Water injection in the aquifer induces deformations in the soil. These mechanical deformations give rise to a change in porosity and permeability, which results in non-linearity of the mathematical problem. Assuming that the deformations are very small, the model provided by Biot’s theory of linear poroelasticity is used to determine the local displacement of the skeleton of a porous medium, as well as the fluid flow through the pores. In this continuum scale model, the Kozeny–Carman equation is commonly used to determine the permeability of the porous medium from the porosity. The Kozeny–Carman relation states that flow through the pores is possible at a certain location as long as the porosity is larger than zero at this location in the aquifer. However, from network models it is known that percolation thresholds exist, indicating that the permeability will be equal to zero if the porosity becomes smaller than these thresholds. In this paper, the relationship between permeability and porosity is investigated. A new permeability-porosity relation, based on the percolation theory, is derived and compared with the Kozeny–Carman relation. The strongest feature of the new approach is related to its capability to give a good description of the permeability in case of low porosities. However, with this network-inspired approach small values of the permeability are more likely to occur. Since we show that the solution of Biot’s model converges to the solution of a saddle point problem for small time steps and low permeability, we need stabilisation in the finite element approximation.

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During the primary oil recovery, wells are drilled into the reservoir, and due to natural driving forces, the oil flows to the surface through these production wells. Unfortunately, the production of oil in this first stage is typically between only five and ten percent of the oil in place in the reservoir. In the secondary oil recovery some wells (used in the previous stage for production) are converted into injection wells and water or gas are injected into the reservoir to displace the oil to the surface. However, even after primary and secondary oil recovery about 60% of the oil-in-place remains in the reservoir. Microbial Enhanced Oil Recovery (MEOR) is a tertiary enhanced oil recovery technique used to extract the remaining oil after the secondary recovery. MEOR technique was proposed since the 1920’s however it was until 1940’s that it was considered seriously. In MEOR, bacteria and the resulting bioproducts are used to increase the mobilization of oil in the reservoir. Bacterial growth can produce gases that increase the pressure of the reservoir and decrease the viscosity of oil. Biosurfactants decrease the oil viscosity which may lead to an increase of the mobility of oil. Furthermore, bacteria can selectively plug the high permeability zones which changes the direction of water flow to the areas where the oil is still trapped. Selective plugging by bacteria is a process that is used simultaneously with a waterflooding operation. Among all the effects of biofilm growth, selective plugging and interfacial tension reduction are thought to have the greatest impact on oil recovery. The applicability of selective plugging to divert the flow of water has been shown in laboratory experiments. However, on the field scale the applicability of MEOR techniques is still under investigation since the MEOR techniques in pilot fields have produced different outcomes. In this study, we present a new 2D microscopic pore network biofilm growth model that takes into account that nutrients might not be able to penetrate the biofilm completely. This phenomenon occurs if the consumption of nutrients is faster than the diffusion of them. We incorporate in the model a characteristic volume related to the penetration depth of the nutrients within the biofilm. This inclusion allows a more accurate description of the biofilm growth in porous media. In addition, we model the continuous spreading of the biofilm through the whole network, which is a phenomenon that has been observed experimentally. Our numerical experiments show that the nutrients spread fast throughout the whole network during the early stages of the process. Since the nutrients are present in the whole network, the biofilm grows and spreads to the neighbouring tubes. For a longer period of time the biofilm grows uniformly through the network, however after this, the depletion of nutrients is observed and the biofilm grows preferentially near the inlet of the network causing the complete blockage of the network. Our model describes the transition between uniform biofilm growth and heterogeneous biofilm growth. @en

Compressing a porous material or injecting fluid into a porous material can induce changes in the pore space, leading to a change in porosity and permeability. In a continuum scale PDE model, such as Biot’s theory of linear poroelasticity, the Kozeny–Carman equation is commonly used to determine the permeability of the porous medium from the porosity. The Kozeny–Carman relation assumes that there will be flow through the porous medium at a certain location as long as the porosity is larger than zero at this location. In contrast, from discrete network models it is known that percolation thresholds larger than zero exist, indicating that the fluid will stop flowing if the average porosity becomes smaller than a certain value dictated by these thresholds. In this study, the difference between the Kozeny–Carman equation and the equation based on the percolation theory, is investigated.@en

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.

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In this work, we model the biofilm growth at the microscale using a rectangular pore network model in 2D and a cubic network in 3D. For the 2D network, we study the effects of bioclogging on porosity and permeability when we change parameters like the number of nodes in the network, the network size, and the concentration of nutrients at the inlet. We use a 3D cubic network to study the influence of the number of nodes in the z direction on the biofilm growth and on upscalability. We show that the biofilm can grow uniformly or heterogeneously through the network. Using these results, we determine the conditions for upscalability of bioclogging for rectangular and cubic networks. If there is uniform biofilm growth, there is a unique relation between permeability and porosity, K ∼ ϕ2, this relation does not depend on the volume of the network, therefore the system is upscalable. However, if there is preferential biofilm growth, the porosity-permeability relation is not uniquely defined, hence upscalability is not possible. The Damköhler number is used to determine when upscalability is possible. If the Damköhler number is less than 101, the biofilm grows uniformly and therefore the system is upscalable. However, if the Damköhler number is greater than 103, the biofilm growth exhibits a deviation from uniform biofilm growth and heterogeneous growth is observed, therefore upscalability is not possible. There is a transition from uniform growth to preferential growth if the Damköhler number is between 101 and 103.@en

After the primary extraction in oil reservoirs up to 60 % of the oil remains trapped in the reservoir (Sen, 2008). Therefore, different mechanisms have been developed to get the oil out to the reservoir. One of these techniques is Microbial Enhanced Oil Recovery (MEOR) which is a technique used to produce more oil in a secondary extraction by using microbes in the reservoir. The main effects caused by microbes in oil recovery is the reduction of the interfacial tension between oil and water, wettability change of the rock and bioclogging caused by the growth and development of biofilm. Among these mechanisms, interfacial tension reduction and biclogging is thought to have the greatest impact on recovery (Sen, 2008). In this work, we describe the growth of biofilm, the growth of the microbial population and the transport of nutrients using a pore network model. We follow the previous models of Thullner et al. (Thullner, 2008) and Ezeuko et al. (Ezeuko, 2011) in which the biofilm is considered as a permeable layer. We consider the biofilm and the bacteria separately. Additionally, we assume that once a tube is full with biofilm, this biofilm can spread to the neighboring tubes. Finally, we study the changes in the hydrodynamic properties of the medium caused by the plugging of the pores and we study the flow diversion of water caused by plugging of the high permeability zones. @en