Transport in Porous Media

, Volume 101, Issue 3, pp 477–505 | Cite as

Thermodynamics, Kinetics, and Hydrodynamics of Mixed Salt Precipitation in Porous Media: Model Development and Parameter Estimation

  • Hossein Safari
  • Mohammad Jamialahmadi


Demands for hydrocarbon production have been increasing in recent years. Today many oilfields around the world are afflicted by the problem of scaling leading to severe formation damage and hampering of petroleum production from hydrocarbon reservoirs. In current study, a mathematical model for prediction of permeability reduction due to scale deposition is developed based on thermodynamics, kinetics, and hydrodynamics of mixed salt precipitation during flow through porous media. Model predictions are compared with sound experimental data for single deposition of barium sulfate and most importantly, for simultaneous precipitation of barium sulfate and strontium sulfate onto rock surface. Owing to high nonlinearity of the proposed model, kinetic parameters embedded in the mathematical model were tuned employing a new approach based on a hybrid algorithm consisting of particle swarm optimization (PSO) technique and pattern search (PS) algorithm. The average absolute deviations ranging from 1.03 to 9.3 % were observed between model forecasts and experimental data which corroborate the suitability and applicability of the model and also confirm the capability of PSO–PS hybrid algorithm as a highly efficient optimization tool. Estimated values for kinetic parameters are also in accordance with collision theory of chemical reactions.


Mixed salt precipitation Permeability reduction Mathematical model Particle swarm optimization Pattern Search 

List of symbols





\({a}_\mathrm{M}\), \({a}_\mathrm{X}\)

Activity of cationic and anionic compound, respectively (mol L\(^{-1})\)

\({A}^{\emptyset }\)

Debye–Huckel coefficient


Total surface area of grains \((\mathrm{{m}}^{2})\)


Second virial coefficient for single electrolyte ij


Third virial coefficient for single electrolyte ij


Saturation concentration \((\mathrm{{kg~m}}^{-3})\)


Bulk concentration, \((\mathrm{{kg~m}}^{-3})\)


Interface concentration which is somewhat between bulk (\({C}_{\mathrm{sb}} )\) and saturation concentration (\({C}_\mathrm{s}^{*} )\), \((\mathrm{{kg~m}}^{-3})\)


Diffusion coefficient of salt, (\(\mathrm{{m}}^{2}~\mathrm{{s}}^{-1})\)

\({D}_{\mathrm{AB}}^{\circ }\)

Diffusion coefficient at infinite dilution, (\(\mathrm{{cm}}^{2}~\mathrm{{s}}^{-1})\)


Particle diameter (m)


Activation energy \((\mathrm{{Kj~mol}}^{-1})\)


Faraday, \((\mathrm{{C~g{\text{- }}equiv}}^{-1})\)


Global best of particle i at iteration j


Chilton and Colburn J-factor


Permeability \((\mathrm{{m}}^{2})\)


Initial permeability \((\mathrm{{m}}^{2})\)


Pre-exponential factor \((\mathrm{{m}}^{4}~\mathrm{{kg}}^{-1}\) \(\mathrm{{s}}^{-1})\)


Reaction rate constant \((\mathrm{{m}}^{4}~\mathrm{{kg}}^{-1}\) \(\mathrm{{s}}^{-1})\)


Solubility product \((\mathrm{{kg}}^{2}~\mathrm{{m}}^{-6})\)

\({K}_\mathrm{sp}^{\circ }\)

Thermodynamic solubility product \((\mathrm{{kg}}^{2}~\mathrm{{m}}^{-6})\)

\({m}_\mathrm{M}, {m}_\mathrm{X}\)

Concentration of cationic and anionic compound, respectively \((\mathrm{{mol~l}}^{-1})\)

\({M}^{\mathrm{m}+}, {X}^{\mathrm{n}-}\)

Chemical representation of cationic and anionic compound, respectively


Chemical representation of scaling mineral


Precipitation rate \((\mathrm{{kg~m}}^{-2}~\mathrm{{s}}^{-1})\)


Net precipitation rate \((\mathrm{{kg~m}}^{-2}~\mathrm{{s}}^{-1})\)


Removal rate \((\mathrm{{kg~m}}^{-2}~\mathrm{{s}}^{-1})\)


Order of chemical reaction


Total number of grains


Number of experimental data points


Pressure (bar)


Personal best of particle i at iteration j


Peclet number


Universal gas constant \((\mathrm{{J~mol}}^{-1}~\mathrm{{K}}^{-1})\)


Reynolds number


Modified Reynolds number


Ionic strength (\(\mathrm{{mol~L}}^{-1}\))


Schmidt number


Time (s)


Temperature (K)


Flow velocity (\(\mathrm{{m~s}}^{-1})\)


Velocity vector of particle i at iteration j


Bulk volume \((\mathrm{{m}}^{3})\)


Grains volume \((\mathrm{{m}}^{3})\)


Volume of scale \((\mathrm{{m}}^{3})\)


Total volume of scale \((\mathrm{{m}}^{3})\)


Inertia weight


Position vector of particle i at iteration j

\({z}_+, {z}_-\)

Valences of cation and anion

Greek letters

\({\alpha }_{1,2}\)

Constants \(\in [0,1]\)

\({\beta }\)

Mass transfer coefficient, \((\mathrm{{m~s}}^{-1})\)

\({\gamma }_\mathrm{M}, \gamma _\mathrm{X}\)

Activity coefficient of cationic and anionic compound, respectively

\(\theta _{ij}\)

Second virial coefficient for two ions of the same sign

\(\rho _\mathrm{p}\)

Precipitant density \((\mathrm{{kg~m}}^{-3})\)

\(\varphi \)

Total number of molecules or ions given by one mole of electrolyte

\(\emptyset \)

Bed porosity

\(\emptyset _0\)

Initial porosity

\(\psi _{ijk}\)

Third virial coefficient for ions i, j, and k not of the same sign

Subscripts and superscripts








Particle index


Iteration index




Scaling bulk


Scaling interface


Total scaling




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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Petroleum EngineeringPetroleum University of TechnologyAhvazIran

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