Quantifying Dominant Remaining Oil Distribution in Displacement Units of High-Water-Cut Reservoirs

Abstract

Remaining oil in high-water-cut reservoirs becomes increasingly dispersed during long-term waterflooding, while preferential flow paths cause severe ineffective water circulation and reduce the efficiency of further oil displacement. To improve the quantitative identification of remaining oil enrichment and water-flushed regions, this study proposes a displacement-unit-based classification and evaluation method for dominant remaining oil distribution. The method integrates dynamic allocation of injected water in multilayer reservoirs, time-varying characterization of reservoir physical properties, streamline-based delineation of displacement units, and saturation tracking using the φ-function. Two quantitative indicators, the remaining oil abundance index (I_so) and the water flushing intensity coefficient (C_f), were introduced to classify displacement units into strongly dominant, weakly dominant, and non-dominant types. The method was applied to a high-water-cut block of the W Oilfield, where 902 displacement units were identified from 65 oil and water wells and 36 sublayers. The results show that strongly dominant, weakly dominant, and non-dominant displacement units accounted for 37.9%, 33.7%, and 28.4% of the total, respectively. In 15 sublayers, the proportion of strongly dominant units exceeded 50%, indicating severe preferential water flow and limited remaining oil potential in these layers. Strongly dominant units were characterized by high water flushing intensity and low remaining oil abundance, whereas weakly dominant units showed remaining oil enrichment mainly at the margins of displacement units. The proposed method couples injection–production dynamics with seepage-field evolution and provides a quantitative basis for fine-scale adjustment of injection–production patterns in high-water-cut reservoirs.

1. Introduction

As major oilfields worldwide have entered the late stage of development, the exploitation of high-water-cut reservoirs faces unprecedented challenges. At this stage, remaining oil distribution in reservoirs exhibits strong heterogeneity and dispersion. The effectiveness of traditional development schemes gradually diminishes, oil recovery efficiency declines significantly, and the potential for enhanced oil recovery becomes limited [1,2,3]. High-water-cut reservoirs typically undergo the evolution of water-injection dynamics, complex seepage-field changes, and fluid property variations. These factors cause reservoirs to display pronounced heterogeneity during long-term injection–production processes, resulting in more spatially dispersed remaining oil and the formation of locally enriched zones [4]. Therefore, traditional remaining oil prediction models and development strategies can no longer meet the demands of high-water-cut reservoir exploitation. How to reveal the distribution patterns of remaining oil in high-water-cut reservoirs and propose effective development optimization strategies has become a critical issue in reservoir engineering.

Currently, the research focus in reservoir development has gradually shifted from purely physical reservoir description to fine characterization of seepage-field evolution and injection–production dynamics. As oilfield development enters the high-water-cut stage, the distribution characteristics of remaining oil become increasingly complex. Traditional remaining oil prediction methods usually focus on static reservoir analysis and statistical evaluation of historical data, and therefore cannot accurately reflect the evolution process of remaining oil under the influence of injection–production dynamics [5,6]. For example, Guo et al. [7] quantitatively described the formation process of preferential flow paths using the flow intensity index. However, this method mainly focused on the static distribution of the seepage field and did not fully consider the time-varying characteristics of injection–production dynamics. Gao et al. [8] proposed a seepage-field evaluation system based on fuzzy comprehensive evaluation. Although this system could reflect flow characteristics to a certain extent, it still failed to effectively reveal the complex evolution process of the seepage field in high-water-cut reservoirs, particularly how injection–production dynamics influence reservoir development optimization. During the development process of high-water-cut reservoirs, the injection–production system exhibits strong dynamic characteristics. These dynamics are reflected not only in changes in water injection intensity and displacement efficiency, but also in the evolution of reservoir physical properties and fluid characteristics over time [9,10,11]. Such dynamics lead to nonlinear evolution in reservoir development, significantly affecting the distribution and production efficiency of remaining oil. For instance, during long-term water injection, the pore structure of the reservoir may change, thereby influencing the distribution pattern of the seepage field [12,13,14]. However, most existing studies are limited to analyses under static injection–production conditions and overlook the profound influence of injection–production dynamics on seepage-field evolution and remaining oil distribution [15,16,17]. How to accurately capture the influence of injection–production dynamics on the seepage field and remaining oil distribution, and to perform reliable classification and evaluation, has become a key issue for enhanced oil recovery. Current remaining oil evaluation methods mostly rely on single-dimensional physical property indicators, such as water-cut and porosity, and therefore cannot fully describe the complex characteristics of remaining oil distribution in high-water-cut reservoirs [18,19,20]. Although some recent studies have attempted to introduce advanced methods such as displacement characteristic curves and multiphase flow models [21,22,23], these methods rely on field data acquisition and numerical simulation accuracy, and still face limitations in data availability and computational precision. Consequently, they cannot fully reveal the dynamic distribution and evolution patterns of remaining oil [24,25]. Therefore, the existing classification and evaluation systems have not effectively integrated the complex relationship between seepage-field characteristics and injection–production dynamics, nor have they thoroughly considered the influence of different displacement units on remaining oil distribution [26,27,28]. Reservoir heterogeneity strongly affects sweep efficiency, breakthrough behavior, and oil recovery during displacement processes. Recent numerical simulations of water-alternating-gas injection in heterogeneous reservoirs have shown that permeability contrast and flow-field evolution play important roles in controlling displacement performance [29]. In addition, displacement-unit-based methods have been proposed to characterize waterflooded low-permeability reservoirs and provide a finer description of reservoir dynamics than conventional static flow-unit methods [30]. However, the coupling among dynamic injected-water allocation, time-varying petrophysical properties, remaining-oil abundance, and classified evaluation of dominant remaining oil has not been fully established. Previous studies on remaining-oil identification and waterflood optimization can be broadly divided into three categories. The first category is based on static reservoir parameters, such as porosity, permeability, net pay thickness, water-cut, and current oil saturation. These methods are easy to implement and have been widely used in field-scale remaining-oil evaluation. However, they mainly reflect the static geological and production state of the reservoir and are unable to describe the dynamic migration of preferential flow paths during long-term waterflooding. The second category includes streamline simulation and interwell-connectivity methods. Streamline-based methods are effective for visualizing flow paths, evaluating injection efficiency, and optimizing waterflooding schemes, while interwell-connectivity models can quantitatively describe the dynamic response between injectors and producers. Nevertheless, these methods usually focus on flow-path identification or well connectivity, and the coupling between flow intensity, remaining-oil abundance, and displacement-unit-scale classification remains insufficient. Recent streamline simulation studies have shown good performance in waterflooding management and injection optimization, but their main emphasis is still on flow-path prediction rather than classified remaining-oil evaluation. The third category focuses on displacement-unit analysis and time-varying petrophysical characterization. For example, analytical displacement-unit methods have been proposed to characterize waterflooded low-permeability reservoirs, and time-varying petrophysical models have been developed to identify preferential flow paths in ultra-high-water-cut reservoirs. These studies indicate that both displacement-unit division and time-varying reservoir properties are important for describing the evolution of waterflooding performance. However, existing displacement-unit methods are mainly used to describe waterflooded reservoir features, whereas time-varying preferential-flow-path methods mainly identify high-flow-capacity channels. A unified framework that simultaneously considers dynamic displacement-unit boundaries, injected-water allocation, time-varying physical properties, remaining-oil saturation tracking, and classified remaining-oil dominance has not been fully established. In summary, existing research methods generally exhibit the following shortcomings. On the one hand, they fail to comprehensively consider the profound influence of injection–production dynamics on seepage-field evolution and remaining oil distribution. On the other hand, existing classification and evaluation systems lack the integration of multi-dimensional seepage characteristics and injection–production dynamics, making it difficult to provide fine-scale optimization schemes for the development of high-water-cut reservoirs. Therefore, a new method is needed to accurately reveal the distribution patterns of remaining oil in high-water-cut reservoirs and to achieve classified evaluation and development optimization based on injection–production dynamics.

To address the above issues, this study proposes a displacement-unit-based dynamic evaluation framework for dominant remaining oil in high-water-cut reservoirs. The novelty of this work lies not in redefining streamline bundles or interwell connectivity units, but in integrating dynamic displacement-unit delineation, injected-water allocation, time-varying petrophysical characterization, and φ-function-based saturation tracking into a unified classification framework. The main objectives and contributions of this study are as follows: (1) a dynamic displacement-unit framework is established to characterize the time-varying seepage regions between injection and production wells during long-term waterflooding; (2) an injected-water allocation method is developed by considering interlayer heterogeneity, interwell pressure response, and relative-permeability effects, so that the water flushing intensity of each displacement unit can be quantified; (3) φ-function-based saturation tracking is coupled with time-varying petrophysical characterization to evaluate the remaining oil abundance within displacement units; (4) a dual-index classification criterion is constructed based on the remaining oil abundance index and water flushing intensity coefficient, and its feasibility is validated through a field case study in a high-water-cut reservoir block.

2. Seepage Characteristics and Characterization Methods of Displacement Units

During the development process of ultra-high-water-cut reservoirs, injected water frequently channels rapidly to production wells through high-permeability zones or natural fractures. This results in a large volume of injected water failing to effectively displace remaining oil. This phenomenon is termed “ineffective water circulation” and significantly affects the overall recovery factor of reservoirs [31,32]. As reservoirs enter the high-water-cut stage, the distribution of remaining oil becomes increasingly complex, the production efficiency of injected water decreases, and the waterflooding efficiency between injection wells and production wells declines significantly. Therefore, how to identify and optimize the seepage paths between injection and production wells to form effective displacement zones has become a key issue for enhancing recovery in high-water-cut reservoirs. To address this, the concept of “displacement units” is proposed in this study to systematically describe and analyze the effective displacement zones of injected water in reservoirs. By identifying and optimizing preferential flow paths, the influence of ineffective water circulation can be effectively reduced, thereby improving reservoir recovery efficiency.

2.1. Dynamic Evolution Patterns of the Seepage Field

In this study, a displacement unit is defined as a dynamically evolving reservoir region within an individual reservoir layer, located along each hydraulically connected direction between an injection well and a production well, in which effective oil–water seepage exchange occurs during reservoir development. Based on the principles of seepage mechanics, the seepage field describes the flow range of fluids in three dimensional porous media, while the streamline field serves as a mathematical representation of fluid flow paths. Streamlines are curves consistent with fluid motion trajectories, and a streamline field composed of multiple streamlines illustrates the overall flow pattern of fluids within the reservoir. Although the displacement unit is related to streamline bundles and interwell connectivity units, it has a different emphasis in this study. Streamline bundles mainly describe flow trajectories, and interwell connectivity units mainly characterize hydraulic communication between injectors and producers. By contrast, the displacement unit used here serves as a quantitative evaluation unit that couples streamline defined flow regions with injected-water allocation, water flushing intensity, time-varying petrophysical properties, and remaining oil abundance. Through analysis of the streamline field, the seepage characteristics between injection and production wells can be effectively characterized, thereby helping to optimize seepage paths and classify remaining oil distribution in reservoir development [33,34].

To investigate the flow characteristics of displacement units in reservoirs and address water invasion issues, the concept of “preferential flow field” was introduced in this study to characterize the seepage regions between injection and production wells in detail and to optimize water injection schemes. As shown in Figure 1, the planar schematic of displacement units shows multiple sub-regions. Fluid is injected through injection wells, while fluids are produced from production wells under pressure drawdown. The pressure difference between injection and production wells drives fluid movement. The direction and density of streamlines are closely related to reservoir permeability and pressure gradients. Therefore, the region enclosed by injector-producer streamlines was used to delineate the spatial boundary of a displacement unit, while its dynamic evaluation was further performed by coupling water allocation, flushing intensity, and remaining oil abundance.

During reservoir displacement, the streamline envelope of displacement units evolves dynamically under the combined influence of reservoir heterogeneity and injection–production pressure differences. To characterize this process, a heterogeneous numerical simulation model was established using the reservoir physical property parameters of the W Oilfield, and the typical evolution patterns are shown in Figure 2. Before water injection, fluid migration was mainly controlled by natural formation energy. After waterflooding began, injected water preferentially flowed through high-permeability regions, while low-permeability regions gradually lost effective flow channels. With prolonged water injection, the streamline distribution tended to stabilize and formed preferential flow paths. When water invasion further intensified, the streamline field reorganized locally, indicating the possible formation of new preferential flow paths and uneven sweep regions.

2.2. Quantitative Characterization Methods for the Seepage Field

During long-term waterflooding in high-water-cut reservoirs, significant fluid flushing occurs among displacement units as water injection continues. This leads to fluid redistribution and changes in flow intensity within the seepage field. To quantitatively characterize the seepage field flushing intensity among displacement units, the cumulative injection pore volume multiple (N_PV) is introduced in this study as a dynamic indicator. N_PV is defined as the ratio of injected water volume to the pore volume within the displacement unit and is expressed as follows:

$$N_{PV}={\displaystyle \frac{W}{V_{\varphi }}}={\displaystyle \frac{W}{\varphi \cdot A\cdot h}}$$

(1)

where W is the cumulative injected water volume in m³, V is the pore volume, ϕ is the porosity, A is the displacement area in m², and h is the effective thickness in m.

To calculate N_PV within a displacement unit, the cumulative injected water volume needs to be determined first. In multilayered reservoirs, the injection–production relationship between wells requires consideration of fluid pressure balance and flow coupling effects among layers. Through a multilayered coupling model, the pressure distribution in different layers can be determined, which allows the flow contribution of fluids in each layer to be evaluated. Traditional allocation formulas mainly rely on permeability and thickness as key parameters. However, in reservoirs with strong heterogeneity, permeability differences between layers can lead to the formation of distinct preferential flow channels, making it difficult to accurately describe actual fluid flow conditions using only permeability- and thickness-based allocation methods [29].

In this study, the oil–water two-phase flow in the displacement unit was described using the mass conservation equation and Darcy’s law. For phase α (α = o, w), the governing equation can be written as:

$${\displaystyle \frac{𝜕}{𝜕t}}\left(\varphi {\rho }_{\alpha }{S}_{\alpha }\right)+\nabla ·\left({\rho }_{\alpha }{\mathrm{u}}_{\alpha }\right)=q_{\alpha }$$

(2)

where ϕ is porosity, ρ_α is phase density, S_α is phase saturation, u_α is Darcy velocity, and q_α is the source/sink term. The Darcy velocity of each phase is expressed as:

$$u_{\alpha }=-{\displaystyle \frac{\mathrm{K}{k}_{r\alpha }}{{\mu }_{\alpha }}}\nabla p_{\alpha }$$

(3)

where K is the absolute permeability tensor, k_rα is the relative permeability of phase α, μ_α is phase viscosity, and p_α is phase pressure. In this study, an isothermal incompressible oil–water system was assumed, and gravity and capillary pressure effects were neglected to focus on the influence of reservoir heterogeneity and injection–production dynamics. Therefore, the phase saturations satisfy:

$$S_o+S_w=1$$

(4)

where S_o and S_w are oil saturation and water saturation, respectively.

To more accurately reflect the actual flow distribution of fluids between layers, a flow heterogeneity factor was introduced into the allocation formula. Meanwhile, by incorporating dynamic production data, such as the production pressure difference and instantaneous flow rate of each well, together with static parameters such as permeability, the allocation coefficient was dynamically adjusted to reflect the time-varying characteristics of the injection–production relationship. During water injection, the relative permeability of water and oil, as well as displacement efficiency, significantly influence fluid flow. Therefore, an adjustment factor based on the dynamic variation of the saturation field was further introduced to characterize the effect of saturation variation on interlayer and interwell flow allocation. Assuming that the injection–production relationship is mainly influenced by formation permeability, layer thickness, interwell pressure difference, and fluid properties, a partitioning formula based on the allocation coefficient of effective displacement units was constructed in this study.

The total injected water volume Q_i of each injection well i is allocated to each reservoir layer according to the characteristics of different layers. The allocation coefficient f_ik for each layer is determined through weighted calculation based on interlayer permeability and thickness, as follows:

$$f_{ik}={\displaystyle \frac{k_k·h_k·a_k·S_{ok}}{{\displaystyle {\sum }_{m=1}^L\left(k_m·h_m·a_m·S_{om}\right)}}}$$

(5)

where:

$$a_k={\displaystyle \frac{k_{ro}\left(S_{wc}\right)}{k_{rw}\left(S_{or}\right)}}$$

(6)

where k_k is the permeability of layer k, h_k is the thickness of layer k, L is the total number of layers penetrated by injection well, α_k is the heterogeneity factor of layer k, S_ok is the oil saturation of layer k, k_ro(S_wc) is the relative permeability of the oil phase at irreducible water saturation, k_rw(S_or) is the relative permeability of the water phase at residual oil saturation, m is the layer index used in the summation, S_ok and S_om are oil saturations of layer k and layer m, respectively.

After calculation, the injected water volume of injection well i, in layer k was obtained as follows:

$$Q_{ik}=f_{ik}·Q_i.$$

(7)

Within a single layer k, the injected water volume Q_ik is further allocated to each production well $j$ connected to injection well i according to the interwell transmissibility coefficient T_ijk, which is defined as follows:

$$T_{ijk}={\displaystyle \frac{k_k·h_k}{{\mu }_w·d_{ij}}}$$

(8)

where u_w is the water viscosity, and d_ij is the distance between injection well i and production well j.

By introducing the dynamic pressure adjustment factor β_ij and the relative permeability influence factor k_rwk, together with the interwell transmissibility coefficient T_ijk, the improved allocation coefficient f_ijk from injection well i in layer $k$to production well j is obtained as follows:

$$f_{ijk}={\displaystyle \frac{T_{ijk}·{\beta }_{ij}·k_{rwk}}{{\displaystyle {\sum }_{j=1}^{N_p}\left(T_{ijk}·{\beta }_{ij}·k_{rwk}\right)}}}$$

(9)

where:

$${\beta }_{ij}={\displaystyle \frac{\Delta P_{ij}}{{\displaystyle {\sum }_{j=1}^{N_p}\Delta P_{ij}}}}$$

(10)

where N_p is the number of production well i connected to injection well in layer k, β_ij is the dynamic pressure difference adjustment factor between injection well i and production well j, and k_rwk is the relative permeability influence factor of layer k.

The parameters α_k and β_ij were determined independently from different data sources. The layer heterogeneity factor α_k was determined from static reservoir and laboratory data, including layer permeability, effective thickness, oil saturation, and endpoint relative permeability obtained from core relative permeability experiments. Therefore, α_k reflects the intrinsic interlayer differences in flow capacity and was not calibrated using interwell pressure data. In contrast, the dynamic pressure adjustment factor β_ij was calculated from the instantaneous pressure difference between injection well i and production well j, and then normalized among all production wells connected to the same injection well in layer k. Thus, β_ij reflects the dynamic injection-production pressure response and was updated with production performance. Because α_k and β_ij were derived from static data and dynamic pressure data, respectively, the two factors were independently determined in the allocation calculation.

Finally, the injected water volume Q_ijk from injection well i to production well j within layer k is obtained, representing the injected water volume within a single displacement unit:

$$Q_{ijk}=f_{ijk}·Q_{ik}=f_{ijk}·f_{ik}·Q_i.$$

(11)

In numerical simulation, the reservoir is discretized into multiple three-dimensional grid cells, with each cell representing a specific reservoir volume. Within these grid cells, fluid flow occurs along the x-, y-, and z-directions, corresponding to the two horizontal directions and the vertical direction. Based on Darcy’s law, the fluid velocity in porous media is used to derive the cumulative injected water volume in the x-, y-, and z-directions, which is expressed as follows:

$$N_{PVx}={\displaystyle \frac{{\displaystyle {\int }_{t_0}^t{Q}_{x}dt}}{V_{\varphi ,x}}}={\displaystyle \frac{{\displaystyle {\int }_{t_0}^t{Q}_{x}dt}}{\varphi ·D_{\mathrm{y}}·D_z·D_x}},$$

(12)

$$N_{PVy}={\displaystyle \frac{{\displaystyle {\int }_{t_0}^t{Q}_{y}dt}}{V_{\varphi ,y}}}={\displaystyle \frac{{\displaystyle {\int }_{t_0}^t{Q}_{y}dt}}{\varphi ·D_x·D_z·D_y}},$$

(13)

$$N_{PVz}={\displaystyle \frac{{\displaystyle {\int }_{t_0}^t{Q}_{z}dt}}{V_{\varphi ,z}}}={\displaystyle \frac{{\displaystyle {\int }_{t_0}^t{Q}_{z}dt}}{\varphi ·D_x·D_y·D_z}}.$$

(14)

In the three-dimensional reservoir model, water flow through a grid cell is controlled by the vectorial fluxes across its cell faces. Therefore, the cumulative injected pore-volume multiple should not be obtained by directly summing the directional values in the x, y, and z directions. Instead, it was calculated from the positive incoming water-phase fluxes across all faces of a grid cell. For a given grid cell $g$, the cumulative injected pore-volume multiple was expressed as:

$$N_{PV g}(t)={\displaystyle \frac{1}{P{V}_g}}{\displaystyle {\int }_0^t{\displaystyle \sum _{f=1}^{N_f}\max }}\left(q_{w,f}(\tau ),0\right)d\tau$$

(15)

where N_{pv g}(t) is the cumulative injected pore-volume multiple of grid cell g; PV_g is the pore volume of grid cell g; q_w,f is the water-phase flux across face f; N_f is the total number of grid-cell faces, t is the injection time, and τ is the integration time variable. The operator max(q_w,f, 0) is used to include only the incoming water-phase fluxes, thereby accounting for the directionality of flow across each cell face. This formulation avoids the direct scalar summation of x-, y-, and z-direction cumulative injected pore volumes and better represents vectorial flow dynamics in the three-dimensional reservoir model.

The total flow rate describes the overall fluid flow condition within each grid cell and characterizes the flow distribution and pathways of fluids in three-dimensional space. N_PV serves as an important tool for identifying preferential flow channels, particularly during the development of high-water-cut reservoirs. Through analysis of N_PV, the variation patterns of seepage-field flushing intensity in displacement units can be effectively characterized. The dynamic evolution of N_PV not only helps to identify potential preferential flow paths, but also provides a quantitative basis for optimizing injection well-pattern configuration and enhancing oil recovery.

3. Time-Varying Laws of Reservoir Physical Properties

During waterflooding development in high-water-cut reservoirs, reservoir physical properties exhibit obvious time-varying characteristics. To investigate this phenomenon, water injection experiments were conducted on 28 core samples from the W Oilfield. The changes in the proportion of cores within different permeability intervals before and after waterflooding were statistically analyzed, and the influence of water injection on reservoir physical properties and their variation patterns was examined.

As shown in Figure 3, the effect of flushing intensity on permeability variation differs significantly among cores with different initial permeabilities. At the initial stage of waterflooding, cores with low initial permeability (0–200 mD) account for a large proportion, and the proportion of cores decreases rapidly with increasing permeability. Meanwhile, the proportion of cores with initial permeability greater than 350 mD is significantly reduced, indicating that low-permeability cores are more easily displaced. In the late stage of waterflooding, the proportion of high-permeability cores increases, especially in the permeability interval above 400 mD, where the core proportion shows an upward trend. These experimental results reveal the influence of water injection on seepage characteristics and reservoir physical properties in layers with different permeabilities, and demonstrate the time-varying patterns of reservoir physical properties during waterflooding. These time-varying patterns are helpful for identifying effective displacement regions that evolve with the development process, and further improve the accurate characterization and optimization of fluid volume in displacement units.

Based on the above experimental results, a numerical simulation model with a five-spot well pattern consisting of one injector and four producers was constructed using reservoir physical property parameters from an experimental block in the W Oilfield, as shown in Figure 4. In this model, two high-permeability zones (② and ③), one medium-permeability zone (④), and one low-permeability zone (①) were designed to simulate the variations in seepage-field flushing intensity and oil saturation at different water-cut stages.

A numerical simulation was performed using high efficiency numerical algorithm to solve the oil-water two-phase flow problem. The algorithm was developed based on Darcy’s law and mass conservation equations and was used to calculate pressure distribution, water-phase flux, cumulative injected pore-volume multiple, and oil saturation evolution during waterflooding. An isothermal oil–water two-phase flow model was adopted. The model used a five-spot well pattern with one injector and four producers. Reservoir heterogeneity was represented by assigning two high-permeability zones, one medium-permeability zone, and one low-permeability zone. The heterogeneous permeability field was generated based on the permeability distribution of the W Oilfield and the core waterflooding experimental results. According to the permeability contrast observed in the study block, the model was divided into low-, medium-, and high-permeability zones. Representative permeability values were assigned to these zones to construct a conceptual heterogeneous field for analyzing preferential flow path formation, water flushing intensity, and oil saturation evolution. The injector was controlled by a constant water-injection rate, while the producers were controlled by fixed production constraints. Gravity and capillary pressure effects were neglected to highlight the influence of permeability heterogeneity and injection–production dynamics on water flushing intensity and oil saturation distribution.

The governing equations were discretized using a finite-difference formulation on a structured Cartesian grid. The pressure and saturation terms were approximated by finite differences in space, and an implicit time-discretization scheme was adopted to improve numerical stability during high-water-cut flow simulation. The nonlinear pressure–saturation equations were solved iteratively until the convergence tolerance was satisfied. The injector and producers were represented by source/sink terms under prescribed injection and production constraints, and the outer boundaries were treated as no-flow boundaries. The initial pressure and saturation fields were assigned according to the reservoir parameters of the W Oilfield and the relative permeability model. Post-processing was performed to obtain pressure distribution, water-phase flux, $N_{PV}$, streamline evolution, and oil saturation distribution for displacement-unit classification.

The distribution of N_PV under different water-cuts was analyzed, as shown in Figure 5. In high-permeability zones, N_PV increases rapidly with rising water-cut. When the water-cut reaches 98%, regions with N_PV greater than 1000 can be regarded as strong flushing-intensity zones. In low-permeability zones, by contrast, N_PV changes only slightly with increasing water-cut, and N_PV in these regions is less than 30, which is recognized as weak flushing-intensity zones. The remaining regions are classified as moderate flushing-intensity zones.

The oil saturation distribution in the model under different water-cut stages is shown in Figure 6. By comparing the results of the four scenarios, it can be found that at the low- and medium-water-cut stages, the oil saturation distributions in low-, medium-, and high-permeability zones are relatively similar, and corresponding seepage channels can be formed. However, at the high-water-cut stage, the performance of each zone differs significantly. In particular, a large amount of remaining oil remains effectively untapped in the low-permeability zone and its adjacent areas. The main reason is that obvious preferential flow channels have formed in medium- and high-permeability zones, which restrict water flow into the low-permeability zone. This leads to the gradual weakening of the seepage field in the low-permeability zone, so that the oil saturation in low-permeability regions cannot be sufficiently reduced.

Combining experimental and numerical simulation approaches, the seepage characteristics and oil saturation variation patterns in different permeability ranges during waterflooding were revealed, and the formation mechanism of preferential flow paths within displacement units was further clarified. Especially during the high-water-cut period, the formation of preferential flow channels was found to inhibit the mobilization efficiency of remaining oil in low-permeability zones. This phenomenon indicates that, during waterflooding development in the high-water-cut stage, the dynamic evolution of seepage channels exerts a significant influence on remaining oil mobilization, particularly in low-permeability regions. Through detailed analysis of the seepage characteristics within displacement units, potential remaining oil enrichment areas can be identified, thereby providing support for the optimization of reservoir development plans.

4. Characterization and Classified Evaluation Method of Remaining Oil in Displacement Units

4.1. Quantitative Characterization Method of Remaining Oil

In simplified waterflooding analyses, linear relative permeability assumptions are sometimes used for preliminary evaluation. However, oil–water relative permeability in real reservoirs generally exhibits a strong nonlinear dependence on saturation. Therefore, nonlinear Corey-type or Brooks–Corey-type models are more commonly used to describe the variation of oil–water two-phase flow capacity with saturation [35]. In the high-water-cut stage, this nonlinearity becomes more significant because water-phase flow capacity increases rapidly while oil-phase mobility decreases continuously. Accordingly, based on experimental data fitting, the $\phi$-function theory [36] provides a more accurate model for describing the nonlinear relationship between relative permeability and oil saturation during waterflooding. By introducing this function, the time-varying evolution of water saturation under long-term water injection can be quantitatively characterized.

Based on the Buckley–Leverett theory, assuming negligible gravity and capillary forces, constant fluid viscosity, and incompressible fluids, the partial differential equation describing the variation of water saturation S_w at position $x$ and time $t$ can be derived as follows:

$${\displaystyle \frac{dx}{dt}}={\displaystyle \frac{Q}{\varphi A}}{\displaystyle \frac{𝜕f_w}{𝜕S_W}},$$

(16)

$$\phi \left(S_W\right)=f_w^{\prime }\left(S_w\right)={\displaystyle \frac{𝜕f_w}{𝜕S_W}}.$$

(17)

According to the Buckley–Leverett fractional-flow theory and Welge’s method for determining the waterflood front, at time t after the start of injection, if the cumulative injection volume is W_i(t), the relationship for water saturation S_w at position x can be expressed as [37,38]:

$$x-x_0={\displaystyle \frac{\phi \left(S_W\right)}{\varphi A}}{\int }_0^{t}Qdt,$$

(18)

$$\phi \left(S_W\right)={\displaystyle \frac{\varphi A\left(x-x_0\right)}{W_i(t)}}={\displaystyle \frac{1}{N_{PV}}}$$

(19)

where x₀ is the coordinate of the injection well origin, φ is the porosity, A is the cross-sectional area of the pay zone, and φ(S_w) is the derivative of the water fractional flow f_w with respect to water saturation S_w.

Relative permeability curves of high-water-cut reservoirs are important tools in reservoir engineering for describing the variation of oil–water two-phase flow capacity with saturation. These curves are usually determined experimentally, but can also be fitted using empirical correlations. In this study, the Brooks–Corey model was adopted to fit the oil–water relative permeability curves:

$$k_{ro}=k_{ro}^{end}·{\left({\displaystyle \frac{S_o-S_{or}}{1-S_{or}-S_{wi}}}\right)}^{no},$$

(20)

$$k_{rw}=k_{rw}^{end}·{\left({\displaystyle \frac{S_w-S_{wi}}{1-S_{or}-S_{wi}}}\right)}^{nw}$$

(21)

where S_wi is irreducible water saturation. The relationship between water-cut (f_w) and water saturation (S_w) can be derived from the relative permeability curves:

$$f_w\left(S_w\right)={\displaystyle \frac{k_{rw}/{\mu }_w}{k_{ro}/{\mu }_o+k_{rw}/{\mu }_w}},$$

(22)

$$\phi \left(S_W\right)=f^{\prime }\left(S_{\mathrm{w}}\right)=\left[f\left(S_{\mathrm{w}}\right)-f^2\left(S_{\mathrm{w}}\right)\right]\left({\displaystyle \frac{n_o}{1-S_{\mathrm{w}}-S_{or}}}+{\displaystyle \frac{n_w}{S_{\mathrm{w}}-S_{wi}}}\right)$$

(23)

where $k_{ro}^{end}$ is final relative permeability of oil phase at residual saturation; $k_{rw}^{end}$ is final relative permeability of water phase at residual saturation; μ_o and μ_w are oil and water viscosities.

The nonlinear relative permeability relationship affects the proposed calculation process mainly through the water fractional flow and the φ(S_w)-function. As k_rw increases and k_ro decreases nonlinearly with increasing water saturation, f_w rises rapidly during the high-water-cut stage. Since φ(S_w) is derived from the derivative of f_w, this nonlinearity further influences water-front movement, saturation tracking, and the calculated remaining oil abundance. Therefore, the nonlinear Corey/Brooks–Corey-type description is necessary for capturing the rapid increase in water-phase flow capacity and the decline in oil-phase mobility during long-term waterflooding.

Based on the oil–water relative permeability curves measured from reservoir experiments, the relationships among water saturation S_w, water-cut f_w, and φ(S_w) can be further derived, as shown in Figure 7.

In actual reservoir development, water saturation increases continuously only when the oil-water front advances to a given location; otherwise, the corresponding water saturation remains below the irreducible water saturation regardless of fluid volume changes. In this case, the calculated value of the φ(S_w) function will be greater than that corresponding to the front water saturation. Therefore, based on this theoretical assumption, a stricter expression for calculating oil saturation S_o using the φ(S_w) function can be derived:

$$S_o=\left\{\begin{array}{ll}1-{\phi }^{-1}\left(S_{\mathrm{w}}\right),\hfill & \phi \left(S_{\mathrm{w}}\right)⩽\phi \left(S_{\mathrm{wf}}\right)\hfill \\ 1-S_{\mathrm{wc}},\hfill & \phi \left(S_{\mathrm{w}}\right)>\phi \left(S_{\mathrm{wf}}\right)\hfill \end{array}\right.$$

(24)

4.2. Classified Evaluation Method of Remaining Oil

To quantitatively characterize remaining oil enrichment and water-sweeping differences within displacement units, a classification method combining the remaining oil abundance index and water flushing intensity coefficient was established in this study. The core idea of this method is to first determine the physical thresholds of remaining oil abundance based on oil–water relative permeability and water-cut characteristics, and then characterize the water flushing degree according to the injected pore-volume difference at the displacement-unit scale. Finally, the displacement units are classified into strongly dominant, weakly dominant, and non-dominant types by jointly considering remaining oil abundance and water flushing intensity.

Based on the core waterflooding experiments and relative permeability tests of the W Oilfield, the relationships among the oil–water relative permeability ratio, water-cut, and oil saturation were plotted, as shown in Figure 8. The curve reflects the nonlinear variation in oil-phase and water-phase flow capacities during waterflooding. As oil saturation decreases, the oil-phase flow capacity gradually weakens, whereas the water-phase flow capacity and water-cut increase rapidly.

As shown in Figure 8, two characteristic points with clear physical significance can be identified: the co-flow point ① and the relative-permeability inflection point ②. The co-flow point corresponds to an oil saturation of 0.458 and a water-cut of 82.8%, at which the oil and water phases begin to exhibit comparable flow capacity. The relative-permeability inflection point corresponds to an oil saturation of 0.377 and a water-cut of 96.4%, indicating that the water-phase flow capacity increases rapidly, while the oil-phase flow capacity further decreases. These two characteristic points represent the critical transition from high-abundance to low-abundance remaining oil and from low-abundance to hard-to-produce remaining oil, respectively. Therefore, they were used as the physical basis for determining the remaining oil abundance thresholds.

To eliminate the influence of differences in initial oil saturation among different displacement units, the remaining oil abundance index $I_{so}$ was defined as follows:

$$I_{so}={\displaystyle \frac{S_o-S_{o,\min }}{S_{o,\max }-S_{o,\min }}}$$

(25)

where I_so is the remaining oil abundance index; S_o is the oil saturation of the current zone; S_o,min and S_o,max are the minimum and maximum values of oil saturation in the reservoir, respectively.

According to this normalization method, the remaining oil abundance indices corresponding to the co-flow point ① and the relative-permeability inflection point ② are 0.53 and 0.21, respectively. Therefore, I_so = 0.53 was used as the boundary between high-abundance and low-abundance remaining oil zones, while I_so = 0.21 was used as the boundary between low-abundance and hard-to-produce remaining oil zones. The corresponding classification criteria are listed in

Table 1. Basis for determining the classification thresholds of the remaining oil abundance index.

Remaining Oil Abundance Zones	High Abundance Zone	Low Abundance Zone	Hard-to-Recover Zone
Range of Remaining Oil Abundance Index	0.53	0.21–0.53	0–0.21

.

In addition to remaining oil abundance, water flushing intensity is another key indicator for evaluating the development state of displacement units. During the high-water-cut stage, the cumulative injected pore-volume multiple N_PV may differ by several orders of magnitude among displacement units because injected water preferentially flows through high-permeability channels and well-connected flow paths. If the original N_PV values are directly used, a few extremely high values in strongly flushed preferential flow paths may dominate the evaluation results and obscure the differences among weakly and moderately swept displacement units. Therefore, logarithmic normalization was used to define the water flushing intensity coefficient C_f. This treatment compresses the numerical range of N_PV, reduces the influence of extreme high values, and preserves the relative differences in flushing intensity among displacement units. Physically, the logarithmic transformation also reflects the diminishing marginal effect of injected water during long-term waterflooding: an increase in N_PV at a low flushing level has a stronger effect on remaining oil mobilization than the same increase after a preferential flow channel has already been strongly flushed.

$$C_f={\displaystyle \frac{\ln (N_{PV})-\ln \left(N_{PV\min }\right)}{\ln \left(N_{PV\max }\right)-\ln \left(N_{PV\min }\right)}}$$

(26)

where C_f is the water flushing intensity coefficient; N_PV is the cumulative injected pore-volume multiple of the displacement unit; and N_PVmin and N_PVmax are the minimum and maximum cumulative injected pore-volume multiples among all displacement units, respectively. To avoid the logarithm of zero, N_PVmin represents the minimum positive value of N_PV in the evaluated displacement units.

This normalization constrains C_f to a dimensionless range of 0–1. A value of C_f close to 1 indicates that the displacement unit has been strongly flushed by injected water and is more likely to relate to preferential flow paths. In contrast, a value of C_f close to 0 indicates insufficient water sweep and relatively high remaining oil potential. Based on the normalized distribution of flushing intensity in the study block, C_f < 0.3, 0.3 ≤ C_f < 0.7, C_f ≥ 0.7 were used to represent weak, moderate, and strong water flushing, respectively.

By jointly considering I_so and C_f, a classified evaluation criterion for remaining oil in displacement units was established. Displacement units with low I_so and high C_f were classified as strongly dominant units, indicating strong water flushing and limited remaining oil potential. Units with intermediate I_so and C_f were classified as weakly dominant units, indicating moderate water flushing and partially retained remaining oil. Units with high I_so and low C_f were classified as non-dominant units, indicating weak water sweep and relatively high remaining oil enrichment.

Affected by reservoir heterogeneity, water flushing intensity shows obvious spatial differences, as shown in Figure 9. According to the simulation results for a block of the W Oilfield, region ① has a water flushing intensity coefficient greater than 0.7, indicating strong water flushing. Injected water mainly flows along preferential flow paths, with high utilization of pore space, which may lead to sparse remaining oil distribution or difficult mobilization of residual oil. Region ② has a coefficient between 0.3 and 0.7, representing moderate flushing intensity and moderate remaining oil abundance. Region ③ has a coefficient below 0.3, corresponding to weak flushing, high remaining oil abundance, and great development potential.

To verify the stability of the selected water-flushing intensity thresholds (Figure 10), a threshold sensitivity analysis was further conducted. As illustrated in Figure 10, the frequency distribution of the water-flushing intensity coefficient (C_f) across the 902 displacement units exhibits a distinctly polarized, bimodal feature. A prominent concentration of units is observed at both the weak flushing (C_f ≤ 0.3) and strong flushing (C_f ≥ 0.7) extremes. This macroscopic polarization of the seepage field structurally underscores the severe channelling characteristics typical of the late high-water-cut stage: injected fluid predominantly channels through mature preferential flow paths, resulting in intense but ineffective water circulation, while simultaneously leaving considerable regions bypassed. To verify the stability of the selected water-flushing intensity thresholds, a threshold sensitivity analysis was further conducted. The baseline scheme used I_so = 0.21/0.53 and C_f = 0.30/0.70, while relaxed-threshold and strict-threshold schemes were also designed for comparison. The results are shown in

Table 2. Results of threshold sensitivity analysis.

Scheme	Thresholds of Iso	Thresholds of Cf	Strongly Dominant Units	Weakly Dominant Units	Non-Dominant Units
S1: Relaxed thresholds	0.19/0.50	0.25/0.65	360 (39.9%)	301 (33.4%)	241 (26.7%)
S2: Baseline thresholds	0.21/0.53	0.30/0.70	342 (37.9%)	304 (33.7%)	256 (28.4%)
S3: Strict thresholds	0.23/0.56	0.35/0.75	320 (35.5%)	307 (34.0%)	275 (30.5%)

. Under small perturbations of the threshold values, the proportions of the three types of displacement units changed only slightly. The classification results of strongly dominant, weakly dominant, and non-dominant displacement units remained relatively stable, indicating that the proposed classification criteria have good robustness.

By comprehensively considering the remaining oil abundance index and water flushing intensity coefficient of actual oilfield blocks, a classified evaluation criterion for remaining oil in displacement units is proposed. According to this criterion, displacement units are divided into three types, as shown in

Table 3. Classified evaluation criteria for remaining oil in displacement units.

Classification	Remaining Oil Abundance Index	Water Flushing Intensity Coefficient
Strongly dominant displacement unit	Iso < 0.21	Cf ≥ 0.7
Weakly dominant displacement unit	0.21 ≤ Iso < 0.53	0.3 ≤ Cf < 0.7
Non-dominant displacement unit	Iso ≥ 0.53	Cf ≤ 0.3

.

5. Field Application

In this section, the proposed method for classified evaluation and quantitative characterization of remaining oil in displacement units was applied to a typical oilfield block. The selected block is located in the W Oilfield and represents a typical medium- to high-permeability reservoir. After years of development, this block has entered the high-water-cut stage.

The test area is characterized by consistent oil–gas and oil–water contacts, forming a unified hydrodynamic system. The oil–gas contact is located at an elevation of approximately 770 m, and the oil–water contact is located at approximately 1050 m. The structural oil column height in this block is approximately 280 m, and the gas column height is approximately 90 m. In the oil zone, the irreducible water saturation is 23.5%, and the initial oil saturation is 76.5%. Development commenced in 1974, and after years of production, four sets of waterflooding layer series have been established. By December 2020, the cumulative injection–production ratio in the block was 1.0, the annual natural decline rate was 8.06%, and the comprehensive decline rate was 2.89%. At that time, the water–oil ratio in the block approached 40, and severe water flooding was observed in the main oil layers. As multiple layer series entered the ultra-high-water-cut stage, the distribution of remaining oil gradually exhibited highly dispersed characteristics, which presented significant challenges for remaining oil potential tapping. Therefore, based on fine characterization of remaining oil distribution, it is necessary to explore efficient waterflooding development technologies suitable for ultra-high-water-cut reservoirs to address the complex issues arising from the current development stage.

Based on the aforementioned method, a displacement-unit model was constructed, and quantitative characterization and evaluation of remaining oil were performed for displacement units involving 65 oil and water wells and 36 sublayers in this block. A three-dimensional numerical model containing multiple displacement units was built using the geological data of the block. In this model, dynamic waterflooding changes, spatial permeability distribution, and oil–water contact evolution were considered. By simulating the waterflooding process at different time stages, the displacement efficiency and productivity variation of the reservoir were analyzed.

According to the characterization method for dynamic seepage regions of displacement units, the envelope ranges of displacement units between injection and production wells during reservoir development were delineated, and a total of 902 interconnected displacement units were identified from perforated layers. Figure 11 and Figure 12 show the planar and vertical schematic diagrams of displacement units for two main layers, respectively, further illustrating the seepage regions of these units.

Using the seepage-field flushing intensity characterization method, the water injection rate Q_ijk within each displacement unit was calculated based on the allocation formula of the displacement-unit allocation coefficient. Subsequently, N_PV and the water flushing intensity coefficient C_f of each displacement unit were obtained. Meanwhile, the remaining oil abundance index of each displacement unit was further derived using the oil saturation calculation method. According to the classification and evaluation criteria for remaining oil, the proportion of strongly dominant displacement units in each sublayer within the block was identified. The results show that a total of 342 strongly dominant displacement units, 304 weakly dominant displacement units, and 256 non-dominant displacement units were recognized. As illustrated in Figure 13, in 15 sublayers of the Northeast Block II, the proportion of strongly dominant displacement units exceeded 50%.

Based on the established classification and evaluation criteria for remaining oil in displacement units, the simulation results were systematically analyzed by classification. The results indicate that strongly dominant displacement units accounted for 37.9% of the total. Figure 14 shows the distribution characteristics of remaining oil in a typical strongly dominant displacement unit. From a vertical perspective, the remaining oil abundance in each sublayer within this unit was low, while the water flushing intensity was high, making effective mobilization of the remaining oil difficult. Moreover, Figure 15 shows the remaining oil distribution characteristics in a typical weakly dominant displacement unit. The remaining oil abundance was high at the edges of the sublayer, whereas in the central region, the remaining oil was more dispersed due to the formation of obvious preferential flow paths.

To further clarify the methodological contribution of the proposed framework, a comparison between the proposed method and representative existing approaches is summarized in

Table 4. Comparison between existing methods and the proposed method.

Method Type	Main Basis	Strength	Main Limitation	Improvement in This Study
Static remaining-oil evaluation	Porosity, permeability, water-cut, oil saturation	Simple and suitable for rapid field diagnosis	Cannot capture dynamic seepage-field evolution during long-term waterflooding	Introduces dynamic displacement units and water-flushing intensity
Streamline simulation	Flow paths, time of flight, streamline density	Effective for visualizing flow paths and waterflood sweep	Mainly focuses on flow-path description; remaining-oil abundance is not explicitly classified	Couples streamline-defined units with remaining-oil abundance evaluation
Interwell connectivity methods	Production–injection response, connectivity coefficients	Quantifies injector–producer communication	Usually lacks direct coupling with saturation evolution and remaining-oil classification	Incorporates interwell pressure response into injected-water allocation
Time-varying petrophysical methods	Dynamic permeability, apparent viscosity, flow resistance	Considers reservoir-property evolution during high-water-cut development	Mainly identifies preferential flow paths, but does not classify displacement-unit-scale remaining oil	Combines time-varying properties with φ-function-based saturation tracking
Existing displacement-unit methods	Analytical or geological division of displacement units	Provides a unit-scale description of waterflooded reservoirs	Dynamic injection–production response and remaining-oil dominance are insufficiently integrated	Establishes a dual-index dynamic classification framework based on Iso and water flushing intensity

. Existing static remaining-oil evaluation methods mainly rely on geological and production indicators and are therefore limited in describing dynamic seepage-field evolution. Streamline simulation and interwell-connectivity methods can characterize flow paths and injector–producer communication, but they generally do not explicitly couple flow intensity with remaining-oil abundance at the displacement-unit scale. In contrast, the proposed method integrates dynamic displacement-unit delineation, injected-water allocation, time-varying petrophysical characterization, and φ-function-based saturation tracking, thereby enabling classified evaluation of dominant remaining oil under high-water-cut conditions.

It should be noted that reservoir water salinity was not explicitly included in the present classification framework. Previous numerical simulation studies have shown that salinity-related effects, such as salt precipitation, may influence near-wellbore and reservoir flow behavior; therefore, future extensions of the proposed method should incorporate salinity-dependent flow parameters to evaluate their influence on water flushing intensity and remaining oil classification [39].

6. Conclusions

This study proposed a displacement-unit-based method for quantitatively identifying dominant remaining oil in high-water-cut reservoirs. Its feasibility was demonstrated through a field application in the W Oilfield. The main conclusions are as follows:

(1)

Displacement units can effectively characterize the dynamic seepage regions between injection and production wells. During long-term waterflooding, reservoir heterogeneity and injection-production pressure differences jointly control the migration and reorganization of streamline enclosed displacement regions, leading to preferential flow paths in high permeability and well-connected zones.

(2)

The combination of the remaining oil abundance index I_so and the water flushing intensity coefficient C_f provides a quantitative basis for classified evaluation of remaining oil. I_so reflects the remaining oil enrichment degree, while C_f characterizes the cumulative water flushing intensity within each displacement unit.

(3)

In the W Oilfield block, 902 displacement units were identified, including 342 strongly dominant units, 304 weakly dominant units, and 256 non-dominant units, accounting for 37.9%, 33.7%, and 28.4%, respectively. In 15 sublayers, strongly dominant units accounted for more than 50%, indicating significant preferential water flow in these layers.

(4)

Strongly dominant units were characterized by high water flushing intensity and low remaining oil abundance, whereas weakly dominant units retained more remaining oil near the margins of displacement units. The proposed method was applied to one representative field block; future work should further verify the method using tracer tests, production logging data, and additional field cases with different reservoir conditions.