non-Darcy Flow Behavior Research Articles

Comprehensive analysis of gas slippage and Non-Darcy flow in carbonate rock samples under stress conditions

This study examines the influence of stress on gas slippage and non-Darcy flow behavior in carbonate porous media, using core samples from two large carbonate reservoirs in the Middle East. Permeability measurements under varying stress conditions revealed significant increases in both the gas slippage factor and the non-Darcy coefficient with rising stress, indicating a related decrease in permeability. A new mathematical model, incorporating effective tortuosity and slippage radius into the Kozeny-Carman equation, was developed to accurately describe the stress-dependent nature of gas flow. Analysis of key pore attributes—such as porosity, pore size, shape, and tortuosity—underscored the critical role of pore structure, especially pore size and connectivity, in governing fluid flow under stress. These findings highlight the need to account for stress effects in reservoir engineering for improved modeling and management of carbonate reservoirs.

A nonlinear mathematical model for fluid flow in low-permeability reservoirs and its effect on well production performance

Fluid flow in low-permeability reservoirs exhibits nonlinear behavior owing to the presence of nano-sized pores and strong fluid-solid interactions. The percolation velocity significantly deviates from that estimated by Darcy's law, particularly in regions with low-pressure gradients. Several modified Darcy's equations have been proposed to describe these nonlinear phenomena. However, existing flow models often suffer from practical limitations, such as complex formulations or challenging determination of parameters. To address this, non-Darcy flow behavior was investigated and a novel low-velocity-nonlinear flow equation that accounts for the boundary layer effect is presented. The curves with different coefficients are plotted and the sensitivity of the corresponding parameters are discussed. Moreover, considering that core flooding experiments fail to accurately reflect actual flow behavior in reservoirs, an inversion method based on genetic algorithm (GA) that utilizes bottomhole pressure (BHP) testing data to estimate the pending coefficients is introduced. Finally, the proposed nonlinear flow equation is applied, using the finite difference method (FDM), to simulate the production performance, including pressure distribution, production rate, and recovery factor of vertical and multi-fractured horizontal wells in low-permeability reservoirs. The results demonstrated significant deviations in reservoir pressure distribution from those estimated using Darcy's law, and the production rate and recovery factor of nonlinear flow were notably lower than those predicted by Darcy's flow law, albeit to varying degrees.

Influence of 3D Fracture Geometry on Water Flow and Solute Transport in Dual-Conduit Fracture

The geometry of the fracture exerts an important impact on the flow of the fractures and the transport of the solutes. Herein, Forchheimer’s law and the weighted-sum ADE (WSADE) model were alternatively employed, and the obtained pressure gradient versus discharge curves for the fitting reveal that Forchheimer’s law adequately described the non-Darcy flow behavior and the robust capability of WSADE in capturing the non-Fickian transport in dual-conduit fractures (DCFs). Different boundary layer effects brought about obvious differences in water flow and solute transport trends between 2D and 3D fractures. Moreover, with the change in the distance between the main conduit and the diversion conduit, the hydraulic parameters were correlated with the fitting parameters in Forchheimer’s law and WSADE. The solute mixing process is dramatically altered by the results, which directly demonstrate major flow patterns at the intersection. The prediction of solute transport in naturally fractured rocks depends primarily on the depicted flow and its effects on mixing. The findings help to increase the understanding of transport processes in such systems, especially for characterizing the dual-peaked BTCs obtained in aquifers.

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.

Statistical analysis of experimental studies of non-Darcy flow in proppant packs

The objective of this study is to determine the most significant variables associated with non-Darcy flow behavior. Statistical analysis of non-Darcy flow reveals that the most significant predictors of mass flow rate are the particular material used, fluid velocity, fluid viscosity, and apparent permeability. Data were collected from various experimental studies conducted at high-flow rates through different proppant packs and using different experimental procedures. The study was conducted in order to validate the Barree and Conway (2004), which yields accurate flow predictions across all flow regimes in porous media. This study examines output data such as pressures, temperature, and mass flow rate given a proppant pack design and specific flowing fluid. Historically, Darcy’s equation (Darcy, 1856) has been used to predict hydrocarbon production; however, it is inaccurate in most reservoirs. Using results based on non-Darcy flow experimentation provides a different perspective to the hydraulic fracturing process.The results of the statistical analysis were compared to previous work done in unconventional reservoir studies. These results show that statistical analysis can recommend the variables to prioritize during the hydraulic fracturing design process. Although this work was conducted under experimentally controlled conditions, it is hypothesized that the results will hold for field conditions.

Flow Characteristics of Deep Fractured Rock Mass under Water-Rock Mass Coupling

The deeply fractured rock mass exists in the environment of in-situ stress and high seepage pressure, and they are prone to seepage-slipping failure. Therefore, it is of excellent engineering value and theoretical significance to study the nonlinear seepage evolution of fractured rock mass under water-rock mass coupling. This paper taking the persistent fracture granite as a tested sample, a series of seepage tests of persistent fracture granite under different seepage pressures were carried out. The nonlinear seepage evolution of fractured granite with different JRC values under water-rock mass coupling was studied, and the 3D reconstruction of the fracture surface was carried out after the experiment. The influence of the roughness degree of fracture surface on the permeability of fractured rock mass is investigated. Based on the experimental results, the non-darcy flow behavior of the fractured granite is analyzed by using the quadratic function-type Forchheimer formula. The results show that the permeability of the fractured granite increases nonlinearly with the increase of seepage gradient when the volume flow rate over 20mm3/s; The research results are of great significance for the study of nonlinear seepage evolution of rock mass with different JRC values under high in-situ stress and high seepage pressure.

Revising Transient-Pressure Solution for Vertical Well Intersected by a Partially Penetrating Fracture with Non-Darcy Flow Effect

The principle purpose of this work is to formulate an accurate mathematical model to evaluate the transient pressure behavior of a well intercepted by a partially penetrating vertical fracture (PPVF) with non-Darcy flow effect. Fracture conductivity is taken into account by coupling the three-dimensional flow in reservoir and the two-dimensional flow within fracture; the Barree-Conway model is incorporated into the model to analyze non-Darcy flow behavior in fracture, which leads to the nonlinearity of the governing equations. A high-effective iterative algorithm using a combined technique of fracture-panel discretization and dimension transform is developed to render the nonlinear equations amenable to analytical linear treatment. On the basis of the solutions, the pressure response and its derivative type curves were generated to identify the evolution of flow regimes with time. Furthermore, the influences of fracture conductivity, penetration ratio, and non-Darcy characteristic parameters on pressure response are investigated. The results show that PPVF exhibits five typical flow regimes, and analytical solutions for each flow regime are similar to that for a fully penetrating vertical fracture (FPVF) that can be correlated with the penetration ratio and apparent conductivity. The non-Darcy flow effect is found to have more significant effect on the low and moderate conductivity, especially in early-stage flow regimes. When the penetration ratio is smaller than 0.5, the pressure behavior exhibit a more remarkable variation with penetration ratio. This study provides a better insight into understanding the influence of non-Darcy flow on flow regime identification.

Experimental Investigation on Non-Darcy Flow Behavior of Granular Limestone with Different Porosity

AbstractStatistics indicate that granular rock mass frequently acts as the flow channel and cause water inrush disasters under high water pressure in underground mining and tunnel engineering. The ...

Numerical Analysis of the Effects of Apparent-Permeability Modeling and Secondary-Fracture Distribution for Hydraulic-Fractured Shale-Gas Production Analysis

Summary Complex flow mechanisms, such as Knudsen diffusion, are encountered in the shale matrix because of the presence of nanopores. Numerous apparent-permeability models have been proposed to capture the ensuing non-Darcy flow behavior. However, these models are not readily available in most commercial reservoir simulators, and ignoring these mechanisms can potentially underestimate the overall matrix conductivity. This work implements an explicit coupling strategy for integrating a pressure-dependent apparent-permeability model in reservoir simulation. The numerical models are subsequently used to study the effects of apparent-permeability modeling and natural-fracture distribution on gas production and water loss during flowback. The effects of multiphase-flow functions on fluid retention are also assessed. A set of 3D reservoir models are constructed using field data obtained from the Horn River shale-gas reservoir. First, stochastic 3D discrete-fracture-network (DFN) models are scaled up into equivalent continuum dual-porosity/dual-permeability models. An apparent-permeability (Kapp) model accounting for contributions of slip flow, Knudsen diffusion, and surface pore roughness is applied at each gridblock. A novel coupling scheme is formulated to facilitate the updating of Kapp after a certain specified time interval, capturing the pressure dependency of the Kapp. The sensitivity of the updating frequency is analyzed. The results reveal that incorporating these additional flow mechanisms by means of the apparent-permeability formulation could potentially increase the overall gas-production prediction by up to 11%, depending on the average pore radius, reservoir pressure, and several other matrix or fluid properties. The implications of Kapp modeling in water-loss mechanisms are further examined through a set of sensitivity analyses, where the effects of multiphase-flow functions and DFN distributions are systematically investigated. The following interesting findings are observed: Ignoring Kapp modeling could overestimate water recovery. Fracturing-fluid propagation and long-term water recovery are strongly affected by the secondary-fracture intensity; increase in secondary-fracture intensity would enhance water loss during flowback. Gas production is highly affected by the amount of water in the near-well region. In a gas/water system, compressibility of the in-situ fluids renders the effects of countercurrent imbibition and water retention to be more complex from those observed in similar water/oil systems. This work offers a novel, yet practical, scheme for representing the pressure-dependent matrix apparent permeability in the flow simulation of shale reservoirs. The proposed method captures the non-Darcy flow behavior caused by the complex transport mechanisms occurring in nanosized pores. Most importantly, this coupling procedure can be implemented in existing commercial reservoir-simulation packages. The results have revealed a few interesting insights regarding the potential implications in fracturing design and estimation of stimulated reservoir volume.

A Semianalytical Model for Predicting Transient Pressure Behavior of a Hydraulically Fractured Horizontal Well in a Naturally Fractured Reservoir With Non-Darcy Flow and Stress-Sensitive Permeability Effects

Summary Non-Darcy flow and the stress-sensitivity effect are two fundamental issues that have been widely investigated in transient pressure analysis for fractured wells. The main object of this work is to establish a semianalytical solution to quantify the combined effects of non-Darcy flow and stress sensitivity on the transient pressure behavior for a fractured horizontal well in a naturally fractured reservoir. More specifically, the Barree-Conway model is used to quantify the non-Darcy flow behavior in the hydraulic fractures (HFs), while the permeability modulus is incorporated into mathematical models to take into account the stress-sensitivity effect. In this way, the resulting nonlinearity of the mathematical models is weakened by using Pedrosa's transform formulation. Then a semianalytical method is applied to solve the coupled nonlinear mathematical models by discretizing each HF into small segments. Furthermore, the pressure response and its corresponding derivative type curve are generated to examine the combined effects of non-Darcy flow and stress sensitivity. In particular, stress sensitivity in HF and natural-fracture (NF) subsystems can be respectively analyzed, while the assumption of an equal stress-sensitivity coefficient in the two subsystems is no longer required. It is found that non-Darcy flow mainly affects the early stage bilinear and linear flow regime, leading to an increase in pressure drop and pressure derivative. The stress-sensitivity effect is found to play a significant role in those flow regimes beyond the compound-linear flow regime. The existence of non-Darcy flow makes the effect of stress sensitivity more remarkable, especially for the low-conductivity cases, while the stress sensitivity in fractures has a negligible influence on the early time period, which is dominated by non-Darcy flow behavior. Other parameters such as storage ratio and crossflow coefficient are also analyzed and discussed. It is found from field applications that the coupled model tends to obtain the most-reasonable matching results, and for that model there is an excellent agreement between the measured and simulated pressure response.

A numerical study of non-Darcy flow in EGS heat reservoirs during heat extraction

Underground non-Darcy fluid flow has been observed and investigated for decades in the petroleum industry. It is deduced by analogy that the fluid flow in enhanced geothermal system (EGS) heat reservoirs may also be in the non-Darcy regime under some conditions. In this paper, a transient 3D model was presented, taking into consideration the non-Darcy fluid flow in EGS heat reservoirs, to simulate the EGS long-term heat extraction process. Then, the non-Darcy flow behavior in water- and supercritical CO2 (SCCO2)-based EGSs was simulated and discussed. It is found that non-Darcy effects decrease the mass flow rate of the fluid injected and reduce the heat extraction rate of EGS as a flow resistance in addition to the Darcy resistance which is imposed to the seepage flow in EGS heat reservoirs. Compared with the water-EGS, the SCCO2-EGS are more prone to experiencing much stronger non-Darcy flow due to the much larger mobility of the SCCO2. The non-Darcy flow in SCCO2- EGSs may thus greatly reduce their heat extraction performance. Further, a criterion was analyzed and proposed to judge the onset of the non-Darcy flow in EGS heat reservoirs. The fluid flow rate and the initial thermal state of the reservoir were taken and the characteristic Forchheimer number of an EGS was calculated. If the calculated Forchheimer number is larger than 0.2, the fluid flow in EGS heat reservoirs experiences non-negligible non-Darcy flow characteristic.

The influence of fracture geometry variation on non-Darcy flow in fractures under confining stresses

To investigate the influence of geometric characteristics of deformable rough fractures under confining stresses on the behaviors of non-Darcy flow, four fractured sandstone specimens were used for hydraulic tests in the experiments. According to the experimental results of the relationships between the hydraulic gradient and the flow rate, it is demonstrated that the Forchheimer's equation can offer a good description of the non-Darcy flow in rough fractures. In addition, the coefficients A and B in Forchheimer's equation are sensitive to the fracture geometric characteristics, and their values also increase as the confining stress rises, mainly owing to the reduction of the hydraulic aperture and the heterogeneous distribution of the interconnected void areas with the confining stress rising. Then, the surface and interior geometric properties of rough fractures were quantitatively characterized with the peak asperity height ξ and the box-counting fractal dimension D of the heterogeneous distribution of the interconnected void areas, respectively. Furthermore, an empirical relationship between the fractal dimension D and the fracture apertures was constructed according to the experimental results. Lastly, a quantitative model was proposed to represent the relationship between the fracture geometric characteristics and the non-Darcy coefficient β. This model was further used to link the non-linear coefficient B of Forchheimer's equation and the critical Reynold number Rec with the fracture geometric characteristics. The proposed models were validated by the experimental data and would be helpful to characterize the non-Darcy flow behavior in rough fractures under various confining stresses.

A Potential Parameter for A Non-Darcy Form of Two-Phase Flow Behaviour, Compressibility Related

Numerous scientists did their studies and conducted various laboratory experiments related to a non-Darcy behavior of a two-phase flow for the past thirty years, and made an effort to clarify the behavior. Non-Darcy flow behavior, phenomena occurred in primary recovery method of reservoirs that have an API degree gravity of less than 20. It was confirmed that it results in greater production. The compressibility of foam fits to be the one of the general fundamental factor that directs the lifetime of a non-Darcy form of two phase flow behavior or also is known as the foamy oil. In the process of usual drive depletion, foamy oil featured of low production GOR and high daily production rate. Foamy oil is more compressible than conventional solution gas due to the oil that gas dispersed in it; as a result, oil formation volume factor is much higher than that in conventional oil. This paper represents a laboratory data followed by some of the analysis related to the properties of non-Darcy form of two phase flow and that is the compressibility parameter. The experimental results showed that at different saturation pressures and at a room temperature, the trends fit the expected behavior above the saturation pressures. Moreover, the measurements of live oil compressibility were also attempted below the saturation pressures. It was concluded that other properties such as the viscosity is added a significant effect rather than compressibility in the behavior of what so called foamy oil compared to the presence or absence of asphaltenes and other polar oil components.

Numerical Modeling of Non-Darcy Flow Behavior of Groundwater Outburst through Fault Using the Forchheimer Equation

AbstractIn a groundwater outburst through a fault, the velocity is generally so high that the relationship between the velocity and the hydraulic gradient deviates from the linear Darcy regime. Thi...

Effects of Non-Darcy Flow and Penetrating Ratio on Performance of Horizontal Wells With Multiple Fractures in a Tight Formation

A novel slab source function has been formulated and successfully applied to examine effects of non-Darcy flow and penetrating ratio on performance of a horizontal well with multiple fractures in a tight formation. The Barree–Conway model is incorporated in the mathematical model to analyze non-Darcy flow behavior in the hydraulic fractures, while the pressure response under non-Darcy flow is determined by two dimensionless numbers (i.e., relative minimum permeability (kmr) and non-Darcy number (FND)). A semi-analytical method is then applied to solve the newly formulated mathematical model by discretizing the fracture into small segments. The newly developed function has been validated with numerical solution obtained from a reservoir simulator. Non-Darcy effect becomes more evident at a smaller relative minimum permeability (kmr < 0.05) and a larger non-Darcy number (FND > 10). The non-Darcy number is found to be more sensitive than the relative minimum permeability, resulting in a larger pressure drop even at a larger kmr. In addition, the non-Darcy flow is found to impose a significant impact on the early-stage bilinear/linear flow regime, resulting in an additional pressure drop that is similar to lowering the fracture conductivity. The pressure response can be classified into two categories by a penetrating ratio of 0.5. When the penetrating ratio is decreased, the early bilinear/linear flow regime occurs, followed by an early radial flow regime.

Grain Size Distribution Effect on the Hydraulic Properties of Disintegrated Coal Mixtures

In order to better understand groundwater influx and protection in coal mining extraction works, an in-house water flow apparatus coupled with an industrial rock testing system, known as MTS 815.02, were used to study the effects of grain size mixtures on the compaction and flow properties of disintegrated, or non-cemented, coal samples. From the Reynolds number evaluation of the samples with different grain mixtures, and the relationship between the water flow velocity and pore pressure gradient differences, it was found that seepage through the mixtures are of non-Darcy flow type. The porosity of coal specimens was found to be highly affected by compaction, and the variations of the porosity were also influenced by the samples’ grain size distribution. It was found that the sample porosity decreases with increasing compaction and decreasing grain sizes. Grain crushing during compaction was observed to be the main cause of the appearance of fine grains, and the washing away of fine grains was consequently the main contributing factor for the weight loss due to water seepage. It was observed that during the tests and with the progression of compaction, permeability k decreases and non-Darcy factor β increases with decreasing porosity φ. The k-φ and β-φ plots show that as the sizes of disintegrated coal samples are getting smaller, there are more fluctuations between the porosity values with their corresponding values of k and β. The permeability value of the sample with smallest grains was observed to be considerably lower than that of the sample with largest grains. Non-Darcy behavior could reduce the hydraulic conductivity. It was found that the porosity, grain breakage and hydraulic properties of coal samples are related to grain sizes and compaction levels, as well as to the arrangement of the grains. At high compaction levels, the porosity of disintegrated coal samples decreased strongly, resulting in a significant decrease of the permeability at its full compression state; Non-Darcy flow behavior has the slightest effect in uniform samples, therefore, indicating that disintegrated coal in uniform grain size mixtures could be treated as an aquicluding (water-resisting) stratum.

A Treatise on Non-Darcy Flow Correlations in Porous Media

Non-Darcy Flow behavior is important for describing fluid flow in consolidated or unconsolidated porous media when abrupt changes in velocity dominates. A criterion or a generalized equation is required to understand this flow behavior in the isotropic/anisotropic carbonate and sandstone reservoirs, and naturally or hydraulically fractured reservoirs. Various correlations and equations have been reviewed in this paper to quantify this non-Darcy coefficient (i.e., beta coefficient) mathematically. It has been observed that this coefficient is highly dependent on rock properties (mainly porosity, permeability and tortuosity). An algorithm to determine the values of the beta coefficient by using the correlations have been presented and coded and converted in to a robust user-friendly simulator. This simulator can take a large amount of data set as input and will generate a large data set of beta values as output. The obtained or calculated beta value is very useful for predicting the change in pressure gradient with respect to velocity and hence can give the best estimate of hydrocarbon production under challenging or adverse pressure drop conditions.

Productivity analysis for a vertically fractured well under non-Darcy flow condition

In this paper, a semi-analytical solution of the pseudosteady-state (PSS) productivity index under non-Darcy flow condition is proposed. Based on the model, a new method of optimization of the fracture conductivity has been developed for non-Darcy flow in the fracture. Meanwhile, the effects of the Reynolds number, the proppant number and the fracture conductivity on the dimensionless productivity index have been discussed in detail.Results show that: (1) the classic UFD (Unified Fracture Design) curves of the Darcy-flow model underestimate the effect of the proppant number and are unsuitable for the non-Darcy-flow fracture optimization. Our discretized model provides a new tool to obtain the optimal fracture parameters for the case of non-Darcy flow condition. (2) for a given penetration ratio, the non-Darcy flow behavior exerts a strong influence on the productivity index with the dimensionless fracture conductivity CfD=0.1−1000 and a considerable productivity index drop can be observed. For the extremely low (CfD<0.1) or high (CfD>1000) dimensionless fracture conductivity, the effect of non-Darcy flow becomes negligible. (3) the effect of the non-Darcy flow on the productivity index becomes pronounced at large value of proppant number, Np. For a given proppant number, the optimal fracture conductivity is slowly increasing as the Reynolds number increases. However, the magnitude of the effect on the productivity index is gradually declining. At the Reynolds number less than 5, the non-Darcy flow has a relatively strong impact on the productivity index and an apparent fall of the maximum productivity index can be noticed, especially for a large proppant number. Beyond the value of 5, the declining trend of the maximum productivity index gradually slows down as the Reynolds number increases.

Analytical method for performance evaluation of fractured horizontal wells in tight reservoirs

This paper presents an analytical model for horizontal well with multiple hydraulic transverse fractures, producing from a tight oil reservoir. The model combines productivity index model (Guo et al.2009) with non-Darcy flow and fracture heterogeneity.Although there are numerous analytical models have been proposed for fractured horizontal wells performance, either pseudo-steady conditions or pressure-transient phenomenon. They rarely consider the influence of non-Darcy flow and fracture heterogeneity on fractured horizontal wells performance. Non-Darcy flow behavior has been proved to exist in tight reservoirs by many previous investigators. In the meantime, fracture heterogeneity should not be neglected in the process of hydraulic fracturing.In this paper, we deduce an analytical production equation in detail, towards fractured wells in tight reservoirs. Based on productivity index model, three flow region are analyzed by seepage characteristics. We make further analysis of pressure distribution in fractured horizontal well by a commercial numerical simulator. In addition, some critical factors on productivity have been considered.Results show that there are strong correlations between productivity and critical factors by the mathematical model. Further verification of the influences on well productivity is demonstrated. The fractured well productivity with different threshold pressure gradient and deformation coefficient has negative correlation. Fractured well productivity increases as fracture number, half of fracture length, and fracture conductivity increases within a certain range. The effect on fractured well productivity by fracture heterogeneity appears to be complex, not only by the end point influence, but also by fractures of internal interference.

Non-Darcy's law-based analytical models for data interpretation of high-pressure packer tests in fractured rocks

High-pressure packer test (HPPT) is an enhanced constant head packer test for characterizing the permeability or coupled hydromechanical properties of fractured rocks under high-pressure groundwater flow conditions. The interpretation of the HPPT data, however, requires further investigations due to the non-Darcy flow prone to occur in the tested rocks at high flow velocities. In this study, two analytical models were presented based on the Forchheimer's law and Izbash's law, respectively, for interpretation of the HPPT data, which can be readily reduced to Hvorslev equation in laminar flow condition. The proposed models were applied to data interpretation and hydraulic parameter estimation for the HPPTs performed in fractured sedimentary rocks located at about 450m depth in central Hainan Island, and validated by numerical simulations on the tests. The results indicate that both of the proposed models well reproduce the non-Darcy flow behavior observed in the HPPTs before the initiation of hydraulic fracturing. Overall, the Forchheimer's law-based model is advantageous for its clear physical meanings, but the Izbash's law-based model could be more convenient when the obtained P–Q curves are divided into different phases according to the flow conditions in the fracture system. Also pointed out were the theoretical flaws in an existing Izbash's law-based analytical model, which lead to a misestimate of hydraulic conductivity and an inconsistency between the analytical solutions and numerical results.