spe-110077-pa (simulation of compositional gravity-drainage processes)

812 December 2011 SPE Journal

Simulation of Compositional Gravity-Drainage ProcessesD.A. DiCarlo, SPE, and M. Mirzaei, SPE, The University of Texas at Austin; and

K. Jessen, SPE, University of Southern California

Copyright © 2011 Society of Petroleum Engineers

This paper (SPE 110077) was accepted for presentation at the SPE Annual Technical Conference and Exhibition, Anaheim, California, 11–14 November 2007, and revised for publication. Original manuscript received for review 2 August 2007. Revised manuscript received for review 6 October 2010. Paper peer approved 11 October 2010.

SummaryThe amount of wetting phase that is recovered by gravity drainage when the displacing fluid is not in chemical equilibrium with the initial fluid involves a complex interplay of gravitational, diffusive, and capillary forces. Previously, in Part 1 in a series of papers, we proposed analytic solutions for capillary/gravity equilibrium (CGE) in compositional gravity drainage and estimated the total recovery of wetting phase (DiCarlo and Orr 2007). In Part 2, we presented the results of experiments in an analog brine/isopropanol (IPA)/isooctane (IC8) system in which the vertical profile of the components was measured destructively after 3 weeks of drainage (DiCarlo et al. 2007). Here, we present numerical simulations of compositional gravity drainage. We find that the CGE solutions are approached asymptotically as the simulation grid is refined for a simplified phase diagram. For vaporizing drainages, a bank of wetting fluid is found to be created from early times in the drain-age because of wetting fluid imbibing back into swept regions. We show that including hysteresis in the capillary pressure curve limits the creation of the wetting-fluid bank. We compare the numerical simulations to the experimental observations and find that the simulations match well for condensing drainages but not for vapor-izing drainages, similar to what is seen for the CGE solutions.

IntroductionGas injection can be an efficient method to displace oil in a reservoir (Hutchinson and Braun 1961; Lake 1989). During the displacement, interaction between the gas and oil components will result in change of the composition and, consequently, a change in the interfacial tension between the phases and in the density and viscosity of both phases along the flow path (Gray and Dawe 1991). For viscous-dominated flows, the composition path and spatial profile of the components are often predicted using the method of characteristics (MOC) (Dumoré et al. 1984; Helfferich 1981; LaForce et al. 2010). In this case, the displacement efficiency will depend on the development of miscibility.

Because of the density difference between the oil and injected gas, gravity also can act as a driving force to displace oil, in a process called gravity drainage (Schechter et al. 1994). However, the composition path and recovery in the gravity-drainage process do not necessarily follow the MOC predictions because the driving force is a nontrivial combination of capillary and gravity forces, which change in space and time. For example, consider gravity drainage with no compositional changes. Here, the amount of oil recovered is determined by the competition between the capillary and gravity forces, whose ratio defined as inverse Bond number is (Schechter et al. 1994)

Nr

gLBt− =1 �

�

/

�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

where � is the interfacial tension (IFT) between the phases, rt is a characteristic throat radius of the porous media, �� is the den-sity difference between the wetting (�w, typically oil for a gas/oil

displacement) and nonwetting (�N, typically gas for a gas/oil displacement) phases, g is the gravitational acceleration, and L is the characteristic length over which the drainage takes place. IfNB

−1 > 1, capillary forces dominate, nonwetting phase cannot enter the porous media, and no wetting phase is recovered. If NB

−1 < 1, gravity dominates, with the ultimate recovery depending on the inverse Bond number and the details of the pressure/saturation curve (Schechter et al. 1994).

However if the injection gas is not in equilibrium with initial oil, which is generally the case in gas-injection processes, IFT and density difference change throughout the composition path and so does the inverse Bond number (DiCarlo and Orr 2007; DiCarlo et al. 2007). For example, the left-hand side of Fig. 1 shows a con-densing drive gravity drainage with initial gas on the extension of the short tie line, which translates to low IFT. Oil is on the exten-sion of a longer tie line with higher IFT. Because the system is drained under gravity, the low-IFT region will appear near the top of the column and high-IFT region will appear near the bottom of the column. The case will be reversed in a vaporizing drive with low IFT occurring near the bottom and high IFT occurring near the top. In the intermediate region, the composition must transition between the two ends, with this transition being determined by the density difference and IFT of the phases. However, these proper-ties are functions of compositions themselves, so calculating the recovery in such cases is much more complicated compared with the viscous-dominated case.

Few studies have been performed to study the recovery from gravity drainage in nonequilibrium systems. Jacquin et al. (1987, 1989) studied the gravity drainage of a C1/C4/C10 fluid system at high temperature and pressure in sandstone cores and concluded that the recovery increases as more C4 is added to either of the phases, but the numerical simulations did not compare well with the experimental results. DiCarlo and Orr (2007) found CGE solutions for a simplified three-component, two-phase system with parallel tie lines (see Fig. 2b) on the basis of the theoretical arguments. Longitudinal diffusion was seen to play a significant role for condensing drainages, and circuitous composition paths were predicted for vaporizing drainages. In a companion work, DiCarlo et al. (2007) performed experiments using an analog fluid system to study the process and showed that the CGE predictions agreed well with observed equilibrium profiles and recoveries for the condensing drainages but showed that the match was not as good for the vaporizing drainages. Instead, a pure-advection solution based on MOC agreed better with observed composition distributions of the vaporizing drainages. It was hypothesized that, for the vaporizing drainage, CGE was not likely to be reached because of capillary hysteresis and countercurrent-flow consider-ations (DiCarlo et al. 2007).

There remains a need to understand the temporal dynamics of compositional gravity drainage because CGE theory uses CGE to predict only the final (equilibrium) composition profiles. The experimental procedure is destructive and provides the composition profile only at a single time for each experiment. Numerical com-positional simulation is the obvious approach to investigate the time evolution of compositional drainages. The dimensions of the experi-ments performed by DiCarlo et al. (2007) suggest that 1D modeling should be appropriate for describing the displacement processes, provided that the essential physics is represented adequately in the simulation model. The essential physical processes that must be included to study the dynamics of the drainage processes of this

December 2011 SPE Journal 813

work are phase equilibrium; gravity-driven advection; capillarity, including hysteresis in the capillary pressure functions; and molecu-lar diffusion. When numerical calculations are applied to analyze displacement behavior, special care must be taken to reduce the numerical artifacts (numerical diffusion/dispersion) to a level such that the physical model is adequately resolved.

The main purposes of the paper are as follows: (1) to use numeri-cal simulation to obtain a temporal understanding of compositional gravity drainage; (2) to compare the simulation model against the experimental results; (3) to compare the simulation to CGE theory for a simplified phase diagram where CGE theory should be exact, and for a real phase diagram where CGE solutions are approxima-tions; (4) to understand the role of capillary pressure hysteresis and countercurrent relative permeabilities in compositional grav-ity drainage; (5) to determine the best approximation method for compositional gravity drainage for both condensing and vaporizing displacements; and (6) to fully determine the role of capillary forces and diffusion in these compositional displacements.

Phase DiagramTo minimize the complexities associated with high-temperature and -pressure experiments, analog fluid systems have been widely

used in the experiments related to compositional effects. Here, we use an analog fluid system containing IPA, IC8, and 10 wt% NaBr brine at room temperature and pressure, and its phase diagram is depicted in Fig. 2a (DiCarlo et al. 2007; Morrow et al. 1988; Schechter et al. 1994). In this system, IC8 acts as the lightest hydrocarbon and primary constituent of the gas phase, while brine acts as the main constitute of the oil phase. IPA represents the intermediate component and transfer between phases.

In addition to the measured phase diagram, a simplified phase diagram, shown in Fig. 2b, was used primarily to study the accu-racy of the CGE solutions (DiCarlo and Orr 2007) and the effect of hysteresis and countercurrent flow. This simplified phase diagram makes use of parallel tie lines, a specified binodal curve, and plait point to allow for simple phase equilibrium calculations. In this model, the phase viscosities are taken to be a constant 10−3 Pa·s. The advantage of this phase diagram is that the CGE solutions should be exact (DiCarlo and Orr 2007).

Experimental SetupThe experimental results were presented in a previous study (DiCarlo et al. 2007); here, we briefly review the relevant details of the experimental setup depicted in Fig. 3. The 60-cm-long experimental column consisted of 20 sections, 3 cm each, which are held together by Teflon shrink tubing. Size-60 sand (0.15–0.3 mm in diameter) was packed into the column and provided a uniform porous medium. In each experiment, the system was first flushed with CO2 and then flooded with the working wetting phase from the bottom. Primary drainage was performed by attaching a burette full of nonwetting displacing fluid (density �n

d) to the top of the column. The composition of initial and displacing phases for different experiments is given in Table 1.

The top boundary condition is given by

P z z g z zn t nd

n t=( ) = −( )� , . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

where zn is the height of the nonwetting phase in the burette and zt is the position of the top of the column.

Likewise, the bottom boundary condition consisted of a connec-tion to a constant-head tank full of the initial wetting phase (density�w

i ), and, thus, the bottom boundary condition is given by

P z z g z zw b wi

w b=( ) = −( )� , . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

where zw is the height of the wetting phase in the burette and zb is the position of the bottom of the column (zt = zb+L, where L is the length of the column). The z = 0 position in the column is defined as the location where the nonwetting-phase pressure would be

Fig. 1—Illustration of the condensing and vaporizing drainages. The amount of total recovery obtained will depend on the in-terplay of capillary and gravitational forces within the region labeled “?,” where the compositions can vary greatly.

)b( )a(

Fig. 2—(a) Phase diagram for the analog brine/IPA/IC8 ternary system at room temperature. (b) Simplified phase diagram.


equal to the wetting-phase pressure if there were no compositional changes during the displacement; i.e.

P z gz gz P zw wi

w nd

n n=( ) = = = =( )0 0� � . . . . . . . . . . . . . . . . . (4)

The tank heights were chosen such that the z = 0 position was tens of centimeters above the bottom of the column to prevent the nonwetting phase from exiting the column and fouling the bottom boundary condition. Table 2 presents the top and bottom boundary conditions for the performed experiments.

After achieving equilibrium in 3 weeks, the column was cut into 3-cm sections and each section had its fluids dissolved in ethanol. The fluid samples from each section were injected into a gas chro-matograph to determine the overall composition in each section.

Numerical ModelWe use an implicit-pressure/explicit-composition (IMPEC) scheme to simulate gravity drainage in the vertical direction. The conservation

equations were discretized in the standard finite-difference frame-work (Aziz and Settari 1979). We assume no volume change on mixing such that the overall flux is constant throughout the column. This is a reasonable approximation for the fluid system used in the experimental work as discussed by DiCarlo et al. (2007). In each timestep, the pressure equation is solved implicitly and the nonwet-ting- and wetting-phase velocities are evaluated from Darcy’s law. The phase velocities are then used to evaluate the individual-com-ponent fluxes. Physical diffusion is added to the advective fluxes by a diffusive-flux approximation. From the overall fluxes (advection and diffusion), the compositions in each gridblock are then updated explicitly to the next time level. Finally, the compositions in each gridblock are equilibrated by flash calculations to obtain the new phase saturations, phase densities, viscosities, IFT, and capillary pressure to form the pressure equation at the new time level.

Crucial to the simulation is performing the flash accurately using the measured phase behavior. The measured phase diagram of the brine/IPA/IC8 system (Fig. 2a) was used in the flash rather than a simplified equation of state (e.g., Peng-Robinson). This was conducted as follows. For each local composition in the two-phase region, the relevant tie line is approximated by a tie line that passes through the local composition and the intersection point of the mea-sured tie lines that surround the local composition (Batycky 1994). From this tie line, the volume fractions of the phases are found using the lever rule, the densities of the phases are found through a linear interpolation of the densities of the measured surrounding tie lines, and the IFT between the phases is found by a logarithmic interpola-tion from the IFT of the surrounding tie lines. Additionally, because brine and IPA have strong hydrogen bonding, the viscosity of the brine/IPA mixture is nonmonotonic with brine fraction. Therefore, the viscosities of the phases were estimated by fitting the data in Morrow et al. (1988) to a second-order polynomial.

For the simplified phase diagram (Fig. 2b), the flash is much simpler, with the details given in DiCarlo and Orr (2007).

For both phase diagrams, capillary pressure is evaluated from Leverett J-function scaling in the form

P P P C J Sc n w i w= − = ( ) ( )� , . . . . . . . . . . . . . . . . . . . . . . . . . (5)

where the IFT, �, for two-phase compositions is determined from the equilibrium phase compositions. J is a modified Leverett J-function for the particular porous medium (Lake 1989; Leverett 1941) and is often modeled by a Corey (Brooks and Corey 1964) or a van Genuchten (van Genuchten 1980) type of pressure/satu-ration curve. We use a single drainage Leverett J-function of the Corey type,

J S r Sw t w( ) = ( )− −1 1* /�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

where rt is the characteristic throat radius, � is the Corey exponent, and Sw

* is the reduced wetting-phase saturation given by

SS S

Sww r

r

* = −−1

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

where Sr is the residual saturation. This model is reasonable if all of the regions undergo drainage at all times. If the composition within a gridblock is in the single-phase region, the block is assumed to

Displacingphase

Initialphase

Sandcolumn

L

z=zb

z=zw

z=zt

z=zn

z=0

z

Fig. 3—Schematic of experimental setup; z = 0 is the height where wetting- and nonwetting-phase pressures are equal given no compositional changes.

TABLE 1—INITIAL- AND DISPLACING-PHASE COMPOSITIONS OF THE FOUR EXPERIMENTAL SETS

Initial phase esahp gnicalpsiD

Drive Brine IPA IC8 Brine IPA IC8

Condensing 1 1.00 0.00 0.00 0.01 0.15 0.84 Condensing 2 1.00 0.00 0.00 0.00 0.48 0.52 Vaporizing 1 0.57 0.43 0.00 0.00 0.00 1.00 Vaporizing 2 0.36 0.64 0.00 0.00 0.00 1.00


have only wetting phase, with the wetting and nonwetting phases having equal pressure (no capillary pressure).

The relative permeabilities were taken to be a Corey type, with

k Srw w= +( )* /3 2 � for wetting phase . . . . . . . . . . . . . . . . . . . . . . (8)

and

k Srn n= *2for nonwetting phase. . . . . . . . . . . . . . . . . . . . . . . . (9)

All relevant parameters used in the simulations are given in Table 3. Within a gridblock, molecular diffusion between the phases is not

accounted for explicitly because the assumption of thermodynamic equilibrium (flash calculation at every timestep) will always dominate the mass transfer between phases. Between gridblocks, molecular dif-fusion is accordingly assumed to take place within each phase only. The explicit treatment of physical diffusion by a diffusive flux used in this work is known to reduce the stable timestep in the IMPEC formulation (Jessen et al. 2008). However, in the 1D computations of this work, the reduced timestep still results in feasible computer requirements. For most of the simulations, we use an approximate and constant value of the molecular-diffusion coefficient of 2×10−10

m2/s (Matthews and Akgerman 1988); however, because the diffu-sion constant varies slightly with concentration (Bird et al. 2007), we perform a set of simulations with different diffusivities to show how molecular diffusion affects the displacement.

The boundary conditions were set as follows. To compare experimental, numerical, and theoretical results, the nonwetting-phase pressure at the top of the porous medium and the wetting-phase pressure at the bottom of the porous medium were set equal to the boundary conditions applied for each experimental run (DiCarlo et al. 2007). For the simulation comparing the simplified phase diagram, there was no concern about fouling the bottom boundary condition, so the z = 0 height was set to be at the bottom of the column (zb = 0). Thus, to compare numerical and theoretical results (DiCarlo and Orr 2007), the nonwetting-phase pressure was set at the top of the column to be

P z z P z z gLn t w b nd=( ) = = =( ) −0 � . . . . . . . . . . . . . . . . . . . (10)

ResultsIn this section, we report a detailed comparison of the experimental observations with analytical and numerical calculations to delin-eate the interplay between capillary and gravity forces.

Simplifi ed Phase Diagram. We fi rst present the results from using the simplifi ed phase diagram. This is done primarily to discuss the roles that numerical diffusion and hysteresis play in the numerical calculations. In the fi rst set of simulations, there is no capillary or relative permeability hysteresis included in the model.

Fig. 4a shows the compositional profile (bottom plot) and com-position path (ternary diagram) for a condensing drainage after 389 hours. In this simulation, 300 gridblocks and a constant timestep of 0.5 seconds were used. On the right-hand side of the profile, the capillary pressure obtained by the simulation is reported. Note that, by setting the nonwetting-phase pressure at the top of the column, the nonwetting-phase pressure is not equal to the wetting-phase pressure (i.e., the capillary pressure is not zero) at the bottom of the column toward the end of the simulation.

Fig. 4b shows the CGE solution for this condensing drainage with the same boundary conditions [which again are slightly differ-ent from those in DiCarlo and Orr (2007)]. This solution assumes that CGE is reached after 389 hours. The agreement is seen to be very good, with only three minor discrepancies: The simulation has (a) a capillary fringe 1 cm higher in the CGE solution, (b) slightly more wetting phase remaining at the top of the column, and (c) a slightly broader zone of transition in the alcohol fraction. The first two discrepancies are a result of the simulation departing slightly from CGE because there is still a driving force for drainage between 40 and 55 cm of the column. This can be seen through the capillary pressure values on the right-hand side of the profile. To reach equilibrium in the simulation, there would be additional drainage of the wetting phase, which allows the alcohol to invade farther into the column and move the capillary fringe slightly down. The third minor discrepancy is because of the numerical diffusion in the simulation and is discussed next.

As expected for the simplified phase diagram, the alcohol component shows an error-function-type behavior with distance (DiCarlo and Orr 2007). To study the effect of the numerical diffusion on the simulation result, a grid-refinement study was performed. Fig. 5 shows the alcohol composition vs. distance for simulations with different numbers of gridblocks along with the CGE solution that includes physical diffusion. Adding more grid-blocks reduces the numerical diffusion, and the curves converge toward the theoretical limit. Also, we see that the 300-gridblock simulation has a diffusion length (6.8 cm) that is only 13% greater than the diffusion length (6.0 cm) of the 1,200-gridblock simula-tion. The differences in the brine and IC8 compositions between the simulations are also very minor. Because the 300-gridblock simula-tion is much less computationally expensive (it is 40 times faster than the 1,200-gridblock simulation) and seems to show all of the relevant features, all of the simulations shown using the simplified phase behavior are performed with 300 gridblocks.

Fig. 6 shows the compositional profile and composition path (ter-nary diagram) for a vaporizing drive after 389 hours, as predicted by numerical calculations and CGE theory. As for the condensing drive, the match is very good, especially considering that the numerical solution has yet to reach CGE. Several features are of note in this example. First, the wetting-phase saturation vs. distance is highly nonmonotonic. Both the simulation and the CGE solution show a bank of brine (wetting phase) behind the initial drainage front.

Fig. 7 shows the development of this bank at different times during the drainage. Clearly, the simulations predict the bank to

TABLE 2—TOP AND BOTTOM BOUNDARY CONDITIONS OF THE FOUR EXPERIMENTAL SETS

Drive Top pressure (Pa) Bottom pressure (Pa)

Condensing 1 1635 5657 Condensing 2 1698 5720 Vaporizing 1 1369 5390 Vaporizing 2 964 4648

TABLE 3—MEASURED WETTING-PHASE RETENTION PARAMETERS FOR THE SAND USED IN THE EXPERIMENTS*

Parameter Symbol eulaV

Inverse J -function J 1 Sw = (Pcrt/ ) (1 Sr)+Sr

Throat radius rt 40 10 6 m Lambda 2.11

Residual saturation Sr 0.09

* The inverse Leverett J-function was of the Corey type and was estimated from two-phase (air/water) -saturation profiles for the sand.


exist from early times (0.5 hours). As the drainage front moves through the porous medium with time, the distance between the bank and the displacement front increases. At later times (5 to 50 hours), the drainage front essentially stops moving but the bank continues to evolve and increase its amount of wetting phase.

Fig. 8 shows the overall flux vs. time for the condensing and vaporizing simulations. Both drainage processes start at the same rate, but the flux for the condensing drainage drops faster than the flux of the vaporizing drainage process. After 8 hours, the mag-nitude of the fluxes cross, and the condensing drive has a higher flux for the rest of the displacement. The insert of Fig. 8 shows the long time-scale development of the overall flux. Note that the vertical axis in the inset is three orders of magnitude smaller than the vertical axis of the main figure. The inset shows that the overall flux becomes negative after 10 hours for the vaporizing drainage but remains positive for all times for the condensing drainage. Thus, 10 hours after the onset of the vaporizing drainage process, wetting fluid is being sucked into the bottom of the column, with nonwetting fluid leaving the top of the column.

Just as importantly, the simulations show that the imbibition processes within the column start well before 8 hours. This can be observed from the earliest time at the top of the column. Fig. 9 shows the saturation vs. time for three selected gridblocks. Wetting phase starts imbibing in the gridblock at z = 50 cm after approximately 20 minutes; gridblocks above this zone begin imbibing even earlier. The gridblock at z = 40 cm goes through two cycles of drainage and imbibition and is draining toward the end of the simulation.

Fig. 10 shows the regions and times where the wetting and non-wetting fluxes are in opposite directions. At early times (less than 10

z, c

m

z, c

m

Pc,

Pa

Pc,

Pa

)b( )a(

Fig. 4—Comparison of (a) numerical calculation and (b) CGE solution for a condensing drainage. In this and many of the following figures, the composition path is shown in the ternary plot with the open square signifying the initial composition and the open circle signifying the displacing composition. The overall profile in space is shown below the ternary plot, with the left-hand axis signifying the position in the column and the right-hand axis signifying the Pc calculated with the densities of the phases at the end of the displacement. The axes are displaced so that z = 0 at the height at which Pc = 0 using the densities of the initial and displacing phases. Note that this is different from the Pc = 0 level in the graphs because the compositions and densities of the phases have changed inside the column during the displacement.

z, c

m

IPA, volume fraction

150 blocks300 blocks600 blocks1,200 blocksTheoretical diffusion

Fig. 5—Results of grid refinement compared to theoretical CGE solution.


hours), most of the column has the wetting and nonwetting phases flowing in the same direction, but there is a region right at the capil-lary fringe where the wetting phase is moving upward and the flow is countercurrent (shaded region). This region extends to the earliest

times at the top of the column. At late times (after 10 hours), the wet-ting phase is moving upward throughout the whole column; but, at these times, the fluxes are quite small (see Fig. 8). Neither imbibition nor countercurrent flow is observed in the condensing drainages.

)b( )a(

z, c

m

z, c

m

Pc,

Pa

Pc,

Pa

Fig. 6—(a) Numerically simulated and (b) CGE solution for a vaporizing drainage.

z, c

m

Wetting-Phase Saturation

t=0.5 hrst=5 hrst=50 hrs

Fig. 7—Simulated wetting-phase saturation vs. height at three different times during the vaporizing drainage.

Time, hours

Ove

rall

Flu

x, m

/s

Fig. 8—Overall flux through the column vs. time for vaporizing and condensing drainages.


To simulate the vaporizing drainages adequately, capillary hysteresis and countercurrent relative permeabilities both need to be included in the model. We do this in the following manner. For countercurrent flow, we decrease the relative permeability of both phases by a factor of 2. Fig. 11a shows the results of the vaporizing drainage when countercurrent relative permeabilities are added. As can be seen, the changes from the case with regular relative perme-abilities (Fig. 6a) are very minimal. This is not surprising; as long as the relative permeabilities are not zero, they only determine the rate at which equilibrium is reached, while the forces (capillary and gravity) determine the equilibrium.

Including capillary hysteresis is more involved. We use a Leverett J-function that is different for imbibition and drainage, which also provides reasonable scanning curves. The formulation is based on the hysteresis model of Scott et al. (1983). The main drainage curve and the main imbibition curve are both given by Corey-type curves in Eq. 6 but with different characteristic radii. For this particular medium, we use rtD = 4×10−6 m (drainage) and rtI = 7×10−6 m (imbibition). The scanning curves mimic the shape of the main curves but are scaled such that the pressure is continuouswhen going from drainage to imbibition or vice versa.

Fig. 11b shows the results of the vaporizing drainage when capillary hysteresis is added. As opposed to countercurrent relative permeabilities, including the capillary hysteresis affects the predic-tion of the composition path. The inclusion of hysteresis results in the removal of the IC8 bank because IC8 now shows a monotonic increase with height. The brine is still nonmonotonic with height, but the bank of brine above the lower fringe is much smaller than what was seen without hysteresis.

Including hysteresis results in a 10% higher recovery of the wetting phase, even though the drainage fronts have the same final height. Clearly, imbibition and hysteresis can play a large role in the dynamics and total recovery of vaporizing drainages.

Experimental Phase Diagram. The results show that the simula-tion method produces reasonable physics with the simplifi ed phase diagram. We now report simulation results using the real brine/IPA/IC8 phase diagram (as shown in Fig. 2a) to compare with the experimental observations (DiCarlo et al. 2007). For the condens-ing drainages, the wetting phase is always continuously decreasing, so the model without capillary hysteresis is appropriate. For the

vaporizing drainages, we report results from simulations both with and without hysteresis.

Fig. 12a compares the simulation results for the first condens-ing drainage with the experimental observations (solid points) after a simulated 722 hours. For this and all the experimental figures, the capillary pressures on the right-hand side of the profile are estimated from the profile being at CGE, the applied boundary conditions, with the density difference given by the measured compositions. The z = 0 level on the left-hand axis is the height at which Pc = 0 using the densities of the initial and displacing phases. Again, this is different from the Pc = 0 level on the right-hand axis because the compositions and densities of the phases have changed inside the column during the displacement. Comparing to the experimental data, we see a generally good agreement, with the transition between states at roughly the same height (18 cm above z = 0 in the simulation vs. 15 cm in the experiment) and with roughly the same path through composition space. Starting from the initial composition (open square), the composition path moves up the wetting side of the binodal curve, followed by a rapid jump to the nonwetting side close to the displacing composition (open circle). For comparison, Fig. 12b shows the CGE solution using the same boundary conditions. The CGE solution is almost identical to the simulation, showing that, for condensing drainages, the CGE solution provides as good an estimate as a numerical calculation, even with a real phase diagram. Both of these solutions predict that the IC8 fraction is not monotonic with height, while monotonic-ity is observed in the experiments. However, this discrepancy is small because the predicted nonmonotonic fraction is at most 0.05, slightly more than the scatter in the data. For further comparison, Fig. 12c shows a pure-advection solution where capillary forces are not considered. As can be seen, the simulated composition path is very different from that predicted by a pure-advection model because pure advection predicts a jump along the initial tie line into the two-phase region. The experiments, the numerical simulation, and CGE theory show that capillary forces act against this jump for condensing drainages.

Fig. 13 shows the effect of the diffusion on the simulated condensing drainage, using three different levels of diffusivity after 722 hours of drainage. Changing the diffusion constant does not qualitatively change the dynamics of the drainages, although they result in slightly different final compositions vs. height. As

Time, hours

Wet

ting-

Pha

se S

atur

atio

nz=50 cmz=40 cmz=30 cm

Fig. 9—Simulated wetting-phase saturation vs. time at three different heights during the vaporizing drainage. Portions of the column are seen to go through multiple cycles of drainage and imbibition.

Time, hours

z, c

m

uw<0, un<0

uw<0, un>0

uw>0, un<0

uw>0, un>0

Fig. 10—Shaded areas represent places and times within the column were the simulation predicts countercurrent flow. The solid lines are the boundaries between positive and negative flow of the wetting phase, and the dashed line is the boundary between positive and negative flow of the nonwetting phase.


)b( )a(

z, c

m

z, c

m

Pc,

Pa

Pc,

Pa

Fig. 11—Numerical simulation of vaporizing drainage with (a) countercurrent relative permeabilities and (b) capillary hysteresis.

expected, increasing diffusion causes the fraction changes with height to become more smeared out. But we also observe that a higher diffusion causes the overall drainage level to move down-ward, with the highest diffusion producing the lowest capillary fringe. This is similar to what is seen for condensing drives in the CGE solutions, in which increased diffusion allows the IPA to enter greater depths, lowering the IFT at these depths (DiCarlo and Orr 2007). Simulations with greater diffusion do not eliminate the predicted nonmonotonic behavior of the IC8 fraction.

Fig. 14a shows the simulation result for the second condens-ing drainage after 722 hours of drainage. Here, we observe that the simulations give the same general composition path as for the previous condensing drainage, with the composition moving along the wetting side of the binodal curve before quickly jumping along a tie line to the nonwetting side near the injected composition. The experimental data show the same general path, but with a jump along the fourth tie line, while the simulation predicts a jump along the third tie line. Similar to the first condensing drainage, the oil fraction is predicted to be nonmonotonic with height; but, for this drainage, the simulation predicts it to have rapid swings around the capillary fringe. The experiments show a slight nonmonotonic behavior but over a much longer length scale.

While the predicted composition path is in reasonable agree-ment, the predicted height of the capillary fringe is much lower than that observed in the experiment. Again, the same general path is seen for CGE theory (Fig. 14b), although the theory is not complete when one of the boundary compositions is far into the single-phase region; and, thus, the CGE solution used a displacing composi-tion that is on the two-phase boundary. The different displacing composition is likely the cause for the capillary-fringe discrepancy between the simulation and the CGE solution. The pure-advection solution (Fig. 14c) predicts a shock between the initial and displac-

ing compositions because they are multicontact miscible, quite different from what is seen on these length scales.

For vaporizing drainages, where we expect imbibition processes to occur, simulations with and without hysteresis were performed. Fig. 15a shows the simulation result for the first vaporizing drain-age without capillary hysteresis included. Here, we observe a jump into the two-phase region along the initial tie line followed by a gradual decrease to the displacing tie line, with a final sharp jump back to the wetting side of the binodal curve. These main features are seen in the experimental data, except for the large jump back to the wetting side. As in the first condensing drainage, the simula-tion and the CGE solution are almost identical. This jump to the wetting side also shows up in the CGE calculations (Fig. 15b) and, in both of these cases, is caused by the IFT (and capillary forces) increasing significantly at the top of the column, which then acts to draw wetting fluid back in. The experiments show a minor move-ment back to the wetting side, but much less so than predicted by the simulation. In contrast, the pure-advection solution (Fig. 15c) matches the measured composition path more accurately. Finally, Fig. 15d shows the simulation result with capillary hysteresis for a shorter time (400 hours); the solutions with and without capillary hysteresis are almost identical.

Fig. 16a shows the simulation results for the second vapor-izing drainage without hysteresis. Here, the composition path is predicted well by the simulations, with a small jump back to the wetting side at the top of the column. As in the second condensing drainage, the capillary fringe is predicted to be much lower than what is observed in the experiment. Also like the second condens-ing drainage, here, the simulation and CGE solution (Fig. 16b) show the same path but do not match up with the capillary fringe. This is caused by the CGE solution requiring both the initial and displacing phases to be within the two-phase region, and a proxy


)b( )a(

z, c

m

z, c

m

Pc,

Pa

z, c

m

Pc,

Pa

Pc,

Pa

)c(

Fig. 12—(a) Numerical calculation, (b) CGE solution, and (c) pure-advection solution compared to measured experimental data for first condensing drainage.


z, c

m

z, c

m

Pc,

Pa

z, c

m

Pc,

Pa

Pc,

Pa

)c(

)b( )a(

Fig. 13—Effect of diffusion on first condensing drainage (a) D = 2×10−10 m2/s, (b) D = 5×10−10 m2/s, and (c) D = 10×10−10 m2/s.


z, c

m

z, c

m

Pc,

Pa

z, c

m

Pc,

Pa

Pc,

Pa

)c(

)b( )a(

Fig. 14—(a) Numerical calculation, (b) CGE solution, and (c) pure-advection solution compared to measured experimental data for second condensing drainage.


z, c

m

z, c

m

Pc,

Pa

Pc,

Pa

)b( )a(

z, c

m

z, c

m

Pc,

Pa

Pc,

Pa

)d( )c(

Fig. 15—(a) Numerical calculation, (b) CGE solution, (c) pure-advection solution, and (d) simulation with hysteresis compared to measured experimental data for first vaporizing drainage.


z, c

m

z, c

m

Pc,

Pa

Pc,

Pa

)b( )a(

z, c

m

Pc,

Pa

)c(

Fig. 16—(a) Numerical calculation, (b) CGE solution, and (c) pure advection solution compared to measured experimental data for second vaporizing drainage.


initial composition was used in the CGE solution to fulfill this requirement. The simulation with hysteresis is not shown because it is very similar to the one without hysteresis for early times. For comparison, the CGE solution and the pure-advection solutions (Fig. 16c) are displayed, with the experiments showing behavior between these two limits.

As expected, none of the numerical simulations matched the experimental results exactly. There is always tuning done to match the experiment to the model. We have attempted to avoid the tuning in these simulations to make them as predictive as possible. Still, it is useful to see if a small amount of tuning can make the fit better. Figs. 17a through 17d show the best-fit simulation results for all four drainage processes where only the boundary conditions were adjusted slightly. We observe that small changes in the boundary condition of the simulations allow us to match both the condens-ing drainage processes and almost all of the features in both the vaporizing drainage processes.

DiscussionThe results and observations from the simulations reported in the preceding sections are summarized in the following list.

• The simulation reproduces the CGE solution when the capil-lary pressure is a unique function of the saturation.

• For vaporizing drainage processes, the wetting phase imme-diately begins to imbibe back into swept regions.

• The addition of capillary pressure hysteresis changes the displacement path for vaporizing drainage processes.

• For condensing drainage processes, the experimental observa-tions and numerical simulations show a very similar path through compositional space but with a different capillary fringe for the second condensing drainage.

• For vaporizing drainage processes, the experimental obser-vations and numerical simulations show a different path through compositional space and a different capillary fringe.

The first result shows that the CGE solution is approached asymptotically for condensing drainage processes. The agreement between the CGE and simulation is excellent, even for the experi-mental phase diagram with nonparallel tie lines when presented with the same input data. This shows that if the equations that are used to describe the capillary, gravitational, and compositional effects are valid for these displacements, then the CGE solution provides as good an estimate of the total drainage as a numerical calculation does. However, the CGE solution is limited in that it currently can handle only cases in which the initial and displacing compositions are both within the two-phase region. Here, this difficulty is over-come by using a composition on the binodal curve that is on the tie line that passes through the composition in the single-phase region. This method produces the same composition path as the simulation but a different capillary fringe. For the CGE solution to be predictive in all cases, it will need to be extended to handle gravity drainages that sample single-phase regions (DiCarlo et al. 2007).

The second result demonstrates that, for vaporizing drainage processes, there is always a portion of the column where the cap-illary forces act to imbibe the wetting phase back into the swept regions. This imbibition region is directly above the drainage front and follows the drainage front to move downward with time. This has two consequences. One, vaporizing drainage processes neces-sarily show imbibition, and this imbibition must be included to model the displacement dynamics adequately. Two, countercurrent flow is expected in this imbibing region. Although the countercur-rent flow does not affect the reported simulation results, counter-current flow is a common cause of flow instabilities because there is a great incentive for the downward flowing nonwetting phase to separate in space from the upward-flowing wetting phase. The simulations and calculations presented here are for 1D displace-ments only but show that the likelihood of preferential flow in two dimensions and three dimensions is present and can be the focus of future studies.

The third result shows that, while countercurrent flow does not alter the composition path greatly for 1D displacements, the addi-tion of capillary hysteresis can alter the composition path. With this change in composition path, there is also a change in the depth of

the drainage front. This is observed to be larger for the simplified phase diagram than it is for the experimental phase diagram.

The fourth result is promising for condensing drainage pro-cesses, with the caveat that the location of the capillary fringe is predicted well only for one of the condensing drainages. Regard-ing this caveat, the depth of the capillary fringe depends greatly on the experimental boundary conditions (i.e., the pressure of the invading and displaced phase in the tanks that are connected to the experimental column). Though conceptually simple, experimen-tally this often causes problems because of possible bubbles in the lines and other seemingly minor difficulties. Often, one performs the same measurement and finds that the capillary fringe moves 5–10 cm even for experiments that do not involve compositional effects. One possible method around this problem is to alter the boundary conditions slightly in the simulation to obtain the best fit with the experiments. Figs. 17a through 17d show the best fits of the data obtained by slightly altering only the bottom boundary condition. One can see that the agreement between experiments and calculations for the condensing drainages is very good, although, for the vaporizing drainage processes, the comparison is not nearly as good.

This observation, along with the fifth observation listed, sug-gests that the model does not get the physics entirely correct for the vaporizing displacements. Care was taken to have all of the relevant physics included in the simulation (e.g., diffusion, capil-larity, hysteresis, the correct phase diagram), yet the simulation still predicted a large amount of wetting phase at the top of the column that was not observed experimentally. On the other hand, a simple pure-advection description reproduces the composition path quite well, although a pure-advection model cannot predict the location of the drainage front. In effect, CGE and the numerical model both overestimate greatly the amount of wetting fluid being drawn back into the swept regions.

One possible explanation can be discerned by looking at the difference between the simulations using both phase diagrams. For the simplified phase diagram, the imbibition of wetting fluid back into the swept region takes place over a wide region of the column. For the experimental phase diagram, the imbibition takes place only at the top edge of the column. This apparently occurs for two reasons. First, the tie lines slant in the direction of the dilution line and, thus, the viscous jump into the two-phase region is very close to the nonwetting side of the binodal curve. At these compositions, the IFT rises extremely fast with decreasing IPA fraction. Thus, in the model, the capillary forces rise quickly at the top of the column as the composition works its way to the displacing tie line. This cre-ates a rapidly increasing pressure gradient at the top of the column, which is mostly within the region of 2–3 cm from the top.

This can also explain why capillary hysteresis appears to play a minor role for this phase diagram. Including capillary hysteresis changes the capillary forces by, at most, a factor of two, while the rapidly changing IFT can change the capillary forces by orders of magnitude. Note also that moderate hysteresis is seen at the top because the gridblock remains at a high wetting-phase satura-tion and, thus, the imbibition curve is not very different from the drainage curve.

While the model is seen to predict this anomalous behavior, it is not surprising that the wetting-phase saturation is much more uniform in the actual experiments. While the model has the effect of diffusion of the species, it does not consider longitudinal disper-sion caused by permeability variations. This would require a multi-dimensional simulator with an appropriate permeability field.

The addition of higher dimensions would require many more parameters, such as the variation and correlation of the perme-abilities and pressure/saturation curves. The only method to produce these parameters would be to fit the experimental data. Thus, this exercise is tautological and not predictive and is not attempted here. It is worth noting that replications of the experiments do not pro-duce exactly the same measured distributions (DiCarlo et al. 2007), and this is another reason that the qualitative comparison between the experiments and the predicted simulations is more interest-ing than attempting to perform an exact match. That being said, by varying one parameter—the bottom boundary condition—the


z, c

m

z, c

m

Pc,

Pa

Pc,

Pa

)b( )a(

z, c

m

z, c

m

Pc,

Pa

Pc,

Pa

)d( )c(

Fig. 17—Best-fit numerical simulation with slightly altered bottom boundary condition: (a) Condensing Drainage 1 with �Pb = −300 Pa, (b) Condensing Drainage 2 with �Pb = −720 Pa, (c) Vaporizing Drainage 1 with �Pb = −130 Pa, and (d) Vaporizing Drain-age 2 with �Pb = 350 Pa.


simulation and the experiments can be made to match much better (Figs. 17a through 17d).

ConclusionsIn summary, we use compositional simulation to study the behavior of compositional gravity-drainage processes. We find that numeri-cal calculation asymptotically approaches the CGE solution with increasing time for both condensing and vaporizing solutions when the composition path remains in the two-phase region. The simula-tions predict that, for vaporizing drainages, the wetting phase is drawn back into the column because of capillary gradients from variations in the IFT, as hypothesized.

On a predictive basis, we find that the simulations and the CGE solutions compare well to the experimental condensing drainages, while the comparison is not nearly as good for the experimental vaporizing drainages. Thus, the description of the physics for condensing drainages seems well understood, but the countervail-ing capillary, gravitational, and hysteretic gradients in vaporizing drainages need to be worked out adequately.

Nomenclature Ci = volume fraction of Component i, dimensionless g = gravitational constant, m/s2

J = modifi ed Leverett J-function, m−1

krj = relative permeability of Phase j, dimensionless L = length over which drainage occurs, m NB

−1 = inverse Bond number, dimensionless Pc = capillary pressure, Pa Pj = pressure of Phase j, Pa rt = throat radius, m Sj = saturation of Phase j, dimensionless S

j = reduced saturation of Phase j, dimensionless

Sr = residual saturation of wetting phase, dimensionless z = vertical distance, m �� = density difference between the wetting and nonwetting

phases, kg/m3

� = Corey exponent, dimensionless �j = density of Phase j, kg/m3

� = interfacial tension, N/m

Subscripts b = bottom i = component j = phase n = nonwetting t = top w = wetting

Superscripts d = displacing phase i = initial phase

AcknowledgmentsWe thank Lynn Orr for helpful discussions and comments on the manu-script. M. Mirzaei acknowledges support from the Industrial Associates Gas Flooding Program at The University of Texas at Austin.

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David DiCarlo is an assistant professor in the Department of Petroleum and Geosystems Engineering at The University of Texas at Austin. His research interests include experimental measurements of multiphase flow, preferential and compo-sitional displacements, and gas-injection processes. He holds a BS degree in physics from Case Western Reserve University and MS and PhD degrees in experimental physics from Cornell University. Mohammad Mirzaei is a PhD student and gradu-ate research assistant at The University of Texas at Austin. Mirzaei holds a BS degree in electrical engineering from Sharif University and an MS degree in reservoir engineering from the University of Tehran. Kristian Jessen is an assistant profes-sor at the Mork Family Department of Chemical Engineering and Materials Science, University of Southern California. His research interests include modeling and simulation of com-positional displacement processes and geological carbon sequestration. He holds a PhD degree in chemical engineering from the Technical University of Denmark and is a cofounder of the company Tie-Line Technology ApS.

spe-110077-pa (simulation of compositional gravity-drainage processes)

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