Klinkenberg Effect Research Articles

Non-Darcy Flow in Hydraulic Flushing Hole Enlargement-Enhanced Gas Drainage: Does It Really Matter?

In recent years, the non-Darcy flow has seen a significant increase in interest in conventional and unconventional gas developments. The nonlinear behavior can be described by adding a quadratic term of the velocity with the permeability-dependent β factor in Darcy’s law. The methods for improving permeability such as the hydraulic flushing hole enlargement (HFHE) technique can remarkably enhance the coal permeability thus changing the β factor. However, few studies have been done previously to seek how this non-Darcy flow impacts the permeability-enhanced gas drainage. In this study, a fully coupled coal deformation and damage, the non-Darcy flow of free methane gas in the fractures, and Fickian diffusion of adsorbed methane in the coal matrix model were developed. The Klinkenberg effect has been taken into account as a reference object. A series of simulation scenarios were carried out to evaluate and compare the influences of the non-Darcy effect, gas-coal interaction effect, and Klinkenberg effect on the HFHE-enhanced gas drainage. Results of the simulation illustrate that the HFHE technique can significantly improve the gas drainage efficiency by permeability enhancement. But the HFHE-enhanced gas drainage is unique to the other gas developments with high or low flow rates; a slight pressure gradient and short drainage time make the non-Darcy effect fail to generate cumulative influence on the HFHE-enhanced gas drainage.

Permeability characteristics of coal containing gangue under the effect of adsorption

In order to find out the influence of adsorption on gas flow characteristics in coal seam, we analyzed the influence of gangue on permeability characteristics of coal, and then deduced the calculation method of the equivalent permeability of the specimen containing gangue. The gas flow characteristics under different conditions of pulverized coal sample, sand sample and coal containing gangue (simulated coal seam containing gangue) were studied using self-developed device “a test device for cracked coal-rock mass of gas migration characteristics”, and then the experimental results were analyzed. The results show that: the coal seam adsorption causes the swelling deformation of coal matrix, reduces the coal seam porosity and the coal seam permeability, and the swelling deformation of coal matrix and gas pressure conform to the rule of quadratic function. The two gangue layers with different porosity both can promote the equivalent permeability. However, the promotion effect is different with the increase of the thickness of the gangue layer. The existence of the gangue layer does not change the relation between permeability and gas pressure, and the both still conform to the quadratic function. The Klinkenberg effect of the gangue coal is weakened, and the slip factor is linearly negative to the thickness of the gangue. With the increase of the gas pressure, the extrusion coefficient of the two phases of coal and gangue first decreases and then increases.

Forecasting of Horizontal Gas Well Production Decline in Unconventional Reservoirs using Productivity, Soft Computing and Swarm Intelligence Models

The most widely used production decline forecasting tools are numerical reservoir simulation, material balance estimates and advanced methods of production decline analysis. Besides, these existing production decline evaluation techniques for unconventional reserves estimations have underlying limitations and assumptions incorporated into their formulation which can result into under- and overestimation. This further raises the debate about which decline curve analysis (DCA) is better than the other for unconventional reservoirs predictions. Currently, data-driven artificial neural network (ANN) has emerged as a new paradigm capable of mapping complex functional relationships. In this present study, ANN technology was used to calibrate tight gas carbonate field historical declining trends which exhibit high early peak production rate and quick decline. Hereafter, making reliable predictions posed as a challenge for the complex target field and hard computing protocols. Therefore, this research applied and tested the capability of backpropagation artificial neural network (BPANN), radial basis function neural network (RBNN) and generalized regression neural network (GRNN) as DCA techniques for predicting the historical production decline trends of an ultra-low porosity and permeability tight gas reservoir. The optimum trained ANN DCA models developed were validated with another surrounding well’s dataset, producing acceptable results in agreement with the actual field data. The GRNN DCA model’s testing performance was poor, and its smoothing parameter was optimized with particle swarm optimization (PSO) algorithm which gave satisfactory results comparable to standalone BPANN and RBFNN DCA models. Furthermore, the optimum ANN DCA models’ generalization strength across the entire field dataset revealed that the developed models’ predictions were robust as compared to the data-driven Arps hyperbolic and power law exponential DCA models. This was evident from the statistical performance criteria employed which indicated that BPANN, RBFNN and PSO-GRNN DCA models are plausible better fit models for matching the target field’s historical decline performances. Also, a novel non-Darcy flow horizontal well productivity evaluation model for the target field was developed based on stress sensitivity coefficient (SSC) of permeability, tortuosity factor, Klinkenberg effect, near-wellbore turbulence effect and threshold pressure gradient (TPG) for validating the ANN DCA models predictions. The productivity model was validated with published horizontal well model with closely matched results. For inflow performances, the horizontal well model with turbulence minimizes negative effects on non-Darcy flow rates than without turbulence. Additionally, pressure drawdown influences the tight gas well productivity such that the lower the pressure drawdown, the smaller the tight gas well productivity. The operating points of the tight gas well were determined through inflow performance relation and tubing performance relation at different SSC and TPG for $$ 3{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}} $$ in tubing size. In another case, synthetic unconventional simulation data with long production history were used for future forecast of 5, 24 and 70 months of production rate and cumulative production which gave pretty close results for all the DCA models, unlike the real field datasets, where the empirical rate-time models under and overestimate. In all, these ANN DCA models and the horizontal well productivity model derived will serve as new computational tools for complementing existing DCA techniques for better understanding of unconventional reservoirs’ production decline performance.

Second-order gas slippage model for the Klinkenberg effect of multicomponent gas at finite Knudsen numbers up to 1

Within microchannels, rarefied gas molecules frequently interact with the channel wall, resulting in non-zero velocity at the wall. This phenomenon is known as the gas slippage effect or Klinkenberg effect in porous media. Although the gas slippage effect of single component gas has been thoroughly investigated, an accurate correlation to account for the gas slippage effect in multicomponent gas flow has not been formulated so far. In this paper, we aim to quantify the multicomponent gas slippage effect by deriving a non-empirical second-order correlation.Our approach is based on kinetic theory. We calculate the mean free path of gas mixtures, and capture the loss of horizontal flux momentum of gas flux after the molecules diffusively reflect at the wall. The horizontal flux momentum acts as shear stress on gas flow. In this sense, the loss of momentum induces reduction of viscosity and enhancement of mass transfer. By quantifying the loss of horizontal momentum as well as the reduction of viscosity, we can solve the gas slippage coefficient for the multicomponent gas flow system.Our model captures the mass transfer mechanism of gas mixtures at low pressure (near-ideal condition). The accuracy of our model has been validated by both molecular-level simulation data and physical experimental data. The difference between our results and benchmark data is within 10% for Knudsen number up to 1.Our model can be readily applied to the numerical simulation of unconventional gas formations.

Numerical Simulation Study on Seepage Theory of a Multi-Section Fractured Horizontal Well in Shale Gas Reservoirs Based on Multi-Scale Flow Mechanisms

Aimed at the multi-scale fractures for stimulated reservoir volume (SRV)-fractured horizontal wells in shale gas reservoirs, a mathematical model of unsteady seepage is established, which considers the characteristics of a dual media of matrix and natural fractures as well as flow in the large-scale hydraulic fractures, based on a discrete-fracture model. Multi-scale flow mechanisms, such as gas desorption, the Klinkenberg effect, and gas diffusion are taken into consideration. A three-dimensional numerical model based on the finite volume method is established, which includes the construction of spatial discretization, calculation of average pressure gradient, and variable at interface, etc. Some related processing techniques, such as boundedness processing upstream and downstream of grid flow, was used to limit non-physical oscillation at large-scale hydraulic fracture interfaces. The sequential solution is performed to solve the pressure equations of matrix, natural, and large-scale hydraulic fractures. The production dynamics and pressure distribution of a multi-section fractured horizontal well in a shale gas reservoir are calculated. Results indicate that, with the increase of the Langmuir volume, the average formation pressure decreases at a slow rate. Simultaneously, the initial gas production and the contribution ratio of the desorbed gas increase. With the decrease of the pore size of the matrix, gas diffusion and the Klinkenberg effect have a greater impact on shale gas production. By changing the fracture half-length and the number of fractured sections, we observe that the production process can not only pursue the long fractures or increase the number of fractured sections, but also should optimize the parameters such as the perforation position, cluster spacing, and fracturing sequence. The stimulated reservoir volume can effectively control the shale reservoir.

Numerical efficiency assessment of the lattice Boltzmann model for digital nano-porous rock applications

Abstract Study of transport properties of low permeable sedimentary clay formations is of growing interest in exploration of non-conventional resources, and for long term radioactive waste repositories where these formations play the role of a natural shield which prevents the spreading of radioactive materials in the environment. Recent advances in imaging techniques allow to obtain 3D volume of compacted clay sample with pixel size of order of few nano-meters which enables us to explore transport properties of these media by means of numerical simulation. In our study, the Lattice Boltzmann Model (LBM) is applied to simulate fluid flow inside the 3D image of a compacted illite sample to explore its transport properties. The impact of different image segmentation methods applied to treat the raw data on the numerical results is addressed. Because of the nano-metric pore size, various physical phenomena may influence fluid flow. In this study, we consider the influence of gas slippage effect (the Klinkenberg effect) on the permeability. The numerically obtained results are compared with experimental data obtained for a reconstructed compacted illite sample. If the original geometry is not strongly modified by the applied image segmentation method, the numerically calculated Klinkenberg factor follows well existing trends and provides predictions close to the experimental values.

Experimental and Numerical Study of the Influence of Gas Pressure on Gas Permeability in Pressure Relief Gas Drainage

Changes in gas pressure or external stress cause changes in permeability; furthermore, the severity of stress disturbance in these situations can easily produce plastic yielding in the coal seam. Variations in coal seam permeability are also different between the elastic and plastic zones during the process of gas drainage. Therefore, it is necessary to consider gas pressure and external stress during pressure relief gas drainage in order to provide guidance for drilling in terms of location, timing, and pressure. An experimental study explored the relationship between external stress, gas pressure, and permeability in elastic and plastic fracture coal samples. This paper then conducts a numerical simulation of gas drainage according to the model, using the FISH programming language embedded in the FLAC3D numerical simulation software. The experimental results demonstrate that permeability in elastic coal samples first decreased and then increased with increasing gas pressure; in contrast, permeability in fractured coal samples decreased with increasing gas pressure and did not appear to increase at all within the tested gas pressure range. Decreases in permeability accompanied increases in gas pressure under constant effective stress; this indicates that decreases in effective stress lead to increases in permeability when the gas pressure exceeds the critical gas pressure under constant external stress. In contrast, when the gas pressure is less than the critical value, the Klinkenberg effect and matrix adsorption expansion play a major role in permeability changes. The influence of the Klinkenberg effect and adsorption on permeability decreases as gas pressure increases under the same effective stress or external stress conditions. Permeability and its sensitivity to gas pressure decrease with increasing effective stress or external stress. Meanwhile, the simulation demonstrates that permeability in the plastic zone increases in a nonlinear fashion as gas pressure decreases. These permeability increases can be divided into four stages: a rapid increase, gradual smoothing, a second slow increase, and ultimate stability.

An Efficient Numerical Hybrid Model for Multiphase Flow in Deformable Fractured-Shale Reservoirs

Summary After hydraulic fracturing, a shale reservoir usually has multiscale fractures and becomes more stress-sensitive. In this work, an adaptive hybrid model is proposed to simulate hydromechanical coupling processes in such fractured-shale reservoirs during the production period (i.e., the hydraulic-fracturing process is not considered and cannot be simulated). In our hybrid model, the single-porosity model is applied in the region outside the stimulated reservoir volume (SRV), and the matrix and natural/induced fractures in the SRV region are modeled using a double-porosity model that can accurately simulate the matrix/fracture fluid exchange during the entire transient period. Meanwhile, the fluid flow in hydraulic fractures is modeled explicitly with the embedded-discrete-fracture model (EDFM), and a stabilized extended-finite-element-method (XFEM) formulation using the polynomial-pressure-projection (PPP) technique is applied to simulate mechanical processes. The developed stabilized XFEM formulation can avoid the displacement oscillation on hydraulic-fracture interfaces. Then a modified fixed-stress sequential-implicit method is applied to solve the hybrid model, in which mixed-space discretization [i.e., finite-volume method (FVM) for flow process and stabilized XFEM for geomechanics] is used. The robustness of the proposed model is demonstrated through several numerical examples. In conclusion, several key factors for gas exploitation are investigated, such as adsorption, Klinkenberg effect, capillary pressure, and fracture deformation. In this study, all the numerical examples are 2D, and the gravity effect is neglected in these simulations. In addition, we assume there is no oil phase in the shale reservoirs, thus the gas/water two-phase model is used to simulate the flow in these reservoirs.

Investigation of the Klinkenberg effect in a micro/nanoporous medium by direct simulation Monte Carlo method

We define effective pore size as a function of the porosity, the tortuosity, the contraction factor, and the intrinsic permeability of the porous medium. The Klinkenberg effect in different porous structures is found to be fully described by the Knudsen number characterized by this pore size.

Experimental Study of Matrix Permeability of Gas Shale: An Application to CO2-Based Shale Fracturing

Because the limitations of water-based fracturing fluids restrict their fracturing efficiency and scope of application, liquid CO2 is regarded as a promising substitute, owing to its unique characteristics, including its greater environmental friendliness, shorter clean-up time, greater adsorption capacity than CH4 and less formation damage. Conversely, the disadvantage of high leak-off rate of CO2 fracturing due to its very low viscosity determines its applicability in gas shales with ultra-low permeability, accurate measurement of shale permeability to CO2 is therefore crucial to evaluate the appropriate injection rate and total consumption of CO2. The main purpose of this study is to accurately measure shale permeability to CO2 flow during hydraulic fracturing, and to compare the leak-off of CO2 and water fracturing. A series of permeability tests was conducted on cylindrical shale samples 38 mm in diameter and 19 mm long using water, CO2 in different phases and N2 considering multiple influencing factors. According to the experimental results, the apparent permeability of shale matrix to gaseous CO2 or N2 is greatly over-estimated compared with intrinsic permeability or that of liquid CO2 due to the Klinkenberg effect. This phenomenon explains that the permeability values measured under steady-state conditions are much higher than those under transient conditions. Supercritical CO2 with higher molecular kinetic energy has slightly higher permeability than liquid CO2. The leak-off rate of CO2 is an order of magnitude higher than that of water under the same injection conditions due to its lower viscosity. The significant decrease of shale permeability to gas after water flooding is due to the water block effect, and much longer clean-up time and deep water imbibition depth greatly impede the gas transport from the shale matrix to the created fractures. Therefore, it is necessary to substitute water-based fracturing fluids with liquid or super-critical CO2 in clay-abundant shale formations.

Investigation of effects of fracturing fluid on hydraulic fracturing and fracture permeability of reservoir rocks: An experimental study using water and foam fracturing

Hydraulic fracturing is a promising technique to enhance gas production from deep geological formations by enhancing reservoir rock permeability. However, permeability enhancement may vary with reservoir conditions and fracturing fluid. The main objective of this study is to investigate the effect of fracturing fluid on the fracture pattern and permeability of fractured rock. Fracturing experiments were conducted on siltstone specimens with water and foam (N2 + water) as the fracturing fluids, using a high-pressure triaxial apparatus. CT scanning and 3-D scanning technology were used to examine the fracture pattern and the fracture surface of the fractured specimens. The fractured specimens were then used to conduct permeability experiments for five different confining pressures (5 – 25 MPa) and six different injection pressures (1 – 6 MPa) with N2, and fracture permeability values were compared, based on the fracturing fluid type as well as the intact rock permeability. The results indicate that foam fracturing can induce a complex twisted fracture with greater surface area and the fracture is throughout the specimen. Moreover, foam fracturing emits more energy than water fracturing during the fracturing process, which is favourable for secondary fracture development. According to the measured permeability values, the fracture permeability of foam fractured specimens is about 5 × 105 times greater than that of intact and water-fractured specimens. With increasing confining pressure, N2 permeability decreases, regardless of the specimen (intact or fractured), and interestingly permeability decreases with increasing injection pressure. These observations are related to the effective stress principle and the Klinkenberg effect.

Laboratory study of proppant on shale fracture permeability and compressibility

Hydraulic fracturing is key for shale gas production and fracture permeability or conductivity is one of the most important parameters for gas production rate. Investigating the proppant distribution and fracture permeability in the field is difficult, therefore, laboratory study is a good alternative. In this work, the effect of the layer number and type of proppant on fracture permeability and compressibility were investigated. A cubic shale sample from the Cambrian Niutitang Formation at Sangzhi, Hunan Province, China, was used in this work. Sands and glass beads of different number of layers were added into an artificial fracture and seven cases, including original sample, non-propped fracture, and four kinds of propped fractures were considered. Permeability at three gas pressure steps and five confining pressure steps were measured in each case at two flow directions. Microscopic X-ray computed tomography was used to detect the distributions of proppant, and the relationship with permeability and its anisotropy was studied. A permeability model combining the stress and Klinkenberg effects was used to match experimental data and a new fracture compressibility model was proposed to predict the change of fracture compressibility with the layer number of proppant. It was found that permeability and compressibility of proppant supported fracture are closely related to proppant packing pattern and layer number, as well as the permeability anisotropy. These results improve our understanding on permeability behaviour for the proppant supported fracture and can assist in the model of fracture permeability and simulation of shale gas production.

Experimental Determination of Gas Relative Permeability Considering Slippage Effect in a Tight Formation

In this paper, the gas relative permeability considering slippage effect has been experimentally examined under various experimental conditions (i.e., ambient, high confining pressure, and high temperature). Experimentally, Klinkenberg permeabilities of 12 core samples have been measured by using steady-state flow experiment. It has been found that the Klinkenberg permeability is independent of the experimental temperature and dramatically decreases as confining pressure is increasing. Furthermore, linear correlations have been newly developed between the Klinkenberg permeability and the gas-measured permeability under various conditions. Subsequently, the developed correlations are correspondingly applied to calibrate the gas relative permeability. It has been found that the gas relative permeability can be overestimated without consideration of the slippage effect, i.e., Klinkenberg effect. In addition, the newly developed correlations have been applied to analyze the sensitivity of gas–water relative permeability to gas-measured permeability, confining pressure, and temperature. It is demonstrated that mobile water greatly alleviates the gas relative permeability in comparison to irreducible water. Although an increased confining pressure simultaneously reduces the effective water phase and gas phase permeability, the gas relative permeability increases and the water relative permeability decreases as the confining pressure increases. It is attributed to the fact that the effective water phase permeability is more sensitive to the confining pressure. Given an elevated experimental temperature, the gas relative permeability is reduced while the water relative permeability is enhanced, implying the significance of temperature effect on gas–water relative permeability measurements.

Laboratory Investigation to Assess the Impact of Pore Pressure Decline and Confining Stress on Shale Gas Reservoirs

Four core samples of outcrop type shale from Mancos, Marcellus, Eagle Ford, and Barnett shale formations were studied to evaluate the productivity performance and reservoir connectivity at elevated temperature and pressure. These laboratory experiments were conducted using hydrostatic permeability system with helium as test gas primarily to avoid potential significant effects of adsorption and/or associated swelling that might affect permeability. It was found that the permeability reduction was observed due to increasing confining stress and permeability improvement was observed related to Knudsen flow and molecular slippage related to Klinkenberg effect. Through the effective permeability of rock is improved at lower pore pressures, as 1000 psi. The effective stress with relatively high flow path was identified, as 100-200 nm, in Eagle Ford core sample. However other three samples showed low marginal flow paths in low connectivity.

Geochemical and petrophysical source rock characterization of the Vaca Muerta Formation, Argentina: Implications for unconventional petroleum resource estimations

The Vaca Muerta Formation is currently regarded as the most prolific source rock interval for unconventional petroleum exploration in Argentina. In this study, we integrate recent CT-scanner, nuclear magnetic resonance (NMR) T1-T2 and Rock-Eval Shale Play® analytical techniques, combined with classical geochemical, petrological and petrophysical methods to characterize Vaca Muerta source rocks along a vertical profile. The proposed analytical workflow was tested here on 4 samples derived from one core: the LJE-1010 borehole drilled in the lowermost Jurassic Vaca Muerta member (Neuquén Basin – Argentina), to evaluate the unconventional hydrocarbon potential and pore network attributes. First, entire rock samples were analyzed by dual energy CT scanner to evaluate possible heterogeneities; this also helped in the selection of representative samples. Then the organic matter properties were investigated by open-system pyrolysis techniques, organic petrography and biomarker analyses as a function of the core depth. A comparative approach between open-system pyrolysis methods (e.g. Rock-Eval Basic/Bulk-Rock & Shale Play® methods vs. Source Rock Analyzer) is also illustrated in this work. Finally, petrophysical characteristics and pore network attributes in shale were estimated by gas permeability (steady-state method corrected for Klinkenberg effect), porosity and NMR methods. The investigated samples have oil-window maturity and are characterized by a network of solid bitumen. Based on new Rock-Eval Shale Play® data, the potentially producible oil present in the rock samples was estimated using the modified oil saturation index (OSI=Sh0+Sh1 peaks×100/TOC). Results indicate that the oil crossover effect and potential productive oils occur within intervals showing higher TOC values (~3 to 8wt%). We also illustrated here that, for unconventional shale play perspectives, Rock-Eval Shale Play® parameters (Sh0 & Sh1) allow to obtain both a better quantification of free and retained hydrocarbons in source-reservoir rock samples and correct original oil in place (OOIP) estimations in early exploration campaign for shale oil and shale gas assessment. For Vaca Muerta rock samples, OOIP estimations range from 60 to 160bbl oil/acre-ft, approximately. We also demonstrated that the solid bitumen, oil and organic matter are clearly distinguished in the studied samples using advanced NMR T1-T2 maps obtained at different temperatures. The solid bitumen was clearly evidenced as a peak with T1/T2 ratio~14 on NMR T1-T2 maps. Concerning the matrix bulk rock permeability, the obtained results showed that the most deep sample is characterized by a permeability of about 140nanoDarcy (nD) whereas the shallower sample containing also higher amounts of organic matter is more permeable (~213nD), indicating that TOC values play a main control on Vaca Muerta poromechanical characteristics. The maximum total porosity values range between 6.5 and 21.8% and were calculated in this work integrating NMR and helium porosity values. Finally, global interpretations of these obtained results suggest that the Vaca Muerta Formation could be considered as a prolific shale play interval for unconventional petroleum exploration in the near future.

Effects of nanoscale pore structure on permeability and relative permeability loss analyzed by GPU enhanced Multiple-Relaxation-Time LBM

Pore scale and the pore structure have significant effects on mass transfer properties through the porous media. In this article the effects of nanoscale pore structure (0.01<Kn<0.1) on permeability and permeability damage were studied by the GPU enhanced Multiple-Relaxation-Time (MRT) Lattice Boltzmann Model (LBM). Six mono-sized-bead packed nanoscale pores and six two-sized-bead packed same porosity different structure (SPDS) nanoscale pores were made to analyze scale and pore structure effects on permeability. The water predicted permeability and air predicted permeability in these nanoscale pores were estimated, and a modified slip boundary condition was implemented to mimic rarified gas effect in this scale. Simulation results are coherent with Klinkenberg effect that air predicted permeability is larger than water predicted permeability in every pore sample. Further analyses showed that pore dimension evenness degree (PDED, the ratio of max pore diameter to min pore diameter) is an essential factor of permeability. Larger PDED increases pore structure complexity, total surface area and tortuosity, but decreases permeability; smaller PDED has opposite effects. PDED is a new pore structure parameter presented in this article. GPU speed up results showed that CUDA parallel significantly accelerated calculation. The highest speed up ratio of CUDA parallel to single CPU was 468.

Gas sorption and non-Darcy flow in shale reservoirs

Gas sorption and non-Darcy flow are two important issues for shale gas reservoirs. The sorption consists of dissolution and adsorption. Dissolved gas and adsorbed gas are different. The former is dissolved in the shale matrix, while the latter is concentrated near the solid walls of pores. In this paper, the Langmuir equation is used to describe adsorption and Henry’s law is used to describe dissolution. The K coefficient in Henry’s law of 0.052 mmol/(MPa g TOC) is obtained by matching experimental data. The amount of dissolved gas increases linearly when pressure increases. Using only the Langmuir equation without considering dissolution can lead to a significant underestimation of the amount of sorbed gas in shales. For non-Darcy gas flow, the apparent permeability model for free gas is established by combining slip flow and Knudsen flow. For adsorbed gas, the surface diffusion effect is also considered in this model. The surface diffusion coefficient is suggested to be of the same scale as the gas self-diffusion coefficient, and the corresponding effective permeability is derived. When frac{1}{p} increases, frac{{k_{text{app}} }}{{k_{text{D}} }} increases, but the relationship is not linear as the Klinkenberg effect suggests. The effect of adsorption on the gas flow is significant in nanopores (r le 2;{text{nm}}). Adsorption increases apparent permeability in shales at low pressures and decreases it at high pressures.

Analytical model of gas transport in heterogeneous hydraulically-fractured organic-rich shale media

Multiple gas transport mechanisms and geological parameters influence total gas production in hydraulically-fractured organic-rich shale reservoirs. The absolute and relative contributions of these processes and parameters vary significantly with the scale of the medium, from pore scale to field scale. The objective of this study is to investigate the impact of these geological parameters and transport processes in organic-rich shale production at field scale.We develop a multi-porosity analytical model which honors pressure- and space-dependent geological properties and gas transport processes in a multi-stage, hydraulically-fractured horizontal well of shale reservoir. The model premises distinct pores in kerogen, inorganic clay, induced fracture, and hydraulic fracture. Multiple flow zones are modeled: unstimulated reservoir volume (USRV), stimulated reservoir volume (SRV), and hydraulic fractures (HF). Compared to the earlier analytical models, the new model accounts for more comprehensive transport mechanisms and medium properties: gas diffusion and desorption in kerogen matrix and organopores, slip flow in inorganic matrix pores, transient Darcy flow from matrix to induced fractures, Darcy flow in both induced and hydraulic fractures, tortuosity of induced fractures, and pressure-dependent permeabilities. These enhanced features introduce important complexities to the analytical solution challenges that are discussed and resolved in detail.Extensive lab and field data for Eagle Ford shale are presented in the new model. Sensitivity analyses are conducted to study the flow contribution from organic and inorganic media. The results show that first, the impacts of gas diffusion, TOC, and organoporosity at pore and core scale are largely eclipsed at field scale by slip flow in inorganic matrix and anomalous diffusion within complex induced-fracture network. Second, fracture tortuosity and the Klinkenberg effect are dominant factors during early- and intermediate-time production. Third, gas diffusion in kerogen matrix is negligible compared to gas desorption and diffusion in organopores. Fourth, permeability variations with pore pressure have a significant impact on late-time production from fractured low-permeability reservoirs.The derived analytical solutions in Laplace space serve as simple but robust tools for studying gas transport in fractured organic shale formation. The field application is validated by matching results for an Eagle Ford shale well. Finally, there is a discussion of the present model’s limitations and potential future extensions.

An improved elastic-tubes model for the correlation of permeability and stress with correction for the Klinkenberg effect

The correlation of permeability and stress is crucial for correctly predicting the coupled process of fluid flow and solid deformation in geophysical systems. Especially in ultra-low-permeability rocks with nanopores, the apparent permeability becomes sensitive to the properties of fluid and solid. This work presents an improved tubes model, where the correlation of permeability and stress is derived with the Klinkenberg effect correction. Both the in-situ stress effect and the Klinkenberg effect have been carefully considered in current correction of permeability prediction based on Darcy's law. The experimental data can be fitted well by the new model, and the Klinkenberg effect of gas flow in the tight porous media is successfully captured. The predictive ability of the current model has been examined as well and the results are acceptable.

Experimental Study of the Klinkenberg Effect on the Gas Permeability of Coal

This paper presents the results of an experimental investigation on gas flow and Klinkenberg effect in coal. An anthracite coal sample is subjected to a range of effective stress conditions in order to investigate a general trend of coal permeability reduction with an increase in effective stress. Based on the measured values of permeability at different mean gas pressures for constant effective stress conditions, intrinsic values of permeability in the range 0.2 – 1.5 mD and gas slip factors are determined using the Klinkenberg plot. Using measured gas permeability and calculated intrinsic permeability values, an assessment of the mean gas pressure required to minimize the Klinkenberg effect is conducted.