Under the macro-constraints of the sedimentary characteristics of Oilfield Q, a heterogeneous pore network model was generated and a microscopic visualization model was fabricated. Microscopic visualization displacement experiments with different displacement directions and fluid viscosities were conducted to reveal the formation mechanism of viscous fingering; parameters such as the number, width, and scale of viscous fingers were quantified, and the influencing factors of viscous fingering in heterogeneous reservoirs were identified. Additionally, the impact of viscous fingering channels on the classification of micro-flow fields during the ultra-high water cut stage was studied.

The fabrication of the heterogeneous pore network model follows the steps below: 1) Engrave the heterogeneous pore network diagram based on the information from casting thin sections. 2) Etch the heterogeneous pore network diagram onto a glass chip using etching technology. 3) Assemble the pore network model into a displaceable chip by bonding technology. 4) Conduct displacement experiments on the heterogeneous pore network using microfluidic technology.

The experimental parameters are as follows:

Simulated formation water: NaHCO₃ type with a salinity of 4505 mg/L; Simulated formation crude oil: with a viscosity range of 22 - 260 mPa∙s; Displacement intensity in microfluidic experiments: controlled within the range of 0.5 - 2.0 mL/min (converted to linear velocity) based on the similarity criteria of physical simulation.

To characterize the viscous fingering phenomenon, a new viscous fingering growth model is proposed to describe the flow law of oil-water two phases. The schematic diagram of this model is shown in Figure 1 . All finger-like flows are combined into an effective finger-like flow, which grows dynamically with water displacement.

The viscous fingering growth function is defined as the following functional equation:

${\lambda }_f=\alpha {\bar{S}}_w^{\beta }$ (1)

In the formula, λ_f is the cross-sectional ratio of the viscous fingering development area; α reflects the size of the maximum cross-section of viscous fingering; β is the growth exponent of viscous fingering.

Based on the viscous fingering modeling, through function fitting, the fitting relationships of the viscous fingering growth function under different experimental conditions are obtained, as shown in Figure 2 ; the fitting coefficients are α and β.

To realize the macroscopic characterization of the viscous fingering law, the viscous fingering number is defined:

$N_{vf}={\left(\frac{{\mu }_o}{{\mu }_w}\right)}^2\frac{{\mu }_w{u}_l}{\sigma }{A}_c{k}^{-1}$ (2)

In the formula, N_vf denotes the viscous fingering number, which is dimensionless with a value range of 10⁹ - 10¹¹; μ_o represents the oil phase viscosity, mPa·s; μ_w stands for the water phase viscosity, mPa·s; u_l is the displacement velocity, mL/min; A_c refers to the cross-sectional area, in m²; k denotes the permeability (permeability anisotropy not considered), mD; and σ represents the interfacial tension (reflecting the magnitude of capillary force), N/m.

Based on the displacement experiment, the relationship between the viscous fingering number and the viscous fingering growth function is established, which reveals the relationship between the development intensity of viscous fingering during the water flooding process in Q Oilfield and reservoir physical properties, fluid physical properties, displacement methods, and displacement velocity from a microscopic perspective, as shown in Figure 3 . This relationship can be applied in numerical simulation to reveal the formation of dominant flow fields and the development process of ineffective circulation channels caused by the development of viscous fingering during the water flooding process.

To characterize the viscous fingering phenomenon in the numerical model, the main steps are as follows:

1) Calculate the viscous fingering number of each grid:

$N_{vf}={\left(\frac{{\mu }_o}{{\mu }_w}\right)}^2\frac{{\mu }_w{u}_l}{\sigma }{A}_c{k}^{-1}$ (3)

2) Calculate the parameters of the viscous fingering function for each grid through interpolation:

${\lambda }_f=a{\bar{S}}_w^{\beta }$ (4)

3) Update the relative permeability curve of each grid.

The viscous growth exponent is embedded into the oil-water relative permeability to modify the relative permeability. The mathematical model is as follows, where the water phase is:

$k_{rw}={\lambda }_f{k}_{rw}^0{\left({\bar{S}}_w\right)}^{n_w}=a{k}_{rw}^0{\left({\bar{S}}_w\right)}^{n_w+\beta }$ (5)

The development characteristics of viscous fingering lead to the enhancement of connectivity between oil and water wells; the larger the viscous fingering number, the more severe the development of viscous fingering, which is reflected in the relative permeability as a significant increase in water phase permeability.

Taking the C3 well group in Oilfield Q as an example, after considering viscous fingering, the fitting error of the model for production performance is significantly reduced. The fitting error of oil production rate is decreased by 62.4%, and the fitting error of water cut is reduced by 71.2%.

Static parameters can indicate the potential for the formation of dominant channels, and the heterogeneity and pore-throat radius of sandstone reservoirs are important factors for the formation of dominant seepage channels [17] . Therefore, it is scientifically based to judge the development level of dominant flow fields by combining the heterogeneity and seepage capacity of the reservoir.

In this paper, the static development index is proposed as the static evaluation indicator for flow field grading. This indicator is based on the idea of AHP (Analytic Hierarchy Process), combined with the actual oilfield production and the experience of reservoir engineers. AHP is a multi-criteria decision-making method used to help people make trade-offs and choices in complex decision-making environments [8] . The key steps are to construct a parameter judgment matrix, conduct qualitative analysis using mathematical methods, and then solve the problem based on the weights calculated for each scheme.

Based on AHP, permeability, pore-throat radius, permeability ratio, and permeability variation coefficient are selected as evaluation parameters, and pairwise comparisons of the above parameters are made. The results are shown in Table 1 .

After the consistency test of the above matrix is passed, the arithmetic mean method, geometric mean method, and eigenvalue method are used to calculate the weights of the three methods respectively, and the static development index (S) is obtained as follows:

$S=0.071\ast A+0.413\ast B+0.316\ast C+0.2\ast D$ (6)

The selection of the four static parameters is closely aligned with the heterogeneity characteristics of the fluvial heavy oil reservoirs in Oilfield Q, enabling a comprehensive characterization of the geological basis for the development of dominant flow fields. The determination of the AHP weight matrix is centered on expert judgment, with the priority of parameter influences clarified through industry experience. Sensitivity analysis of weights verifies that this weight allocation method can accurately reflect the actual contribution of each parameter to the dominant flow field, providing reliable static indicator support for the subsequent hierarchical evaluation of flow fields.

In the practical application of oilfield development, the time-varying displacement flux can only qualitatively analyze the relative distribution of reservoir scouring intensity, and cannot quantitatively divide different flow field regions, nor can it characterize the development location and degree of dominant flow fields. In addition, the time-varying displacement flux includes the combined scouring intensity of the oil phase and water phase, while the dominant channeling channels are controlled by the scouring effect of injected water. Therefore, based on the time-varying displacement flux, this paper further proposes the time-varying water phase displacement flux.

$D_w=\frac{Q_w}{\varphi A}={\displaystyle \underset{i=t_1}{\overset{t_2}{\sum }}\sqrt{{\left(\frac{Q_{wxi}}{\varphi A_x}\right)}^2+{\left(\frac{Q_{wyi}}{\varphi A_y}\right)}^2+{\left(\frac{Q_{wzi}}{\varphi A_z}\right)}^2}}$ (7)

The time-varying water phase displacement flux characterizes the dynamic water phase scouring degree, and the change value of oil saturation characterizes the degree of oil phase displacement within a certain period of time. The two are combined into a new dominant flow field evaluation indicator—the dynamic oil-flux ratio, which refers to the change in grid oil saturation caused by the water flow over a period of time. Its expression is:

$\begin{array}{c}F=\frac{\left(S_{oi}-S_o\right)}{D_w}\end{array}$ (8)

where:

$Q_x$, $Q_y$, $Q_z$—Cumulative fluid volume in x, y, z directions, m³;

$A_x$, $A_y$, $A_z$—Grid cross-sectional area in x, y, z directions, m²;

$Q_{wx}$, $Q_{wy}$, $Q_{wz}$—Water volume passing through the cross-section per unit time in x, y, z directions, m³/s;

$S_o$—Oil saturation; $S_{oi}$—Original oil saturation; $\varphi$—Porosity;

$t_1$—Start time; $t_2$—End time.

Since the dynamic oil-flux ratio and the static development index have different dimensions, to eliminate the influence of dimensions between them and make the model more stable and converge faster, after eliminating the extreme values in the samples that may affect the results, the membership function is selected to normalize the data. The linear normalization method is used to normalize the static development index and the dynamic oil-flux ratio respectively, mapping the data values to the range of [0, 1]. The formula is expressed as:

$x^{\prime }=\frac{x-\min \left(x\right)}{\max \left(x\right)-\min \left(x\right)}$ (9)

Common classification methods include fuzzy comprehensive evaluation and cluster analysis. The fuzzy comprehensive evaluation method has heavy subjective factors in determining indicator boundaries and weights, which will affect the identification results. Cluster analysis is a commonly used data analysis method used to group samples in a dataset according to their similarity. Common clustering methods include K-means clustering, hierarchical clustering, DBSCAN clustering, etc. The K-means clustering method is sensitive to the selection of initial clustering centers, while hierarchical clustering and DBSCAN clustering are not efficient enough in processing large-scale data. Fuzzy C-means clustering analysis has higher objectivity compared with comprehensive evaluation, better robustness compared with other clustering methods, and more efficient data processing, which can greatly improve the identification accuracy. The idea of the fuzzy C-means clustering algorithm is to determine the membership matrix U and cluster centers V such that the objective function J_m(U, V) is minimized. The steps of the algorithm are as follows [18] :

1) Given the sample set $X=\left\{x_1,x_2,\cdots ,x_n\right\}$, determine the number of categories C (2 ≤ C ≤ N), the fuzzy weight index m and an appropriately small iterative stop threshold ε;

2) Set the initial membership matrix U(s), and set the number of iterations s = 0;

3) Calculate the clustering center $v_i^{\left(s\right)}$ at time U(s):

$v_i^{\left(s\right)}=\frac{{\displaystyle \underset{j=1}{\overset{N}{\sum }}{\mu }_{ij}^m{x}_j}}{{\displaystyle \underset{j=1}{\overset{N}{\sum }}{\mu }_{ij}^m}},i=1,2,\cdots ,C$ (10)

4) Modify the membership matrix U(s), and calculate the objective function $J_m^s$:

${\mu }_{ij}^{\left(s\right)}=\frac{1}{{\displaystyle \underset{k=1}{\overset{C}{\sum }}{\left(\frac{d_{ij}}{d_{kj}}\right)}^{\frac{2}{m-1}}}}$ (11)

$J_m^s\left(U^{\left(s\right)},V^{\left(s\right)}\right)={\displaystyle \underset{j=1}{\overset{N}{\sum }}{\displaystyle \underset{i=1}{\overset{C}{\sum }}{u}_{ij}^m}}{d}_{ij}^2$ (12)

5) For a given threshold U > 0, if the objective function $\Vert J_m^s-J_m^{s-1}\Vert \le \epsilon$, the algorithm terminates; otherwise, go to step (3).

The silhouette coefficient is an evaluation indicator of the density and dispersion of clusters. It is an evaluation method to reflect the quality of clustering results. The closer the silhouette coefficient is to 1, the better the clustering effect; if the silhouette coefficient is close to −1, it indicates that the clustering is unreasonable. The silhouette coefficient of a dataset for a certain clustering can be derived from the silhouette coefficient of individual samples:

$s_k=\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{\sum }}\frac{b_i-a_i}{\max \left(a_i,b_i\right)}}$ (13)

Where, a_i is the average distance between the sample and other samples in the same cluster; b_i is the minimum value of the average distance between the sample and all samples in other clusters; n is the number of samples in the dataset; S_k is the number of clusters for the silhouette coefficient.