WANG Yi-wei, ZHANG Shi-cheng

Ministry of Education Key Laboratory of Petroleum Engineering, China University of Petroleum, Beijing 102249, E-mail:wangyw@pepris.com

YAN Jin

Petroleum Exploration and Production Research Institute, SINOPEC ,Beijing 100083, China

CHEN Wen-bin

Sinochem Petroleum Exploitation and Production Co., Ltd, Beijing 100031, china

A NEW CALCULATION METHOD FOR GAS-WELL LIQUID LOADING CAPACITY*

WANG Yi-wei, ZHANG Shi-cheng

Ministry of Education Key Laboratory of Petroleum Engineering, China University of Petroleum, Beijing 102249, E-mail:wangyw@pepris.com

YAN Jin

Petroleum Exploration and Production Research Institute, SINOPEC ,Beijing 100083, China

CHEN Wen-bin

Sinochem Petroleum Exploitation and Production Co., Ltd, Beijing 100031, china

(Received April 8, 2010, Revised June 15, 2010)

This article proposes a new model for calculating the gas-well liquid loading capacity, which is critical to an accurate prediction of gas well production. Based on analysis of flow regime during the gas well production with water, which is regarded as many single particles in the model, with the shape of particles being assumed as disk-like ellipsoid instead of traditional sphere and changing according to the forces exerted on them, the influences of non-Darcy flow, compressibility, and non-sphere shape on friction factor are analyzed. The differences between the new model and other models are discussed and a new formula for calculating the critical flow rate is obtained. The calculation results and a comparison with other two models show that the new model is more consistent with the actual situation and is practical.

gas well, accumulated liquid, drag factor, critical flow rate

1. Introduction

Most gas fields in exploration in China contain some water, so two phases, that with water and that without water, coexist in the whole exploring process. An optimized production, based on a detailed analysis of dynamic process, pressure system and other factors, is desirable to make the production period without water as long as possible. Liquids will begin to accumulate in the well once the carrying capacity is not enough after the beginning of gas-carrying liquids being produced[1], and this will disturb the exploration seriously. The prediction of the gas carrying capacity and to make a good use of this capacity are very important in planning a proper exploration strategy. In this article, the formulas for the critical velocity and flow rate are derived through theoretical analysis of the forces affecting liquid drops[2,3].

The studies of vertical tubing models may be traced back to a quite early time and the popularly accepted and applied models in the prediction of liquid accumulation are the wall film mobility model and the high speed gas drag model[4,5]. There are two different points of views concerning the geometry of the liquid drop which is not continuous[6]. One assumes that the liquid drop is still spherical and of the Newton liquid, and a model can be built based on this assumption. The other assumes that the liquid drop is ellipsoidal due to the pressure differences, a model can be built based on that assumption and a formula can be derived for the critical gas-rate[7-10].

The widely used calculation models are Turner model and Li Min model[11-13], with results apparently different due to factors of turbulent flow, compressibility and non-sphere geometry[14,15]. In this article, the formulas for critical velocity and flow rate are derived through theoretical analysis of the forces affecting liquid drops.

2. Liquid accumulation process in gas well

The most common flow state is the annular-mist flow, with the liquid being continuously brought outby the high speed gas flow. The liquid drop will move in the opposite direction once the velocity of gas is below a critical value and the liquid accumulation will begin (Fig.1). Thus, the continuous liquid movement in the wellbore should also be described by the liquid drop model.

3. Geometry of liquid drop

There are two main forces acting on the moving liquid drop, one is the velocity pressure, the other is the surface force. The two forces jointly make the liquid drop take the shape of sphere or flat ellipsoid during the flowing (Fig.2). The liquid drop is spherical when the surface force is large and the drop is difficult to be brought out, because the effective area in the flow direction will be less than that for an ellipsoid. The moving velocity of gas is quite high, and the velocity pressure on the liquid drop could be neglected.

The geometry of the liquid drop is shown in Fig.3. Three dimensionless numbers, Renault number (Re), Hostaux number (0E) and Morton number (Nm), are used to describe the forces acting on the liquid drop. The Renault number reflects the influence of the liquid property, geometry and moving velocity on the liquid drop. Hostaux number is the ratio between the gravity and the surface tension, which reflects the physical property of the continuous phase. The Renault number usually is more than 1 000 for producing gas well and the Morton number for water and normal low molecular weight organic liquid is usually between 10?10to 10?12, and the geometry of moving liquid drop should be a flat ellipsoid.

4. Calculation of drag coefficient

Drag coefficient can be obtained from the Navier-Stokes equation for incompressible sticky liquid flowing around a sphere. It is however mainly determined by experiments. The relation between drag coefficient and Renault Number, obtained from a large number of experiments, with a single incompressible sphere steadily flowing in a stable, isothermal and infinite medium, is defined as the standard drag coefficient curve (in Fig.4).

The factors, such as turbulence flow, compressibility of gas, non-isothermal condition, non-spherical shape and rotation of liquid drop. are not discussed here. The experiments carried out in 1971 by Baily and Hiatt indicate that the actually measured results of drag coefficient is far deviated from the standard drag coefficient curve. The main factors influencing the moving liquid drop are turbulence, gas compressibility and non-spherical shape, and their joint effect can be described by a modification factor, as

where f(δ) is the turbulence effective modification factor, fc(Mr) the compressibility modification factor, β the non-spherical shape modification factor.

4.1 Turbulence effect

The drag coefficient measured from experiments will be higher than the standard drag coefficient curve when the turbulence reaches 8%, and this differencewill increase as the turbulence increases. The drag coefficient curve will obey the theoretical curve only when the turbulence is less than 1% and Re＜103. So, the influence of the turbulence flow increases as the turbulence degree increases and decreases as the Re decreases.

The accurate relationship between the drag coefficient and the turbulent degree is still not very clear and cannot be described by a formula, so the modification factor for influence of turbulence flow is simply assumed as f(δ)=1.

4.2 Compressibility effect

The influence of the relative Mach (Mr) is related to the Re. When the relative Mach Mris less than 0.4, the drag coefficient varies as the standard drag coefficient, when it is greater than 0.4, CDwill be higher than the standard drag coefficient curve. The CDvariation, calculated for different Re, against Mr.

The compressibility effect modification factor can be described as

4.3 Non-spherical geometry modification

The drag coefficient for non-spherical particle can be calculated as

whereDSPC stands for the drag coefficient based on the volume equivalent spherical particle

The values of β are listed in Table 1.

5. Continuous liquid bring model

Assumptions:

(1) Vertical gas well, (2) liquid phase is not continuous in wellbore, liquid drops will not combine or break, (3) liquid geometry is disk-like ellipsoid, (4) liquid cannot be compressed.

The gravity to make liquid drop or fall down is

where V is the volume of the liquid drop,Lρ and Gρa(bǔ)re the density of liquid and gas, g the gravitational acceleration.

The drag coefficient on the moving liquid drop is

whereDC is the drag coefficient, A the area of the liquid drop along the moving direction,tV the final velocity of the liquid drop while settling free.

When the liquid drop acquires a steady speed, defined as the settling speed, the drag coefficient is just the same as the gravity. This state could be described as

and the settling speed can be calculated as:

It is assumed that the liquid drop moves steadily at the falling speed, with a pressure difference between the front and the back of the liquid drop. Bernoulli equation can be used to calculate this pressure difference.

From the principle of conservation of energy, the sum of the interfacial work caused by the pressure difference and the surface tension variation should be equal to zero, which means

The volume of the disk-like ellipsoid is

From Eqs.(4), (5) and (6), we have

Substituting Eq.(7) into Eq.(3), it follows that

Usually the gas flow will follow the third sector of the standard drag coefficient curve, which means that the CDwill not change significantly with Re. f(δ)=1 is taken in the calculation because the relationship between the drag coefficient and the turbulent degree is not clear, The relative Mach is small when the liquid moves in the wellbore, so fc(Mr)≈1 is also assumed. The geometry is assumed as a disk-like ellipsoid, so β=3.08.

Combining the influence of turbulence flow, compressibility and non-spherical geometry, the drag coefficient is

and the minimum liquid flow velocity, or the critical velocity, is

The minimum liquid bringing gas rate, or the critical gas rate, is

wherescq is the critical gas rate, p the pressure of calculated well depth, T the temperature of calculated well depth, Z the gas compressibility factor, A the tubing area.

6. Comparison of the models

In Turner model, it is assumed that the liquid drop moving along the gas flow is spherical and the derived formula is also based on this assumption. In Li Min model, it is assumesd that the pressure difference will make the liquid drop having the shape of ellipsoid, while the effective area along the flow direction covers nearly 100 percent, so CDis assumed to be equal to 1. In the new model proposed in this article, the liquid drop is assumed to be of flat ellipsoid shape, and the influence of drag coefficient and other factors are considered by taking CD=1.355. A comparison of these models is listed in Table 2, where the following differences can be seen: first , the liquid drop geometry assumption is different, second, the drag coefficient is different, third, the factor in the calculation formula is different. If the calculation results are compared, the result of Li Min model is only 38% of that of Turner model, the new model gives only 35% of Turner model’s result, as is consistent with the commonly accepted knowledge that the actual critical gas rate is only about one third of the Turner model’s result.

7. Application

Liquids will be produced along with gas during the exploitation of a gas field, and will be accumulated in the wellbore once there is not enough energy to bring it out, which would affect, even interrupt the well production.As a serious problem affecting the gas well production, the prediction of the liquid accumulation is critical in order to find a way to solve that problem.

The water producing gas well data (Table 3) are used to calculate the critical gas rate by different models. The results are shown in Table 4. Water begins to accumulate in the wellbore once the actual gas rate is less than the calculation result. Judging bythe calculation results of Turner model, all gas wells should be in the liquid accumulation stage, while the actual tests indicate that most of the wells are in a normal production state. This shows that the calculation result from Turner model is higher than the actual critical gas rate.

The new model calculation result is smaller with the consideration of the liquid drop geometry, and themodified drag coefficient. The results for 14 wells agree with the actual production state.

8. Conclusions

(1) The flow is an annular-mist flow in the gas well production. Liquid exists as liquid drops and is brought out through high speed gas flow. Gas phase is continuous while the liquid phase is not continuous. The liquid can be treated as liquid drops in the gas production.

(2) Liquid drop will have the shape of disk-like ellipsoid due to velocity and pressure difference caused by the gas moving velocity.

(3) The drag coefficient in the new model is larger than that in Turner model and LI Min model.

(4) This new model is more practical, gives results consistent with the actual gas well production, and can be used in determining the gas rate for wells.

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10.1016/S1001-6058(09)60122-0

* Biobraphy: WANG Yi-wei (1973-), Male, Ph. D. Candidate, Senior Engineer