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Modeling and Simulation of Microbial Enhanced Oil Recovery Including Interfacial Area

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Abstract

The focus of this paper is the derivation of a nonstandard model for microbial enhanced oil recovery (MEOR) that includes the interfacial area (IFA) between oil and water. We consider the continuity equations for water and oil, a balance equation for the oil–water interfacial area, and advective–dispersive transport equations for bacteria, nutrients, and biosurfactants. Surfactants lower the interfacial tension (IFT), which improves oil recovery. Therefore, the parametrizations of the IFT reduction and residual oil saturation are included as a function of the surfactant concentration in the model. We consider for the first time in context of MEOR, the role of IFA in enhanced oil recovery. The motivation to include the IFA is to model the hysteresis in the capillary pressure–saturation relationship in a physically based manner, to include the effects of observed bacteria migration toward the oil–water interface and the production of biosurfactants at the oil–water interface. A comprehensive 2D implementation based on two-point flux approximation and backward Euler is proposed. An efficient and robust linearization scheme is used to solve the nonlinear systems at each time step. Illustrative numerical simulations are presented. The differences in the oil recovery profiles obtained with and without IFA are discussed. The presented model can also be used to design new experiments toward a better understanding and eventually optimization of MEOR.

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Abbreviations

A :

Cross-sectional area

\(a_\mathrm{ow}\) :

Specific oil–water IFA

\(C_\mathrm{b},\;C_\mathrm{n},\;C_\mathrm{s}\) :

Bacterial, nutrient, and surfactant concentration

\(C^*_\mathrm{n}\) :

Critical nutrient concentration for metabolism

\(d_1\) :

Bacterial decay rate coefficient

\(D_\mathrm{b}^{\text {eff}},\;D_\mathrm{n}^{\text {eff}},\;D_\mathrm{s}^{\text {eff}}\) :

Effective diffusion coefficients

\(E_\mathrm{ow}\) :

Production rate of specific IFA

\(e_\mathrm{ow}\) :

Strength of change of specific IFA

F :

Source/sink term

\(\mathbf{g }\) :

Gravity

\(g_{1\;{\text {max}}}\) :

Maximum bacterial growth rate coefficient

\(\mathbf{k },\;\mathbf{k }_\mathrm{o},\;\mathbf{k }_\mathrm{w}\) :

Absolute, oil, and water effective permeabilities

\(k_\mathrm{a}\) :

Diffusion coefficient for the chemotaxis

\(k_\mathrm{r,o},\;k_\mathrm{r,w}\) :

Oil and water relative permeabilities

\(\mathbf{k }_\mathrm{ow}\) :

Interfacial permeability

\(K_\mathrm{b/n},\;K_\mathrm{s/n}\) :

Saturation constants for bacteria and surfactants on nutrients

\(K_\mathrm{s/a}\) :

Saturation constant for surfactants on IFA

\(l_1,\;l_2,\;l_3\) :

Fitting parameters for modeling the reduction of IFT

L :

Length of porous medium

\(N_\text {B}\) :

Bond number

\(N_\text {Ca}\) :

Capillary number

\(N_\text {T}\) :

Trapping number

\(p,\;p_\mathrm{o},\;p_\mathrm{w}\) :

Average, oil, and water pressure

\(Q_T\) :

Water injection rate

\(s_\mathrm{o},\;s_\mathrm{w}\) :

Oil and water saturation

\(s_\mathrm{or},\;s^{\text {min}}_\mathrm{or},\;s^{\text {max}}_\mathrm{or}\) :

Residual oil saturation, minimum, and maximum

\(s_\mathrm{wi}\) :

Irreducible water saturation

\(s_\mathrm{w}^*\) :

Effective water saturation

\(T_1,\;T_2\) :

Fitting parameters for modeling the residual oil saturation

\(\mathbf{u },\;\mathbf{u }_{\varSigma }\) :

Volumetric and total flow rate per area

\(v_\mathrm{g}\) :

Settling velocity of bacteria

\(v_\mathrm{ow}\) :

Interfacial velocity

W :

Width of porous medium

\(Y_\mathrm{s/b},\;Y_\mathrm{s/n}\) :

Surfactant yield coefficients per unit bacteria and nutrient

\(\alpha _0\) :

Angle of flow relative to the horizontal

\(\alpha _1,\;\alpha _2,\;\alpha _3,\;\alpha _4\) :

Parameters for the IFA relation

\(\alpha _{\mathrm{b},L},\;\alpha _{\mathrm{n},L},\;\alpha _{\mathrm{s},L}\) :

Longitudinal dispersivities

\(\alpha _{\mathrm{b},T},\;\alpha _{\mathrm{n},T},\;\alpha _{\mathrm{s},T}\) :

Transverse dispersivities

\(\delta \) :

Dirac delta

\(\Delta t\) :

Time step

\(\Delta x\) :

Space step

\(\lambda _\mathrm{o},\;\lambda _\mathrm{w}\) :

Oil and water mobilities

\(\mu \) :

Viscosity

\(\mu _{\mathrm{s}\;\text {max}}\) :

Maximum specific biomass production rate

\(\phi \) :

Porosity

\(\rho \) :

Density

\(\sigma ,\;\sigma _{\text {min}},\;\sigma _{\text {max}}\) :

IFT, minimum and maximum

\(\theta \) :

Contact angle

b :

Bacteria

n :

Nutrient

s :

Surfactant

o :

Oil

w :

Water

BE:

Backward Euler

EOR:

Enhanced oil recovery

IFA:

Interfacial area

IFT:

Interfacial tension

MEOR:

Microbial enhanced oil recovery

ODE(s):

Ordinary differential equation(s)

PDE(s):

Partial differential equation(s)

REV:

Representative elementary volume

TPFA:

Two-point flux approximation

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Acknowledgements

This worked was supported by the Research Council of Norway under the Projects IMMENS No. 255426 and CHI No. 255510.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Bergen, Allégaten 41, P.O. Box 7803, 5020, Bergen, Norway

    David Landa-Marbán, Florin A. Radu & Jan M. Nordbotten

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  1. David Landa-Marbán
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  2. Florin A. Radu
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  3. Jan M. Nordbotten
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Correspondence to David Landa-Marbán.

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Landa-Marbán, D., Radu, F.A. & Nordbotten, J.M. Modeling and Simulation of Microbial Enhanced Oil Recovery Including Interfacial Area. Transp Porous Med 120, 395–413 (2017). https://doi.org/10.1007/s11242-017-0929-6

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