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A fully compositional model considering the effect of nanopores in tight oil reservoirs
Author’s Accepted Manuscript A Fully Compositional Model Considering the Effect of Nanopores in Tight Oil Reservoirs Bicheng Yan, Yuhe Wang, John E. Killough
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S0920-4105(16)30951-2 http://dx.doi.org/10.1016/j.petrol.2017.01.005 PETROL3813
To appear in: Journal of Petroleum Science and Engineering Received date: 8 November 2016 Revised date: 23 December 2016 Accepted date: 4 January 2017 Cite this article as: Bicheng Yan, Yuhe Wang and John E. Killough, A Fully Compositional Model Considering the Effect of Nanopores in Tight Oil R e s e r vo i r s , Journal of Petroleum Science and Engineering, http://dx.doi.org/10.1016/j.petrol.2017.01.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A Fully Compositional Model Considering the Effect of Nanopores in Tight Oil Reservoirs Bicheng Yana, Yuhe Wangb*, John E. Killougha a
Department of Petroleum Engineering, Texas A&M University, College Station, TX, USA b
Department of Petroleum Engineering, Texas A&M University at Qatar, Doha, Qatar
*Corresponding Author: Yuhe Wang, [email protected]
Abstract Conventional compositional simulators are usually difficult to interpret the different gas oil ratio (GOR) from tight oil reservoirs, and this also indicates an unreliable prediction of ultimate hydrocarbon recovery. We realize that there are two issues related to the compositional simulation of production in tight oil reservoirs. Firstly, tight oil reservoirs typically exhibit extremely small matrix pore size in the order of nanometers, so the capillary pressure between vapor and liquid phases is considerable such that the PVT of the confined fluid deviates from that of the bulk fluid with capillary pressure ignored. Secondly, during depletion process, rock compaction causes pore space reduction and brings remarkable changes in rock properties. In this work we implement rigorous confined fluid phase behavior calculation depending on capillary pressure and rock compaction in a fully compositional simulator. Capillary pressure in matrix nanopores is calculated by Leverett J-function. Further, the impact of capillarity on phase equilibrium is taken into account through modifying the stability test and two-phase flash calculation. Dynamic rock compaction is considered in the simulator via rock compaction tables, such that fluid mobility decreases with permeability reduction and capillary effect is simultaneously coupled. The unique implementation in the simulator captures the dynamic behavior of rock and fluid properties in tight oil reservoirs. Typical suppression of bubble point pressure and reduction of oil viscosity and density is observed from our simulation results. Reservoir scale simulation results shows that this model resolves the problem of the inconsistent GOR in tight oil production and greatly facilitate the history matching process. The enhanced compositional simulation will ultimately improve our understanding of tight oil reservoirs and provide better guidance for recovery prediction.
Keywords: Tight oil reservoirs; compositional simulation; vapor-liquid equilibrium; capillary pressure; rock compaction
Nomenclature Acronyms 1
BHP
Bottom-Hole Pressure;
BIC
Binary Interacting Coefficient;
GOR Gas oil ratio; IMPEM
Implicit pressure explicit mass;
SSI
Successive substitution iteration;
VLE
Vapor Liquid Equilibrium;
Variables Fugacity of component in phase ; Vapor mole fraction, fraction; Permeability, md; Component equilibrium ratio, fraction; Number of non-water components in the reservoir; Vapor phase pressure, psia; Liquid phase pressure, psia; Capillary pressure in between liquid and vapor phases, psia; Pore radius in porous medium, m; Gas universe constant, 8.314
;
Fugacity equilibrium residual of component ; Capillarity residual; Residual of Rachford-Rice Equation Reservoir temperature, VLE
;
Vapor liquid equilibrium; Molar fraction in liquid phase, fraction; Molar fraction in vapor phase, fraction; Molar fraction in the overall mixture, fraction; Fugacity coefficient of phase ; Porosity, fraction; Viscosity, cp; Density,
;
2
Chemical potential of component in phase ; Interfacial tension; Contact angle between liquid and vapor phase; Superscripts and subscripts Liquid phase; Vapor phase;
1. Introduction The development of unconventional tight oil resources has achieved great success through the application of horizontal well drilling and hydraulic fracturing. As predicted by EIA (Figure 1), tight oil production worldwide will increase to be more than double from 2015 to 2040 1, and specifically tight oil production in United States will increase from 4.1 million barrels/day in 2016 to 7.1 million barrels/day in 2040. Despite of this great success, our understanding of the production mechanisms in tight oil reservoirs is still very limited.
Figure 1. World tight oil production from 2015 to 2040
1
In tight oil resources, the matrix permeability is extremely low and the average pore size is in the nanometer magnitude. For example, the average pore size in Bakken shale matrix is about 10 to 50 nm. Fluid within such tiny pores exhibits different properties from conventional oil reservoirs, because the capillary pressure in those nanopores is significant and it may induce deviation in phase behavior of tight oil reservoirs. The effect of porous media on the phase behavior of hydrocarbon fluid was investigated in literatures. Sigmund, et al. 2 concluded that the effect of curvature on phase behavior is not significant except at high surface curvatures. However, this would not happen in hydrocarbon reservoirs even with the presence of clay particles, since those finest pores are likely to be filled by connate water. Through implementing a capillarity equation in phase equilibrium, Brusilovsky 3 demonstrated that bubble point pressure decreases but dew point pressure increases. Further, Nojabaei, et al. 4 investigated the impact of nanopores on phase behavior of Bakken shale oil, and the results showed that bubble-point pressure can be supressed significantly, and 3
the same conclusion was also reported in other literatures 5. Nojabaei, et al. 6 implemented a compositionally-extended black-oil model considering capillarity in phase behavior, and the capillary pressure varies with interfacial tension and pore size distribution. This approach is innovative but the extended black-oil formulation is intrinsically limited in terms of compositional characterization. Sandoval, et al. 7 developed an efficient and robust algorithm for calculating fluid phase behavior in the presence of capillarity. They found that the bubble point pressure suppression becomes significant when moving away from the critical point, while the dew point pressure exhibits different variations depending on the location of phase diagram. Rezaveisi, et al. 8 implemented capillarity in flash and stability test in UTCOMP simulator, but still they didn’t apply it for practical field-scale application of tight oil reservoirs. Dong, et al. 9 coupled PengRobinson Equation of State with capillary pressure equation and adsorption theory to investigate phase behavior in cylindrical nanopores, and they found that bubble point pressure is overestimated if neglecting adsorption film. Moreover, Nojabaei, et al. 4 reported that as pore pressure decreases or effective stress increases, the tight matrix is likely to experience a reduction of pore size. This observation can also be inferred from production data in Bakken shale and lab data. In their approach, through PVT analysis they theoretically demonstrated that the suppression of bubble point pressure becomes even more significant when considering pore size reduction through the depletion process, and applied this as input in a commercial reservoir simulator for history matching. Therefore, a rigorous consideration of pore size change throughout the tight oil production processes is also very necessary for general compositional reservoir simulation. Motivated by those points, this work implements the impact capillarity and rock compaction in a general fully compositional reservoir simulator 10 and applies it to predict tight oil reservoir performance. Specifically capillary equation is explicitly implemented in stability test based on Gibbs free energy approach and vapor-liquid equilibrium (VLE). Further, matrix pore size is iteratively calculated during pressure depletion by rock compaction tables. The implementation is designed to rigorously consider the fluid and rock properties variations in the nanopores. The reminder of this paper is organized as follows. The next session illustrates the vapor-liquid equilibrium formulation considering capillary pressure and the overall phase behavior evaluation workflow in compositional simulation. The third session presents our main results based on the new model. Finally, the whole work is concluded and summarized. 2. Numerical Formulation In this work, a fully compositional reservoir simulator 10-11 is extended to accommodate the effect of capillary pressure on vapor-liquid equilibrium. The simulator is designed volume-balance formulation and it is solved by the classic implicit pressure explicit mass (IMPEM) scheme. Since simulator formulation itself is not the focus of this paper, more details with regards to this can be referred to 12. Originally the simulator is able to handle multi-phase multi-component reservoir simulation, and the PVT modules in the simulator include stability test based on Gibbs free energy approach and vapor-liquid phase split calculation 13, but the effect of capillary pressure on phase behavior is not implemented. In this part, we will basically introduce the newly implemented flash and stability test algorithm considering capillarity effect in the reservoir simulator. 2.1 Stability Test Considering Capillary Pressure
4
In this part, the consideration of capillarity effect in stability test is based on the stability criterion of Michelsen 13b. If the original system is stable, Equation (1) will be valid. and are respectively the chemical potential of the incipient phase and the original system. ∑
(1)
Based on Equation (2), chemical potential is usually convenient to be represented in terms of fugacity, given by Equations (3) and (4). Here it is assumed that the original phase is liquid phase and the trial phase is vapor phase. The opposite setting, where the original phase is vapor and the trial phase is liquid, is equivalent. (2) (3) (4) As a result, given system temperature , Equation (1) is transformed into Equation (5). The last term in Equation (5) is usually ignored because of the small capillary pressure between vapor and liquid phases. However, it is taken into account here because capillary pressure is hypothesized to be significant in confined nanopores. ∑ ∑
{
∑
{
{
} } }
(5)
Equation (6) is defined to be independent of , and thus it can be calculated in advance, which is convenient in terms of implementation. Then we have Equation (7). (6) ∑
(7)
The stationary criterion is Equation (8). (8)
Let
, then (9)
5
Based on material balance Equation (10), we can calculate through Equation (11), and this indicates that can be interpreted by phase composition moles. ∑
∑
(10) (11)
∑
Here the solution of Equation (9) can be used to examine the stability test. In terms of , this original system is stable if ∑ and is unstable if ∑ . Equation (9) here is solved by the combination of Successive Substitution Iteration (SSI) and Newton-Raphson method. 2.2 Vapor-Liquid Two-Phase Split Considering Capillary Pressure The consideration of capillarity effect on vapor-liquid equilibrium (VLE) can be readily represented by Equations (12) and (13). In Equation (12), and respectively designate vapor phase pressure and liquid phase pressure, and represents the capillary pressure between those two phases. Equation (13) is the equality of fugacity in liquid and vapor phases, and the two fugacities are defined through Equations (14) and (15). In conventional flash calculation, is considered to be equal to and thus conventional VLE only includes the component fugacity equilibrium in vapor and liquid phases. It’s noted that those two consecutive equations are established based on continuous bulk vapor and liquid phase separated by a curvature interface. In ultra-tight porous media those two phases are not necessarily to be continuous, but it can be shown that pressure and phase compositions in different isolated vapor or liquid regions are equal when the system reaches equilibrium 14. Therefore, the continuous phase prerequisite is assumed to be still reliable here. (12) (13) (14) (15)
To close the nonlinear system in Equations (12) and (13), other constraints are required. The first one is the material-balance constraint with the introduction of vapor mole fraction , as shown in Equation (16). (16)
Additionally component mole fractions of each phase and the overall mixture should sum to unity when equilibrium is reached, as shown in Equation (17). With the definition of equilibrium ratio in Equation (18), the material-balance constraints in Equation (16) can be transformed to the classic Rachford-Rice Equation 15, as shown in Equation (19). ∑
∑
∑
(17) 6
(18) ∑
(19)
Based on Equations (14) and (15), the definition of component equilibrium ratio can be further transformed into Equation (20). Equation (20) shows that the capillary pressure is embedded in this modified component equilibrium ratio, and it can be reduced to conventional format if ignoring the capillary pressure. (20) At a given capillary pressure, Equations (13) and (19) are solved through combining successive substitution iteration (SSI) and Newton-Raphson method. 2.3 Capillary Pressure Evaluation in Tight Porous Media The calculations above require the evaluation of capillary pressure. During the iteration of stability test and VLE phase split calculation, capillary pressure values are calculated using saturation results from the previous iteration. The capillary pressure is fundamentally related to the geometry of the capillary and the wettability of the porous media, and it can be calculated through Young-Laplace equation, shown as Equation (21). This approach has been applied in Nojabaei, et al. 4, Honarpour, et al. 5a, and Pang, et al. 5c etc, and the pore radius is taken into account. (21)
As Nojabaei, et al. 4 stated, capillary pressure calculated through Young-Laplace equation is much lower than the experimental measurement due to the very low interfacial tension predicted by Macleod-Sugden correlation 16. This observation also indicates the importance of reliable capillary pressure prediction for shale/tight rocks. On the other hand, Leverett J-function (Equation (22)) is based on measured reference capillary pressures, and the scaling factor is supposed to be given for a particular rock type based on lab data. Unfortunately so far there is no solid measurement of capillary pressure for the Bakken shale. Because the porous media investigated here is a saturated one, the Leverett J-function approach is assumed to be appropriate to evaluate capillary pressure here. Compared to Young-Laplace Equation, Leverett J-function is friendly to the simulator implementation, since pore size information is embedded in permeability and porosity. √
(22)
The porosity of the Bakken shale is assumed to be 0.06. Further, the corresponding permeability and pore radius can be calculated based on Kozeny-Carman equation 17 and Nelson’s correlation 18. The reference capillary pressures under different pore radii are also 7
presented based on a similar formation from Crain’s petrophysical handbook. Those values are illustrated in Table 1 and are expected to be very close to practical situation.
Table 1. Pore radius, permeability and capillary pressure Pore Radius (nm)
Premeability (md)
Capillary Pressure (psi)
50
0.0070
102.20
40
0.0046
127.75
30
0.0027
170.33
20
0.0012
255.50
10
0.0003
511.00
2.4 Rock Compaction towards Nanopore Space Dynamic rock compaction is considered in the simulator through rock compaction table. Therefore, when pore pressure changes during time-step and newton-step evolution, permeability reduction ratio is interpolated from the rock compaction table. The permeability update further impacts the pore radius and capillary pressure. For Bakken shale investigated in this work, the rock compaction table is presented in Table 2. Table 2: Rock compaction table of Bakken shale Pressure Change (psi)
Premeability Reduction Ratio
-5180
0.489
-4450
0.500
-3700
0.511
-2960
0.532
-2220
0.588
-1480
0.675
-740
0.791
0
1.0
With the consideration of rock compaction, pore size reduces during depletion, and the bubble point pressure for confined fluid will be even more suppressed during production. As a result, reservoir fluid is likely to stay in single phase (oil) with light components still dissolved in it, and oil density and viscosity decrease and oil compressibility increases such that fluid flow capacity and flow driving energy increases. On the other hand, permeability decreases with rock compaction, and thus it negatively impacts the fluid flow capacity. Therefore, when considering rock compaction for fluid flow in tight porous media, it induces greater confinement effect and lower permeability and those two effects compete with each other, as shown in Figure 2.
8
No pressure maintenance
Reservoir Compaction
Lower Permeability
_
+ Flow Capacity
Figure 2. Impact of rock compaction and fluid confinement on flow in tight porous media
2.5 VLE Workflow Considering Capillarity and Rock Compaction In summary, the workflow of vapor liquid equilibrium with the consideration of capillary pressure and rock compaction is shown in Figure 3. At a given thermodynamic condition in a time/newton step, through compaction table rock permeability and porosity are updated and used to evaluate capillary pressure. Meanwhile, if previously the fluid stays in single-phase, stability test is performed and capillarity is considered in the calculation. If the fluid is unstable at single phase or it is previously in two-phase, then two-phase flash calculation considering capillarity is performed. Otherwise, fluid is still single phase. After the phase split, fluid properties and other secondary variables are updated, and capillary pressure is calculated and is resent to stability test and flash calculation, and new secondary variables are updated. This process is iterated until both phase equilibrium and capillarity equality are ensured, and then simulation gets into solving a global non-linear system for updating primary variable(s).
9
Starting point
No
Single-phase previously?
Update K and
Yes Stability test Stable?
No
Yes Two-phase flash
Single-phase liquid
The rest procedures in a time/newton step
Saturation dependence
Figure 3. Workflow for vapor-liquid equilibrium considering capillarity and rock compaction
3. Cases Analysis This part mainly presents the results of fluid PVT in confined nanopores and the application into compositional simulation of tight oil reservoirs. Fluid compositional data is based on Bakken shale oil from Nojabaei, et al. 4, as shown in Table 3 and 4. At the temperature of 240 °F the saturation pressure for this fluid here is the bubble point pressure. The fluid data in Table 3 and 4 and will be used in all cases presented below, and the rock compaction data in Table 2 will be used in those cases below if rock compaction is considered.
Component
Mole Fraction
C1 0.36736 C2 0.14885 C3 0.09334 C4 0.05751 C5-C6 0.06406 C7-C12 0.15854 C13-C21 0.07330 C22-C80 0.03704 Reservoir temperature (°F)
Table 3. Bakken oil composition data Critical Critical Pressure Temperature (°F) (psia) -116.63 667.80 90.09 707.80 206.01 616.30 305.65 550.70 415.81 461.29 593.58 363.34 872.43 249.61 1384.82 190.12 240
Acentri c Factor 0.0130 0.0986 0.1524 0.2010 0.2684 0.4291 0.7203 1.0159
Molar Weight
Paracho r
16.04 30.07 44.10 58.12 78.30 120.56 220.72 443.518
74.8 107.7 151.9 189.6 250.2 350.2 590.2 1216.8
Table 4. Bakken oil binary interaction coefficient (BIC) C1
C2
C3
C4
C5-C6
C7-C12
C13-C21
C22-C80
0.005 0.0035 0.0035 0.0037 C1 0.0050 0.0031 0.0031 0.0031 C2 Other BICs are zero and not presented here for simplicity.
0.0033 0.0026
0.0033 0.0026
0.0033 0.0026
10
3.1 Confined Fluid Phase Behavior The P-T diagram of the Bakken oil is calculated through rigorous stability test and saturation pressure calculation algorithms in session 2. The consideration of capillary pressure is evaluated through Young-Laplace equation under different nanopore radii, and compared with the conventional scenario without capillarity. The results are plotted in Figure 4. Here it shows that at a given temperature the bubble point pressure decreases as pore size reduces. When pore radius is 10 nm, the suppression of bubble point pressure is about 125 psia. 3500
Pressure, psia
3000
2500
No Capillary Effect r = 50 nm r = 40 nm r = 30 nm r = 20 nm r= 10 nm
2000
1500
1000 80
130
180
230
280
330
380
430
Temperature,oF
Figure 4. Bubble point pressure lines of Bakken oil with considering capillarity under different pore radii and without considering capillarity
As previously discussed, capillary pressure predicted by this approach is much lower than the actual one. Therefore, we further evaluate capillary pressure through Leverett J-function approach for all cases, and the results are presented in Table 5. We can see that there are significant differences between bubble point pressures based on Young-Laplace equation and those based on Leverett J-function. The differences here also reveal that capillary pressure data with good quality is very important in phase behavior prediction.
Table 5. Bubble point pressure of Bakken oil at 240 °F Bubble Point Pressure (psia) Pore Radius (nm) Young-Laplace Leverett J-Function 50 2766 2512 40 2761 2450 30 2751 2345 20 2732 2145 10 2641 1588 No Capillary 2788
If compared to the same fluid system with unsuppressed bubble point pressure, a suppression in bubble point pressure physically indicates that there are more light components remaining in the liquid oil phase under the same pressure and temperature condition. More light components in oil phase are expected to bring a reduction of oil density and oil viscosity. Fluid viscosity here is calculated by Lohrenz-Bray-Clark correlation 19. With capillary pressure considered in phase behavior, Figure 5 illustrates the oil density, oil viscosity and their ratio changes with nanopore 11
radius for Bakken oil at 240 and 1500 psia. The results are also compared with bulk fluid properties without considering capillary pressure. In Figure 5 (a) and (b), it clearly shows that oil density and oil viscosity decrease as nanopore radius decreases from 50 nm to 10 nm, and the two properties are smaller than those values when neglecting capillary pressure in phase behavior. Further, since oil mobility is directly proportional to this ratio, the ratio of oil density to oil viscosity is also plotted in Figure 5 (c). We can see that as nanopore radius decreases, the ratio of oil density to viscosity increases, and thus oil becomes much more mobile in confined nanopore space. Besides, more light components in oil phase also indicates that oil phase compressibility increases, which provides more driving energy to extract liquid from tight oil reservoirs. 0.50 Blue dash line: oil viscosity without capillarity
Oil viscosity, cp
Oil density, lb/cu.ft
45 Purple dash line: oil density without capillarity
40
35
0.42 0.34 0.26
0.18 0.10
30 0
10
20 30 40 Pore radii, nm
50
0
60
10
20 30 40 Pore radii, nm
(a)
50
60
(b) oil ρ/µ, lb/cu.ft/cp
250 200 150
100 Green dash line: oil ρ/µ without capillarity
50 0 0
10
20 30 40 Pore radii, nm
50
60
(c) Figure 5. Confined fluid properties change with pore radius at 240 (b) oil viscosity; (c) the ratio of oil density to oil viscosity.
and 1500 psia: (a) oil density;
3.2 1D Core Size Model The first example is a one dimensional core size model, as shown in Figure 6. Other parameters related to this model is presented in Table 6. A producer is located at the first grid block, and it is constrained by a minimum BHP (1500 psia) and maximum oil volume rate (0.001 RB/day). The oil rate is very low since the model is small.
12
Figure 6. 1D core model with a producer
Table 6. Reservoir parameters for 1D core size model 0.5 x 164 x 0.5 Model size ( ) 0.5 x 3.28 x 0.5 Grid size ( ) Grid number 1 x 50 x 1 Porosity 0.06 0.002 Permeability ( ) 6840 Initial pressure ( ) 1500 Producer minimum BHP ( ) Producer oil rate (RB/day) 0.001 240 Initial temperature ( )
Three different scenarios are simulated, respectively Case 1 with no capillarity effect on VLE and no rock compaction, Case 2 with capillarity effect on VLE but no rock compaction, and Case 3 with capillarity effect on VLE and with rock compaction. The results are plotted in Figure 7. Figure 7 presents the cumulative oil production and gas-oil-ratio in the wellbore condition. Through comparison, it clearly shows that Case 2 produces the most oil from the reservoir, while Case 1 produces the least oil from the reservoir. The comparison between those two cases shows that capillary pressure effect on phase behavior favors oil production in the tight oil reservoir, since oil mobility of confined fluid increases. Case 3 considers both capillary pressure in VLE and rock compaction. As expected this case produces median oil production here. This indicates that during pressure depletion rock compaction reduces rock permeability and it offsets the increase of mobility caused by capillary pressure. The GOR at the in-situ reservoir condition tells us that capillary pressure induces lower bubble point pressure and delayed gas phase occurrence, and this is even more exacerbated by further considering rock compaction. Besides, the early zero GOR at the in-situ reservoir conditions indicates that BHP at that period is higher than the bubble point pressure.
13
1.6
4500 4000
Cum Oil Production, RB
3000 2500 0.8
Oil Prod for Case 1
Oil Prod for Case 2
Oil Prod for Case 3
GOR for Case 1
GOR for Case 2
GOR for Case 3
2000
1500
0.4
1000
GOR in well, RCF/RB
3500
1.2
500 0.0
0
0
1
2
3
4
5
Time, days
Figure 7. Cumulative oil production and GOR at in-situ reservoir conditions for three different scenarios in the 1D model: Case 1: with no capillarity effect on VLE and no rock compaction, Case 2 with capillarity effect on VLE but no rock compaction, Case 3 with capillarity effect on VLE and rock compaction.
3.3 Fractured Tight Oil Reservoir with Multiple Hydraulic Fractures Here a tight oil reservoir is perforated by a horizontal well and 4 planar hydraulic fractures traverse through the reservoir in y-direction. The horizontal well is constrained by constant oil rate (50 RB/day) and minimum BHP (1900 psia). Basic reservoir parameters are presented in Table 7. The schematic of the model is illustrated in Figure 8. Therefore, fluid flow through the reservoir follows the sequence of matrix-fracture-well. In hydraulic fractures capillary pressure is not considered since we assume that capillary pressure effect is negligible and thus bulk fluid phase behavior is appropriate here. On the other hand, in the tight oil matrix the effect of capillary pressure on phase behavior and the rock compaction is conditionally considered. Table 7. Reservoir parameters for the 3D tight oil reservoir model 1000 x 1000 x 20 Model size ( ) Grid number 35 x 67 x 10 Matrix porosity 0.06 0.002 Matrix permeability ( ) Fracture number 4 1.0 Fracture grid width ( Fracture porosity 0.02 100 Fracture conductivity ( ) 6380 Initial pressure ( ) 1900 Producer minimum BHP ( ) Producer oil rate (RB/day) 50 240 Initial temperature ( )
14
Confined
Unconfined Figure 8. Tight oil reservoir with horizontal well and multiple hydraulic fractures
Similarly, here we also compare three different scenarios in the reservoir model, and the results at the in-situ reservoir conditions are illustrated in Figure 9. Here it shows that the case with capillary pressure in VLE but no rock compaction (Case 2) has the highest cumulative oil production, and the case with both capillarity and rock compaction (Case 3) has the median oil production. This further demonstrates that at reservoir scale the consideration of capillarity in phase behavior in Case 2 facilitates the oil transport into the wellbore, while the median oil production in Case 3 is caused by the permeability decrease due to rock compaction. There is a decrease of in-situ GOR for cases considering capillary pressure in phase behavior. The insitu GOR for the case with compaction and capillary pressure interestingly exhibits a step-like increase and decrease back to near zero. This might be explained by that nanopore size reduction due to rock compaction further suppresses the bubble point pressure and thus most of the hydrocarbon molecules flow in liquid phase in the wellbore. 14000000
2000
Oil Prod for Case 2
Oil Prod for Case 3
GOR for Case 1
GOR for Case 2
GOR for Case 3
1800 1600
10000000
1400 1200
8000000
1000 6000000
800
600
4000000
400
2000000
GOR in the well, RCF/RB
Cum Oil Production, RB
12000000
Oil Prod for Case 1
200 0 0
200
400
600
800
1000
1200
1400
1600
1800
0 2000
Time, days
Figure 9. Cumulative oil production and GOR at in-situ reservoir conditions for three different scenarios in the reservoir model: Case 1: with no capillarity effect on VLE and no rock compaction, Case 2 with capillarity effect on VLE but no rock compaction, Case 3 with capillarity effect on VLE and rock compaction.
4. Conclusions 15
In this work a fully compositional reservoir model considering the impact of capillary pressure and rock compaction on vapor liquid equilibrium is developed, and Leverett J-function is used to evaluate capillary pressure for tight oil. This model is used to predict the phase behavior in confined nanopore space and tight oil reservoir performance. The capillary pressure effect on vapor-liquid equilibrium is significant when pore size in the matrix is in the nanometer scale. For oil-wet tight oil matrix, fluid confinement in the nanopores suppresses the bubble point pressure, and results in decreases in oil density and viscosity. This ultimately favors the oil phase transport to the wellbore by increasing the oil mobility and more driving energy. Besides, through our compositional model oil production in tight oil reservoirs is demonstrated to very sensitive to rock compaction during reservoir depletion. Rock compaction directly decreases rock permeability and reduces fluid mobility. On the other hand, pore size reduction during rock compaction further exacerbates bubble point pressure suppression of confined fluid in nanopores. Therefore, reservoir simulation of tight oil reservoirs should consider both capillary pressure and rock compaction on production. Reference 1. EIA, Annual Energy Outlook 2016. EIA: International Energy Outlook 2016 2016. 2. Sigmund, P. M.; Dranchuk, P. M.; Morrow, N. R.; Purvis, R. A., Retrograde Condensation in Porous Media. 1973. 3. Brusilovsky, A. I., Mathematical Simulation of Phase Behavior of Natural Multicomponent Systems at High Pressures With an Equation of State. 1992. 4. Nojabaei, B.; Johns, R. T.; Chu, L., Effect of Capillary Pressure on Phase Behavior in Tight Rocks and Shales. 2013. 5. (a) Honarpour, M. M.; Nagarajan, N. R.; Orangi, A.; Arasteh, F.; Yao, Z., Characterization of Critical Fluid PVT, Rock, and Rock-Fluid Properties - Impact on Reservoir Performance of Liquid Rich Shales. Society of Petroleum Engineers: 2012; (b) Du, L.; Chu, L., Understanding Anomalous Phase Behavior in Unconventional Oil Reservoirs. Society of Petroleum Engineers: 2012; (c) Pang, J.; Zuo, J. Y.; Zhang, D.; Du, L., Impact of Porous Media on Saturation Pressures of Gas and Oil in Tight Reservoirs. Society of Petroleum Engineers: 2012. 6. Nojabaei, B.; Siripatrachai, N.; Johns, R. T.; Ertekin, T., Effect of Saturation Dependent Capillary Pressure on Production in Tight Rocks and Shales: A Compositionally-Extended Black Oil Formulation. Society of Petroleum Engineers: 2014. 7. Sandoval, D.; Yan, W.; Michelsen, M. L.; Stenby, E. H., Phase Envelope Calculations for Reservoir Fluids in the Presence of Capillary Pressure. Society of Petroleum Engineers: 2015. 8. Rezaveisi, M.; Sepehrnoori, K.; Pope, G. A.; Johns, R. T., Compositional Simulation Including Effect of Capillary Pressure on Phase Behavior. Society of Petroleum Engineers: 2015. 9. Dong, X.; Liu, H.; Hou, J.; Wu, K.; Chen, Z., Phase Equilibria of Confined Fluids in Nanopores of Tight and Shale Rocks Considering the Effect of Capillary Pressure and Adsorption Film. Industrial & Engineering Chemistry Research 2016, 55 (3), 798-811. 10. Dean, R. H.; Lo, L. L., Simulations of Naturally Fractured Reservoirs. 1988. 11. Tang, D. E.; Zick, A. A., A New Limited Compositional Reservoir Simulator. Society of Petroleum Engineers: 1993. 12. (a) Wong, T. W.; Firoozabadi, A.; Aziz, K., Relationship of the Volume-Balance Method of Compositional Simulation to the Newton-Raphson Method; (b) Young, L. C.; Stephenson, R. E., A Generalized Compositional Approach for Reservoir Simulation. 1983. 13. (a) Michelsen, M. L., The isothermal flash problem. Part II. Phase-split calculation. Fluid Phase Equilibria 1982, 9 (1), 21-40; (b) Michelsen, M. L., The isothermal flash problem. Part I. Stability. Fluid Phase Equilibria 1982, 9 (1), 1-19. 16
14. Bedrikovetsky, P., Mathematical Theory of Oil and Gas Recovery. Kluwer Academic Publishers: The Netherlands, 1993. 15. Rachford, H. H.; Rice, J. D., Procedure for use of electrical digital computers in calculating flash vaporization hydrocarbon equilibrium. JPT 1952, 19, 195. 16. Pederson, K. S.; Christensen, P. L., Phase behavior of petroleum reservoir fluids. Taylor & Francis Group: CRC Press, 2007. 17. (a) Carman, P. C., Fluid flow through granular beds. Chemical Engineering Research and Design 1937, 75, S32-S48; (b) Kozeny, J., Ueber kapillare Leitung des Wassers im Boden. Sitzungsber Akad 1927, 136 (2a), 271-306. 18. Nelson, P. H., Permeability-porosity relationships in sedimentary rocks. Log Analyst 1994, 35 (3), 38-62. 19. Lohrenz, J.; Bray, B. G.; Clark, C. R., Calculating Viscosities of Reservoir Fluids From Their Compositions. 1964.
Highlights
A fully compositional model with rigorous dynamic confined fluid phase behavior.
Unique implementation on a 3D comprehensive compositional simulator.
Realistic tight oil recovery simulation, revealing the necessity of added physics.
17
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PII: DOI: Reference:
S0920-4105(16)30951-2 http://dx.doi.org/10.1016/j.petrol.2017.01.005 PETROL3813
To appear in: Journal of Petroleum Science and Engineering Received date: 8 November 2016 Revised date: 23 December 2016 Accepted date: 4 January 2017 Cite this article as: Bicheng Yan, Yuhe Wang and John E. Killough, A Fully Compositional Model Considering the Effect of Nanopores in Tight Oil R e s e r vo i r s , Journal of Petroleum Science and Engineering, http://dx.doi.org/10.1016/j.petrol.2017.01.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A Fully Compositional Model Considering the Effect of Nanopores in Tight Oil Reservoirs Bicheng Yana, Yuhe Wangb*, John E. Killougha a
Department of Petroleum Engineering, Texas A&M University, College Station, TX, USA b
Department of Petroleum Engineering, Texas A&M University at Qatar, Doha, Qatar
*Corresponding Author: Yuhe Wang, [email protected]
Abstract Conventional compositional simulators are usually difficult to interpret the different gas oil ratio (GOR) from tight oil reservoirs, and this also indicates an unreliable prediction of ultimate hydrocarbon recovery. We realize that there are two issues related to the compositional simulation of production in tight oil reservoirs. Firstly, tight oil reservoirs typically exhibit extremely small matrix pore size in the order of nanometers, so the capillary pressure between vapor and liquid phases is considerable such that the PVT of the confined fluid deviates from that of the bulk fluid with capillary pressure ignored. Secondly, during depletion process, rock compaction causes pore space reduction and brings remarkable changes in rock properties. In this work we implement rigorous confined fluid phase behavior calculation depending on capillary pressure and rock compaction in a fully compositional simulator. Capillary pressure in matrix nanopores is calculated by Leverett J-function. Further, the impact of capillarity on phase equilibrium is taken into account through modifying the stability test and two-phase flash calculation. Dynamic rock compaction is considered in the simulator via rock compaction tables, such that fluid mobility decreases with permeability reduction and capillary effect is simultaneously coupled. The unique implementation in the simulator captures the dynamic behavior of rock and fluid properties in tight oil reservoirs. Typical suppression of bubble point pressure and reduction of oil viscosity and density is observed from our simulation results. Reservoir scale simulation results shows that this model resolves the problem of the inconsistent GOR in tight oil production and greatly facilitate the history matching process. The enhanced compositional simulation will ultimately improve our understanding of tight oil reservoirs and provide better guidance for recovery prediction.
Keywords: Tight oil reservoirs; compositional simulation; vapor-liquid equilibrium; capillary pressure; rock compaction
Nomenclature Acronyms 1
BHP
Bottom-Hole Pressure;
BIC
Binary Interacting Coefficient;
GOR Gas oil ratio; IMPEM
Implicit pressure explicit mass;
SSI
Successive substitution iteration;
VLE
Vapor Liquid Equilibrium;
Variables Fugacity of component in phase ; Vapor mole fraction, fraction; Permeability, md; Component equilibrium ratio, fraction; Number of non-water components in the reservoir; Vapor phase pressure, psia; Liquid phase pressure, psia; Capillary pressure in between liquid and vapor phases, psia; Pore radius in porous medium, m; Gas universe constant, 8.314
;
Fugacity equilibrium residual of component ; Capillarity residual; Residual of Rachford-Rice Equation Reservoir temperature, VLE
;
Vapor liquid equilibrium; Molar fraction in liquid phase, fraction; Molar fraction in vapor phase, fraction; Molar fraction in the overall mixture, fraction; Fugacity coefficient of phase ; Porosity, fraction; Viscosity, cp; Density,
;
2
Chemical potential of component in phase ; Interfacial tension; Contact angle between liquid and vapor phase; Superscripts and subscripts Liquid phase; Vapor phase;
1. Introduction The development of unconventional tight oil resources has achieved great success through the application of horizontal well drilling and hydraulic fracturing. As predicted by EIA (Figure 1), tight oil production worldwide will increase to be more than double from 2015 to 2040 1, and specifically tight oil production in United States will increase from 4.1 million barrels/day in 2016 to 7.1 million barrels/day in 2040. Despite of this great success, our understanding of the production mechanisms in tight oil reservoirs is still very limited.
Figure 1. World tight oil production from 2015 to 2040
1
In tight oil resources, the matrix permeability is extremely low and the average pore size is in the nanometer magnitude. For example, the average pore size in Bakken shale matrix is about 10 to 50 nm. Fluid within such tiny pores exhibits different properties from conventional oil reservoirs, because the capillary pressure in those nanopores is significant and it may induce deviation in phase behavior of tight oil reservoirs. The effect of porous media on the phase behavior of hydrocarbon fluid was investigated in literatures. Sigmund, et al. 2 concluded that the effect of curvature on phase behavior is not significant except at high surface curvatures. However, this would not happen in hydrocarbon reservoirs even with the presence of clay particles, since those finest pores are likely to be filled by connate water. Through implementing a capillarity equation in phase equilibrium, Brusilovsky 3 demonstrated that bubble point pressure decreases but dew point pressure increases. Further, Nojabaei, et al. 4 investigated the impact of nanopores on phase behavior of Bakken shale oil, and the results showed that bubble-point pressure can be supressed significantly, and 3
the same conclusion was also reported in other literatures 5. Nojabaei, et al. 6 implemented a compositionally-extended black-oil model considering capillarity in phase behavior, and the capillary pressure varies with interfacial tension and pore size distribution. This approach is innovative but the extended black-oil formulation is intrinsically limited in terms of compositional characterization. Sandoval, et al. 7 developed an efficient and robust algorithm for calculating fluid phase behavior in the presence of capillarity. They found that the bubble point pressure suppression becomes significant when moving away from the critical point, while the dew point pressure exhibits different variations depending on the location of phase diagram. Rezaveisi, et al. 8 implemented capillarity in flash and stability test in UTCOMP simulator, but still they didn’t apply it for practical field-scale application of tight oil reservoirs. Dong, et al. 9 coupled PengRobinson Equation of State with capillary pressure equation and adsorption theory to investigate phase behavior in cylindrical nanopores, and they found that bubble point pressure is overestimated if neglecting adsorption film. Moreover, Nojabaei, et al. 4 reported that as pore pressure decreases or effective stress increases, the tight matrix is likely to experience a reduction of pore size. This observation can also be inferred from production data in Bakken shale and lab data. In their approach, through PVT analysis they theoretically demonstrated that the suppression of bubble point pressure becomes even more significant when considering pore size reduction through the depletion process, and applied this as input in a commercial reservoir simulator for history matching. Therefore, a rigorous consideration of pore size change throughout the tight oil production processes is also very necessary for general compositional reservoir simulation. Motivated by those points, this work implements the impact capillarity and rock compaction in a general fully compositional reservoir simulator 10 and applies it to predict tight oil reservoir performance. Specifically capillary equation is explicitly implemented in stability test based on Gibbs free energy approach and vapor-liquid equilibrium (VLE). Further, matrix pore size is iteratively calculated during pressure depletion by rock compaction tables. The implementation is designed to rigorously consider the fluid and rock properties variations in the nanopores. The reminder of this paper is organized as follows. The next session illustrates the vapor-liquid equilibrium formulation considering capillary pressure and the overall phase behavior evaluation workflow in compositional simulation. The third session presents our main results based on the new model. Finally, the whole work is concluded and summarized. 2. Numerical Formulation In this work, a fully compositional reservoir simulator 10-11 is extended to accommodate the effect of capillary pressure on vapor-liquid equilibrium. The simulator is designed volume-balance formulation and it is solved by the classic implicit pressure explicit mass (IMPEM) scheme. Since simulator formulation itself is not the focus of this paper, more details with regards to this can be referred to 12. Originally the simulator is able to handle multi-phase multi-component reservoir simulation, and the PVT modules in the simulator include stability test based on Gibbs free energy approach and vapor-liquid phase split calculation 13, but the effect of capillary pressure on phase behavior is not implemented. In this part, we will basically introduce the newly implemented flash and stability test algorithm considering capillarity effect in the reservoir simulator. 2.1 Stability Test Considering Capillary Pressure
4
In this part, the consideration of capillarity effect in stability test is based on the stability criterion of Michelsen 13b. If the original system is stable, Equation (1) will be valid. and are respectively the chemical potential of the incipient phase and the original system. ∑
(1)
Based on Equation (2), chemical potential is usually convenient to be represented in terms of fugacity, given by Equations (3) and (4). Here it is assumed that the original phase is liquid phase and the trial phase is vapor phase. The opposite setting, where the original phase is vapor and the trial phase is liquid, is equivalent. (2) (3) (4) As a result, given system temperature , Equation (1) is transformed into Equation (5). The last term in Equation (5) is usually ignored because of the small capillary pressure between vapor and liquid phases. However, it is taken into account here because capillary pressure is hypothesized to be significant in confined nanopores. ∑ ∑
{
∑
{
{
} } }
(5)
Equation (6) is defined to be independent of , and thus it can be calculated in advance, which is convenient in terms of implementation. Then we have Equation (7). (6) ∑
(7)
The stationary criterion is Equation (8). (8)
Let
, then (9)
5
Based on material balance Equation (10), we can calculate through Equation (11), and this indicates that can be interpreted by phase composition moles. ∑
∑
(10) (11)
∑
Here the solution of Equation (9) can be used to examine the stability test. In terms of , this original system is stable if ∑ and is unstable if ∑ . Equation (9) here is solved by the combination of Successive Substitution Iteration (SSI) and Newton-Raphson method. 2.2 Vapor-Liquid Two-Phase Split Considering Capillary Pressure The consideration of capillarity effect on vapor-liquid equilibrium (VLE) can be readily represented by Equations (12) and (13). In Equation (12), and respectively designate vapor phase pressure and liquid phase pressure, and represents the capillary pressure between those two phases. Equation (13) is the equality of fugacity in liquid and vapor phases, and the two fugacities are defined through Equations (14) and (15). In conventional flash calculation, is considered to be equal to and thus conventional VLE only includes the component fugacity equilibrium in vapor and liquid phases. It’s noted that those two consecutive equations are established based on continuous bulk vapor and liquid phase separated by a curvature interface. In ultra-tight porous media those two phases are not necessarily to be continuous, but it can be shown that pressure and phase compositions in different isolated vapor or liquid regions are equal when the system reaches equilibrium 14. Therefore, the continuous phase prerequisite is assumed to be still reliable here. (12) (13) (14) (15)
To close the nonlinear system in Equations (12) and (13), other constraints are required. The first one is the material-balance constraint with the introduction of vapor mole fraction , as shown in Equation (16). (16)
Additionally component mole fractions of each phase and the overall mixture should sum to unity when equilibrium is reached, as shown in Equation (17). With the definition of equilibrium ratio in Equation (18), the material-balance constraints in Equation (16) can be transformed to the classic Rachford-Rice Equation 15, as shown in Equation (19). ∑
∑
∑
(17) 6
(18) ∑
(19)
Based on Equations (14) and (15), the definition of component equilibrium ratio can be further transformed into Equation (20). Equation (20) shows that the capillary pressure is embedded in this modified component equilibrium ratio, and it can be reduced to conventional format if ignoring the capillary pressure. (20) At a given capillary pressure, Equations (13) and (19) are solved through combining successive substitution iteration (SSI) and Newton-Raphson method. 2.3 Capillary Pressure Evaluation in Tight Porous Media The calculations above require the evaluation of capillary pressure. During the iteration of stability test and VLE phase split calculation, capillary pressure values are calculated using saturation results from the previous iteration. The capillary pressure is fundamentally related to the geometry of the capillary and the wettability of the porous media, and it can be calculated through Young-Laplace equation, shown as Equation (21). This approach has been applied in Nojabaei, et al. 4, Honarpour, et al. 5a, and Pang, et al. 5c etc, and the pore radius is taken into account. (21)
As Nojabaei, et al. 4 stated, capillary pressure calculated through Young-Laplace equation is much lower than the experimental measurement due to the very low interfacial tension predicted by Macleod-Sugden correlation 16. This observation also indicates the importance of reliable capillary pressure prediction for shale/tight rocks. On the other hand, Leverett J-function (Equation (22)) is based on measured reference capillary pressures, and the scaling factor is supposed to be given for a particular rock type based on lab data. Unfortunately so far there is no solid measurement of capillary pressure for the Bakken shale. Because the porous media investigated here is a saturated one, the Leverett J-function approach is assumed to be appropriate to evaluate capillary pressure here. Compared to Young-Laplace Equation, Leverett J-function is friendly to the simulator implementation, since pore size information is embedded in permeability and porosity. √
(22)
The porosity of the Bakken shale is assumed to be 0.06. Further, the corresponding permeability and pore radius can be calculated based on Kozeny-Carman equation 17 and Nelson’s correlation 18. The reference capillary pressures under different pore radii are also 7
presented based on a similar formation from Crain’s petrophysical handbook. Those values are illustrated in Table 1 and are expected to be very close to practical situation.
Table 1. Pore radius, permeability and capillary pressure Pore Radius (nm)
Premeability (md)
Capillary Pressure (psi)
50
0.0070
102.20
40
0.0046
127.75
30
0.0027
170.33
20
0.0012
255.50
10
0.0003
511.00
2.4 Rock Compaction towards Nanopore Space Dynamic rock compaction is considered in the simulator through rock compaction table. Therefore, when pore pressure changes during time-step and newton-step evolution, permeability reduction ratio is interpolated from the rock compaction table. The permeability update further impacts the pore radius and capillary pressure. For Bakken shale investigated in this work, the rock compaction table is presented in Table 2. Table 2: Rock compaction table of Bakken shale Pressure Change (psi)
Premeability Reduction Ratio
-5180
0.489
-4450
0.500
-3700
0.511
-2960
0.532
-2220
0.588
-1480
0.675
-740
0.791
0
1.0
With the consideration of rock compaction, pore size reduces during depletion, and the bubble point pressure for confined fluid will be even more suppressed during production. As a result, reservoir fluid is likely to stay in single phase (oil) with light components still dissolved in it, and oil density and viscosity decrease and oil compressibility increases such that fluid flow capacity and flow driving energy increases. On the other hand, permeability decreases with rock compaction, and thus it negatively impacts the fluid flow capacity. Therefore, when considering rock compaction for fluid flow in tight porous media, it induces greater confinement effect and lower permeability and those two effects compete with each other, as shown in Figure 2.
8
No pressure maintenance
Reservoir Compaction
Lower Permeability
_
+ Flow Capacity
Figure 2. Impact of rock compaction and fluid confinement on flow in tight porous media
2.5 VLE Workflow Considering Capillarity and Rock Compaction In summary, the workflow of vapor liquid equilibrium with the consideration of capillary pressure and rock compaction is shown in Figure 3. At a given thermodynamic condition in a time/newton step, through compaction table rock permeability and porosity are updated and used to evaluate capillary pressure. Meanwhile, if previously the fluid stays in single-phase, stability test is performed and capillarity is considered in the calculation. If the fluid is unstable at single phase or it is previously in two-phase, then two-phase flash calculation considering capillarity is performed. Otherwise, fluid is still single phase. After the phase split, fluid properties and other secondary variables are updated, and capillary pressure is calculated and is resent to stability test and flash calculation, and new secondary variables are updated. This process is iterated until both phase equilibrium and capillarity equality are ensured, and then simulation gets into solving a global non-linear system for updating primary variable(s).
9
Starting point
No
Single-phase previously?
Update K and
Yes Stability test Stable?
No
Yes Two-phase flash
Single-phase liquid
The rest procedures in a time/newton step
Saturation dependence
Figure 3. Workflow for vapor-liquid equilibrium considering capillarity and rock compaction
3. Cases Analysis This part mainly presents the results of fluid PVT in confined nanopores and the application into compositional simulation of tight oil reservoirs. Fluid compositional data is based on Bakken shale oil from Nojabaei, et al. 4, as shown in Table 3 and 4. At the temperature of 240 °F the saturation pressure for this fluid here is the bubble point pressure. The fluid data in Table 3 and 4 and will be used in all cases presented below, and the rock compaction data in Table 2 will be used in those cases below if rock compaction is considered.
Component
Mole Fraction
C1 0.36736 C2 0.14885 C3 0.09334 C4 0.05751 C5-C6 0.06406 C7-C12 0.15854 C13-C21 0.07330 C22-C80 0.03704 Reservoir temperature (°F)
Table 3. Bakken oil composition data Critical Critical Pressure Temperature (°F) (psia) -116.63 667.80 90.09 707.80 206.01 616.30 305.65 550.70 415.81 461.29 593.58 363.34 872.43 249.61 1384.82 190.12 240
Acentri c Factor 0.0130 0.0986 0.1524 0.2010 0.2684 0.4291 0.7203 1.0159
Molar Weight
Paracho r
16.04 30.07 44.10 58.12 78.30 120.56 220.72 443.518
74.8 107.7 151.9 189.6 250.2 350.2 590.2 1216.8
Table 4. Bakken oil binary interaction coefficient (BIC) C1
C2
C3
C4
C5-C6
C7-C12
C13-C21
C22-C80
0.005 0.0035 0.0035 0.0037 C1 0.0050 0.0031 0.0031 0.0031 C2 Other BICs are zero and not presented here for simplicity.
0.0033 0.0026
0.0033 0.0026
0.0033 0.0026
10
3.1 Confined Fluid Phase Behavior The P-T diagram of the Bakken oil is calculated through rigorous stability test and saturation pressure calculation algorithms in session 2. The consideration of capillary pressure is evaluated through Young-Laplace equation under different nanopore radii, and compared with the conventional scenario without capillarity. The results are plotted in Figure 4. Here it shows that at a given temperature the bubble point pressure decreases as pore size reduces. When pore radius is 10 nm, the suppression of bubble point pressure is about 125 psia. 3500
Pressure, psia
3000
2500
No Capillary Effect r = 50 nm r = 40 nm r = 30 nm r = 20 nm r= 10 nm
2000
1500
1000 80
130
180
230
280
330
380
430
Temperature,oF
Figure 4. Bubble point pressure lines of Bakken oil with considering capillarity under different pore radii and without considering capillarity
As previously discussed, capillary pressure predicted by this approach is much lower than the actual one. Therefore, we further evaluate capillary pressure through Leverett J-function approach for all cases, and the results are presented in Table 5. We can see that there are significant differences between bubble point pressures based on Young-Laplace equation and those based on Leverett J-function. The differences here also reveal that capillary pressure data with good quality is very important in phase behavior prediction.
Table 5. Bubble point pressure of Bakken oil at 240 °F Bubble Point Pressure (psia) Pore Radius (nm) Young-Laplace Leverett J-Function 50 2766 2512 40 2761 2450 30 2751 2345 20 2732 2145 10 2641 1588 No Capillary 2788
If compared to the same fluid system with unsuppressed bubble point pressure, a suppression in bubble point pressure physically indicates that there are more light components remaining in the liquid oil phase under the same pressure and temperature condition. More light components in oil phase are expected to bring a reduction of oil density and oil viscosity. Fluid viscosity here is calculated by Lohrenz-Bray-Clark correlation 19. With capillary pressure considered in phase behavior, Figure 5 illustrates the oil density, oil viscosity and their ratio changes with nanopore 11
radius for Bakken oil at 240 and 1500 psia. The results are also compared with bulk fluid properties without considering capillary pressure. In Figure 5 (a) and (b), it clearly shows that oil density and oil viscosity decrease as nanopore radius decreases from 50 nm to 10 nm, and the two properties are smaller than those values when neglecting capillary pressure in phase behavior. Further, since oil mobility is directly proportional to this ratio, the ratio of oil density to oil viscosity is also plotted in Figure 5 (c). We can see that as nanopore radius decreases, the ratio of oil density to viscosity increases, and thus oil becomes much more mobile in confined nanopore space. Besides, more light components in oil phase also indicates that oil phase compressibility increases, which provides more driving energy to extract liquid from tight oil reservoirs. 0.50 Blue dash line: oil viscosity without capillarity
Oil viscosity, cp
Oil density, lb/cu.ft
45 Purple dash line: oil density without capillarity
40
35
0.42 0.34 0.26
0.18 0.10
30 0
10
20 30 40 Pore radii, nm
50
0
60
10
20 30 40 Pore radii, nm
(a)
50
60
(b) oil ρ/µ, lb/cu.ft/cp
250 200 150
100 Green dash line: oil ρ/µ without capillarity
50 0 0
10
20 30 40 Pore radii, nm
50
60
(c) Figure 5. Confined fluid properties change with pore radius at 240 (b) oil viscosity; (c) the ratio of oil density to oil viscosity.
and 1500 psia: (a) oil density;
3.2 1D Core Size Model The first example is a one dimensional core size model, as shown in Figure 6. Other parameters related to this model is presented in Table 6. A producer is located at the first grid block, and it is constrained by a minimum BHP (1500 psia) and maximum oil volume rate (0.001 RB/day). The oil rate is very low since the model is small.
12
Figure 6. 1D core model with a producer
Table 6. Reservoir parameters for 1D core size model 0.5 x 164 x 0.5 Model size ( ) 0.5 x 3.28 x 0.5 Grid size ( ) Grid number 1 x 50 x 1 Porosity 0.06 0.002 Permeability ( ) 6840 Initial pressure ( ) 1500 Producer minimum BHP ( ) Producer oil rate (RB/day) 0.001 240 Initial temperature ( )
Three different scenarios are simulated, respectively Case 1 with no capillarity effect on VLE and no rock compaction, Case 2 with capillarity effect on VLE but no rock compaction, and Case 3 with capillarity effect on VLE and with rock compaction. The results are plotted in Figure 7. Figure 7 presents the cumulative oil production and gas-oil-ratio in the wellbore condition. Through comparison, it clearly shows that Case 2 produces the most oil from the reservoir, while Case 1 produces the least oil from the reservoir. The comparison between those two cases shows that capillary pressure effect on phase behavior favors oil production in the tight oil reservoir, since oil mobility of confined fluid increases. Case 3 considers both capillary pressure in VLE and rock compaction. As expected this case produces median oil production here. This indicates that during pressure depletion rock compaction reduces rock permeability and it offsets the increase of mobility caused by capillary pressure. The GOR at the in-situ reservoir condition tells us that capillary pressure induces lower bubble point pressure and delayed gas phase occurrence, and this is even more exacerbated by further considering rock compaction. Besides, the early zero GOR at the in-situ reservoir conditions indicates that BHP at that period is higher than the bubble point pressure.
13
1.6
4500 4000
Cum Oil Production, RB
3000 2500 0.8
Oil Prod for Case 1
Oil Prod for Case 2
Oil Prod for Case 3
GOR for Case 1
GOR for Case 2
GOR for Case 3
2000
1500
0.4
1000
GOR in well, RCF/RB
3500
1.2
500 0.0
0
0
1
2
3
4
5
Time, days
Figure 7. Cumulative oil production and GOR at in-situ reservoir conditions for three different scenarios in the 1D model: Case 1: with no capillarity effect on VLE and no rock compaction, Case 2 with capillarity effect on VLE but no rock compaction, Case 3 with capillarity effect on VLE and rock compaction.
3.3 Fractured Tight Oil Reservoir with Multiple Hydraulic Fractures Here a tight oil reservoir is perforated by a horizontal well and 4 planar hydraulic fractures traverse through the reservoir in y-direction. The horizontal well is constrained by constant oil rate (50 RB/day) and minimum BHP (1900 psia). Basic reservoir parameters are presented in Table 7. The schematic of the model is illustrated in Figure 8. Therefore, fluid flow through the reservoir follows the sequence of matrix-fracture-well. In hydraulic fractures capillary pressure is not considered since we assume that capillary pressure effect is negligible and thus bulk fluid phase behavior is appropriate here. On the other hand, in the tight oil matrix the effect of capillary pressure on phase behavior and the rock compaction is conditionally considered. Table 7. Reservoir parameters for the 3D tight oil reservoir model 1000 x 1000 x 20 Model size ( ) Grid number 35 x 67 x 10 Matrix porosity 0.06 0.002 Matrix permeability ( ) Fracture number 4 1.0 Fracture grid width ( Fracture porosity 0.02 100 Fracture conductivity ( ) 6380 Initial pressure ( ) 1900 Producer minimum BHP ( ) Producer oil rate (RB/day) 50 240 Initial temperature ( )
14
Confined
Unconfined Figure 8. Tight oil reservoir with horizontal well and multiple hydraulic fractures
Similarly, here we also compare three different scenarios in the reservoir model, and the results at the in-situ reservoir conditions are illustrated in Figure 9. Here it shows that the case with capillary pressure in VLE but no rock compaction (Case 2) has the highest cumulative oil production, and the case with both capillarity and rock compaction (Case 3) has the median oil production. This further demonstrates that at reservoir scale the consideration of capillarity in phase behavior in Case 2 facilitates the oil transport into the wellbore, while the median oil production in Case 3 is caused by the permeability decrease due to rock compaction. There is a decrease of in-situ GOR for cases considering capillary pressure in phase behavior. The insitu GOR for the case with compaction and capillary pressure interestingly exhibits a step-like increase and decrease back to near zero. This might be explained by that nanopore size reduction due to rock compaction further suppresses the bubble point pressure and thus most of the hydrocarbon molecules flow in liquid phase in the wellbore. 14000000
2000
Oil Prod for Case 2
Oil Prod for Case 3
GOR for Case 1
GOR for Case 2
GOR for Case 3
1800 1600
10000000
1400 1200
8000000
1000 6000000
800
600
4000000
400
2000000
GOR in the well, RCF/RB
Cum Oil Production, RB
12000000
Oil Prod for Case 1
200 0 0
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Figure 9. Cumulative oil production and GOR at in-situ reservoir conditions for three different scenarios in the reservoir model: Case 1: with no capillarity effect on VLE and no rock compaction, Case 2 with capillarity effect on VLE but no rock compaction, Case 3 with capillarity effect on VLE and rock compaction.
4. Conclusions 15
In this work a fully compositional reservoir model considering the impact of capillary pressure and rock compaction on vapor liquid equilibrium is developed, and Leverett J-function is used to evaluate capillary pressure for tight oil. This model is used to predict the phase behavior in confined nanopore space and tight oil reservoir performance. The capillary pressure effect on vapor-liquid equilibrium is significant when pore size in the matrix is in the nanometer scale. For oil-wet tight oil matrix, fluid confinement in the nanopores suppresses the bubble point pressure, and results in decreases in oil density and viscosity. This ultimately favors the oil phase transport to the wellbore by increasing the oil mobility and more driving energy. Besides, through our compositional model oil production in tight oil reservoirs is demonstrated to very sensitive to rock compaction during reservoir depletion. Rock compaction directly decreases rock permeability and reduces fluid mobility. On the other hand, pore size reduction during rock compaction further exacerbates bubble point pressure suppression of confined fluid in nanopores. Therefore, reservoir simulation of tight oil reservoirs should consider both capillary pressure and rock compaction on production. Reference 1. EIA, Annual Energy Outlook 2016. EIA: International Energy Outlook 2016 2016. 2. Sigmund, P. M.; Dranchuk, P. M.; Morrow, N. R.; Purvis, R. A., Retrograde Condensation in Porous Media. 1973. 3. Brusilovsky, A. I., Mathematical Simulation of Phase Behavior of Natural Multicomponent Systems at High Pressures With an Equation of State. 1992. 4. Nojabaei, B.; Johns, R. T.; Chu, L., Effect of Capillary Pressure on Phase Behavior in Tight Rocks and Shales. 2013. 5. (a) Honarpour, M. M.; Nagarajan, N. R.; Orangi, A.; Arasteh, F.; Yao, Z., Characterization of Critical Fluid PVT, Rock, and Rock-Fluid Properties - Impact on Reservoir Performance of Liquid Rich Shales. Society of Petroleum Engineers: 2012; (b) Du, L.; Chu, L., Understanding Anomalous Phase Behavior in Unconventional Oil Reservoirs. Society of Petroleum Engineers: 2012; (c) Pang, J.; Zuo, J. Y.; Zhang, D.; Du, L., Impact of Porous Media on Saturation Pressures of Gas and Oil in Tight Reservoirs. Society of Petroleum Engineers: 2012. 6. Nojabaei, B.; Siripatrachai, N.; Johns, R. T.; Ertekin, T., Effect of Saturation Dependent Capillary Pressure on Production in Tight Rocks and Shales: A Compositionally-Extended Black Oil Formulation. Society of Petroleum Engineers: 2014. 7. Sandoval, D.; Yan, W.; Michelsen, M. L.; Stenby, E. H., Phase Envelope Calculations for Reservoir Fluids in the Presence of Capillary Pressure. Society of Petroleum Engineers: 2015. 8. Rezaveisi, M.; Sepehrnoori, K.; Pope, G. A.; Johns, R. T., Compositional Simulation Including Effect of Capillary Pressure on Phase Behavior. Society of Petroleum Engineers: 2015. 9. Dong, X.; Liu, H.; Hou, J.; Wu, K.; Chen, Z., Phase Equilibria of Confined Fluids in Nanopores of Tight and Shale Rocks Considering the Effect of Capillary Pressure and Adsorption Film. Industrial & Engineering Chemistry Research 2016, 55 (3), 798-811. 10. Dean, R. H.; Lo, L. L., Simulations of Naturally Fractured Reservoirs. 1988. 11. Tang, D. E.; Zick, A. A., A New Limited Compositional Reservoir Simulator. Society of Petroleum Engineers: 1993. 12. (a) Wong, T. W.; Firoozabadi, A.; Aziz, K., Relationship of the Volume-Balance Method of Compositional Simulation to the Newton-Raphson Method; (b) Young, L. C.; Stephenson, R. E., A Generalized Compositional Approach for Reservoir Simulation. 1983. 13. (a) Michelsen, M. L., The isothermal flash problem. Part II. Phase-split calculation. Fluid Phase Equilibria 1982, 9 (1), 21-40; (b) Michelsen, M. L., The isothermal flash problem. Part I. Stability. Fluid Phase Equilibria 1982, 9 (1), 1-19. 16
14. Bedrikovetsky, P., Mathematical Theory of Oil and Gas Recovery. Kluwer Academic Publishers: The Netherlands, 1993. 15. Rachford, H. H.; Rice, J. D., Procedure for use of electrical digital computers in calculating flash vaporization hydrocarbon equilibrium. JPT 1952, 19, 195. 16. Pederson, K. S.; Christensen, P. L., Phase behavior of petroleum reservoir fluids. Taylor & Francis Group: CRC Press, 2007. 17. (a) Carman, P. C., Fluid flow through granular beds. Chemical Engineering Research and Design 1937, 75, S32-S48; (b) Kozeny, J., Ueber kapillare Leitung des Wassers im Boden. Sitzungsber Akad 1927, 136 (2a), 271-306. 18. Nelson, P. H., Permeability-porosity relationships in sedimentary rocks. Log Analyst 1994, 35 (3), 38-62. 19. Lohrenz, J.; Bray, B. G.; Clark, C. R., Calculating Viscosities of Reservoir Fluids From Their Compositions. 1964.
Highlights
A fully compositional model with rigorous dynamic confined fluid phase behavior.
Unique implementation on a 3D comprehensive compositional simulator.
Realistic tight oil recovery simulation, revealing the necessity of added physics.
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