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Influence of Considering the Sorption Effect in the Betti-Maxwell Reciprocal Theorem on Gas Transport Capacity in Unconventional Reservoirs

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Abstract

Pore bulk modulus in continuum apparent permeability models was generally treated as an independent parameter or deduced from the Betti-Maxwell reciprocal theorem neglecting the sorption effect, which is not appropriate for the unconventional reservoirs with strong gas sorption. In this work, pore bulk modulus was firstly derived from the Betti-Maxwell theorem with the consideration of the gas sorption strain and was incorporated into the solid–fluid coupling apparent permeability model. Numerical simulation was then conducted under the uniaxial strain condition and the stress confined condition to investigate the gas transport characteristics. The results showed that when taking into account for the sorption effect in the Betti-Maxwell theorem, the pore bulk modulus dramatically decreased with the falling pore pressure during the gas transport process under two loading conditions. Moreover, gas transport ability under the stress confined condition was greatly degraded while it was hardly changed under the uniaxial strain condition. Both the pore bulk modulus and the stress distribution in the reservoirs were found responsible for the variation of gas flow capacity. Finally, the effects of the parameters associated with the pore bulk modulus were examined to further discuss the impact of involving the sorption effect in the Betti-Maxwell theorem.

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Abbreviations

α :

Biot's coefficient of the bulk medium

α p :

Biot's coefficient of the pores

β :

Dimensionless rarefaction coefficient

C admax :

Maximum concentration of adsorbed gas

D ad0 :

Surface diffusion coefficient

E :

Young's modulus of the bulk medium

ε ad :

Sorption strain of the bulk medium

\(\varepsilon_{{{\text{ad}}}}^{{\text{p}}}\) :

Sorption strain of the pores

ε L :

Langmuir volumetric strain

ε :

Strain tensor

ε v :

Volumetric strain of the bulk medium

\(\varepsilon_{{\text{v}}}^{{\text{p}}}\) :

Volumetric strain of the pores

η :

Ratio of the sorption strains

f i :

Components of body force

ϕ :

Pore porosity

ϕ 0 :

Initial pore porosity

J :

Gas mass flux per volume of bulk medium

K :

Bulk modulus of the bulk medium

K p :

Bulk modulus of the pores

K s :

Bulk modulus of solid grain

K n :

Knudsen number

k app :

Apparent permeability

\(k_{\infty }\) :

Intrinsic permeability

\(k_{\infty 0}\) :

Initial intrinsic permeability

M g :

Molar mass of gas molecules

μ :

Gas viscosity

ν :

Poisson's ratio of the bulk medium

p :

Pore pressure

p 0 :

Initial pore pressure

p w :

Outer gas pressure on the boundary

p c :

External confining stress

P L :

Langmuir pressure constant

Q 0 :

External gas resource

R :

The universal gas constant

ρ :

Gas mass per volume of bulk medium

ρ g :

Gas density

ρ ga :

Gas density under standard conditions

ρ b :

Density of the bulk medium

σ :

Stress tensor

σ kk :

First stress invariant

\(\overline{\sigma }\) :

Mean compressive stress

T :

Temperature of the stratum

τ h :

Tortuosity factor

u i :

Components of displacement

V :

Total volume

V p :

Pore volume

V L :

Langmuir volume constant

ω bu :

Weight of free gas flow

ω ad :

Weight of adsorbed gas flow

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Acknowledgements

The authors would like to thank the financial supports from the National Natural Science Foundation of China (Grant No. 11872324) and Education Department of Sichuan Province of China (Grant No. 18ZA0491).

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Authors and Affiliations

  1. School of Manufacturing Science and Engineering, Key Laboratory of Testing for Manufacturing Process, Ministry of Education, Southwest University of Science and Technology, Mianyang, 621010, Sichuan, People’s Republic of China

    Qin Ying, Jing Zhao, Wenbo Wu & Bin Gu

  2. School of Civil Engineering and Architecture, Hainan University, Haikou, 570228, Hainan, People’s Republic of China

    Hongbin Zhang

  3. Shock and Vibration of Engineering Materials and Structures, Key Laboratory of Sichuan Province, School of Manufacturing Science and Engineering, Southwest University of Science and Technology, Mianyang, 621010, Sichuan, People’s Republic of China

    Bin Gu

  4. Institute of Biomechanics and Medical Engineering, Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, People’s Republic of China

    Xi-Qiao Feng

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  1. Qin Ying
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Ying, Q., Zhang, H., Zhao, J. et al. Influence of Considering the Sorption Effect in the Betti-Maxwell Reciprocal Theorem on Gas Transport Capacity in Unconventional Reservoirs. Transp Porous Med 137, 451–469 (2021). https://doi.org/10.1007/s11242-021-01565-7

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