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Deterministic/Probabilistic Model as Strategy to Study Nanofluid Transport in Porous Media

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Abstract

We developed a model for simulating scalar transport and non-equilibrium interfacial mass transfer in porous media based on a hybrid probabilistic/deterministic approach. The probabilistic model formulation accounts for different mass transfer mechanisms, such as interfacial mass transfer and attachment/detachment phenomena, occurring under equilibrium or non-equilibrium conditions. Mass transport equations are solved using both finite volume method (FVM) and stochastic particle method (SPM). Specifically, the SPM allows to solve the probabilistic component of the hybrid method. The impact of the number of particles and the mesh size cell for computing the ensemble average is analyzed in this work. The core flooding setup of an inert tracer is initially simulated and compared to experimental data reported in the literature displaying a good agreement. Values of root mean square less than 0.088 were obtained for all the cases studied. Besides, the non-equilibrium mass transfer capabilities of the model are appraised by simulating the injection of a nanoparticle dispersion in the core and comparing the simulation results with reported experimental data. The probabilistic model shows advantages with respect to the deterministic description at localization of “sharp” profile or high gradients and reduction in complexity of the transport equation described by SPM, allowing to obtain additional information such as the standard deviation of the field scalar variables of the transport process, which is directly related to equilibrium/non-equilibrium state of the system.

Article Highlights

  • The stochastic particle model is extended to consider compressible particles, making the description of the phases more realistic.

  • The model is applied to the description of nanoparticle transport considering additional phenomena.

  • This method allows one to describe the deterministic transport model in a simpler way.

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Abbreviations

ADE:

Advection–diffusion equation

D:

Dimension

EOR:

Enhanced oil recovery

Eq.:

Equation

FVM:

Finite volume method

IMPIS:

Implicit pressure, implicit saturation

MBE:

Mass balance equation

PDF:

Probability density function

REV:

Representative elementary volume

RMSNV:

Root mean square normalized values

SPM:

Stochastic particle method

St. Dev.:

Standard deviation

A :

Cross-flow area

B :

Formation volumetric factor

b :

Random number

C :

Component concentration

\(c_{{\text{t}}}\) :

Total compressibility

D :

Effective dispersion coefficient

Dx:

Cell size

\({\text{d}}{\mathcal{F}}_{\alpha li}\) :

Differential of Darcian flux

\({\text{d}}{\mathcal{F}}_{\alpha li}^{\prime }\) :

Differential of dispersion flux

\({\text{d}}\Gamma\) :

Surface differential

\({\varvec{k}}\) :

Permeability

\(k_{{\text{r}}}\) :

Relative permeability

\(k_{c}\) :

Mixing model parameter

\(k_{{{\text{ra}}}}\) :

Retention parameter at the site 2

\(k_{{{\text{rd}}}}\) :

Mobilization parameter at the site 2

\(k_{{{\text{irr}}}}\) :

Irreversible parameter at the site 1

\(L\) :

Porous medium length

\(\dot{m}\) :

Mass rate

\(m\) :

Mass

\({\text{Np}}\) :

Total number of particles in the grid cell

\({\text{Np}}_{{\text{m}}}\) :

Number of particles pairs that participate in the mixing process

\(n\) :

Time

\(\hat{n}\) :

Unitary vector

\(n_{\alpha li}^{{\Omega^{\prime } }}\) :

Number of particle of \(\alpha\)-component in \(l\)-phase inside \(\Omega^{\prime }\)-domain

\(\dot{n}_{m\alpha i}^{{\Omega^{\prime } }}\) :

Rate of particles of \(\alpha\)-component in \(l\)-phase inside \(\Omega^{\prime }\)-domain transferred

\(\dot{n}_{\alpha lq}\) :

Rate of particle of \(\alpha\)-component in \(l\)-phase that leaves or enters of the porous medium through sources or sinks

\(\dot{n}_{\alpha lr}\) :

Rate of particle of \(\alpha\)-component in \(l\)-phase that transferred to rock phase

\(\dot{n}_{{\alpha ll^{\prime } }}\) :

Rate of particle of \(\alpha\)-component in \(l\)-phase that transferred to reference phase

\(P\) :

Pressure

\({\text{Pe}}\) :

Peclet number

\(q\) :

Source/sink

\({\varvec{r}}\) :

Position vector

\(S\) :

Saturation

\(t\) :

Time

\(V\) :

Cell volume

\(V_{{\text{p}}}\) :

Porous volume

\({\varvec{v}}\) :

Darcian’s velocity vector

\({\varvec{v}}_{{\text{T}}}\) :

Total deterministic velocity

\({\varvec{W}}\) :

White noise

\(w\) :

Weight factor

\(x\) :

Retention concentration on the rock

\(\rho\) :

Density

\(\rho_{l}^{{{\text{pn}}}}\) :

Particle number density

\(\sigma_{C\alpha l}\) :

Standard deviation

\(\phi\) :

Porosity

\(\mu\) :

Viscosity

\(\tau\) :

Time scale

\(\Omega\) :

Integration domain o REV

\(\Omega^{\prime }\) :

Particles integration domain

\(1\) :

Site 1

\(2\) :

Site 2

g:

Gas

\(i\) :

Number of particle

\(j\) :

Number of particle

\(k\) :

Number of cell

\(l\) :

Phase

\(l^{\prime }\) :

Reference phase

\(m\) :

Mean value

o:

Oil

p:

Pore

r:

Rock

w:

Water

\(x\) :

Coordinate

\(\alpha\) :

Component

\(\varepsilon\) :

Partition coefficient

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Acknowledgements

The authors acknowledge COLCIENCIAS and ANH for the support provided in contract 272-2017, to the Project “Strategy of transformation of the Colombian energy sector in the horizon 2030” funded by the call 788 of Minciencias Scientific Ecosystem, Contract number FP44842-210-2018 and Universidad Nacional de Colombia for logistical and financial support.

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Authors and Affiliations

  1. Departamento de Procesos y Energía, Facultad de Minas, Universidad Nacional de Colombia, Medellín Campus, Medellín, Colombia

    E. A. López, J. M. Mejía & F. Chejne

  2. Instituto de Física, Universidad de Antioquia, Medellín, Colombia

    E. A. López

Authors
  1. E. A. López
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  2. J. M. Mejía
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  3. F. Chejne
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Correspondence to E. A. López.

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López, E.A., Mejía, J.M. & Chejne, F. Deterministic/Probabilistic Model as Strategy to Study Nanofluid Transport in Porous Media. Transp Porous Med 139, 357–380 (2021). https://doi.org/10.1007/s11242-021-01669-0

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