Abstract
The highly compressible fluid flow through a three-scales rigid porous medium (pore, fracture, macroscopic sample) is investigated using a homogenization method. The macroscopic description is strongly dependent on the separation of the different scales, and three cases are considered. The pores either play the role of a compressible fluid reservoir, introduce a memory effect, or are ignored, respectively. The homogenization result is compared to classical phenomenological models that are available in the case of slightly compressible fluids. Pseudo-steady state models are shown to give a rough description of the phenomenon.
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Abbreviations
- C*:
-
gas compressibility coefficient
- f:
-
subscript for the fractures
- F :
-
nonlinear function
- h :
-
positive integer
- ĩ :
-
identity tensor
- k :
-
subscript taking on the values p for the pores and f for the fractures
- \(\tilde k_{{\text{p,}}} \tilde k_{\text{f}} \) :
-
particular solutions for the velocity fields in the pores and the fractures, respectively
- \(\tilde K_{{\text{p,}}} \tilde K_{\text{f}} \) :
-
pore and fracture permeability tensors, respectively
- l,l′,l″ :
-
characteristic lengths for the pore scale, the fracture scale and the macroscopic medium, respectively
- m :
-
positive integer
- n,n′ :
-
pore porosity and fracture porosity, respectively
- n′ :
-
normal unit vector
- p:
-
subscript for the pores
- P 0 :
-
initial pressure
- P p,P f :
-
pore and fracture pressures, respectively
- q :
-
interporosity flow
- Q :
-
dimensionless number
- r :
-
positive integer
- s :
-
characteristic coefficient of a fractured rock
- S :
-
Strouhal number
- t,T :
-
time variables for the pores and the fractures, respectively
- T p,T f :
-
characteristic times for the pores and the fractures, respectively
- u p,u f :
-
pore and fracture fluid displacements, respectively
- v p,v f :
-
pore and fracture fluid velocities, respectively
- v p k :
-
order of magnitude ofV k , due to the macroscopic pressure gradient
- v t k :
-
order of magnitude ofv k , due to the temporal change of pressure
- x, x′, x″ :
-
space variables for the pore, fracture and macroscopic scales, respectively
- α,Β,γ :
-
ratios between the different characteristic lengths
- γ, γ′ :
-
boundaries of the pores and the fractures, respectively
- δ:
-
Laplace operator
- ∇:
-
gradient operator
- ε :
-
small parameter
- λ,Μ :
-
fluid viscosities
- ρ0:
-
initial density
- ρ p,ρf:
-
pore and fracture densities, respectively
- Τ :
-
particular solution for the pressure
- Τ p,Τf:
-
characteristic times for the pores and the fractures, respectively
- Ω :
-
pulsation
- Ω, Ω′:
-
periods at the pore and fracture scales, respectively
- Ωp, Ω′sp, Ω′f :
-
parts of the period occupied by the pores, the solid plus the pores and the fractures
- 〈Φ〉Ω, 〈Φ〉Ω′ :
-
volume averages of the quantityΦ on Ω, Ω′, respectively
- 〈Φ〉eff :
-
particular volume average on Ω′
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Royer, P., Auriault, JL. Transient quasi-static gas flow through a rigid porous medium with double porosity. Transp Porous Med 17, 33–57 (1994). https://doi.org/10.1007/BF00624049
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DOI: https://doi.org/10.1007/BF00624049