1 Introduction

Microbially induced calcite precipitation (MICP) has the potential to replace many conventional soil improvement methods involving cement as a nature-based and non-polluting alternative. The basis of the technique consists in catalysing the hydrolysis of urea with urease, which is produced by certain bacteria, to produce carbonate and ammonia. When dissolved calcium is available, it reacts with the carbonate and precipitates calcium carbonate (CaCO3) minerals, with calcite being the most efficient phase due to its properties and very low solubility. When calcite precipitates between soil particles, soil strength and stiffness are increased as a result of the bonds created [6, 34, 35].

A variety of multiphysical phenomena are involved in this process, most of which are intrinsically coupled. Water flow transports solutes and bacteria, which react producing calcite crystals and thus reducing the pore space available for flow. Likewise, water pressure is influenced by mechanical deformation which, in turn, depends on its stiffness and strength. These latter properties are significantly modified after calcite precipitation [7]. Therefore, given the complexity of the involved reactive, transport and mechanical phenomena in MICP, the design of treatment strategies is believed to be greatly enhanced via numerical tools that allow considering the interaction of all relevant factors and optimize the resources for a given target application [23].

Transport models have been developed that address the processes of bacteria, solute transport, reaction and precipitation in soils [4, 8, 13, 29, 32]. These numerical tools provide insight into how coupled phenomena, such as (i) the extent and amount of expected precipitated CaCO3 [31]; (ii) the relative importance of different reaction rates [14]; (iii) the coupled effects between precipitation and permeability reduction [32]; and (iv) the impact of bacteria attachment to the grains surface in different conditions [16], which ultimately interact.

While bio-chemical transport processes have been thoroughly studied with numerical models, only few studies reported in the literature have included mechanical deformation in their formulations [8, 18, 33]. All of these consider an elastic stress–strain relationship, which is a limiting assumption not only for the analysis of limit states but also for service states in partially cemented structures. This is because the behaviour of bio-cemented sands, depending on its cementation level, can exhibit a significant increase in its stiffness and strength with respect to the untreated state [6, 9, 21, 24]. Because the extent of the cemented zone is limited, in engineering applications the treatment process will result in heterogeneous mechanical properties, even if initially the material is relatively homogeneous, as observed from large-scale tests [12, 25, 29, 37, 38]. It can be expected that non-cemented zones will locally yield at lower stress levels than the cemented zones, while the structure may remain far from global failure. Therefore, elasticity might not provide realistic results if the aim is to assess the strength and deformation of bio-cemented soils under both service and ultimate conditions.

It is therefore necessary, for practical applications, to include elastoplastic modelling in the existing numerical frameworks in order to evaluate and predict the strength and deformation response of geotechnical works. In this context, a number of constitutive models have been proposed that can describe the mechanical behaviour of sands with varying bio-cementation levels [10, 11]. Gai and Sanchez [10] presented a model based on a modified Cam-Clay framework, to which the addition of two parameters demonstrated to qualitatively reproduce the main responses of bio-cemented sands. While that model does not consider an eventual increase in elastic stiffness due to cementation, this is overcome by the use of a sub-loading surface. However, a key feature of soil improvement, which is the development of tensile strength with cementation, is neglected, and the flow rule is associated, resulting in an overestimation of dilatancy upon shearing.

Gajo et al. [11] developed a constitutive model for cemented soils with reactive bonds that accounts for bond creation, dissolution and degradation. The model was formulated based on microstructural considerations and lead to a good reproduction of experimental results involving triaxial testing of bio-cemented sands. Although the microstructural basis allows for a physical interpretation of the material behaviour, it requires a significant amount of model parameters that might preclude its application in practice.

The objective of this work is to unify the bio-chemo-hydraulic processes and the elasto-plastic stress–strain response of bio-cemented soils in a single and comprehensive numerical model for the analysis of geotechnical problems. The model is developed using the framework presented by Laloui and Fauriel [8] for the FEM analysis of MICP processes. The model has been modified and enhanced to reproduce new experimental evidence available on the behaviour of bio-cemented soils, among the main modifications these include:

  • The consideration of water mass reacting to precipitate calcite.

  • A model of bacterial transport, with a limit on the bacteria concentration that can be attached.

  • A new elastoplastic constitutive model that accounts for the precipitated calcite-dependent behaviour of soils.

The elastoplastic model is based on the well-established Mohr coulomb, efficient for engineering practice, and has been developed to simulate the stress–strain behaviour of bio-cemented soils. The model is calibrated with laboratory experiments from different experimental campaigns on Ottawa sand targeting transport processes and mechanical behaviour separately. The capabilities of the complete model are demonstrated by simulating a synthetic application consisting on a shallow foundation. In particular, the impact of different injection strategies on the settlement and stress distribution below the foundation are evaluated and discussed.

2 Model formulation

2.1 Conceptual approach

The core principle of MICP relies on the hydrolysis of urea, which produces carbonate and ammonium according to:

$${\text{CO}}{\left({{\text{NH}}}_{2}\right)}_{2}+2{{\text{H}}}_{2}{\text{O}}\stackrel{{\text{urease}}}{\to }2{{\text{NH}}}_{4}^{+}+{{\text{CO}}}_{3}^{2-}$$
(1)

The carbonate produced from this reaction can precipitate into calcium carbonate when calcium is available to react according to:

$${{\text{Ca}}}^{2+}+{{\text{CO}}}_{3}^{2-}\to {{\text{CaCO}}}_{3}\downarrow$$
(2)

Assuming that the precipitation of calcium carbonate is significantly faster in comparison to the hydrolysis rate, the above chemical reactions can be expressed as a single reaction [30], such that:

$${\text{CO}}{\left({{\text{NH}}}_{2}\right)}_{2}+2{{\text{H}}}_{2}{\text{O}}+{{\text{Ca}}}^{2+}\to 2{{\text{NH}}}_{4}^{+}+{{\text{CaCO}}}_{3}$$
(3)

The model is formulated considering this single reaction. When calcite precipitation is the main chemical process of interest, it has been shown to give satisfactory results when compared to considering both hydrolysis and precipitation rates [14].

The derivation of the field equations follows from Fauriel and Laloui [8] although some assumptions have been revisited (see “Appendix A”). The conceptual framework is based on the compositional approach that consists in establishing the balance of components in the system of interest [19]. A water-saturated porous medium is considered in which the state variables are the displacements of the solid skeleton, \(\mathbf{u}\), pore fluid pressure, \({p}_{{\text{w}}}\), attached bacteria concentration, \({c}^{{\text{B}^{\prime}}}\), calcite concentration, \({c}^{{\text{C}}}\), suspended bacteria concentration \({c}^{{\text{B}}}\), suspended urea \({c}^{{\text{U}}}\) and calcium \({c}^{{\text{R}}}\) (reactants) concentration, and suspended ammonium (by-product) concentration \({c}^{{\text{P}}}\). For further details on the conceptual approach, the reader is referred to Fauriel and Laloui [8].

2.2 Balance equations

The general form from which the field equations are derived follow from the formulation of mass balance for each species, which can be expressed as:

$$\frac{\partial }{\partial t}\left({\theta }_{\alpha }{\rho }_{\alpha }^{\gamma }\right)=-\nabla \cdot {\theta }_{\alpha }{(}{\rho }_{\alpha }^{\gamma }{\mathbf{q}}_{\alpha }+{\mathbf{j}}_{\alpha }^{\gamma })-{\Lambda }_{\alpha \beta }^{\gamma }+{\Omega }_{\alpha }^{\gamma }$$
(4)

where \(t\) is time, subscript \(\alpha =s,w\) refers to the phase, superscript \(\gamma =C,U,B,B^{\prime}\) to the species, \(\theta\) is the volume fraction, \(\rho\) refers to density \(\mathbf{q}\) is the advective flux, \(\mathbf{j}\) is the sum of dispersive and diffusive flux, \({\Lambda }_{\alpha \beta }^{\upgamma }\) refers to the phase change from \(\alpha\) to \(\beta\) phase and \({\Omega }_{\alpha }^{\gamma }\) accounts for the production of \(\gamma\) in \(\alpha\) phase (e.g. sink, sources or bio-chemical reactions).

The mass balance equation of water is formulated considering that water mass is consumed during the hydrolysis of urea, in consistency with the overall chemical reaction (Eq. 3). This process is usually neglected in existing models [8, 18, 32]. Although water is always available in a saturated porous medium, accounting for the water consumption is necessary to avoid unrealistic generation of water flux or pressures due to the decrease in porosity by calcite precipitation. Some additional assumptions have also been adopted for the present study: fluid density variation with concentration is neglected, the volume of bacteria is assumed negligible and it is assumed that the solid phase is incompressible. With these premises, the final mass balance of the mixture is given by:

$$\frac{\partial {\epsilon }_{{\text{v}}}}{\partial t}-\nabla \cdot {\mathbf{q}}_{{\text{w}}}+n{\beta }_{{\text{w}}}\frac{\partial {p}_{{\text{w}}}}{\partial t}=\left(\frac{2{m}^{{\text{w}}}+{m}^{{\text{r}}}-2{m}^{{\text{p}}}}{{\rho }_{{\text{w}}}}-\frac{{m}^{{\text{c}}}}{{\rho }_{{\text{c}}}}\right){k}_{{\text{rea}}}n$$
(5)

where \({\epsilon }_{{\text{v}}}\) is the volumetric strain, \(t\) is time, \({\mathbf{q}}_{{\text{w}}}\) is the fluid (water) flux, \(n\) is the porosity, \({\beta }_{{\text{w}}}\) is the water compressibility, \({m}^{{\text{w}}}\), \({m}^{{\text{r}}}\) and \({m}^{{\text{c}}}\) are water, reactant and calcite molecular mass, respectively, \({\rho }_{{\text{w}}}\) and \({\rho }_{{\text{c}}}\) are the fluid and calcite density, respectively, and \({k}_{{\text{rea}}}\) is the hydrolysis and precipitation rate.

The bacteria are assumed to attach to the soil grains following an attachment rate, \({k}_{{\text{att}}}^{{\text{B}}}\), and to decay with time following a decay rate \({k}_{{\text{d}}}^{{\text{B}}}\).The equations describing the solute and attached bacteria are, respectively, given by:

$$\frac{\partial {c}^{{\text{B}}}}{\partial t}=-{\mathbf{v}}_{{\text{w}}}\nabla {{\text{c}}}^{{\text{B}}}-\nabla \cdot {\mathbf{j}}^{{\text{B}}}-\left(\frac{1}{n}\frac{\partial n}{\partial t}+{k}_{{\text{att}}}^{{\text{B}}}+{k}_{{\text{d}}}^{{\text{B}}}\right){c}^{{\text{B}}}$$
(6)
$$\frac{\partial {{c}^{{\text{B}^{\prime}}}}}{\partial t}=\frac{1}{{\rho }_{{\text{c}}}}\left(-{k}_{{\text{rea}}}\frac{n}{1-n}{m}^{{\text{c}}}\right){{c}^{{\text{B}^{\prime}}}}+\frac{n}{1-n}{k}_{{\text{att}}}^{{\text{B}}}{c}^{{\text{B}}}-{k}_{{\text{d}}}^{{\text{B}}}{c}^{{{\text{B}^{\prime}}}}$$
(7)

where \({\mathbf{v}}_{{\text{w}}}={\mathbf{q}}_{{\text{w}}}/n\) is the advective velocity and \({\mathbf{j}}^{{\text{B}}}\) is the non-advective flux of bacteria.

Transport (composed by advection, dispersion and diffusion) and reaction of solutes, which are \(\gamma =C,U,R\) and \(P\), are expressed by the following equation:

$$\left(1+{K}_{{\text{d}}}^{\gamma }\right)\frac{\partial {c}^{\gamma }}{\partial t}=-{\mathbf{v}}_{{\text{w}}}\nabla {c}^{\gamma }-\nabla \cdot {\mathbf{j}}^{\gamma }-\frac{1}{n}\frac{\partial n}{\partial t}\left(1+{K}_{{\text{d}}}^{\gamma }\right){c}^{\gamma }-{r}^{\gamma }n{k}_{{\text{rea}}}$$
(8)

where \({K}_{{\text{d}}}^{\gamma }\) is the partitioning coefficient between mobile and immobile concentration of the \(\gamma\) solute, \({c}^{\gamma }\) is the solute concentration and \({r}^{\gamma }\) is the stoichiometry coefficient of the solute in the reaction expressed by Eq. 3, i.e. \({r}^{{\text{U}}}={r}^{{\text{C}}}=+1\), and \({r}^{{\text{P}}}=-2\).

The porosity rate \(\frac{\partial n}{\partial t}\), which appears in the bacteria and solute transport equations, depends on the deformation of the porous skeleton and the precipitated calcite according to:

$$\frac{\partial n}{\partial t}=\left(1-n\right)\frac{\partial {\epsilon }_{{\text{v}}}}{\partial t}-\frac{n{k}_{{\text{rea}}}{m}^{{\text{c}}}}{{\rho }_{{\text{c}}}}$$
(9)

It is assumed that the produced calcite remains attached at the reaction location. Its evolution is given by:

$$\frac{\partial {c}^{\text{C}}}{\partial t}=\frac{1}{{\rho }_{{\text{s}}}}\left(-{k}_{{\text{rea}}}\frac{n}{1-n}{m}^{{\text{c}}}\right){{c}^{{\text{C}}}}+{k}_{{\text{rea}}}n$$
(10)

where the first term on the right-hand side expresses the change in concentration due to the increase of grain surfaces. Finally, the momentum balance of the mixture is included in the system, which establishes the equilibrium of stresses:

$$\nabla \cdot \left({{\varvec{\upsigma}^{\prime}}}-\mathbf{m}{p}_{{\text{w}}}\right)+\rho \mathbf{g}=0$$
(11)

where \({{\varvec{\upsigma}}}^{\prime}\) is the effective stress tensor, \(\mathbf{m}\) is the identity tensor, and \(\mathbf{g}\) is the gravity.

2.3 Constitutive equations for transport

The constitutive equations used to express the balance equations in terms of the primary variables are described in this section. Water flow is expressed through Darcy’s law:

$${\mathbf{q}}_{{\text{w}}}=-{\mathbf{K}}_{{\text{w}}}\left(\nabla {p}_{{\text{w}}}+{\rho }_{{\text{f}}}g\nabla z\right)$$
(12)

where \({\mathbf{K}}_{{\text{w}}}\) is the hydraulic conductivity tensor and \(z\) is the coordinate in the direction of gravity. The hydraulic conductivity evolves with the porosity following the Kozeni–Carman equation:

$${\mathbf{K}}_{{\text{w}}}={\mathbf{K}}_{0}\frac{{\left(1-{n}_{0}\right)}^{2}}{{n}_{0}^{3}}\frac{{n}^{3}}{{\left(1-n\right)}^{2}}$$
(13)

where \({\mathbf{K}}_{0}\) is the initial hydraulic conductivity and \({n}_{0}\) is the initial porosity.

To keep the formulation general enough to encompass sands and fine-grained soils, non advective transport is considered. Non-advective fluxes are modelled as \({\mathbf{j}}^{\gamma }=-{\mathbf{D}}^{\gamma }\nabla {c}^{\gamma }\), where \({\mathbf{D}}^{\gamma }\) is the hydrodynamic dispersion coefficient of component \(\gamma\). This coefficient accounts for diffusive and dispersive fluxes as:

$$\mathbf{D}={D}_{\gamma }\mathbf{I}+\frac{\left({\alpha }_{{\text{L}}}-{\alpha }_{{\text{T}}}\right)\left({\mathbf{v}}_{{\text{w}}}\cdot {\mathbf{v}}_{{\text{w}}}\right)}{\left|{\mathbf{v}}_{w}\right|}+{\alpha }_{L}\left|{\mathbf{v}}_{w}\right|\mathbf{I}$$
(14)

where \({D}_{\gamma }\) is the molecular diffusion coefficient and \({\alpha }_{{\text{L}}}\) and \({\alpha }_{{\text{T}}}\) are the lateral and transversal dispersion coefficients, respectively.

The bacterium involved in most MICP applications is Sporosarcina pasteurii for which Tobler et al. [26] showed the importance of attachment and retention on their transport dynamics. Attachment of bacteria is assumed to follow an equilibrium isotherm with a maximum attached concentration, \({B}^{\prime}_{max}\), according to the model proposed by Torkzaban et al. [27]:

$${k}_{{\text{att}}}={k}_{a,{\text{max}}}\left(1-\frac{{c}^{{B}^{\prime}}}{{B}^{\prime}_{{\text{max}}}}\right)$$
(15)

As shown by Torkzaban et al. [27], \({k}_{{\text{a}},{\text{max}}}\) and \({B}^{\prime}_{\text{max}}\) are attachment coefficients that depend on the ionic strength of the solution, which increases the bacteria retention, as well as on the pore structure. The reaction rate, \({k}_{{\text{rea}}}\) is taken as a Michaelis–Menten form:

$${k}_{{\text{rea}}}={U}_{{\text{max}}}\frac{{c}^{R}}{{{c}^{R}+K}^{m}}$$
(16)

where \({K}^{m}\) establishes the concentration at which the reaction rate reaches 50% of \({U}_{{\text{max}}}\), which is the maximum urease activity. \({U}_{{\text{max}}}\) depends on the bacteria concentration as:

$${U}_{{\text{max}}}={u}_{sp}\left(\frac{n}{1-n}{{c}^{{B}^{\prime}}}+{c}^{B}\right)$$
(17)

No additional decay term with time is included in the urease activity as this is already accounted for by the constant \({k}_{{\text{d}}}\).

2.4 Elasto-plastic constitutive model

In this section, an elastoplastic model for the behaviour of bio-cemented sands is described. As an alternative to the available models based on critical state theory, simplicity is considered an essential aspect for its use in engineering applications. Calcite precipitation modifies the mechanical response of soils as a result of the inter-particle bonding and the densification. Figure 1 shows the strain–stress response measured in a drained, conventional triaxial compression test (CTC) of an untreated and treated sand [24]. The figure displays the main trends that are observed when comparing the typical axial strain—deviatoric stress relation. With respect to the untreated state, the following differences become readily visible: (i) increase of stiffness in the initial loading phase; (ii) a higher peak strength for the bio-cemented samples; (iii) brittle response upon failure of the bio-cemented sand; and (iv) similar strength at residual state.

Fig. 1
figure 1

Typical stress–strain response in drained triaxial tests of bio-cemented sand compared to the untreated state. Data from Terzis and Laloui [24]

Full size image

The constitutive equations are expressed in terms of the stress invariants mean effective stress \({p}^{\prime}={\text{tr}}\left({{\varvec{\upsigma}^{\prime}}}\right)/3\), deviatoric stress \(q=\sqrt{3}J\), and Lode’s angle \(\theta = \frac{1}{3}\sin^{ - 1} \left( {{\raise0.7ex\hbox{${3\sqrt 3 \det {\mathbf{s}}}$} \!\mathord{\left/ {\vphantom {{3\sqrt 3 \det {\mathbf{s}}} {2J^{3} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${2J^{3} }$}}} \right)\), where \({\mathbf{s}} = {\mathbf{\sigma^{\prime}}} - p^{\prime}{\mathbf{I}}\), and \(J=\sqrt{{\text{tr}}({\mathbf{s}}^{2})/2}\). Similarly, the volumetric strain \({\epsilon }_{{\text{v}}}={\text{tr}}\left({\varvec{\upepsilon}}\right)\) and deviatoric strain \({\epsilon }_{q}=\sqrt{2{\text{tr}}({{\varvec{\upgamma}}}^{2})/3}\) invariants are defined, where \({\varvec{\upgamma}}={\varvec{\upepsilon}}-{\epsilon }_{{\text{v}}}\mathbf{I}/3\).

In view of the response of treated samples and for the purposes of the present work, linear elasticity is used:

$${\text{d}}{\epsilon }_{{\text{v}}}=\frac{{\text{d}}{p}^{\prime}}{K},\quad {\text{d}}{\epsilon }_{q}=\frac{{\text{d}}q}{3G}$$
(18)

where \(K\) is the bulk modulus and \(G\) is the shear modulus for a given calcite content. The increase in stiffness of treated specimens with respect to the untreated sand is not linearly proportional to the amount of calcite. Indeed, the effect of bio-cementation is known to increase non-linearly for increasing calcite content in the range of tested calcite content [17, 21, 24]. Accordingly, a generalization of the expression proposed by Fauriel & Laloui [8] is used:

$$G={{G}_{0}(1+{\omega }_{{\text{c}}}{m}^{{\text{c}}}{c}^{{\text{C}}} )}^{2}, K={{K}_{0}(1+{\omega }_{{\text{c}}}{m}^{{\text{c}}}{c}^{{\text{C}}} )}^{2}$$
(19)

where \({G}_{0}\) and \({K}_{0}\) are the shear and bulk moduli of the untreated state and \({\omega }_{c}\) is a material parameter. \({\omega }_{c}\) expresses the efficiency of the precipitated calcite in bonding the soil particles and improving the elastic properties. Based on shear wave measurements of \(G\) values, Fauriel and Laloui [8] proposed \({\omega }_{{\text{c}}}=0.062\), which can be used as a first approximation. Otherwise, in case experimental results at different calcite contents are available, it can be calibrated with the elastic response during triaxial tests.

Most MICP treatments target sands and silty-sands. For the present study and in view of simplicity for its use in engineering applications, the yield function is represented by a cohesive-frictional expression:

$${f}_{Y}=q-\eta \left({p}^{\prime}+{p}^{\prime}_{{\text{t}}}\right)$$
(20)

where \(p^{\prime}_{{\text{t}}}\) is the isotropic tensile strength and \(\eta\) defines the slope of the shear strength envelope in the \(\left( {q,p^{\prime}} \right)\) plane. In order to consider the effects of the stress path on the strength, the parameter \(\eta\) is established as a function of Lode’s angle, \(\theta\) following Ref. [28]:

$$\eta ={a}_{{\text{L}}}{\left[1-{b}_{{\text{L}}}{\text{sin}}\left(3\theta \right)\right]}^{-0.229}$$
(21)

where \({a}_\text{L}\) and \({b}_\text{L}\) can be defined as a function of the shear strength angle \(\phi^{\prime}\):

$$a_{{\text{L}}} = \frac{{6\sin \phi^{\prime}}}{{3 - \sin \phi^{\prime}}}\frac{1}{{\left( {1 + b_{{\text{L}}} } \right)^{ - 0.229} }}, \quad b_{{\text{L}}} = \frac{{\left( {{\raise0.7ex\hbox{${3 + \sin \phi^{\prime}}$} \!\mathord{\left/ {\vphantom {{3 + \sin \phi^{\prime}} {3 - \sin \phi^{\prime}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${3 - \sin \phi^{\prime}}$}}} \right)^{1/ - 0.229} - 1}}{{\left( {{\raise0.7ex\hbox{${3 + \sin \phi^{\prime}}$} \!\mathord{\left/ {\vphantom {{3 + \sin \phi^{\prime}} {3 - \sin \phi^{\prime}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${3 - \sin \phi^{\prime}}$}}} \right)^{1/ - 0.229} + 1}}$$
(22)

The material peak strength usually increases exponentially with calcite content [9, 17, 24]. Experimental evidence suggests that this increase is mostly associated to a change in cohesion, whereas the shear strength angle at failure does not vary significantly within ranges of moderate cementation [3, 36]. For practical purposes, in the present study the shear strength angle is assumed to remain unchanged due to bio-cementation.

The increase in cohesion is accounted with an empirical exponential equation by making the isotropic tensile strength a function of the current calcite content as:

$$p^{\prime}_{{\text{t}}} = p^{\prime}_{{{\text{t0}}}} \left( {m_{{\text{c}}} c^{{C}} } \right)^{{n_{{\text{t}}} }} d$$
(23)

where \(p^{\prime}_{{{\text{t}}0}}\) and \(n_{{\text{t}}}\) define the evolution of tensile strength with calcite and \(d\) is a variable that accounts for the strength decrease of treated samples once the peak strength has been attained.

As an irreversible process, similar to damage, it is assumed that \(d\) is controlled by plastic deviatoric strains following an exponential decay as:

$$d=1-{\beta }_{{\text{d}}}\left[1-{\text{exp}}\left(-\frac{{\chi }^{2}}{{\alpha }_{{\text{d}}}^{2}}\right)\right]$$
(24)

where \({\beta }_{{\text{d}}}\) establishes the residual strength of treated samples (\(0\le {\beta }_{{\text{d}}}\le 1\)), \({\alpha }_{{\text{d}}}\) the rate of damage (which represents the debonding process) with plastic strains, which can be calibrated with triaxial tests, and \(\chi ={\int }_{t}\Vert {\text{d}}{\epsilon }_{{\text{q}}}^{{\text{p}}}\Vert {\text{d}}t\) accounts for the dependency of accumulated plastic strains. Equation 24 is similar to the expression proposed in [20] to model the damage process in shales, and it allows reproducing the post-peak response of cemented samples, in the \(({\epsilon }_{{\text{q}}},q)\) plane, as observed experimentally in triaxial shearing tests.

While untreated samples may show either dilatant or contractive responses upon shearing, the post-peak response of cemented samples until the residual strength is reached is frequently associated with dilatancy [9, 24]. To encompass both responses, the flow rule is not associated and it takes the following form:

$$\frac{{{\text{d}}\epsilon_{{\text{v}}}^{{\text{p}}} }}{{{\text{d}}\epsilon_{{\text{q}}}^{{\text{p}}} }} = - \left( {A + A_\text{c} \frac{{p_{{\text{t}}} }}{{p_{{{\text{t}}0}} }}} \right)$$
(25)

where \(A\) is the dilatancy angle of the untreated material (here defined as \(- \frac{{{\text{d}}\varepsilon_{{\text{v}}}^{{\text{p}}} }}{{{\text{d}}\varepsilon_{{\text{q}}}^{{\text{p}}} }}\)) and \(A_{{\text{c}}}\) is a material parameter used to quantify the increase in dilatancy due to the cementation. This empirical expression allows reproducing different post-peak dilatancy responses and their dependency on the cementation level by means of \({p}_{{\text{t}}}\). According to Eq. 25, as tensile strength decreases due to plastic strains, the dilatancy exhibited tends towards that of the untreated sand, as observed in [9, 25]. The influence of post-peak and dilatancy parameters \({\alpha }_{d}\) and \({A}_{c}\) for a treated specimen is shown in Fig. 2 together with the reference response for the untreated state. The ranges of values shown in Fig. 2 for \({\alpha }_{{\text{d}}}\) and \({A}_{{\text{c}}}\) include a wide variety of responses and can be used as a reference for calibration purposes.

Fig. 2
figure 2

Influence of parameters \({A}_{{\text{c}}}\) and \({\alpha }_{{\text{d}}}\) on the post-peak and dilatant response predicted by the model

Full size image

The elastoplastic model requires 10 input parameters to be specified (\(K_{0} , G_{0} , \omega_{{\text{c}}} ,p^{\prime}_{{{\text{t}}0}} , n_{{\text{t}}} ,\phi^{\prime}, \alpha_{{\text{d}}} ,\beta_{{\text{d}}} ,A,A_{{\text{c}}} )\) all of which can be identified from drained triaxial tests. Parameters \({K}_{0}, {G}_{0}, {\phi }{\prime}\) and \(A\) are derived from tests on untreated specimens, whereas the remaining parameters require the availability of results from at least one treated sample for which the calcite content is known. Further details on the calibration procedure are given in Sect. 3.

2.5 Finite element implementation

The finite element formulation of the problem is outlined in Fauriel and Laloui [8], and only a brief description is presented in this section including the main differences. Two types of finite element are used to solve the equations, one for the HM problem and another for the BC problem. The Bubnov–Galerkin method is applied for the flow and the equilibrium equations (HM), whereas the Petrov–Galerkin method is applied to the reactive transport equations of subsystem BC [1]. From an implementation point of view, the main difference with respect to Fauriel and Laloui [8] is that the mechanical constitutive model is elastoplastic. The integration of elastoplasticity is solved with an explicit Euler algorithm with a substepping and error control scheme, based on that presented by Sloan [22]. In this class of schemes, the error is defined as the difference between a first order and a second-order Euler solution. If for a given time step the difference (defined as error) is above the specified tolerance, the step is divided into smaller steps until the error is below tolerance (according to Ref. [22] typically 10–3). Such substepping algorithm allows a flexible time step for the mechanical problem.

For most problems in which the concentrations of urea and calcium are injected equimolarly, a single equation can represent both components alleviating computational resources [8]. In these cases, the final system of algebraic equations that describe the entire system is:

$${\left[\begin{array}{cc}\mathbf{S}+\theta \Delta t\mathbf{H}& {\mathbf{Q}}^{\mathbf{T}}\\ \theta \Delta t\mathbf{H}& 0\end{array}\right]}_{{\varvec{n}}+{\varvec{\theta}}}{\left\{\begin{array}{c}\mathbf{u}\\ {\mathbf{p}}_{\mathbf{w}}\end{array}\right\}}_{{\varvec{n}}+1}+{\left[\begin{array}{cc}0& 0\\ 0& \Delta t{\mathbf{F}}_{\mathbf{i}\mathbf{n}\mathbf{t}}\end{array}\right]}_{{\varvec{n}}+{\varvec{\theta}}}={\left[\begin{array}{cc}\mathbf{S}-\left(1-\theta \right)\Delta t\mathbf{H}& {\mathbf{Q}}^{\mathbf{T}}\\ -\left(1-\theta \right)\Delta t\mathbf{Q}& 0\end{array}\right]}_{n+\theta }{\left\{\begin{array}{c}\mathbf{u}\\ {\mathbf{p}}_{\mathbf{w}}\end{array}\right\}}_{n}+{\left\{\begin{array}{c}\Delta t{\mathbf{f}}_{{{\varvec{p}}}_{{\textbf{w}}}}\\\Delta t{\mathbf{f}}_{{\textbf{u}}}\end{array}\right\}}_{n+\theta }$$
(26)
$$\begin{aligned} & \left[ {\begin{array}{*{20}c} {{\varvec{M}}_{{\textbf{b}}} + 0.5{{\varvec{\Delta}}}{\varvec{tK}}_{{\textbf{b}}} + 0.5{\Delta }t{\varvec{L}}_{{\textbf{b}}} } & 0 & 0 \\ 0 & {{\varvec{M}}_{{\textbf{r}}} + 0.5{{\varvec{\Delta}}}{\varvec{tK}}_{{\textbf{r}}} + 0.5{\Delta }t{\varvec{L}}_{{\textbf{r}}} } & 0 \\ 0 & 0 & {{\varvec{M}}_{{\textbf{p}}} + 0.5{{\varvec{\Delta}}}{\varvec{tK}}_{{\textbf{p}}} + 0.5{\Delta }t{\varvec{L}}_{{\textbf{p}}} } \\ \end{array} } \right]_{n + \theta } \left\{ {\begin{array}{*{20}c} {{\mathbf{C}}_{{\textbf{b}}} } \\ {{\mathbf{C}}_{{\textbf{r}}} } \\ {{\mathbf{C}}_{{\textbf{p}}} } \\ \end{array} } \right\}_{n + 1} \\ & \quad = \left[ {\begin{array}{*{20}c} {{\varvec{M}}_{{\textbf{b}}} - 0.5{{\varvec{\Delta}}}{\varvec{tK}}_{{\textbf{b}}} + 0.5{\Delta }t{\varvec{L}}_{{\textbf{b}}} } & 0 & 0 \\ 0 & {{\varvec{M}}_{{\textbf{r}}} - 0.5{{\varvec{\Delta}}}{\varvec{tK}}_{{\textbf{r}}} + 0.5{\Delta }t{\varvec{L}}_{{\textbf{r}}} } & 0 \\ 0 & 0 & {{\varvec{M}}_{{\textbf{p}}} - 0.5{{\varvec{\Delta}}}{\varvec{tK}}_{{\textbf{p}}} + 0.5{\Delta }t{\varvec{L}}_{{\textbf{p}}} } \\ \end{array} } \right]_{n + \theta } \left\{ {\begin{array}{*{20}c} {{\mathbf{C}}_{{\textbf{b}}} } \\ {{\mathbf{C}}_{{\textbf{r}}} } \\ {{\mathbf{C}}_{{\textbf{p}}} } \\ \end{array} } \right\}_{n} \\ & \quad \quad + \left\{ {\begin{array}{*{20}c} {{\Delta }t{\mathbf{f}}_{{\textbf{b}}} } \\ {{\Delta }t{\mathbf{f}}_{{\textbf{r}}} } \\ {{\Delta }t{\mathbf{f}}_{{\textbf{p}}} } \\ \end{array} } \right\}_{n + \theta } \\ \end{aligned}$$
(27)

where \(\mathbf{S}\) is the compressibility matrix; \(\mathbf{H}\) is the permeability matrix; \({\mathbf{F}}_{\mathbf{i}\mathbf{n}\mathbf{t}}\) is the internal force rate vector; \(\mathbf{Q}\) is the hydro-mechanical coupling matrix; \(\mathbf{u}\) is the vector of displacements; \({\mathbf{f}}_{{\textbf{u}}}\) and \({\mathbf{f}}_{{{\varvec{p}}}_{{\textbf{w}}}}\) are the external force and flux rates; \({\mathbf{M}}_\text{b}, {\mathbf{M}}_\text{r}\) and \({\mathbf{M}}_{p}\) are the bacterial and the chemical mass matrices; \({\mathbf{K}}_\text{b}, {\mathbf{K}}_\text{r}\) and \({\mathbf{K}}_\text{p}\) are the bacterial and the chemical stiffness matrices; \({\mathbf{L}}_\text{b}, {\mathbf{L}}_\text{r}\) and \({\mathbf{L}}_\text{p}\) are the bacterial and chemical matrices; \({\mathbf{C}}_\text{b}, {\mathbf{C}}_\text{r}\) and \({\mathbf{C}}_\text{P}\) are the bacteria, reactants (calcium and urea) and by-product (ammonium) concentration vectors; and \({\mathbf{f}}_\text{b}, {\mathbf{f}}_\text{r}\) and \({\mathbf{f}}_\text{p}\) are the bacterial and the chemical force vectors. This scheme is solved with the finite element code LAGAMINE [2]. The HM subproblem is solved monolithically and the subproblem for each solute species (BC) sequentially, all using a Newton–Raphson scheme. Then, once convergence is achieved for both subproblems, the global BC-HM solution is obtained iteratively for each time step. When the same mesh and integration points are used for the HM and the BC problem, the results of each problem are directly used in the other problem. If a different mesh is used for the transport and HM problems, the values of the state variables are interpolated according to the method presented in [1]. In the present study, 8-node elements have been used for the HM problem and 4-node elements for the BC problem (whose coordinates coincide with the corner nodes of the HM elements), both types of elements with 4 integration points.

3 Model calibration and performance

This section presents the estimation of reactive transport and mechanical parameters as well as typical responses of the model. Two experimental campaigns reported in the literature were used to separately calibrate the reactive transport parameters and the elastoplastic parameters.

The first application case is based on the study reported by Martinez et al. [15] in which five pairs of columns were tested by applying different injection protocols. The treated material was Ottawa 50–70 sand fully saturated. The columns were approximately 0.5 m height with an internal diameter of 0.05 m and were extensively monitored along the height giving a significant amount of information. A scheme of the test is shown in Fig. 3. This information allowed for a better understanding of the multiphysical processes occurring at different stages of the treatment protocol. The initial information for defining the input parameters is shown in Table 1, as reported in Ref. [15] for the sand and Ref. [8] for the components.

Fig. 3
figure 3

Scheme of the column tests reported in Martinez et al. [9] and sequence followed in the analysed test

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Table 1 Basic parameters from the literature for the chemo-hydraulic model
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The results of test 3B were used to calibrate the remaining bio-chemo-hydraulic parameters of the numerical model. The treatment protocol consisted of a first injection of bacterial solution from the top side of the column at a flow rate of 10 mL/min during 1 h after which 8 h of retention period (i.e. no flow) was allowed. Subsequently the cementation solution, consisting of 0.05 M of urea and 0.05 M calcium chloride, was injected from the bottom side of the column at a flow rate of 2.2 mL/min for 10 h.

A qualitative curve-fitting was used to calibrate the required bio-chemical parameters, which are \({u}_{{\text{sp}}}, {K}_{{\text{att}}}, {K}_{{\text{dec}}}\) and \({B}^{\prime}_{{\text{max}}}\). The calibration procedure starts assuming that no reaction nor attachement takes place. The bacteria transport parameters, which are the attachment and decay rates, are determined from the first phase by matching the breakthrough curve at the beginning of the cementation solution injection as shown in Fig. 4. Afterwards, the maximum ureolysis rate, \({u}_{{\text{sp}}}\), was calibrated to fit the final calcite precipitated inside the column, as shown in Fig. 5. Overall, given the qualitatively satisfactory match that was achieved with the model, the processes of interest (bacteria transport, solution transport and reaction rate) can be well reproduced, with parameter values that are in line with those reported in the literature (Table 2). The quantitative discrepancy between the simulated and the measured breakthrough curve is probably due to the fact that during the time that elapsed between the end of injection and the flushing stage, the bacteria did not tend to attach in the experiment, whereas the model overestimates the retention during this phase.

Fig. 4
figure 4

Model calibration of bacteria transport parameters against the breakthrough curve measured from the top side by Martinez et al. [15]

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Fig. 5
figure 5

Final profile of calcite precipitated in the column. Model results are compared to the experimental measurements by Martinez et al. [15]

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Table 2 Bio-chemical parameters calibrated from test 3B
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The mechanical parameters that govern the stress–strain and strength responses were calibrated against results from triaxial drained tests reported by Feng and Montoya [9]. The tests were performed on Ottawa 50–70 sand, the same material used in the column experiment studied for the reactive transport. Three levels of cementation, denoted in [9] as lightly, moderately and heavily treated, were tested at confining pressures of 100 kPa, 200 kPa and 400 kPa. The untreated material was also tested at the same confinement pressures for comparison purposes.

Figure 6 shows the model calibration to the experimental results. The parameters were first calibrated for the untreated tests at 100 and 200 kPa of confinement to obtain the behaviour at untreated state. The parameters defining the behaviour of treated samples (\(\omega_{{\text{c}}} ,p^{\prime}_{{{\text{t}}0}} , n_{{\text{t}}} ,\alpha_{{\text{d}}} ,\beta_{{\text{d}}} ,A_{{\text{c}}}\)) were calibrated using the tests performed on heavily cemented (defined in Ref. [9] as 4.3% to 5.3% CaCO3 content) samples at the same confining pressures of 100 and 200 kPa. As shown in Fig. 6, a good match was obtained for both the stress–strain response and the dilatant behaviour of the samples. It is of particular interest the increase in dilatancy that is exhibited by the treated samples, which occurs mostly during the post-peak softening response. The calibrated parameters are summarised in Table 3.

Fig. 6
figure 6

Model calibration of the stress–strain response of Ottawa sand against experimental results by Feng and Montoya [9]

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Table 3 Bio-mechanical parameters calibrated against Feng and Montoya [9] tests
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The predictive potential of the model was assessed against test results of samples treated at lower calcite contents as well as higher confining pressures. Tests at confining pressures of 100, 200 and 400 kPa with samples treated between 2.4 and 5% of calcite content were considered. The results are shown in Fig. 7. Globally the tests were well reproduced, in spite of quantitative discrepancies for the test at 400 kPa. This is because the post-failure brittle response of treated samples, which was calibrated at low confinements, becomes less pronounced when failure occurs at higher confinements. The discrepancy in terms of volumetric strains for the test at 2.4% of calcite content could be explained by a different initial relative density of the material, which implies that the relatively low cementation content did not have the same effect as it would have in an initially denser material. It should be noted that in case more precise results were desired other reference models for sand, incorporating for instance the effects of relative density, could be used to better fit the material response.

Fig. 7
figure 7

Model predictions of the stress–strain response at different confinement and cementation levels. Experimental results obtained by Feng and Montoya [9] are shown for comparison purposes

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4 Application to a predictive case

The application of the formulation for geotechnical applications is demonstrated in this section with a synthetic case of a shallow foundation. The modelling set-up and finite element mesh is shown in Fig. 8. The foundation has a 1 m width and lies on a homogeneous sand formation. The material parameters calibrated for Ottawa sand 50–70 (Tables 1, 2, 3) are used to simulate the soil layer. A constant load of 1 kPa is imposed at the surface of the entire model. Three injection points are used, located at 0.25 m below the foundation surface, one located at the symmetry axis, coincident with the centre of the foundation, and the other two, placed symmetrically, at 0.5 m from the centre. Different injection set-ups are tested, equivalent to typical column experiments reported in the literature. The domain is discretized into 2100 8-nodes hydro-mechanical elements, with smaller elements near the foundation, and the same space discretization is used for the bacteria and chemical transport problem.

Fig. 8
figure 8

Finite element mesh and boundary conditions of the modelling set-up

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The target volume of soil to be treated is assumed to correspond to a square zone with a side of 1.5 times the foundation width, equivalent to 2.25 m2 per linear meter, which for a porosity of 0.38, involves a pore volume of PV = 0.86 m2 per linear meter to be injected. Given that the injection point is located at a shallower depth with respect to the radius of the reference volume, some loses are expected, although these compensate the increase in strength by having the maximum calcite content closer to the foundation. Detailed studies can be performed to further optimize this configuration, which were left beyond the scope of this study.

Four different injection protocols, as shown in Fig. 9, were modelled with the aim of studying the settlement response to different calcite distributions. Case #1 considers 4 PVs of injection for both bacteria and reactants (i.e. calcium and urea) at a reference flow rate of 0.21 m3/h for bacteria and 0.07 m3/h for reactants only from the central well. Cases #2 and #3 also involve 4 PV injected but they attempt to better distribute the bacteria and reactants by using both the central and lateral wells to maximize the extent of the injected volumes. Case #4 is used to evaluate the increase in strength with increasing injection volume. To ensure consistency with the calibrated parameters, in all cases the species are injected at the same concentration as that from the column test performed by Martinez et al. (2013), which is 0.4OD of bacteria and 0.05 M of urea and calcium. Once the injection of reactants is completed, a vertical displacement on top of the loading area (shallow foundation) is imposed until the model fails to obtain convergence.

Fig. 9
figure 9

Cases studied of different treatment protocols

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4.1 Results

Figure 10 shows the distribution of calcite precipitated for each case. Since the soil is considered as saturated, the injected fluid tended to flow towards the surface as a result of the pressure gradient. Overall the expected treated zone is achieved in all cases although with significant differences in the distribution and homogeneity of the treatment. Case 1 resulted in higher precipitation towards the injection point, which was expected since both bacteria and the cementation solution are injected from the same source. It appears that the precipitated calcite does not exceed 1.3%, which is a result of the bacteria attachment reaching the maximum value calibrated. In Case 2, a zone of low treatment is obtained at the midpoint between the central and the outer injection points, a consequence of the flow pattern that does not allow the delivery of reactants at that region. This effect is overcome in Cases 3 and 4, where a zone of higher calcite precipitation is obtained between the injection points as a result of the alternating scheme of injections. The resulting distribution of calcite is more heterogeneous than in the other two cases. Consequences of these patterns are further discussed in the following.

Fig. 10
figure 10

Predicted distribution of calcite precipitated in the 4 cases

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The load–displacement curves that were obtained after simulating the foundation loading up to failure (i.e. untin no convergence could be obtained) are shown in Fig. 11. For comparison purposes, the untreated case at failure is also shown in Fig. 11 indicating that without treatment, a load of 115 kN leads to a settlement of 7 mm. Results from this case are taken as a benchmark to assess the increase in bearing capacity that is obtained after the different MICP treatment strategies. It appears that while there is an apparent increase of bearing capacity between the treated cases and the untreated one, only small differences exist between the treated cases. Case 3 lead to the least increase in bearing capacity with 100 kN inducing a settlement of 4 mm. In spite of doubling the amount of injected pore fluids in Case 4, the increase in bearing capacity is similar to that obtained with Cases 1 and 2, indicating that for the sole purpose of increasing bearing capacity, injection protocol is not the most optimal.

Fig. 11
figure 11

Load–displacement curves obtained from the modelling results of the 4 cases compared to the untreated state

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Figure 12 shows the distribution of vertical load in Case 1 at the end of the linear phase of the load–displacement curve and at 150 kN of loading. As it can be observed, although the load tends to be distributed towards the corner of the foundation, as it is expected for a homogeneous soil layer, the stiffness of the zones with higher calcite content tend to concentrate higher stresses, leading to the higher stiffness observed compared to the untreated case. After the shear strength of the untreated soil is reached, the stresses are mostly transferred through the treated domain and a zone with localised plastic strains is obtained. The plastic zone is aligned at the interface between the untreated and the treated zone, although this is very close to the corner of the foundation.

Fig. 12
figure 12

Case 1 results. a Load-displacment curve. b Vertical stress distribution at a load of 85 kN before plastic zones develop. c Vertical stress distribution after the development of plastic strains. The contour lines indicate the calcite content

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While in Case 1 the treated zone coincides with the foundation width, this is not the case in the other cases. Figures 13 and 14 show the plastic zones that develop during loading in Cases 2 and 4, respectively. Two states are shown, at the inflection point and at the end of the inflection point in the load–displacement curves as mentioned earlier. As it can be seen from the development of plastic strains, this transition phase corresponds to the development of the plastic strains. In Case 2 (Fig. 13), plastic points develop at the interface below the treatment zone and vertically at a certain distance from the foundation corner. As the load increases, it is the latter mechanism that fully develops, by exceeding the shear strength of the treated zone. As a consequence of the heterogeneous profile in calcite content that precipitated below the corner of the foundation, the failure mechanism in Case 2 is different from that observed in Case 4 (Fig. 14) where a clear vertical line of plastic state is reached as a result of the more homogeneous distribution of calcite. It is also noted that in this case the inflection point in the load–displacement curve is sharper than in Case 2, probably a consequence of the limited capacity of the stress to redistribute to higher strength zones. Nevertheless, the progressive yielding remains ductile, without any drop in vertical stress as displacements increase.

Fig. 13
figure 13

Case 2 results. a Load–displacement curve. b Plastic strains distribution at a load of 112 kN. c Plastic strains distribution after a load of 127 kN. The contour lines indicate the calcite content in % of mass

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Fig. 14
figure 14

Case 4 results. a Load-displacment curve. b Plastic strains distribution at a load of 95 kN. c Plastic strains distribution after a load of 104 kN. The contour lines indicate the calcite content in % of mass

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Table 4 presents a summary of the main results from the different cases. Because failure was reached at different load levels for each case, a comparison between cases is better done in terms of the settlement at a given load. At 80 kN of loading, the highest settlement obtained for the treated cases corresponds to Case 3 with 3.25 mm which represents a reduction of 19% with respect to the untreated case. The difference between the treatment cases is however much lower. While Case 4 resulted in 5.08 kg/m3 of precipitated calcite compared to 2.30 kg/m3 in Case 3, this only translated to a modest decrease of 0.3 mm in settlement at 80 kN, which represents a reduction of 10% of settlement due to an increase of 120% in precipitated calcite. In view of the higher efficiency of Case 2, it appears that the most optimal treatment for this set-up is to target a high calcite content below the foundation corner, where the failure mechanism tends to develop.

Table 4 Quantitative comparison of the different cases
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Although in this particular case the interest is in increasing bearing capacity, comparison between cases highlights that the model can be used to target an optimal treatment strategy depending on the intended geotechnical application. Indeed, assuming that the foundation is significantly stiffer than the soil layer, the degree of homogeneity in the treatment is not a priority. Similarly, other applications where a global failure mechanism is to be prevented, such as in some landslides, the failure surface can be targeted using similar injection schemes. For these applications, the formulation can be further enhanced towards better reproducing strain localisation which is a consequence of the particularly brittle response exhibited by bio-cemented soils. However, in other cases such as erosion protection, homogenisation might be one of the central criteria to be achieved by the treatment in order to minimise the existence of preferential flow paths. In this context, Cases 3 and 4, with alternating injection points would be preferable to Case 2, where a low calcite content zone remained between the injection points.

5 Conclusion

A complete bio-chemo-hydro-mechanical formulation has been presented for predicting the cementation extent, strength and deformation of MICP-treated soils. The formulation involves elasto-plastic stress–strain modelling, through a simple and efficient model that accounts for the stiffness and strength increase of sands due to calcite precipitation. The complete model is implemented using a finite element formulation allowing the analysis of boundary value problems. The formulation has been verified and calibrated with laboratory experiments for both reactive transport and mechanical behaviour of the treated soils.

The model was applied to study the load–displacement response of a shallow foundation on a soil treated with different bio-chemical injection protocols. Depending on the treatment protocol, different distributions of cementation is obtained which yield different mechanical properties such as stiffness and strength. Ultimately, the response upon loading depends on the chosen treatment strategy. In particular, stresses tend to concentrate towards highly cemented zones, and, as the bearing capacity is approached, the less cemented or untreated zones tend to reach a plastic state before the cemented areas. In some cases, the distribution of plastic strains is modified depending on the distribution of precipitated calcite, demonstrating the importance of considering the variable strength distribution to assess the treatment performance.

The model presented herein aims to serve as a robust tool to iterate between various bio-cementation contents and the associated spatial distributions to ultimately achieve the maximum in terms of desired geotechnical behaviour by minimizing the necessary resources to do so (i.e. reactant volumes, number of injection points etc.). The framework used in this study is considered to be general in the sense that more refined bio-chemical and elastoplastic models can be integrated into it depending on the problem to be analysed. It should be noted that the bio-chemical parameters used to perform the large-scale simulations come exclusively from one column experiment. It is very likely that scale effects or three-dimensional flow patterns play a role in particular for the attachment and the reaction rate as already pointed out in other studies. Further work should be directed towards evaluating the scale dependency of these and other parameters.