A Smooth Particle Hydrodynamics (SPH) method is employed to simulate the two-phase flow of oil and water in a reservoir. It is shown that, in comparison to the classical finite difference approach, this method is more stable and effective at capturing the complex evolution of this category of two-phase flows. The influence of several smooth functions is explored and it is concluded that the Gaussian function is the best one. After 200 days, the block water cutoff for the Gaussian function is 0.3, whereas the other functions have a block water cutoff of 0.8. The effect of various injection ratios on real reservoir production is explored. When 14 and 8 m^{3}/day is employed, the water breakthrough time is 130 and 170 days, respectively, and the block produces 9246 m^{3} and 6338 m^{3} of oil cumulatively over 400 days.

The low flow capacity, complex fluid storage mode, and interacting oil-water two-phase flows in shale oil reservoirs present a slew of issues for the oil industry and economic calculations and analyses [

The majority of scholars who study reservoir numerical calculation begin with numerical solutions [

Due to the aforementioned drawbacks of the grid-based technique, researchers have begun to address the problem by deleting the grid or using nodes directly. SPH is the earliest continuum point-based method for solving partial differential equations that is fully mess-free and does not require a backdrop mesh [

Lucy proposed the SPH method [

Yang used the average density to update the variable smooth length using the SPH approach in order to maintain the symmetry of the particle-particle interaction. A pair-search approach called global pairing search will be employed, as well as a leapfrog method called time integration. A numerical model can be used to simulate single-phase fluid flow in reservoir porous media. To check the accuracy of the simulation findings, one-dimensional shock tubes are used. It provides a theoretical foundation for using the meshless particle approach to analyze reservoir porous media flow [

The capacity of the SPH approach to model the migration law of multiphase fluids is unrivaled. In this study, the SPH approach is used to simulate two-phase oil-water flow and is compared to traditional numerical simulation. The impacts of various smooth functions, injection production ratios, and formation pressures are examined in the simulation.

The following are some basic assumptions: (1) The fluid in the reservoir only flows in one direction; (2) The reservoir contains just two phases of oil and water, both of which are miscible; (3) The fluid flow in the reservoir is isothermal; (4) Ignore the influence of capillary force; and (5) Ignore the influence of gravity.

Continuity equation of oil phase and water phase:

where: ^{3}; ^{3}/day;

The formation fluid flow obeys Darcy’s law. The expressions of the water phase motion equation and oil phase motion equation are as follows:

where:

Saturation equation:

SPH is a meshless algorithm in which the computational domain is discrete in comparison to the number of particles. Two critical steps in the derivation of the SPH approach are the kernel approximation and the particle approximation. In the SPH formula, the general field variable _{i} can be calculated as

where Ω is the computational domain. _{i}. The sum of the domain m_{j} is the mass and density of particle

The gradient of the field variable _{i} can be calculated by the gradient transferred to the kernel function.

where

When the particle approaches the computational domain’s boundary, issues arise as a result of the reduced support domain, resulting in a loss of accuracy and consistency. Mirror and dynamic particles are used in this work to assure the uniformity of fluid particles near the border. The references contain additional information on these two particles [

In this paper, the Gaussian kernel function is selected. The Gaussian kernel function is sufficiently smooth. For the case of irregular particle distribution, the calculation is stable, the accuracy is high, and the numerical speed tends to 0 is very fast.

where,

The variables defined on the fluid particles are updated by the standard second-order jump time integral, and the velocity and density of particle

where:

The boundary treatment method adopts the dynamic coupling boundary conditions proposed by Shao et al. [

According to the SPH discretization rules and combined with the problem characteristics in this paper, the oil phase pressure and water phase saturation in

where: _{j} is the mass of particle _{ij} is the smooth function of particle

The relative permeability curve is shown in ^{3}/day, the recovery rate is 10 m^{3}/day, and the overall manufacturing duration is 400 days. The permeability distribution of the model is depicted in

The physical properties of the material are listed in

Parameters | Value |
---|---|

Pressure (MPa) | 10 |

Temperature (°C) | 70 |

Porosity (%) | 30 |

Effective thickness (m) | 10 |

Water volume factor | 1.05 |

Water saturation (%) | 20 |

Water viscosity (cp) | 2 |

Oil viscosity (cp) | 1 |

SPH method itself has high accuracy, but its results are affected by smooth kernel function, which directly determines the calculation efficiency and accuracy. Lucy [

As illustrated in

The injection production ratio is crucial in calculating the actual production levels of a reservoir. The injection production ratio is critical for maximizing economic gain. To ensure that the simulation’s other physical parameters are consistent with the above calculation example, the injection volume is maintained at 40 m^{3}/day, and the production volumes for each production well are set to 8 m^{3}/day, 10 m^{3}/day, 12 m^{3}/day, and 14 m^{3}/day, respectively, in this paper’s example of the above calculation.

As illustrated in ^{3}/day, the block’s cumulative oil production can reach 9246 m^{3} in 400 days. If production is increased to 8 m^{3}/day, however, the water breakthrough time is lowered to 170 days, and the block’s cumulative oil production can reach 6338 m^{3} in 400 days. As can be shown, the choice of an appropriate fixed liquid production method has a direct effect on the economic benefits of a piece of real estate within a given range. This is compatible with conventional wisdom and the method given in this paper, which illustrates its resilience.

The initial pressure is a critical factor in determining daily oil production. The initial formation pressure can be used to evaluate the reservoir’s development potential and more intuitively reflect the reservoir’s oil phase conductivity. The initial formation pressures of 15 MPa, 20 MPa, 25 MPa, and 30 MPa are used in this paper for simulation, and the examples above are used to ensure that the other physical parameters match those in

As illustrated in ^{3}/day. This is consistent with conventional wisdom, demonstrating the validity of the SPH method in this paper.

(1) This is the first time the SPH method has been applied to two-phase oil-water simulations. In comparison to the finite difference method, the example results show that this approach is correct, but it has a higher degree of stability because it does not require the development of a grid.

(2) This article analyzes the smooth kernel functions computed using various SPH algorithms. The Gaussian function has a block water cutoff of 0.3 at 200 days, while the other three functions have a block water cutoff of 0.8, with noticeable inaccuracy. It indicates that in this circumstance, the Gaussian function is more appropriate for the simulation.

(3) This article will evaluate the influence of various injection production ratios on real reservoir output. When 14 m^{3}/day is established, the water breakthrough time is 130 days, and the cumulative oil production of the block can reach 9246 m^{3} in 400 days. When 8 m^{3}/day is established, the period required for water breakthrough is decreased to 170 days, and the cumulative oil production of the block can reach 6338 m^{3} in 400 days. This is consistent with conventional wisdom, proving the method’s robustness in this research.

(4) This article examines the effect of initial formation pressure on daily oil production. A single well’s daily oil production is consistent regardless of initial pressure; well P1’s water breakthrough time is always around 150 days, and subsequent daily oil production is always 0.5 m^{3}/day. This is consistent with the objective understanding that the major driver of oil-water two-phase flow is differential pressure in the reservoir.