# Modeling Oil Recovery in Mixed-Wet Rocks: Pore-Scale Comparison Between Experiment and Simulation

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## Abstract

To examine the need to incorporate in situ wettability measurements in direct numerical simulations, we compare waterflooding experiments in a mixed-wet carbonate from a producing reservoir and results of direct multiphase numerical simulations using the color-gradient lattice Boltzmann method. We study the experiments of Alhammadi et al. (Sci Rep 7(1):10753, 2017. https://doi.org/10.1038/s41598-017-10992-w) where the pore-scale distribution of remaining oil was imaged using micro-CT scanning. In the experiment, in situ contact angles were measured using an automated algorithm (AlRatrout et al. in Adv Water Resour 109:158–169, 2017. https://doi.org/10.1016/j.advwatres.2017.07.018), which indicated a mixed-wet state with spatially non-uniform angles. In our simulations, the pore structure was obtained from segmented images of the sample used in the experiment. Furthermore, in situ measured angles were also incorporated into our simulations using our previously developed wetting boundary condition (Akai et al. in Adv Water Resour 116(March):56–66, 2018. https://doi.org/10.1016/j.advwatres.2018.03.014). We designed six simulations with different contact angle assignments based on experimentally measured values. Both a constant contact angle based on the average value of the measured values and non-uniform contact angles informed by the measured values gave a good agreement for fluid pore occupancy between the simulation and the experiment. However, the constant contact angle assignment predicted 54% higher water effective permeability after waterflooding than that estimated for the experimental result, whereas the non-uniform contact angle assignment gave less than 1% relative error. This means that to correctly predict fluid conductivity in mixed-wet rocks, a spatially heterogeneous wettability state needs to be taken into account. The novelty of this work is to provide a direct pore-scale comparison between experiments and simulations employing experimentally measured contact angles, and to demonstrate how to use measured contact angle data to improve the predictability of direct numerical simulation, highlighting the difference between the contact angle required for the simulation of dynamic displacement process and the contact angle measured at equilibrium after waterflooding.

## Keywords

Direct numerical simulation Lattice Boltzmann method Wettability Mixed-wet Carbonates## 1 Introduction

Multiphase flow in porous media has a wide range of applications including oil recovery, carbon storage and water flow in the unsaturated zone (Blunt 2017; Pruess and García 2002). To improve the predictive power of pore-scale models describing these phenomena, it is of great importance to implement realistic spatial information on wettability.

Recently, advances in imaging techniques have made it possible to directly observe contact angles on a pore-by-pore basis inside porous media. Andrew et al. (2014) measured in situ contact angles for a supercritical \(\hbox {CO}_2\)-brine-carbonate system at high temperatures and pressures. Khishvand et al. (2016) studied two- and three-phase flow experiments on Berea sandstone rock samples and measured contact angles based on micro-CT images. Furthermore, Alhammadi et al. (2017) conducted waterflooding experiments on carbonate rock samples saturated with crude oil taken from a producing hydrocarbon reservoir. The phase distributions were characterized with micro-CT imaging, and the in situ contact angles for three samples were measured using an automated algorithm (AlRatrout et al. 2017). The measured contact angles showed a wide distribution with values both above and below \(90^{\circ }\). Since the measurements of wettability were obtained on a pore-by-pore basis, it is of interest to determine how to incorporate this information in direct numerical simulation of two-phase flow.

Ramstad et al. (2012) conducted direct numerical simulations on micro-CT images of Berea and Bentheimer sandstone using the color-gradient lattice Boltzmann method. They computed the relative permeability for both steady-state and unsteady-state simulations. The results were compared with experimentally obtained relative permeabilities (Oak et al. 1990; Øren et al. 1998). Since the experiments were conducted on strongly water-wet outcrop sandstones, a good agreement with the experiment was obtained by assigning a constant contact angle of \(\theta =35^{\circ }\) in the simulations. Raeini et al. (2015) performed direct numerical simulations on micro-CT images of Berea sandstone using a volume-of-fluid-based finite-volume method. There was a good agreement in the capillary trapping curve between their simulations and published experimental measurements. This was achieved by assigning a constant contact angle of \(\theta =45^{\circ }\) in the simulations. Although there have been several other works on two-phase flow in 3D porous media using direct numerical simulation methods, most studies have assumed a constant contact angle (Pan et al. 2004; Li et al. 2005; Boek et al. 2017; Leclaire et al. 2017).

However, the experimental findings previously described strongly suggest that in natural crude oil and reservoir rock systems, it is common to have a wide range of contact angles as a result of wettability alteration caused by the sorption of surface active compounds to the solid surface. The degree of alteration depends on pore geometry, pore size, surface roughness and mineralogy (Buckley et al. 1998). Therefore, to better understand oil recovery from mixed-wet rocks, a spatial variation in contact angle should be taken into account.

Although several studies investigating a spatially heterogeneous wettability state using pore network modeling can be found in literature (McDougall and Sorbie 1995; Øren et al. 1998; Valvatne and Blunt 2004), there are few studies on direct numerical simulation of 3D porous media considering a distribution of contact angles from experimental measurements. There are several benefits of using direct numerical simulations, as opposed to network models. Direct simulation avoids uncertainty in pore network extraction and allows direct comparison of fluid distribution between experiments and simulations. Landry et al. (2014) investigated the impact of a mixed-wet state in a bead pack on relative permeability using two-phase lattice Boltzmann simulations. Using the approach of Hazlett et al. (1998), by altering the wettability of solid surfaces in contact with non-wetting phase after a drainage simulation, they calculated the decrease in non-wetting phase relative permeability as a result of the wettability alteration. In their work, a degree of wettability alteration was used as a variable in their sensitivity simulations. Jerauld et al. (2017) showed a comparison of steady-state relative permeability on a reservoir sandstone sample between experiments and simulations. The experiments were performed at reservoir temperatures and pressures after more than 3 weeks of aging in crude oil. They conducted three simulations with different wettability states assuming Gaussian distributions with different mean contact angles: \(\theta =45^{\circ } \pm 15^{\circ }\), \(\theta =90^{\circ } \pm 15^{\circ }\) and \(\theta =135^{\circ } \pm 15^{\circ }\). While the impact of wettability on relative permeability was modest, they concluded that the weakly oil-wet simulation (the simulation with \(\theta =135^{\circ } \pm 15^{\circ }\)) gave the best agreement with the relative permeability obtained from the experiments. In their work, although a range of the contact angle values was considered, their spatial distribution was not taken into account and the values were simply assumed: There was no independent measurement of contact angle.

In this paper, we show comparisons between waterflooding experiments in a mixed-wet carbonate from a producing reservoir (Alhammadi et al. 2017) and the results of direct numerical simulations using the color-gradient lattice Boltzmann method. We input spatially distributed experimentally measured contact angles (AlRatrout et al. 2017) in a direct numerical simulation on a 3D micro-CT image of a carbonate rock. The simulation results are compared to the experimental results in terms of local fluid configuration and fluid conductance (i.e., relative permeability). The key idea is to assign wettability information measured in the experiment to the numerical model. For this purpose, we design: (a) three simulations with the same contact angle for every pore, where the contact angle values represent an average for water-wet, weakly oil-wet and strongly oil-wet conditions, and (b) three simulations with different contact angles assigned to different pores informed by the measured values.

The paper is organized as follows: Firstly in Sect. 2, our direct numerical simulation method and the experimental data are described. Then in Sect. 2.3.4, the simulation results are compared with results obtained from waterflooding experiments on the same sample in terms of local fluid occupancy and water effective permeability. Conclusions are drawn in Sect. 3.

## 2 Materials and Methods

### 2.1 The Multiphase Lattice Boltzmann Method

The no-slip boundary condition is implemented based on the full-way bounce back scheme at the solid boundary lattice nodes. In this scheme, the particle distributions at boundary lattice nodes are bounced back into flow domain instead of performing the collision step.

To properly model the wettability of the fluids, the wetting boundary condition described in Akai et al. (2018) is used. The basic idea of this boundary condition is to modify the direction of the interface normal vector at the boundary according to a specified contact angle. This method allows the accurate assignment of contact angle in arbitrary 3D geometries, with lower spurious currents than the commonly applied fictitious density boundary condition (Akai et al. 2018). For the inlet and outlet boundaries of a simulation domain, we use a constant velocity and a constant pressure boundary condition, respectively (Zou and He 1997).

### 2.2 Pore-Scale Waterflooding Experiments

Alhammadi et al. (2017) conducted three waterflooding experiments using three carbonate samples drilled from the same core plug and saturated with two types of crude oil (a light crude oil from the same reservoir and an Arabian medium crude oil that is relatively heavier). The sample was mainly composed of calcite (\(96.5 \pm 1.9\) weight %) with minor amounts of dolomite, kaolinite and quartz. The helium porosity and permeability measured on the core plug were 27.0% and 6.8 \(\times \) \(10^{-13}\) \(\hbox {m}^2\) (686 mD), respectively (Alhammadi et al. 2017). Through applying three aging protocols, three distinct wettability states were established after primary drainage. The distributions of initial oil after drainage and remaining oil after waterflooding were imaged with a micro-CT scanner at subsurface conditions. The in situ contact angles were measured at the three-phase contact line from the dot product of vectors representing the oil–brine interface and the brine–rock interface using the automated algorithm developed by AlRatrout et al. (2017). The measured contact angles had a wide distribution with different mean contact angles for the three samples.

### 2.3 Pore Structures Used for the Simulations and Measured Contact Angles

#### 2.3.1 Upscaling of Micro-CT Data

*i*,

*j*,

*k*) in the coarsened grid system and \(V^{i,j,k}_{\alpha }\) is the volume fraction of phase \(\alpha \) (

*s*for solid,

*o*for oil and

*w*for water, respectively) in the grid block at (

*i*,

*j*,

*k*). Based on the resultant label data, \(L^{i,j,k}\), consisting of 256 \(\times \) 256 \(\times \) 200 \(\hbox {voxel}^3\) (i.e., 1.28 \(\times \) 1.28 \(\times \) 1.00 \(\hbox {mm}^3\)), the void spaces were extracted. The label data, \(L^{i,j,k}\), were also used to compare experimentally measured local fluid occupancy after waterflooding to the simulated results. After removing isolated void spaces in the sub-volume, the connected void space had a porosity of 17.8%.

#### 2.3.2 Partitioning of the Void Space into Individual Pore Regions

We partitioned the void space into pore regions (pores) for two reasons: firstly, to assign different contact angles to each pore region; and secondly to analyze the simulation results in terms of local fluid occupancy. The measured contact angles were available as 3D data points along three-phase contact lines observed in the experiments, whereas our simulation requires input of a contact angle for all grid voxels at solid and pore boundaries. We do not have sufficient experimental data to assign contact angles on a voxel-by-voxel basis, and in any event, this requires an unnecessarily detailed characterization of wettability. Instead, we assign a single contact angle to a pore region, but allow different pores to have different contact angles. This approach is conceptually similar to that adapted in pore network modeling (Blunt 2017). Moreover, our comparison between the simulation and experimental results will be made in terms of local fluid occupancy, i.e., we compare the fluid occupancy of each individual pore region.

#### 2.3.3 Measured Contact Angles

Summary of the wettability of each pore region

Pore type | Criteria | No. pores | Pore volume (%) | Mean \(\theta _{p}\) |
---|---|---|---|---|

Water-wet (WW) | \( 0^{\circ } \le \theta _{p} < 70^{\circ }\) | 5 | 0.40 | \(61^{\circ }\) |

Neutrally wet (NW) | \( 70^{\circ } \le \theta _{p} < 110^{\circ }\) | 212 | 63.30 | \(101^{\circ }\) |

Oil-wet (OW) | \(110^{\circ } \le \theta _{p} \le 180^{\circ }\) | 130 | 36.21 | \(114^{\circ }\) |

Undefined | – | 13 | 0.09 | – |

Total | – | 360 | 100.00 | \({{106}^{\circ }}^{\mathrm{a}}\) |

#### 2.3.4 Simulation Conditions

The experimental sample had a helium porosity of 31.7%, while its segmented porosity based on the micro-CT images at a resolution of 2 \(\upmu \hbox {m}/\mathrm{voxel}\) was 20.4%. This implies that the sample has micro-porosity whose pore size is below the resolution of the micro-CT imaging. However, in this paper, we only consider resolvable macro-porosity: We assume that the micro-porosity remained water-filled in the experiments.

According to the images taken before waterflooding, the initial water saturation is estimated at 6% of which only a saturation of 1% is in the connected pore space. In this work, we assigned an initial water saturation of 1% in the locations where water was imaged in the experiments after primary drainage. In reality, more water was present in unresolved micro-porosity and it is likely that the water was connected, but through layers that were not resolved in the images. Higher-resolution imaging and simulations are required to assess the impact of this water and micro-porosity on the displacement behavior.

In the simulations, as in the experiments, the main flow direction was vertical, in the *z* direction. Ten lattice nodes as a buffer zone (0.05 mm) was attached to the inlet and outlet; therefore, the model used for the simulations consisted of 256 \(\times \) 256 \(\times \) 220 \(\hbox {voxel}^3\) at 5 \(\upmu \hbox {m}/\mathrm{voxel}\) (i.e., 1.28 \(\times \) 1.28 \(\times \) 1.10 \(\hbox {mm}^3\)). The pore structure used for the simulations is shown in Fig. 4. Water was injected from the inlet face at \(z=0\) mm with a constant velocity, while the outlet face at \(z=1.10\) mm had a constant pressure. We note that these boundary conditions imposed on the cropped sub-volume do not exactly reproduce the experimental waterflooding conditions since in the experiment the inlet and outlet faces of the sub-volume were neither a constant flow nor constant pressure condition. This uncertainty associated with boundary conditions could be reduced by increasing the size of a simulation domain.

*Ca*of order \(10^{-5}\), which is two orders of magnitude higher than the experimental value since computational time significantly increases as the capillary number decreases lower than \(10^{-5}\). Chatzis and Morrow (1984) reported that an average capillary number below which mobilization of residual oil occurs was \(Ca =3.8 \times 10^{-5}\) based on core flooding experiments on various sandstone cores. Later, Raeini et al. (2014a) showed that using direct numerical simulations on a single pore throat geometry, the threshold capillary number below which snapped-off droplets become trapped is \(Ca^\text {throat} = {\mu _w {\bar{u}}_\text {throat}}/{\sigma }= 9.3 \times 10^{-4}\), where \(Ca^\text {throat}\) is the pore-scale capillary number defined using the average velocity in a throat (\({\bar{u}}_\text {throat}\)). Assuming a cylindrical pore structure in which the maximum velocity at the center is two times higher than the average velocity, our Darcy-scale capillary number used for simulations can be translated to \(Ca^\text {throat}=2{\mu _{w}}{q_{w}}/{\phi \sigma } \approx 3 \times 10^{-4}\), where \(\phi \) is the porosity, \(20\%\) in our case. Since this capillary number is lower than the threshold capillary number reported in Raeini et al. (2014a), we assume the simulations and the experiment are comparable. Nevertheless because recent experimental work indicates in mixed-wet conditions dynamic effects can occur even for a capillary number of order \(10^{-6}\) (Zou et al. 2018), this assumption has to be further investigated.

Comparison between experimental and simulation conditions

Experiment | Simulation | |
---|---|---|

Oil/water viscosity | 5.64 | 5.00 |

Capillary number | 3.0 \(\times 10^{-7}\) | 3.3 \(\times 10^{-5}\) |

Amount of water injected | 20 PV | 10 PV |

Case descriptions

Case names | Description |
---|---|

Case 1 | Constant contact angle \(\theta = 30^{\circ }\) |

Case 2 | Constant contact angle \(\theta =107^{\circ }\) |

Case 3 | Constant contact angle \(\theta =150^{\circ }\) |

Case 4 | Average angles for each pore were applied |

Case 5 | \(\theta =150^{\circ }\) for OW\(^{\mathrm{a}}\), \(\theta =100^{\circ }\) for NW\(^{\mathrm{b}}\) and \(\theta =30^{\circ }\) for WW\(^{\mathrm{c}}\) were applied |

Case 6 | \(\theta =150^{\circ }\) for OW\(^{\mathrm{a}}\), \(\theta =80^{\circ }\) for NW\(^{\mathrm{b}}\) and \(\theta =30^{\circ }\) for WW\(^{\mathrm{c}}\) were applied |

### 2.4 Fluid Saturation During Waterflooding

After 10 PVs of waterflooding, the simulations were continued while stopping water injection as in the experiment. After stopping water injection, the average fluid velocity within the simulation domain continued to decrease. We continued the simulations until the average fluid velocity became 10 times lower than the water velocity used for waterflooding. This equilibrium process was conducted to compare the simulation results with the experimental result which was imaged 2 h after the end of water injection. In fact, there was no appreciable change in the average fluid saturation between the end of 10 PVs of waterflooding and the equilibrium process. However, in oil-wet cases (case 2–6), we observed intermittent water pathways in the later part of waterflooding. This intermittent change in water phase connectivity could affect the water effective permeability which will be discussed in Sect. 2.7. Thus, the equilibrium simulations were continued to completely disconnect these unstable water pathways which did not exist in the experimentally obtained fluid distribution.

### 2.5 Local Fluid Occupancy Based on the Pore Size

*RF*of the \(n-\hbox {th}\) bin obtained from the simulation and experimental results, respectively. As shown in Table 4, cases 2 and 5 show the smallest error in recovery factors for each pore region. This means that the right amount of fluid was properly placed in the correct pore sizes in these cases.

### 2.6 Local Fluid Occupancy at the Sub-Pore Scale

Summary of the quantitative comparison between the simulations and experiment

\(\varDelta {RF}\) | \(E_{\mathrm{local}}\) | \(S_w\) | \(k_w\) | |||
---|---|---|---|---|---|---|

(%) | (%) | (%) | Diff.\(^\mathrm{a}\) (%) | (mD) | Diff.\(^\mathrm{a}\) (%) | |

Experiment | – | – | 65 | – | 537 | |

Case 1 | 22 | 52 | 75 | 15.3 | 717 | 33.5 |

Case 2 | 6 | 31 | 61 | − 6.0 | 829 | 54.4 |

Case 3 | 9 | 37 | 62 | − 4.7 | 484 | − 9.8 |

Case 4 | 9 | 37 | 57 | − 11.8 | 686 | 27.7 |

Case 5 | 6 | 35 | 64 | − 0.3 | 533 | − 0.7 |

Case 6 | 10 | 40 | 64 | − 1.5 | 496 | − 7.6 |

### 2.7 Fluid Conductance

The comparison of water effective permeability after waterflooding is summarized in Table 4. Case 5 showed the best agreement with the computed water effective permeability of the experiment with only 0.7% difference. Note that if a constant contact angle \(\theta =107^{\mathrm{o}}\) (case 2) is used, which also had the best agreement in recovery factors for each pore and is a reasonable assumption when measured contact angles are not available, the water effective permeability was overestimated by 54% although the predicted water saturation was lower than that of the experiment. As discussed in the previous sections, in case 5, even though the voxel-by-voxel prediction of occupancy has an error of 35%, a proper placement of fluid in correct pore sizes as shown in Figs. 7, 8 and a proper representation of fluid connectivity result in an accurate prediction of the water effective permeability.

Moreover, we compare the simulated fluid velocity distributions between the experiment and simulation for case 5. Figure 11 shows the spatial distribution of the normalized fluid velocity, \(U_{\mathrm{abs}}/U_{\mathrm{avg}}\), where \(U_{\mathrm{abs}}\) is the magnitude of the computed fluid velocities at each voxel and \(U_{\mathrm{avg}}\) is the average value and a histogram of the \(U_{\mathrm{abs}}/U_{\mathrm{avg}}\) sampled uniformly in 200 bins of \(\log (U_{\mathrm{abs}}/U_{\mathrm{avg}})\) (Bijeljic et al. 2013). Although the experiment shows more channels with low fluid velocity than the simulation, the main flowing channels with higher velocity are well captured by the simulation.

## 3 Conclusions

We have performed direct numerical simulations on pore space images employing a direct assignment of local contact angle obtained from experimental measurements. As opposed to pore network models, use of direct numerical simulations allows direct comparison of fluid occupancy between experiments and simulations avoiding uncertainty in pore network extraction.

Six simulations with different contact angle assignments were performed. These cases were designed based on the experimentally measured values. We considered three cases with constant contact angles, and three cases where the contact angle varied across different pore regions in accordance with the experimentally measured values. Then, the local fluid occupancy of each pore region was analyzed for these simulation results and compared with that obtained from the experiment.

In the experiment, the larger pores showed greater local recovery, as they were preferentially filled first during waterflooding, with oil retained in the smaller pores and as layers. This is indicative of oil-wet or mixed-wet conditions.

Applying a constant contact angle of \(\theta =107^{\circ }\), which was the average value of the in situ measured angles, gave a good agreement in the local fluid occupancy based on the pore size. However, this case did not accurately predict the water effective permeability. This means that the spatial heterogeneity of the contact angle distribution observed in the experiment has to be taken into account to accurately predict fluid connectivity. However, directly applying pore-averaged measured values to each pore region did not improve the quality of the match. Applying a higher contact angle than the pore-averaged measured value for oil-wet pore regions improved the agreement between the experiment and simulations. In a process when non-wetting phase displaces wetting phase, it is locally the largest contact angle that determines the threshold capillary pressure at which one phase can advance and displace another. As a consequence, using contact angle values near the maximum observed within each pore provided the most accurate reproduction of the experimental results.

For the cases where the wetting boundary conditions of all solid and fluid boundaries were informed from the measured angles, but using near-maximum values in pores whose average contact angles indicated oil-wet behavior, the fluid occupancy for each pore region after waterflooding was reasonably well predicted. The simulated water effective permeability also gave a good agreement with the experimental result.

Overall, we have demonstrated how to use micro-CT image based experimentally measured contact angles in direct numerical simulations to improve the characterization of multiphase flow displacement in mixed-wet rocks. In particular, a spatially heterogeneous wettability state needs to be taken into account to obtain accurate predictions of fluid occupancy and flow properties.

## Notes

### Acknowledgements

We thank Japan Oil, Gas and Metals National Corporation (JOGMEC) for their financial support. We also thank Abu Dhabi National Oil Company (ADNOC) and ADNOC Onshore (previously known as Abu Dhabi Company for Onshore Petroleum Operations Ltd) for sharing the experimental data used in this work. Ahmed A. AlRatrout is acknowledged for preparing the experimentally measured contact angle data. Also, we thank anonymous reviewers for their constructive comments. The experimental data used here are available at Alhammadi et al. (2018).

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