# Study of Multi-phase Flow in Porous Media: Comparison of SPH Simulations with Micro-model Experiments

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## Abstract

We present simulations and experiments of drainage processes in a micro-model. A direct numerical simulation is introduced which is capable of describing wetting phenomena on the pore scale. A numerical smoothed particle hydrodynamics model was developed and used to simulate the two-phase flow of immiscible fluids. The experiments were performed in a micro-model which allows the visualization of interface propagation in detail. We compare the experiments and simulations of a quasistatic drainage process and pure dynamic drainage processes. For both, simulation and experiment, the interfacial area and the pressure at the inflow and outflow are tracked. The capillary pressure during the dynamic drainage process was determined by image analysis.

## Keywords

SPH Micro-model Surface tension Multi-phase flow Porous media## 1 Introduction

In this work, we present a comparison between direct numerical simulations of immiscible two-phase flow in a porous structure based on smoothed particle hydrodynamics (SPH) and the results obtained from micro-model experiments.

First SPH simulations of multi-phase flow in porous structures on, the pore scale have been performed by Tartakovsky and Meakin (2006) and Tartakovsky et al. (2015). Still comparisons with experiments are very rare and focus mostly on the dynamics of a single interface, like a capillary rise. The need for an experimental work where a number of capillaries, like a porous structure, would be used became evident.

Both quasi-static and dynamic drainage experiments were performed in a poly-dimethyl-siloxane (PDMS) micro-model (Karadimitriou et al. 2012). The fluids used as the wetting and the non-wetting phase were Fluorinert FC-43 and water dyed with ink, respectively. The dynamic drainage experiments were performed under constant pressure conditions. Flow through the micro-model was visualized with the use of a custom open-air microscope. The acquired images were stored in a computer for further processing (Karadimitriou et al. 2013).

The images acquired by the visualization setup were subsequently processed to obtain information on the interfacial area between the wetting and the non-wetting phase, as well as between the two fluids and the solid phase. Phase saturation and the pressure in the reservoir at the inlet were also captured. In this way, a direct comparison of experiments and simulations was obtained.

For the simulations a modified version of the Lagrangian SPH method (Gingold and Monaghan 1977) is used to easily track the moving interfaces of the multi-phase flow. To account for interface and contact line phenomena, the Navier–Stokes equations are extended by the Continuum Surface Force model (CSF) (Brackbill et al. 1992) to describe the forces acting on the interface and the contact line force model (CLF) (Huber et al. 2013) to describe wall–fluid–fluid interactions. These models allows us to track the extension of all interfaces and contact lines.

## 2 Experiment

### 2.1 Experimental Setup

The experimental setup is the one used in various studies by Karadimitriou et al. (2012, 2013). The micro-model used for the experiment had a rectangular shape with dimensions \(2.5\times 1\,\hbox {mm}\). It consists of a big void, which corresponds to the size of the micro-model, with a random distribution of round pillars. This configuration creates a pore network with average pore diameter of \(160\,\upmu \hbox {m}\) (Fig. 1). The micro-model was fabricated according to the work of Karadimitriou et al. (2013). First a silicon wafer with the imprinted micro-model was created using photolithography. Subsequently, this wafer was used as mold on one PDMS slab. A second PDMS slab was then used to cover the micro-model.

### 2.2 Two-Phase Flow Experiments

The two-phase flow experiment has been performed according to Karadimitriou et al. (2013). The micro-model is connected with the two reservoirs. Each reservoir contains a different liquid phase, one being the wetting phase and one the non-wetting phase. In this experiment, due to the hydrophobicity of PDMS, water (blue ink) is the non-wetting phase and Fluorinert FC-43 is the wetting phase. Figure 1 shows the micro-model fully wetted with Fluorinert. In all experiments, we acquired images of the phase propagation. From those images, all interfacial areas and the capillary pressure at the wn interface was determined.

#### 2.2.1 Dynamic Drainage Experiment

The dynamic drainage experiment consisted of one full drainage process. During drainage a constant external pressure was applied to the reservoir containing the non-wetting phase. The pressure is controlled by a pressure valve (Bronkhorst) and maintained a stable pressure during the experiment. In order to achieve dynamic conditions, the applied pressure was kept at \(1.86\,\text {kPa}\) during the drainage process. Images of the process were taken with a rate of 5 fps.

#### 2.2.2 Quasistatic Drainage Experiment

The quasistatic experiment has been performed using a flow control. The drainage process was paused for eight times to let the system relax. One image was taken in each relaxed condition, and the current pressure in the reservoir was measured.

## 3 Model Description

### 3.1 Implementation in SPH

#### 3.1.1 SPH Formulation for Multi-phase Flow

#### 3.1.2 Surface Tension in SPH

*c*.

*c*at the interface. In SPH Eq. (8) can be written as

*kl*-interface and are orthogonal to the contact line itself. \(\delta _{wns}\) is the volume reformulation parameter for the CLF (Huber et al. 2013).

### 3.2 Corrected SPH

- 1.
The calculation of the normal vector \(\mathbf {n}\) and the curvature \(\kappa \) does not have full support in the domain (Morris 2000; Adami et al. 2010a) (see Fig. 3b, c).

- 2.
The simulations are performed with open boundaries. This means that new particles can enter the system and others leave it. In this inlet and outlet zone, the particle distribution is not perfect. In order to minimize the errors at the open boundaries the application of corrected kernels and their gradients is crucial.

- 3.
The convergence with increasing particle resolution in a disturbed system is only guaranteed with higher-order methods (Fatehi and Manzari 2011; Tartakovsky et al. 2015).

*i*. The gradient of this corrected kernel becomes

### 3.3 Incompressibility Treatment and Time Integration

### 3.4 Boundary Conditions

#### 3.4.1 Solid Wall Boundary Conditions

#### 3.4.2 Inflow/Outflow Condition

### 3.5 Implementation

The presented SPH models are implemented in SiPER.^{1} SiPER is an ISPH code developed by our group at the University of Stuttgart. It is specialized on multi-phase flow in complex porous media. It is a MPI parallel implementation in standard C language. We use the boomerAMG preconditioner and BiCGStab solver from HYPRE and PETSc library for solving the PPE.

## 4 Simulation

### 4.1 Validation of the 2D Simulation

The imbibition and drainage process in a porous structure is driven by interfacial forces and a pressure gradient, respectively. Moreover the propagation speed is constrained by viscous forces. Before performing the simulations of the micro-model, the model was validated by two test cases. In the first test case, the flow profile in a single-phase Poiseuille flow is validated and in the second test case a dynamic capillary rise is compared with a reduced model.

#### 4.1.1 Poiseuille Flow

The Poiseuille flow is a simple test case to verify a correct implementation of no-slip boundaries at the solid wall and viscous forces in general. In most literature of SPH, Poiseuille flow is applied by using periodic boundary conditions in the direction of the flow and applying a body force to accelerate the fluid. In this work we apply a pressure gradient between inlet and outlet with Dirichlet boundary conditions for the pressure as driving force (see Sect. 3.4.2). In this way we can use the Poiseuille flow also as test case for the pressure field and the acceleration term arising from a pressure gradient.

We consider a two-dimensional, rectangular domain with a length of \(L_x = 3D = 480\,\upmu \hbox {m}\) and a height of \(L_y = D\). In *y*-direction we apply no-slip wall boundary conditions. The fluid has a density of \(\rho = 1000\,\hbox {kg}/\hbox {m}^3\) and a dynamic viscosity of \(\mu = 0.001\,\text {Pa s}\). The Reynolds number is \(Re = \frac{\rho \mathbf {u}_\mathrm{avg} D}{\mu } = 7.1\). The dimensionless velocity is defined as \(\mathbf {u}^* = \frac{\mathbf {u}}{\mathbf {u}_\mathrm{max}}\). The dimensionless positions are \(x^* = \frac{x}{D}\) and \(y^* = \frac{y}{D}\). The corresponding pressure difference between in- and outlet is \(\Delta p^* = \frac{\Delta p}{\mathbf {u}_\mathrm{avg}^2 \rho } = 5.06\). Initially the system is at rest. The domain size is set to 7500 particles.

#### 4.1.2 Capillary Rise

*U*in the vicinity of the static contact angle \(\theta _0\):

*U*are the same. With this equation one can solve the reduced model with the information of the dynamic contact angle. To stay close to the respective experiment, we chose water (\(\rho _n = 1000\,\hbox {kg}\), \(\mu _n = 1.0\times 10^{-3}\,\hbox {Pa}\,\hbox {s}\)) as light non-wetting phase and Fluorinert (\(\rho _w = 1800\,\hbox {kg}\), \(\mu _w = 4.7\times 10^{-3}\,\hbox {Pa}\,\hbox {s}\)) as denser wetting phase. The static contact angle is the same as the one in the experiment \(\theta _0 = 45^{\circ }\), and the surface tension is \(\sigma _{wn} = 0.055\, \hbox {N}/\hbox {m}\), which corresponds to the surface tension between water and Fluorinert. The pressure boundary conditions where chosen to be \(p_\mathrm{top} = 0\,\text {Pa}\) and \(p_\mathrm{bottom} = \rho _n\,g\,H = 91.2\,\text {Pa}\). In Fig. 7 the good correlation of the evolution of the capillary rise between the reduced model and the SPH simulation is shown.

### 4.2 Simulation of a Drainage Processes in a Porous Structure

The geometry of the porous structure in the simulation is the same as the one of the PDMS micro-model in the experiment (comp. Fig. 1). The length of the simulation domain including the inflow channel is \(l=3.48\,\hbox {mm}\) with a resolution of the whole porous structure of \(1200\times 362\) particles. So each particle represents an area of \(2.9\times 2.9\,\upmu \hbox {m}\). We perform two different types of simulations. The first type is a dynamic drainage process with a fixed pressure difference between inlet and outlet. The other type is a quasistatic drainage process, where the displacement of the wetting phase is performed in multiple steps with relaxation times in between to measure system properties under static conditions.

#### 4.2.1 Dimension Reduction

All simulations of the micro-model were performed in 2D in order to limit the computational load. By reducing the vertical dimension, we assumed a parabolic flow profile in vertical direction, to take the additional viscous friction into account and a constant curvature of the interface in vertical direction.

*Viscous Force*

*b*of the duct is larger than the height

*a*. For this reason an additional viscous force is applied to take care of the dimension reduction in the simulation:

Comparison of theoretic values of the permeability of rectangular ducts and simulation results using the dimension reduction

Width \((\upmu \hbox {m})\) | Height \((\upmu \hbox {m})\) | \(\kappa _\mathrm{duct} \,(\hbox {m}^2)\) | \(\kappa _{sim} \,(\hbox {m}^2)\) | Error (%) |
---|---|---|---|---|

80 | 100 | \(2.748\times 10^{-10}\) | \(3.035\times 10^{-10}\) | 11.45 |

160 | 100 | \(5.093\times 10^{-10}\) | \(5.360\times 10^{-10}\) | 6.21 |

500 | 100 | \(7.283\times 10^{-10}\) | \(7.382\times 10^{-10}\) | 2.31 |

160 | \(\infty \) | \(2.073\times 10^{-9}\) | \(2.071\times 10^{-9}\) | 0.12 |

*Surface Force*

*a*and

*b*is the height and the width of the channel and \(\theta \) is the static contact angle. The surface tension between Fluorinert and water is \(\sigma _{wn} = 0.055 \,\hbox {N}/\hbox {m}\). The static contact angle in the micro-model was identified as \(\theta \approx 45^{\circ }\). In this case Eq. 28 simplifies to:

#### 4.2.2 Dynamic Drainage Process

For the dynamic drainage process, we performed several simulations with the pressure difference between inlet and outlet \(\Delta p\) ranging from 1.678 to \(5.778\,\text {kPa}\). Then we compared the simulated and experimental drainage process with \(\Delta p = 1.86\,\text {kPa}\).

*F*and the saturation of the wetting phase \(S_w\). In an isothermal and isochoric system the change of the Helmholtz free energy,

*F*is only affected by interface formation:

#### 4.2.3 Quasistatic Drainage Process

The quasistatic drainage process in the simulation is setup as follows. The system is initially fully wetted with Fluorinert. The non-wetting phase is pushed with a certain velocity in the system. The boundary conditions are still Dirichlet boundary conditions for the pressure and homogeneous Neumann boundary conditions for the velocity as described in Sect. 3.4.2. We use a PI controller for the inlet pressure to obtain an average velocity in the inflow channel \(\bar{v}_\mathrm{dyn}\). The setpoint of the average inflow velocity is set to \(\bar{v}_\mathrm{dyn} = 1 \times 10^{-2}\,\hbox {m}/\hbox {s}\). The pressure at the outlet is fixed at \(p=1.0\,\text {kPa}\).

The Fluorinert is displaced until the saturation \(S_w\) has dropped by \(5\,\%\) in the domain. After that the setpoint of the velocity is set to \(\bar{v}_\mathrm{stat} = 0\,\hbox {m}/\hbox {s}\) for a certain time interval \(\Delta t_\mathrm{stat} = 1\times 10^{-2}\,\hbox {s}\) to let the system relax. The capillary pressure was measured from the pressure difference between inlet and outlet in the relaxed system. Then the setpoint of the average inlet velocity is switched back to \(\bar{v}_\mathrm{dyn}\).

## 5 Discussion of Results

The SPH simulation can give very interesting insights into the dynamic drainage process (see Fig. 8). The variation of the local capillary pressure at the different interfaces can be monitored in detail. Also the interfacial area in 2D can be determined with an easy summation over the SPH particles. Nonetheless, the measured interfacial areas show a significant difference in both the dynamic and quasistatic drainage process. The comparison of the interfacial areas are shown in Fig. 13b for the dynamic drainage and in Fig. 14b for the quasistatic process. The main part of this difference results from the larger number of small intrusions of wetting fluid captured in the SPH simulations. Another part of the difference could also be explained by the different evaluation processes. In the simulation the interfacial area is computed by a simple product of the measured interface in 2D and a constant contribution of the height of the micro-model, which will overestimate the interfacial area by some percent.

In the simulations, we performed several drainage processes with increasing pressure difference (Fig. 9). Apart from a faster decrease in the saturation of the wetting phase, also the total change in saturation at breakthrough varies. In Fig. 12 a comparison between the final states before breakthrough of the simulated drainage process with an applied pressure difference of \(\Delta p = 1.86\,\text {kPa}\) and \(\Delta p = 2.778\,\text {kPa}\) is shown. With a pressure difference of only \(\Delta p = 1.86\,\text {kPa}\) one can see a clear effect of capillary fingering, both in the simulation and the experiment (compare final states in Fig. 10). This effect vanishes at pressure gradients of \(\Delta p = 2.778\,\text {kPa}\) and higher. This observation corresponds to the work of Joekar-Niasar (2012), who showed in a non-equilibrium pore network simulation that pressure–saturation curves depend on the capillary number.

*Ca*). While the CLF model properly captures the dynamic contact angle at high

*Ca*-numbers, it can’t capture the “stick–slip” behavior related to contact angle hysteresis on rough surfaces. To get an idea how this “stick–slip” behavior could change the drainage process a second resistance is added. This resistance at the contact line (RCL) increases with lower volume fluxes. The combined resistance of single-phase flow and the contact line could be expressed by a power law of the form:

*Ca*-numbers is necessary for a better understanding of the “stick–slip” behavior of the system of interest. This study illustrates that a detailed knowledge of the dynamic contact angle is also important when simulating flow with low

*Ca*numbers, where static contact angles can be assumed constant most of the time. With a model to describe the dynamics of the contact line separated from the interface, like the CLF model, it is possible to include information on roughness in an extended model description in the future.

## 6 Conclusion

We performed simulations and experiments of a drainage process in a small micro-model. The implemented SPH model is able to describe the effects of surface tension in a multi-phase system. We found good agreement between simulations and experiments for the flow path and the evolution of the capillary pressure. Combined with the easy tracking of interfacial areas, SPH simulations are an appropriate tool to capture \(p_c\)–\(S_w\)–\(a_{wn}\) relationships for larger geometries in the future. The uniqueness of those relationships could be studied during drainage and imbibition, which was proposed by Hassanizadeh and Gray (1993). Apart from those good correlations a major difference in the dynamic evolution of the drainage process was found. We assume surface roughness and sticking effects play a major role when displacing a wetting fluid from a porous structure. These effects need further investigations.

## Footnotes

## Notes

### Acknowledgments

Financial support by the International Research Training Group NUPUS funded by the German Research Foundation is gratefully acknowledged.

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