# A network model for the biofilm growth in porous media and its effects on permeability and porosity

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## Abstract

Successful microbial enhanced oil recovery depends on several factors like reservoir characteristics and microbial activity. In this work, a pore network is used to study the hydrodynamic evolution over time as a result of the development of a biofilm in the pores. A new microscopic model is proposed for biofilm growth which takes into account that nutrients might not fully penetrate the biofilm. An important novelty in this model is that acknowledges the continuous spreading of the biofilm over the network. The results from the current study can be used to obtain a new relation between the porosity and permeability which might be used as an alternative to the Kozeny Carman relation.

## Keywords

Biofilm growth model Pore network Bioclogging Permeability Porosity## 1 Introduction

The production of oil from the reservoir is initially accomplished by the internal pressure of the reservoir. However, when the primary production declines some external forces have to be applied, hence waterflooding or gas injection techniques are implemented to extract oil from the reservoir. These injection schemes are called the secondary oil recovery production. Nevertheless, even after primary and secondary recovery two-thirds of the oil are still trapped in the ground [4]. The tertiary oil recovery extraction aims to increase the mobility of the remaining oil. One of the tertiary (or enhanced) oil recovery techniques is the microbial enhanced oil recovery (MEOR) which uses the growth of bacteria and the resulting by-products in order to increase the oil production. Microbial growth may enhance oil displacement by increasing the efficiency of the waterflooding process, by reducing interfacial tension and by changing the rock wettability [1, 14]. It has been observed that interfacial tension reduction and the increase of waterflooding efficiency caused by selective plugging are the mechanisms that have the largest impact on oil recovery [24].

Since it is hard to quantify the relation between the successful application of MEOR and parameters like the individual reservoir characteristics and the microbial activity, the development of computational models is of vital importance. These models are used to predict the bacterial growth and the in-situ regeneration of bioproducts in order to develop a proper field strategy [24]. The influence of biofilm growth on porous media characteristics such as permeability and porosity has been modeled in several studies [5, 6, 9, 12, 16, 18, 29]. The mathematical description is based on a theoretical framework and phenomenological relations obtained from experimental results [5, 9, 16, 18, 22, 29]. Biofilm growth models include Darcy continuum models [27, 33], bacterially-based models [16], Lattice Boltzmann based simulations [17, 32] and Pore Network Models (PNM) [5, 9, 18, 21, 26, 31]. Usually, in biofilm growth models the porous medium consists of three components: the grains, the biofilm which grows on the walls of the solid grains and the liquid in the pore space. The grains are assumed to be impermeable to the liquid and the nutrients, therefore hydrodynamic model equations are written only for the liquid and biofilm [17].

Cunningham et al. [8] showed experimentally the effect of the accumulation of biofilm on the porosity, permeability and friction factor of the porous media. The porosity of the media decreased between 50 and 96% due to the accumulation of biofilm, while permeability decreased between 92 and 98%. Taylor and Jaff [28] obtained an analytic expression to describe changes in the porous media as a result of biofilm growth in the continuum scale. However, in Taylor and Jaff [28] it is assumed that biofilm growth proceeds uniformly through the network which is an oversimplification according to laboratory experiments [15]. Clement et al. [6] model the biofilm growth using a macroscopic approach. This model does not assume any specific pattern for biofilm accumulation, instead it is based on macroscopic estimates of average biomass concentrations. Seki and Miyazaki [23] proposed a mathematical model for bioclogging that takes into account the nonuniform microbial distribution of colonies which ranges from micro-colonies to biofilm. However assuming uniform biofilm thickness in their model gives an overestimation of the bioclogging process [36]. Therefore, pore network models and pore-scale models are needed to describe the growth of biomass and its effects on the macroscopic properties of the media properly [35].

In PNMs, the porous medium is modeled by cylindrical interconnected tubes in which water or any fluid can flow. The biofilm development is stimulated by the injection of nutrients into the network. Transport of nutrients takes place within an aqueous phase and is described by a convection–diffusion–reaction equation. The reaction term models the consumption of nutrients caused by bacterial population growth. The bacterial population will determine the development of biofilm in the pores of the medium. This biofilm will grow and will change the radii of the pores, leading to porosity and permeability reduction and hence to a modification in the flow pattern dynamics of the fluid that carries the nutrients through the network [5, 26, 31]. Kim and Fogler [12] studied the effects of biofilm growth on porosity under starvation conditions. They show a good agreement with experimental results and show the existence of a critical shear stress. Raoof and Hassanizadeh [19] used a pore network model to describe two-phase flow in a porous media. They took into account the influence of the nodes of the network on the effective resistance of the fluids. They used a coordination number distribution which allows a maximum coordination number of 26. Additionally, they assigned a variety of cross-sectional shapes including circular, rectangular and triangular. They claimed that the inclusion of the volume of the nodes of the network affects the relation between the relative permeability and the saturation of the fluids. Despite the relevance of their work, in their model, they did not include the development of biofilm in the porous medium. In this study,as an approximation, we disregard the volume of the nodes to avoid additional complications in the model. Arns et al. [2] studied the effect of topology in the relative permeability of the networks. They found that the relative permeability curves obtained with stochastic networks are in good agreement with the ones obtained from imaged rock networks. The bacteria and Extracellular Polymeric Substance (EPS) in porous media are often lumped together and are represented as a continuous uniform layer of biomass attached on the surface of the solid grains of the porous media [9, 26, 31]. This uniform layer of biomass is referred to as biofilm. Furthermore, the biofilm growth rate is usually assumed to be proportional to the volume of biomass. Nevertheless, the nutrients might not be available in the entire volume of the biofilm. This phenomenon occurs if the consumption of nutrients is faster than the diffusion rate within the biofilm so that the (diffusion) penetration of the nutrients into the biofilm proceeds at a slower rate than the other processes [10, 13, 25]. Hence, the hypothesis that the nutrients are distributed over the whole volume of biofilm is questionable. Therefore, we assume that biofilm growth occurs only in a limited volume where the concentration of nutrients is maximal.

Usually, in PNMs the microbial activity is assumed to exist only within the tubes and no spread of biomass between neighboring tubes is described [5, 18, 21, 26, 31]. However, experiments show that the biomass or biofilm continuously grows, extending through the whole medium [15]. To model the inter-pore transport, Ezeuko et al. [9] consider a spreading potential among neighboring tubes. The spreading of the biofilm is allowed once the biomass has completely saturated the host pore. Thullner et al. [31] modeled the colony growth by assuming that a tube in the network was completely full or empty. Hence a binary switch mechanism is used to describe the spreading of biomass. The switch to completely filled tubes is determined by the size of the tubes. However, they did not consider any exchange of biomass between neighboring tubes. In our model, we describe the continuous spreading of the biofilm between adjacent tubes by computing the spreading of biomass from one pore to its neighbors, if there is a difference of volume of biomass between neighboring tubes.

In this study, we present a new biofilm growth model which takes into account that nutrients cannot fully penetrate the biofilm since consumption of nutrients is faster than the diffusion rate through the biofilm. We take into account that the biofilm growth is limited within a thin penetration layer, in which bacteria are in direct contact with the nutrients. In our model, there are two types of biofilm development: growth in the interior of the tube and growth at the extremes of the tube. Biofilm growth in the extremes of the tube will lead to the spreading of the biofilm to the neighboring tubes and through the whole network. The currently proposed biofilm growth model approach has several advantages over other models. Firstly, we incorporate the likely non-homogeneous distribution of the nutrients within the biofilm. Secondly, since biofilm growth takes place mainly in the boundary between water and biofilm, the internal biofilm growth will naturally stop if the tube is full of biofilm. Finally, the biofilm growth in the extremes of the tubes leads to spreading of biomass through the whole network. In this model there is no need to seed initially all the tubes in the network to observe the clogging of the network. This paper is focused on the presentation of biofilm growth model in a pore network. Future research might be the used of these results to obtain an alternate relation between porosity and permeability. The up-scaling of these results is beyond the scope of this paper.

## 2 Mathematical model

*l*. The number of tubes connected in each node is four for interior nodes, three for boundary nodes and two for the nodes in the corners of the network.

*R*(see Fig. 1). The volumetric flow of the water phase \(q_{ij}\) in the tube \(t_{ij}\) is described by a modified form of the Poiseuille equation [30],

*l*is the length of the tube and the dimensionless number \(\beta \) is the ratio between the viscosity of water flowing through the biofilm and the viscosity of water flowing through the bulk. We use \(\beta = 10^7\) which according to Thullner and Baveye [30] is a good approximation for an impermeable biofilm. Mass conservation is imposed on each of the nodes. For the node \(n_i\) we have

*C*, this gives

*A*denotes the area of the cross-section of the tube and

*D*is the diffusion coefficient of water. Further \(b^+\) represents the concentration of biofilm that grows as a result of consumption of nutrients (no detachment term is taken into account in this equation). In general, the concentration of nutrients

*b*is linked to the volume of biofilm, \(V_{bf} \) by

In this model we assume that nutrients might not penetrate completely through the biofilm since the reaction is faster than the diffusion rate within the biofilm. Therefore we propose that there exists a maximal distance (from the water biofilm interface) that the nutrients can travel within the biofilm. The maximal distance is called the penetration layer, \({\varGamma }\), and implicitly defines a maximal volume in which the nutrients can diffuse. This volume is called the penetration volume \(V_p\) and it is assumed to be constant during the whole process of biofilm growth. If the volume of biofilm is smaller than the penetration volume, the nutrients can penetrate the whole biofilm volume and hence the biofilm growth rate is proportional to the volume of biofilm. However, if the biofilm volume is much larger than the penetration volume, the nutrients react with the biofilm only within this penetration volume, which is adjacent to the water-biofilm interface. In this case, the biofilm growth rate is proportional to the area of the water biofilm interface. Further, since in general there are two regions in the tube where the biofilm encounters the nutrients, we model two kinds of biofilm growth: internal biofilm growth and biofilm growth at the extremes of the tube (see Fig. 2). Firstly, we describe the internal growth.

Secondly, we describe the biofilm that grows in the extremes of the tube. Since the penetration volume in the extremes is very small compared to the whole volume of biofilm, the biofilm growth rate in the extremes of the tubes is proportional to the area of the interface between water and biofilm (see Fig. 2). We assume binary interactions with the neighboring tubes. The area of the interface between water and biofilm \(A^e_{wbf}\) between the tube \(t_{ij}\) and the tube \(t_{jk}\) can be written in terms of the difference between biofilm volumes of neighboring tubes. The biofilm grows in the extreme of the tube with a larger volume of biofilm and it is given to the neighboring tube which has a smaller volume of biofilm.

*H*because detachment occurs only when there is biofilm within the tube. In case there is no biofilm in the tube, \(H=0\), which means the detachment rate is zero. In Eq. (12) the first term is the interior biofilm growth, the second and third term describes the biofilm which grows in the extremes of the neighboring tubes and the fourth term is a term for the detachment of the biofilm.

*p*, subject to

*x*direction and \(L_y\) the size in

*y*direction. Next to this \(r_{w_{ij}}\) decreases as a result of deposition of biofilm, which grows under the presence of nutrients. The balance of nutrients is given by,

## 3 Numerical method

The numerical approach and the computational procedure used in this work are described in this section. When mass conservation, Eq. (1), is combined to Eq. (2) a linear system for the pressures at the nodes \(p_i\) arises. After solving this system, the flux \(q_{ij}\) in each of the tubes is computed.

## 4 Simulation results

In this section we describe the numerical experiments and the results obtained for the biofilm growth in a pore network. Firstly, in order to validate the advection–diffusion part of our code, we compare our results with an analytic solution and with a Continuous Time Random Walk (CTRW) transport model [7]. Secondly, we studied the biofilm growth effects on the outflux and porosity. For this study, the biofilm growth rate \(k_1\) is fixed but three different detachment rates \(k_2\) are used. Finally, we compare our results with the Kozeny–Carman relation and with two quasi-steady biofilm growth models.

Parameters for the simulation without biofilm growth

Name | Symbol | Value |
---|---|---|

Tube length | | \( 3.5 \times 10^{-5} \;\text {(m)} \) |

Network size in the | \(L_x\) | \(1.9 \times 10^{-2}\;\text {(m)}\) |

Network size in the | \(L_y\) | \(9.5 \times 10^{-4}\;\text {(m)}\) |

Number of tubes in the network | \(N_a\) | 4210 |

Radius of the tubes | | \( 3.5338\times 10^{-6}\;\text {(m)}\) |

Global pressure gradient | \({\varDelta } p\) | \(1.6\;\text {(kPa/m)} \) |

Viscosity of water | \(\mu \) | \(4.7 \times 10^{-5}\;\text {Pa min}\) |

Density of water | \(\rho \) | \(1000 (\text { kg/m}^3)\) |

Diffusion coefficient of water | \(D*\) | \(3.971 \times 10^{-8}\;(\text {m}^2/\text {min}\) |

Velocity | \(v*\) | \(5.32 \times 10^{-5}\;(\text {m}^2/\text {min}\) |

Inlet concentration | \(C_{in}\) | \(1\;(\text {kg/m}^3)\) |

*x*direction and 11 nodes in

*y*direction. The number of tubes is determined implicitly by the number of nodes and by the topology of the network. Further, we assume that all the tubes in the network have the same radius. We use the volumetric flows through the pores from the network model for the solution of the concentration of nutrients. Under these conditions for the size of the mesh and the uniform size of the radii in all the tubes, we can compare the results with a model based on CTRW and with an analytic solution in one dimension [34]. The analytic solution of the advection–diffusion equation (Eq. 4 without reaction term) in 1-D is given by:

Parameters for the second series simulation

Name | Symbol | Value |
---|---|---|

Mean pore radius | | \(12.2 \times 10^{-6}\;\text {(m)}\) [9] |

Global pressure gradient | \({\varDelta } P\) | 1.6 \(\;\text {(kPa/m)}\) |

Viscosity of water | \(\mu \) | \(1.66 \times 10^{-5}\;\text {(Pa min)}\) |

Density of water | \(\rho _w\) | 1000 \((\text {kg/m}^{3})\) |

Density of biofilm | \(\rho _{bf}\) | 20 \((\text {kg/m}^{3})\) [20] |

Yield coefficient | | 0.34 [3] |

Half saturation constant for biofilm | \(E_{sb}\) | \(2 \times 10^{-3} (\text { kg/m}^{3})\) [3] |

Inlet concentration | \(C_{in}\) | 1 \((\text {kg/m}^{3})\) |

Initial biomass concentration | \(b_{0}\) | \(1 \times 10^{-4} (\text {kg/m}^{3})\) |

Biofilm/bulk water viscosity ratio | \(\beta \) | \(10^{7}\) [30] |

For each pair of biofilm growth \(k_1\) and detachment rate factor \(k_2\), we performed ten simulations where we fixed all the parameters, except the initial distribution of tubes seeded with biofilm. The normalized flux \(Q_n\) is defined as, \( Q_n= \frac{Q}{Q_{0}}\), where \(Q_0\) is the initial flux in the network (i.e. before biofilm growth). We compute the average of the normalized flux and we observe that the 95% confidence interval is very close to the average value of the normalized flux, therefore the initial random biofilm distribution does not have a significant effect on the results.

Finally, in addition to the full model which considers the transport of nutrients and the biofilm growth as two coupled phenomena, two quasi-steady state models of biofilm growth are also considered in this work. In these models we set an amount of volume of biofilm in the network, then we compute the effect of the volume of biofilm in the radius available for water and finally we compute the flux through the network. Note that the transport-diffusion equation is not solved in these models.

In the first model we consider that initially biofilm is present in all the tubes of the network and that the biofilm grows at the same rate in all the tubes. Therefore we refer to this model as uniform biofilm growth.

*K*and it is given by the following equation,

*L*and the viscosity \(\mu \) are constant during the process of biofilm growth, we have that

Note that in order to derive Eq. (35) the parameter \(C_k\) has been taken constant. However, since the porous medium channels are changed by the non-uniform accumulation of biomass, the assumption of taking \(C_k\) constant is probably inappropriate. Hence, our results may deviate from the results predicted by the Kozeny–Carman model.

In Fig. 8 the numerical results for the porosity \(\phi \) versus the normalized flux are shown for the two different detachment rates studied in this work \(k_2=10^{-7} \,(1/\hbox {s})\) and \(k_2=0 \, (1/\hbox {s})\), the two cases of quasi-steady-state biofilm growth models and the Kozeny–Carman relation [Eq. (35)].

The uniform growth model and the full model overlap from the initial porosity to 0.35 approximately where a sudden decrease in the normalized flux is described in the full-model for \(k_2 =0 \, (1/\hbox {s})\) and \(k_2 =10^{-7} \, (1/\hbox {s})\). This is explained as follows: in the beginning in the full model, the biomass starts spreading through the network and since the thickness of the biomass in the tubes is still small, the influence of the biomass on the permeability is insignificant at this stage. However, since the nutrients are transported through the network and the biomass is spread continuously, uniform biofilm growth is stimulated in the network, causing a decrease in the permeability due to the accumulation of biomass. Afterwards, the nutrients are consumed by the bacteria and the biofilm starts growing and clogging the pores, therefore there is a reduction of the flux of the nutrients in the whole network. Hence, the nutrients are present preferentially near the inlet which causes a preferential growth of biofilm near the inlet and at the final stage causes the total decrease of the flux. The random growth model shows a linear decay of the normalized flux. For high porosity the slope of the decay of the normalized flux predicted by the random growth model is similar to the slope of the normalized flux predicted by the full model. The linear behavior of the random growth model deviates from the full model for lower porosity. The random biofilm growth predicts a plugging of the network when the porosity is about 0.2. The porosity is approximately half the initial porosity, which is in accordance with the percolation threshold for a rectangular network, [11]. The fact that the full model stays in accordance with the uniform growth model seems to indicate that at the beginning the time evolution of the flux is predominantly determined by the localized growth kinetics of the biofilm, rather than the kinetics of spreading over the network. Finally, the Kozeny–Carman relation shows a behavior that is similar to the uniform biofilm growth, but the decrease of the normalized flux with the decrease in porosity is faster than the uniform growth.

## 5 Conclusions and outlook

In this work, we simulate biofilm growth, in particular its effects on the porous medium characteristics such as porosity and permeability. We use a two-dimensional pore network model to represent the porous medium. We develop a new model for biofilm growth, which predicts that the nutrients are not able to penetrate fully in the biofilm if the reaction term is dominant over the diffusion of nutrients within the biofilm. In addition, our model is able to simulate the spreading of the biofilm through the whole network which is a phenomenon that has been observed experimentally [15]. The proposed model shows that at early stages biofilm growth is mostly uniform through the whole network, however eventually the biofilm will grow preferentially near the inlet of the network, plugging the pores at the inlet and causing a cease of the flux through the network. The modifications in porosity and permeability caused by biofilm growth might be beneficial for a microbial enhanced oil recovery technique, especially in the first stage before the plugging of the network. Since we see that uniform growth provides a relatively good correspondence with the full model for high porosity, we conclude that the clogging of the porous medium in high permeability layers is feasible without blocking the inlet. For this reason, we propose to stop injection of nutrients in order to avoid plugging the medium. This behavior is not described by the uniform growth model, the random growth nor the Kozeny–Carman relation.

Since we consider a 2D rectangular pore network model consisting of cylindrical tubes with the same radii, this model could be too simplified to describe a real reservoir field. Interesting further research is to find the representative elementary volume in order to upscale these results to the macroscale. In addition, future plans entail the study of the effects of biofilm growth in porosity and permeability in more complex topologies in 2D and 3D.

## Notes

### Acknowledgements

We thank the Mexican Institute of Petroleum (IMP) for financially supporting this research through the Programa de Captación de Talento, Reclutamiento, Evaluación y Selección de Recursos Humanos (PCTRES) Grant.

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