# A criterion for evaluating the effect of shale-matrix dual-continuum flow on gas production

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## Abstract

Because of the large range of pore size distribution and the chemical/mineralogical heterogeneity within shale matrix, gas flow in it may exhibit dual-continuum flow behavior even though rock matrix is not fractured. The relative importance of dual-continuum flow has considerable implications to the shale reservoir characterization and gas production. A matrix-dominant gas flow model is developed to evaluate the impacts of dual-continuum gas flow on gas production. The simulation results reveal three regimes with regard to the relative importance of the dual-continuum flow. When the mass transfer coefficient between the mobile and immobile continua is relatively high, the dual-continuum behavior becomes insignificant within the context of production rate prediction. When the mass transfer coefficient is very small, the dual continuum gas flow behavior also becomes insignificant because gas flow between the two continua is essentially negligible. Thus, the dual-continuum gas flow behavior becomes important to the long-term gas production rate only for the intermediate range of mass transfer coefficient that is shown to depend on porosity and gas adsorption properties. Based on the physical reasoning and numerical simulation results, we propose a criterion to determine the relative importance of the dual-continuum gas flow. The criterion is expressed with a dimensionless parameter that combines several parameters including the mass transfer coefficient, gas viscosity and densities and matrix porosity. The numerical simulation results show that when the dimensionless parameter is less than a critical value, the dual-continuum gas flow becomes important under a variety of different conditions.

## Keywords

Shale gas Unconventional reservoir Dual-continuum flow Gas flow Stress-dependent permeability## List of symbols

*A*_{0}A constant in Eq. (16)

*A*_{F}Fracture surface area (L

^{2})*a*_{v}Interfacial areas between mobile and immobile continua within a unit volume of bulk rock matrix (L

^{−1})*α*Dimensionless rarefaction coefficient in Eq. (16)

*α*_{0}A constant in Eq. (16)

- \(\alpha_{k}\)
The pressure-sensitivity factor for permeability

- \(\alpha_{\phi }\)
The pressure-sensitivity factor for porosity

*B*_{0}A constant in Eq. (16)

*B*Apparent mass transfer coefficient defined in Eq. (10) (T

^{−1})*B**Mass transfer coefficient defined in Eq. (8) (L

^{2}T)*d*_{F}Average spacing of fractures that have significantly higher permeability values than the matrix (L)

*f*_{c}A factor that defined in Eq. (13)

*k*_{B}Boltzmann constant

*K*_{n}Knudsen number

*k*_{i}Local gas permeability in the immobile continuum (L

^{2})*k*_{m}Matrix permeability (L

^{2})*k*_{0}Matrix permeability under the initial reservoir pressure (L

^{2})*L*_{i}Characteristic length of immobile packets (L)

*M*Molecular mass

*p*Gas pressure (M L

^{−1}T^{−2})*p*_{i}Pore pressure in the immobile continuum (M L

^{−1}T^{−2})*p*_{m}Pore pressure in the mobile continuum (M L

^{−1}T^{−2})- ∆
*p*_{m} Pressure difference between the current reservoir pore pressure and the initial reservoir pore pressure for the mobile continuum (M L

^{−1}T^{−2})*p*_{L}Gas pressure when \(\rho_{a,i} = \frac{{\rho_{L} }}{2}\) (M L

^{−1}T^{−2})*p*_{m}Pressure in the mobile continuum (M L

^{−1}T^{−2})*p*_{i}Pressure in the immobile continuum (M L

^{−1}T^{−2})*p*_{well}Pressure in the horizontal well (M L

^{−1}T^{−2})*p*_{0}Initial reservoir pressure, (M L

^{−1}T^{−2})*π*_{c}A dimensionless parameter defined in Eq. (24)

*q*_{M}Gas mass flux at the fracture-matrix interface (M L

^{−3}T^{−1})*q*_{im}Gas mass transfer rate (per unit volume of the porous medium) from the immobile continuum to the mobile continuum (M L

^{−3}T^{−1})*R*Universal gas constant in Eq. (11)

*r*Pore throat radius (L)

*t*Time (T)

- T
Reservoir temperature

*x*Spatial coordinate and has zero value at the fracture-matrix interface (L)

*μ*_{m}Gas viscosity in the mobile continuum (M T

^{−1}L^{−1})*μ*_{i}Gas viscosity in the immobile continuum (M T

^{−1}L^{−1})*ρ*Gas density (M L

^{−3})*ρ*_{i}Free-gas density in the pores of immobile continuum (M L

^{−3})*ρ*_{m}Gas density in the mobile continuum (M L

^{−3})*ρ*_{L}Adsorbed gas density for \(\rho_{i} \to \infty\) (M L

^{−3})*ρ*_{a,i}Adsorbed gas density in the immobile continuum (or the mass of adsorbed gas divided by the bulk volume of the porous medium) (M L

^{−3})- \(\phi_{i}\)
Porosity for the immobile continuum

- \(\phi_{\text{m}}\)
Porosity for the mobile continuum

- \(\phi_{\text{t}}\)
Total porosity including both mobile and immobile continuum

- \(\phi_{\text{h}}\)
Portion of the porosity that is not stress sensitive

- \(\phi_{\text{s}}\)
Stress-sensitive portion of the matrix porosity under the ambient condition

- λ
Mean free path

- z
Gas compressibility factor

## Notes

### Compliance with ethical standards

### Conflict of interest

All authors declares that they have no conflict of interest.

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