Low-velocity non-Darcy Flow Research Articles

Rate Transient Analysis for Multi-Fractured Wells in Tight Gas Reservoirs Considering Multiple Nonlinear Flow Mechanisms

Making rate transient analysis (RTA) and formation evaluation for multi-fractured tight gas wells has always been a difficult problem. This is because the fluid flow in the formation has multiple nonlinear flow mechanisms, including gas-water two-phase flow, gas slippage, low-velocity non-Darcy flow, and stress-dependent permeability. In this paper, a novel RTA method is proposed for multi-fractured wells in tight gas reservoirs incorporating nonlinear flow mechanisms. The RTA method is based on an analytical model, which is modified from the classical trilinear flow model by considering all the nonlinear flow mechanisms. The concept of material balance time and normalized rate is used to process the production data for both water and gas phases. The techniques of approximate solutions in linear/bilinear flow regimes and type curve fitting are combined in the proposed RTA method. After that, the rate transient behaviors and influencing factors of multi-fractured tight gas wells are analyzed. A field case from Northwestern China is used to test the efficiency and practicability of the proposed RTA method. The results show that six flow regimes for both gas and water production performances are exhibited on the log-log plots of normalized production rate against material balance time. The rate transient responses are sensitive to the nonlinear flow mechanisms, and formation and fracture properties. The medium flow regimes are significantly affected by fracture number, fracture conductivity, fracture half-length, stress-dependent permeability, gas-water two-phase flow, and formation permeability, which should be considered in making RTA of fractured tight gas wells. The field case shows that both gas and water production performances can be well-fitted using the proposed RTA method. The major innovation of this paper is that a novel RTA method is proposed for fractured tight gas wells considering multiple nonlinear flow mechanisms, and it can be used to make reasonable formation and fracturing evaluations in the field.

A coupled matrix-fracture productivity calculation model considering low-velocity non-Darcy flow in shale reservoirs

The flow behavior of shale reservoirs is extremely complex. The mainly used development method of large-scale volume fracturing of horizontal wells is associated with a series of problems such as low productivity in general, fast decrease in productivity, and difficulty to increase productivity. Therefore, it is necessary to evaluate the productivity of shale oil wells and determine the key control factors. Considering low-velocity non-Darcy flow, this paper establishes an analytical solution model for productivity calculation of multi-fracture shale oil horizontal wells with coupled matrix-fracture flow. Calculating the oil production rate by considering low-velocity non-Darcy flow in shale oil reservoirs obtains values closer to the actual oil production rate on site. The average relative error of the oil production rate of the H8 shale oil well is 4.27%. The average relative error of the oil production rate of the TMS shale oil wells is 34.97%, which exceeds the rate of the semi-analytical model by 47.84%. Sensitivity analysis shows that the oil production rate of multi-fracture shale oil horizontal wells is most sensitive to fracture cluster spacing and flowing bottom hole pressure. With decreasing fracture cluster spacing and flowing bottom hole pressure, both the matrix pressure gradient and the apparent permeability of the matrix increase. These increases not only increase the oil production rate but also reduce the influence of the non-Darcy effect on the oil production rate caused by boundary effect. In addition, the limits of the impact of non-Darcy effects caused by boundary effect on oil production rate have been clarified. The results enable not only the prediction of the oil production rate, cumulative oil production, and recovery factor of multi-fracture horizontal shale oil wells, but also the finding of the shale oil sweet spot according to the influence of matrix permeability and crude oil viscosity on the oil production rate.

Investigation of low water recovery based on gas-water two-phase low-velocity Non-Darcy flow model for hydraulically fractured horizontal wells in shale

Various mechanisms are employed to interpret the low water recovery during the shale-gas production period, such as extra-trapped water in the fracture network, water imbibition due to osmotic pressure and capillary pressure. These lead to the difficulty of water flow, which could be described by low-velocity non-Darcy's law known as threshold pressure gradient (TPG). In this paper we firstly employ the low-velocity non-Darcy's law to describe the water flow and use Darcy flow accounting for slip flow and free molecular flow mechanisms to model gas flow in the shale formation. The sensitive study using numerical simulation shows that the proposed flow model could model the low fracturing liquid recovery and that large pseudo TPG leads to lower fracturing liquid recovery. Thus, the proposed model would give new insight to model the low water recovery in shale formations.

Viscoplastic effects of Newtonian fluids in nanopores: a molecular dynamics study

Understanding the influence of fluid–solid interactions on low-velocity non-Darcy flow in ultra-low permeability reservoirs is essential for enhanced oil recovery. The Poiseuille flows of C12H26 and argon under different pressure gradients in a nanopore are simulated using the molecular dynamics method. The results demonstrate that the confined fluids behave as elastic solids under lower pressure gradient but as viscous fluids under higher pressure gradient in the simulation. With increase of the external pressure gradient, the generated shear stress exceeds the yield stress related to the fluid-solid interaction and leads to a transition. This phenomenon indicates that the confined fluids in the nanopore exhibit apparent viscoplasticity. According to the viscoplastic hypothesis, the confined fluid can be divided into elastic and viscous layers, depending on the competition between the shear and yield stresses. The simulation results also indicate the yield stress increases near the wall. Assuming that the yield stress is inversely proportional to the distance from the wall, a non-Darcy flow model is proposed to calculate the flow rate by pressure gradient. The predictions of the model are consistent with the simulations for both C12H26 and argon, indicating the similarity of viscoplastic effects on the seepage of both fluid-nanopore systems.

Integration of Multi-Region Material Balance Equation with Binomial Productivity Equation for Performance Prediction

In order to overcome the limitations of conventional material balance methods, a new multi-region material balance method, considering the strong heterogeneity and low-velocity non-Darcy flow characteristics of bio-reef gas reservoir, is proposed to estimate the development performance of reef reservoirs. Firstly, the reef reservoir is divided into two regions, the reef cover high-permeability development area and the reef edge low-permeability supplying area. Then, based on one-dimensional non-linear seepage mechanism and mass conservation principle, the multi-region supplying material balance equation is established. Finally, combined with the binomial productivity equation, the development performance can be easily estimated at differentP/Zcurves. The application results indicate that the low-permeability replenishment area plays an important role, especially after entering the decline stage, on the development performance of high-permeability region. Compared with the traditional material balance method, the new method can help for better understanding of unbalanced development in different regions and the replenishment effect of low-permeability region.

A FRACTAL MODEL OF LOW-VELOCITY NON-DARCY FLOW CONSIDERING VISCOSITY DISTRIBUTION AND BOUNDARY LAYER EFFECT

Seepage flow in low permeability reservoirs is strongly influenced by the boundary layer, which is characterized by a low-velocity non-Darcy flow phenomenon, and a nonlinear relationship between velocity and pressure gradient. The viscosity is not uniformly distributed in the pore throat and has a great effect on the nonlinear flow behavior. In this paper, a low-velocity non-Darcy flow model is proposed. By quantitatively characterizing the boundary layer thickness and considering the non-uniform distribution of viscosity for the first time, a modified Hagen–Poiseuille equation combining the boundary layer effect and viscosity distribution in a single capillary is derived. Then, the analytical expression of the nonlinear relationship between the flow rate and pressure gradient in the fractal porous media is established, assuming that the pores in the tight formation are fractal distribution. Finally, the validity of the model is verified by experimental data. The relative error between the model and experimental data, with a permeability smaller than 3mD, is 3.27%. And the parameters affecting low-velocity non-Darcy flow are also analyzed. The results showed that the influence of boundary layer is one of the causes of low-velocity non-Darcy flow in tight reservoir. The study has also revealed that the non-uniform distribution of viscosity in the pore throat has a great influence on the nonlinear flow, which is often neglected in previous works.

ANALYTICAL STUDY ON A ONE-DIMENSIONAL MODEL COUPLING BOTH DARCY FLOW AND LOW-VELOCITY NON-DARCY FLOW WITH THRESHOLD PRESSURE GRADIENT IN HETEROGENEOUS COMPOSITE RESERVOIRS

ANALYTICAL STUDY ON A ONE-DIMENSIONAL MODEL COUPLING BOTH DARCY FLOW AND LOW-VELOCITY NON-DARCY FLOW WITH THRESHOLD PRESSURE GRADIENT IN HETEROGENEOUS COMPOSITE RESERVOIRS

Embedded discrete fracture modeling: Flow diagnostics, non-Darcy flow, and well placement optimization

Low-velocity non-Darcy flow often occurs in low permeability reservoirs. In this paper, we discuss how to incorporate a non-Darcy flow correction into embedded discrete fracture modeling (EDFM). The resulting method is validated by reproducing type curves containing the characteristic no-flow segment, followed by a nonlinear and a linear segment. Simulation results are also validated against a direct simulation method that uses local grid refinement to (almost) resolve hydraulic fractures. Second, we discuss how grid-based flow diagnostics can be formulated on top of the new EDFM method and used to provide computationally inexpensive predictions of flow patterns, interwell connections, and measures of dynamic heterogeneity. We then demonstrate how such flow diagnostics can be used to optimize well placement and improve oil recovery.

Temporal Scale Analysis of Gas Flow in Tight Gas Reservoirs considering the Nonequilibrium Effect

The fractal geometry, anisotropy, discontinuity, and non-Darcy flow of tight reservoirs exert a significant effect on well production performance. In this study, the reservoir fractal geometry is represented by exponential functions on the basis of microseismic data, while the discontinuity of the fractures is presented as a nonequilibrium effect. The impact of the nonequilibrium effect and the low velocity non-Darcy flow on the temporal scale of the wellbore pressure is predicted herein. Results showed that the time scale analysis accurately simulates gas flow in a tight reservoir. The wellbore pressure gradually increases, whereas the pressure in the matrix lags when the nonequilibrium effect is considered. The wellbore pressure is affected in the early period by the nonequilibrium effect. However, at the later stage, the pressure in the matrix is mainly affected by the non-Darcy flow. When the non-Darcy flow is dominant, the pores without gas flowing through are better presented.

Fluid flow through anisotropic and deformable double porosity media with ultra-low matrix permeability: A continuum framework

Fractured porous media or double porosity media are common in nature. At the same time, accurate modeling remains a significant challenge due to bi-modal pore size distribution, anisotropy, multi-field coupling, and various flow patterns. This study aims to formulate a comprehensive coupled continuum framework that could adequately consider these critical characteristics. In our framework, fluid flow in the micro-fracture network is modeled with the generalized Darcy's law, in which the equivalent fracture permeability is upscaled from the detailed geological characterizations. The liquid in the much less permeable matrix follows a low-velocity non-Darcy flow characterized by threshold values and non-linearity. The fluid mass transfer is assumed to be a function of the shape factor, pressure difference, and (variable) interface permeability. The solid deformation relies on a thermodynamically consistent effective stress derived from the energy balance equation, and it is modeled following anisotropic poroelastic theory. The discussion revolves around generic double porosity media. Model applications reveal the capability of our framework to capture the crucial roles of coupling, poroelastic coefficients, anisotropy, and ultra-low matrix permeability in dictating the pressure and displacement fields.

Flow Model Considering the Influence of Low Velocity Non-Darcy Flow and Reservoir Connectivity and its Application

During the development of low and ultra-low permeability sandstone reservoir, due to the starting pressure gradient of poor reservoir physical properties, it is difficult to establish effective displacement between oil and water wells, and high voidage replacement ratio and low formation pressure are common phenomena. So it is necessary to analyze the influential factor of high voidage replacement ratio of low and ultra-low permeability reservoir. A one source and one sink flow model considering low velocity non-Darcy flow and unconnected reservoirs was established by using the flow theory of unidirectional flow and radial flow, and the model was solved by numerical method. The results show that the unconnected layers of water well will lead to the increase of invalid water injection; the larger the starting pressure gradient is and the larger the volume proportion of unconnected layer is, the smaller the maximum annual liquid production rate is and the higher the cumulative voidage replacement ratio corresponding to maintaining the maximum liquid production rate is. It is theoretically proved that the existence of unconnected layers and starting pressure gradient are the factors leading to the existence of high voidage replacement ratio in low and ultra-low permeability reservoirs. The study provides a theoretical basis for the adjustment of high-efficiency water injection in reservoirs.

The electroviscous flow of non-Newtonian fluids in microtubes and implications for nonlinear flow in porous media

This paper aims to interpret the low-velocity nonlinear flow occurring in low-permeability reservoirs based on the theories of electrokinetic transport and non-Newtonian rheology of fluids. To achieve this end, we simulate the steady-state electroviscous flow of Bingham-Papanastasiou (BP) fluids in circular microtubes by simultaneously solving the Poisson-Boltzmann and the modified Navier-Stokes equations. The induced electrical field strength Es, velocity profiles, and the transport capacity of the non-Newtonian fluid under the effects of various factors (such as capillary radius R, zeta potential ζ, yield stress τ0, and stress growth index m) were examined. The results show that the generated Es of the BP fluid is highly affected by the fluid rheology, which is quite different from that of the Newtonian liquid. The velocity profiles become lower and flatter as m or τ0 increases, and this is more remarkable in smaller microtubes. The apparent viscosity of non-Newtonian fluid declines monotonically with increasing c∞, yet non-monotonically with R, m, τ0, and ζ. In addition, the low-velocity nonlinear flow in microtubes can be successfully captured when considering the electrokinetic flow of the non-Newtonian fluid rheology. While for the Newtonian fluid, only involving the electroviscous effect fails to generate the nonlinear flow behavior. The contributions of electrokinetic parameters versus rheological properties to the degree of flow nonlinearity are also discussed. The impact of electrokinetic parameters (ζ, c∞) on the flow characteristics is significant at high-pressure gradients and becomes trivial when the pressure gradient is relatively low. In contrast, the fluid rheological parameters (m, τ0) greatly determine the magnitude of the flow nonlinearity occurring at the low-pressure gradients. In sum, the electroviscous flow of BP fluids in microchannels provides a possible explanation of the low-velocity non-Darcy flow in porous media.

Pressure Analysis for Volume Fracturing Vertical Well considering Low-Velocity Non-Darcy Flow and Stress Sensitivity

In general, there is stress sensitivity damage in tight reservoirs and fractures. Furthermore, the flow in tight reservoirs is the low-velocity non-Darcy flow. Currently, few researches of pressure analysis for volume fracturing vertical well are conducted simultaneously considering the low-velocity non-Darcy flow and stress sensitivity. In the paper, a novel flow model of a volume fractured vertical well is proposed and solved numerically. Firstly, the threshold pressure gradient, permeability modulus, and experimental data are, respectively, utilized to characterize the low-velocity non-Darcy flow, matrix stress sensitivity, and fracture stress sensitivity. Then, a two-region composite reservoir is established to simulate the vertical well with volume fracturing. After that, the logarithm meshing method is used to discrete the composite reservoir, and the flow model is solved by the method of finite difference and IMPES. Finally, the model verification is conducted, and the effects of the low-velocity non-Darcy flow and stress sensitivity on the pressure and pressure derivative are analyzed. The six flow regimes are identified by the dimensionless pressure and pressure derivative curve. They are, respectively, the fracture linear flow regime, early transition flow regime, radial flow regime, crossflow regime, advanced transition flow regime, and boundary controlling flow regime. The stress sensitivity and threshold pressure gradient have a great effect on the dimensionless pressure and pressure derivative. With the increase of reservoir stress sensitivity, the pressure and pressure derivative are upward at the advanced transition flow and boundary controlling regimes. However, the pressure and pressure derivative are downward at the advanced transition flow and boundary controlling regimes when the fracture sensitivity increases. An increase in the threshold pressure gradient results in a high dimensionless pressure and pressure derivative. This work reveals the effects of low-velocity non-Darcy flow and stress sensitivity on pressure and provides a more accurate reference for reservoir engineers in pressure analysis when developing a tight reservoir by using the volume fracturing vertical well.

Representation of a new physics-based non-Darcy equation for low-velocity flow in tight reservoirs

There has been a growing interest in the study of low-velocity non-Darcy flow in tight reservoirs. Most existing non-Darcy equations, however, are flawed in model assumption, equation derivation or field practicability. For another, current non-Darcy simulations are based on the traditional methods, which are incapable of describing the fracture-rich feature of tight reservoir geology. In light of this, this paper introduces a new physics-based non-Darcy equation for low-velocity flow from four aspects: equation derivation, experimental validation, flow simulation and flow analysis. The rigorous derivations of the new equation are presented with a modified Buckingham-Reiner equation in which the boundary layer effect and the Bingham fluid behavior are described. The non-Darcy coefficient controlled by the boundary layer parameter and threshold pressure gradient (TPG) is incorporated into Darcy's equation. The equation validations are performed by liquid column experiment and core flow experiment, which yields good matching relationship between modeled values and observed values. The non-Darcy flow simulation is based on the triangle-gridded discrete fracture-matrix model (DFM) which can model arbitrary fractures. The upstream weighting scheme for computing the non-Darcy transmissibility is proposed to improve numerical robustness. Then the scheme is implemented by constructing non-Darcy operators, consisting of half-face upstream weighting operator and gradient operator, and further setting up the Automatic Differentiation (AD)-style discrete equations. The studies on non-Darcy flow in matrix coupled with Darcy flow in fracture are performed by modeling two types of fractured reservoirs. Detailed analyses are illustrated with pressure transient, long-term well bottom pressure and pressure map.

Analytical Solutions for Non-Darcy Transient Flow with the Threshold Pressure Gradient in Multiple-Porosity Media

Low-velocity non-Darcy flow can be described by using the threshold pressure gradient (TPG) in low-permeability porous media. The existence of the TPG yields a moving boundary so that fluid starts to flow inside this boundary when the pressure gradient overcomes the viscous forces, and beyond this boundary, there will be no flow. A mathematical model of considering the TPG is developed to describe the flow mechanism in multiple-porosity media. By defining new dimensionless variables, the nonlinear mathematical model can be solved analytically. This new approach has been validated with several approximate formulas and numerical tools. The diffusion of the moving boundary varying with time is analyzed in detail in multiple-porosity media, and then the effect of the moving boundary on pressure transient response is investigated and compared with that of the traditional three boundary types (closed boundary, infinite-pressure boundary, and constant-pressure boundary). Sensitivity analysis is conducted to study the effect of the TPG on pressure and pressure derivative curves and rate decline curves for single-porosity media, dual-porosity media, and triple-porosity media, respectively. The results show that the moving boundary exerts a significant influence on reservoir performance at a relatively early time, unlike the other three boundary types, and only a boundary-dominated effect at the late time. The larger the threshold pressure gradient, the smaller the diffusion distance of the moving boundary and the rate of this well at a given dimensionless time. At the same time, the pressure transient response exhibits a higher upward trend because of a larger TPG. All behavior response might be explained by more pressure drop consumed in low-permeability reservoirs. The finding is helpful to understand the performance of low-permeability multiple-porosity media and guide the reasonable development of low-permeability reservoirs.

Enrichment factors and sweep spot evaluation of Jurassic tight oil in central Sichuan Basin, SW China

The secondary migration mechanism, enrichment factors of Jurassic tight oil in central Sichuan Basin were well investigated through physical simulation experiment of reservoir formation, casting and fluorescent thin sections, field emission scanning electron microscope (FESEM) and environment scanning electron microscope (ESEM). The results show that migration of Jurassic tight oil in central Sichuan Basin is a low-velocity non-Darcy flow through low-efficient migration path under the huge migration driving force, and has three migration and seepage stages, i.e. viscous flow stage, nonlinear seepage stage, and quasi-liner seepage stage. Microscopically, the migration pathway of tight oil is the pore-fracture composite conduction; macroscopically, the migration mode of tight oil is the large-scale short–distance migration. Distribution of favorable zones of tight oil is controlled by distribution of high-quality source rocks. The hydrocarbon-generation strength of 0.4 × 106 t/km2 can be as a threshold to determine favorable zone of tight oil in the study area. The reservoirs with high permeability and high porosity can form tight oil sweet spots, and the development degree of fractures is closely related to well with high yield of tight oil well.

Effect of dynamic pseudo threshold pressure gradient on well production performance in low-permeability and tight oil reservoirs

Pseudo threshold pressure gradient (PTPG) is a significant parameter that characterizes fluid flow in low permeability and tight reservoirs, which would illustrate pressure derivative curves anomaly caused by low-velocity non-Darcy flow. During the development of low-permeability and tight reservoirs, PTPG is a dynamic parameter influenced by effective stress. A series of core flooding experiments have been conducted in this paper to conclude the relationship between effective stress and PTPG with the unique experimental method. In the experiment, the precision of pressure measurement is greatly improved by a novel device and the influence of the core-end effect and the defects of the core holder on the experimental results is eliminated. The results show that PTPG increase along with the increase of effective stress. Based on the conclusion, a dynamic PTPG model is established, and the corresponding parameters are derived by fitting the experimental data. The well performance of a vertical well estimated by this model shows the production rate and recovery factor of non-Darcy flow considering dynamic PTPG are smaller than that of Darcy flow but larger than that of conventional non-Darcy flow. The recovery factor of non-Darcy flow considering dynamic PTPG is 21.2% higher than that of conventional non-Darcy flow when the production time is 800 days. The production rate and recovery factor curves of different overburden pressure show that the higher the overburden pressure is, the worse the productivity of the well is. The recovery factor at an overburden pressure of 50 MPa is 11.1% lower than that at an overburden pressure of 30 MPa.

A FRACTAL MODEL FOR LOW-VELOCITY NON-DARCY FLOW IN TIGHT OIL RESERVOIRS CONSIDERING BOUNDARY-LAYER EFFECT

Flow in nanoscale pore-throats of tight oil reservoirs is strongly affected by boundary-layers, and exhibits low-velocity non-Darcy flow phenomena. The relationship between flow velocity and pressure gradient is highly nonlinear and difficult to be modeled mathematically. This paper proposed a low-velocity non-Darcy flow model which can account for boundary-layer effect in tight oil reservoirs. First, a modified Hagen–Poiseuille equation coupled with boundary-layer effect in a single capillary tube was derived. Then, assuming pores in tight formations following fractal distribution, an analytical expression of nonlinear correlation between flow velocity and pressure gradient in fractal porous media was developed. Finally, the proposed model was validated with experiment data, and parameters influencing low-velocity non-Darcy flow were quantitatively evaluated. The research results show that the decreasing boundary-layer thickness with the increase pressure gradient is the main reason of low-velocity non-Darcy flow in tight oil reservoirs. Our model can effectively describe the nonlinear relationship between flow velocity and pressure gradient. The relationship between threshold pressure gradient (TPG) and pseudo threshold pressure gradient (PTPG) can also be predicted using our model. Fluid viscosity has great impact on nonlinear flow behavior, and with fluid viscosity increasing TPG and PTPG increase significantly. TPG is the function of fluid type, fluid viscosity and maximum pore diameter, and decreases exponentially with the increasing maximum pore size.

A multi-linear transient pressure model for multistage fractured horizontal well in tight oil reservoirs with considering threshold pressure gradient and stress sensitivity

In general, tight oil reservoirs has a numerous tiny and complicated throats, and ultra-low permeability, which leads to a low-velocity non-Darcy flow when the fluid flows in tight reservoirs. The permeability of tight oil reservoirs is no longer a constant, but a variable, which is dependent on reservoir pressure. In other word, the tight oil reservoir is stress sensitive media. The low-velocity non-Darcy flow and stress sensitivity will greatly affect the bottom hole pressure response. So, the low-velocity non-Darcy flow and stress sensitivity should be taken into consideration when researching the transient pressure behaviors of multistage fractured horizontal wells. Currently, there are many researches on transient pressure behaviors of the multistage fractured horizontal well, but none of these researches has taken the low-velocity non-Darcy flow and stress sensitivity into consideration simultaneously. In this paper, threshold pressure gradient and permeability modulus are used to characterize the low-velocity non-Darcy flow and stress sensitivity, respectively. And, a multi-linear transient pressure model of multistage fractured horizontal well with considering low-velocity non-Darcy flow and stress sensitivity is established. The Pedrosa's transformation, perturbation theory, Laplace transformation and stehfest numerical inverse are used to solve this transient pressure model. The novel model is verified by comparing the result of proposed model with that of Ekaterina Stalagorova's model. According to the dimensionless pressure derivative curve, the flow process can be identified as six regimes: ①the main fracture linear flow stage, ②the first transition flow stage, ③dual-linear flow stage, ④the second transition flow stage, ⑤the formation linear flow stage, ⑥the boundary flow stage. Finally, the effects of threshold pressure gradient, permeability modulus and fracture density of stimulated reservoir on dimensionless pressure derivative are analyzed. This research provides theoretical support for correctly understanding the flow law of multistage fractured horizontal well.

Pore network modeling of the Non-Darcy flows in shale and tight formations

Pore network modeling is a powerful tool to simulate multiphase flow in porous media. Quasi-static model is used in this work and the drainage displacement process is simulated. In shale and tight formations, there are proposed non-Darcy flow mechanisms: gas non-Darcy flow and liquid low velocity non-Darcy flow. The gas flow in shale and tight formations is generally classified in slip flow and transitional flow regimes according to the Knudsen number values. The BK model (Beskok and Karniadakis, 1999) is used for gas non-Darcy flow in this study and the liquid low velocity non-Darcy flow is mainly based on our previous work (Wang and Sheng, 2017a, b). Both effects are incorporated into our pore network model separately. For gas-water flow, gas is the non-wetting phase with gas non-Darcy flow and water is the wetting phase with Darcy flow. For oil-water flow, the low velocity non-Darcy flow is considered for both water and oil phases. Then our model is applied in 3 cases. Case 1 is the Berea sandstone (Valvatne, 2004), which is the benchmark for conventional pore network modeling. In this case, no non-Darcy flows is considered in our pore network model and it is totally Darcy flow. The absolute permeability and relative permeability are both matched with the experimental data. Case 2 is the Bossier tight gas sandstone (Rushing et al., 2003). Gas apparent permeability vs. pressure was measured at different water saturations in their experiments. In this way, the gas non-Darcy flow in two-phase conditions are verified. Case 3 is the Barnett shale (Moghaddam and Jamiolahmady, 2016), which is our major focus. The two types of non-Darcy flows are studied and further discussed in this case. Specifically, the effect of gas non-Darcy flow enhances the gas permeability 2.66 times of the Darcy permeability when pressure is 10 MPa, while the effect of liquid low velocity non-Darcy flow decreases the liquid permeability to 40% of the Darcy permeability when pressure gradient is 0.1 MPa/m. Two types of relative permeability (kr) definition are presented in this study: Darcy permeability based kr and normalizedkr where non-Darcy permeability is used as the base permeability. In the results, the normalized relative permeability doesn't change much for both non-Darcy flows, which implies that we can probably assume the relative permeability unchanged and just consider the effect on absolute permeability, when we deal with these non-Darcy flow mechanisms.