# Similarity Analysis in Scaling a Gas Hydrates Reservoir

^{1}

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## Abstract

**:**

## 1. Introduction

^{16}−2.1 × 10

^{16}m

^{3}, which is twice the amount of carbon to be found in all known fossil fuels (coal, oil, and natural gas) on Earth.

## 2. Experimental Section

#### 2.1. Experimental Apparatus

**Figure 2.**Distributions of temperature, resistance measuring points and production wellhead of each layer within the three-dimensional reactor.

_{I}wellhead in the layer C along the centerline of the reactor, and the outlets for the gas and water production are the V

_{1}–V

_{4}wellheads in the layer A.

#### 2.2. Experimental Method

_{i}= 130 °C) in the pre-heater. After preheating, the hot water is injected through the inlet valve with the hot water injection rate (q

_{i}= 40 mL/min). The outlet valves of the wells open simultaneously, and then the gas production starts. The gas production pressure controlled by the back-pressure regulator keeps steady at 6.5 MPa. After more than 1 h hot water injection process, the rate of the gas production drops to approximately 0. We believe that no more hydrate decomposition in the CHS. Finally, the system is closed for more than 2 h, and then the system pressure starts to drop to atmosphere gradually. During these processes, the temperatures and pressures in the vessel, the gas production rate, the water injection/production rates are recorded at 10 s intervals.

## 3. Scaling Criteria

#### 3.1. Mathematical Model

_{g}and K

_{w}are the permeabilities of the gas and water, respectively; P

_{g}and P

_{w}are pressures of the water and gas, respectively; µ

_{g}and µ

_{w}are the viscosities of gas and water, respectively; ρ

_{g}, ρ

_{w}and ρ

_{h}are the densities of gas, water, and hydrate, respectively; s

_{g}, s

_{w}, and s

_{h}are the saturations of gas, water, and hydrate, respectively; ${\dot{m}}_{g}$, ${\dot{m}}_{w}$, and ${\dot{m}}_{h}$ are the masses of gas, water, and hydrate formation or dissociation; q

_{g}and q

_{w}are the boundary changes of gas and water, which can be expressed as follows:

_{p}and y

_{p}are the coordinates of the production well, respectively; x

_{I}, y

_{I}are the coordinates of the injection well, respectively; P

_{gp}and P

_{wp}are the production pressures of gas and water, respectively; r

_{0}is the well radius; r

_{e0}is the effective radius of well; q

_{I}is the rate of the water injection, H is the thickness of the hydrate reservoir. The power-law model [15] Equation (6) is used to describe the local absolute permeability:

_{0}is the total porosity; K

_{0}is the maximum absolute permeability corresponding to ø

_{0}; ø

_{e}is the effective porosity defined as ø

_{e}= ø

_{0}(1−S

_{h}); β is the index parameter.

_{w}+ S

_{g}+ S

_{k}= 1

_{g}, h

_{w}, h

_{h}and h

_{r}are the specific heats of gas, water, hydrate, and rock, respectively; λ

_{g}, λ

_{w}, λ

_{h}, and λ

_{r}are conductivity coefficients of gas, water, hydrate, and rock, respectively; T is the temperature; T

_{I}is the temperature of the injected water; t is the time; ∆H is the enthalpy change of hydrate decomposition and can be expressed as follows [16]:

^{3}

_{w}and M

_{g}are the molecular weight of the water and gas, respectively; N

_{h}is the coefficient of dissociation reaction.

_{eq}are the local gas fugacity and the equilibrium gas fugacity which are usually replaced by local gas pressure P

_{g}and P

_{eq}; k

_{d}is the dissociation constant; A

_{s}is the specific surface area of porous media bearing gas hydrate, which is calculated as follows:

_{eq}is the equilibrium temperature of gas hydrate; P

_{eq}is the equilibrium pressure of gas hydrate; T

_{0}is 273.15 K; a, b, c are constant.

#### 3.2. Calculation

_{D}, y

_{D}, z

_{D}, and t

_{D}are the dimensionless coordinates and dimensionless time, respectively; L, W, and H are the length, width, and thickness of the hydrate reservoir, respectively; S

_{hi}is the initial saturation of hydrate.

_{rwg}is the effective permeability of the gas with the irreducible water; K

_{rgw}is the effective permeability of the water with the residual gas; s

_{rg}and s

_{rw}are the saturations of the residual gas and residual water, respectively.

- π
_{1}and π_{2}are the dimensionless permeabilities of water and gas, respectively; - π
_{3}is the dimensionless absolutely permeability; - π
_{4}–π_{11}are the similarities of geometry, well position, and well radius, respectively; - π
_{12}–π_{14}are the density ratios of hydrate to gas, rock to gas, and water to hydrate, respectively; - π
_{15}–π_{17}are the conductivity coefficient ratios of gas to water, hydrate to water, and rock to water, respectively; - π
_{18}–π_{20}are the specific heat ratios of gas to water, hydrate to water, and rock to water, respectively; - π
_{21}is the dimensionless dissociation heat of hydrate; - π
_{22}and π_{23}are the ratios of hydrate equilibrium pressure to gas production pressure, and initial gas pressure to gas production pressure, respectively; - π
_{24}is the dimensionless temperature in the hydrate reservoir; - π
_{25}is the dimensionless injection temperature; - π
_{26}is the dimensionless initial gas saturation; - π
_{27}is the initial hydrate saturation; - π
_{28}and π_{29}are the saturation of residual gas and residual water, respectively; - π
_{30}is the total porosity; - π
_{31}is the mobility ratio of the water in residual gas and the gas in residual water; - π
_{32}is the amount ratio of gas flow per unite area to gas production per unite area in hydrate sediment; - π
_{33}is the ratio of conduction heat to hydrate dissociation heat per unite time; - π
_{34}is the ratio of capillary force to gas production pressure; - π
_{35}is the amount ratio of water flow per unite area to water production per unite area in hydrate sediment; - π
_{36}is the dimensionless gravity.

## 4. Results and Discussion

#### 4.1. Production Process

**Figure 3.**Cumulative volumes of produced gas/water and injected water during hydrate dissociation with thermal stimulation method.

**Figure 4.**Three-dimensional spatial temperature distributions during the hydrate dissociation with thermal stimulation method. (

**a**) 0 min; (

**b**) 25 min; (

**c**) 50 min; (

**d**) 100 min.

#### 4.2. Similar Model

_{i}, equilibrium pressure P

_{eq}, maximal porosity ø

_{0}, initial pressure p

_{i}, hydrate density ρ

_{h}, and initial hydrate saturation s

_{hi}. According to the scaling criteria, these parameters should keep the same in either the model or the prototype.

_{32}, π

_{35}, and the non-dimensional time t

_{D}, some important physical parameters (K

_{0}, q

_{I}, and t) ratio existing between the model and the prototype can be obtained. The detail calculation is shown as follows:

_{m}/L

_{f}= H

_{m}/H

_{f}= W

_{m}/W

_{f}.

_{32}for the model and prototype are same, which can be expressed as:

Parameters | L/m | H/m | W/m | q_{I} /m^{3}s^{−1} | r_{0}/m | S_{h} |
---|---|---|---|---|---|---|

CHS | 0.18 | 0.18 | 0.18 | 6.7 × 10^{−7} | 2 × 10^{−3} | 0.310 |

Scaling model | 18 | 18 | 18 | 3.1 × 10^{−2} | 1 × 10^{−1} | 0.310 |

Parameters | T_{I}/°C | ø _{0} | P_{gp}/MPa | g/ms^{−2} | t/min | Q/m^{3} |

CHS | 130 | 0.46 | 6.5 | 9.8 | 100 | 0.152 |

Scaling model | 130 | 0.46 | 6.5 | 9.8 | 2154 | 1.52 × 10^{5} |

^{−2}m

^{3}/s for 2154 min, leading to the final cumulative volume of the produced gas of 1.52 × 10

^{5}m

^{3}, which means that the final cumulative volume of the produced gas is enlarged 10

^{6}times. The experimental results and the scaling criteria could be used to evaluate the hydrate dissociation strategies and the gas production potential of the hydrate reservoir.

## 5. Conclusions

^{−2}m

^{3}/s for 2154 min in the enlarged hydrate reservoir, and the final cumulative volume of the produced gas is 1.52 × 10

^{5}m

^{3}.

## Nomenclature

## Abbreviation

CHS | Cubic Hydrate Simulator |

## Symbols

V | wellhead |

q_{i} | rate of hot water (mL/min) |

T_{i} | temperature of hot water (K) |

x, y, z | coordinates |

ø | porosity |

ø_{0} | total porosity |

ø_{e} | effective porosity |

s | saturation |

P | pressure (MPa) |

µ | viscosity (Pa S) |

ρ | density (kg m ^{−3}) |

K | permeability (m ^{−2}) |

K_{0} | maximum absolute permeability (m ^{−2}) |

$\dot{m}$ | mass rate (m ^{3} s^{−1}) |

h | specific heat (J kg ^{−1}K^{−1}) |

λ | conductivity coefficient (w m ^{−1} K^{−1}) |

q | heat changes on boundary (J) |

g | the gravitational acceleration (m s ^{−2}) |

x_{p}, y_{p} | coordinates of the production well (m) |

x_{I}, y_{I} | coordinates of the injection well (m) |

r_{0} | well radii (m) |

r_{e0} | effective radii of well (m) |

∆H | enthalpy change of hydrate decomposition (J) |

M | molecular weight |

N_{h} | coefficient of dissociation reaction (5.8) |

A_{s} | specific surface area of porous media (m ^{2}) |

f | gas fugacity (Pa) |

k_{d} | the dissociation constant |

L | length (m) |

H | thickness (m) |

W | width (m) |

Q | volume of gas production (m ^{3}) |

σ | gas throttle coefficient |

## Subscripts

i | initial |

p | production |

g | gas |

w | water |

h | hydrate |

r | rock |

eq | phase equilibrium |

D | dimensionless |

m | model |

f | prototype |

## Acknowledgments

## Conflict of Interest

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**MDPI and ACS Style**

Wang, Y.; Xu, C.-G.; Li, X.-S.; Li, G.; Chen, Z.-Y.
Similarity Analysis in Scaling a Gas Hydrates Reservoir. *Energies* **2013**, *6*, 2468-2480.
https://doi.org/10.3390/en6052468

**AMA Style**

Wang Y, Xu C-G, Li X-S, Li G, Chen Z-Y.
Similarity Analysis in Scaling a Gas Hydrates Reservoir. *Energies*. 2013; 6(5):2468-2480.
https://doi.org/10.3390/en6052468

**Chicago/Turabian Style**

Wang, Yi, Chun-Gang Xu, Xiao-Sen Li, Gang Li, and Zhao-Yang Chen.
2013. "Similarity Analysis in Scaling a Gas Hydrates Reservoir" *Energies* 6, no. 5: 2468-2480.
https://doi.org/10.3390/en6052468