Buckley-Leverett Theory Research Articles

A Buckley-Leverett Theory Based Lattice Boltzmann Method for Immiscible Two-Phase Flow with Viscous Coupling in Porous Media

A Buckley-Leverett Theory Based Lattice Boltzmann Method for Immiscible Two-Phase Flow with Viscous Coupling in Porous Media

Study on the Viscosity Optimization of Polymer Solutions in a Heavy Oil Reservoir Based on Process Simulation

Polymer flooding has been proved by many scholars for use in heavy oil reservoirs. However, due to mobility control effects and injectivity, selecting the appropriate solution viscosity is essential. It is difficult to form a deep understanding of the effect and mechanism of polymer flooding using conventional experimental methods with oil recovery as the reference standard, so it is necessary to conduct further study with the aid of simulation methods. In this study, a one-dimensional displacement mathematical model based on the Buckley–Leverett theory was established, and in the range of water–oil viscosity ratio from 0.1 to 0.6, the variation of water saturation along the flow caused by polymer solution was studied. The research results showed that under the action of a polymer solution, compared with water flooding, there was a decreasing region of water saturation along the flow due to oil phase accumulation. The larger the water–oil viscosity ratio, the larger the area of water saturation decline and the greater the degree of water saturation decline, resulting in a better the displacement effect. However, under the condition of oil–phase viscosity of 70 mPa·s, when the water–oil viscosity ratio reached 0.4, the range and degree of water saturation decline along the way no longer changed, all the crude oil that could be swept had been displaced, and the outlet end was close to producing only water, not oil; therefore, further increasing the water–oil viscosity ratio could not increase the oil recovery. At the same time, the increase in pressure had not changed, that is, the increase in pressure had not resulted in the same increase in oil recovery. When the viscosity of oil phase increased to 140 mPa·s, the same rule was displayed, the appropriate water–oil viscosity ratio was also 0.4. The paper analyzed the action mechanism of polymer solution through process description, and the results provided a clear selection method for selecting reasonable polymer solution concentration, as well as a reference for polymer solution range under different crude oil viscosity conditions.

Experimental and Analytical Investigation of an Immiscible Displacement Process in Real Structure Micromodels

The recovery of oil from a reservoir can be accomplished with various methods, one of the most commonly applied types being waterflooding. A common theory used to describe immiscible displacement is the Buckley–Leverett theory. A brand new type of micromodel, generated and fabricated by using a micro-computer tomography (μCT) image stack of a real sandstone core, was used to conduct immiscible displacement experiments. Critical logging data were recorded, and a high-resolution camera took pictures of the displacement process. In an image processing tool (MATLAB), an algorithm was developed to evaluate the pictures of the experiment and to examine the changes in the saturations of the displacing and the displaced fluid. The main objective of the displacement experiment was to validate the new microchip in two-phase displacement experiments and to assess the feasibility of the image processing algorithm. This was performed by comparing the results of the experimental to the analytical solutions, which were derived from the Buckley–Leverett theory. The comparison of the results showed a good match between the two types of solutions. The applicability of the analytical results to the experimental procedures was observed. Additionally, the usage of the newly fabricated micromodel and its potential to visualize the fluid flow behavior in porous media were assessed.

Parameter Effect Analysis of Non-Darcy Flow and a Method for Choosing a Fluid Flow Equation in Fractured Karstic Carbonate Reservoirs

Fractured karstic carbonate reservoirs have obvious multi-scale characteristics and severe heterogeneity due to the development of abundant karst caves and fractures with different scales. Darcy and non-Darcy flows coexist due to this property. Therefore, selecting the appropriate flow equations for different regions in the numerical simulation of fluid flows, particularly two-phase and multiphase flows, is a critical topic. This paper compares and analyses the displacement distance differences of waterfront travel using the Darcy, Forchheimer and Barree–Conway equations, as well as analyzes the influence of the Forchheimer constant, fluid viscosity, flow rate and absolute permeability on inertia action based on the Buckley–Leverett theory. The results show that the Forchheimer number/Reynolds number of water/oil two-phase flow is not a constant value and varies with water saturation, making it difficult to determine whether the inertial action should be considered solely based on these values; the influence of inertial action can be measured well by comparing the difference between the displacement distances of the waterflood front, and the quantitative standard is given for the selection of the flow equation of different regions by calculating the allowable error of the displacement distance of the waterflood front. The magnitude of the inertial effect is affected by the physical properties of the fluid and reservoir medium and the fluid velocity. The smaller the difference in the viscosity of the oil/water fluid, the smaller the inertial effect is. This technique was used a preliminary attempt to analyze the fractured karstic carbonate reservoirs at Tarim, and the results confirmed the validity of the method described in this article.

Determination of relative permeability curve under combined effect of polymer and surfactant

Relative permeability of surfactant-polymer (SP) flooding is determined by an inversion method based on numerical simulation and unsteady-state coreflooding experiment. Compared with the common methods, this method is more time-saving and simpler than steady-state method, and considers more physicochemical properties (adsorption and diffusion) of SP than unsteady-state method based on Buckley-Leverett theory. It is also more applicable to reservoir engineering than pore-scale simulation. After the reliability is confirmed, the superiority of the method is strongly proved by studying the influence of adsorption and diffusion of SP on relative permeability. Comparing relative permeability with and without effect of adsorption and diffusion of SP, the findings show adsorption and diffusion of SP significantly reduce the average inversion error of relative permeability curve from 11.9 to 3.7 for oil and from 10.7 to 1.8 for water. On this basis, the variations of SP-flooding relative permeability under the effect of polymer and surfactant are discussed in turn. In the existing researches, the quantitative characterization of variation of SP-flooding relative permeability curves is limited by curve crossing and endpoint saturation changes. In this work, normalized relative permeability is introduced to eliminate the adverse effect caused by curve crossing and endpoint saturation changes. The findings show that although the rise of aqueous-phase viscosity reduces water relative permeability, it has little effect on normalized water relative permeability. Normalized oil and water relative permeability increase in logistic functions with the decrease of IFT in semi-logarithmic plot. Water relative permeability at residual oil endpoint decreases linearly as aqueous-phase viscosity increases, and increases exponentially as IFT decreases in semi-logarithmic plot. Ultimately, the functional models based on normalized relative permeability and cubic B-spline are built through multiple regression, which can calculate relative permeability of SP flooding in term of IFT and aqueous-phase viscosity.

Buckley–Leverett Theory for a Forchheimer–Darcy Multiphase Flow Model with Phase Coupling

This paper is dedicated to the modeling, analysis, and numerical simulation of a two-phase non-Darcian flow through a porous medium with phase-coupling. Specifically, we introduce an extended Forchheimer–Darcy model where the interaction between phases is taken into consideration. From the modeling point of view, the extension consists of the addition to each phase equation of a term depending on the gradient of the pressure of the other phase, leading to a coupled system of differential equations. The obtained system is much more involved than the classical Darcy system since it involves the Forchheimer equation in addition to the Darcy one. This model is more appropriate when there is a substantial difference between the phases’ velocities, for instance in the case of gas/water phases, and applications in oil recovery using gas flooding. Based on the Buckley–Leverett theory, including capillary pressure, we derive an explicit expression of the phases’ velocities and fractional water flows in terms of the gradient of the capillary pressure, and the total constant velocity. Various scenarios are considered, and the respective numerical simulations are presented. In particular, comparisons with the classical models (without phase coupling) are provided in terms of breakthrough time among others. Eventually, we provide a post-processing method for the derivation of the solution of the new coupled system using the classical non-coupled system. This method is of interest for industry since it allows for including the phase coupling approach in existing numerical codes and software (designed for solving classical models) without major technical changes.

Experimental investigation of solute transport in variably saturated porous media using x-ray computed tomography

Solute transport through variably saturated porous media is ubiquitous in multiple subsurface flows, piquing the geoscience community's interest. This study adopts a novel experimental approach using microfocus x-ray computed tomography for real-time imaging of a three-dimensional NaI tracer plume in a partially saturated packing column. A stabilized two-phase flow field is achievable through continuous co-injection of two-phase fluids: NaCl solvent and pump oil. Thus, the critical role of the NaCl saturation Sw and Péclet number on dispersion can be fully studied by controlling the NaCl fractional flow rate and the total flow rate from the Buckley–Leverett theory. Furthermore, we study solute transport behavior based on statistical moments, the dispersion coefficient, the dilution index, and the mean scalar dissipation rate. Experimental results indicate that the solute transport is Fickian for high Sw ≥ 0.34. In contrast, anomalous transport behavior is found for Sw < 0.34, where the concentration distribution is initially left-tailed and leptokurtic before reaching a well-dispersed regime. The dispersion coefficient is 2–10 times larger for partially saturated cases compared with the fully saturated case and shows a non-monotonical dependency on Sw. Finally, the analysis of the dilution index indicates that the overall mixing strength increases when Sw decreases, whereas the mean scalar dissipation rate reveals that the time scaling of transverse mixing is the largest at an intermediate Sw. The results can be used to elucidate the solute transport behavior in a two-phase system.

Buckley–Leverett Theory for Two-Phase Immiscible Fluids Flow Model with Explicit Phase-Coupling Terms

The theory of two-phase immiscible flow in porous media is based on the extension of single phase models through the concept of relative permeabilities. It mimics Darcy’s law for a fixed average saturation through the introduction of saturation-based permeabilities to model the momentum exchange between the phases. In this paper, we present a model of two-phase flow, based on the extension of Darcy’s law including the effect of capillary pressure, but considering in addition the coupling between the phases modeled through flow cross-terms. In this work, we extend the Buckley–Leverett theory to the subsequent model, and provide numerical experiments shading the light on the effect of the coupling cross-terms in comparison to the classical Darcy’s approach.

A physically-supported data-driven proxy modeling based on machine learning classification methods: Application to water front movement prediction

By defining an optimum injection/production strategy in the water flooding process, the water front movement is controlled and an early breakthrough is avoided, and as a result, the sweep efficiency is increased. The most important part of injection/production planning is determining the injection fluid front position. However, using a commercial reservoir simulator comes up with a considerable time and CPU effort, particularly in the case of large and complex reservoirs. Several proxy models, either physics-based or Data-Driven, have been developed to predict water front movement with low computational time and cost, each offering some advantages and shortcomings such as error accumulation and short-term predictions. In this paper, we use a hybrid view in developing a classification based smart proxy model at the grid-block level (CSPMG) for front prediction to benefit from the advantages of both physics-based and Data-Driven proxies. The idea is to formulate the problem by using the physical principles underlying the problem (physical-view) and then use machine learning classification models to capture the pattern between inputs and the target feature (Data-Driven view). Based on the Buckley-Leverett theory, which is a physics-based method for front advancement in porous media, water front prediction was formulated as a classification problem in which the grid-blocks behind and ahead of the front were considered as separate classes and a label was assigned to each class. Then, artificial neural networks (ANNs) were trained on a training database to predict the class label of each grid block. The water front was considered as the boundary between two adjacent classes. A binary and a ternary classification problem were formulated and two proxy models were developed. A blind test was carried out to compare their results with each other, with a Data-Driven regression model, and with those of a reservoir simulator. The results showed that the CSPMG matches the reservoir simulator results and outperforms the regression model and makes longer predictions.

ESTIMATION OF FIELD PRODUCTION PROFILES IN CASE OF ASPHALTENE DEPOSITION

In this work, we aimed to predict possible field production scenarios in case of asphaltene deposition based on field data as well as recommend remediation and stimulation measures to mitigate the risks of asphaltene deposition in the reservoir. We considered the influence of asphaltene formation in the near-wellbore of producers on the production data without reservoir pressure maintenance system in one of the oil fields. The asphaltene envelope in the reservoir oil was obtained, and the operating conditions of the field were evaluated under the possibility of asphaltene deposition. According to the results of dynamic modeling, the pressure map was plotted and the low-pressure areas in the near-wellbore were shown, which contributes to the aggravation of the problem associated with the asphaltene envelope. Based on the geometrical features of the low-pressure area, the dependence of the permeability reduction in the near-wellbore of the production well on the operating time was obtained using the asphaltene deposition model proposed by Wang and Civan. Based on the Buckley-Leverett theory, the field production profiles were calculated with and without asphaltene deposition. A decrease in the oil rate and consequently, the decrease in cumulative oil production in the field is expected due to the damage formation by solids. Maintenance of the production level will be facilitated by treating the near-wellbore with aromatic solvents and maintaining the reservoir pressure above the asphaltene onset pressure.

Study on the Theoretical Method of Determining Reasonable Water Injection Intensity

Currently, water flooding is the main method to maintain reservoir pressure and improve oil production in many oilfields. A calculation procedure is developed based on the Buckley-Leverett theory (φ function) to determine the reasonable water injection rate, which directly improve the effectiveness of field development. φ function (the derivative of water fractional flow with respect to water saturation) is the reciprocal of PV number, which relates to the swept area, effective reservoir thickness, water cut and cumulative water injection. Considering all these factors, the proposed method can ensure the rationality of water injection rate.

Establishment and Application of a Water-drive Curve Based on Relative Permeability Following Power Function

Based on Buckley-Leverett theory of water flooding in non-piston way, fractional flow equation, Welge average water saturation equation and the relative permeability curve, a new relationship between water cut and recovery efficiency and the corresponding water drive characteristic curve are derived making use of the general solution of ordinary differential equation formula with the initial condition. The ratio of oil-water relative permeability follows the power function in the process of derivation. It is different from the previous derived methods of water displacing curves to consider the effect of water-free oil production period on water drive type curve in this paper. The practical application in K oilfield indicates that the method is rational and effective.

Evaluation of the Vertical Producing Degree of Commingled Production via Waterflooding for Multilayer Offshore Heavy Oil Reservoirs

Recently, commingling production has been widely used for the development of offshore heavy oil reservoirs with multilayers. However, the differences between layers in terms of reservoir physical properties, oil properties and pressure have always resulted in interlayer interference, which makes it more difficult to evaluate the producing degree of commingled production. Based on the Buckley–Leverett theory, this paper presents two theoretical models, a one-dimensional linear flow model and a planar radial flow model, for water-flooded multilayer reservoirs. Through the models, this paper establishes a dynamic method to evaluate seepage resistance, sweep efficiency and recovery percent and then conducts an analysis with field data. The result indicates the following: (1) the dynamic difference in seepage resistance is an important form of interlayer interference during the commingled production of an offshore multilayer reservoir; (2) the difference between commingled production and separated production is small within a certain range of permeability ratio or viscosity ratio, but separated production should be adopted when the ratio exceeds a certain value.

A further study on Welge equation

Buckley–Leverett theory and Welge equation is one of the most widely used methods for predicting fluid transport in porous media. However, the average water saturation in Welge equation is a function of both water saturation and water cut, which is inconvenient to be used. While the linear relationship (the slope is 2/3) of the average water saturation and outlet water saturation proposed by Зфрос provides a simple method the oil–water viscosity ratio greatly limits the application. In this paper, based on the analytical relative permeability model and Welge equation, a new average water saturation equation with a variable (the outlet water saturation) was derived and applied in some cases under different relative permeability parameters and oil–water viscosity. Moreover, a simplified equation was proposed through certain data regression. Thus, the coefficient calculation method of the linear function was introduced based on the actual production data. Results showed a nonlinear relationship between the average water saturation and outlet water saturation, which is mainly related to water and oil relative permeability, and oil–water viscosity ratio. For the outlet water saturation higher than frontal saturation, the equation can be simplified to a linear function, which is the derivation of Зфрос equation. However, it is a function of both relative permeability and oil–water viscosity ratio instead of a constant slope of 2/3. In view of the fact that oil–water viscosity is mostly bigger than 10 in most reservoirs, the new model is an important supplement to Buckley–Leverett theory and Welge equation.

Nanofluid pre-treatment, an effective strategy to improve the performance of low-salinity waterflooding

Low-salinity water (LSW) has been confirmed as a promising improved oil recovery technique. Implementing this technique, in some cases, induces fines migration and has posed great challenges on well injectivity. The aim of this work is to establish a mathematical foundation and explain the improved performance (both EOR and well injectivity) through nanofluid pre-treatment in single-layered and multi-layered heterogeneous reservoirs.We modify the fractional flow function while considering for fines straining effects in one-dimensional radial system, where fines migration varies with distance from the injection well owing to different drag forces. The interplay among nanoparticles, fines, and rocks is described through the maximum retention concentration of fine particles that depend upon both fluid quality and velocity. Our modification of the classical Buckley-Leverett theory leads to the characteristic curves along which water saturation varies. As low-salinity waterflooding continues, the induced fines migration delays the breakthrough of injected water, and bring uniform control of injected fluid entering each layer. On the other hand, fines migration may lead to the issue of well injectivity decline.Nanofluid pre-treatment can significantly mitigate fines migration near wellbore while it may occur far from the injection well. We discuss cases with different nanofluid treatment schemes prior to low-salinity waterflooding by comparing water-saturation profile, water-cut history plot, injection pressure, and oil recovery. A good agreement is obtained between semi-analytical solution and finite-difference simulation. It was observed that although nanofluid treatment slightly accelerate breakthrough of the injected water, it can help maintain long-term well injectivity and sweep efficiency. In practice, our work provides valuable insight to design of nanofluid utilization and improve efficiency of low-salinity waterflooding.

Modeling two-phase flow of immiscible fluids in porous media: Buckley-Leverett theory with explicit coupling terms

Continuum models that describe two-phase flow of immiscible fluids in porous media often treat momentum exchange between the two phases by simply generalizing the single-phase Darcy law and introducing saturation-dependent permeabilities. Here we study models of creeping flows that include an explicit coupling between both phases via the addition of cross terms in the generalized Darcy law. Using an extension of the Buckley-Leverett theory, we analyze the impact of these cross terms on saturation profiles and pressure drops for different couples of fluids and closure relations of the effective parameters. We show that these cross terms in the macroscale models may significantly impact the flow compared to results obtained with the generalized Darcy laws without cross terms. Analytical solutions, validated against experimental data, suggest that the effect of this coupling on the dynamics of saturation fronts and the steady-state profiles is very sensitive to gravitational effects, the ratio of viscosity between the two phases, and the permeability. Our results indicate that the effects of momentum exchange on two-phase flow may increase with the permeability of the porous medium when the influence of the fluid-fluid interfaces become similar to that of the solid-fluid interfaces.

Non-Darcy displacement by a non-Newtonian fluid in porous media according to the Barree-Conway model

An analytical solution for describing the non-Darcy displacement of a Newtonian fluid by a non-Newtonian fluid in porous media has been developed. The two-phase non-Darcy flow is described using the Barree-Conway model under multiphase conditions. A power-law non-Newtonian fluid, whose viscosity is a function of the flow potential gradient and the phase saturation, is considered. The analytical solution is similar to the Buckley-Leverett theoretical solution, which can be regarded as an extension of the Buckley-Leverett theory to the non-Darcy flow of non-Newtonian fluids. The analytical results revel how non-Darcy displacement by a non-Newtonian fluid is controlled not only by relative permeabilities but also by non-Darcy flow coefficients as well as non-Newtonian rheological constitutive parameters and injection rates. The comparison among Darcy, Forchheimer and Barree-Conway models is also discussed. For application, the analytical solution is then applied to verify a numerical simulator for modeling multi-phase non-Darcy flow of non-Newtonian fluids.

Desarrollo de una aplicación para implementar un modelo predictivo de inyección química

Se presenta una aplicación desarrollada en Visual Basic que permite determinar el comportamiento de campos petrolíferos donde se aplican procesos de inyección de agua, polímeros, surfactantes y surfactantes-polímeros, a través de un modelo analítico que considera la teoría de Buckley- Leverett y el método de tubos de corriente. La aplicación no reemplaza la simulación numérica, porque no tiene en cuenta el modelo estático del reservorio y, en general, requiere poca información de entrada; sin embargo, es una aproximación rápida de predicción para obtener tablas y gráficas de tasas de producción y acumulados de petróleo y agua y factor de recobro de petróleo y comparar los resultados con los obtenidos por otros tipos de herramientas predictivas.

On the Buckley-Leverett Equation With Constant-Pressure Boundary Conditions

Summary The classical Buckley-Leverett theory applies to 1D linear flow of two immiscible phases under the fundamental assumption that the flow rate (or total volumetric flux) is constant as a function of time. One phase is injected into the medium at a constant rate, thereby displacing the other phase. If the displacing phase is instead injected at constant pressure and the outlet pressure is also constant, the problem is still well-defined; however, the classical Buckley-Leverett theory does not apply. This is because the injected phase and the in-situ phase have different properties such as viscosities. If the boundary pressures are kept constant, the flow rate will, therefore, vary over time. The main result of this paper is to show that the solution of the displacement problem can be obtained from the constant-rate solution through an analytical determination of the flow rate as a function of time, given constant-pressure boundaries. The theory developed in this paper also provides an analytical solution for the location of the displacement front at any given time, the time for frontal breakthrough at the outlet end, and the pressure distribution as a function of time inside the medium. It is demonstrated through computed examples that the constant-flow-rate solution, in general, cannot be used to approximate the corresponding solution for constant-pressure boundaries because the variation in the flow rate is very significant. A standard numerical method has also been applied and compared with the analytical solutions, demonstrating that fine numerical-simulation grids are required for acceptable comparison with the analytical solution.

Experimental and Analytical Investigation of Spontaneous Imbibition in Water-Wet Carbonates

We perform co-current spontaneous imbibition ambient-condition experiments in three carbonates with a wide range of permeability under strongly water-wet conditions. We measure water saturation profiles as a function of distance and time in air-filled rocks with no initial water saturation using X-ray CT scanning. We demonstrate that the saturation profiles are functions of distance divided by the square root of time. We also demonstrate that the profiles are consistent with analytical solutions for imbibition in one dimension, and using reasonable estimates of relative permeability and capillary pressure, we can match the experimental results. We discuss how, in combination with conventional measurements of relative permeability (steady-state or using Buckley–Leverett theory in an unsteady-state experiment), the capillary pressure can be determined, or how the relative permeability can be determined from the spontaneous imbibition experiment and the capillary pressure.