A Pore-Scale Model for Permeable Biofilm: Numerical Simulations and Laboratory Experiments

Abstract

In this paper, we derive a pore-scale model for permeable biofilm formation in a two-dimensional pore. The pore is divided into two phases: water and biofilm. The biofilm is assumed to consist of four components: water, extracellular polymeric substance (EPS), active bacteria, and dead bacteria. The flow of water is modeled by the Stokes equation, whereas a diffusion–convection equation is involved for the transport of nutrients. At the biofilm–water interface, nutrient transport and shear forces due to the water flux are considered. In the biofilm, the Brinkman equation for the water flow, transport of nutrients due to diffusion and convection, displacement of the biofilm components due to reproduction/death of bacteria, and production of EPS are considered. A segregated finite element algorithm is used to solve the mathematical equations. Numerical simulations are performed based on experimentally determined parameters. The stress coefficient is fitted to the experimental data. To identify the critical model parameters, a sensitivity analysis is performed. The Sobol sensitivity indices of the input parameters are computed based on uniform perturbation by ± 10% of the nominal parameter values. The sensitivity analysis confirms that the variability or uncertainty in none of the parameters should be neglected.

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Abbreviations

c :

Nutrient concentration

D :

Nutrient diffusion coefficient

d :

Biofilm thickness

J :

Nutrient flux

k :

Permeability

\(k_\mathrm{res}\) :

Bacterial decay rate coefficient

\(k_\mathrm{str}\) :

Stress coefficient

\(k_{n}\) :

Monod half-velocity coefficient

L :

Pore length

p :

Pressure

q :

Water velocity

S :

Tangential shear stress

T :

Time

U :

Reference water velocity

u :

Velocity of the biomass

W :

Pore width

Y :

Growth yield coefficient

\(\mu \) :

Dynamic viscosity

\(\mu _\mathrm{n}\) :

Maximum rate of nutrient utilization

\(\nu \) :

Unitary normal vector

\(\nu _n\) :

Interface velocity

\(\varPhi \) :

Growth velocity potential

\(\rho \) :

Density

\(\tau \) :

Unitary tangential vector

\(\theta \) :

Volume fraction

a :

Active bacteria

b :

Biofilm

d :

Dead bacteria

i :

Input

o :

Output

e :

EPS

w :

Water

ALE:

Arbitrary Lagrangian–Eulerian

EPS:

Extracellular polymeric substance

MEOR:

Microbial enhanced oil recovery

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Acknowledgements

The work of DLM, NL, KK, PP, GB, TS, and FAR was partially supported by GOE-IP and the Research Council of Norway through the projects IMMENS No. 255426 and CHI No. 255510. ISP was supported by the Research Foundation-Flanders (FWO), Belgium, through the Odysseus programme (Project G0G1316N) and the Akademia grant of Equinor.

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Appendix A: Sensitivity Analysis Method

Appendix A: Sensitivity Analysis Method

In this Appendix, we describe the theory behind the performed sensitivity analysis. The variation is assumed uniform in the sense that each parameter varies within a range where all values are equally likely. The sensitivity analysis relies on the Hoeffding or Sobol decomposition of the quantity of interest, here denoted q, as a series expansion in subsets of all possible combinations of the n input parameters \(\pmb {y} = (y_1,...,y_n)\),

$$\begin{aligned} q(\pmb {y}) = q^{\lbrace \emptyset \rbrace } + \sum _{i=1}^{n}q^{\lbrace i\rbrace }(y_i) + \sum _{i=1,j>i}^{n} q^{\lbrace i,j\rbrace }(y_i,y_j)+\ldots + q^{\lbrace 1,\ldots , n\rbrace }(\pmb {y}). \end{aligned}$$

The Sobol decomposition terms are defined recursively as integrals over subsets of the range of \(\pmb {y}\), denoted \(\pmb {Y}\). We introduce a uniform weight function \(w(\pmb {y})=w_1(y_1)...w_{n}(y_n)\) with \(w_i = 1/(\max (y_i)-\min (y_i))\) and the subscript notation \(\sim i\) to denote all parameters except parameter i. The decomposition terms are then determined by

$$\begin{aligned} q^{\lbrace \emptyset \rbrace }&= \int \limits _{\pmb {Y}}q(\pmb {y})w(\pmb {y})\mathrm{d}\pmb {y},\\ q^{\lbrace i\rbrace }(y_i)&= \int \limits _{\pmb {Y}_{\sim i}}q(\pmb {y})w_{\sim i}(\pmb {y}_{\sim i})\mathrm{d}\pmb {y}_{\sim i} - q^{\lbrace \emptyset \rbrace }, \quad 1\le i \le n, \\ q^{\lbrace i, j\rbrace }(y_i,y_j)&= \int \limits _{\pmb {Y}_{\sim i, j}}q(\pmb {y})w_{\sim i,j}(\pmb {y}_{\sim i,j})\mathrm{d}\pmb {y}_{\sim i,j} - q^{\lbrace i\rbrace }(y_i) - q^{\lbrace j \rbrace }(y_j) - q^{\lbrace \emptyset \rbrace },\;1\le i < j \le n \end{aligned}$$

and so on for higher-order terms.

The Sobol index for the s-parameter combination \(\{ y_{i_1},y_{i_2},...,y_{i_s}\}\) is given by

$$\begin{aligned} S_{ \{ i_1,...,i_s \rbrace } = \frac{1}{\mathrm{Var}(q)}\int \limits _{\pmb {Y}_{i_1,...,i_s}} (q^{\lbrace i_1,...,i_s\rbrace }(y_{i_1},...,y_{i_s}))^2 w_{i_1}(y_{i_1})...w_{i_s}(y_{i_s}) dy_{i_1}... dy_{i_s}. \end{aligned}$$

The total variability of variable i is obtained by summing over all subsets of parameters including parameter i, which yields the total Sobol index for parameter i,

$$\begin{aligned} S_{\lbrace i \}} = \sum _{i \in I} S_{i}. \end{aligned}$$
(3)

In this work, the Sobol decomposition terms are computed from a generalized polynomial chaos expansion in Legendre polynomials (Sudret 2008), where the expansion coefficients are obtained from sparse quadrature rules using the Smolyak algorithm (Smolyak 1963). This quadrature rule is very sparse but assumes high regularity on the quantity of interest as a function of the input parameters.

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Landa-Marbán, D., Liu, N., Pop, I.S. et al. A Pore-Scale Model for Permeable Biofilm: Numerical Simulations and Laboratory Experiments. Transp Porous Med 127, 643–660 (2019). https://doi.org/10.1007/s11242-018-1218-8

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Keywords

  • Biofilm
  • Numerical simulations
  • Laboratory experiments
  • Microbial enhanced oil recovery
  • Porosity