**Experiment and Evaluation of Natural Gas Hydration in a Spraying Reactor**

Wenfeng Hao

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[30] R. Barati, S. Johson, S. McCool, D. Green, G. Willhite and J.-T. Liang, Fracturing fluid cleanup by controlled release of enzymes from polyelectrolyte complex nanoparticles,

[31] J. Tannich, Liquid removal from hydraulically fractured gas wells, Journal of Petroleum

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[34] R. Snellings, M. G. and E. J., Supplementary cementitious materials. Reviews in Mineralogy and Geochemistry 74, 2012, pp. 211-278. DOI: 10.2138/rmg.2012.74.6. [35] R. B. Ghahfarokhi, Fracturing Fluid Cleanup by Controlled Release of Enzymes from Polyelectrolyte Complex Nanoparticles, PhD thesis, University of Kansas, Lawrence,

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Technology, 1975.

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Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68458

### Abstract

1L spraying reactor with a heat exchanger outside was used to investigate the effect of spraying hydration process on storage capacity of methane in hydrate and on a methane storage rate in hydrate to solve a problem of lower gas molecular transfer rate and worse heat transfer rate. Some results showed that ethanol as a promoter had better spraying hydration rate under the liquid spraying pressure 4–5 MPa, 0.46Vg VH -1 min-1, which had been approximately 10 times when conventional additive, sodium dodecyl sulfate, was added to reaction system. Others showed that the spraying hydration reactor in advantage had lain in achieving higher hydration rate at lower operational pressure of gas phase compared with semi-continuous stirred tank reactor. Furthermore, evaluation investigation on spraying hydration reaction showed that energy consumption had been 0.41kJ, while methane hydrates containing 1kJ heat were produced, and that the capital efficiency in economy for the hydration process had been 0.41 under perfect competition. Finally, the process evaluation parameter used had become a measure instrument for the prospect of resource utilization efficiency or for venture forecasting of capital investment.

Keywords: natural gas hydration, spraying reactor, experiment, evaluation, economics

### 1. Introduction

A natural gas hydrate is a crystalline compound in which certain compounds stabilize the cages formed by hydrogen-bonded water molecules under favorable conditions of pressure and temperature [1]. Natural gas hydrates possess exceptional gas storage characteristic, as the hydrates can contain 150–180 V V<sup>1</sup> (standard temperature, pressure) natural gas [2, 3]. Utilizing the storage properties of natural gas in hydrates, natural gas storage and transportation

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

will turn to be more economical than conventional ways such as liquefied natural gas transportation and pipeline transportation in the near future, thus middle- or small-scale natural gas fields also become valuable exploitable resources in the forthcoming times [4]. To improve such a technology and to turn to be a reality as soon as possible, many laboratories have studied the synthesis of natural gas hydrates during recent decades. These studies are mainly divided into two groups: one group consists of fundamental research and the other group consists of applied background research. In fundamental studies, natural gas hydrates are synthesized in gas and liquid reaction systems when the conditions of the reactants or mediums are gases of different compositions [5], liquids of different compositions [6–8], and different combinations of liquid-solid systems [9, 10]. In applied background studies, natural gas hydrate formations and process are evaluated in reactors of varying scales and types [11–15]. In all the above studies, the economic efficiency of natural gas hydrate synthesis is the crucial problem that needed to be solved. At present, the gas capacity in hydrates and the hydrate rate remain the main factors to improve the technical levels. Generally, the mass transfer and heat transfer are enhanced to promote the hydrate process in a reactor. However, none of the endeavors for natural gas hydrate transportation currently show economical advantages over liquefied natural gas transportation and pipeline transportation. These endeavors merely have theoretical significance in a laboratory and are worthless to natural gas fields with middle- or small-scale commercial exploitation. To allow natural gas hydrate transportation to compete with liquefied natural gas transportation and pipeline transportation and promote the effective utilization of natural gas resources, natural gas hydration in a spaying reactor under liquids of different compositions is carried out and experimental results received are compared to other reaction systems in current investigations. Moreover, the hydration process is evaluated to provide an effective way to natural gas hydrate formation in a spraying reactor and to give a reference for optimal resource or capital utilization.

#### 2. Experimental

#### 2.1. Apparatus and material

An experimental apparatus, as shown in Figure 1, is built to investigate the storage capacity of methane hydration and to evaluate the methane gas hydrate process. A cylindrical highpressure reactor made of stainless steel with available volume 1.072 L is used to generate the gas hydrate. Designing pressure of the hydration reactor is from 0 to 40 MPa with the temperature in the range of 263.15–323.15 K. In order to ensure the stability of the reactor flow and prevent air backflow, a buffer tank is arranged in the experimental device. The pressure regulator is used to retain constant pressure in the reactor when the experiments are carried out. Volume of the buffer tank and maximum working pressure of the buffer tank are 12 L and 15 MPa, respectively. A water bath is used to provide temperature control of the experiments. There is a canella around the exterior of the reactor that circulates a cooling ethylene glycol water solution. A J2-63/7-type piston pump is used as a circulating pump, which drives and cools the liquid in the outer circulation loop by the external water bath DC-2080. At the reactor inlet and piston inlet, two filters are installed to prevent pipe blockage. Besides, a bypass is Experiment and Evaluation of Natural Gas Hydration in a Spraying Reactor http://dx.doi.org/10.5772/intechopen.68458 77

Figure 1. Liquid spraying experimental apparatus of gas hydrates formation.

will turn to be more economical than conventional ways such as liquefied natural gas transportation and pipeline transportation in the near future, thus middle- or small-scale natural gas fields also become valuable exploitable resources in the forthcoming times [4]. To improve such a technology and to turn to be a reality as soon as possible, many laboratories have studied the synthesis of natural gas hydrates during recent decades. These studies are mainly divided into two groups: one group consists of fundamental research and the other group consists of applied background research. In fundamental studies, natural gas hydrates are synthesized in gas and liquid reaction systems when the conditions of the reactants or mediums are gases of different compositions [5], liquids of different compositions [6–8], and different combinations of liquid-solid systems [9, 10]. In applied background studies, natural gas hydrate formations and process are evaluated in reactors of varying scales and types [11–15]. In all the above studies, the economic efficiency of natural gas hydrate synthesis is the crucial problem that needed to be solved. At present, the gas capacity in hydrates and the hydrate rate remain the main factors to improve the technical levels. Generally, the mass transfer and heat transfer are enhanced to promote the hydrate process in a reactor. However, none of the endeavors for natural gas hydrate transportation currently show economical advantages over liquefied natural gas transportation and pipeline transportation. These endeavors merely have theoretical significance in a laboratory and are worthless to natural gas fields with middle- or small-scale commercial exploitation. To allow natural gas hydrate transportation to compete with liquefied natural gas transportation and pipeline transportation and promote the effective utilization of natural gas resources, natural gas hydration in a spaying reactor under liquids of different compositions is carried out and experimental results received are compared to other reaction systems in current investigations. Moreover, the hydration process is evaluated to provide an effective way to natural gas hydrate formation

in a spraying reactor and to give a reference for optimal resource or capital utilization.

An experimental apparatus, as shown in Figure 1, is built to investigate the storage capacity of methane hydration and to evaluate the methane gas hydrate process. A cylindrical highpressure reactor made of stainless steel with available volume 1.072 L is used to generate the gas hydrate. Designing pressure of the hydration reactor is from 0 to 40 MPa with the temperature in the range of 263.15–323.15 K. In order to ensure the stability of the reactor flow and prevent air backflow, a buffer tank is arranged in the experimental device. The pressure regulator is used to retain constant pressure in the reactor when the experiments are carried out. Volume of the buffer tank and maximum working pressure of the buffer tank are 12 L and 15 MPa, respectively. A water bath is used to provide temperature control of the experiments. There is a canella around the exterior of the reactor that circulates a cooling ethylene glycol water solution. A J2-63/7-type piston pump is used as a circulating pump, which drives and cools the liquid in the outer circulation loop by the external water bath DC-2080. At the reactor inlet and piston inlet, two filters are installed to prevent pipe blockage. Besides, a bypass is

2. Experimental

2.1. Apparatus and material

76 Advances in Natural Gas Emerging Technologies

used to adjust spaying pressure and liquid flow. External heat exchange pipeline size and length are φ6 1 and 2 m, respectively. Spraying water diagram in an idiographic reactor is shown in Figure 2 when the maximum flow is 2.5 L min<sup>1</sup> . Moreover, there are two platinum resistance thermometers with an accuracy of 0.1 K. One extends into the bottom of the reactor, which is used to measure temperatures of the reaction liquids, while the other extends into the gas phase at the top, which is used to measure temperatures of the inlet methane gas. A model D07-11 M/ZM mass gas flow meter is used to measure the gas added to reactor during hydrate formation. The flow meter has a capacity of 0–1000 sccm at an accuracy within 2% of full scale and is repeatability of within 0.2% of the flow rate. There is a data collector to record the temperature of the reactor, the gas flow meter and the total gas volume of the consumed gas in the process of hydrate formation as a function of time. An electronic balance with a readability of 0.1 mg and an electronic balance with a readability of 0.01 g are used in weighing. The experimental materials used in this study are provided in Table 1.

Figure 2. Photograph of water spraying by nozzle.


Table 1. Experimental material used in this work.

#### 2.2. Procedure

### 2.2.1. Determination of working conditions

In order to study the hydration process between methane gas and atomizing liquid sprayed and to ensure hydrate formation mainly in spraying droplet rather than in the main liquid phase, the temperature and pressure of the spraying liquid must meet the phase equilibrium conditions of methane hydrate formation, and the main liquid phase temperature and pressure condition do not meet the conditions of the phase equilibrium of methane, the formation of gas hydrate or seldom hydration occurs in the main body of liquid phase.

By adjusting the temperature of the water bath and the valve, the spray liquid is kept at a state with a low temperature (determined equilibrium pressure) and high pressure. Herein, the outlet pressure nozzle experiment always is higher than the equilibrium pressure of 1–3 MPa, which ensures that the initial impetus is always higher. Then, identify the gas phase pressure, which is slightly lower than the phase equilibrium pressure, to ensure that the hydrate formed mainly in spraying droplet instead of in the main body of liquid phase.

The temperature of the spraying liquid is set at 273.7 K in the experiment while the phase equilibrium pressure is 2.64 MPa for methane hydration at the temperature. Liquid injection pressure and methane gas pressure are from 4 to 5 and 2.4 MPa, respectively. Under these conditions, methane hydrate formation is compared by using pure water, sodium dodecyl sulfate solution, ethanol solution as a spraying liquid to investigate the effect of additive on methane hydrate formation.

In addition, in order to test the effect of gas phase pressure on the spraying hydration process, spraying hydration formation is also compared when the methane gas pressure is 0.5 and 2.4 MPa.

#### 2.2.2. Process


mixture reacted from piston pump to the reactor. Under the experimental flow rate, as shown in Figure 2, cooled liquid temperature could approach external cooler temperature, 273.7 K, after they flowed through the nozzle. The temperature was selected as a hydration temperature.


#### 2.3. Calculation of storage capacity of methane hydrate

The volume [3] of gas stored in a unit volume of hydrate under the hydrate formation conditions of pressure and temperature is expressed as

$$\mathcal{C} = \frac{V\_{\rm NG}}{V\_{\rm NGH}} = \frac{V\_{\rm NG}}{V\_L \* (1 + \Delta V)} \tag{1}$$

where C is the volume of gas stored in a unit volume of hydrate, VNG is the volume of gas consumed, VNGH is the volume of hydrate when the reaction ends, VL is the volume of water added and ΔV is the molar volume change of water turned into hydrate. Herein ΔV of methane hydrate is 4.6 cm<sup>3</sup> mol�<sup>1</sup> .

The hydration rate of hydrate formation can be calculated by the following equation:

$$r = \frac{\mathbf{C}}{t} \tag{2}$$

where r, C and t are hydration rate, gas hydrate capacity and reaction time, respectively.

### 3. Results and discussion

2.2. Procedure

2.2.1. Determination of working conditions

Table 1. Experimental material used in this work.

78 Advances in Natural Gas Emerging Technologies

methane hydrate formation.

experimental run.

2.2.2. Process

In order to study the hydration process between methane gas and atomizing liquid sprayed and to ensure hydrate formation mainly in spraying droplet rather than in the main liquid phase, the temperature and pressure of the spraying liquid must meet the phase equilibrium conditions of methane hydrate formation, and the main liquid phase temperature and pressure condition do not meet the conditions of the phase equilibrium of methane, the formation of gas

Sodium dodecyl sulfate ≧98 Guangzhou Chemical Reagent Co. Ethanol ≧99.9 Guangzhou Chemical Reagent Co.

By adjusting the temperature of the water bath and the valve, the spray liquid is kept at a state with a low temperature (determined equilibrium pressure) and high pressure. Herein, the outlet pressure nozzle experiment always is higher than the equilibrium pressure of 1–3 MPa, which ensures that the initial impetus is always higher. Then, identify the gas phase pressure, which is slightly lower than the phase equilibrium pressure, to ensure that the hydrate formed

The temperature of the spraying liquid is set at 273.7 K in the experiment while the phase equilibrium pressure is 2.64 MPa for methane hydration at the temperature. Liquid injection pressure and methane gas pressure are from 4 to 5 and 2.4 MPa, respectively. Under these conditions, methane hydrate formation is compared by using pure water, sodium dodecyl sulfate solution, ethanol solution as a spraying liquid to investigate the effect of additive on

In addition, in order to test the effect of gas phase pressure on the spraying hydration process, spraying hydration formation is also compared when the methane gas pressure is 0.5 and 2.4 MPa.

1. The reactor was cleaned by water and experimental gas twice before preparing for an

2. Six hundred and fifty grams of 0.001 mol L<sup>1</sup> sodium dodecyl sulfate solutions were charged into the empty reactor. Afterwards, the constant bath was run and its temperature was maintained at 272.2 K. An external cooler was set in 273.7 K to cool liquid

hydrate or seldom hydration occurs in the main body of liquid phase.

Component Purity/composition Supplier

(%)

Water Distilled

Methane ≧99.99 Fushan Kede Gas Co.

mainly in spraying droplet instead of in the main body of liquid phase.

#### 3.1. Effect of liquid composition on the hydration process of methane

The capacity and reaction rate of methane hydrate under different liquid compositions are plotted in Figure 3. Three results were given at different spraying times. Figure 3 shows or deduces the following results under gas pressure 2.4 MPa: methane storage capacity, reaction

Figure 3. Effect of liquid composition on methane hydrate formation (T ¼ 273.7 K, P ¼ 2.4 MPa).

time, and the average hydration rate were 6.4 Vg VH �1 , 229 min, and 0.028 Vg VH �<sup>1</sup> min�<sup>1</sup> , respectively, when reaction liquid did not have any additives; methane storage capacity, reaction time, and the average hydration rate were 6.9 Vg VH �1 , 143 min, and 0.048 Vg VH �<sup>1</sup> min�<sup>1</sup> , respectively, when 0.001 mol L�<sup>1</sup> sodium dodecyl sulfate solutions were reaction liquid; methane storage capacity, reaction time, and the average hydration rate were 10.5 Vg VH �1 , 23 min, and 0.46 Vg VH �<sup>1</sup> min�<sup>1</sup> , respectively, when 0.018 mol L�<sup>1</sup> ethanol solutions were reaction liquid.

By analysis Figure 3, the following deductions could have been drawn:


#### 3.2. Effect of gas pressure on the hydration process of methane

Effects of gas pressure on methane hydrate formation in a spraying reactor with a closed loop are given in Figure 4. The capacity and reaction rate of methane hydrate under two different

Figure 4. Effect of pressure of gas phase on methane hydrate formation (T ¼ 273.7 K).

time, and the average hydration rate were 6.4 Vg VH

time, and the average hydration rate were 6.9 Vg VH

hydration rate reaches 0.46 Vg VH

0.46 Vg VH

�<sup>1</sup> min�<sup>1</sup>

80 Advances in Natural Gas Emerging Technologies

sulfate as an additive.

�1

�1

, respectively, when 0.018 mol L�<sup>1</sup> ethanol solutions were reaction liquid.

respectively, when reaction liquid did not have any additives; methane storage capacity, reaction

respectively, when 0.001 mol L�<sup>1</sup> sodium dodecyl sulfate solutions were reaction liquid; methane

1. Liquid spraying with a higher pressure and lower temperature could increase the driving force of the hydration reaction, which had reduced the pressure of the gas phase.

2. The additive would affect the hydration reaction rate: without additives, hydration rate was slower and operation time was also longer; if additive was used, hydration rate and operation time were shorten obviously. In the experiments, sodium dodecyl sulfate and ethanol as additives on the hydration rate increase were given. Ethanol as an additive,

3. Because the spraying system was a closed circuit device, the hydrate particles were apt to block the reaction device. As a result, the gas hydrate slurry had a lower gas storage

Effects of gas pressure on methane hydrate formation in a spraying reactor with a closed loop are given in Figure 4. The capacity and reaction rate of methane hydrate under two different

capacity under this state. Thus, this device still had greater space to be improved.

�<sup>1</sup> min�<sup>1</sup>

storage capacity, reaction time, and the average hydration rate were 10.5 Vg VH

Figure 3. Effect of liquid composition on methane hydrate formation (T ¼ 273.7 K, P ¼ 2.4 MPa).

By analysis Figure 3, the following deductions could have been drawn:

3.2. Effect of gas pressure on the hydration process of methane

, 229 min, and 0.028 Vg VH

, 143 min, and 0.048 Vg VH

, which was about 10 times sodium dodecyl

�<sup>1</sup> min�<sup>1</sup>

�<sup>1</sup> min�<sup>1</sup>

, 23 min, and

�1

,

,

gas pressures were compared. On the one hand, when methane gas pressure was 0.5 MPa, and there were no additives in liquid reagent, gas storage capacity in hydrate, operation time, and hydration rate were 1.4 Vg VH �1 , 80 min, and 0.0175 Vg VH �<sup>1</sup> min�<sup>1</sup> , respectively. On the other hand, when methane gas pressure was 2.4 MPa, and there were no additives in liquid reagent, gas storage capacity in hydrate, operation time, and hydration rate were 6.4 Vg VH �1 , 229 min, and 0.028 Vg VH �<sup>1</sup> min�<sup>1</sup> , respectively.

Analysis of Figure 4 showed that the hydration rate had depended not only on the liquid pressure and temperature, but also on the gas pressure as an important factor. If a higher hydration rate needed to be kept, an appropriate gas pressure must have been maintained.

#### 3.3. Comparison of a hydration rate between two kinds of reactors

In order to show the characteristics of methane hydration process in the spraying reactor, the methane hydration rate in the spray reactor was compared with that of the semi-continuous stirred tank reactor, and the results obtained are shown in Table 2.


Table 2. Comparison of two kinds of reactors on hydration rate.

The methane hydrate rate was 0.43 Vg VH <sup>1</sup> min<sup>1</sup> in a semi-continuous stirred tank reactor at 5.0 MPa and sodium dodecyl sulfate being additives. However, the methane spraying hydration rate reached 0.46 Vg VH <sup>1</sup> min<sup>1</sup> at 2.4 MPa and under liquid spraying pressure 4–5 MPa with the assistance of ethanol as additives. The compared results showed that the advantages of methane hydration process in a spraying reactor had lain in lower gas pressure and higher hydration rate could have been obtained.

### 4. Evaluation of hydration process in a spraying reactor

### 4.1. Mechanism of spraying hydration process

In order to explain and evaluate the hydration formation in the spraying reactor, the following procedures were assumed:


In the experiment, an additive as a hydration promoter reduced the surface tension of solution and had some functions such as wetting, penetration, emulsification, and solubilization; thus, the surface gas-liquid mass transfer rate was improved, accelerating the implementation of the above process. Moreover, the phase equilibrium conditions of methane gas and water were only considered in the experimental design, neglecting the effects of the additives on the phase equilibrium change. The understanding of the mechanism still had limitations, which could not fully have explained the spraying hydration process. Thus, phase equilibrium data from different components of the hydration system still needed to have been added and other pieces of evidence had also been needed.

#### 4.2. Evaluation of the reactor performance evaluation

The production capacity and energy consumption for the spraying reactor would have become key points as a basis for the reactor amplification in this section. The daily production capacity and daily energy consumption of the reactor were calculated for evaluating the reactor efficiency.

#### 4.2.1. Mass balance

The methane hydrate rate was 0.43 Vg VH

82 Advances in Natural Gas Emerging Technologies

hydration rate could have been obtained.

4.1. Mechanism of spraying hydration process

surface and had stayed in the nucleation stage.

piston pump could not be continued to run.

tion rate reached 0.46 Vg VH

procedures were assumed:

buoyant force.

of evidence had also been needed.

<sup>1</sup> min<sup>1</sup> in a semi-continuous stirred tank reactor at

<sup>1</sup> min<sup>1</sup> at 2.4 MPa and under liquid spraying pressure 4–5 MPa

5.0 MPa and sodium dodecyl sulfate being additives. However, the methane spraying hydra-

with the assistance of ethanol as additives. The compared results showed that the advantages of methane hydration process in a spraying reactor had lain in lower gas pressure and higher

In order to explain and evaluate the hydration formation in the spraying reactor, the following

1. Methane gas molecules with a certain pressure quickly diffused to spraying and atomization liquid droplet surface with a higher pressure and lower temperature from the nozzle afterwards were dissolved in it. The temperature condition was less than phase equilibrium temperature at the given pressure and had a greater degree of super-cooling.

2. Methane gas molecules around spherical droplets diffused toward the internal liquid droplets and formed an unstable cluster. Afterwards, they began to nucleate and to form a collective cluster. After that collective clusters had reached a critical size and they began to grow rapidly and formed a stable crystal releasing the heat of reaction. At the same time, the liquid pressure drops and liquid droplets temperature increased. When gas pressure was too low, because driving force from methane gas molecules to liquid droplet diffusion was insufficient, the hydration reaction process had occurred only on the droplet

3. With the decrease of the reaction degree of super-cooling, crystal growth rate declined. When the droplet temperature was higher than the equilibrium temperature, crystal growth ceased and crystal was suspended in the liquid under the action of gravity and of

4. During the hydration process, the hydrate particles in the liquid increased rapidly and the viscosity of hydrate slurry increased gradually. Moreover, the resistance from the gas hydrate slurry that inhaled into the circulating pipeline was gradually increased until the

In the experiment, an additive as a hydration promoter reduced the surface tension of solution and had some functions such as wetting, penetration, emulsification, and solubilization; thus, the surface gas-liquid mass transfer rate was improved, accelerating the implementation of the above process. Moreover, the phase equilibrium conditions of methane gas and water were only considered in the experimental design, neglecting the effects of the additives on the phase equilibrium change. The understanding of the mechanism still had limitations, which could not fully have explained the spraying hydration process. Thus, phase equilibrium data from different components of the hydration system still needed to have been added and other pieces

4. Evaluation of hydration process in a spraying reactor

For this experiment, 4.32 g methane gas was stored when 650 g solution or pure water was added to the reactor for a batch operation. Process reaction time was 0.5 h and supplementary time was 0.5 h, allowing 20 runs per day.

Throughput calculation was the first step of a mass balance. The mass of methane hydrates produced during the hydration process, which consisted of mass of water solution and mass of methane gas reacted, was calculated. The mass balance equation for the produced methane hydrate slurry could thus have been expressed as

$$m = m\_w + m\_{\rm CH\_4} \tag{3}$$

where m, mw, and mCH4 are mass of hydrates formed, mass of water solution added, and mass of methane gas reacted, respectively, during a run.

If the run time of the reactor was τ, then the mass of methane hydrates slurry produced could have been written as follows:

$$m\_t = m\*\tau\tag{4}$$

where mt is the mass of hydrates formed and τ is the daily run time.

The methane gas fraction of the methane hydrates slurry might have been expressed as

$$
\Theta = \frac{m\_{\rm CH\_4}}{m} \times 100\% \tag{5}
$$

where θ is the methane gas mass fraction of the methane hydrates slurry.

In terms of these equations, a mass balance was calculated and is shown in Table 3.

#### 4.2.2. Energy balance

For convenience in calculation, the temperature changes of the inlet gas and the inlet water could have been considered to have a negligible effect on their consumption of the hydration


Table 3. Throughput and methane gas fraction of methane hydrates.

process, or else their temperature could have been controlled. Then, the total energy consumption during a run could have been expressed, including energy consumption of the compression process, energy consumption of the cooling process, and the power for driving the plunger pump. In each run the equation for the total energy consumption could have been written as

$$Q^\* = (1+\zeta) \times \left(Q\_{\text{cp}} + W\_r + W\right) \tag{6}$$

where Q� , Qcp, Wr, W and ζ are the total energy consumption, energy consumption of the compression process, energy consumption of the refrigeration unit, work for driving the plunger pump, and an coefficient of other auxiliary operation energy consumption to operation process energy consumption, respectively. Here ζ value was 0.01 when the calculation of the total energy consumption was carried out implemented.

1. Compression of methane gas

Here, a general assumption and conventional calculation were used [16]. Assume that the initial pressure of the feed gas was set to be P<sup>1</sup> and the initial temperature was set to be T1. The feed gas was pressurized to the hydrate operation pressure P<sup>2</sup> by an adiabatic compression process with efficiency factor ηad. The final temperature T<sup>2</sup> after compression could have been calculated from the initial temperature using the following equation:

$$T\_2 = \left(\frac{1 + \phi^{\left(\lambda - 1/\sqrt{\lambda}\right)} - 1}{\eta\_{\rm ad}}\right) T\_1 \tag{7}$$

where λ is the ratio of the heat capacity at constant pressure to the heat capacity at constant volume, expressed as

$$
\lambda = \mathfrak{c}\_p / \mathfrak{c}\_{\nu} \tag{8}
$$

where cp is the heat capacity at constant pressure and cv is the heat capacity at constant volume.

φ, the ratio of the final pressure P<sup>2</sup> to the initial pressure P<sup>1</sup> of the compression process, could have been expressed as

$$
\phi = P\_2/P\_1. \tag{9}
$$

The temperature of the compression process was calculated from the above equation. Results for the model parameters are given in Table 4.

Assuming that the work performed on methane gas was Wcp, the compression process energy consumption Qcp could have been expressed as


Table 4. Calculation of final temperature of compress process.

$$\mathcal{W}\_{\text{cp}} = \mathcal{Q}\_{\text{cp}} \eta\_{\text{ad}} \tag{10}$$

where ηad is the efficiency factor under adiabatic conditions.

One was that the internal energy change ΔU could have been expressed as

$$
\Delta \mathcal{U} = \mathcal{Q}\_{\text{cp}} - \mathcal{W}\_{\text{cp}} = \left(1 - \eta\_{\text{ad}}\right) \mathcal{Q}\_{\text{cp}}.\tag{11}
$$

That was:

ð6Þ

T<sup>1</sup> ð7Þ

λ ¼ cp=cv, ð8Þ

φ ¼ P2=P1: ð9Þ

process, or else their temperature could have been controlled. Then, the total energy consumption during a run could have been expressed, including energy consumption of the compression process, energy consumption of the cooling process, and the power for driving the plunger pump. In each run the equation for the total energy consumption could have been written as

Q� ¼ ð Þ� 1 þ ζ Qcp þ Wr þ W

compression process, energy consumption of the refrigeration unit, work for driving the plunger pump, and an coefficient of other auxiliary operation energy consumption to operation process energy consumption, respectively. Here ζ value was 0.01 when the calculation of

Here, a general assumption and conventional calculation were used [16]. Assume that the initial pressure of the feed gas was set to be P<sup>1</sup> and the initial temperature was set to be T1. The feed gas was pressurized to the hydrate operation pressure P<sup>2</sup> by an adiabatic compression process with efficiency factor ηad. The final temperature T<sup>2</sup> after compression could have been calculated from the initial temperature using the following equation:

where λ is the ratio of the heat capacity at constant pressure to the heat capacity at

where cp is the heat capacity at constant pressure and cv is the heat capacity at constant

φ, the ratio of the final pressure P<sup>2</sup> to the initial pressure P<sup>1</sup> of the compression process,

The temperature of the compression process was calculated from the above equation.

Assuming that the work performed on methane gas was Wcp, the compression process

Parameter T<sup>1</sup> (K) (MPa) (MPa) ηad φ λ T<sup>2</sup> (K) Value 298 0.1 7 0.8 70 1.29 894

<sup>T</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>φ</sup>ð<sup>λ</sup>�<sup>1</sup>=<sup>λ</sup><sup>Þ</sup> � <sup>1</sup> ηad !

the total energy consumption was carried out implemented.

1. Compression of methane gas

84 Advances in Natural Gas Emerging Technologies

constant volume, expressed as

could have been expressed as

Results for the model parameters are given in Table 4.

energy consumption Qcp could have been expressed as

Table 4. Calculation of final temperature of compress process.

volume.

where Q�

� �

, Qcp, Wr, W and ζ are the total energy consumption, energy consumption of the

$$Q\_{\rm cp} = \frac{\Delta U}{1 - \eta\_{\rm ad}}.\tag{12}$$

The other was that the internal energy change ΔU could also have been expressed as

$$
\Delta \mathcal{U} = n \int\_{T\_1}^{T\_2} \mathbb{C}\_v dT \tag{13}
$$

where n is the molecular number of the methane gas.

The heat capacity at constant volume cv could have been expressed using the heat capacity at constant pressure cp, which in turn was related to absolute temperature T. So, the heat capacity at constant volume cv was related to absolute temperature T. The relationship between the heat capacity at constant pressure and absolute temperature could have been expressed as follows:

$$
\sigma\_p = a + bT + cT^2,\tag{14}
$$

where a, b, and c are the parameters of heat capacity at constant pressure.

But the relationship between the heat capacity at constant volume and the heat capacity at constant pressure was

$$
\mathcal{L}\_v = \mathcal{c}\_p - \mathcal{R}\_\prime \tag{15}
$$

where R is the gas constant.

Therefore, substituting Eqs. (14) and (15) into Eq. (13), ΔU became

$$
\Delta II = n \int\_{T1}^{T2} \left( a + bT + cT^2 - R \right) dT \tag{16}
$$

Integrating the right-hand side of Eq. (16), the internal energy change ΔU became

$$
\Delta \Delta I = n \left[ (a - R)(T\_2 - T\_1) + \frac{b}{2} \left( T\_2^2 - T\_1^2 \right) + \frac{c}{3} \left( T\_2^3 - T\_1^3 \right) \right]. \tag{17}
$$

In Eq. (17), values of T<sup>1</sup> and T<sup>2</sup> are presented in Table 4, and values of a, b, c and R are presented in Table 5.


Table 5. Parameter of heat capacity at constant pressure and gas constant [17].

Substituting these data into Eq. (17), the internal energy change ΔU was written simply as

$$
\Delta \mathcal{U} = \mathcal{Z} 6.17 n\_{\prime} \tag{18}
$$

Substituting Eq. (18) into Eq. (11) or Eq. (12), the compression process energy consumption Qcp became

$$Q\_{\rm cp} = 130.85n \tag{19}$$

2. Cooling of the methane hydration process

In the methane hydration process, substantial heat of the reaction, 54.2 kJ mol�<sup>1</sup> [18], was released by a chemical reaction, which could have been expressed as

$$\text{CH}\_4 + 5.75\text{H}\_2\text{O} \rightarrow \text{CH}\_4 \cdot 5.75\text{H}\_2\text{O} + 54.2 \text{ kJ } mol^{-1} \tag{20}$$

Therefore, the heat of reaction released could have been expressed as

$$Q\_{\rm rh} = 54.2n \tag{21}$$

where Qrh is the heat of reaction released and n is the molecular number of the methane gas.

According to principle of heat balance, heat exchanged in the cooling system was equal to the heat of reaction released that was,

$$Q\_{\varepsilon} = 54.2n \tag{22}$$

where Qe is the heat exchanged in the cooling system.

Work consumption of the refrigeration unit Wr could have been expressed as

$$W\_r = \frac{54.2n}{\text{CO}\_2} \tag{23}$$

where Wr is the work consumption of the refrigeration unit and CO<sup>P</sup> is the coefficient of performance.

3. Power for driving the plunger pump

The power for driving the plunger pump in terms of experimental determination was expressed as

$$W = (P\_o - P\_i)Vt\tag{24}$$

where W, Pi, Po, V, and t are the power for driving the plunger pump, inlet pressure of plunger pump, outlet pressure of plunger pump, liquid volume flow rate, and operation time, respectively.

During a run, power for driving the plunger pump was calculated and is shown in Table 6.

4. Total energy consumption of the methane hydration process

Substituting these data into Eq. (17), the internal energy change ΔU was written simply as

Substituting Eq. (18) into Eq. (11) or Eq. (12), the compression process energy consump-

In the methane hydration process, substantial heat of the reaction, 54.2 kJ mol�<sup>1</sup> [18], was

where Qrh is the heat of reaction released and n is the molecular number of the methane

According to principle of heat balance, heat exchanged in the cooling system was equal to

Work consumption of the refrigeration unit Wr could have been expressed as

Wr <sup>¼</sup> <sup>54</sup>:2<sup>n</sup> CO<sup>P</sup>

where Wr is the work consumption of the refrigeration unit and CO<sup>P</sup> is the coefficient of

The power for driving the plunger pump in terms of experimental determination was

CH4 <sup>þ</sup> <sup>5</sup>:75H2O ! CH4 � <sup>5</sup>:75H2O <sup>þ</sup> <sup>54</sup>:<sup>2</sup> kJ mol�<sup>1</sup> <sup>ð</sup>20<sup>Þ</sup>

released by a chemical reaction, which could have been expressed as

) <sup>b</sup>�10<sup>3</sup> (J mol�<sup>1</sup> <sup>K</sup>�<sup>2</sup>

Table 5. Parameter of heat capacity at constant pressure and gas constant [17].

Methane 14.15 75.496 �17.99 8.314

Therefore, the heat of reaction released could have been expressed as

tion Qcp became

Parameter a (J mol�<sup>1</sup> K<sup>1</sup>

86 Advances in Natural Gas Emerging Technologies

gas.

performance.

expressed as

2. Cooling of the methane hydration process

the heat of reaction released that was,

3. Power for driving the plunger pump

where Qe is the heat exchanged in the cooling system.

ΔU ¼ 26:17n, ð18Þ

) R (J mol�<sup>1</sup> K�<sup>1</sup>

)

) <sup>c</sup> � <sup>10</sup><sup>6</sup> (J mol�<sup>1</sup> <sup>K</sup>�<sup>3</sup>

Qcp ¼ 130:85n ð19Þ

Qrh ¼ 54:2n ð21Þ

Qe ¼ 54:2n ð22Þ

W ¼ ð Þ Po � Pi Vt ð24Þ

, ð23Þ

Total energy consumption per day for the methane hydration process could have been calculated from Eq. (6). The total energy consumption per day for the methane hydration process Qt was expressed as

$$Q\_t = Q^\* \ast \tau Q^\*,\tag{25}$$

where Qt is the total energy consumption per day and τ is the run time per day.

Since the total mass of methane hydrate slurry produced per day was mt, the energy consumption for each 1 kg methane hydrate slurry produced could have been written as

$$Q\_0 = \frac{Q\_t}{m\_t} \,\prime \tag{26}$$

where Q<sup>0</sup> is the energy consumption per 1 kg methane hydrate produced, and mt is the total mass of methane hydrates produced in a day.

The parameter values for the methane hydration process are given in Table 7.

#### 4.2.3. Resource efficiency for utilization in a spraying reactor

In order to evaluate the resource efficiency for utilization of the methane hydration process in a spray reactor, introducing a dimensionless parameter Ω, energy consumption evaluation parameter [11], which was expressed as the ratio between the energy consumption per 1 kg methane hydrate slurry produced to heat value of the 1 kg methane hydrate slurry.



Table 6. Power calculation for driving the plunger pump.

Table 7. Energy consumption calculation for hydration process in a spraying reactor.

The expression was

$$
\Omega = Q\_0 / Q\_{c'} \tag{27}
$$

where Q<sup>0</sup> is the energy consumption per 1 kg methane hydrate produced, and Qc is the heat value of 1 kg methane hydrate slurry.

The heat value of 1 kg methane hydrate slurry was expressed as

$$Q\_{\mathfrak{c}} = 1 \times \theta \times \mathfrak{q} \tag{28}$$

where q is the combustion heat of methane [19].

The dependent data of heat value of methane hydrate slurry and energy consumption of hydration process are given in Table 8.

According to the thoughts of the energy consumption evaluation parameter, assessment of the parameter could have been used as reference data to evaluate the process quality. The size of its value depends on the complexity of the process, energy consumption level of the auxiliary process, and specific factors of the level of science and technology.

For the methane hydration process in a spraying reactor, if experimental gas directly came from a small-scale natural gas field, then the energy consumption of methane gas compression could have been neglected. Thus, the total energy consumption in such a run could have been replaced by the energy consumption of the cooling process and the power of driving plunger pump. Calculated results are given in Table 9.

As shown in Table 9, the energy consumption evaluation parameter had a value of 0.246. Compared to the process for hydration of compressed methane gas, the energy consumption decrease was 39%. If the management level was improved or the auxiliary energy consumption ζ was reduced to 0.005, the energy consumption evaluation parameter would have continued to decline, the calculation results are shown in Table 10. As shown in Table 10, the process evaluation parameter had a value of 0.245. Compared to spraying hydration of this laboratory scale, the decrease was 40%. If further decrease of the process evaluation parameter needed to be done, then specific aspects of the scientific and technological levels, such as a reactor with a superior performance, optimal operation condition, and production with a large scale should have been excavated. Under current states, the parameter value still was at a high level


Table 8. Heat value of the hydrate slurry and evaluation for the hydration process.


Table 9. Data of the process evaluation after process simplification.


Table 10. Data of the process evaluation with smaller auxiliary energy consumption.

compared with references data reported [4–14]. Therefore, the investigation still would have had a long way to go if this technology could have been applied to industrial production. Only when the energy consumption parameter has been controlled into an appropriate level and has had some advantages compared to the operational mode, the technology would have had possibility to implement practice in industry.

Moreover, an energy consumption evaluation parameter was converted into a process evaluation parameter to represent another meaning in application. Here, it meant that social resource in economy was used to produce new resources from nature or other areas, holding efficiency for a capital utilization process in economy [11]. The capital efficiency for utilization in economy was 0.41 in terms of principle of process evaluation in this experimental work. In other words, 0.41 United States dollar must be consumed when 1 United States dollar was produced under perfect competition. It was thus clear that the parameter was a measure for the prospect of capital economic analysis and of venture forecasting [20–23].

### 5. Conclusions

The expression was

value of 1 kg methane hydrate slurry.

88 Advances in Natural Gas Emerging Technologies

hydration process are given in Table 8.

where q is the combustion heat of methane [19].

pump. Calculated results are given in Table 9.

Parameter q (kJ kg�<sup>1</sup>

The heat value of 1 kg methane hydrate slurry was expressed as

process, and specific factors of the level of science and technology.

Ω ¼ Q0=Qc, ð27Þ

Qc ¼ 1 � θ � q, ð28Þ

) <sup>Q</sup>0(kJ kg�<sup>1</sup>

) <sup>Q</sup><sup>0</sup> (kJ kg�<sup>1</sup>

) Ω

) Ω

where Q<sup>0</sup> is the energy consumption per 1 kg methane hydrate produced, and Qc is the heat

The dependent data of heat value of methane hydrate slurry and energy consumption of

According to the thoughts of the energy consumption evaluation parameter, assessment of the parameter could have been used as reference data to evaluate the process quality. The size of its value depends on the complexity of the process, energy consumption level of the auxiliary

For the methane hydration process in a spraying reactor, if experimental gas directly came from a small-scale natural gas field, then the energy consumption of methane gas compression could have been neglected. Thus, the total energy consumption in such a run could have been replaced by the energy consumption of the cooling process and the power of driving plunger

As shown in Table 9, the energy consumption evaluation parameter had a value of 0.246. Compared to the process for hydration of compressed methane gas, the energy consumption decrease was 39%. If the management level was improved or the auxiliary energy consumption ζ was reduced to 0.005, the energy consumption evaluation parameter would have continued to decline, the calculation results are shown in Table 10. As shown in Table 10, the process evaluation parameter had a value of 0.245. Compared to spraying hydration of this laboratory scale, the decrease was 40%. If further decrease of the process evaluation parameter needed to be done, then specific aspects of the scientific and technological levels, such as a reactor with a superior performance, optimal operation condition, and production with a large scale should have been excavated. Under current states, the parameter value still was at a high level

) <sup>θ</sup> Qc (kJ kg�<sup>1</sup>

Table 8. Heat value of the hydrate slurry and evaluation for the hydration process.

Parameter <sup>Q</sup>� (kJ) Qt (kJ) Qc (kJ kg�<sup>1</sup>

Table 9. Data of the process evaluation after process simplification.

Value 50010 0.0066 330.07 135.69 0.41

Value 53.1 1062 330.07 81.1 0.246

Through the methane hydration experiment in a spraying reactor and analysis of the result received, the following conclusions were drawn:


hydrate slurry with 1 kJ heat was produced in this work. The derivative result only was equal to 40% of this experimental apparatus. If the natural gas was from natural gas and the better management mode was used, then the energy consumption was 0.245 kJ when methane hydrate slurry production with 1 kJ heat value, whose decrease was 40% compared to the experimental scale.

5. The efficiency for capital utilization in economy was 0.41 in this work. Compared to data reported, the capital efficiency for utilization in economy still was at lower level. The spraying hydration process still had larger space to be improved.

### Acknowledgements

The financial support from the Chinese Natural Science Foundation (nos. 50176051, 090410003 and 20490207), the Natural Science Foundation of Liaoning Province (no. 2013020150), and the Program for Liaoning Excellent Talents in University (no. LJQ2011134) are gratefully acknowledged. Support for the publication of this research from Association of Science and Technology in Croatia is acknowledged gratefully and synchronously.

### Nomenclature


#### Experiment and Evaluation of Natural Gas Hydration in a Spraying Reactor http://dx.doi.org/10.5772/intechopen.68458 91


### Author details

#### Wenfeng Hao

hydrate slurry with 1 kJ heat was produced in this work. The derivative result only was equal to 40% of this experimental apparatus. If the natural gas was from natural gas and the better management mode was used, then the energy consumption was 0.245 kJ when methane hydrate slurry production with 1 kJ heat value, whose decrease was 40% com-

5. The efficiency for capital utilization in economy was 0.41 in this work. Compared to data reported, the capital efficiency for utilization in economy still was at lower level. The

The financial support from the Chinese Natural Science Foundation (nos. 50176051, 090410003 and 20490207), the Natural Science Foundation of Liaoning Province (no. 2013020150), and the Program for Liaoning Excellent Talents in University (no. LJQ2011134) are gratefully acknowledged. Support for the publication of this research from Association of Science and Technology

)

)

)

)

)

)

)

spraying hydration process still had larger space to be improved.

in Croatia is acknowledged gratefully and synchronously.

a Parameters of heat capacity at constant pressure (J mol<sup>1</sup> K<sup>1</sup>

b Parameters of heat capacity at constant pressure (J mol<sup>1</sup> K<sup>2</sup>

c Parameters of heat capacity at constant pressure (J mol<sup>1</sup> K<sup>3</sup>

C Volume of gas stored in a unit volume of hydrate (V V<sup>1</sup>

C Heat capacity at constant volume or pressure (J mol<sup>1</sup> K<sup>1</sup>

)

)

1 )

Q Energy consumption, heat or heat value (kJ) (kJ kg<sup>1</sup>

q Combustion heat of methane hydrates (kJ kg<sup>1</sup>

)

COp Coefficient of performance (W W<sup>1</sup>

P Pressure or power (MPa) (W)

R Gas constant (J mol<sup>1</sup> K<sup>1</sup>

r Hydration rate (V V<sup>1</sup> s

T Absolute temperature (K)

V Volume or volume velocity (m3

W Work or work consumption (kJ)

Δ Change value of a parameter (–)

U Internal energy (kJ)

η Efficiency factor (–)

t Time (s)

Special characters

n Molecular number of the methane gas (mol)

pared to the experimental scale.

90 Advances in Natural Gas Emerging Technologies

Acknowledgements

Nomenclature

Address all correspondence to: haowenfeng@163.com

1 Department of Chemical Energy for Energy, School of New Energy, Shenyang Institute of Engineering, Shenyang, China

2 Center for Natural Gas Hydrate Research, Guangzhou Institute of Energy Conversion, Chinese Academy of Sciences, Guangzhou, China

3 Institute of Adsorption and Inorganic membrane, State Key Laboratory of Fine Chemicals, Dalian University of Technology, Dalian, China

### References


[15] Brown TD, Taylor CE, Bernardo MP. New natural gas storage and transportation capabilities utilizing rapid methane hydrate formation techniques. In: Proceeding of 2010 AIChE Spring Meeting and 6th Global Congress on Process Safety the AIChE 2010 Spring National Meeting (San Antonio, TX 3/21-25/2010) 2010, 7p

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[2] Makogon YF. Hydrates of Hydrocarbons. Oklahoma: PennWell Publishing Company; 1997 [3] Khokhar AA, Gudmundsson JS, Sloan ED. Gas storage in structure H hydrates. Fluid

[4] Hao WF, Wang JQ, Fan SS, Hao WB. Study on methane hydration process in a semicontinuous stirred tank reactor. Energy Conversion and Management. 2007;48(3):954–960

[5] Azmi N, Mukhtar H, Sabil KM. Purification of natural gas with high CO2 content by formation of gas hydrates: Thermodynamic verification. Journal of Applied Sciences.

[6] Li W-Q, Kou Z-L, Li W-Y, Wang Z, Zhang W, He D-W. Experimental study of methane hydrate prepared through reaction of Al4C3 with H2O. Chinese Journal of High Pressure

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[8] Veluswamy HP, Wong AJH, Babu P, Kumar R, Kulprathipanja S, Rangsunvigit P, Linga P. Rapid methane hydrate formation to develop a cost effective large scale energy storage

[9] Gholipour Zanjani N, Zarringhalam Moghaddam A, Nazari K, Mohammad-Taheri M. Enhancement of methane purification by the use of porous media in hydrate formation

[10] Linga P, Daraboina N, Ripmeester JA, Englezos P. Enhanced rate of gas hydrate formation in a fixed bed column filled with sand compared to a stirred vessel. Chemical

[11] Hao WF, Wang JQ, Fan SS, Hao WB. Evaluation and analysis method for natural gas hydrate storage and transportation processes. Energy Conversion and Management.

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[13] Linga P, Daraboina N, Ripmeester JA, Englezos P. Enhanced rate of gas hydrate formation in a fixed bed column filled with sand compared to a stirred vessel. Chemical

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natural gas transport and storage. Energy. 2010;35:2717–2722


**Chapter 5**

## **Gas Well Testing**

### Freddy Humberto Escobar

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67620

#### Abstract

Modeling liquid flow for well test interpretation considers constant values of both density and compressibility within the range of dealt pressures. This assumption does not apply for gas flow case in which the gas compressibility factor is also included for a better mathematical representation. The gas flow equation is normally linearized to allow the liquid diffusivity solution to satisfy gas flow behavior. Depending upon the viscosity-compressibility product, three treatments are considered for the linearization: square of pressure squared, pseudopressure, or linear pressure. When wellbore storage conditions are insignificant, drawdown tests are best analyzed using the pseudopressure function. Besides, since the viscosity-compressibility product is highly sensitive in gas flow; then, pseudotime best captures the gas thermodynamics. Buildup pressure tests, for example, require linearization of both pseudotime and pseudopressure. The conventional straight-line method has been customarily used for well test interpretation. Its disadvantages are the accuracy in determining of the starting and ending of a given flow regime and the lack of verification. This is not the case of the Tiab's Direct Synthesis technique (TDS) which is indifferently applied to either drawdown or buildup tests and is based on features and intersection points found of the pressure and pressure derivative log-log plot.

Keywords: TDS technique, pseudotime, pseudopressure, rapid flow, viscosity, rate transient analysis, pressure transient analysis

### 1. Introduction

Contrary to liquids, a gas is highly compressible and much less viscous. In general, gas viscosity is about a 100 times lower than the least viscous crude oil. It is important, however, to try to provide the same mathematical treatment to oil and gas hydrocarbons, so interpretation methodologies can easily be applied in a more practical way. Then, the gas flow equation is normally linearized to allow the liquid diffusivity solution to satisfy the gas behavior when analyzing transient test data of gas reservoirs. Depending on the values of reservoir pressure,

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

viscosity, and gas compressibility factor, the gas flow behavior can be treated as a function of either pressure to the second power or linear pressure with a region which does not correspond to any of these and it is better represented by a synthetic function call pseudopressure. Pseudopressure is a function that integrates pressure, density, and compressibility factor. The gas system's total compressibility highly depends on gas compressibility which for ideal gases changes inversely with the pressure. Then, another artificial function referred as pseudotime is included to further understand the transient behavior of gas flow in porous media. For instance, when wellbore storage conditions are insignificant, drawdown tests are best analyzed using the pseudopressure function. On the other hand, buildup pressure tests require linearization of both pseudotime and pseudopressure.

This chapter will be devoted to provide both fundamental of gas flow in porous media as well as interpretation of pressure and rate data in gas reservoirs. The use of the oil flow equations and interpretation techniques is carefully extended for gas flow so that reservoir permeability, skin factor, and reservoir area can be easily estimated from a gas pressure or gas rate test by using conventional analysis and characteristic points found on the pressure derivative plot (TDS technique). Conventional analysis—the oldest pressure transient test interpretation technique is based upon understanding the flow behavior in a given reservoir geometry, so the pressure versus time function is plotted in such way that a linear trend can be obtained. Both slope and intercept of such linear tendency are used to characterize the reservoir. Conventional analysis has two main drawbacks: (1) difficulty of finding a given flow regime and (2) absence of parameter verification. On the other hand, TDS technique—is strongly based on the log-log plot of pressure and pressure derivative versus time curves which provide the best way for flow regime identification; then, it uses the "fingerprints" or characteristic points found in such plot which are entered in practical and direct analytical equations to easily find reservoir parameters. Moreover, the same parameters can be obtained from different sources for verification purposes. Such is the case, for instance, of the reservoir area in elongated systems which can be estimated five times.

The chapter will include both interpretation techniques TDS and conventional in two cases: (1) infinite and (2) finite reservoirs. Channels or elongated systems in which reservoir hemilinear, parabolic or linear flow regimes developed once radial flow regime vanishes are reported in Refs [8, 13, 14]. This formation of linear flow regime normally occurs in fluvial deposits (channels), sand lens, parallel faulting, terrace faulting, and carbonate reefs. Then, such systems are worth of transient pressure analysis characterization. Latest researches on the determination of drainage area in constant-pressure-bounded systems using either conventional analysis or TDS technique are also reported by Escobar et al. [10].

It is convenient to mention some other important aspects concerning gas well testing which have appeared recently. The first case is the transient rate analysis in hydraulically fractured wells which was presented by [19] for both oil and gas wells. The traditional model for elliptical flow included the reservoir area as a variable. Handling the interpretation using TDS Technique may be little difficult for unexperienced interpreters. Therefore, [20] introduced a model excluding the reservoir drainage area and avoiding the necessity of developing pseudosteady-state regime. When a naturally fractured reservoir is subjected to hydraulic fracturing, the interpretation should be performed according to the presented by [21]. [35]

presented the pressure behavior of finite-conductivity fractured wells in gas composite systems. As far a horizontal wells, the recent works by [23] and [24] included off-centered wells for transient-rate or transient-pressure cases, respectively. [29] presented a study of production performance of horizontal wells when rapid flow conditions are given.

Practical exercises will provide in the chapter provide a better understanding and applicability of the interpretation techniques.

The purpose of this chapter is two folded: (1) to present the governing equation for gas flow used in well test interpretation and (2) to use both conventional and TDS Techniques as valuable tools for well test interpretation in both transient rate and transient pressure analysis. Some detailed examples will be given for demonstration purposes.

### 2. Transient pressure analysis

Transient pressure analysis is performed measuring the bottom-hole pressure while the flow rate is kept constant.

#### 2.1. Fluid flow equations

viscosity, and gas compressibility factor, the gas flow behavior can be treated as a function of either pressure to the second power or linear pressure with a region which does not correspond to any of these and it is better represented by a synthetic function call pseudopressure. Pseudopressure is a function that integrates pressure, density, and compressibility factor. The gas system's total compressibility highly depends on gas compressibility which for ideal gases changes inversely with the pressure. Then, another artificial function referred as pseudotime is included to further understand the transient behavior of gas flow in porous media. For instance, when wellbore storage conditions are insignificant, drawdown tests are best analyzed using the pseudopressure function. On the other hand, buildup pressure tests require linearization of both

This chapter will be devoted to provide both fundamental of gas flow in porous media as well as interpretation of pressure and rate data in gas reservoirs. The use of the oil flow equations and interpretation techniques is carefully extended for gas flow so that reservoir permeability, skin factor, and reservoir area can be easily estimated from a gas pressure or gas rate test by using conventional analysis and characteristic points found on the pressure derivative plot (TDS technique). Conventional analysis—the oldest pressure transient test interpretation technique is based upon understanding the flow behavior in a given reservoir geometry, so the pressure versus time function is plotted in such way that a linear trend can be obtained. Both slope and intercept of such linear tendency are used to characterize the reservoir. Conventional analysis has two main drawbacks: (1) difficulty of finding a given flow regime and (2) absence of parameter verification. On the other hand, TDS technique—is strongly based on the log-log plot of pressure and pressure derivative versus time curves which provide the best way for flow regime identification; then, it uses the "fingerprints" or characteristic points found in such plot which are entered in practical and direct analytical equations to easily find reservoir parameters. Moreover, the same parameters can be obtained from different sources for verification purposes. Such is the case, for instance, of the reservoir area in elongated systems which can be estimated five times. The chapter will include both interpretation techniques TDS and conventional in two cases: (1) infinite and (2) finite reservoirs. Channels or elongated systems in which reservoir hemilinear, parabolic or linear flow regimes developed once radial flow regime vanishes are reported in Refs [8, 13, 14]. This formation of linear flow regime normally occurs in fluvial deposits (channels), sand lens, parallel faulting, terrace faulting, and carbonate reefs. Then, such systems are worth of transient pressure analysis characterization. Latest researches on the determination of drainage area in constant-pressure-bounded systems using either conventional

analysis or TDS technique are also reported by Escobar et al. [10].

It is convenient to mention some other important aspects concerning gas well testing which have appeared recently. The first case is the transient rate analysis in hydraulically fractured wells which was presented by [19] for both oil and gas wells. The traditional model for elliptical flow included the reservoir area as a variable. Handling the interpretation using TDS Technique may be little difficult for unexperienced interpreters. Therefore, [20] introduced a model excluding the reservoir drainage area and avoiding the necessity of developing pseudosteady-state regime. When a naturally fractured reservoir is subjected to hydraulic fracturing, the interpretation should be performed according to the presented by [21]. [35]

pseudotime and pseudopressure.

96 Advances in Natural Gas Emerging Technologies

The gas diffusivity equation in oil-field units is given by:

$$\frac{1}{r}\frac{\partial}{\partial r}\left(\frac{P}{\mu(P)Z(P)}r\frac{\partial P}{\partial r}\right) = \frac{\phi}{0.0002637}\frac{\partial}{\partial t}\left(\frac{P}{Z(P)}\right) \tag{1}$$

Which can be modified to respond for three-phase flow (oil, water, and gas):

$$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial P}{\partial r}\right) = \frac{\phi c\_t}{0.0002637\lambda\_t}\frac{\partial P}{\partial t} \tag{2}$$

where, the total compressibility, ct, and total mobility, λt, are given by:

$$\mathbf{c}\_t \approx \mathbf{c}\_\mathcal{S} \mathbf{s}\_\mathcal{S} + \mathbf{c}\_o \mathbf{S}\_o + \mathbf{c}\_w \mathbf{S}\_w + \mathbf{c}\_f \tag{3}$$

$$
\lambda\_t = \frac{k\_\mathcal{g}}{\mu\_\mathcal{g}} + \frac{k\_o}{\mu\_o} + \frac{k\_w}{\mu\_w} \tag{4}
$$

As can be inferred from Eq. (3), the total compressibility varies significantly when dealing with monophasic gas flow since the gas compressibility varies along with the pressure. Agarwal [1] introduced the pseudotime function to alleviate such problem. This function accounts for the time dependence of gas viscosity and total system compressibility:

$$t\_a = \int\_{t\_{rf}}^t \frac{dt}{\mu(t)c\_t(t)}\tag{5}$$

Pseudotime is better defined as a function of pressure as a new function given in hr psi/cp:

$$t\_a(P) = \int\_{P\_{nf}}^{P} \frac{(dt/dP)}{\mu(P)c\_t(P)} dP \tag{6}$$

Notice that μ and ct are now pressure-dependent properties.

As expressed by Eq. (1), viscosity and gas compressibility factor are strong functions of pressure; then, to account for gas flow behavior, Al-Hussainy et al. [2] introduced the pseudopressure function which basically includes the variation of gas viscosity and compressibility into a single function which is given by:

$$m(P) = 2\int\_{p\_{\rm ref}}^{P} \frac{P}{\mu(P)Z(P)}dP\tag{7}$$

After replacing Eqs. (6) and (7) into Eq. (1), it yields:

$$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial m(P)}{\partial r}\right) = \frac{\phi}{0.0002637k}\frac{\partial m(P)}{\partial t\_a(P)}\tag{8}$$

Contrary to liquid well testing, rapid gas flow has a strong influence on well testing, [32]. As the flow rate increases, so does the skin factor, then:

$$\mathbf{s}\_d = \mathbf{s} + D\boldsymbol{\eta} \tag{9}$$

Eq. (9) shows that the apparent skin factor is a function of the mechanical skin factor—which is assumed to be constant during the test— and the product of the flow rate with the turbulence factor or non-Darcy term. This implies that two flow test ought to be run at different flow rates to find mechanical skin factor and the turbulence factor from:

$$(\mathbf{s}\_{\mathfrak{u}})\_1 = \mathbf{s} + D\mathfrak{q}\_1 \tag{10}$$

$$(\mathbf{s}\_{\mathfrak{u}})\_2 = \mathbf{s} + D\mathfrak{q}\_2 \tag{11}$$

Solving the simultaneous equations:

$$D = \frac{(s\_a)\_1 - (s\_a)\_2}{q\_1 - q\_2} \tag{12}$$

$$s = (s\_a)\_1 - \frac{(s\_a)\_1 - (s\_a)\_2}{q\_1 - q\_2} q\_1 \tag{13}$$

where, the skin factors 1 and 2 are estimated from each pressure test. However, there is a need of estimating the turbulence factor by empirical correlations for buildup cases or when a single test exists. Then, the non-Darcy flow coefficient is defined by [26]:

$$D = 2.222 \times 10^{-15} \frac{\nu\_g k h \beta}{\mu\_g r\_w h\_p^2} \tag{14}$$

The above equation is also applied to partially completed or partially penetrated wells. hp is the length of the perforated interval. For fully perforated wells, hp = h.

Parameter β is called turbulence factor or inertial factor can be found by correlations. The correlation proposed by Geertsma [21] is given by:

$$\beta = \frac{4.851 \times 10^4}{\phi^{5.5} \sqrt{k}} \tag{15}$$

The consideration on the skin factor effect on gas testing was recognized by Fligelman et al. [25] who provided correction charts to account for apparent skin factor values.

#### 2.2. Conventional analysis

Pseudotime is better defined as a function of pressure as a new function given in hr psi/cp:

ð P

ðdt=dPÞ μðPÞctðPÞ

dP (6)

dP (7)

<sup>∂</sup>taðP<sup>Þ</sup> (8)

q<sup>1</sup> (13)

(12)

Pref

As expressed by Eq. (1), viscosity and gas compressibility factor are strong functions of pressure; then, to account for gas flow behavior, Al-Hussainy et al. [2] introduced the pseudopressure function which basically includes the variation of gas viscosity and compressibility into a single

> ð P

P μðPÞZðPÞ

<sup>¼</sup> <sup>φ</sup> 0:0002637k

Contrary to liquid well testing, rapid gas flow has a strong influence on well testing, [32]. As

Eq. (9) shows that the apparent skin factor is a function of the mechanical skin factor—which is assumed to be constant during the test— and the product of the flow rate with the turbulence factor or non-Darcy term. This implies that two flow test ought to be run at different flow rates

> <sup>D</sup> <sup>¼</sup> <sup>ð</sup>saÞ<sup>1</sup> � ðsaÞ<sup>2</sup> q<sup>1</sup> � q<sup>2</sup>

<sup>s</sup> ¼ ðsaÞ<sup>1</sup> � <sup>ð</sup>saÞ<sup>1</sup> � ðsaÞ<sup>2</sup>

where, the skin factors 1 and 2 are estimated from each pressure test. However, there is a need of estimating the turbulence factor by empirical correlations for buildup cases or when a single

q<sup>1</sup> � q<sup>2</sup>

∂mðPÞ

sa ¼ s þ Dq (9)

ðsaÞ<sup>1</sup> ¼ s þ Dq<sup>1</sup> (10)

ðsaÞ<sup>2</sup> ¼ s þ Dq<sup>2</sup> (11)

Pref

taðPÞ ¼

mðPÞ ¼ 2

Notice that μ and ct are now pressure-dependent properties.

After replacing Eqs. (6) and (7) into Eq. (1), it yields:

the flow rate increases, so does the skin factor, then:

Solving the simultaneous equations:

1 r ∂ ∂r r ∂mðPÞ ∂r � �

to find mechanical skin factor and the turbulence factor from:

test exists. Then, the non-Darcy flow coefficient is defined by [26]:

function which is given by:

98 Advances in Natural Gas Emerging Technologies

The solution to the transient diffusivity equation, Eq. (8), is given by:

$$m(P)\_D(1, t\_{\rm Da}) = -\frac{1}{2} Ei\left(-\frac{1}{4t\_{\rm Da}}\right) \tag{16}$$

The dimensionless parameters used in this chapter are given below. The rigorous dimensionless time is:

$$t\_D = \frac{0.0002637kt}{\phi(\mu c\_t)r\_w^2} \tag{17}$$

Including the pseudotime function, ta(P), the dimensionless pseudotime is:

$$t\_{\rm Dt} = \left(\frac{0.0002637k}{\phi r\_w^2}\right) t\_a(P) \tag{18}$$

Notice that the viscosity-compressibility product is not seen in Eq. (16) since they are included in the pseudotime function. However, if we multiply and, then, divide by (μct)i, a similar equation to the general dimensionless time expression will be obtained.

$$t\_{\rm Da} = \left(\frac{0.0002637k}{\phi(\mu c\_l)\_i r\_w^2}\right) [(\mu c\_t)\_i \times t\_a(P)] \tag{19}$$

The dimensionless pseudopressure and pseudopressure derivatives are:

$$t^\* \Delta m(P)\_D^{'} = \frac{\hbar k[t^\* \Delta m(P)^{'}]}{1422.52 q\_{sc} T} \tag{19a}$$

$$m(P)\_D = \frac{hk[m(P\_i) - m(P)]}{1422.52qT} \tag{20}$$

$$\left(t\_a(P)^\* \Delta m(P)\_D\right)' = \frac{\hbar k [t\_a(P)^\* \Delta m(P)']}{1422.52 q\_{sc} T} \tag{21}$$

And the dimensionless wellbore storage coefficient is given by:

$$\mathbf{C}\_{D} = \left(\frac{0.8935}{\phi h c\_{l} r\_{w}^{2}}\right) \mathbf{C} \tag{22}$$

The dimensionless radii are given:

$$r\_D = \frac{r}{r\_w} \tag{23}$$

$$r\_{Dt} = \frac{r\_e}{r\_w} \tag{24}$$

For practical purposes, Eq. (16) will end up in a semilog behavior of pseudopressure drops against time. After replacing the respective dimensionless quantities into the mentioned straight-line semilog expression, it is obtained [4]:

$$m(P\_i) - m(P\_{wf}) = \frac{1.422 \times 10^6 qT}{k\hbar} \left[ 1.1513 \text{log} \left( \frac{kt}{1688 \phi(\mu\_g c\_i)\_i r\_w^2} \right) + s' + D \right] \tag{25}$$

$$m(P\_i) - m(P\_{wf}) = \frac{1.422 \times 10^6 qT}{kh} \left[ 1.1513 \log \left( \frac{kt\_d(P)}{1688 \phi r\_w^2} \right) + s' + D \right] \tag{26}$$

The above equations are applied during transient or radial flow regime. They are used to find reservoir transmissibility and apparent skin factor from the slope and intercept, respectively, of a semilog plot of well-flowing pressure versus time. After applying the superposition principle, the above equations for the buildup case are converted into:

$$m(P\_i) - m(P\_{wf}) = \frac{1.422 \times 10^6 qT}{kh} \log\left(\frac{t\_p + \Delta t}{\Delta t}\right) \tag{27}$$

$$m(P\_i) - m(P\_{wf}) = \frac{1.422 \times 10^6 qT}{kh} \log\left(\frac{t\_a(P)\_p + \Delta t\_a(P)}{\Delta t\_a(P)}\right) \tag{28}$$

From a semilog plot of pseudopressure versus time (or pseudotime), its slope allows calculating the reservoir permeability and the intercept is used to find the pseudoskin factor, respectively:

$$k = \frac{1637.74 \eta T}{mh} \tag{29}$$

$$s' = \left[\frac{m(P\_i) - m(P\_{1hr})}{m} - \log\left(\frac{k}{\phi(\mu c\_t)\_i r\_w^2}\right) - 3.227 + 0.8686\right] \tag{30}$$

Notice that for the pseudotime case, (µct)<sup>i</sup> product in the above equation will be set as the unity. The gas pseudoskin factor is estimated for the buildup case as:

$$s' = \left[\frac{m(P\_{1lr}) - m(P\_{wf})}{m} - \log\left(\frac{k}{\phi(\mu c\_t)\_i r\_w^2}\right) - 3.227 + 0.8686\right] \tag{31}$$

The governing dimensionless pressure equation during pseudosteady-state period is given by [28]:

$$m(P)\_D = \frac{2t\_D}{r\_{eD}^2} + \ln r\_D - 0.75 + s'\tag{32}$$

By replacing the dimensionless quantities, changing the log base, the above equation leads to:

$$m(P\_i) - m(P\_{wf}) = \frac{0.2395qTt}{Ah\phi} + \frac{3263qT}{kh} \left[ \log \frac{0.472r\_\varepsilon}{r\_w} + \frac{s'}{2.303} \right] \tag{33}$$

A Cartesian plot of m(Pwf) versus time or pseudotime during pseudosteady state will yield a straight line in which slope, m\*, is useful to find the well drainage area:

$$A = \frac{0.23395(5.615)qT}{\phi l m^\*} \tag{34}$$

Such deliverability tests as backpressure, isochronal, modified isochronal, and flow after flow are conducted for the purpose of determining the flow exponent n (n = 1 is considered turbulent flow and 0.5 < n < 1 is considered to be rapid flow) and the performance coefficients. They assumed that stabilization is reached during the testing which is not true in most of the cases. Then, they are not included in this chapter but can be found in Chapter 4 of Ref. [4].

#### 2.3. TDS technique

taðPÞ � ΔmðPÞ<sup>D</sup>

And the dimensionless wellbore storage coefficient is given by:

The dimensionless radii are given:

100 Advances in Natural Gas Emerging Technologies

straight-line semilog expression, it is obtained [4]:

<sup>m</sup>ðPiÞ � <sup>m</sup>ðPwfÞ ¼ <sup>1</sup>:<sup>422</sup> � <sup>106</sup>

s

<sup>0</sup> <sup>¼</sup> <sup>m</sup>ðPiÞ � <sup>m</sup>ðP1hr<sup>Þ</sup>

<sup>m</sup>ðPiÞ � <sup>m</sup>ðPwfÞ ¼ <sup>1</sup>:<sup>422</sup> � <sup>106</sup>

ple, the above equations for the buildup case are converted into:

<sup>m</sup>ðPiÞ � <sup>m</sup>ðPwfÞ ¼ <sup>1</sup>:<sup>422</sup> � <sup>106</sup>

<sup>m</sup>ðPiÞ � <sup>m</sup>ðPwfÞ ¼ <sup>1</sup>:<sup>422</sup> � <sup>106</sup>

<sup>0</sup> <sup>¼</sup> hk½taðP<sup>Þ</sup>

� �

CD <sup>¼</sup> <sup>0</sup>:<sup>8935</sup> φhctr<sup>2</sup> w

> rD <sup>¼</sup> <sup>r</sup> rw

rDe <sup>¼</sup> re rw

For practical purposes, Eq. (16) will end up in a semilog behavior of pseudopressure drops against time. After replacing the respective dimensionless quantities into the mentioned

kh <sup>1</sup>:1513log kt

kh <sup>1</sup>:1513log ktaðP<sup>Þ</sup>

qT kh log tp <sup>þ</sup> <sup>Δ</sup><sup>t</sup>

qT kh log

From a semilog plot of pseudopressure versus time (or pseudotime), its slope allows calculating the reservoir permeability and the intercept is used to find the pseudoskin factor, respectively:

<sup>k</sup> <sup>¼</sup> <sup>1637</sup>:74qT

φðμctÞ<sup>i</sup> r2 w

� �

� �

<sup>m</sup> � log <sup>k</sup>

1688φðμgctÞ<sup>i</sup>

!

1688φrw<sup>2</sup> � �

> Δt � �

taðPÞ<sup>p</sup> þ ΔtaðPÞ ΔtaðPÞ !

mh (29)

� 3:227 þ 0:8686

� �

" #

rw<sup>2</sup>

þ s <sup>0</sup> þ D

þ s <sup>0</sup> þ D

qT

qT

The above equations are applied during transient or radial flow regime. They are used to find reservoir transmissibility and apparent skin factor from the slope and intercept, respectively, of a semilog plot of well-flowing pressure versus time. After applying the superposition princi-

� ΔmðPÞ 0 �

<sup>1422</sup>:52qscT (21)

C (22)

(23)

(24)

(25)

(26)

(27)

(28)

(30)

Tiab [33] proposed a revolutionary technique which is very useful to interpret pressure tests using characteristics points found on the pressure and pressure derivative versus time log-log plot. He obtained practical analytical solutions for the determination of reservoir parameters.

$$m(P)\_{Dr} = \left(\frac{7.029 \times 10^{-4} \text{k}\text{h}}{qT}\right) \left(\frac{m(P\_i) - m(P\_{wf})(t\_n)}{q\_n}\right) = \frac{1}{2}(\text{Int}\_D + 0.80907 + 2\text{s})\tag{35}$$

From a log-log plot of pseudopressure and pseudopressure derivative against pseudotime, Figure 1, several main characteristics are outlined:

1. The early unit-slope line originated by wellbore storage is described by the following equation:

$$(m(P)\_D = \frac{t\_{Da}}{C\_D} \tag{36}$$

Replacing the dimensionless parameters in Eq. (36), a new equation to estimate the wellbore storage coefficient is obtained:

Figure 1. Log-log plot of pseudopressure and pseudopressure derivative versus pseudotime. After Ref. [7].

$$\mathcal{C} = (0.419qTc\_t) \left(\frac{t\_a(P)}{\Delta m(P)}\right) \tag{37}$$

2. The intersection of the early unit-slope line with the radial horizontal straight line gives:

$$
\left(\frac{t\_{Da}}{\mathbb{C}\_D}\right)\_i = 0.5\tag{38}
$$

From this, an equation to estimate either permeability or wellbore storage is obtained once the dimensionless parameters are replaced.

As presented by Tiab [33], the governing equation for the well pressure behavior during radial flow reformulated by Escobar et al. [7] in terms of pseudofunctions is expressed by:

$$t\_a(P)\_i = \frac{1695c\_tC}{kh} \tag{39}$$

3. According to Ref. [28], another form of Eq. (35) is obtained when wellbore storage and skin factor are included:

$$m(P)\_{D^r} = \frac{1}{2} \left\{ \ln \left( \frac{t\_{Dd}}{\mathbb{C}\_D} \right)\_r + 0.80907 + \ln(\mathbb{C}\_D e^{2s}) \right\} \tag{40}$$

From the above equation, the derivative of pseudopressure with respect to the natural log of tDa/CD is given by:

$$\left[\frac{t\_{Du}}{\mathbb{C}\_D} m(P)\_D'\right]\_r = 0.5\tag{41}$$

From Eq. (21), the dimensionless pseudopressure derivative with respect to the natural log of log tDa/CD gives:

Gas Well Testing http://dx.doi.org/10.5772/67620 103

$$
\left[\frac{t\_{\rm Dt}}{\mathbf{C}\_{D}} m(P)\_{D}^{\prime}\right]\_{r} = \left[7.029 \times 10^{-4} \frac{k\hbar}{qT}\right] [t\_{\rm d}(P) \ast m(P)^{\prime}] \tag{42}
$$

Combination of Eqs. (41) and (42) will result into an equation to estimate permeability:

$$k = \frac{711.26qT}{h[t\_a(P) \* \Delta m(P)]\_r} \tag{43}$$

3. Dividing Eq. (40) by Eq. (41), replacing the dimensionless quantities and, then, solving for the pseudoskin factor will yield:

$$s'=0.5\left[\frac{[\Delta m(P)]\_r}{[t\_d(P)\*\Delta m(P)']\_r} - \ln\left(\frac{k\left(t\_d(P)\right)\_r}{\phi r\_w^2}\right) + 7.4316\right] \tag{44}$$

Finally, the pressure derivative during the pseudosteady-state flow regime of closed systems is governed by:

$$t\_{\rm Dt} \* m(P)'\_{\rm D} = 2\pi t\_{\rm Dt} \tag{45}$$

The intersection point of the above straight line and the radial flow regime straight line is:

$$t\_{aDARP\_i} = \frac{1}{4\pi} \tag{46}$$

After substituting the dimensionless pseudotime function into Eq. (46), a new equation for the well drainage area is presented:

$$A = \frac{k t\_a (P)\_{rpi}}{301.77 \,\phi} \tag{47}$$

Further applications of gas well test can be found in the literature. Escobar et al. [12] introduced the mathematical expressions for interpretation of pressure tests using the pseudopressure and pseudopressure derivative as a function of pseudotime for hydraulically fractured wells and naturally fractured (heterogeneous) formations. Fligelman [30] presented an interpretation methodology using TDS technique for finite-conductivity fractured wells. They used pseudopressure and rigorous time. In 2012, Escobar et al. [16] implemented the transient pressure analysis on gas fractured wells in bi-zonal reservoirs. Moncada et al. [31] extended the TDS for oil and gas flow for partially completed and partially penetrated wells. As far as horizontal wells, it is worth to mention the work performed in Refs. [11] and [15] on homogeneous and naturally fractured reservoirs.

#### 2.4. Example 1

<sup>C</sup> ¼ ð0:419qTct<sup>Þ</sup> taðP<sup>Þ</sup>

Figure 1. Log-log plot of pseudopressure and pseudopressure derivative versus pseudotime. After Ref. [7].

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09

*t* , psi hr/cp *<sup>a</sup>*

2. The intersection of the early unit-slope line with the radial horizontal straight line gives: tDa CD 

i

From this, an equation to estimate either permeability or wellbore storage is obtained once the

As presented by Tiab [33], the governing equation for the well pressure behavior during radial

taðPÞ<sup>i</sup> <sup>¼</sup> <sup>1695</sup>ctC

3. According to Ref. [28], another form of Eq. (35) is obtained when wellbore storage and skin

r

From the above equation, the derivative of pseudopressure with respect to the natural log of

From Eq. (21), the dimensionless pseudopressure derivative with respect to the natural log of

r

þ 0:80907 þ lnðCDe

flow reformulated by Escobar et al. [7] in terms of pseudofunctions is expressed by:

<sup>2</sup> ln tDa CD 

> tDa CD mðPÞ 0 D

dimensionless parameters are replaced.

1.E+06

1.E+07

*m*(*P*) and *t* \*

*a*

*m*(*P*)', psi /cp

2

1.E+08

102 Advances in Natural Gas Emerging Technologies

Unit-slope line

<sup>m</sup>ðPÞDr <sup>¼</sup> <sup>1</sup>

factor are included:

tDa/CD is given by:

log tDa/CD gives:

ΔmðPÞ 

Infinite-acting line

¼ 0:5 (38)

Pseudostable-state line

( ) \* ( )' Δ *<sup>a</sup>* Δ *m P t mP*

kh (39)

¼ 0:5 (41)

2s Þ (37)

(40)

Chaudhry [4] presented a reservoir limit test for a gas reservoir (example 5-2 of Ref. [4]). However, once the pressure derivative was taken to the test data, no late pseudosteady state regime was observed. Then, the input data given below were used to simulate a pressure test given in Table 1.

#### 104 Advances in Natural Gas Emerging Technologies




Table 1. Pressure, pseudopressure, time, and pseudotime data for example 1.

Sg = 70% Sw = 30% q = 6184 MSCF/D h = 41 ft k = 44 md Bg = 0.00102 ft<sup>3</sup>

rw = 0.4271 ft φ = 10.04% ct = 0.0002561 psi<sup>1</sup> ω<sup>g</sup> = 0.0992 md/cp γ<sup>g</sup> = 0.732 Pcr = 380.16 psia Tcr = 645.06 R T = 710 R re = 2200 ft (349 Ac)

t, hr P, psi t, hr P, psi 0 3965 1.2713 3677.2527 0.001 3960.629 1.6005 3670.3779 0.002 3956.4313 2.0148 3663.602 0.003 3952.3774 2.5365 3656.8388 0.004 3948.4516 3.1933 3650.1477 0.005 3944.6431 4.0202 3643.5144 0.006 3940.9438 5.0107 3637.2126 0.007 3937.3469 6.0107 3632.0323 0.008 3933.8466 7.0107 3627.6664 0.009 3930.4382 8.0107 3623.8936 0.0113 3922.8277 9.0107 3620.5718 0.0143 3913.8402 10.0107 3617.6047 0.018 3903.2573 12.0107 3612.4479 0.0226 3891.1325 21.0107 3596.5482 0.0285 3877.5006 30.0107 3586.4447 0.0358 3862.4816 39.0107 3579.0231 0.0451 3846.2054 48.0107 3573.1536 0.0568 3829.2592 57.0107 3568.2967 0.0715 3812.1329 66.0107 3564.1453 0.09 3795.2335 75.0107 3560.4805 0.1133 3779.2686 84.0107 3557.2312 0.1271 3771.7594 93.0107 3554.3136 0.16 3757.7512 102.0107 3551.6672 0.2015 3745.1803 179.5107 3535.5164 0.2537 3734.0438 269.5107 3523.5847 0.3193 3724.1258 359.5107 3514.1663 0.402 3715.1658 449.5107 3505.545 0.5061 3706.8112 539.5107 3497.2804

m(Pi) = 340920304.2 psi<sup>2</sup>

/cp

104 Advances in Natural Gas Emerging Technologies

/STB

Estimate permeability, skin factor, and drainage area by both conventional analysis and TDS technique.

#### 2.4.1. Solution by conventional analysis

Figure 2 presents a semilog pressure of pseudopressure versus pseudotime. The slope and intercept of the radial flow regime straight line in such plot are given below:

$$m = -3995147.42 \ (\text{psi}^2/\text{cp})/(\text{log hr} - \text{psi}/\text{cp})$$

$$m(P)\_{1\,\text{hr}} = 342125555.5 \ \text{psi}^2/\text{cp}$$

Use of Eqs. (27) and (28) allows finding reservoir permeability and pseudoskin factor, respectively:

$$k = \frac{1637.74qT}{mh} = \frac{1637.74(6184)(710)}{(3995147.42)(41)} = 43.45 \text{ md}$$

$$s' = \left[\frac{340920304.25 - 342125555.5}{-3995147.42}\right]$$

$$s' = \left[\frac{-3995147.42}{(0.1004)(0.4271^2)}\right] - 3.227 + 0.8686$$

To find the well drainage area, the Cartesian plot given in Figure 3 was built. Its slope, m\* = 0.0914 (psi<sup>2</sup> /cp)/(hr�psi/cp), is plugged into Eq. (34):

$$A = \frac{0.23395qT}{\phi l m^\*} = \frac{0.23395(6184)(710)}{(0.1004)(41)(0.0914)(43560)} = 336.4 \text{ Ac}$$

Figure 2. Semilog plot for example 1.

Figure 3. Cartesian plot for example 1.

Estimate permeability, skin factor, and drainage area by both conventional analysis and TDS

Figure 2 presents a semilog pressure of pseudopressure versus pseudotime. The slope and

<sup>m</sup>ðPÞ1 hr <sup>¼</sup> <sup>342125555</sup>:5 psi<sup>2</sup>

mh <sup>¼</sup> <sup>1637</sup>:74ð6184Þð710<sup>Þ</sup>

Þ

<sup>φ</sup>hm� <sup>¼</sup> <sup>0</sup>:23395ð6184Þð710<sup>Þ</sup>

*<sup>m</sup>*(*P*) = 342125556 psi /cp <sup>2</sup> 1 hr

To find the well drainage area, the Cartesian plot given in Figure 3 was built. Its slope,

ð0:1004Þð41Þð0:0914Þð43560Þ

1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

*t* (*P*)*,* hr-psi/cp

Use of Eqs. (27) and (28) allows finding reservoir permeability and pseudoskin factor, respec-

=cpÞ=ðlog hr � psi=cpÞ

<sup>ð</sup>3995147:42Þð41<sup>Þ</sup> <sup>¼</sup> <sup>43</sup>:45 md

� 3:227 þ 0:8686

=cp

3 7 7

<sup>5</sup> ¼ �0:<sup>5172</sup>

¼ 336:4 Ac

*m* = -3995147.42 psi \*hr/log psi-hr/cp

2

intercept of the radial flow regime straight line in such plot are given below:

<sup>m</sup> ¼ �3995147:<sup>42</sup> <sup>ð</sup>psi<sup>2</sup>

340920304:25 � 342125555:5 �3995147:42

<sup>ð</sup>0:1004Þð0:4271<sup>2</sup>

/cp)/(hr�psi/cp), is plugged into Eq. (34):

� �

<sup>k</sup> <sup>¼</sup> <sup>1637</sup>:74qT

�log <sup>43</sup>:<sup>45</sup>

<sup>A</sup> <sup>¼</sup> <sup>0</sup>:23395qT

technique.

tively:

2.4.1. Solution by conventional analysis

106 Advances in Natural Gas Emerging Technologies

s <sup>0</sup> ¼

m\* = 0.0914 (psi<sup>2</sup>

*m*(*P*), psi /cp 2

3.05E+08 3.10E+08 3.15E+08 3.20E+08 3.25E+08 3.30E+08 3.35E+08 3.40E+08 3.45E+08

Figure 2. Semilog plot for example 1.

#### 2.4.2. Solution by TDS technique

Figure 4 presents the pseudopressure and pressure derivative versus pseudotime log-log plot in which wellbore storage, radial flow regime, and late pseudosteady-state regimes are clearly observed. The following characteristic points were read from Figure 4:

Figure 4. Pseudopressure drop and pseudopressure derivative versus time log-log plot for example 1.

Permeability and pseudoskin factor are respectively estimated from Eqs. (42) and (44),

$$k = \frac{711.26qT}{h[t\_a(P) \* \Delta m(P)]\_r} = \frac{711.26(6184)(710)}{(41)(1735066.96)} = 43.9 \text{ md}$$

$$s' = 0.5 \left[ \frac{23918367.9}{1735066.96} - \ln \left( \frac{43.9(1694705.5)}{(0.1004)(0.4271^2)} \right) + 7.4316 \right] = -0.454$$

and well drainage area is found with Eq. (47):

$$A = \frac{kt\_a(P)\_{rpi}}{301.77 \,\phi} = \frac{(43.9)(10113641.48)}{301.77 \,(0.1004)} = 336.4 \text{ Ac}$$

Finally, the inertial factor and the non-Darcy flow coefficient are estimated with Eqs. (14) and (15):

$$\beta = \frac{4.851 \times 10^4}{\phi^{5.5} \sqrt{k}} = \frac{4.851 \times 10^4}{(0.1004)\sqrt{43.9}} = 2265091235.63 \text{ ft}^{-1}$$

$$D = 2.222 \times 10^{-15} \frac{(0.732)(41)(43.9)(2265091235.63)}{(0.0992)(0.4271)(41^2)} = 9 \times 10^{-5} \text{ D/Msff}$$

The true skin factor is found with Eq. (9):

$$s\_a = s + D\eta = -0.454 + 9 \times 10^{-5} \, (6184) = 1.42 \, \text{J}$$

It can be seen that the simulated parameters closely match the results obtained from the examples.

#### 3. Transient rate analysis

Transient rate analysis is performed by recording the continuous changing flow rate under a constant bottom-hole pressure condition. This procedure is normally achieved in very low gas formations and shale gas systems.

#### 3.1. Basic flow and dimensional equations

The Laplace domain, the rate of solution for a well producing against a constant bottom-hole well-flowing pressure was given by [34]:

$$\eta\_D = \frac{1}{\mu \mathcal{K}\_0(\sqrt{\mu})} \tag{48}$$

The solution for a bounded reservoir was presented by [5]:

$$\overline{q}\_{D} = \frac{I\_{1}(r\_{\varepsilon D}\sqrt{\overline{u}})K\_{1}(\sqrt{\overline{u}}) - K\_{1}(r\_{\varepsilon D}\sqrt{\overline{u}})I\_{1}(\sqrt{\overline{u}})}{\sqrt{\overline{u}}[I\_{0}(\sqrt{\overline{u}})K\_{1}(r\_{\varepsilon D}\sqrt{\overline{u}}) + K\_{0}(\sqrt{\overline{u}})I\_{1}(r\_{\varepsilon D}\sqrt{\overline{u}})]} \tag{49}$$

For considerable longer times, Ref. [27] showed that the qD function in Eq. (48) may be approximated by:

$$\frac{1}{q\_D} = \frac{1}{2} \left[ \ln t\_D + 0.80907 \right] \tag{50}$$

where the dimensionless reciprocal rate and reciprocal rate derivative are given by:

$$1/q\_D = \frac{kh \, \Delta m(P)}{1422.52 \, Tq} \tag{51}$$

$$t\_D \* (1/q\_D)' = \frac{kh\left[t \* \Delta m(P)'\right]}{1422.52\ Tq} \tag{52}$$

Including pseudoskin effects in Eq. (49),

$$\frac{1}{q\_D} = \frac{1}{2} \left[ \ln t\_D + 0.80907 + 2s' \right] \tag{53}$$

#### 3.2. Conventional analysis

Permeability and pseudoskin factor are respectively estimated from Eqs. (42) and (44),

<sup>¼</sup> <sup>711</sup>:26ð6184Þð710<sup>Þ</sup>

<sup>ð</sup>0:1004Þð0:4271<sup>2</sup>

� �

� �

<sup>301</sup>:<sup>77</sup> <sup>φ</sup> <sup>¼</sup> <sup>ð</sup>43:9Þð10113641:48<sup>Þ</sup>

<sup>p</sup> <sup>¼</sup> <sup>4</sup>:<sup>851</sup> � <sup>104</sup>

<sup>D</sup> <sup>¼</sup> <sup>2</sup>:<sup>222</sup> � <sup>10</sup>�<sup>15</sup> <sup>ð</sup>0:732Þð41Þð43:9Þð2265091235:63<sup>Þ</sup>

Finally, the inertial factor and the non-Darcy flow coefficient are estimated with Eqs. (14) and (15):

<sup>ð</sup>0:1004<sup>Þ</sup> ffiffiffiffiffiffiffiffiffi

<sup>ð</sup>0:0992Þð0:4271Þð412

sa <sup>¼</sup> <sup>s</sup> <sup>þ</sup> Dq ¼ �0:<sup>454</sup> <sup>þ</sup> <sup>9</sup> � <sup>10</sup>�<sup>5</sup> <sup>ð</sup>6184Þ ¼ <sup>1</sup>:<sup>42</sup>

It can be seen that the simulated parameters closely match the results obtained from the examples.

Transient rate analysis is performed by recording the continuous changing flow rate under a constant bottom-hole pressure condition. This procedure is normally achieved in very low gas

The Laplace domain, the rate of solution for a well producing against a constant bottom-hole

<sup>u</sup> <sup>p</sup> Þ � <sup>K</sup>1ðreD

<sup>u</sup> <sup>p</sup> <sup>Þ</sup> ffiffiffi

ffiffiffi <sup>u</sup> <sup>p</sup> Þ þ <sup>K</sup>0<sup>ð</sup> ffiffiffi

ffiffiffi <sup>u</sup> <sup>p</sup> <sup>Þ</sup>I1<sup>ð</sup> ffiffiffi

<sup>u</sup> <sup>p</sup> <sup>Þ</sup>I1ðreD

qD <sup>¼</sup> <sup>1</sup> uK0<sup>ð</sup> ffiffiffi

ffiffiffi <sup>u</sup> <sup>p</sup> <sup>Þ</sup>K1<sup>ð</sup> ffiffiffi

<sup>u</sup> <sup>p</sup> <sup>Þ</sup>K1ðreD

<sup>ð</sup>41Þð1735066:96<sup>Þ</sup> <sup>¼</sup> <sup>43</sup>:9 md

þ 7:4316

¼ �0:454

Þ

<sup>301</sup>:<sup>77</sup> <sup>ð</sup>0:1004<sup>Þ</sup> <sup>¼</sup> <sup>336</sup>:4 Ac

<sup>43</sup>:<sup>9</sup> <sup>p</sup> <sup>¼</sup> <sup>2265091235</sup>:63 ft�<sup>1</sup>

<sup>Þ</sup> <sup>¼</sup> <sup>9</sup> � <sup>10</sup>�<sup>5</sup> <sup>D</sup>=Mscf

<sup>u</sup> <sup>p</sup> <sup>Þ</sup> (48)

<sup>u</sup> <sup>p</sup> Þ� (49)

ffiffiffi

0 � r

<sup>1735066</sup>:<sup>96</sup> � ln <sup>43</sup>:9ð1694705:5<sup>Þ</sup>

<sup>k</sup> <sup>¼</sup> <sup>711</sup>:26qT h½taðPÞ � ΔmðPÞ

<sup>A</sup> <sup>¼</sup> ktaðPÞrpi

<sup>β</sup> <sup>¼</sup> <sup>4</sup>:<sup>851</sup> � <sup>10</sup><sup>4</sup> φ<sup>5</sup>:<sup>5</sup> ffiffi k

<sup>0</sup> <sup>¼</sup> <sup>0</sup>:<sup>5</sup> <sup>23918367</sup>:<sup>9</sup>

and well drainage area is found with Eq. (47):

The true skin factor is found with Eq. (9):

3. Transient rate analysis

formations and shale gas systems.

3.1. Basic flow and dimensional equations

well-flowing pressure was given by [34]:

The solution for a bounded reservoir was presented by [5]:

qD <sup>¼</sup> <sup>I</sup>1ðreD

<sup>u</sup> <sup>p</sup> <sup>½</sup>I0<sup>ð</sup> ffiffiffi

s

108 Advances in Natural Gas Emerging Technologies

After replacing the dimensionless quantities and changing the logarithm base, it yields:

$$\frac{1}{q} = \frac{1.422 \times 10^6 qT}{k \hbar \Delta m(P)} \left[ 1.1513 \log \left( \frac{k t\_a(P)}{1688 \phi r\_w^2} \right) + s' \right] \tag{54}$$

As for the case of pressure transient analysis, from a semilog plot of pseudopressure versus time (or pseudotime), its slope allows calculating the reservoir permeability and the intercept is used to find the pseudoskin factor, respectively:

$$k = \frac{1637.74T}{mh\ \Delta m(P)}\tag{55}$$

$$s' = \left[\frac{(1/q)\_{1hr}}{m} - \log\left(\frac{k}{\phi(\mu c\_t)\_i r\_w^2}\right) - 3.227 + 0.8686\right] \tag{56}$$

Considering approximation for large time to the analytical Laplace inversion of Eq. (49), the following expression is obtained:

$$q\_D = \frac{1}{\ln r\_{eD} - 0.75} \exp\left[\frac{-2t\_D}{r\_{eD}^2(\ln r\_{eD} - 0.75)}\right] \tag{57}$$

For tD ≥ tDpss, this flow period is known as the exponential decline period. tDpss is the time required for the development of true pseudosteady state at the producing well for constant rate production case. Eq. (57) concerns only the circular reservoir. The solution can be generalized for other reservoir shapes by using the Dietz shape factor [6], CA,

$$q\_D = \frac{2}{\ln\left(\frac{4A\_D}{\chi\mathcal{C}\_A}\right)} \exp\left[\frac{-4\pi t\_D}{A\_D \ln\left(\frac{4A\_D}{\chi\mathcal{C}\_A}\right)}\right] \tag{58}$$

where, AD (dimensionless area) and reD (dimensionless radius) are given by:

$$A\_D = \frac{A}{r\_w^2} \tag{59}$$

$$r\_{eD} = \frac{r\_e}{r\_w e^{-s}} = \frac{r\_e}{r\_{weff}}\tag{60}$$

Eq. (58) suggests that a plot of log(q) versus time will yield a straight line with negative slope Mdecline,

$$M\_{\text{decline}} = \frac{2(0.0002637) \text{k}}{r\_{eD}^2 (\ln r\_{eD} - 0.75) \phi \mu c\_l r\_w^2} \tag{61}$$

and intercept at (t = 0):

$$\eta\_{\rm int} = \frac{kh \Delta m(P)}{1637.74 B \mu (\ln r\_{cD} - 0.75)} \tag{62}$$

The reservoir area can be determined by solving the Eq. (62) for reD:

$$r\_{eD} = \exp\left(\frac{1637.74 \text{B}\mu}{kh \Delta m(P)(\ln r\_{eD} - 0.75)} q\_{\text{int}} + 0.75\right) \tag{63}$$

#### 3.3. TDS technique

Escobar et al. [9] extended the TDS Technique for gas well in homogeneous and naturally fractured formations using rigorous time. The equations they presented for wellbore storage coefficient and permeability are given below:

$$C = 0.4196 \frac{Tq \ t\_N}{\mu \ \Delta m(P)\_N} = 0.4198 \frac{T}{\mu \ \Delta m(P)} \left[ \frac{t}{t \* (1/q)'} \right]\_N \tag{64}$$

$$k = 711.5817 \frac{T}{h \, \Delta m(P) [t \times (1/q)']\_r} \tag{65}$$

Using a procedure similar to the pressure transient case, Escobar et al. [9] found an expression to estimate the pseudoskin factor:

$$s'=0.5\left\{\frac{(1/q)\_r}{[t\times(1/q)']\_r}-\ln\left(\frac{kt\_r}{\phi\,\mu\,c\_lr\_w^2}\right)+7.43\right\}\tag{66}$$

For the estimation of reservoir area, Escobar et al. [9] also presented an equation that uses the starting time of the pseudosteady-state period, tspss.

Gas Well Testing http://dx.doi.org/10.5772/67620 111

$$r\_{\varepsilon}^{2} = \left(\frac{0.0015k \, t\_{\text{sys}}}{\phi \mu \, c\_{t}}\right)^{1/2} \tag{67}$$

As treated in pressure transient analysis, Eq. (41), the reciprocal rate derivative takes a value of 0.5 during radial flow. The intercept of this with the reciprocal rate derivative of Eq. (57) will provide:

$$t\_{D\_{\eta i}} = \frac{1}{2} r\_{eD}{}^2 [\ln(r\_{eD}) - 0.75] \tag{68}$$

in which numerical solution gives:

$$r\_{eD} = 1.0292 \,\, t\_{D\_{\text{pri}}}^{0.4627} \,\, \, \, \, \tag{69}$$

After replacing the dimensionless quantities, we obtain:

$$r\_{\varepsilon} = 22.727 \times 10^{-3} r\_{\text{weff}} \left( \frac{k}{\phi \,\mu \, c\_l r\_{\text{weff}} \text{f}^2} \right)^{0.4627} t\_{\text{rpi}}^{0.4627} \tag{70}$$

Refs. [13] and [14] presented rate transient analysis for long homogeneous and naturally fractured oil reservoirs using TDS technique and conventional analysis, respectively. Equations can be easily translated to gas flow.

#### 3.4. Example 2

qD <sup>¼</sup> <sup>2</sup> ln <sup>4</sup>AD γCA

Mdecline,

and intercept at (t = 0):

110 Advances in Natural Gas Emerging Technologies

3.3. TDS technique

where, AD (dimensionless area) and reD (dimensionless radius) are given by:

� � exp �4πtD

AD <sup>¼</sup> <sup>A</sup> r2 w

Eq. (58) suggests that a plot of log(q) versus time will yield a straight line with negative slope

Mdecline <sup>¼</sup> <sup>2</sup>ð0:0002637Þ<sup>k</sup> r2

<sup>q</sup>int <sup>¼</sup> khΔmðP<sup>Þ</sup>

1637:74Bμ khΔmðPÞðln reD � 0:75Þ

Escobar et al. [9] extended the TDS Technique for gas well in homogeneous and naturally fractured formations using rigorous time. The equations they presented for wellbore storage

Using a procedure similar to the pressure transient case, Escobar et al. [9] found an expression

For the estimation of reservoir area, Escobar et al. [9] also presented an equation that uses the

<sup>¼</sup> <sup>0</sup>:<sup>4198</sup> <sup>T</sup>

hΔmðPÞ½t � ð1=qÞ

� ln ktr

( )

φ μ ctrw<sup>2</sup> � �

μΔmðPÞ

0 � r

þ 7:43

The reservoir area can be determined by solving the Eq. (62) for reD:

reD ¼ exp

<sup>C</sup> <sup>¼</sup> <sup>0</sup>:<sup>4196</sup> Tq tN

<sup>0</sup> <sup>¼</sup> <sup>0</sup>:<sup>5</sup> <sup>ð</sup>1=qÞ<sup>r</sup>

½t � ð1=qÞ

μ ΔmðPÞ<sup>N</sup>

<sup>k</sup> <sup>¼</sup> <sup>711</sup>:<sup>5817</sup> <sup>T</sup>

0 � r

coefficient and permeability are given below:

s

starting time of the pseudosteady-state period, tspss.

to estimate the pseudoskin factor:

rwe�<sup>s</sup> <sup>¼</sup> re rwef f

eDðln reD � 0:75Þφμctr<sup>2</sup>

� �

reD <sup>¼</sup> re

2 4

ADln <sup>4</sup>AD γCA � �

3

w

<sup>1637</sup>:74Bμðln reD � <sup>0</sup>:75<sup>Þ</sup> (62)

qint þ 0:75

t t � ð1=qÞ 0

N

� �

5 (58)

(59)

(60)

(61)

(63)

(64)

(65)

(66)

Escobar et al. [9] presented an example for a homogeneous bounded reservoir. Figure 5 and Table 2 present the reciprocal rate and reciprocal rate derivative versus rigorous time for this exercise. Other relevant data for this example are given below:

Figure 5. Reciprocal rate and reciprocal rate derivative for example 2—homogeneous bounded reservoir. After Ref. [9].


Find reservoir permeability, skin factor, and drainage radius for this example using the TDS Technique.


Table 2. Reciprocal rate, reciprocal rate derivative versus time data for example 2.

#### 3.4.1. Solution

The following characteristic points were read from Figure 5:


Eqs. (65), (66), and (70) are used to obtain permeability, skin factor, and drainage.

$$k = \frac{711.5817T}{h \Delta m(P)[t \times (1/q)']\_r} = \frac{711.5817(670)}{(80)(30976300)(7.293 \times 10^{-6})} = 26.37 \text{ mol}$$

$$s' = 0.5 \left\{ \frac{(5.76 \times 10^{-5})}{(7.293 \times 10^{-6})} - \ln \left( \frac{(25)(0.0472)}{(0.25)(0.0122)(0.00187)(0.3)^2} \right) + 7.43 \right\} = 0.68$$

$$\sigma\_c = 22.727 \times 10^{-3} (0.3) \left( \frac{(25)}{(0.25)(0.0122) \left(0.00187 \right)(0.3)^2} \right)^{0.4627} (0.6)^{0.4627} = 19.5 \text{ ft}$$

Notice that the results closely match the permeability and external reservoir radius as presented by Ref. [9].

Finally, it is worth to mention that nowadays, conventional shale-gas reservoirs have become very attractive in the oil industry. Then, their characterization via well test analysis is very important. Shale-gas reservoir is normally tested under constant well-flowing pressure conditions—transient rate analysis—then, the recent studies performed in Refs. [17] and [22] should be read. If such wells are tested under constant rate conditions—pressure transient analysis then the reader should refer to the works by Bernal et al. [3] and Escobar et al. [18].

### Nomenclature

Find reservoir permeability, skin factor, and drainage radius for this example using the TDS

h = 80 ft k = 25 md rw = 0.3 ft φ = 25% ct = 0.00187 psi�<sup>1</sup> μ<sup>g</sup> = 0.0122 md/cp γ<sup>g</sup> = 0.85 T = 670 R re = 30 ft (0.065 Ac)

3.11E-04 43189.24959 179751.1071 4.72E-02 17360.95346 135561.954 3.89E-04 40834.91569 162199.4397 6.09E-02 16811.48278 135417.1412 4.67E-04 39060.52002 163534.7645 7.58E-02 16365.45213 135450.8332 5.45E-04 37655.94848 160065.4877 0.094767643 15935.44127 134654.7122 7.01E-04 35537.19664 157149.8777 0.122175588 15464.07533 130033.9989 8.72E-04 33849.9472 154708.1815 0.152075165 15058.4209 120446.6934 1.07E-03 32361.46026 152593.8776 0.176991479 14766.60107 110524.7186 1.35E-03 30871.32659 150543.0499 0.224830802 14266.98732 91414.11215 1.66E-03 29601.36073 148839.1511 0.284629956 13699.34978 71972.81156 1.97E-03 28617.55908 147555.5061 0.354395635 13079.44168 55862.58323 2.36E-03 27646.46771 146325.5009 0.450074281 12279.66271 41400.85561 2.80E-03 26785.03603 145257.0868 0.569672588 11348.65545 30197.97654 3.42E-03 25823.21027 144105.0598 0.709203946 10349.64392 22079.63438 4.04E-03 25068.84401 143224.7896 0.900561238 9119.41797 15300.5905 4.73E-03 24398.17164 142461.4954 1.139757852 7783.903591 10316.16432 5.54E-03 23753.19888 141745.6607 1.315168702 6930.721305 7965.441894 6.63E-03 23056.85049 140996.5214 1.522472435 6043.090594 6009.327954 7.87E-03 22423.58559 140330.625 1.801535152 5027.292996 4239.056471 9.12E-03 21908.28203 139803.0353 2.120463971 4077.321632 2935.619806 1.15E-02 21141.51732 139045.8455 2.630750081 2923.310968 1712.191843 1.49E-02 20332.18915 138280.5426 3.28455416 1917.14852 909.5234175 1.86E-02 19683.45322 137688.6861 4.241340618 1040.993708 385.7970016 2.34E-02 19065.59817 137131.4759 5.261912839 543.216581 162.6885651 3.02E-02 18406.42719 136508.2532 6.569520997 230.4829168 41.88651512 3.77E-02 17872.96521 135981.9447 9.12095155 27.98082076 0.548915189

, MSCF�<sup>1</sup> t, hr 1/q, MSCF�<sup>1</sup> t\*(1/q)

0 , MSCF�<sup>1</sup>

0

/cp ΔP = 580 psi

Technique.

Δm(P) = 340920304.2 psi<sup>2</sup>

t, hr 1/q, MSCF�<sup>1</sup> t\*(1/q)

112 Advances in Natural Gas Emerging Technologies

3.4.1. Solution

The following characteristic points were read from Figure 5:

Table 2. Reciprocal rate, reciprocal rate derivative versus time data for example 2.



### Greek


### Suffices



### Author details

PD Dimensionless pressure Pwf Well-flowing pressure, psi q Gas flow rate, MSCF

114 Advances in Natural Gas Emerging Technologies

r Radius, ft

t Time, hr

Greek

Δ Change, drop φ Porosity, fraction

γ <sup>g</sup>Gas gravity λ Mobility, md/cp μ Viscosity, cp

Suffices

1 hr One hour

e External eff Effective g Gas

D Dimensionless

rw Radio del pozo, ft

sa Total skin factor

1/q Reciprocal of the flow rate, D/Mscf

s , ft

re External reservoir radius, ft

rweff Effective wellbore radius, rwe

tDpss Exponential decline period t\*(1/q)' Reciprocal rate derivative, D/Mscf tD\*(1/qD)' Dimensionless reciprocal rate derivative

tp Horner or producing time tpss Exponential decline period, hr

α Turbulence factor or inertial factor

γ Euler's constant—1.781 or e0.5772

cr Condition at critical point

DA Dimensionless referred to drainage area Da Dimensionless referred to pseudotime

De Dimensionless referred to external

s' Apparent or pseudoskin factor

tD\*PD' Dimensionless pressure derivative

tspss Time to initiate pseudosteady state, hr u Argument for a Bessel function Z Gas supercompressibility factor

Freddy Humberto Escobar

Address all correspondence to: fescobar@usco.edu.co

Universidad Surcolombiana, Colombia

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