non-Darcy Flow Research Articles

Integrated flow model for evaluating maximum fracture spacing in horizontal wells

Multi-stage fractured horizontal wells are extensively used in unconventional reservoir; hence, optimizing the spacing between these hydraulic fractures is essential. Fracture spacing is an important factor that influences the production efficiency and costs. In this study, maximum fracture spacing in low-permeability liquid reservoirs is studied by building an integrated flow model incorporating key petrophysical characteristics. First, a kinematic equation for non-Darcy seepage flow is constructed using the fractal theory to consider the non-homogeneous characteristics of the stimulated rock volume area (StRV) and its stress sensitivity. Then, the kinematic equation is used to build an integrated mathematical model of one-dimensional steady-state flow within the StRV to analytically determine the pressure distribution in StRV. The resultant pressure distribution is utilized to propose an optimal value for the maximum fracture spacing. Finally, the effects of fractal index, initial matrix permeability, depletion, and stress sensitivity coefficient on the limit disturbed distance and pressure distribution are studied. This study not only enriches the fundamental theory of nonlinear seepage flow mechanics but also provides some technical guidance for choosing appropriate fracture spacing in horizontal wells.

Bejan’s heatline and mulitphysics analysis of double diffusive free convection in a doubly stratified non-Darcian porous enclosure with heat generation

The current article illustrates the characteristics of the MHD free convective transport phenomenon in the thermal and mass stratified fluid-saturated Darcy-Forchheimer porous square cavity under the action of the cross-diffusion (Soret and Dufour effects) heat generation and chemical reaction effects. To acquire the spatial distribution of thermal fluxes along with directional propagation in chemically reactive and heat generative non-Darcy fluid flow, a newly developed heatline model for multi-force effect following the heat function theory is proposed in this study. In addition to the thermal profile visualization tools, the solutal free convective fluid flow fields are visually demonstrated by the massline contours curves using a novel mass-function model analogous to the heat function methodology. On the other hand, for the numerical solution of the non-dimensional form of the partial differential equations governing the study, the finite element method is applied. Moreover, the non-Darcy fluid flow attributes subjected to the multiple forces are graphically portrayed by the streamlines, isotherms, iso-concentrations, cumulative Nusselt/Sherwood number plots, heatlines, and masslines for the related dimensionless parameter. Here, these visuals delineate the contra-rotating fluid flow transition under both the thermal and concentration stratification forces. Especially, forceful retardation in both the thermal and concentration flux’s movements is noticed when magnetic forces are intensified. An upward geometrical shift of the higher concentration region is seen with increasing chemical reaction parameter.

Non-Darcy nanoliquid buoyant flow and entropy generation analysis in an inclined porous annulus: Effect of source-sink arrangement

In several engineering and industrial applications, the primary challenge in the design of thermal devices is to minimize the entropy production with maximum energy dissipation so that the system efficiency could be enhanced. In particular, an inclined partially heated annulus is far challenging due to the presence of both tangential and normal gravity components, which produces major impacts on the fluid movement and associated transport rates as well as entropy production. From several important applications point of view, the current investigation is mainly focused to address the combined impacts of source-sink arrangements as well as the annular inclination on buoyant nanoliquid motion, thermal dissipation along with irreversibility distribution in a tilted porous annulus. The governing model equations are numerically solved by utilizing the finite difference based time-splitting and relaxation techniques. Using second-order finite difference approximation, the partial derivatives are approximated and the resulting system of algebraic equations are inverted using Thomas algorithm. The influence of various controlling parameters such as geometric inclination angle, nanoparticle shape, Darcy number, length and position of source-sink on fluid movement, thermal and entropy distribution are scrutinized in detail. Through a wide range of numerical experiments, it has been found that the annular inclination angle and location of source-sink arrangement plays a vital role on fluidity and flow patterns of nanoliquid in the porous saturated annulus. For smaller size of source-sink pair, higher thermal dissipation with minimal entropy production could be achieved, which is an important observation for the design of thermally efficient systems. We have also noticed that a better thermal transport rate with minimum irreversibility rate could be attained for a particular source-sink arrangement, in particular for γ=0∘, the staggered positioning of source-sink pair is beneficial rather than in-line arrangement. Among the various parameters arising in this analysis, the nanoparticle shape has been one of the key parameters to test the thermal efficiency. It has been predicted that the blade shaped nanoparticle induces better thermal efficiency, among the five different shapes of nanoparticle. The outcomes of this analysis could be utilized in applications, such as solar systems, electronic equipment cooling and heat exchangers.

Investigation of the free convection of nanofluid flow in a wavy porous enclosure subjected to a magnetic field using the Galerkin finite element method

This paper provides a numerical simulation to explore natural convection and fluid flow in a three-dimensional wavy cubical enclosure subjected to a magnetic field along the z-axis. The main emphasis of this research is to discuss entropy generation inside a wavy porous cavity. Brinkman's extended non-Darcy flow model is utilized for studying the free convection in the permeable medium. The equations of the mathematical model were solved using Galerkin finite element method. Isotherms, streamlines, velocity profile, and variation of average Nusselt number are calculated for various values of Ra (103–106), Ha (0–100), and φ (0–0.08). The results show that the streamlines and isotherms increase with the increase of Darcy and Rayleigh numbers. Besides, entropy generation is sensitive to the values of Darcy number, Rayleigh number, Hartmann number, and the number of undulations. Besides, the variation of the Nusselt number is affected by the Rayleigh number and Darcy number. The highest Nusselt number can be obtained for a maximum number of undulations only at the small values of the Rayleigh number. Otherwise, this pattern will be changed in the opposite trend at large values of the Rayleigh number. At the highest values of Ra, increasing Ha from 0 to 100 decreased Nuavg by 23%, while increasing Da from 10−5 to 10−2 enhanced it by 177%. Finally, it was found that the values of Beavg are only being enhanced by only one parameter, namely the Hartmann number. The obtained results from the current non-Darcy flow model agree well with the available literature results.

Analytical Model for the Pressure Performance Analysis of Multi-Fractured Horizontal Wells in Volcanic Reservoirs

Multi-fractured horizontal well (MFHW) technology is a key technology for developing unconventional reservoirs, which can generate a complex fracture network called a stimulated reservoir volume (SRV). Currently, there are many relative analytical models to describe the fluid seepage law, which are not suitable for volcanic reservoirs as of yet. The reasons are as follows: (1) due to the development of natural fractures, multi-scaled flow (matrix, natural fractures, SRV) should be considered to characterize MFHW flow in volcanic reservoirs; (2) non-Darcy flow and stress sensitivity should be considered simultaneously for seepage in volcanic reservoirs. Thus, this paper presents a novel MFHW analysis model of volcanic reservoirs that uses a multi-scale dual-porosity medium model and complex flow mechanisms. Laplace transformation, the Duhamel principle, the perturbation method and Stehfest numerical inversion are employed to solve the model to obtain dynamic pressure response curves. The results show that the pressure response curve can be divided into eight stages. Sensitivity analysis shows that the parameters of hydraulic fractures mainly affect the early flow stage. The parameters of the SRV region mainly affect the middle flow stage. The parameters of unreconstructed regions, non-Darcy flow and stress sensitivity mainly affect the late flow stage.

A NEW SEVEN-REGION FLOW MODEL FOR DELIVERABILITY EVALUATION OF MULTIPLY-FRACTURED HORIZONTAL WELLS IN TIGHT OIL FRACTAL RESERVOIR

Deliverability evaluation plays an important role in the reservoir exploitation. In this study, a new seven-region semi-analytical mathematical model considering the influences of fractal, imbibition and non-Darcy flow is proposed to evaluate the deliverability of multiply-fractured horizontal wells in tight oil reservoirs. The Laplace transformation, perturbation method and Stehfest numerical inversion are employed to solve the model. The reliability and accuracy of the analytical solution are verified by the field example. The sensitivity analysis of the major influencing factors on the deliverability is specifically analyzed. The numerical results indicate that the seven-region semi-analytical model can better explain the heterogeneity of fracture network, and its solution can provide an effective algorithm for the deliverability evaluation. It is found that the fractal plays a predominant influence on the productivity of tight oil reservoirs. The larger the fractal dimension and the smaller the fractal index, the higher the accumulative production rate. The imbibition also has an important effect on the deliverability of tight oil reservoirs. As the rising of wetting angle, both daily and accumulative production rates can obviously decrease. The imbibition has a positive impact on the production rate in the water-wet formations, while it has a negative impact on the production rate in the oil-wet formations. Compared with the fractal and imbibition, the threshold pressure gradient has less influence on the production of tight oil reservoirs. There exists a negative correlation between the threshold pressure gradient and the production performance. This work provides a new approach to understand the fractal tight oil reservoirs, which is of great significance for the deliverability evaluation.

Absolute permeability assessment of porous structures under different boundary conditions using lattice Boltzmann method

Digital rock physics is now recognized as a satisfactory solution for studying fluid dynamics phenomena within porous structures. On the other hand, the lattice Boltzmann method shown its capabilities in this field well. However, choosing the suitable boundary condition with the most negligible effect on flow behavior is a challenge. Therefore, this study aims to introduce the most common boundary conditions, such as periodic, constant pressure, and periodic-pressure, so it focuses on investigating their performance in predicting the permeability of synthetic and natural structures. Hence, three benchmark tests are initially considered to validate the model. First, the Poiseuille flow test has been addressed, so the observations reveal that the results are in close agreement with the analytical solution. In the following, synthetic models are introduced as square and hexagonal arrangements. The details have shown that using different boundary conditions in permeability prediction is associated with a good agreement between the results and the analytical data. Finally, the digital image obtained by x-ray tomography of a real porous sample has been studied. Also, the sensitivity analysis for permeability estimation has been done on pressure gradient and relaxation time. Detailed investigations show that the increase in pressure gradient under the pressure-periodic boundary condition is associated with an increase in permeability, but it has practically no effect on the result of the periodic boundary condition. In contrast, increasing this parameter under constant pressure did not affect the permeability curve, though it grows slightly with further increase, which is quite reliable considering the compressibility error and non-Darcy flow. On the other hand, increasing the relaxation time has resulted in increased permeability. However, it is recommended to use periodic and constant pressure boundary conditions because of the lower rate of change.

Finding preferential paths by numerical simulations of reactive non-darcy flow through porous media with the Lattice Boltzmann method

Finding preferential paths by numerical simulations of reactive non-darcy flow through porous media with the Lattice Boltzmann method

A new model simulating the development of gas condensate reservoirs

A new simulation model for the development of gas condensate reservoirs is introduced based on the influence that phase change, non-Darcy flow, and capillary pressure have on the production of gas condensates. The model predicts well performance, including bottom-hole pressure, oil/gas production rate, oil/gas recovery, gas–oil ratio, and the change in produced fluid composition. It also calculates dynamic characters, such as the change of pressure field and oil/gas saturation field during the development of gas condensate reservoirs. The model is applicable to different boundary conditions (both constant-pressure and sealed boundary) and different production modes (both constant-pressure and constant-volume production modes). Model validation attempted using numerical simulation results for sealed boundary conditions with constant-pressure production mode has shown a relatively good match, proving its validity. For constant-pressure boundary conditions with constant-volume production mode, four stages are defined according to the dynamic behavior of production performance in the development of gas condensate reservoirs.

Seepage-Fractal Characteristics of Fractured Media Rock Materials Due to High-Velocity Non-Darcy Flow

Under the influence of internal and external factors, a fracture network is easily generated in concrete and rock, which seriously endangers project safety. Fractal theory can be used to describe the formation and development of the fracture network and characterize its structure. Based on the flow balance in the node balance field, Forchheimer’s law is introduced to derive the control equation of high-velocity non-Darcy flow in the fracture network. The fracture network is established according to the geological parameters of Sellafield, Cumbria, England. A total of 120 internal fracture networks are intercepted according to 10 dimensions (1 m, 2 m, …, 10 m) and 12 directions (0°, 30°, …, 330°). The fractal dimension, equivalent hydraulic conductivity (K), and equivalent non-Darcy coefficient (β) of the fracture network are calculated, and the influence of the fractal dimension on K and β is studied. The results indicate that the fractal dimension of the fracture network has a size effect; with the increase in the size, the fractal dimension of the fracture network undergoes three stages: rapid increase, slow increase, and stabilization. In the rapid increase stage, K and β do not exist. In the slow increase stage, K exists and is stable, and β does not exist. In the stabilization stage, K and β both exist and are stable. The principal axes of the fitted seepage ellipses of K and β are orthogonal, and the main influencing factors are the direction and continuity of the fracture.

Sign-preserving second-order IMPEC time discretization and its application in compressible miscible displacement with Darcy-Forchheimer models

In this paper, we construct sign-preserving second-order IMplicit Pressure Explicit Concentration (IMPEC) time methods for generalized coupled non-Darcy flow and transport problems in petroleum engineering, extending the algorithm given in [10] which is only applicable to Darcy flows. We use interior penalty discontinuous Galerkin (IPDG) methods for spatial discretization, and develop bound-preserving technique to obtain physically relevant numerical approximations. The sign-preserving second-order IMPEC method is an important innovation. The method is based on the framework of the second-order strong-stability-preserving Runge-Kutta (SSP-RK2) method. The basic idea is to treat the pressure equation implicitly and the concentration equation explicitly so as to obtain a first-order time integration. Then we introduce a correction stage to compensate the accuracy, maintaining the physical bounds of the numerical cell averages. Unfortunately, the above algorithm is not applicable to non-Darcy problems. There are two main difficulties. Firstly, since the velocity equation is nonlinear and time-independent, all variables in the equation must be calculated at the same time level. However, the treatment in [10] will yield an extremely complicated algorithm and significantly large computational cost, and some iterations whose convergence may not be available if the solutions are not smooth. In our scheme, we linearize the velocity equation and the numerical solutions are not at the same time level, leading to first-order accurate solutions. Therefore, we adopt a completely different approach from [10] to derive the correction stage, keeping the physical bounds of the numerical solutions. Secondly, though with the new correction stage, it is still not easy to solve velocity equations in some stages. In this paper, we construct a direct solver for the velocity equation to save computational cost. Numerical experiments will be given to demonstrate that the improved sign-preserving second-order IMPEC scheme can reduce the computational cost significantly compared with explicit schemes if the diffusion coefficient D is small in the concentration equation. The proposed method also yields much larger CFL number compared with first-order IMPEC schemes. Moreover, the effectiveness of the bound-preserving technique will also be verified.

A Novel Equivalent Numerical Simulation Method for Non-Darcy Seepage Flow in Low-Permeability Reservoirs

The low permeability and submicron throats in most shale or tight sandstone reservoirs have a significant impact on microscale flow. The flow characteristics can be described with difficultly by the conventional Darcy flow in low-permeability reservoirs. In particular, the thickness of the boundary layer is an important factor affecting the formation permeability, and the relative permeability curve obtained under conventional conditions cannot accurately express the seepage characteristics of porous media. In this work, the apparent permeability and relative permeability were calculated by using non-Darcy-flow mathematical modeling. The results revealed that the newly calculated oil–water relative permeability was slightly higher than that calculated by the Darcy seepage model. The results of the non-Darcy flow based on the conceptual model showed that the area swept by water in non-Darcy was smaller than that in Darcy seepage. The fingering phenomenon and the high bottom hole pressure in the non-Darcy seepage model resulted from the larger amount of injected water. There was a large pressure difference between the injection and production wells where the permeability changed greatly. A small pressure difference between wells resulted in lower variation of permeability. Consequently, the non-Darcy simulation results were consistent with actual production data.

Analysis of steady flow in radial porous media

Abstract The outflow depth from the radial porous media (inflow to the well) is very useful as the downstream boundary condition and the starting point for water surface profile calculations. Based on the studies, unlike the Stephenson's hypothesis (the outflow depth is equal to the critical depth), the outflow depth from the rockfill media is a coefficient (Γ) of the critical depth. In the present study, using several (large scale and almost real) experimental data in the radial non-Darcy flow condition, dimensional analysis and the particle swarm optimization (PSO) algorithm, an equation was presented to calculate the mentioned coefficient based on upstream water depth (h) and distance between the well center and the upstream (R). Then, using the calculated outflow depth and the 1D flow analysis equations, the water surface profile in the radial non-Darcy condition was calculated for the first time. The results showed that considering an outflow depth equal to the critical depth and using the proposed solution in the present study, the mean relative error (MRE) values of 83.43% and 3.53% were obtained, respectively. In addition, using the proposed solution for different experimental conditions, an average MRE of 2.58% was calculated for the water surface profile.

Characterization of a Non-Darcy Flow and Development of New Correlation of NON-Darcy Coefficient

Darcy’s law has long been used to describe the flow in porous media. Despite the progress that took place in oil production industry research, it became clear that there is a loss of pressure, especially in the area near the wellbore region, where Darcy’s law is not applicable. For this reason, Forchheimer presented his equation in 1910, where he added a new term to Darcy’s law dealing with pressure loss due to inertial forces by introducing a new term, the β coefficient, into the equation. This paper presents a study of fluid flow through porous media, where water was used as a working fluid. Furthermore, the characteristics of the non-Darcy flow were analyzed by presenting the corresponding pressure and velocity gradient curves for each pressure. Extensive analysis indicates that many of the correlations available in the literature either have defective units or are the product of a small number of experiments. In this study, we benefit from relatively large samples, the radial flow, and the perforation in the middle of the samples. The properties of the samples were measured using mercury intrusion porosimetry. It was found that there is a direct relationship between the porosity and the grain’s size; the greater the size of the grains, the greater the porosity, and vice versa. The non-Darcy coefficient term, β, is found to be inversely proportional to the porosity and permeability. In a previous study, the β was investigated for compressible flow scenarios; however, this study calculated it for an incompressible flow. Finally, by analyzing the β values of both studies, we could deduce new novelty correlations for the β coefficient term, where the permeability, porosity, and tortuosity are included.

Non-Darcy flows in layered porous media (LPMs) with contrasting pore space structures

Compared to single layer porous media, fluid flow through layered porous media (LPMs) with contrasting pore space structures is more complex. This study constructed three-dimensional (3-D) pore-scale LPMs with different grain size ratios of 1.20, 1.47, and 1.76. The flow behavior in the constructed LPMs and single layer porous media was numerically investigated. A total of 178 numerical experimental data were collected in LPMs and single layer porous media. In all cases, two different flow regimes (i.e., Darcy and Non-Darcy) were observed. The influence of the interface of layers on Non-Darcy flow behavior in LPMs was analyzed based pore-scale flow data. It was found that the available correlations based on single layer porous media fail to predict the flow behavior in LPMs, especially for LPM with large grain size ratio. The effective permeability, which incorporated the influence of the interface is more accurate than the Kozeny-Carman equation for estimating the Darcy permeability of LPMs. The inertial pressure loss in LPMs, which determines the onset of the Non-Darcy flow, was underestimated when using a power law expression of mean grain size. The constant B, an empirical value in the classical Ergun equation, typically equals 1.75. The inertial pressure loss in LPMs can be significantly different from it in single lager porous media. For Non-Darcy flow in LPMs, it is necessary to consider a modified larger constant B to improve the accuracy of the Ergun empirical equation.

Numerical study of non-Darcy hybrid nanofluid flow with the effect of heat source and hall current over a slender extending sheet

The current evaluation described the flow features of Darcy Forchhemier hybrid nanoliquid across a slender permeable stretching surface. The consequences of magnetic fields, second order exothermic reaction, Hall current and heat absorption and generation are all accounted to the fluid flow. In the working fluid, silicon dioxide (SiO2) and titanium dioxide (TiO2) nano particulates are dispersed to prepare the hybrid nanoliquid. TiO2 and SiO2 NPs are used for around 100 years in a vast number of diverse products. The modeled has been designed as a nonlinear set of PDEs, Which are degraded to the dimensionless system of ODEs by using the similarity transformation. The reduced set of nonlinear ODEs has been numerically estimated through bvp4c package. The outcomes are tested for validity and consistency purpose with the published report and the ND solve technique. It has been noted that the energy curve lessens with the influence of thermodiffusion, Brownian motion and rising number of nanoparticles, while boosts with the result of magnetic field. Furthermore, the concentration outline of hybrid nanoliquid improves with the upshot of chemical reaction.

Forced convection of non-darcy flow of ethylene glycol conveying copper(II) oxide and titanium dioxide nanoparticles subject to lorentz force on wedges: Non-newtonian casson model.

The topic of two-dimensional steady laminar MHD boundary layer flow across a wedge with non-Newtonian hybrid nanoliquid (CuO-TiO2/C2H6O2) with viscous dissipation and radiation is taken into consideration. The controlling partial differential equations have been converted to non-linear higher-order ordinary differential equations using the appropriate similarity transformations. It is demonstrated that a number of thermo-physical characteristics govern the transmuted model. The issue is then mathematically resolved. When the method’s accuracy is compared to results that have already been published, an excellent agreement is found. While the thermal distribution increases with an increase in Eckert number, radiation and porosity parameters, the velocity distribution decreases as porosity increases.

Repeated Uniaxial Compression Test of Permeable Friction Courses Mixtures under Seepage Field and Stress Field Coupling

In order to study the variations of the non-Darcy flow permeability coefficient and the porosity of permeable friction courses (PFCs), as well as the effects of the coupled seepage and stress fields on them, repeated uniaxial compression tests were carried out under the coupled action of water and a load. A set of water and load coupling tests were conducted, and a non-Darcy flow permeability coefficient tester was also made. After the PFC-13 specimen was carried out by the repeated uniaxial compression test under the water and load coupling, the total air void ratio and effective air void ratio were measured by the vacuum sealing method, and the non-Darcy flow permeability coefficients were obtained by a non-Darcy flow permeability coefficient tester. It was found that the coupled action of water and a load caused the total air void ratio, effective air void ratio, and permeability coefficient to sharply increase and reduced the number of repeated uniaxial compression cycles. These results are helpful for the design, construction, and maintenance of PFC mixtures.

The inflow performance relationship in fractured, vertical, and dewatered coalbed-methane wells

Although not greatly studied, the inflow performance relationship (IPR) in dewatered and vertical coalbed methane (CBM) wells is essential in the development of a CBM reservoir. The dynamics of the stress sensitivity effect (SSE) and the matrix shrinkage effect (MSE), as well as the hydraulic fracture propagation, have all been neglected, especially for the exterior region, which is larger than the drainage radius. A novel IPR model has now been built that integrates dynamic SSE–MSE and hydraulic fracture propagation into the skin factor, and is validated with real production data from the Qinshui Basin, China. The absolute open flow rate given the SSE–MSE is 60.5% larger than without SSE–MSE. If the positive effect of the SSE–MSE on the permeability in the exterior region is neglected, the absolute open flow rate is reduced by 21%. The MSE has a greater effect on the fracturing skin factor than the SSE and tends to lower the fracturing skin factor, benefitting the production of CBM. Moreover, in light of the SSE or MSE, the non-Darcy flow effect is weaker due to restraints on the velocity of flow or the permeability. Useful and essential theoretical guidance for real CBM production can thus be gleaned from this novel IPR model.

Development of production decline curves for non-Darcy oil flow in low-permeability reservoirs

The purpose of work is creating a new technique for result interpretation of long-term well monitoring by production decline curve analysis in low-permeability reservoirs. For these reservoirs oil flow does not obey Darcy law, in some cases threshold pressure gradient exists. Standard techniques for well test results interpretation do not allow to determine reservoir parameters reliably. Currently, most of works devoted to the interpretation of well test results for nonlinear flow use the approach of numerical modeling and the method of type-curve matching to determine reservoir parameters. Of course, these methods are powerful tool, but analytical solutions of the considered problems are also of practical interest. Obtained solutions make it possible to determine not only reservoir parameters but also flow law parameters and to diagnose characteristic regimes of non-Darcy flow. That is why, it becomes necessary to develop new approaches for reservoir parameters and flow law determination, which was implemented in the proposed paper. A one-dimensional isothermal problem of single-phase non-Darcy flow is considered in two geometries: plane parallel (flow from fracture) and plane radial (flow from single well). Well operates with constant bottomhole pressure. Flow equation is characterized by two parameters: threshold pressure gradient and coefficient describing the change in flow rate near the threshold gradient value. To obtain analytical solutions, integral equation method was applied. The behavior of flow rate at different times was investigated, the results showed good agreement with numerical simulation. Decline flow rate curves were analyzed when changing parameters of flow equation. Also, the possibility of reservoir parameters determination was investigated.