## Abstract

The work aims to numerically investigate the quasi-static response of partially fluid-saturated concrete under two-dimensional uniaxial compression at the mesoscale. We investigated how the impact of free pore fluid content (water and gas) affected the quasi-static strength of concrete. The totally and partially fluid-saturated concrete behavior was simulated using an improved pore-scale hydro-mechanical model based on DEM/CFD. The fluid flow concept was based on a fluid flow network made up of channels in a continuous region between discrete elements. A two-phase laminar fluid flow was postulated in partially saturated porous concrete with very low porosity. Position and volumes of pores/cracks were considered to correctly track the liquid/gas content. In both dry and wet conditions, a series of numerical simulations were performed on bonded granular specimens of a simplified spherical mesostructure that mimicked concrete. The effects of fluid saturation and fluid viscosity on concrete strength and fracture, and fluid pore pressures were investigated. It was found that each of those effects significantly impacted the hydro-mechanical behavior of concrete. Due to the rising fluid pressure in pores during initial specimen compaction under compressive loading that promoted a cracking process, the compressive strength increased as fluid saturation and fluid viscosity decreased.

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## Acknowledgements

The present study was supported by the research project “*Fracture propagation in rocks during hydro-fracking-experiments and discrete element method coupled with fluid flow and heat transport*” (years 2019–2023), and financed by the National Science Centre (NCN) (UMO-2018/29/B/ST8/00255).

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MK: Conceptualization, Methodology, Investigation, Writing— original draft, Writing—review & editing, JT: Conceptualization, Methodology, Writing—original draft, Writing—review & editing, MN: Investigation.

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## Appendices

### Appendix A

The DEM equations are listed below:

where \({\overrightarrow{F}}_{n}\)—the normal contact force, *U—*the overlap between discrete elements, \(\overrightarrow{N}\)—the unit normal vector at the contact point, \({\overrightarrow{F}}_{s}\)—the tangential contact force, \({\overrightarrow{F}}_{s,prev}\)—the tangential contact force in the previous iteration, \({\overrightarrow{X}}_{s}\)—the relative tangential displacement increment, *K*_{n}—the normal contact stiffness, *K*_{s}—the tangential contact stiffness, *E*_{c}*—*the elastic modulus of the particle contact, *υ*_{c}—the Poisson’s ratio of particle contact,* R*—the particle radius, *R*_{A} and *R*_{B} contacting particle radii, *μ*_{c}*—*the Coulomb inter-particle friction angle, \({F}_{max}^{s}\)—the critical cohesive contact force, \({F}_{min}^{n}\)—the minimum tensile force, *C*—the cohesion at the contact (maximum shear stress at zero pressure), and *T*—the tensile strength of the contact, \({\overrightarrow{F}}_{damp}^{k}\)—the dampened contact force, \({\overrightarrow{F}}^{k}\) and \({\overrightarrow{v}}_{p}^{k}\)-*k*th—the components of the residual force and translational particle velocity *v*_{p} and *α*_{d}—the positive damping coefficient smaller than 1 (sgn(\(\cdot\)) that returns the sign of the *k*th component of velocity).

### Appendix B

The hydraulic aperture *h* of the artificial channels 'S2S' is computed by a modified empirical formula developed by Hökmark et al. [85]:

where \({h}_{inf}\)—the hydraulic aperture for the infinite normal stress, \({h}_{0}\)—the hydraulic aperture for the zero normal stress, \({\sigma }_{n}\)—the effective normal stress at the particle contact and *β*—the aperture coefficient. The geometry of the nearby triangles has a direct bearing on the hydraulic aperture of the actual channels ‘T2T’ (Fig. 22) as

where *e* is the edge length between two adjacent triangles, *ω* denotes the angle between the edge with the length *e* and the center line of the channel that connects two adjacent triangles, and *γ* is the reduction factor established to maintain the maximum Reynolds number *Re* along the main flow route at a value below the critical value for laminar flow (*Re* = 2100). The parameter *γ* was determined in parametric tests at fluid pressure up to 140 MPa.

Following Barmak et al. [86], the continuity and momentum equations are developed that are rendered dimensionless

where \({\text{u}}_{j}=\left({u}_{j},{v}_{j}\right)\) and \({p}_{j}\) are the velocity and pressure of the fluid phase *j*, \({\rho }_{j}\) and \({\mu }_{j}\) are the corresponding density and dynamic viscosity. The Reynolds number is \({\text{Re}}_{p}={\rho }_{p}{u}_{i}{h}_{p}/{\mu }_{p}\) and the density and viscosity ratios are \(r={\rho }_{q}/{\rho }_{p}\) and \(m={\mu }_{q}/{\mu }_{p}\). The lower and upper phases in the dimensionless formulation, are located in the regions \(-{n}_{d}\le y\le 0\) and \(0\le y\le 1\), where \({n}_{d}={h}_{q}/{h}_{p}\). The velocities satisfy the no-slip boundary condition at the channel walls.

The continuity of velocities and tangential stresses is required by the boundary conditions at the interface at *y* = 0 [86]

and

The mass flow rates \({M}_{q,x}\) and \({M}_{p,x}\) for both fluid phases are derived by solving Eqs. B3 and B4 with the boundary conditions (Eqs. B5–B7), as well as the shear stress \({\tau }_{f0}\) at the channel surfaces.

The fluid in VPs (triangular cells), in contrast to the model of fluid flow in the channels, is presumed to be compressible. The discretized form of the mass conservation equation for the liquid phase is

with

where \({V}_{i}^{n+1}\) and \({V}_{i}^{n}\) are the volume of VP_{i} at a time increment *n* + 1 and *n* (the third dimension is the unit dimension), respectively, *f* is the face (edge) number, \({U}_{f}^{n}\) denotes the volume flux through the face (m^{3}/s), based on the average velocity in the channel, \({\alpha }_{q,f}^{n}\) is the face value of the liquid phase volume fraction (–), *t* is the time step (s), *n* denotes the time increment and *i* is the VP number (–). The same equations are defined for the gas phase.

VPs in contrast to the channel flow model assume that the fluid is compressible. The mass flow rates of fluid phases in channels are only calculated to estimate the mass flow rate of fluid flowing through the cell faces. The fluid pressure can exceed 70 MPa in specific situations, such as during the hydraulic fracturing process. The gas phase exceeds the critical point and becomes a supercritical fluid under these conditions. Therefore, for both fluid phases in VPs, the Peng-Robinson equation of state [69] is used to describe fluid behavior above the critical point

where *P* is the pressure (Pa), *R* denotes the gas constant (*R* = 8314,4598 J/(kmol K)), \({V}_{q/p}\) is the molar volume of liquid (*q*) and gas (*p*) fraction (m^{3}/kmol) and *T* denotes the temperature (K). The parameters in Eq. B10 are:

where \({T}_{q/p,c}\) is the critical temperature of phase (K), \({P}_{q/p,c}\) denotes the critical pressure of phase (Pa), \({\omega }_{q/p}\) is the acentric factor of phase (–), and *T*_{r} denotes the reduced temperature \(\frac{T}{{T}_{c}}\). When *c*_{1} = *c*_{2} = *c*_{3} = 0, the original model is obtained. The extra factors help connect vapor pressure data from highly polar liquids like water and methanol. For most substances, Equations B11–B16 provide a good fit for the vapor pressure, however predicting molar volumes for liquid phase can be very inaccurate. The forecast of saturated liquid molar quantities might deviate by l0–40% [87]. Peneloux and Rauzy [88] proposed an effective correction term

where *s* is the small molar volume correction term that is component dependent; \({V}_{q}\) is the molar volume predicted by Eq. B16 and \({V}_{q}^{corr}\) refers to the corrected molar volume. The value of *s* is negative for higher molecular weight non-polar and essentially for all polar substances. The molar volume correction term is considered to be 0.0 m^{3}/kmol and − 0.0034 m^{3}/kmol for the gas phase and liquid phase (water), respectively. The Peng-Robinson equation of state has the advantage of being able to describe the behavior of supercritical fluids at extremely high fluid pressures and temperatures. For each phase, the mass conservation equation is used. The mass transfer between phases and the grid velocity is ignored when there is no internal mass source. The discretized form of the mass conservation equation for the liquid phase is

with

where *f* is the face (edge) number, \({U}_{f}^{n}\) denotes the volume flux through the face (m^{3}/s), based on the average velocity in the channel, \({\alpha }_{q,f}^{n}\) is the face value of the fluid phase volume fraction (–), *t* is the time step (s), *n* denotes the time increment and *i* is the VP number (–). The explicit formulation is used instead of an iterative solution of the transport equation during each time step since the volume fraction at the current time step is directly computed from known quantities at the previous time step. Similarly, the mass conservation equation for a gas phase is introduced. The product \({\rho }_{q}{U}_{f}^{n}{\alpha }_{q,f}^{n}\) in Eq. B18 is the mass flow rate \({M}_{q,f}\) of the liquid phase flowing through the face *f* (edge of a triangle) of VP_{i}. The density of the liquid phase can be calculated by solving the mass conservation equation for both phases

The density of the gas phase can also be computed in the same way. It should be noted that the molar volume *V* (*q/p*) is related to the gas density.

and to the liquid density

where *w*_{p} and *w*_{q} are molar weights of gas and liquid phases, respectively. Due to the fact that the fluid phases share the same pressure

the fluid phase fractions are computed. Inserting Eq. B21 for the gas phase and Eq. B22 for the liquid phase into Eq. B23, a polynomial equation is obtained with respect to the liquid fraction \({\alpha }_{q,i}^{n+1}\). The gas-phase fraction is computed as \({\alpha }_{p,i}^{n+1}=1-{\alpha }_{q,i}^{n+1}\). Equation B10 is used to calculate the new pressure \({P}_{i}^{n+1}\) in VP_{i}. The explicit formulation is utilized instead of an iterative solution of the transport equation during each time step.

Because of the passage of a viscous fluid, there is shear stress along the channel's edge. The shear stress profile in the fluid is triangular for immovable parallel plates with no-slip boundary conditions (zero velocity). The shear stress \({\tau }_{f0}\) at the interface between liquid and gas is

The fluid pressures in VPs and ‘S2S’ channels are converted into the forces \({\overrightarrow{F}}_{P,j}\) and \({\overrightarrow{F}}_{S,j}\) acting on spheres. For simplicity, the fluid pressure forces acting on the sphere are calculated by assuming that the fluid–solid contact area is multiplied by the pressure in the cell (\({\overrightarrow{F}}_{P,j}\)) or channel (\({\overrightarrow{F}}_{S,j}\)). The contact area is calculated as a section of the cylinder's surface with a height equal to the diameter of the sphere, and not as a section of the sphere's surface. This simplifies the computational algorithm and only slightly overestimates the forces

where \(\overrightarrow{n}\)—the unit vector normal to the discretized sphere’s edge, \({P}_{i}\)—the pressure in VP, *i*—the VP number, *j*—the sphere number, and \({A}_{k}\)—the contact area between the fluid in VP_{i} and sphere

with \({r}_{j}\)—the sphere radius and \({e}_{k}\)—the sphere edge length. The shear stresses are finally converted into the forces acting on spheres as

where \(\overrightarrow{I}\)—the unit vector parallel to the channel wall and oriented in the fluid flow direction, \({\tau }_{f0,i}\)—the shear stress in the channel, *i*—the channel number, *j*—the sphere number and \({A}_{k}\)—the contact area between the channel and sphere, and \({L}_{k}\)—the channel length.

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Krzaczek, M., Tejchman, J. & Nitka, M. Effect of free water on the quasi-static compression behavior of partially-saturated concrete with a fully coupled DEM/CFD approach.
*Granular Matter* **26**, 38 (2024). https://doi.org/10.1007/s10035-024-01409-3

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DOI: https://doi.org/10.1007/s10035-024-01409-3