A quantitative model for evaluating the impact of volatile oil non-equilibrium phase transition on degassing

PETROLEUM EXPLORATION AND DEVELOPMENT Volume 39, Issue 5, October 2012 Online English edition of the Chinese language journal Cite this article as: PETROL. EXPLOR. DEVELOP., 2012, 39(5): 636–643.

RESEARCH PAPER

A quantitative model for evaluating the impact of volatile oil non-equilibrium phase transition on degassing WU Keliu1,*, LI Xiangfang1, WANG Haitao2, GUAN Wenlong3, WANG Xing4, LIAO Zongbao4, LI Wuguang1 1. Key Laboratory for Petroleum Engineering of the Ministry of Education, China University of Petroleum, Beijing 102249, China; 2. Sinopec Research Institute of Petroleum Engineering, Beijing 100101, China; 3. PetroChina Research Institute of Petroleum Exploration and Development, Beijing 100083, China; 4. CNOOC Research Institute, Beijing 100027, China

Abstract: To quantitatively evaluate the impact of non-equilibrium phase transition on degassing in an oil-gas system and accurately determine the bubble point pressure of the system with different oil rates, a model, which takes the influence of pressure drop and non-equilibrium phase transition into consideration, is established based on the Henry model to calculate the gas solubility. To measure the degassing speeds with non-equilibrium phase transition and equilibrium phase transition respectively, degassing experiments are done at different pressure drop speeds with the oil-gas system composed of transformer oil and carbon dioxide. A function characterizing the non-equilibrium nature of the oil and gas system is derived after calculation and matching, based on which, degassing speeds and bubble point pressures at different pressure drop speeds are calculated. The impact of non-equilibrium phase transition on degassing and the deviation degree of the bubble point pressure are evaluated quantitatively. The model considering the non-equilibrium phase transition has higher computational accuracy than the Henry model not considering the non-equilibrium phase transition. The computation results indicate that the non-equilibrium phase transition reduces the bubble point pressure in the reservoir and slows down the degassing process of the volatile oil during the development of volatile oil reservoirs. The extent to which the non-equilibrium phase transition impacts degassing depends on the speed of pressure drop and the value of pressure drop: the bigger the pressure drop value and the faster the pressure drop speed, the more significant impact the non-equilibrium phase transition exerts on degassing, the lower the bubble point pressure and the slower the degassing speed. Key words: oil-gas system; volatile oil; non-equilibrium phase transition; degassing; model; solubility

Introduction Volatile oil is an important part of oil-gas resources[13]. During the exploitation of volatile reservoirs, both reservoir pressure and wellbore pressure decrease gradually. Degassing happens as soon as reservoir pressure falls below the bubble point of oil in place, which induces the change of single-phase flow into gas-oil two-phase flow. As a result, the solubility of gas in volatile oil keeps changing in every producing step [46]. As for the calculation of gas solubility, Henry (1803) concluded an empirical law of gas solubility under low pressure for the first time[7, 8]. Afterward Standing (1947) established the “Standing Graph” based on 105 experimental data, which, however, could hardly reach the industrial accuracy, especially for light oil[9]. Lasater (1958) conducted 158 experiments with 137 gas-oil systems and set up the equations on

gas solubility[10], which was more accurate than Standing’s in predicting the gas solubility of light oil. Vasquez and Begs (1980) proposed an improved gas solubility model via regression of 5008 experimental data[11], which further improves the prediction accuracy. However, these models are fitting formulas based on the experimental data of a specific fluid and not suitable for predicting the gas solubility of different fluid systems. In China, scholars such as Han Buxing have individually studied on the gas solubility of heavy oil, light oil, pure water and saline and proposed models for them[1219]. Unfortunately all of those models were established by means of multiple regression under the assumption of instant thermodynamic balance, which would result in big errors in cases such as predicting the gas solubility during the non-equilibrium phase transition when open or shut-in the well sharply. Chen et al held the idea that solubility is related to the mo-

Received date: 08 Mar. 2012; Revised date: 04 Jun. 2012. * Corresponding author. E-mail: [email protected] Foundation item: Supported by National Science and Technology Major Project Program (2011ZX05030-005-04) and the National Natural Science Foundation of China (50974128). Copyright © 2012, Research Institute of Petroleum Exploration and Development, PetroChina. Published by Elsevier BV. All rights reserved.

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lecular polarity of both solute and solvent, surface tension, temperature and pressure of the system[2025], while they haven’t taken the impact of non-equilibrium phase transition into consideration. On the foundation of non-equilibrium phase transition mechanism[2632], this paper proposes a gas solubility model of gas-oil system under non-equilibrium phase transition, which can serve to calculate the rate of degassing and bubble point pressure of volatile oil under different pressure drop.

1 1.1

Establishment and solution of the model Establishment of the model

The balance of a specific system would be broken under changing conditions and the transition from one balance to another is not instantaneous. In other words, a certain amount of time would be taken here, during which the system is in non-equilibrium state[27, 28]. For a system with volatile oil and gas, the feature of non-equilibrium state occurs when the speed of temperature and pressure transition exceeds that of phase transition[2932]. Even though the temperature of volatile oil reservoirs keeps constant during exploitation, the rapid pressure variation plays quite a vital role in oil-gas phase transition, especially in the vicinity of the wellbore with high flow rate. To study the non-equilibrium of oil-gas phase transition, for gas-oil system composed by fixed components, under different pressure drop rate, expressing the relationship of oil-gas volume change with mathematical equations, we can get the gas solubility model considering non-equilibrium phase transition of volatile oil-gas system. Henry’s law, which is suitable for the equilibrium state of volatile oil-gas system—without temperature or pressure change, can be expressed as[7]: w Dp (1) Solution factor is the function of temperature only when the volatile oil-gas system is under a process of pressure drop at constant temperature, which is to say the solution factor  in Henry’s law is constant[33]. The gas solubility of the volatile oil-gas system at constant pressure is a pan-function of pressure. The impact of pressure drop rate on gas solubility is significant and the gas solubility becomes bigger as the pressure drop rate increases. The corresponding model for gas solubility calculation is: t ª dp º dt» ] D « p  ³ K (t ) (2) dt ¼ 0 ¬ K(t) in the equation stands for the degree of non-equilibrium of phase transition. It decreases as the pressure drop time increases. That is to say, the non-equilibrium phase transition of oil-gas system is remarkable in the early pressure drop stage, becoming weaker gradually then. A specific oil-gas system has a unique corresponding weight function, which is an inherent property of the system and is only related to the components of the oil-gas system.

Equation (2) shows that the gas solubility conforms to Henry’s law when dp/dt0; while significant deviation occurs as pressure drop velocity dp/dt increases, the non-equilibrium transition affects gas solubility remarkably. Once the weight function is be determined by experiments, the gas solubility of the non-equilibrium phase transition of volatile oil-gas system can be obtained from Equation (2). 1.2

The solution of the model

According to Equation (2), the material balance equation of contact degassing can be expressed as: t ª dp º (3)  ] o  D « p  ³ K (t ) dt» dt ¼ 0 ¬ Considering that the pressure and volume of the system in an experiment need to be measured at the same time, and the pressure drops at a constant rate, dp/dt =v, baed on Equation (3), the degassing rate at the nth time point can be given by the equation below: n § ·  n ] o  D ¨ pn  lim v't ¦ K i ¸ (4) 't o 0 i 1 © ¹ Substituting the tested data of n(n)= n(p(n)) in Equation (4), n

we can calculate the value of weight function

¦K

i

at ti.

i 1

Equation (4) can be rewritten as:  p · § ] lim ¨ o  n  n ¸ (5) DQ DQ Q ' ' 't ¹ t t © i 1 Substitute the tested data in Equation (5) to calculate the value of weight function, then draw them into a graph of K–t, which is a curve like “S”. The fitted expression of the S curve can be: K t Hmr t (6) n

 ¦ Ki

't o0

The weight function can be obtained once H, m and r in equation (6) are determined. Substitute Equation (6) into Equation (2), the gas solubility under different pressure and pressure drop rate of a volatile oil-gas system in non-equilibrium phase transition can be calculated. 1.3

Analysis of the model

To analyze the reliability of the new model, on the basis of the material balance law, the system pressure is calculated, and compared with that from actual measurement then. During pressure draws down in the system, the gas phase volume equals to the change of dissolved gas volume in the liquid phase. The corresponding material balance equation is:  pVg / poVL (7)

The calculation expression of the system gas phase volume is: Vg

V  ( po  p )ZVL

(8)

Substituting Equation (7) and Equation (8) into Equation (3) gives t ª º p ªV dp º dt» (9) «  ( po  p)Z » ] o  D « p  ³ K (t ) dt ¼ po ¬VL ¼ 0 ¬

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Table 1 Analyzed data sheet for the components of transformer oil Name of Mass fraction Name of components of components /% components

Mass fraction of components /%

C13+C13ˉ

3.10

C19

16.99

C14

0.88

C20

14.10

C15

3.04

C21

7.78

C16

8.27

C22

4.04

C17

16.11

C23

1.41

C18

15.70

C24+C24+

8.58

Substitute the weight function and the system volume and the liquid volume at different time into Equation (9), and then obtain the numerical solution pressure by iteration and compare it with the ones from tests to verify the reliability of the model.

2

Laboratory experiment

2.1 2.1.1

Experiment design Experiment samples

To determine the impact of pressure drop rate, namely nonequilibrium phase transition, on gas solubility in oil phase, we carried out experiments with the liquid-gas system composed by transformer oil and CO2 of 99.9% purity[34, 35]. The component of transformer oil is shown in Table 1. 2.1.2

Experiment instruments

The experiment apparatus mainly includes the step-down pump, the PVT reactor and the incubator (shown in Fig. 1). (1) The step-down pump has a temperature range of 20–70 qC, pressure control accuracy of ±0.001 MPa, and the highest work pressure of 25MPa; (2) The PVT reactor can work at the temperature of 20–400 qC, with the highest work pressure of 25MPa; (3) The incubator can work at 20–180 qC, and control the temperature at ±1 qC, it is used to conduct constant temperature experiment at 40 qC.

Fig. 1

Chart of the experimental apparatus

2.1.3

Experiment methods

Through the minimum miscibility pressure experiment, determine the maximum solubility of CO2 in a certain volume of transformer oil at a certain pressure, namely the initial solubility. Load the experiment fluid prepared by PVT meter into a PVT reactor, lower the pressure of PVT reactor with the step-down pump and conduct contact degassing at a certain pressure drop rate, which keeps increase in the four groups of experiments in order to investigate the impact of non-equilibrium on degassing, the pressure drop rates of the four groups of experiments are 0.02, 0.04, 0.06, 0.08 MPa/min, respectively. Equal-interval tests are needed to obtain the weight function, so we measure the system pressure P (n) and rate of degassing n(n) every five minutes. To verify the reliability of the new model at rapid decompression, a rapid decompression experiment was conducted, the numerical solution for the system pressure at given degassing velocity is calculated and compared with the experiment value from tests. The experiment started at an initial pressure of 5.5 MPa, and an initial total volume of 5.2 cm3. The pressure was lowered to 1.5 MPa quickly, and the changes of liquid volume and system pressure during the system volume increases from 5.2 cm3 to 7.4 cm3 were recorded. Equilibrium experiment was conducted to compare the equilibrium phase transition and the non-equilibrium phase transition. During the experiment, the system pressure draw down slowly (pressure draw down once every 10 minutes) until the oil and gas are well separated. Record the system pressure and degassing velocity during the whole process. 2.1.4

Experiment procedures

(1) Determine the maximum gas-oil ratio parameter. Through the minimum miscibility pressure experiment, determine the maximum solubility of CO2 in a certain volume of transformer oil at a certain pressure. (2) Prepare the experiment fluid with PVT meter according to the parameters determined by step (1): 10.6 cm3 CO2 can dissolve in the 200 cm3 liquid at saturated pressure 3.6 MPa, and the initial gas solubility o is 53 cm3/cm3. (3) Load the experiment fluid into a PVT reactor, lower the pressure with the step-down pump, the pressure drop rates of the four groups of experiments are 0.02 MPa/min, 0.04 MPa/min, 0.06 MPa/min, 0.08 MPa/min, respectively. (4) Conduct contact degassing at the metering cylinder. (5) Record the experiment data. Record the time, corresponding degassing velocity and oil volume. (6) Conduct validation experiment. Repeat steps (1)-(5), and pressure draw down rapidly in step (3), the system volume increases to 7.4 cm3 rapidly. (7) Conduct equilibrium experiment. Repeat steps (1)-(5), and pressure draws down once every 10 minutes in step (3), stop until the oil and gas are well separated.

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2.1.5

Key Points of Experiment

(1) When lowering the system pressure with the step-down pump at constant pressure drop rate, the pressure drop rate is difficult to keep constant because the water flow velocity in pump is hard to control; (2) At the exit end of PVT reactor, the fluid flow out instantly at high velocity, the pressure of PVT reactor drawdown is hard to control at the desired experiment pressure. Therefore, to stabilize the pressure of PVT reactor, back pressure is needed at the same time to let oil and gas flow out steadily; (3) During the experiment, the volume of transformer oil is changing. During experimental data processing, we can calculate the CO2 solubility in transformer oil in reactor according to matter conservation; (4) Pressure draws down slowly in equilibrium experiment, oil and gas get fully separated, while pressure draws down rapidly in non-equilibrium phase transition experiment, part of the dissolved gas wouldn’t get well separated. 2.2 2.2.1

Results and analysis Results of the experiment

Comparing the degassing dynamics of the four processes, the n= n(p) curves are drawn at different pressure drop rates

Fig. 2

(Fig. 2). From Fig. 2, we can see there are big discrepancies between the values of equilibrium and non-equilibrium phase transition experiments. The calculated value of Henry Law is close to the experiment value of equilibrium phase transition and Henry Law can accurately calculate the gas solubility of equilibrium phase transition. The calculated value of the new model is in good agreement with the experiment value of non-equilibrium phase transition and the new model can accurately describe the physical process of non-equilibrium phase transition. Fig. 2 illustrates that the deviation of n=n(p) between equilibrium and non-equilibrium transition is fortified by greater pressure drop rate and the influence of the degree of non-equilibrium is more remarkable. Such a phenomenon is also valid as the pressure drop degree increases even though the pressure drop rate is constant. 2.2.2

Analysis of the experiment

Substitute the tested pressure and amount of degassing measured by the experiments into Equation (5), then calculate the weight function K(ti) at different time ti, shown in Fig. 3. Then the function expression of the curve is as follows. K˙1.32u0.750.06 t (10)

Relation curve of degassing velocity and system pressure under different pressure drop rates

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Fig. 5 P-t relation curve built on the experiment value and the calculated value Fig. 3 Relation curve of weight function and time in transformer oil and CO2 system

Input the weight function K(t) and pressure drop rate into the Henry’s model (1) and the new model (2), calculate the amount of degassing separately and then compare the errors of both models taking the tested data as reference, errors are expressed as follows. G E J ×100% (11)

E

It can be seen from Fig.4 that (1) the new model, whose error is less than 5%, is 25%–80% more accurate than Henry’s model when the pressure is under bubble point pressure and can describe the degassing progress of volatile oil-gas system more accurately; (2) the error of the new model increases as the pressure drop accelerates in the case of the same pressure, and such a difference augments as pressure approaches the bubble point pressure, 3.6MPa. The reasons can be that the faster the pressure draws down, the more significant influence of non-equilibrium phase transition on degassing will be and the non-equilibrium feature of weight function is not enough to describe this process; (3) when the pressure drop rate holds constant, the biggest error occurs just below the bubble point

pressure and gets smaller afterwards. Because when the pressure is slightly under the bubble point pressure, the gas just comes out, and the phase changes dramatically. Consequently the process is too complicated to be described well by the model, and the calculation would not be accurate accordingly. Meanwhile the relative measurement error of tested values is higher. To verify the applicability of the model, the weight function, the system total volume and the liquid volume at different time are substituted into Equation (9), and then obtain the numerical solution p (ti) by iteration (Fig. 5). It is clear from Fig. 5 that the new model can fit the degassing of non-equilibrium phase transition within a quite wide range of pressure drop rate.

3

Application of the model

3.1 Quantitative description of the deviation of degassing

By using the model given in this paper, the effect of non-equilibrium phase transient on degassing degree in a compound of transformer oil and CO2 at different pressure drop rates can be quantitatively evaluated. Taking dp/dt=v as constant, the equation (3) can be written as: t ª º  ] o  D « p  v ³ K (t )d t » (12) 0 ¬ ¼ Assuming when pressure is p+p, the amount of degassing at the equilibrium degassing curve equals to that under a pressure of p at the non-equilibrium curve, then:  ] o  D ( p  'p ) (13) Subtract Equation (12) from (13), t

'p

v ³ K (t )d t

(14)

0

Define the deviation index , to assess the influence of non-equilibrium phase transition on degassing. t

K Fig. 4 Calculated error of degassing velocity of non-equilibrium transition

 640 

'p u 100% p  'p

v ³ K (t )d t 0

t

u 100%

(15)

p  v ³ K (t )d t 0

As Equation (15) indicates, both the pressure drop rate and

WU Keliu et al. / Petroleum Exploration and Development, 2012, 39(5): 636–643

Fig. 6 Effect of non-equilibrium phase transition on degassing at different pressure drop speeds

its extent determine the effect of non-equilibrium phase transient on degassing degree. The quantitative description of the non-equilibrium’s effect on gas solubility of the system with transformer oil and CO2 is shown in Fig. 6. As shown in Fig. 6, the influence is more obvious for a larger pressure drop as the pressure drop rate keeps constant. The amount of degassing of non-equilibrium process is much less than that of equilibrium and such a deviation enlarges as pressure drops. Besides, the influence is also more obvious for a faster pressure drop rate. That is the amount of degassing of a faster process is less than that of the slower, while both are much less than that of the equilibrium process. The accurate description of the deviation of degassing at different pressure and pressure drop rate would be conducive to the establishment of a more accurate oil well productivity models to provide theoretical basis for determining a more rational production. 3.2 Quantitative evaluation of the bubble point pressure deviation

Given the expression of weight function (10), the bubble point pressure pvb at different pressure drop rate can be derived. The amount of degassing n0 as the system is at bubble point. From Equation (3) we have: t ]o =p  v ³ K (t )d t D 0

(16)

Rearrange Equation (16), f ( p)

t ]o  p  v ³ K (t )d t D 0

(17)

Solve Equation (17) by iteration: input a small value to p1, say p1= 0.1 MPa, and substitute it into equation (17) to calculate f1  p1 , then set p2= p1+0.001, find f2  p2 using the same way and repeat until fn1  pn1 , fn  pn , where pnİ3.6 MPa. The corresponding pressure pj of the minimum value in f j  p j (j=1–n) is bubble point pressure pvb. The bubble point pressure pvb at different pressure drop rate is smaller that that of equilibrium process pb. Compare the

Fig. 7 Relationship of bubble point pressure deviation and pressure drop speed

bubble point pressures from the tests with calculated ones, the bubble point pressure deviation is expressed as follows pvb  pb H ×100% (18) pb From Fig. 7, the deviation of bubble point pressure increases as pressure drop accelerates, that is for a system the pressure should drop to a lower level for a bubble to come out as the pressure drops faster. When the pressure drop speed is 0.08 MPa/min the deviation is about 41.67% and the influence of non-equilibrium is significant. In order to prevent degassing, gas or water injection are necessary to keep reservoir pressure in developing volatile oil reservoirs. The bubble point is different for different production rates, namely the timing to keep reservoir pressure at different pressure drop rate is different accordingly. Without taking the influence of non-equilibrium effect on bubble point pressure into account, water or gas could be injected too early, rendering huge waste of money. Only by considering the feature of non-equilibrium phase transition can more rational development strategies be plotted.

4

Conclusions

Degassing would take place in volatile oil reservoirs as the pressure decreases below the bubble point pressure. A model of gas solubility is proposed considering the features of non-equilibrium phase transition, which makes it possible to establish more accurate oil well productivity models, providing basis for selecting a more rational production. Lab experiments and calculation results show that the model proposed here, whose error is within 5%, is more accurate than Henry’s model which doesn’t consider the non-equilibrium process. The model can describe the degassing dynamics of oil-gas systems more accurately. The non-equilibrium phase transition brings with a lower bubble point and slower degassing process in volatile oil reservoir compared with the equilibrium process. The proposed model can calculate the bubble point of volatile oil and the amount of degassing during different time period, which

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WU Keliu et al. / Petroleum Exploration and Development, 2012, 39(5): 636–643

serves as the basis of determining water/gas injection timing and the scientific management and exploitation of volatile oil reservoirs.

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Nomenclature

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w —calculated gas solubility by Henry Law, mL/mL;

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—solution factor, cm3/ (cm3·MPa);

1955, 204: 156–159.

p—pressure of the system, MPa;

[7]

—calculated gas solubility by the new model, mL/mL;

Mackay D, Shiu W Y, Sutherland R P. Determination of air-water Henry’s Law constants for hydrophobic pollutants.

t—time consumed by pressure drop, min;

Environmental Science & Technology, 1979, 13(3): 333–337.

K(t)—weight function of the system at t;

[8]

—amount of degassing, mL/mL;

Sotelo J L, Beltran F J, Benitez F J, et al. Henry’s Law constant for the ozone-water system. Water Research, 1989,

o—initial gas solubility of the system,mL/mL;

23(10): 1239–1246.

n—amount of degassing at n, mL/mL;

[9]

Standing M B. A pressure-volume-temperature correlation for

pn—pressure of the system at n, MPa;

mixtures of California oil and gases. In: API Drilling and

H, m, r—fitted constants;

Production Practice Conference. Washington: American Pe-

po—initial pressure of the system,MPa;

troleum Institute, 1947.

VL—liquid volume of the system, mL;

[10] Lasater J A. Bubble point pressure correlation. Journal of Pe-

Vg—gas volume of the system, mL;

troleum Technology, 1958, 10(5): 65–67.

V—total volume of the system, mL;

[11] Vasquez M, Beggs H D. Correlations for fluid physical propˉ

—liquid compressibility factor, MPa 1;

erty prediction. Journal of Petroleum Technology, 1980, 32(6):

—calculated errors of degassing, %;

968–970.

—measured degassing rate, mL/mL;

[12] Han Buxing, Yan Haike, Hu Riheng. Gas-solubility, viscosity

—calculated degassing rate, mL/mL;

and density measurements for Karamai heavy oil saturated

p—system pressure difference between equilibrium phase transitions and non-equilibrium phase transitions at the same degassing

with methane. Oilfield Chemistry, 1990, 7(2): 188–190. [13] Ke Jie, Han Buxing, Yan Haike, et al. Correlation and calculation of gas solubility for heavy oil in Karamai oil field. Acta

rate, MPa; —deviate degree of degassing,%;

Petrolei Sinica, 1994, 15(3): 91–94.

pb—bubble point pressure of equilibrium phase transitions, MPa;

[14] Pan Jingjun, Han Buxing, Yan Haike. Hydrocarbon gas solu-

pvb—bubble point pressure of non-equilibrium phase transitions,

bility, viscosity and density measurements for Fengcheng heavy crude oil. Oilfield Chemistry, 1999, 16(3): 268–272.

MPa; f(p)—transfer function for calculating the bubble point pressure of

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non-equilibrium phase transition;

Acta Petrolei Sinica, 2000, 21(3): 89–94.

g—total iterative number; j—the j th iteration;

[16] Xue Haitao, Lu Shuangfang, Fu Xiaotai, et al. Prediction of

pj—system pressure at the j-th iteration, MPa;

model accuracy of the oil-gas volume ratio in crude oil. Jour-

|fj(pj)|—absolute value of transform function at the j th iteration;

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—deviation degree of bubble point pressure,%.

modeling of gas-liquids-solids multi-phase equilibrium in

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