Mathematical Modeling, Laboratory Experiments, and Sensitivity Analysis of Bioplug Technology at Darcy Scale

Abstract

In this paper, we study a Darcy-scale mathematical model for biofilm formation in porous media. The pores in the core are divided into three phases: water, oil, and biofilm. The water and oil flow are modeled by a generalized version of Darcy's law, and the substrate is transported by mechanical dispersion, diffusion, and convection in the water phase. Initially, there is biofilm on the pore walls. The bio-film consumes substrate for production of biomass and modifies the pore space, which changes the rock permeability. The model includes detachment of biomass caused by water flux and death of bacteria, and it is implemented in the MATLAB Reservoir Simulation Toolbox (MRST). We discuss the capability of the numerical simulator to capture results from laboratory experiments. We perform a novel sensitivity analysis based on sparse-grid interpolation and multiwavelet expansion to identify the critical model parameters. Numerical experiments using diverse injection strategies are performed to study the impact of different porosity/permeability relationships in a core saturated with water and oil. Introduction After primary and secondary production, up to 85% of the oil remains in the reservoir (Patel et al. 2015). Microbial improved and enhanced oil recovery (MIEOR) is one of the secondary and tertiary methods to increase the oil production using microorganisms (Wood 2019). Bioplug technology is an MIEOR strategy that comprises plugging the most permeable zones in the reservoir, which provokes water to flow through new paths, and recovering the oil in these new zones. However, microorganisms could also form biofilms in undesirable zones in the reservoir, leading to negative effects such as a decrease in water injectivity. Therefore, understanding the mechanisms involved in the development of biofilms is important to control their formation. The bioplug technology is intended for use on the field scale but to perform field-scale experiments is both time consuming and economically infeasible. Experiments in microsystems allow us to observe processes in greater detail, which leads to improvement of the experimental methods in core-scale experiments before field applications. For example, in Liu et al. (2019), the effects of flow velocity and substrate (also referred to as nutrients/food) concentration on biofilm in a microchannel was studied, finding values of substrate concentration and flow velocity for a strong plugging effect. Core samples from reservoirs can be used to study changes in permeability because of biofilm formation; for example, in Suthar et al. (2009), two-phase flow experiments were performed to study the selective plugging strategy for MIEOR. In that study, the MIEOR effects increased the oil recovery by approximately 25%. Mathematical models of bioplug technology are important because they help to predict the applicability of this MIEOR strategy and to optimize its benefits. In Kim (2006), a mathematical model for single-phase flow was proposed that includes changes of rock porosity and permeability as a result of biofilm growth. The author calibrated the model using data from experiments in silica sand columns and performed a simple sensitivity analysis (one-at-a-time technique) of a few model parameters. Li et al. (2011) built a mathematical model for two-phase flow including the effects of bio-surfactants and biomass on improving oil recovery. The authors also performed a simple sensitivity analysis of a few model parameters and compared the numerical results for two different porosity/permeability relationships. They concluded that MIEOR could enhance the oil recovery substantially if a larger capillary number is achievable. Nielsen et al. (2016) built a two-phase-flow mathematical model for MIEOR that included a decrease in oil/water interfacial tension by produced surfactants and selective plugging by microbes and metabolic products. The authors studied the oil recovery for diverse injection strategies, changing the pore volumes injected and substrate concentration at a fixed-flow rate. In Dzianach et al. (2019), the authors present a recent review of mathematical models of biofilms for diverse purposes. They concluded that cooperation between various disciplines is required to develop novel models. In this work, we present a two-phase core-scale model of bioplug technology. To our knowledge, this is the first mathematical model for two-phase flow and permeable biofilm. This mathematical model is the result of a research project where microbiologists, physicists, chemists, and mathematicians were involved. A detailed description of this project and previous publications can be found in Landa-Marbán (2019). In contrast to Li et al. (2011), in this work, we perform simulations to find at which part (low, medium, or high porosity) of five porosity/permeability relationships the oil recovery is more sensitive. Unlike Nielsen et al. (2016), we study the oil recovery for several injection strategies by changing the substrate concentration, flow rate, and injection direction. Sensitivity studies of mathematical models are of great interest because they provide estimates of the influence of the inputs (e.g., physical parameters) on a quantity of interest (e.g., biofilm formation). In Brockmann et al. (2006), a regional steady-state sensitivity analysis was performed to identify parameters with the largest impact on a mathematical model for deammonification in biofilm systems. Sensitivity analysis by means of Sobol decomposition provides rigorous estimates of parameter dependencies but are prohibitively expensive to compute if the number of parameters is large. This is remedied for smooth problems by first computing spectral (generalized polynomial chaos) expansions in the parameters, which then leads to efficient evaluation of the sensitivity indices via post-processing of spectral coefficients (Sudret 2008). The latter method was used in Landa-Marbán et al. (2019), where a global sensitivity analysis was performed using Sobol indices to identify the critical parameters of a pore-scale model for permeable biofilm. In this paper, we introduce a different approach that can also be used for nonsmooth mathematical models in the dependent parameters, where spectral expansions with global smooth-basis functions are not a robust choice. We propose a two-stage method where we first use