# Micro-continuum Framework for Pore-Scale Multiphase Fluid Transport in Shale Formations

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## Abstract

A micro-continuum simulation framework is proposed to study the complex pore-scale dynamics associated with hydrocarbon recovery from shale gas. The model accounts for the presence of immiscible fluid phases and for transport mechanisms in the nanoporous structures including slip flow, adsorption, surface and Knudsen diffusion. We employ the concept of sub-grid models to simulate the transport phenomena in shale gas. Specifically, we use high-resolution FIB–SEM images that provide information on the spatial distribution of the minerals, resolved pore space, and sub-resolution porous regions. The model is used to investigate several production scenarios at the pore-scale. In one setting, the organic matter is in direct contact with a micro-crack; in the other setting, clay regions are sandwiched between the organic matter and the “open” crack. The simulations show that it is important to account for the presence of multiple immiscible fluid phases because they can play a critical role in hydrocarbon production from shale-gas formations both in terms of production rate and in terms of residual mass of hydrocarbon. Moreover, we show that, because of wettability conditions, the rate of hydrocarbon recovery, as well as the ultimate recovery, depends strongly on the spatial distribution of the kerogen and clay in the vicinity of the micro-cracks.

## Keywords

Micro-continuum Source rocks Two-phase Pore-scale Adsorption## Abbreviations

- \(\bar{\mathsf {v}}\)
Single-field velocity (\(\hbox {m/s}\))

- \(\bar{\mathsf {v}}_r\)
Compression velocity of the gas/liquid interface (\(\hbox {m/s}\))

- \(\bar{C}_{A}\)
Single-field concentration for species

*A*(\(\hbox {kg/m}^{3}\))- \(\bar{p}\)
Single-field pressure (\(\hbox {Pa}\))

- \(\varvec{n}_{\mathrm{wall}}\)
Normal vector to the solid surface

- \(\varvec{t}_{\mathrm{wall}}\)
Tangent vector to the solid surface

- \(\chi _s\)
Indicator function for phase

*s*- \(\Delta P\)
Depletion pressure, \(\Delta P = P^* - P_0\) (\(\hbox {Pa}\))

- \(\hat{\varvec{n}}_{\mathrm{lg}}\)
Normal vector to water/gas interface

- \(\lambda \)
Mean free path (\(\hbox {m}\))

- \(\varvec{\Phi }_{A}\)
Continuous species transfer function (\(\hbox {kg/m}^2\))

- \(\mathsf {F}_A\)
Mass flux of species

*A*(\(\hbox {kg/m}^2/\hbox {s}\))- \(\mathsf {F}_c\)
Surface tension forces (\(\hbox {kg/m}^2/\hbox {s}^2\))

- \(\mu \)
Single-field viscosity (\(\hbox {kg/m/s}\))

- \(\mu _\mathrm{g}\)
Gas viscosity (\(\hbox {kg/m/s}\))

- \(\mu _\mathrm{l}\)
Water viscosity (\(\hbox {kg/m/s}\))

- \(\phi \)
Porosity

- \(\phi _s\)
Porosity of phase

*s*- \(\rho \)
Single-field fluid density (\(\hbox {kg/m}^{3}\))

- \(\rho _\mathrm{g}\)
Gas density (\(\hbox {kg/m}^{3}\))

- \(\rho _\mathrm{l}\)
Water density (\(\hbox {kg/m}^{3}\))

- \(\sigma \)
Surface tension (\(\hbox {kg/s}^{2}\))

- \(\theta \)
Contact angle

- \(A_e^s\)
Specific surface area of phase

*s*(\(\hbox {m}^{-1}\))- \(C_{\mathrm{g}}\)
Cumulative mass of gas produced (\(\hbox {kg}\))

- \(D_A\)
Single-field diffusivity of species

*A*(\(\hbox {m}^2/\hbox {s}\))- \(d_A\)
Diameter of molecules

*A*(\(\hbox {m}\))- \(D_{g,A}\)
Diffusivity of species

*A*in gas (\(\hbox {m}^2/\hbox {s}\))- \(D_{{Kn}}\)
Knudsen diffusion coefficient (\(\hbox {m}^2/\hbox {s}\))

- \(D_{l,A}\)
Diffusivity of species

*A*in water (\(\hbox {m}^2/\hbox {s}\))- \(D_{s,A}\)
Surface diffusion coefficient (\(\hbox {m}^2/\hbox {s}\))

*k*Permeability field, \(k=\sum _s \chi _s k_s\) (\(\hbox {m}^2\))

- \(k_0\)
Permeability related to the nanopores geometry (\(\hbox {m}^2\))

- \(k_\mathrm{B}\)
Boltzmann constant (\(k_\mathrm{B}=1.38\times 10^{-23}\hbox {J/K}\))

- \(k_\mathrm{r}\)
Single-field relative permeability

- \(k_s\)
Permeability of phase

*s*(\(\hbox {m}^2\))- \(k_{\mathrm{nano}}\)
Permeability correction due to slip and nanoscale effects

- \(k_{\mathrm{r,g}}\)
Relative permeability to gas

- \(k_{\mathrm{r,l}}\)
Relative permeability to water

*Kn*Knudsen number

- \(M_A\)
Molar mass of species

*A*(\(\hbox {g/mol}\))- \(m_{\mathrm{ads}}\)
Mass of gas adsorbed on the surface of the pores (\(\hbox {kg}\))

- \(m_{\mathrm{tot,gas}}\)
Total mass of gas in the system (\(\hbox {kg}\))

- \(m_{\mathrm{vol}}\)
Mass of gas contained in the volume of the pores (\(\hbox {kg}\))

- \(P^*\)
Initial pressure in the domain (\(\hbox {Pa}\))

- \(P_0\)
Pressure at the outlet (\(\hbox {Pa}\))

- \(p_c\)
Capillary pressure (\(\hbox {Pa}\))

- \(P_L^s\)
Pressure at which half of the adsorbed molecules is released (\(\hbox {Pa}\))

- \(q^s_{\mathrm{ads},A}\)
Quantity of molecules

*A*adsorbed per unit of surface (\(\hbox {kg/m}^{2}\))- \(q_L^s\)
Adsorption capacity (\(\hbox {kg/m}^{2}\))

- \(Q_{g}\)
Gas mass flow rate at the outlet (\(\hbox {kg/s}\))

*r*Pore radius (\(\hbox {m}\))

- \(R_A^s\)
Desorption rate of molecules A (\(\hbox {kg/m}^{2}/\hbox {s}\))

- \(R_{\mathrm{g}}\)
Ideal gas constant (\(R_\mathrm{g}=8.314\hbox {J/mol/K}\))

- \(S_{\mathrm{l}}\)
Water saturation

*T*Temperature (\(\hbox {K}\))

*x*Percentage of the mass of methane adsorbed

## Notes

### Acknowledgements

We acknowledge the TOTAL STEMS project for financial support and Isabelle Jolivet and Regis Lasnel from TOTAL BGM for the FIB–SEM dataset. We thank the Stanford Center for Computational Earth & Environmental Sciences (CEES) for computational support.

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