Mathematical Modelling and Simulation of Biogrout

Abstract

Biogrout is a method to reinforce sand and gravel by the production of calcium carbonate. This calcium carbonate is produced using micro-organisms that are either present in the subsoil or injected into it. The micro-organisms are supplied with urea and calcium. Subsequently, they catalyse the hydrolysis of urea, by which carbonate is formed. In the presence of calcium, the carbonate precipitates as calcium carbonate. Ammonium is the unwanted by-product of this reaction. The calcium carbonate crystals are formed in the pores and they connect the grains. In this way, the strength of the material is increased. Biogrout can be applied on locations where soil improvement is desired. Upon doing so, one needs to have a reliable prediction of the effect of the Biogrout treatment. Therefore, a thorough understanding of the process is necessary and a sound mathematical model is dispensable. In this thesis we focus on the modelling of the Biogrout process. We start with a mathematical model for the hydrolysis-precipitation reaction (Chapters 2 and 3). As a result of the precipitation of the solid calcium carbonate, the porosity decreases. Therefore, the permeability decreases as well. Due to the precipitation reaction, chemicals disappear from the solution causing a decrease in liquid volume. On the other hand, there is less void space available due to the decreasing porosity. These phenomena cause a net outflow out of the pores. The chemicals urea, calcium and ammonium are dissolved in the fluid. The concentrations are modelled with an advection-dispersion-reaction-equation. The density of the fluid evolves over time as a result of the altering composition, which gives a density-driven component to the flow. It is assumed that the solid calcium carbonate is not transported. Therefore, the differential equation for calcium carbonate only contains an accumulation and a reaction term. The reaction rate depends on the amount of micro-organisms present in the soil. In these chapters, it is assumed that the micro-organisms are homogeneously distributed. The Finite Element Method (FEM) is used to solve the model equations. Since high flow rates are not desirable in the Biogrout process, since such a high flow rate will flush out the micro-organisms, advection is not dominating. Hence, the Standard Galerkin FEM can be used. The Backward Euler method is used for the time discretisation and Newton’s method is applied to deal with the non-linearities. Chapter 4 describes a model for the placement of micro-organisms and considers three concentrations of micro-organisms: suspended micro-organisms, (temporarily) adsorbed micro-organisms and fixated micro-organisms. This fixation takes place after contact between the fixation fluid and the micro-organisms. The resulting microbial concentrations can be used as input for the reaction rate in the hydrolysis-precipitation model. This is done in Chapter 5 by combining the models. In Chapter 6 several differential equations for the fluid are compared. This leads to an adaptation of the differential equation for the flow that is used in the first chapters. Often, a hydrostatic pressure is used as a boundary condition. Chapter 7 explains how this pressure can be calculated in case of dynamically evolving fluid densities. Due to the dissolved chemicals, the fluid is denser than water. If such a dense fluid is injected, front instabilities in the form of fingers might occur. In Chapter 8 the front instabilities are induced by an initial variation of the porosity in the spatial domain. The effect of front instabilities on the Biogrout process is considered. In the last chapter, several experimental results are compared to the numerical results of simulations with the model. It appears that the model can describe the experimental results reasonably well.