Analysis of the Effect of Coordination Number on Permeability of the Three-Dimensional (3D) Rock Model Using the Lattice Boltzmann Method (LBM)

Understanding the correlation between coordination number and permeability is crucial for predicting fluid flow behavior in hydrocarbon reservoirs. In this study, we investigated the influence of coordination number on permeability in three-dimensional (3D) rock models. The research methodology involved the creation of synthetic 3D rock models, incorporating pores networks with varying coordination numbers. Utilizing the Lattice Boltzmann Methods (LBM), a computational fluid dynamics approach, we simulated fluid flow through these synthetic rock models and quantified their permeability. Our findings demonstrated a strong dependency of permeability on the coordination number of synthetic rock models. The application of the Lattice Boltzmann Method (LBM) proved to be an effective tool for understanding fluid flow behavior in porous rock formations and can serve as a basis for further optimization of reservoir management strategies to maximize hydrocarbon exploitation.


Introduction
Permeability is a crucial petrophysical property of sedimentary rocks that plays a significant role in assessing reservoir quality.It quantitatively defines the fluid flow behavior through porous materials.Permeability is also a factor that can affect the economic value of existing fluid reserves.Therefore, research on fluid flow in hydrocarbon reservoir rocks based on their permeability is one of the important studies that need to be conducted in determining the characteristics of a reservoir.Fluid flow analysis in 3D rock models resulting from micro-CT scans can be carried out based on the coordination number to see the connectivity between pores in reservoir rocks [1].
Pore network models, employed to comprehend and simulate fluid flow in intricate porous material geometries, rely on the coordination number, which signifies the average connectivity of pore bodies within a specific pore.This parameter, fundamental to pore networks, significantly influences the hydraulic conductance of porous rocks, with estimation methods falling into two main categories: forward methods, involving microscopic imaging and morphological analysis [2] [3], and backward methods, which leverage macroscopic porous media characteristics to back-calculate the underlying pore network model [4].In this research, a forward modeling method approach will be used.
A coordination number can be interpreted as a number that shows the number of branching meetings with a node in the digital rock image model [5].This parameter is one of the fundamentals of the pore network and has a visible impact on the hydraulic conductance of porous rocks.The coordination number is a crucial factor that impacts the topology of porous rocks, consequently affecting the rock's permeability value [6].The distribution of coordination numbers can exhibit variations within porous media, contingent upon factors such as porosity, localized fluctuations in grain size distribution, and rock type [7].Coordination number estimation can be done using a microscopic approach.Research on coordination number estimation has been carried out by utilizing digital 3-dimensional (3-D) rock images from real and synthetic rock samples to introduce two mathematical correlations that estimate 3-D coordination number distribution parameters using one 2-D cross-sectional image [8].
In digital image analysis techniques, the permeability of rocks can be computed using the Lattice Boltzmann Method (LBM).The Lattice Boltzmann Method stands as one of the numerical methods employed to numerically describe the motion of fluids in space, derived from the Boltzmann equation [9].Lattice Boltzmann models incompressible fluids, wherein fluid particles can exclusively move in alignment with the lattice velocity vectors.The LBM approach to modeling fluid flow is rooted in collision models.Through this approach, fluid flow modeling becomes more straightforward and can accommodate highly intricate porous medium structures.In this study, we created three-dimensional rock models by varying their coordination numbers while keeping the porosity of each sample constant.This approach allows porosity to remain a consistent variable throughout the research.Therefore, by the end of the study, we will be able to examine how the coordination number influences the permeability value of a three-dimensional rock model.

Digital Samples
The research started by creating synthetic rock models with the help of computer modeling.The advantage of computer modeling is that one can easily adjust the required parameters accordingly, and various models of porous media can be generated, ranging from a simple capillary cylindrical tube to a complex structured sedimentary rock model.To distinguish between them, the models have different numbers of branches and coordination numbers which are determined at the beginning when creating the model.The size of each sample is 100 × 100 × 100 with a bit depth of 8 which has a cylindrical shaped pipe as the pore space.This synthetic rock model is created by varying the coordination number.The branching variations in this synthetic model also indicate that there are differences in the number of coordination numbers in each model.In this study, we created two conditions.The first was the porosity of each sample made almost the same value, which had a value of about 8%, henceforth this is referred to as condition 1.We also made the pore sizes of each sample the same, which was 24 pixels, henceforth this is referred to as condition 2. By looking at these two conditions, we wanted to see how the coordination number affects permeability.

Permeability
In 1856, Darcy conducted an experiment to quantify permeability.This experiment established a relationship between the pressure gradient of a fluid-saturated porous medium and the corresponding fluid flow rate as follows: where   represents the rate of fluid flow in the  direction,  is the medium's permeability,  is the fluid's dynamic viscosity,  is the fluid pressure, and  is the cross-sectional area perpendicular to the pressure gradient.The minus sign (−) in the equation above indicates that the fluid flow will move from an area with a high hydraulic gradient to an area with a low hydraulic gradient.In other words, the flow tends to follow the pressure decrease or decreasing hydraulic gradient.To simulate Darcy's experiment, LBM applies a pressure gradient to the inlet and outlet of a saturated porous medium.We use Palabos, a parallel Lattice Boltzmann solver perform the simulation [10].Palabos utilized the D3Q19 scheme which describes motion in three dimensions and 19 associated velocity vectors.Permeability is then calculated using Darcy's law: the rate of fluid flow in the  direction can be expressed as the rate volume of fluid passing through the cross-sectional area  perpendicular to the flow direction as: where  is the length of the medium.Equation (2) can then be rewritten as: Equation ( 4) is practically modified to conform with Palabos as follows: where 〈〉 denotes the average magnitude of the intrinsic velocity over the entire system volume.Darcy's law relies on certain assumptions: only one fluid phase exists in the pore, the fluid system is at a constant temperature, the fluid is not compressible, the flow is smooth, the fluid behaves according to Newton's laws, there is no interaction between the fluid and the pore walls, and the flow remains constant over time.Permeability has units of area ( 2 in SI units), but sometimes in terms of the geophysical aspect, it is more common to use the Darcy where 1  = 0.987 × 10 −12  2 or also millidarcy (mD) where 1  = 0.001 .To convert the permeability value into mD units, the following equation is used [11]:   is the permeability in mD units,  is the permeability in lattice square units ( 2 ), and  is the image resolution in / units.

Result and Discussion
Synthetic 3D rock models in the form of simple pipes with varying branching were created.The z-axis was used as the direction of fluid flow for all models in this study.As previously explained, permeability calculations in this study were performed using the Lattice Boltzmann Method (LBM).To obtain the permeability values using LBM, the 3D synthetic models were inputted into the Palabos program by providing the model size and pressure input.The pressure gradient used for all models in this study was 0.00005 atm/cm.This value was chosen to be very small to ensure that the flow is not turbulent and satisfies the laminar flow conditions.Permeability calculations using LBM were performed separately for each branching variation.Darcy's law, which is used in palabos relies on several assumptions: only one fluid phase is present in the pore, the fluid system maintains a constant temperature (isothermal), the fluid is not compressible, the flow is smooth (laminar), the fluid behaves according to Newton's laws, there is no interaction between the fluid and the pore walls, and the fluid flow remains constant (steady state).Talking about the relationship between coordination number and permeability certainly involves various other physical parameters such as porosity, pore size, and pore distribution.Previous research has discussed the relationship between coordination number and porosity, as well as the relationship between coordination number and tortuosity, and found that the greater the coordination number, the porosity of the rock will increase.Additionally, if the coordination number increases, the tortuosity value of the rock will decrease [12] [5] [13].
From Figure 2, it can be observed that in condition 1 or when porosity is almost the same value, the relationship between coordination number and permeability becomes non-linear.Increasing the coordination number will be inversely proportional to the permeability value, resulting in a decrease in permeability.In the creation of synthetic rock models, a model with a higher coordination number is constructed with a smaller pore size to maintain the same porosity value.The pore network structure will significantly differ depending on the configuration of the pore network, even if the porosity remains the same [14].Pore size and connectivity play a crucial role in influencing permeability.With an increase in coordination number, necessitating a reduction in pore size to maintain the porosity were almost the same value.It's possible that's the factor that caused permeability decreases because flow predominantly occurs through smaller pores, as observed in condition 1 in this study.However, in condition 2 or when pore sizes are made uniform, an increase in coordination number leads to an enlargement of both porosity and inter-pore connectivity, contributing to the increased permeability of the rock model.The trend of changes in permeability in each model can be seen in Figure 2. It is important to note that the limitations of these conditions may result in different trends in natural rock samples.There needs to be further analysis of the influence of pore diameter on permeability, which is not addressed in this study.It may represent the complexity of the transport mechanism.

Conclusion and Future Work
From the performed analysis, we can draw several conclusions.Pore size and porosity in synthetic rock models, created by varying the coordination number significantly, have a substantial impact on permeability.Permeability tends to decrease when the porosity of each sample is nearly the same.It is possible that samples with identical porosity with varying coordination numbers exhibit distinct permeability trends, whether it's a decrease or even the possibility of a stable permeability trend.In the future, research should be conducted by varying branching forms in synthetic models while maintaining the same coordination numbers to assess the influence of flow patterns, which is one of the factors influencing permeability.

Acknowledgment
This research was funded and facilitated by the Rock Physics laboratory, Earth Physics and Complex System Research Group, Institute Technology of Bandung.

References
[1] S.-C.Chen, E. K. Lee and Y.-I.Chang, "Effect of the coordination number of the pore-network on the transport and deposition of particles in porous media," Separation and Purification Technology, pp.11-26, 2002.

Figure 1 .
Figure 1.Binary Image of Synthetic Rock Samples.The black color on the model represents the solid and the white color represents the pores.

Figure 2 .Figure 3 .
Figure 2. Visualized the trend of decreasing permeability values in condition 1 and increasing permeability in condition 2 for each model.

Table 1 .
Synthetic model permeability with coordination number variations for condition 1 and condition 2.