3,

respectively) is the non-negative ‘ ’; otherwise, a three-phase contact line exists. In the non-spreading case, oil is wetting

relative to gas and either water or gas could be the wetting phase depending on the gas-water contact angle. The difference

between oil-water and gas-oil interfacial tension identifies the intermediate-wetting phase in non-water-wet rocks. This

complex fluid configuration and flow in porous media delineates regions in saturation space where commonly used relative

permeability models fail to replicate the physical behavior. There are commonly used correlations based on an assumed fluid

configuration that are not easy to generalize to other fluid configurations.

Three-Phase Relative Permeability Models

Based on an extension of Marle’s (1981) description, there are numerous parameters that impact relative permeability:

saturations, wettability, pore shape and size, trapping number (interaction of viscous, gravitational, and capillary forces (Pope

et al., 2000), saturation direction and saturation path, and composition. In three-phase flow, except for certain special

conditions, relative permeability measurements are incomplete. More precisely, for a specified phase with a fixed saturation,

there are an infinite number of combinations of the saturations for three-phase flow. A conventional approach is to estimate

2 SPE 165324-MS

three-phase flow relative permeability based on measured two-phase relative permeabilities. Table 1 provides a

chronological list of some of the three-phase relative permeability models. It implies that saturation-averaged interpolation

method is a common choice in more recent models in lieu of other mathematical models, viz. capillary models, statistical

models, and network models. Several investigators (Fayers and Matthews (1984); Baker (1988); Delshad and Pope (1989);

Oak ( (1990) and (1991)); Skjæveland and Kleppe (1992); Hustad and Holt (1992); Balbinski et al. (1999); Pejic and Maini

(2003); Spiteri et al. (2006); Ahmadloo et al. (2009)) have identified various limitations of classical three-phase relative

permeability models such as Stone I and Stone II, especially for non-water wet conditions and/or at low oil saturation.

Three-Phase Residual Saturation and Its Impact on Three-Phase Relative Permeability

Two- and three-phase residual saturations are not unique and depend on wettability and saturation history, i.e. saturation

direction and saturation path. Some of the published three-phase residual saturation models are summarized in Table 2.

One of the shortcomings of widely used three-phase relative permeability models to predict the actual behavior in three-

phase flow initiates from the residual saturation prediction. Balbinski et al. (1999) investigated the impact of residual oil

saturation in Stone I and the implied residual oil saturation by Stone II model and realized that there is a problem of

consistency in choice of residual oil saturation for Stone I; besides, the Stone II formulation results in mathematically

singular residual oil saturation values near the two-phase axis that are much larger than two-phase values.

In extreme wettability conditions, the wetting fluid, i.e. water or oil, shows minimal trapping and hysteretic behavior.

The advancing and receding wetting films formed by saturation increasing and decreasing processes are stable leading to a

hydraulically continuous wetting phase even at very low saturations (Dullien et al., 1989). Non-wetting phase trapping

causes the wetting phase to flow in larger pores than it would otherwise have done in the absence of trapping. The wetting

and non-wetting phase relative permeability depends only on the wetting and non-wetting phase saturation, respectively. The

non-wetting phase relative permeability is saturation-history dependent. The intermediate-wetting phase relative permeability

depends on the saturation of two phases and is impacted by the presence of an immobile wetting phase. In an extensive study

of Prudhoe Bay intermediate-wet sandstone, Jerauld (1997) observed the direct impact of initial water saturation on oil

relative permeability. Kalaydjian et al. (1995) reported experiments on water-wet samples showing that relative permeability

is affected by the spreading coefficient and by the presence of connate water.

For oil-wet Tensleep sandstone and Grayburg carbonate samples, Schneider and Owens (1970) recognized that in the

absence of a water film the water relative permeability decreased by increasing gas entrapment. The water relative

permeability decreased during secondary drainage compared to that of a waterflood at a given water saturation. Even low

residual oil saturations following a miscible flood drastically decreased the water relative permeability. Baker (1993) reported

gas flood experiments on oil-wet samples from the Tensleep formation showing a correlation between residual and initial

water saturations. DiCarlo et al. (2000) aged water-wet sand to measure three-phase gravity drainage on oil-wet sand and

showed that water relative permeability for oil-wet sand is similar to oil relative permeability for water-wet sand.

For the intermediate-wet state, water, oil and gas may all show hysteresis. The initial-trapped saturation dependency may

exist for all phases. Jerauld (1997) found systematic differences between water relative permeability during waterfloods and

miscible water alternating gas (WAG) floods in Prudhoe Bay intermediate-wet sandstone. Oak (1991) treated water-wet

Berea sandstone to make it intermediate wet and found a slight dependency of water relative permeability on the saturations

of the other two phases implying water entrapment in non-water wet rocks.

Based on results of capillary bundle model by van Dijke et al. (2001), gas is always non-wetting to oil calling for a

hysteretic behavior leading to gas entrapment during oil floods. For both spreading and non-spreading systems, gas-water

displacement occurs through a double displacement mechanism. In an oil-wet pore, gas and water are not in direct contact,

but gas is not necessarily the non-wetting phase. If the oil-water IFT is larger than the gas-oil IFT, then gas and water behave

as intermediate- and non-wetting phases, respectively. Vizika and Lombard (1996) showed that within three-phase gas

drainage in oil-wet sands, the gas relative permeability does not change both for spreading and non-spreading oil. DiCarlo et

al. (2000) showed that gas relative permeability in an oil-wet sand is smaller than in a water-wet sand and concluded that gas

is not necessarily the non-wetting phase.

Initial-Trapped Saturation

Phase trapping depends on porosity, microposity, and the maximum phase saturation attained in the increasing process.

Several empirical models describe the initial-trapped saturation relationship. Table 3 summarizes different models used to

estimate the trapped saturation. Following the pioneering work of Land [(1968) and (1971)] to estimate the trapped non-

wetting saturation and to introduce the definition for flowing saturation, different adaptations of his approach have been

widely used. Ma and Youngren (1994) generalized the Land correlation to account for sharp level-off of the trapped

saturation at high initial saturations:

 

 1

m a x

t

j

j

mj d

a x

SS

a S

(2)

where ‘ ’ and ‘ ’ are empirically derived constants.

SPE 165324-MS 3

Table 3 implies that except for the parabola shape of the Spiteri et al. (2008) correlation, the trapped saturation

monotonically increases by higher initial saturation attained in the saturation increasing process. Some investigators stated

that the hyperbolic laws, i.e. Land or its modified formulations, cannot represent the trapping behavior within their

experiments. Two-phase pore-network model results showed that the Land equation does not capture the trend of initial-

trapped saturation for high initial saturation in non-water-wet rocks (Spiteri et al., 2008). Pentland et al. (2010) measured oil

trapping as a function of its initial saturation in unconsolidated sandpacks. Their results were in agreement with the Aissaoui

(1983) and the Spiteri et al. (2008) trapping models. Suzanne et al. (2003) exaimined the two-phase initial-trapped gas

saturation for a large set of sandstone samples and concluded that the piecewise linear model, i.e. Aissaoui (1983), represents

the initial-trapped gas saturation relationship in shaly sandstone rock. They described that the constant trapped gas saturation

plateau corresponding to the high initial gas saturation is due to the microporosity in which gas dos not trap.

Hysteresis Models

There are numerous empirical models to capture saturation direction and path dependency in relative permeability and/or in

capillary pressure. Table 4 lists some of the available two- and three- phase hysteresis models. A common feature proposed

by two-phase hysteresis models is the assumption of reversibility of the relative permeability curves where the primary

saturation decreasing curve is representative of any subsequent saturation increasing processes. However, this assumption is

not validated with experimental results for multi-cycle processes.

The three-phase hysteresis models, however, predict cycle-dependent relative permeability curves where each cycle has

the scanning curves for the saturation increasing and decreasing processes. These models predict an irreversible hysteresis

behavior imposed by saturation path dependency. It is observed that the both gas and water relative permeabilities decrease in

each WAG cycles at the same saturation resulting in lowering the well injectivity. Larsen and Skauge (1998) published an

empirical model based on their immiscible- and miscible- WAG experiments. Their model relaxes the assumption of the

scanning curves’ reversibility resulting in a cycle-dependent relative permeability model for the gas phase. They also

proposed a three-phase water relative permeability model for water-wet media based on saturation-averaged interpolation

between primary and secondary waterflood relative permeabilities. For oil-wet media, the water entrapment by gas is only

considered in the Egermann three-phase hysteresis model (Egermann et al., 2000). Oil relative permeability in subsequent

WAG cycles increases because trapped oil saturation decreases with a higher likelihood of simultaneous water and gas

trapping.

A major deficiency of the existing models is the limitation in accounting for hysteresis when one phase

disappears/appears because of mass transfer between the phases. Moreover, the currently available three-phase hysteresis

models, by their nature, add complexity to compositional simulations.

Proposed Relative Permeability Model (UTKR3P) A three-phase relative permeability model should capture the essential physical behavior and be as simple as possible to be

useful in compositional reservoir simulators. Equations with a minimum of parameters have advantages over tables.

Therefore, we generalized the approach that Jerauld (1997) used for the gas phase to all of the phases. For phase ‘ ’ flowing

with phases ‘ ’ and ‘ ’, the relative permeability is given by:

1

1

2

2

2

0

11

1

1 j

j

j

C

CC

r

j j

j j

j j

r

Sk k

S

C

C

j= phase 1, 2, or 3 (3)

where normalized saturation is defined as:

     

33  

     

1

m in ,

1

j j j t

jP

iri

S S SS

S

j= phase 1, 2, or 3 (4)

Three-phase parameters, i.e. endpoint relative permeability ( ) and curvatures (

and ), are estimated using a linear

saturation-weighted interpolation between the two-phase parameters as follows:

   

 

1

m m mc c

c

l l l

c

j

j

lm

j

j

S S F S S FF

S S S

(5)

where ‘ ’ and ‘

’ are two-phase parameters ( ,

, or where ‘i’=’m’ or ‘l’).

This relative permeability formulation is inherently saturation-path dependent and multiple relative permeability curves

can be generated for each phase based on the current three-phase residual saturation values. The trapped saturation ( ) in the

normalized saturation (Eq. 4) is defined based on the implemented hysteresis model and [ ]. To calculate the

normalized saturation for each phase, three sets of three-phase residual saturations ( ) are needed. The ‘

’ is modeled

4 SPE 165324-MS

using a modified correlation proposed by Yuan and Pope (2012). We have modified the definition of saturation from original

model to an effective saturation. For phase ‘ ’, the three-phase residual saturation defined as:

23

, , (1 ( )( ))    P

r l lj c c c

P

mj mj jrjm in S m a x S S b S SS S S m l j

(6)

The two-phase residual saturation assumed either as the minimum of two-phase residual saturations corrected for

the effect of compositional changes using Eq. 8 ( and

) or estimated by a saturation-weighted interpolation between

‘ ’ and ‘

’ (use Eq. 5 for linear interpolation). If the optional compositional consistency method (Eq. 11) is not used,

the latter method can mitigate discontinuities associated with the appearance or disappearance of a phase during

compositional simulations, as noted by Fayers (1989) and Balbinski et al. (1999).

The fitting parameter ‘ ’ depends on the rock wettability and phase composition and directly impacts the relative

permeabilities. For example, by allocating phase indices 1, 2, and 3 for water, oil, and gas phases, respectively, Fig. 1 depicts

the impact of ‘ ’ on the curvature of the oil-isoperms. Large positive values of ‘ ’ result in convex oil-isoperms while

negative values of ‘ ’ result in concave oil-isoperms. The effect of ‘b’ is more pronounced in the higher oil-isoperm values.

Many other models do not have this essential flexibility.

Eq. 3 results in an S-shape curve which can be transformed to an exponential Corey-type model based on the exponents

‘ ’ and ‘

’. At small phase saturations, the model approaches Corey-type behavior leading to good agreement with

experimental results. The existence of a continuous oil layer, for example, in intermediate-wet rocks can be modeled

effectively using the proposed approach. At high saturations, however, the second term in the denominator of Eq. 3

dominates and dampens the sharp increase in relative permeability. The power exponent of the second term in the

denominator of Eq. 3 is chosen to be the maximum acceptable value resulting in a non-negative slope of the relative

permeability curve.

Compositional effects Relative permeability is in general a function of phase compositions as well as phase saturations. Furthermore, the relative

permeability model should be continuous as phases appear or disappear both for physical realism and to prevent numerical

instabilities and other problems in numerical reservoir simulators. We incorporate these features in two ways: the trapping

number effect (Pope et al., 2000) and by direct and continuous compositional dependency following Yuan and Pope (2012).

Some previous models have used density to identify phases but this is not a general solution to the problem. Jerauld

(1997) proposed a gas-like parameter based on the parachor-weighted molar density of immiscible oil, gas, and the current

condition. Hustad (2002) related the endpoint saturation to the ratio of IFT under the current state to an immiscible state and

also scaled the two-phase relative permeabilities. Yuan and Pope (2012) stated that to avoid any discontinuities in the relative

permeability it should be a function of intensive thermodynamic properties that are continuous at local thermodynamic

equilibrium. They proposed the molar Gibbs Free Energy (GFE) of each phase available from flash calculations in

compositional simulations, and used a GFE-averaged interpolation scheme between the reference and current states for each

phase. This method calls for relative permeability equations that can be interchangeably used for all of the phases. It

alleviates labeling the phases as water/oil/gas, resolves the phase-flipping issue due to incorrect phase labeling, and results in

a continuous relative permeability values.

a) Trapping number effect

Two-phase parameters, denoted as ‘ ’ in Eq. 7 represent any of the parameters of ‘ ’, ‘

’, or ‘ ’ and are modeled

using the capillary-desaturation results:

  1j j j

L H L

i i i i ijQ Q QQ i= phase or (7)

Two-phase residual saturations are correlated using the following equation:

*

( )H L H

r i r i r i r ij j j j jS S S S i= phase or (8)

where ‘ ’ and ‘ ’ denote high and low trapping number conditions, and is the trapping coefficient and a function of the

trapping number ( ). Note that two different trapping coefficients are used in estimation of the two-phase residual

saturation compared to the other two-phase parameters (Eqs. 7 and 8).

 

1   

1j

j

j T jT N

(9)

T

.

N    j

j

i

i

j

i ik g D

i= phase or (10)

SPE 165324-MS 5

In Eq. 9, and are obtained by fitting the residual saturation data for phase ‘ ’ displaced by phase . Trapping

number effect can be considered as an additional option for the three-phase relative permeability model.

b) Compositional consistency

The initial value of for the different phases are calculated at the initial pressure, temperature and composition and sorted as

[minimum GFE , intermediate GFE

, and maximum GFE and are input parameters to the model. Linear

interpolation, then, is used to calculate relative permeability parameters based on for new values of pressure, temperature,

and composition.

 

 m in  

   

 

 

   

   

   

m in

m in in t m in in t

R R Rin t

in t

in t m a x in t in t

R R R

j

j

j

j

jm a x in t

G GF F F if G G

G GF

G GF F F if G G

G G

(11)

where ‘ ’ refers to any of the two-phase flow relative permeability parameters ( ,

, ,

, or ). ‘ ’,’

’, and

‘ ’ are relative permeability parameters associated with aforementioned reference GFE values, respectively.

Compositional consistency can be considered as an additional option for the three-phase relative permeability model.

Proposed Three-Phase Hysteresis Model (UTHYST) The proposed model offers a simple approach to calculate irreversible hysteresis behavior for any phase. The hysteretic

relative permeability uses the following equation in multi-cycle processes where the impact of saturation direction and/or

saturation path is considered.

 

     

 

,  j j jr r t

n

jk k S S

j= phase 1, 2, or 3 (12)

This method is applicable to any relative permeability model that includes the trapped saturation. The monotonically

increasing trapped saturation for different phases is preserved even in miscible multi-cycle processes as follows:

   

   

  , 1

j

j

c

t t t In

m

n Ej

m

j

SS S S

S

(13)

where the ‘( )

’ is the trapped saturation corresponding to the end of previous cycle, ‘ ’ is the trapped saturation for

the current cycle, and set of parantheses introduces the effect of conjugate phase saturation at the start of the current cycle on

phase entrapment. Water, oil, and gas are defined as the conjugate phase for the gas, water, and oil phase, respectively.

We adopt a modified Jerauld’s model (Jerauld, 1997) to estimate phase trapping for the non-wetting phases:

 

   

1   

m a x ( , )  ,

1 m a x ( , )      

j

j

j j j

j j

j j j

m a x

cm a x

t c

m a

j

x

j j

L

c

S S SS S L S

L S S S

(14)

The trapped saturation in different cycles is calculated as follows:

 

     

, 1, 1

,j j j

m a x

t tE n

jE n

S S S L

(15)

 

     

, 1 

,j j j j

m a x

t tnE n

jS S S S L

(16)

The trapping behavior during multi-cycle WAG processes may not be reproduced using constant Land’s coefficient.

Element et al. (2003) used in-situ saturation monitoring to mitigate laboratory artifacts within their experiments on multi-

cycle water-alternate-CO2 floods in the aged Berea sandstone. They showed that Land’s coefficient varies during different

hysteresis cycles and that none of available hysteresis models can accurately predicts the observed three-phase hysteresis.

Spiteri and Juanes (2006) analyzed the two-phase samples of Oak data (1991) with a wide range of permeability and found

that generally Land’s coefficient depends on rock permeability and the pair of fluids used. They attributed this behavior to the

difference in advancing contact angle for different fluids. Shahverdi et al. (2011) carried out miscible WAG floods using

water- and mixed-wet cores and stated that using two-phase Land’s coefficient to calculate trapped gas saturation results in

poor agreement with the measured data. The proposed model, thereby, addresses a cycle-dependent Land coefficient in a

multi-cycle process.

6 SPE 165324-MS

3*

1

1 1

1j

j jr c ici

LS S S

(17)

where is the multiphase residual saturation corrected for compositional changes at high trapping number in which the

trapped saturation decreases due to mass transfer between the trapped and the displacing fluids (refer to Eq. 8). The

denominator of the second part of Eq. 17 refers to the maximum phase saturation based on the minimum attainable saturation

of the other two phases flowing with phase ‘ ’ ( and ).

Testing the Models We first show the proposed UTKR3P model captures the trends in two- and three-phase measured data. Then, we present

compositional simulation studies.

Experimental three-phase data We tested the proposed model using the relative permeability data of Donaldson (1966), Oak (1990), Oak (1991), Maini et al.

(1990) and Baker (1993). In general, the UTKR3P model captures the relative permeability trends for water- and oil- wet

rocks even in saturation range where phase relative permeability is a function of its own saturation. It is due to applying an

interpolation scheme between the two-phase relative permeability parameters. We present the results of the predicted and

measured relative permeability data of intermediate samples (Oak, 1991) where all relative permeabilities were functions of

two saturations. Fig. 2 shows six measured two-phase relative permeabilities. The two-phase data provided six sets of

endpoints, curvature, and residual saturations. The ternary diagram shows only the measured three-phase gas isoperms in the

decreasing water saturation, decreasing oil saturation and increasing gas saturation (DDI) direction.

To quantify the effectiveness of the UTKR3P model compared to measured three-phase data, we used the standard error

of estimate (SEE) as

j

2

 S E E

c a lc e

j

x p r

r

j

r jk k

N

(18)

where the superscripts and denote calculated and measured relative permeability and ‘N’ is number of

experimental data points for each phase. Fig. 3 shows the trend of measured and calculated relative permeability for different

phases. There is a good agreement between the measured and calculated relative permeability values considering only one

parameter ( ) is used and a very narrow range of saturations was experienced in the experiment. Matching ‘ ’ parameters

and the SEEs are given for each phase in Fig. 3.

While this test provide promising results, further examination of different aspects of the proposed model at different

wettability conditions is indispensable.

Numerical simulation Results and Discussion In this section we discuss the simulation results for a synthetic, two-dimensional reservoir. The effect of relative permeability

and hysteresis on oil recovery was simulated for a homogenous reservoir. The simulations have been performed using an

IMPEC compositional gas flooding simulator. We consider a 2D homogenous reservoir and replicate the high-perm,

intermediate-wet sample (Sample 15) of Oak (1991). Eight-components were used to represent light oil with Peng-Robinson

Equation-of-State. The simulation model has two wells, an injector and a producer, which are located at the opposite ends of

the reservoir model. A water-alternating-gas injection scheme is employed with 3 cycles of 15 days of CO2 injection followed

by 15 days of water injection. The cumulative injection after 90 days is around 1.0 HCPV. Both the injection and the

production wells are pressure constrained and the reservoir pressure is maintained at the initial pressure throughout the

simulation. A summary of the reservoir properties are given in Table 5a.

To study the impact of the parameters in the UTKR3P model, the values of some of the parameters were varied. The

values of the parameters for the base case are given in Table 5a. For the sensitivity cases, only the varied parameters are

summarized. In all the analyses below, since the UTKR3P model is phase independent, the phases have been numbered ‘1’,

‘2’, and ‘3’. For these simulations, however, they represent water, oil and gas, respectively. The relative permeabilities for

the grid cell closest to the injector have been plotted in all the figures. The impact of the saturation path dependency on the

relative permeability has been incorporated by a couple of parameter sets in the UTKR3P model. One is the parameter set of

‘C1’, ‘C2’, and ‘ ’ for each of the two-phase combinations denoted as 12, 13, 21, 23, 31, and 32. The second is the ‘b’

parameter for each phase (b1, b2, b3), which relates the two-phase residual saturations to the three-phase values. The impact of

each set of parameters is studied independently.

Sensitivity to parameter set (C1, C2,

): To simplify the analysis, only the two phase end point relative permeability has

been assumed to be different to show the impact of saturation path dependency on three phase relative permeabilities. Two

models, Case 2A1 and Case 2A2, as described in Table 5a and 5b are constructed for this purpose. Fig. 4a from Case 2A1

shows the saturation-averaged three phase relative permeability values for the hysteretic phase and discretized by time for the

SPE 165324-MS 7

first WAG cycle (30 days). Also plotted in Fig. 4a are the corresponding two phase curves. Tracking the gas saturation close

to the wellbore shows that from the start of gas injection, three phase flow occurs and the gas saturation in the gridlock

increases and reaches a maximum within the first 2 days. The water saturation decreases and reaches the connate water

saturation as shown by the gas relative permeability following the two-phase gas oil curve and producing most of the oil

during 2-15 days. Thereafter, as water is injected, the gas saturation starts to decrease and follows the two phase gas-water

curve until the gas saturation goes to zero at 30 days. This analysis shows how a simple difference in end point permeability

in the two phase dataset can manifest the saturation path dependency. The capability of the model to switch seamlessly from

two phases to three-phase and back to two-phase prediction is depicted by the simulation run. The above analyses are for

showing the saturation path dependency by varying the end-point values. Once the other parameters (C1, C2) are included in

the analysis, the impact becomes more pronounced.

Sensitivity to parameter ‘b’: Fig. 4b shows the results from the base case (Case 1A) and Case 2B. The only difference in the

models is the ‘b’ parameter, which is non-zero in Case 2B for one of the phases. The ‘b’ parameter impacts the three phase

residual saturations which in-turn modifies the three phase relative permeability. The first cycle overlaps for both the cases

and the relative permeability shifts slightly in the next cycles as the saturation direction is reversed.

Sensitivity to trapped gas: Trapped gas is one of the major contributors of hysteretic behavior in WAG injection. In this

model (Case 3), we have switched on the hysteretic model (UTHYST). The difference between this model (Case 3) and the

base case (Case 1A) is the trapped gas which is taken into account in Case 3. By following the red vertical line in Fig. 4c, one

can see that by going through the increasing and decreasing phase saturation cycles, the relative permeability decreases

because of the phase trapping. Fig. 4e shows the phase saturation and the trapped saturation as a function of time. The

trapped saturation increases during the decreasing saturation cycle due to capillary trapping at the pore level. The trapped

saturation remains constant during the increasing saturation cycle. This hysteretic behavior is also one of the major causes for

the reduced water injectivity seen in field cases of WAG injection.

Impact on oil recovery: Hysteresis may improve oil recovery by reducing the gas mobility and also freeing more oil in

exchange for the trapped gas. Fig. 4d shows the recovery factor for the Case 1A, Case 2B, and Case 3. The impact of

saturation path dependency alone has not resulted in a change in the oil recovery. Quantifying the effect of saturation path

alone is not straight-forward and depends on the direction and magnitude of saturation changes of each phase. In Case 3, the

oil recovery is higher as expected due to the additional oil mobilized from the pores where gas is trapped. The magnitude of

this could be much higher depending on the values for the hysteresis parameters , , and .

Testing the hysteresis model (UTHYST)

We further tested the UTHYST model by simulating a three-cycle WAG flood in sandstone sample with an initial oil

saturation of 75%. The maximum gas saturation in drainage processes constantly increases in this study. Figs. 5a and 5b

show the saturation path in a ternary diagram and the experienced gas saturation, respectively. Figs. 6a and 6b describe the

reversibility and irreversibility behavior of hysteretic phase in a saturation history dependent process using both Carlson and

UTHYST models incorporated in the UTKR3P model, respectively. The cycle-dependent relative permeability within

different WAG cycles constantly decreases gas relative permeability.

Summary and conclusions A new three-phase relative permeability model (UTKR3P) incorporating hysteresis and compositional dependency has been

developed, tested against experimental data for different wettability conditions, and implemented into a compositional

reservoir simulator. The new model is significantly more complete and flexible than previously published models.

The new hysteresis model is an extension of the Land trapping model but with a dynamic Land coefficient. The model

overcomes some of the limitations of published three-phase hysteresis models for non-water wet rocks and mitigates some of

the complexity and difficulties associated with commonly applied models in numerical reservoir simulators. The model is

validated using multi-cyclic three-phase water-alternating-gas experimental data for non-water wet rocks. Numerical

simulations of a carbonate reservoir with and without hysteresis were used to assess the impact of the saturation direction and

saturation path on gas entrapment and oil recovery. The hysteresis models of Carlson (1981) for intermediate and non-

wetting phases, Fayers and Matthews (1984) for the oil phase, and Larsen and Skauge (1998) for water and gas in water-wet

rock were also incorporated in UTKR3P, implemented into a compositional reservoir simulator, and successfully tested.

Acknowledgements The authors gratefully acknowledge the financial support provided by the DOE-NETL under contract number

DE-FE0005952.

8 SPE 165324-MS

Nomenclature

= Constant in general initial-trapped saturation formulation (Eq. 2)

= A wettability-dependent constant (Table 2)

= Parameter for relating three-phase to two-phase residual saturation (Eq. 6)

and

= Relative permeability curvature parameters (Eq. 3)

= Spreading coefficient,

= Constant in general initial-trapped saturation formulation (Eq. 2)

= Depth,

= Dimensionless oil-like parameters based on parachor-weighted molar density (Table 1)

= Gravitational constant,

= Molar Gibbs free energy, GFE/RT

= Gibbs free energy,

= Permeability tensor,

= End-point relative permeability

= Phase relative permeability

= Land coefficient

= Constant parameter impacts oil-isoperm curvature (Table 1)

= Trapping number

= Capillary pressure,

= Mean hydraulic radius of the pores occupied by a specific phase, (Table 1)

= Saturation,

= Reduced saturation (Table 1) ,

= Normalized saturation

= Residual saturation corrected for compositional changes (Eq. 8),

= Trapping model parameter (Eq. 9)

= Trapped saturation in current cycle,

= Permeability reduction factor (Eq. 13)

β = Constant in trapped saturation equation (Eq. 14)

and = Fitting parameters of zero-oil-isoperm curvature (Table 2)

and = Contact angle dependent coefficient (Table 3)

= Constant to change the curvature of ROS (Table 2)

ε = Pore size distribution factor in Land’s imbibition relative permeability equation (Table 4)

= Trapping coefficient (Eq. 9)

𝜆 = Empirical constant in the Killough’s parametric interpolation method (Table 4)

= Density,

= Interfacial tension (IFT),

= Trapping model parameter (Eq. 9)

= Phase potential,

= Fractal dimension of the porous medium

Subscripts and superscripts: = Two-phase

= Three-phase

= Connate water, Critical oil, or Critical gas saturation (minimum attainable saturation in multi-phase flow)

cr = Critical non-wetting saturation corresponding to the point where maximum trapped saturation is first reached (Table 3)

= Decreasing saturation process

= End of cycle

= Flowing saturation

= Initial

= Increasing saturation process

= Intermediate

= Total liquid

= Cycle number

= Value to oil at connate water saturation (Table 1)

= Primary decreasing

= Primary increasing

= Residual

= Phase ‘ ’ residual saturation to phase ‘ ’ = Trapped

= Value at irreducible water saturation (Table 1)

= Water-hydrocarbon (Table 1)

SPE 165324-MS 9

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SPE 165324-MS 11

of Water Alternate Gas (WAG) Injection.

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12 SPE 165324-MS

Table 1: Chronological list of some of three-phase relative permeability models

Model

Basis o

f

Inte

rpo

lation

We

ttability

Ph

ase

Ap

plicab

ility

Composition Dependency

Main Feature Nc or

NT Comp. Consis

Corey et al. (1956) C.P+B W 3P NO NO Requires at specified to reproduce all of the

and ;

;

Fluid flow in separate channel with no fluid-fluid interaction

Wyllie and Gardner (1958) P.S.D W oil & gas NO NO Water as part of the rock matrix

Naar and Wygal (1961) C.P+R W 3P NO NO 3-phase imbibition assuming

where =constant; Adding trapping gas

mechanism; is the only model’s adjusting parameter

Land (1968) C.P+B W 3P NO NO Imbibition 2- and 3- phase KR based on Corey model (1956) and modified Naar

and Wygal model (1961); Defining ‘free saturation’

Stone I (1970) and

Stone II (1973) C.F W 3P NO NO

Oil blockage by water and gas; and depend on their sat

Stone I: ( ) ; Stone II:

does not appear

explicitly; (assumption: segregated flow);

Stone II requires four 2P KR as opposed to only 2 for stone I

Modified Stone I (Hirasaki)

and Modified Stone II Dietricht and Bondor (1976)

C.F W 3P NO NO

Hirasaki: Representing all model 2-phase permeabilities relative to

Dietricht and Bondor: Renormalizing Stone II model to represent and

based on combination of KR to oil at and absolute permeability.

Modified Stone I and Stone II

(Aziz & Settari, 1979) C.F W oil NO NO Smooth reduction to 2-phase data due to added normalized sat.

Parker et al. (1987) C.P + Pc-S

W oil NO NO Based on van Genuchten’s Pc-S model and Mualem’s hydraulic conductivity

model (Mualem, 1976); Does not require 2-Phase KR data;

Alemań and Slattery (1988) C.F W oil NO NO Using a statistical structure model in local volume-averaged method; requires four

2P KR and reducing to Stone I if or is linear function of sat.

Baker 1 (1987) and

Baker 2 (1988) S.A W oil NO NO

1: ‘True linear interpolation’ by straight lines between equal KR on ternary diagram;

2: Interpolation between 2-phase KR values to find 3-phase oil KR

Delshad and Pope (1989) S.A W oil NO NO 2-phase KR does not appear explicitly

Modified Stone I

(Kokal & Maini, 1990) C.F W oil NO NO

Improved modified Stone I (Aziz & Settari, 1979) by adding one more

normalizing factor

Modified Stone I

(Hustad & Holt, 1992) C.F W oil NO NO where , , 0 1;

Dria et al. (1993) C.P+B W 4P NO NO Extension of Corey’s et al. model (1956) for 4P flow based on residual sat.

Robinson and Slattery (1994) C.F W oil NO NO Adding hysteresis and to Alemań and Slattery (1988) model

Bradford et al. (1997) C.P+B FW W & oil NO NO Adding the influence of wettability to Brudine model; gas is non-wetting phase

Jerauld (1997) P.L int. 3P YES Par

Inherent hysteresis;

; S.A of and which in turn are

based on S.A of oil and gas phase parameters using ; ;

;

; ;

;

and are general form of Stone I

Moulu (1997)

Moulu (1999) F.R ALL 3P NO NO

Extension of Vizika et al. model (1993); 3P in each fractal pore; each phase sat.

calculated as the relative area occupied; KR calculated by Poiseuille-type flow

; ;

Goodyear and Townsley (Balbinski et al. , 1999)

Pc-S W oil NO NO Extension of Brooks-Corey method (1966);

and reduces to Corey form

DiCarlo et al. (2000) L.D W, int. oil NO NO at ; Does not require 2-Phase KR data

Modified Baker (Blunt (2000)) S.A ALL 3P YES Den General Baker (1988) model based on layer drainage model; Inherent hysteresis

Modified Baker (Fayers

(2000)) S.A ALL 3P YES Den Introducing sat. rescaling: ( )

Hustad & Hansen (1996);

Hustad (2002);

Hustad & Browning (2010)

S.A ALL 3P YES NO

2-Phase process-dependent normalized sat. to look up 2-phase KR values and

estimate three-phase KR based on S.A between 2-phase residual sat.;

Inherent hysteresis

Shahverdi and

Sohrabi (2012) S.A ALL 3P NO NO

Independent impact of 2-phase KR on 3-phase KR; each phase in contact with 2

other phases in 3-phase fluid distribution

Yuan and Pope (2012) T.H ALL 3P YES GFE Adding compositional consistency to any 3-phase KR model

UTKR3P

(Developed model, 2013) S.A ALL 3P YES GFE

( : KR parameters); Inherent hysteresis (sat. path dependence)

but other hysteresis method can be added

*Abbreviations: 3P: Water/oil/gas; 4P: Water/oil/gas/2nd hydrocarbon (CO2-oil mixing at low temperature and reservoir pressures); ALL: Water-wet, intermediate-wet, and oil-wet; B: Burdine pore-size distribution model based on tortuous bundle of parallel capillary tubes (Burdine, 1953); C.A: Capillary tubes; C.F: channel flow theory(=probability-based model) considering at most one mobile fluid in any flow channel; Comp. Consis: Compositional consistency

(Property, if applicable); Den: Density; F.R: Fractal pore; FW: Fractional wet; Int.: Intermediate-wet; KR: Relative permeability; NAPL: Light-non-aqueous-phase

liquid; Par: Parachor-weighted molar density; Nc or NT: Capillary or Trapping number; Pc-S: Capillary pressure-saturation; P.L: Pore-level mechanisms interpretation; R: Random interconnection of straight capillaries; ROS: Residual oil saturation; S.A: Saturation average; sat.: Saturation; T.H: Intensive

thermodynamic properties, e.g. molar Gibbs free energy (GFE); W: Water

SPE 165324-MS 13

Table 2: Three-phase residual saturation correlations

Model Rock

Wettability Phase

Applicability Residual Saturation Correlation

Fayers and Matthews (1984) water oil where

Alemań (1986) ALL oil

(

)

where

Fayers (1989) ALL oil

Fayers et al. (2000) mixed oil

where

Yuan and Pope (2012) ALL water/oil/gas [

( )] where =1, 2, and 3 and ≠ ≠

Table 3: Trapping-initial saturation models

Model Fluid and Rock System Trapping Formulation

Land (1968) gas/oil in consolidated sandstone In Eq. 2:

,

Carlson (1981) Conceptual extension of Land’s eq. assuming parallel scanning curves

is shift in scanning waterflood curves

Jerauld (1997) gas/oil and gas/water in consolidated

intermediate-wet sandstone In Eq. 2:

,

and

Aissaoui (1983) and modified Kleppe et al. (1997)

gas/water in consolidated sandstone

Spiteri et al. (2008) pore-network model 𝜆 (

) 𝜆

Table 4: Chronological list of some of hysteresis models

Hysteresis Models Applicable to

Wettability Main Feature Water Oil Gas

Two-Phase Hysteresis (Saturation direction dependent):

Land (1971) NO NO YES Water

( ) ( ), , ;

Calls ε to be used in Corey-type equation to calculate imbibition KR

Killough (1976) NO NO YES Water Calls for bounding drainage and imbibition curves; needs λ parameter to

calculate imbibition KR; ( )

( )

( ) (

);

Carlson (1981) YES YES YES Water Needs a bounding drainage curve and a one point on an imbibition curve to produce imbibition curves; Parallel imbibition scanning curves

Fayers and Matthews (1984) NO YES NO ALL Oleic phase hysteresis when hysteresis is also applied to gaseous phase

Beattie et al. (1991) YES YES NO Water

Defining

,

, and

; Assuming a

specified saturation dependent form: and

estimating scanning curves for oil and water relative permeability

Delshad et al. (2003) YES YES N.A. Mixed Main drainage Pc and relative permeability, Brooks-Corey function scanning curve Pc, modified van Genuchten function, relative permeability prediction by integrating a pore-distribution model.

Three-Phase Hysteresis (Saturation history dependent: saturation direction and path cycle dependent):

Parker and Lenhard (1987)

YES YES YES ALL

Applying a KR-saturation-Pc model using 1) 2P wetting-fluid saturation-capillary head pair for primary increasing and decreasing processes and 2) three-parameters to define saturation paths

Larsen and Skauge (1998)

YES NO YES Water

Water: gas-saturation-interpolation-scheme between 2P (water-oil) and 3P (water-oil-trapped gas) water relative permeability;

Gas: ( )

[ ],

( ) ( )

Egermann et al. (2000)

YES YES YES ALL Application of fractal theory, Land formulation, and defining trapping and untrapping coefficients; Different Land constant at different cycles

UTHYST (Developmed model, 2013)

YES YES YES ALL Monotonically increasing trapped saturation but including capillary-desaturation effect at high trapping numbers; Dynamic Land coefficient

*Abbreviations: ALL: Water/Mixed/Oil; KR: Relative permeability; Pc: Capillary pressure

14 SPE 165324-MS

Table 5a: Reservoir and fluid properties Table 5b: Base case 2D simulation

used in 2D simulation cases parameters (without hysteresis)

Initial Pore Volume= 1.18 MMSTB

0.20 0.02

Horizontal Permeability=1010 md and kv/kh=0.2

0.23

2.82

=220°F and =1500 psia

0.33

3.90

=0.45

0.95

3.16

Initial Oil Composition:

0.82

0.00

CO2=%1.14; C1-N2=%14.54; C2-C3=%17.51; C4-C6=%12.5;

0.56

0.00

C7+=%12.61; C10+=%13.98; C14+=%12.37; C20+= %15.36

0.10

0.00

Light Oil (API=30.I and µo=1.9 cp; Pb=1091 psia)

0.00 0.00

Minimum Miscibility Pressure= 2850 psia

: Sat-weighted interpolation between

and

Table 5c: Summary of 2D simulations

Physical process Case description Case no. Property varied

Path dependency

Base case +

; saturation weighted 3P Kr03

Case2A1

0.56; 0.8;

IHYST 0 ; 0

Base case +

; explicitly specified

Case2A2

0.56;

0.8;

0.68

Base case + ≠ 0 Case2B 5.0;

0.0; 0.0

Trapped gas effect Base case + IHYSTPH3=4

(UTHYST model) Case3 1.05; 1.0

Figure 1: Effect of parameter 'b2' on the curvature of phase 2 isoperms

SPE 165324-MS 15

Figure 2: Measured 2- and 3-phase relative permeability in intermediate-wet sandstone (sample 15, Oak, (1991)). Blue triangle, red

circle, and green diamond show 2-phase relative permeabilities for phase 1 (water), phase 2 (oil), and phase 3 (gas), respectively.

Three sides of the triangle show six pairs of measured 2-phase relative permeabilities and the measured 2-phase residual saturation

are noted at the sides. Inside the triangle shows the measured 3-phase gas isoperms for the DDI process

Figure 3: Calculated three-phase relative permeability using proposed model vs. measured data (Sample 15, Oak, 1991)

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007

0

0.002

0.004

0.006

0.008

0.01

0 0.002 0.004 0.006 0.008 0.01

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Water (1) Oil (2)

Gas (3)

( Constant)

Phase-3 relative permeability in DDI process (inside the ternary diagram):

b1= 4.4

SEE1=0.088

b2=15

SEE2=0.028

b3=5.5

SEE3=0.005

Gas

Flood

kr31

kr13

kr12 kr21

kr32

kr23

( Constant)

16 SPE 165324-MS

Figure 4a: UTKR3P model-Path dependency (Kr031 ≠ Kr032) Figure 4b:UTKR3P model - Path dependency (bj≠0)

Figure 4c: UTKR3P model with UTHYST –Trapped phase 3 only Figure 4d: Oil recovery comparison among simulation cases

Figure 4e: Phase 3 saturation and trapped saturation in three WAG cycles (G: CO2 inj., W: Water inj.)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1

Ph

ase

-3 r

ela

tive

pe

rme

abili

ty

Phase-3 saturation

2P Kr32

3P Kr3: 0-2days

3P Kr3: 2-15days

3P Kr3: 15-15.5days

3P Kr3:15.5-30days

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1

Ph

ase

-3 r

ela

tive

pe

rme

abili

ty

Phase-3 saturation

Base case - No hysteresis (bj=0)

Path dependency (b1=5.0; b2=0.0; b3=0.0)

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1

Ph

ase

-3 r

ela

tive

pe

rme

abili

ty

Phase-3 saturation

0-15days

15-30days

30-45days

45-60days

60-75days

75-90days

63.0% 65.9%

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0 15 30 45 60 75 90

Oil

reco

very

fra

ctio

n

Time (Days)

Base case (No hysteresis)

Path dependency (bj≠0)

Trapped gas modeling (UTHYST)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 15 30 45 60 75 90

Ph

ase

-3 T

rap

pe

d S

atu

rati

on

Ph

ase

-3 S

atu

rati

on

Time (days)

G1 G2 G3 W1 W2 W3

SPE 165324-MS 17

Figure 5a: Ternary diagram for 3-cycle WAG Figure 5b: Saturation history in a 3-cycle WAG

Figure 6a: Calculated phase 3 relative permeability Figure 6b: Calculated phase 3 relative permeability

using Carlson’s model using UTHYST model

Phase 2

Phase 3

Cycle 1 Cycle 2 Cycle 3

Cycles 1,2, and 3

Cycle 1

Cycle 2

Cycle 3

S1 S2

S3

Time (days) P

has

e S

atu

rati

on

P

has

e-3

re

lati

ve p

erm

eab

ility

Ph

ase

-3 r

ela

tive

pe

rme

abili

ty

Phase-3 saturation Phase-3 saturation