The reservoir is the networked rock skeleton of an oil and gas trap, as well as the generic term for the fluid contained within pore fractures and karst caves. Heterogeneity and a complex internal pore structure characterize the reservoir rock. By introducing the fractal permeability formula, this paper establishes a fractal mathematical model of oil-water two-phase flow in an oil reservoir with heterogeneity characteristics and numerically solves the mathematical model using the weighted least squares meshless method. Additionally, the method's correctness is verified by comparison to the exact solution. The numerical results demonstrate that the fractal oil-water two-phase flow mathematical model developed using the meshless method is capable of more accurately and efficiently describing the flow characteristics of the oil-water two-phase migration process. In comparison to the conventional numerical model, this method achieves a greater degree of convergence and stability. This paper examines the effect of varying the initial viscosity of the oil, the initial formation pressure, and the production and injection ratios on daily oil production per well, water cut in the block, and accumulated oil in the block. For 10 and 60 cp initial crude oil viscosities, the water cut can be 0.62 and 0.80, with 3100 and 1900 m^{3} cumulative oil production. Initial pressures have little effect on production. In this case, the daily oil production of well PRO1 is 1.7 m^{3} at 7 and 10 MPa initial pressure. Block cumulative oil production is 3465.4 and 2149.9 m^{3} when the production injection ratio is 1.4 and 0.8. The two-phase meshless method described in this paper is essential for a rational and effective study of production dynamics patterns in complex reservoirs and the development of reservoir simulations of oil-water flow in heterogeneous reservoirs.

The reservoir is the network of rock skeletons that surround a sealed oil and gas trap, as well as the generic term for the fluid contained in pore fractures and karst caves. The heterogeneity and complexity of reservoir rock have a significant effect on reservoir physical properties such as flowability, reservoir capacity, and rock particle adsorption capacity. To maximize recovery, it is necessary to master the situation of underground reservoirs as much as possible during oil field development [

Mandelbrot pioneered the concept of a fractal, which is used to describe the geometry found in nature that is self-similar and has no feature length [

The majority of the research on fractal reservoirs discussed above makes use of numerical models based on grid systems, such as the finite difference method (FDM) and the finite element method (FEM) [

This Paper introduces the fractal permeability expression, establishes a numerical model of fractal oil reservoir two-phase, and solves it using the weighted least squares meshless method (MWLS). Relevant examples are provided to demonstrate the accuracy and benefits of this method.

(1) Assume that the outer boundary of the reservoir is closed; (2) Fluid and rock micro-compressibility; (3) Isothermal flow event, ignoring heat dissipation; (4) The model applies to oil-water two-phase, and the two are not mixed; (5) Ignore the influence of capillary force.

The distribution of reservoir porosity and permeability conforms to the following rules:

In the formula,

The meshless method can be theoretically justified using the weighted residual method. The Moving Least Square method (SLS) is used to construct approximate functions in this paper, followed by the Least Square method to discretize the governing equations and establish the corresponding meshless method, namely the weighted least squares meshless method (MWLS). For more information about the weighted least squares meshless method, see reference [

The time items of the above

Among them,

The above

In

For residual changes:

The MWLS form of the oil phase pressure equation can be obtained as follows:

Considering

which:

Water saturation MWLS is calculated as follows:

Considering

which:

Based on the theoretical research described above, a two-dimensional oil-water two-phase numerical simulation model with fractal properties is established. The model is 100 m ∗ 100 m ∗ 10 m in size, with the injection well IN1 in the center, the production wells distributed around it, an injection rate of 20 m^{3}/day, a recovery rate of 5 m^{3}/day, and a production time of 200 days. The model's permeability distribution is depicted in

The Buckley-Leverett [

Properties | Values | Properties | Values |
---|---|---|---|

Porosity | 0.3 | Oil volume factor | 1.2 |

Reservoir size | 100 m ∗ 100 m ∗ 10 m | Water viscosity | 1 cp |

Rock compressibility | 1.07e-4 MPa^{−1} |
Water volume factor | 1.000 |

Oil compressibility | 3.02e-3 MPa^{−1} |
Oil density | 820 kg/m^{3} |

Water compressibility | 5e-4 MPa^{−1} |
Water density | 1000 kg/m^{3} |

Oil viscosity | 20 cp | Initial oil saturation | 0.80 |

Initial pressure | 7 MPa | Initial water saturation | 0.20 |

Fractal index | −0.15 | Fractal mass dimension | 2.0 |

Sensitivity analysis is performed on the initial crude oil viscosity, initial water saturation, initial pressure, and other parameters that affect the production dynamic data, and then the correctness of this method is verified in the numerical solution of fractal reservoir oil-water two-phase flow using the above-mentioned theories and methods.

The initial formation pressure is the critical factor affecting the reservoir's daily oil production. The development benefit of the reservoir can be determined using the initial formation pressure, which more intuitively reflects the reservoir's crude oil diversion capacity. In this paper, the initial formation pressures of 7, 8, 9, and 10 MPa are used for simulation to ensure that the other physical parameters are consistent with

^{3}/day of oil. When the initial viscosity of the crude oil is 60 cp, however, well PRO1 produces only 1.23 m^{3}/day of oil. The higher the initial crude oil viscosity, the shorter the time required for water to penetrate the reservoir. As illustrated in ^{3}. However, the block's cumulative oil production is limited to 1900 m^{3} due to the initial crude oil viscosity of 60 cp. The lower the initial viscosity, the greater the accumulation of oil in the block, which is primarily determined by the two-phase flow mobility ratio.

The initial formation pressure is the critical factor affecting the reservoir's daily oil production. The development benefit of the reservoir can be determined using the initial formation pressure, which more intuitively reflects the reservoir's crude oil diversion capacity. In this paper, the initial formation pressures of 7, 8, 9, and 10 MPa are used for simulation to ensure that the other physical parameters are consistent with

The pressure change value for well PRO1 at various initial formation pressures is depicted in

In actual reservoir production, injection-production parameters are critical for maximizing economic benefits. The injection volume is kept constant in this paper, and simulations using the production-injection ratios of 0.8, 1.0, 1.2, and 1.4 are used to ensure that all other physical parameters are consistent with

From ^{3}, but when the injection ratio is 0.8, the FWCT after 300 days is only 0.6872 and the FOPT is only 2149.9 m^{3}, while when the injection ratio is 0.8, the FWCT after 300 days is only 0.6872 and the FOPT is only 2149.9 m^{3}. Therefore, if the water cost of processing production is low, increasing the injection ratio can result in a higher FOPT, which can be used to guide production.

The weighted least squares meshless method is introduced in this paper to solve an oil reservoir oil-water two-phase flow model with fractal characteristics. The fractal permeability formula is used to establish a fractal mathematical model of oil-water two-phase flow in an oil reservoir that takes heterogeneity into account, and the mathematical model is numerically solved using the weighted least squares meshless method. The following are the major conclusions:

According to the model developed in this paper, the difference between this method and the analytical model is only 2%. The numerical results demonstrate that using the meshless method, the fractal oil-water two-phase flow mathematical model can more accurately and efficiently describe the flow characteristics of the oil-water two-phase migration process.

The effects of varying the initial crude oil viscosity on the daily oil production of a single well, water cut, and cumulative oil production are examined in this article. When the initial crude oil viscosity is 10 and 60 cp, the water cut can reach 0.62 and 0.80, respectively, resulting in cumulative oil production of 3100 and 1900 m^{3}. According to objective facts, this method's stability and convergence are also demonstrated from the side.

This paper investigates the effect of different initial formation pressures on the daily output of a single well and the pressure of a single well, concluding that different initial formation pressures do not affect the output. This is consistent with the objective understanding that differential pressure drives oil-water two-phase flow in reservoir production. In this example, the daily oil production of well PRO1 is 1.7 m^{3} in 200 days when the initial pressure is 7 and 10 MPa.

This paper relates the effects of various production injection ratios on daily oil production from a single well, block water cut, and cumulative oil production from a block. When the production injection ratio is 1.4 and 0.8, respectively, the block water cut is 0.7322 and 0.6872, and the cumulative oil production of the block is 3465.4 and 2149.9 m^{3}, respectively. As a result, during actual production, the injection ratio should be rationally planned to maximize economic benefit.

Density of oil and water respectively (kg/m^{3});

Fractal permeability (mD);

Relative permeability of water phase and oil phase (mD);

Viscosity of oil phase and water phase mPa⋅s;

Pressure of oil phase and water phase in Reservoir (MPa);

Fractal porosity;

Saturation of water phase and oil phase;

production time (day);

Source-sink phase under standard conditions (m^{3}/day).