Investigation of the Critical Velocity Required for a Gravity-Stable Surfactant Flood

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Investigation of the Critical VelocityRequired for a Gravity-Stable Surfactant

FloodShayan Tavassoli,SPE,Jun Lu,SPE,Gary A. Pope,SPE, andKamy Sepehrnoori,SPE, The University of Texas at Austin

Summary

Classical stability theory predicts the critical velocity for a misci-ble fluid to be stabilized by gravity forces. This theory was testedfor surfactant floods with ultralow interfacial tension (IFT) andwas found to be optimistic compared with both laboratory dis-placement experiments and fine-grid simulations. The inaccurateprediction of instabilities on the basis of available analytical mod-els is because of the complex physics of surfactant floods. First,we simulated vertical sandpack experiments to validate the nu-merical model. Then, we performed systematic numerical simula-tions in two and three dimensions to predict formation ofinstabilities in surfactant floods and to determine the velocityrequired to prevent instabilities by taking advantage of buoyancy.

The 3D numerical grid was refined until the numerical resultsconverged. A third-order total-variation-diminishing (TVD) fi-nite-difference method was used for these simulations. We inves-tigated the effects of dispersion, heterogeneity, oil viscosity,relative permeability, and microemulsion viscosity. These resultsindicate that it is possible to design a very efficient surfactantflood without any mobility control if the surfactant solution isinjected at a low velocity in horizontal wells at the bottom of thegeological zone and the oil is captured in horizontal wells at thetop of the zone. This approach is practical only if the vertical per-meability of the geological zone is high. These experiments andsimulations have provided new insight into how a gravity-stable,low-tension displacement behaves and in particular the impor-tance of the microemulsion phase and its properties, especially its

viscosity. Numerical simulations show high oil-recovery efficien-cies on the order of 60% of waterflood residual oil saturation(ROS) for gravity-stable surfactant floods by use of horizontalwells. Thus, under favorable reservoir conditions, gravity-stablesurfactant floods are very attractive alternatives to surfactant/poly-mer floods. Some of the worlds largest oil reservoirs are deep,high-temperature, high-permeability, light-oil reservoirs, and thuscandidates for gravity-stable surfactant floods.

Introduction

In this section, we briefly discuss the classical stability theory andphysics of surfactant floods.

Classical Stability Theory. There is a broad literature on the sta-

bility analysis of fluid flow through porous media with the empha-sis on miscible displacements (Koval 1963; Todd and Longstaff1972; Hickernell and Yortsos 1986; Tan and Homsy 1986; Homsy1987; Tan and Homsy 1988; Araktingi and Orr 1993; Riaz andMeiburg 2003). Engelberts and Klinkenberg (1951) were two ofthe first to study the stability of immiscible displacement processes.Hill (1952) noticed fingering in vertical miscible displacements inpacked beds and based the stability criterion on the pressure onboth sides of the interface. Instabilities at the interface between dis-placing and displaced fluids are responsible for the creation andgrowth of viscous fingers (Hill 1952; Chouke et al. 1959).

More recently, research on linear-stability analysis of immisci-ble displacements shows the importance of considering fluidproperties across the shock instead of considering the full Buck-ley-Leverett profile (Buckley and Leverett 1942). The stabilitycriterion has been found to be dependent on the viscosity ratio atthe front and the gravity number. Furthermore, recent results ofthis research imply that an unstable flow with an unfavorable mo-bility ratio could become stable as buoyancy dampens the growthof fingering at the shock front. However, it is not easy to stabilizea front under the effect of unfavorable gravitational forces bychanges in the mobility ratio (Riaz and Tchelepi 2004). Riaz andTchelepi (2006) also found the importance of considering theshape of relative permeability curves instead of merely taking into

account the endpoint relative permeabilities.To analyze the stability of surfactant floods on the basis of the

classical stability theory, we applied a conditional stability theoryin which viscous forces destabilize and gravity forces stabilize thedisplacement. Comparing with the classification in Lake (2008),as given in Table 1, surfactant flooding in a dipping reservoir isconditionally stable (Type I), whereas polymer flooding with afavorable mobility ratio in the same reservoir is unconditionallystable. The conditional stability is most useful in determining amaximum rate in a dipping reservoir withM

>1. For Type I sta-bility, the stability criterion is an upper bound for superficial ve-locity, which is shown inFig. 1and calculated by

ucriticalkk

rl0 ql0 qlgsina

M 1 ; 1

wherel0 andlare the displacing and displaced phase, respectively.The critical velocity, defined as the velocity above which the frontbecomes unstable, is calculated by use of Eq. 1. The mobility ratiois calculated by dividing the mobility of the displacing fluid bythe mobility of the displaced fluid. The oil bank is the displacedfluid for a surfactant flood. Its total mobility is given by

krl;T kr1 kr2 kr1

l1

kr2

l2; 2

where kr1 and kr2 are the water and oil relative mobility, respec-tively. The total mobility of the oil bank (krl;T) at different watersaturations is plotted inFig. 2on the basis of relative permeabilitydata available in Table 2,and the minimum value was calculated

to be 0.123 cp1. On the basis of the properties of the experiment,kz 5,500 md ( 5.4310

12 m2),krl0 1, Dqg 0.084 psi/ft (1900.13 Pa/m), ll0 0.77cp ( 7.710

-4 Pa.s), sina 1 (verticaldisplacement), and / 0.35, the critical Darcy velocity (ucritical)was calculated to be 0.397 ft/D(1.40110-6 m/s). Therefore, thecritical interstitial velocity or critical frontal velocity (Vcritical ucritical//) is equal to 1.134 ft/D, which is more than five times largerthan the critical velocity observed in experiment and simulationresults (0.2 ft/D). This study shows that good agreement betweentheory and experiments can be obtained by modifying the stabilitytheory to take into account the formation of microemulsion and itsviscosity.

Physics of Surfactant Floods. Recent advances in the develop-

ment and understanding of high-performance surfactants haveresulted in significant improvements in the efficiency of chemical

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CopyrightVC 2014 Society of Petroleum Engineers

This paper (SPE 163624) was accepted for presentation at the SPE Reservoir SimulationSymposium, The Woodlands, Texas, USA, 1820 February 2013, and revised forpublication. Original manuscript received for review 12 March 2013. Revised manuscriptreceived for review 3 August 2013. Paper peer approved 16 October 2013.

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flooding (Adkins et al. 2010; Yang et al. 2010; Adkins et al. 2012;Solairaj et al. 2012; Lu et al. in press). The development andapplication of large-hydrophobe Guerbet alkoxy sulfates arereported by Adkins et al. (2012) and Yang et al. (2010). Solairajet al. (2012) recently developed a new correlation to study the sur-factant structure and performance relationship on the basis of anextensive data set. However, high-performance surfactant floodscan be achieved only with the development of a surfactant withspecific characteristics, knowledge of reservoir properties andconditions, and careful engineering design of the process.

Consideration of the flow properties such as viscosity and den-sity of each liquid as a function of composition adds to the com-plexity of the physics of surfactant floods. Among the flowproperties, microemulsion viscosity is especially important. Microe-mulsion viscosity should be controlled because low viscosities arerequired for reasonable flow rates and to prevent high surfactantretention, high pressure gradients, and reduced sweep efficiency.Microemulsion viscosity depends on the oil concentration in themicroemulsion, among other variables. For the special case of grav-ity-dominated flow, which is the focus of this paper, microemulsiondensity is also important.

Surfactants reduce the IFT between oil and water to values ofapproximately 103 dyne/cm. The reduction in IFT reduces thecapillary pressure and ROS, which then results in an increase inthe water relative permeability. The mobility of the surfactant so-lution is then greater than the mobility of the oil bank that it ispushing. This unfavorable mobility ratio can lead to hydrody-

namic instabilities (fingering). Fingering should not be confusedwith channeling, which happens in reservoirs with large correla-tion-length permeability fields. Channeling means the displacingfluid flows preferentially in high-permeability layers, resulting inpoor sweep efficiency.

Fingering can be reduced or prevented by increasing the vis-cosity of the surfactant solution by adding polymer to it, by creat-ing in-situ foam, or by using gravity to stabilize the displacement.Mobility reduction prevents the formation of fingers and thusincreases sweep efficiency. However, there are some limitations

and challenges with the use of either polymers or foam for mobil-ity control. Limitations with polymers include the following: sta-bility of the polymer at high temperature, high pH, and highsalinity; compatibility with the surfactant; and injectivity. Com-monly used polymerssuch as copolymers of polyacrylamideand acrylatehydrolyze at high temperatures or high pH valuesand therefore become less tolerant of calcium and other multiva-lent cations in hard brines. Polymers also must be compatiblewith the surfactant (i.e., form a stable aqueous phase). High-vis-

cosity polymer floods might also cause injectivity problems. Lim-itations with foam include foam stability and gravity segregation.

Experimental Results

To study and understand the stability of surfactant floods, a seriesof experiments were performed by Lu et al. (2014). A brief sum-mary of these experiments is given here. Detailed results can befound in Lu et al. (2014). The experiments were carried out in aglass column packed with sand to provide visual observation ofthe fluids and fronts. The experimental results were used to evalu-ate oil cut and oil recovery by continuous upward injection of sur-factant solution without polymer at different injection rates. Inthese experiments, the injection rate, which is the most importantfactor in stability of the process, has been considered as the mainvariable with the corresponding interstitial (frontal) velocity val-ues of 0.2, 0.4, and 0.8 ft/D.

The sandpack was saturated with brine and the permeabilitymeasured. A tracer test showed the sandpack was nearly homoge-neous. Oil was injected and its permeability measured. Water wasinjected until the ROS was reached. Table 2 gives the propertiesof the sandpack. A 1 wt% surfactant solution was injected contin-uously for two pore volumes (PV) at 388C. The surfactant solu-tion consisted of 0.5% C1313 propylene oxide-sulfate, 0.5%C2024 internal olefin sulfonate, 2% isobutyl alcohol, and 0.5%Na2CO3. The sodium carbonate was used to reduce surfactantadsorption. The phase behavior, IFT, and microemulsion viscositywere measured and can be found in Lu et al. (2014). The sandpackwas cleaned and used again for the second and third experiments.

The relative permeability endpoints listed in Table 2 are the

VolumetricSweepEfficiency

Darcy Velocityucritical

Fig. 1Type I conditional stability (Lake 2008).

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1

TotalMobility(cp1)

Sw

Min. = 0.123 cp 1

Fig. 2Total mobility of the displaced phase and the minimumvalue.

TABLE 2SUMMARYOF EXPERIMENTPROPERTIES

Porous Medium F95 Ottawa Sand

Diameter (ft) 0.157

Length (ft) 0.82

Porosity 0.35

Permeability (md) 5,500

Temperature (F) 100

k

r2 0.873

Sr1 0.17

k

r1 0.365

Sr2 0.15

TABLE 1STABILITY CONDITIONS BASEDONRANGES FOR

MOBILITY RATIO ANDGRAVITYDIFFERENCE (LAKE 2008)

Case

Number

Mobility

Ratio

Condition

Gravity

Difference

Condition Description

1 M

< 1 Dqgsina> 0 Stable

2 M

> 1 Dqgsina> 0 Conditionally Stable (Type I)

3 M

< 1 Dqgsina< 0 Conditionally Stable (Type II)

4 M

> 1 Dqgsina< 0 Unstable

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average values for the three experiments. The oil-breakthroughtime and oil bank cut for each experiment are given in Table 3.

Photographs were made during each flood to observe the effectof increasing velocity. Figs. 3 through 5 are photographs at dif-ferent PV of injection for each velocity. The oil bank can clearlybe seen in these photographs because the oil is black. For clarityof explanation,Fig. 6is a schematic of what the four flow regimeswould look like if there were no mixing and fingering. The firstinterface from the top is between the initial condition (water andresidual oil) and the oil bank. The second interface from the top isbetween the oil bank and microemulsion that forms when the sur-factant mixes with oil and water. The third interface is betweenthe microemulsion and the aqueous surfactant solution pushing itupward.

As can be seen in Fig. 3, the displacement is stable or nearly

so at 0.2 ft/D and becomes unstable at higher velocities of 0.4 and0.8 ft/D. A very important and interesting observation from theseexperiments is the formation of fingers at the interface between

the surfactant solution and the microemulsion phase caused bythe lower viscosity of the surfactant solution compared with themicroemulsion.

Chemical-Flooding Simulator

UTCHEM is a multicomponent, multiphase, 3D numerical simu-lator formulated to mechanistically model various chemical-flooding processes (Delshad et al. 2011). Aqueous, oleic, and

microemulsion phases may appear depending on the salinity andother parameters affecting the phase behavior. First, the aque-ous-phase pressure is calculated on the basis of the overall mass

TABLE 3DESCRIPTIONOF EXPERIMENTS

Number

Frontal

Velocity

(ft/D)

Oil-Bank

Cut

Oil-Bank-Breakthrough

Time (PV)

Experiment No. 1 0.2 0.7 0.65



V = 0.2 ft/D

PV = 0.22 PV = 0.46 PV = 0.65

Pore Volume of Injection

PV = 1 PV = 1.24

Fig. 3Surfactant-flood progress in vertical direction at differ-ent PV with frontal velocity equal to 0.2 ft/D.

V = 0.4 ft/D

PV = 0.18 PV = 0.47 PV = 0.64 PV = 1


PV = 1.21 PV = 1.48

Fig. 4Surfactant-flood progress in vertical direction at different PV with frontal velocity equal to 0.4 ft/D.

V = 0.8 ft/D

PV = 0.22 PV = 0.42 PV = 0.66 PV = 1 PV = 1.25


PV = 1.45 PV = 1.6

Fig. 5Surfactant-flood progress in vertical direction at different PV with frontal velocity equal to 0.8 ft/D.

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balance. Second, the mass-conservation equations are solved for theconcentration of each species (Delshad et al. 1996). The flow equa-

tions are provided in Appendix A. These equations are solved im-plicitly for pressure and explicitly for concentrations by use of ablock-centered finite-difference scheme (Appendix B). The finite-difference method uses second- and third-order approximations forthe spatial derivatives. To increase the stability and robustness ofthe discretizations, a flux limiter that is TVD was implemented inthe simulator (Liu 1993; Liu et al. 1994).

Relative Permeability. The UTCHEM simulations used Corey-type relative permeability equations (Brooks and Corey 1966;Delshad and Pope 1989) as a function of trapping number tomodel the effects of both gravity and viscous forces on the mobili-zation of trapped oil:

krl k

rlSnlnl ; forl 1 ; ; np; 3

where the normalized saturation (Snl) is defined as

Snl Sl Slr

1 X3l1

Slr

; forl 1 ; ; np; 4

where k

rl, nl, and S lrare the relative permeability endpoint, expo-nent, and residual saturation for phase l . The dependence of resid-ual saturations on IFT is modeled as a function of trapping number.The trapping number becomes more important in this study be-cause it includes both gravity and viscous forces. The trappingnumber is defined as (Pope et al. 2000):

NTl

~~k ~rUl0 gql0 ql~rDh i

rll0; 5

where rll0 is the IFT between displaced (l) and displacing (l0)

phases. Residual saturations are then computed as a function oftrapping number as

Slr min Sl; Shighlr

Slowlr Shighlr

1 TlNTl

!; forl 1 ; ; np;

6

whereTl is the trapping parameter for phase l . The endpoints and

exponents in relative permeability functions are computed as alinear interpolation (Delshad et al. 1987; Delshad and Pope 1989)

between the given input values at low and high trapping numbersk

rllow; k

rlhigh; nlowl ; n

highl :

k

rl k

rllow

Sl0rlow Sl0 r

Sl0rlow Sl0rhighk

rlhigh k

rllow; forl 1; ; np:

7

nl nllow

Sl0rlow Sl0r

Sl0 rlow Sl0rhighnl

high nllow; forl 1; ; np:

8

SimulationModel

Numerical simulations were performed to capture and study insta-

bilities in surfactant floods. Grid-refinement studies were per-formed for each case, and results are given here. To ensure theconvergence and the accuracy of the solution, we used a flux-lim-ited TVD scheme in all the simulations. TVD schemes providehigher-order accuracy in the smooth regions along with oscilla-tion-free solutions across discontinuities. Application of a fluxlimiter imposes constraints on the gradients of the flux functionsto minimize the oscillations across shocks. To decrease the simu-lation time for our field case studies, we compared different solv-ers and found the PETSc package (PETSc 2010) significantlyreduced the simulation time. Field case studies are performedwith the use of this solver package.

Fingering patterns are affected by the permeability distribu-tion. Krueger (1989) and Chang et al. (1994) found that a smallperturbation in the permeability was enough to trigger viscous fin-gers for adverse-mobility-ratio displacements. In this study, ran-dom, uncorrelated, normally distributed permeability was used asthe perturbation to trigger fingers rather than correlated perme-ability, which causes channeling as well as fingering. Randompermeability is calculated by use of the following equation foreach gridblock in the direction of flow:

ki kavg1 ksRi; 9

where ki is the gridblock permeability, kavg is the average perme-ability,Riis a random number from a set of random numbers witha mean of zero and the standard deviation of unity, and ks is thepermeability coefficient of variation (Krueger 1989). A randompermeability field with an average permeability of 5,500 md was

used to match the results of the sandpack experiment shown inFig. 7. On the other hand, we used a Dykstra-Parsons coefficient

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. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

SurfactantSolution

Oil Bank

Microemulsion

z

Initial

Condition

Fig. 6Schematic of four idealized flow regimes.

0

0.16

6500

6000

5500

PER

M

(md)

5000

4500

0.33

0.5

0.65

0.82

Fig. 7Generated random permeability field.

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and correlation lengths to represent heterogeneities in the fieldsimulations performed in this study. The Dykstra-Parsons coeffi-cient is the most commonly used measure of permeability varia-tion and is calculated as (Dykstra and Parsons 1950)

VDPk0:50 k0:16

k0:50; 10

where k0:50 is the median permeability and k0:16 is the permeabil-ity one standard deviation (r) below k0:50 on a log-probability

plot. All permeability fields for field case studies are generated byPetrel (Petrel 2011), which will be discussed later.

Results andDiscussion

In this section, we first discuss the validation of the model basedon the sandpack experiments. Next, we present the results of asensitivity study to determine the effect of different parameters onthe stability of surfactant floods. Last, the feasibility of gravity-stable floods is discussed and compared with surfactant/polymerfloods.

Experiments at three different velocities were simulated andcompared with the experimental oil-cut and oil-recovery data (Ta-ble 3). The simulations were used to predict the critical velocityrequired for a gravity-stable surfactant flood. Fig. 8 shows that a10150 grid is adequate for simulating the sandpack experi-ments. The corresponding gridblock sizes are 0.014, 0.14, and0.016 ft in thex-,y-, andz-direction, respectively.

We compared numerical and experimental results at differenttimes to distinguish different interfaces at each velocity and investi-gate instabilities at different fronts. To specify different interfacesbetween immiscible phases, we plotted the total oil concentrationin each gridblock to locate the oil bank and the microemulsion vis-cosity in each gridblock to locate the microemulsion phase. Fig. 9shows the simulation results for the case of a frontal velocity of 0.4ft/D. The simulated oil bank is more extended compared with that

of the experiments. However, oil-saturation values at the gridblocksrepresenting the oil-bank region from simulation results are alsoless than the average value observed from experimental results

. . . . . . . . . . . . . . . . . . . . . . . .

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

Cum.

OilRecovery(fractionofIOIP)

PV

10x1x50

10x10x50

20x20x80

Fig. 8Grid refinement for the sandpack experiments.


0

0.4

43

0.3

32

0.2

21

0.1

11

0.0

00 15

.00

11.25

7.50

3.75

0.00

0.5

16

0.00

0.00

3.75

7.50

11.25

15.00

3.75

7.50

11.25

15.00

0.0

00

0.1

32

0.2

64

0.3

95

0.5

28

0.7

00 0

.00

3.75

3.50

11.25

15.00

0.53

4

0.3

56

0.7

91

0.5

93

0.3

96

0.1

98

0.0

00

0.1

78

0.0

00

0.3

87

0.2

58

0.1

29

0.0

00

0.16

0.33

0.5

0.65

0.82

C2-total ME VISCC2-total ME VISC C2-total ME VISC C2-total ME VISC C2-total ME VISC

0

0.16

0.33

0.5

0.65

0.82

0.16

0

0.33

0.5

0.65

0.82

0.16

0

0.33

0.5

0.65

0.82

0.16

00

5

10

15

20

0.33

0.5

0.65

0.82

Fig. 9Comparison between experimental and numerical results at frontal velocity equal to 0.4 ft/D. The model provides oil-con-centration and microemulsion-viscosity profiles for the corresponding PV of injection in the experiment.

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providing the same average oil-saturation value. The comparisonbetween the experimental and simulation results confirms the accu-racy of the fine-grid numerical model used in this study. The circles

show the presence of fingers in the experimental results and thoseproduced in the numerical model. This confirms that the model iscapable of capturing small-scale fingers.

Several parameters were varied to match the oil cut and totaloil recovery obtained in the experimental results at differentvelocities.Fig. 10shows that an increase in the oil relative perme-ability exponent increased the oil-breakthrough time. However,the oil-breakthrough time is not sensitive to the water relative per-meability exponent. The oil relative permeability exponent wasincreased from 2 to 4 to match the oil cut and oil-breakthroughtime at higher velocities. Cumulative oil recovery and oil cut forthe case of interstitial velocity equal to 0.4 ft/D are plotted in

Figs. 11 and 12, respectively. The simulation results match theexperimental results with the higher value of oil relative perme-ability exponent.

The feasibility of a gravity-stable surfactant flood and theeffect of several parameters on its stability are discussed in thissection. The simulation cases in this section are modeled after a2D cross section of a vertical surfactant displacement. The geom-etry and properties are different from the experiments presentedearlier. A grid-refinement study showed almost no differencebetween 40140 grid compared with finer grids of 50150and 80180. The oil saturation for a 2D cross section of a verti-cal surfactant/polymer flood with a favorable mobility ratio is pre-sented inFig. 13.The blue color zone shows the formation of theoilbank. Implementation of a mobility-control agent changes anunstable displacement to an unconditionally stable displacementwhere it is no longer rate dependent. The ratio of the polymer vis-cosity to water viscosity needed to achieve oil recoveries close to100% at different injection rates is presented in Fig. 14. Theamount of polymer needed for a stable displacement increases asthe velocity increases until it reaches a plateau, which verifies thatthe displacement becomes unconditionally stable at high polymerconcentrations.

Figs. 15 and 16 show the behavior of an unstable and a stablesurfactant flood. The oil saturation for a 2D cross section of a verti-cal surfactant displacement is shown in Fig. 15. The displacementfront is unstable at high velocity (0.5 ft/D) and is characterized bythe formation of fingers. By lowering the velocity to maintain sta-bility conditions, the surfactant displacement becomes stable with-out the use of polymer. Oil saturation for a 2D cross section of a

vertical surfactant displacement is presented in Fig. 16. The dis-placement front is stable at low velocity (0.02 ft/D). Fig. 17shows

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

1 2 3 4 5

OilBreakthroughTime(PV)

Oil Relative Permeability Exponent

nw = 2 ,V = 0.4 ft/day

nw = 2 ,V = 0.2 ft/day

Fig. 10Effect of oil relative permeability exponent on oil-breakthrough time at different velocities.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Cum.

OilRecovery

PV

Cum Oil - UTCHEMCum Oil - Experimental

Fig. 11Cumulative oil-recovery comparison between experi-mental and simulation results.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2

OilCUT

PV

Oil CUT - UTCHEM

Oil CUT - Experimental

Fig. 12Oil-cut comparison between experimental and simula-tion results.

0.00 0

16

32

48

64

800 16 32 48 64 80

0.15 0.30 0.45 0.60

Fig. 13Oil-saturation profile at 0.3 PV with polymer (V50.5 ft/D).

0

2

4

6

8

10

12

1416

0 0.5 1 1.5 2

polymer

/water

V (ft/D)

Fig. 14Polymer viscosity needed for a stable displacement asa function of velocity.

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the effect of velocity on oil recovery. The displacement becomes

stable, with oil recovery close to 100% at this low velocity andremains stable at lower velocities.

Effect of Surfactant Adsorption. Oil recovery in vertical surfac-tant displacements is presented in Fig. 18 for two different cases:one considering surfactant adsorption and one neglecting it.Higher values of surfactant adsorption retard surfactant propaga-tion and reduce oil recovery. Surfactant adsorption is modeled bya Langmuir-type isotherm, and it is considered to be irreversiblewith concentration and reversible with salinity. The adsorbed con-centration of surfactant C3 is given by

^C3 min

~C3;

a3 ~C3 C3

1 b3 ~C3 C3" #

; 11

where the concentrations are normalized by the water concentra-tion and the minimum is taken to guarantee that the adsorption isnot greater than the total surfactant concentration. Adsorptionincreases linearly with effective salinity and decreases as the per-meability increases, as follows:

a3 a31 a32CSEkref=k0:5; 12

whereCSEis the effective salinity and the ratio ofa3/b3representsthe maximum level of adsorbed surfactant and b3 controls the cur-vature of the isotherm. The adsorption-model parameters a31,a32,and b3 are found by matching laboratory surfactant-adsorption

data. The reference permeability (kref) is the permeability at whichthe input adsorption parameters are specified. The measured sur-factant adsorption in the sand was so close to zero that it wasneglected. However, field-scale simulations were performed with0.24-mg/g rock adsorption, a typical field value.

Effect of Buoyancy. Oil recovery in vertical surfactant displace-ments is presented in Fig. 19 for two different cases: one consid-ering favorable buoyancy force and one neglecting it. In theabsence of a favorable buoyancy force, oil recovery is low andremains unchanged even at low velocities. Hence, favorable buoy-ancy force dampens the creation of instabilities at the front andassists in attaining a stable front. This result implies the benefitsof the injection of surfactant solution at the bottom of the geologi-

cal zone and oil production in horizontal wells at the top of thezone.

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. . . . . . . . . . . . . . . . .

0.00 0.150

16

32

48

64

800 32 48 64 80

16

0.30 0.45 0.60

Fig. 15Oil-saturation profile at 0.3 PV without a mobility-con-trol agent (V50.5 ft/D).

0.00 0.150

16

32

48

64

800 16 32 48 64 80

0.30 0.45 0.60

Fig. 16Oil-saturation profile at 0.3 PV without a mobility-con-trol agent (V50.02 ft/D).

0.6

0.7

0.8

0.9

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6


V (ft/D)

Fig. 17Conditional stability of surfactant flood (velocitydependence).

0.6

0.7

0.8

0.9

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6OilRecovery(fractio

nofIOIP)

V (ft/D)

Surfactant ADS = 0

Fig. 18Effect of surfactant adsorption on oil recovery.

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6


V (ft/D)

No Favorable Buoyancy

Fig. 19Effect of favorable buoyancy force on oil recovery atdifferent velocities.

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Effect of Permeability. Oil recovery in vertical surfactant dis-placements is presented in Fig. 20 at 0.02 ft/D and 0.5 ft/D as afunction of vertical permeability. As expected, high vertical per-

meability is favorable and oil recovery increases as permeabilityin the flow direction increases.

Effect of Dispersion (Longitudinal and Transverse). Longitu-dinal and transverse dispersivities appear in the calculations ofdispersion tensor and are calculated as

~~KklijDkl

s dij

aTl

/Slj~uljdij

aLl aTl

/Sl

uliulj

j~ulj ; 13

where aLl and aTl are the longitudinal and transverse dispersivityof phase l, respectively. Cumulative oil recovery and oil cut for

vertical surfactant displacements at a velocity of 0.5 ft/D for dif-ferent values of longitudinal dispersivity are shown in Figs. 21and 22,respectively. The oil recovery decreases slightly for large

values of longitudinal dispersivity. Figs. 23 and 24 show that anincrease in the transverse dispersivity, which results in crossflowand mixing, improves the oil recovery by attenuating the fingers.

Effect of Relative Permeability. Different water/oil relativepermeability curves used in the sensitivity study are presented inFig. 25.The relative permeability curves are calculated as a func-tion of trapping number (Eqs. 3 through 8). Oil recovery in verti-cal surfactant displacement vs. injection rate corresponding tothese pairs of relative permeability curves is presented in Fig. 26.The oil recovery increases as the relative permeability curvesapproach a straight line, which occurs at high trapping numbers(ultralow IFTs).

. . . . . . .

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500


Permeability (md)

V = 0.02 ft/D

V = 0.5 ft/D

Fig. 20Effect of permeability on cumulative oil recovery attwo different velocities.

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

CumulativeOilRecovery

PV

Longitudinal Dispersivity = 0.1 ftLongitudinal Dispersivity = 1 ftLongitudinal Dispersivity = 10 ft

Fig. 21Effect of longitudinal dispersion on cumulative oil re-covery (V50.5 ft/D).

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2

OilCUT

PV

Longitudinal Dispersivity = 0.1 ftLongitudinal Dispersivity = 1 ft

Longitudinal Dispersivity = 10 ft

Fig. 22Effect of longitudinal dispersion on oil cut (V50.5 ft/D).

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

CumulativeOilRecovery

PV

Transverse Dispersivity = 10 ftTransverse Dispersivity = 1 ftTransverse Dispersivity = 0.1 ftTransverse Dispersivity = 0.01 ft

Fig. 23Effect of transverse dispersion on cumulative oil re-covery (V50.5 ft/D).

0

0.2

0.4

0.6

0.8

0 0.5 1 1.5 2

OilCUT

PV

Transverse Dispersivity = 10 ftTransverse Dispersivity = 1 ft

Transverse Dispersivity = 0.1 ftTransverse Dispersivity = 0.01 ft

Fig. 24Effect of transverse dispersion on cumulative oil re-covery (V50.5 ft/D).

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

kr

Water Saturation

krw, Low Ncap kro, Low Ncapkrw, Case 1 kro, Case 1krw, Case 2 kro, Case 2krw, Case 3 kro, Case 3krw, Case 4 kro, Case 4

Fig. 25Different relative permeability curves in the sensitivitystudy.

8 2014 SPE Journal


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3DField CaseStudy

In this section, we discuss the results of a simulation study of a

field case using reservoir characteristics favorable for a successfulgravity-stable surfactant flood without the use of any mobility-control agent. High vertical permeability, low oil viscosity, andlarge thickness were found to be favorable. Petrel (Petrel 2011)was used to generate the permeability field for the simulationmodel shown inFig. 27.

The idea is to inject the surfactant solution at a low velocity inhorizontal wells at the bottom of the geological zone and to cap-ture the oil at the top of the zone. As shown in Fig. 28, the hori-zontal injector is at the bottom edge of the reservoir model andthe producer is at the diagonally opposite edge of the reservoir

model. The simulation model is 1,0001,000210 ft with an av-erage permeability of 2,100 md and a porosity of 0.23. Fig. 29

shows that a 203050 grid is a sufficiently fine grid for a goodnumerical approximation of these field-scale displacements. Thecorresponding gridblock sizes are 50, 33.33, and 4.2 ft in the x-,

y-, and z-direction, respectively. The VDP is 0.7, and the dimen-sionless correlation lengths in the x-, y-, and z-direction are 3, 3,and 0.1, respectively.

Oil-recovery results for different injection rates and well spac-ings are presented in Fig. 30. The oil recovery increases as theinjection rate and well spacing decrease. This result shows thepotential for a gravity-stable surfactant flood to have a high oil re-covery on the order of 60% of waterflood ROS even in a heteroge-neous oil reservoir.

Summary andConclusions

Surfactant-flood experiments in sandpacks without polymer formobility control were simulated to understand the conditionsneeded for a gravity-stable surfactant flood. Fine-grid simulationswere performed to capture and study instabilities in surfactantfloods and estimate the critical velocity for a stable displacement.A TVD scheme was used in the UTCHEM simulator to ensure theconvergence and accuracy of the numerical solution by use of afine grid. The numerical results were compared with visual resultsof the experiments to locate different interfaces and instabilities.Creation of instabilities at higher injection velocities was pre-dicted correctly by simulation results in comparison with the ex-perimental results.

Simulations were performed to study the effect of differentparameters on the stability of surfactant floods. High verticalpermeability is extremely important because it benefits gravity-

stable floods in two ways: The critical velocity increases and theproject life decreases. Longitudinal dispersion was found to

0.7

0.75

0.8

0.85

0.9

0.95

1

0 0.1 0.2 0.3 0.4 0.5 0.6

OilRecovery(FractionofIOIP)

V (ft/D)

Case 4Case 3Case 2Case 1Low Ncap

Fig. 26Effect of relative permeability curve on oil recovery atdifferent velocities.

Permeability (mD)10000

1000

100

10

Fig. 27Permeability field generated by Petrel on the basis ofall the data points (50350380).

Producer

Injector

yz

x

Fig. 28Horizontal-well locations.

0

20

40

60

80

0 0.5 1 1.5

Cum.OilRe

covery(%IOIP)

PV

505050

203050

505080

Fig. 29Grid refinement for the field-scale simulation models.

0.3

0.4

0.5

0.6

0.7

0.8

0 10000 20000 30000 40000 50000

OilRecovery(fractionofIOIP

)

Injection Rate (ft 3/D)

Well Spacing = 500 ftWell Spacing = 1000 ftWell Spacing = 2000 ft

Fig. 30Oil recovery for field cases with different wellspacings.

2014 SPE Journal 9


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increase the growth of instabilities because it extends the mixingzone, which leads to the reduction in effectiveness of surfactantfloods. Transverse dispersion, on the other hand, was found todampen the instabilities and increase the effectiveness of thedisplacement.

These results indicate that it is possible to design a very effi-cient surfactant flood without any mobility control if the surfac-tant solution is injected at a low velocity in horizontal wells at thebottom of the geological zone and the oil is captured in horizontalwells at the top of the zone. This approach is practical only if thevertical permeability of the geological zone is high. These experi-ments and simulations have provided new insight into how a grav-ity-stable, low-tension displacement behaves and, in particular,the importance of the microemulsion phase and its properties,especially its viscosity. Numerical simulations show high oil-recovery efficiencies on the order of 60% of waterflood ROS forgravity-stable surfactant floods using horizontal wells. Thus,under favorable reservoir conditions, gravity-stable surfactantfloods are very attractive alternatives to surfactant/polymer floods.Some of the worlds largest oil reservoirs are deep, high-tempera-ture, high-permeability, light-oil reservoirs, and thus are candi-dates for gravity-stable surfactant floods.

Nomenclature

a3,b3 adsorption parameter for surfactant

a31 adsorption parameter for surfactant, (L2

)0.5

a32 adsorption parameter for surfactant, (L2)0.5 (eq/L3)1

C

k compressibility of species k, (m L1 t2)1

Ck adsorbed concentration of speciesk, L3/L3 PV

~Ck overall concentration of species kin the mobile andstationary phases, L

3/L

3PV

Ckl concentration of specieskin phasel,L3/L

3

Cr rock compressibility, (m L1 t2)1

CSE effective salinity, eq/L3

Ct total compressibility, (m L1 t2)1

D depth, LDkl diffusion coefficient of specieskin phasel, L

2 t1

~~Dkl dispersive flux of species kin phaselg gravitational constant, L t2

k

permeability, L2

~~k permeability tensor, L2

krl relative permeability of phasel

klowrl ; k

highrl Phase l endpoint relative permeability at low and

highNc,respectively~~Kkl dispersion coefficient tensor for species kin phasel,L2 t1

M mobility ratioM

endpoint mobility rationc number of components

ncv volume-occupying species (water, oil, surfactant)nl relative permeability exponent for phaselnp number of phases

nwbc number of wellblocksnwell number of wells

NBL number of gridblocksNTl trapping number of phase l

Pcl1 capillary pressure between phases l and water, mL1 t2

Pl pressure of phasel, m L1 t2

Qk source/sink for species kper bulk volume, L3

t1

/L3

rkl reaction rate for species kin phasel,m L3 t1

rks reaction rate for species kin solid phase, m L3 t1

Rk total source/sink for speciesk, m L3 t1

Sl saturation of phasel,L3/L3 PV

Slr residual saturation of phasel,L3/L3 PV

t time, tTl trapping parameter for phase l~ul Darcy flux of phasel,L t

1

V interstitial (frontal) velocity, L t1

a dip angle

aLl,aTl longitudinal and transverse dispersivity, respec-tively, of phase l,L

dij Kronecker delta functionkrl relative mobility of phasel, (m L

1 t1)1

k

rl endpoint relative mobility of phase l,(m L1

t1

)1

krTc total relative mobility, (m L1

t1

)1

ll viscosity of phasel,m L1 t1

qk density of species kat PRrelative to its density at 1atm, m L3

ql density of phasel,m L3

rll0 interfacial tension between phases land l0,m t2

s

tortuosity factor/ porosityUl potential of phasel,m L

1 t2

Dx,Dy,Dz gridblock size inx-,y-,z-direction, respectively, LDt timestep, t

Subscriptsk species number (1 water, 2 oil, 3 surfactant,

and so forth)l phase number (1 aqueous, 2 oleic, 3

microemulsion)ref reference condition

Superscripts

low low capillary-number extremehigh high capillary-number extreme

Acknowledgments

We would like to acknowledge the support by the industrial spon-sors of the Chemical Enhanced Oil Recovery Industrial AffiliatesProject in the Center for Petroleum and Geosystems Engineeringat The University of Texas at Austin.

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AppendixAFlowEquations

This appendix provides details of flow equations solved implicitlyfor pressure and explicitly for concentrations, by use of a block-

centered finite-difference scheme.The mass-conservation equation for component k is expressed

in terms of overall volume of component k/unit PV ( ~Ck) as

@

@t/ ~Ckqk ~r

Xnpl1

qkCkl~ul ~~Dkl

" # Rk; A-1

where the overall volume of component k /unit PV is the sumover all phases including the adsorbed phases,

~Ck 1 Xncvk1

Ck

!Xnpl1

SlCkl Ck; for k 1; ; nc:

A-2

We assume ideal mixing and small and constant compressibilitiesC

k:

qk 1 C

kPref Pref;0; A-3

and the source terms Rkare expressed as

Rk /Xnpl1

Slrkl 1 /rks Qk: A-4

The dispersive flux is assumed to have a Fickian form:

~~Dkl;x /Sl~~Kkl ~rCkl; A-5

where the dispersion tensor ~~Kkl is calculated as follows (Bear

2007):

~~KklijDkl

s dij

aTl

/Slj~uljdij

aLl aTl

/Sl

uliulj

j~ulj ; A-6

and phase flux is expressed on the basis of Darcys law,

~ul krl~~k

ll ~rPl qlg~rD: A-7

The pressure equation is developed by summing the mass-balanceequations over all the volume-occupying components. The pres-sure equation in terms of the reference phase pressure (Phase 1)becomes:

/Ct@P1

@t

~r ~~k krTc~rP1

~r Xnpl1

~~k krlc~rDXnpl1

~~k krlc~rPcl1Xncvk1

Qk;

A-8

where

krlckrl

ll

Xncvk1

qkCkl; A-9

total relative mobility is expressed as

krTcXnpl1

krlc; A-10

and Ct is the total compressibility of rock (Cr) and componentcompressibilities (C

k),

. . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

2014 SPE Journal 11
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Ct Cr Xncvk1

C

k~Ck: A-11

AppendixBTimestep Selection

The automatic timestep selection is derived from the method ofrelative changes for the first three components (water, oil, and sur-factant). The next time-level timestep size will be selected on thebasis of the following calculations (Delshad et al. 2011):

Dtn1 Dtnmin DClim

maxNBL

i1jDCi;kj

0B@ 1CA; k 1; 2; 3; B-1where DClim is the concentration tolerance and Dt

n1 is limited toDtmin Dt Dtmax. Dtmin and Dtmaxare calculated on the basis ofthe minimum and maximum Courant number (CN), respectively.CNis defined as

CN QDt

DxDyDz/; B-2

where Q is the maximum injection/wellblock or production/well-block. The minimum and maximum timesteps are calculated as

Dtmin CNMIN

minnwell

M1max

nwbc

i1

Qi

Dxi Dyi Dzi/i

; B-3

Dtmax CNMAX

minnwell

M1maxnwbc

i1

Qi

Dxi Dyi Dzi/i

; B-4

where CN is limited to CNMIN CN CNMAX. CNMIN andCNMAXare the minimum and maximum input parameters in thesimulator, respectively, and are specified on the basis of the pro-cess and the heterogeneity of the case. Here we used 0.001 and0.01 for the minimum and the maximum Courant numbers,respectively, in our simulation cases. For more heterogeneouscases, we used 0.0001 and 0.001 for these parameters, respec-

tively. DClim is a fraction of the initial or injected concentration(whichever is larger) for the first three components (water, oil,and surfactant). We used 0.01, 0.01, and 0.001 as a fraction of thetotal water, oil, and surfactant concentration, respectively, forDClim.

Shayan Tavassoli is a PhD degree candidate in the PetroleumEngineering Department at The University of Texas at Austin.Since 2010, he has served as graduate research assistant forenhanced-oil-recovery (EOR) and reservoir-simulation groupsat the Center for Petroleum and Geosystems Engineering, TheUniversity of Texas at Austin. Tavassolis current research focus

includes EOR, reservoir engineering, reservoir simulation, andunconventional resources. He is an active member of SPE.Tavassoli holds a bachelors degree in chemical engineeringfrom Sharif University of Technology and a masters degree inpetroleum engineering from The University of Texas at Austin.

Jun Lu is a PhD candidate in petroleum engineering at the Uni-versity of Texas at Austin. His research interests include improvedoil recovery, surfactant development, groundwater remedia-tion, and produced-water purification. Lu holds an MS degreein petroleum engineering from New Mexico Tech, and MSand BS degrees in environmental engineering from NanjingAgricultural University and Suzhou University of Science andTechnology, respectively. He is an SPE member. email: [email protected]

Gary A. Pope is the Texaco Centennial Chair in Petroleum En-gineering at The University of Texas at Austin, where he hastaught since 1977. Pope is Director of the Center for Petroleumand Geosystems Engineering at The University of Texas at Aus-tin. Previously, he worked in production research at Shell De-velopment Company for 5 years. Popes teaching andresearch are in the areas of EOR, reservoir engineering, natu-ral-gas engineering, and reservoir simulation. He holds a PhDdegree from Rice University and a bachelors degree fromOklahoma State University, both in chemical engineering.Pope was elected to the National Academy of Engineering in1999 for his contributions to understanding multiphase flowand transport in porous media and applications of these prin-ciples to improved oil recovery and aquifer remediation. Hisawards include SPE Honorary Member status, AIME Environ-mental Conservation Distinguished Service Award, SPE IOR Pio-

neer Award, the Lohmann Medal, SPE/AIME Anthony F. LucasGold Medal, SPE John Franklin Carll Award, SPE DistinguishedAchievement Award, SPE Distinguished Member Award, andSPE Reservoir Engineering Award.

Kamy Sepehrnoori is the W. A. Monty Moncrief CentennialEndowed Chair in Petroleum Engineering at The University ofTexas at Austin. His research and teaching interests includecomputational methods, reservoir simulation, parallel compu-tations, EOR modeling, inverse problems, naturally fracturedreservoirs, and unconventional resources. Sepehrnoori holds aPhD degree in petroleum engineering from The University ofTexas at Austin.

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SIMetricConversion Factors

ft 3.048* E01 m

md (9.8691016) m2

cp 1.0* E03 Pas

psi 6.894757 E00 Pa

D 86400 s

*Conversion factor is exact.

Investigation of the Critical Velocity Required for a Gravity-Stable Surfactant Flood

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