Mathematical Model of Coalbed Gas Flow with Klinkenberg Effects in Multi-Physical Fields and its Analytic Solution

The deep-mining coal seam impacted by high in situ stress, where Klinkenberg effects for gas flow were very obvious due to low gas permeability, could be regarded as a porous and tight gas-bearing media. Moreover, the Klinkenberg effects had a significant effect on gas flow behavior of deep-mining coal seam. Based on the gas flow properties of deep-mining coal seams affected by in situ stress field, geothermal temperature field and geo-electric field, a new mathematical model of coalbed gas flow, which reflected the impact of Klinkenberg effects on coalbed gas flow properties in multi-physical fields, was developed by establishing the flow equation, state equation, and continuity equation and content equation of coalbed gas. The analytic solution was derived for the model of one-dimensional steady coalbed gas flow with Klinkenberg effects affected by in situ stress field and geothermal temperature field, and a sensitivity analysis of its physical parameters was carried out by comparing available analytic solutions and the measured values. The results show that the analytic solutions of this model of coalbed gas flow with Klinkenberg effects are closer to the measured values compared to those without Klinkenberg effects, and this model can reflect more accurately gas flow of deep-mining coal seams. Moreover, the analytic solution of this model is more sensitive to the change of Klinkenberg factor b and temperature grad G than depth h.     

分形理论在煤层气非平衡吸附模型中的应用

传统的煤层气动力学模型均是建立在欧几里得几何基础上的,难以描述煤层孔隙结构的复杂性及形状的不规则性。本文以分形理论为基础,通过引入分形维数来刻画煤层孔隙结构的复杂性并考虑煤层的吸附特性、双重介质特征及介质的变形,建立基于Fick第二定律的分形介质煤层气非稳态渗流数学模型。由于流动方程的强非线性,结合各类边界条件用正则摄动法和Laplace变换得到模型在拉氏空间上的近似解析解,再利用Laplace数值反演求得实空间上的数值解。对参数进行敏感性分析并绘制了典型压力曲线,这些结果为煤层气开采提供了理论依据和试井方法。     

Laboratory Investigation of LNAPL Migration in Double-Porosity Soil Under Fractured Condition Using Digital Image Analysis

Gas production from shale gas reservoirs plays a significant role in satisfying increasing energy demands. Compared with conventional sandstone and carbonate reservoirs, shale gas reservoirs are characterized by extremely low porosity, ultra-low permeability and high clay content. Slip flow, diffusion, adsorption and desorption are the primary gas transport processes in shale matrix, while Darcy flow is restricted to fractures. Understanding methane diffusion and adsorption, and gas flow and equilibrium in the low-permeability matrix of shale is crucial for shale formation evaluation and for predicting gas production. Modeling of diffusion in low-permeability shale rocks requires use of the Dusty gas model (DGM) rather than Fick’s law. The DGM is incorporated in the TOUGH2 module EOS7C-ECBM, a modified version of EOS7C that simulates multicomponent gas mixture transport in porous media. Also included in EOS7C-ECBM is the extended Langmuir model for adsorption and desorption of gases. In this study, a column shale model was constructed to simulate methane diffusion and adsorption through shale rocks. The process of binary \(\hbox {CH}_{4}{-}\hbox {N}_{2}\) diffusion and adsorption was analyzed. A sensitivity study was performed to investigate the effects of pressure, temperature and permeability on diffusion and adsorption in shale rocks. The results show that methane gas diffusion and adsorption in shale is a slow process of dynamic equilibrium, which can be illustrated by the slope of a curve in \(\hbox {CH}_{4}\) mass variation. The amount of adsorption increases with the pressure increase at the low pressure, and the mass change by gas diffusion will decrease due to the decrease in the compressibility factor of the gas. With the elevated temperature, the gas molecules move faster and then the greater gas diffusion rates make the process duration shorter. The gas diffusion rate decreases with the permeability decrease, and there is a limit of gas diffusion if the permeability is less than \(1.0\,\times \,10^{-15}\, \hbox { m}^{2}\). The results can provide insights for a better understanding of methane diffusion and adsorption in the shale rocks so as to optimize gas production performance of shale gas reservoirs.     

Numerical and experimental investigation of convective drying in unsaturated porous media with bound water

The heat and mass transfer in an unsaturated wet cylindrical porous bed packed with quartz particles was investigated theoretically for relatively low convective drying rates. Local thermodynamic equilibrium was assumed in the mathematical model describing the multi-phase flow in the unsaturated porous media using the energy and mass conservation equations to describe the heat and mass transfer during the drying. The drying model included convection and capillary transport of the free water, diffusion of bound water, and convection and diffusion of the gas. The numerical results indicated that the drying process could be divided into three periods, the temperature rise period, the constant drying rate period and the decreasing drying rate period. The numerical results agreed well with the experimental data verifying that the mathematical model can evaluate the drying performance of porous media for low drying rates. The effects of drying conditions such as the ambient temperature, the relative humidity, and the velocity of the drying air, on the drying process were evaluated by numerical solution.     

A fractional non-linear creep model for coal considering damage effect and experimental validation

Accurate prediction of coal׳s creep behavior is of great significance to coalbed methane extraction. In this study, taking into account the visco-elastic–plastic characteristics and the damage effect, a fractional non-linear model is proposed to describe the creep behavior of coal. The constitutive and creep equations of the proposed fractional non-linear model are derived via the Boltzmann superposition principle and discrete inverse Laplace transform. Furthermore, uniaxial creep tests under different axial stress conditions were carried out to validate the proposed model. It is found that the present model can describe the experimental data from creep tests with better accuracy than classical models. Particularly, the present model can predict the accelerating creep deformation of coal which classical models fail to reproduce. Finally, the parametric sensitivity analysis is performed to investigate the effects of model parameters on the creep strain. It is verified that the introduction of fractional parameters and damage factor in the present model is essential to accurate prediction of the full creep stage of coal.     

煤系页岩瓦斯吸附?解吸迟滞效应核磁共振谱实验研究

煤系页岩瓦斯主要以吸附态和游离态形式存在, 其解吸过程相对吸附过程具有普遍滞后现象, 因此从微细观角度定量研究其吸附?附解吸迟滞规律对页岩气井后期稳产增产具有重要意义. 在前人研究基础上结合核磁共振谱理论推导出能够准确表征煤系页岩瓦斯吸附?解吸迟滞效应微细观评价模型, 并采用核磁共振谱测试技术, 以双鸭山盆地东保卫煤矿三采区36# 煤层底板煤系页岩为研究对象, 进行煤系页岩瓦斯吸附?解吸迟滞效应核磁共振谱实验, 模拟不同储层原位应力状态煤系页岩瓦斯迟滞效应发生全过程, 进一步对吸附态瓦斯、游离态瓦斯以及微细观方法测定的宏观瓦斯迟滞规律进行定量化研究, 并对其发生机理以及其对深部煤系页岩瓦斯开采影响进行了初步探究. 结果表明: 应力状态下吸附态和游离瓦斯均有滞后效应; 瓦斯宏观迟滞系数与平均有效应力呈幂函数关系, 而瓦斯宏观迟滞效应中由吸附态或游离态瓦斯引起的迟滞系数与平均有效应力关系均可采用二次多项式拟合; 孔裂隙应力损伤和微孔隙瓦斯扩散受限耦合或许是煤系页岩瓦斯吸附?解吸迟滞效应产生根本原因之一.      

Injection of liquid carbon dioxide into a reservoir partially saturated with methane hydrate

A mathematical model is proposed to describe methane–carbon dioxide replacement in gas hydrate by injecting liquid carbon dioxide into a porous medium initially saturated with methane and its hydrate. Self-similar solutions of the axisymmetric problems are constructed that describe the distribution of the main parameters of the reservoir. It is shown that there exist solutions according to which the process can occur both with and without boiling of carbon dioxide. Diagrams of the existence of each type of solution are constructed.     

A Computational Approach to Determine Average Reservoir Pressure in a Coalbed Methane (CBM) Well Flowing Under Dominant Matrix Shrinkage Effect

Desorption of gas from coal matrix alters the pore volume of fracture network. Consequently, cleat porosity and permeability of reservoir changes as pressure depletes. The method of standard pressure analysis calculations produces incorrect results in the case of coalbed methane reservoirs producing under dominant matrix shrinkage effect. The change in cleat porosity and permeability due to shrinkage of coal matrix following gas desorption with pressure depletion invalidates the underlying assumptions made in the derivation of diffusivity equation. Consequently, equations of pseudo-steady state commonly used in conventional reservoirs no longer remain valid as the porosity and permeability values change with pressure depletion. In this paper, effort has been made to describe pseudo-steady-state flow in coalbed methane reservoirs in the form of a new equation that accounts for pressure dependency of cleat porosity and permeability due to shrinkage of coal matrix. The concept of Al-Hussainy et al. (1966) has been extended to define a new pseudo-pressure function which assimilates within itself the pressure dependence of porosity and permeability Palmer and Mansoori (1998). Equation has been used to relate the cleat porosity with pressure. The equation-based computational method suggested in this paper finds its usefulness in estimating average reservoir pressure for any known flowing bottom hole pressure and thus reducing the frequency of future pressure buildup tests. The new equation is also useful in predicting reservoir pressure under the situation when coal matrix shrinks below desorption pressure. The equation used in the computational method has been validated with the help of numerical simulator CMG-GEM.     

The Use of Fick's Law for Modeling Trace Gas Diffusion in Porous Media

Two models for combined gas-phase diffusion and advection in porous media, the advective-diffusive model (ADM) and the dusty-gas model (DGM), are commonly used. The ADM is based on a simple linear addition of advection calculated by Darcy's law and ordinary diffusion using Fick's law with a porosity–tortuosity–gas saturation multiplier to account for the porous medium. The DGM applies the kinetic theory of gases to the gaseous components and the porous media (or dust) to develop an approach for combined transport due to diffusion and advection that includes porous medium effect. The ADM and Fick's law are considered to be generally inferior for gas diffusion in porous media, and the more mechanistic DGM is preferred. Under trace gas diffusion conditions, Fick's law overpredicts the gas diffusion flux compared to the DGM. The difference between the two models increases as the permeability decreases. In addition, the difference decreases as the pressure increases. At atmospheric pressure, the differences are minor (<10%) for permeabilities down to about 10^–13 m². However, for lower permeabilities, the differences are significant and can approach two orders of magnitude at a permeability of 10^–18 m². In contrast, at a pressure of 100 atm, the maximum difference for a permeability of 10^–18 m² is only about a factor of 2. A molecule–wall tortuosity coefficient based on the DGM is proposed for trace gas diffusion using Fick's law. Comparison of the Knudsen diffusion fluxes has also been conducted. For trace gases heavier than the bulk gas, the ADM mass flux is higher than the DGM. Conversely, for trace gases lighter than the bulk gas, the ADM mass flux is lower than the DGM. Similar to the ordinary diffusion variation, the differences increase as the permeability decreases, and get smaller as the pressure increases. At atmospheric pressure, the differences are small for higher permeabilities (>10^–13 m²) but may increase to about 2.7 for He at lower permeabilities of about 10^–18 m². A modified Klinkenberg factor is suggested to account for differences in the models.     

Laboratory and Simulation Investigation of Enhanced Coalbed Methane Recovery by Gas Injection

Methane/carbon dioxide/nitrogen flow and adsorption behavior within coal is investigated simultaneously from a laboratory and simulation perspective. The samples are from a coalbed in the Powder River Basin, WY. They are characterized by methane, carbon dioxide, and nitrogen sorption isotherms, as well as porosity and permeability measurements. This coal adsorbs almost three times as much carbon dioxide as methane and exhibits significant hysteresis among pure-component adsorption and desorption isotherms that are characterized as Langmuir-like. Displacement experiments were conducted with pure nitrogen, pure carbon dioxide, and various mixtures. Recovery factors are greater than 94% of the OGIP. Most interestingly, the coal exhibited ability to separate nitrogen from carbon dioxide due to the preferential strong adsorption of carbon dioxide. Injection of a mixture rich in carbon dioxide gives slower initial recovery, increases breakthrough time, and decreases the volume of gas needed to sweep out the coalbed. Injection gas rich in nitrogen leads to relatively fast recovery of methane, earlier breakthrough, and a significant fraction of nitrogen in the produced gas at short times. A one-dimensional, two-phase (gas and solid) model was employed to rationalize and explain the experimental data and trends. Reproduction of binary behavior is characterized as excellent, whereas the dynamics of ternary systems are predicted with less accuracy. For these coals, the most sensitive simulation input were the multicomponent adsorption–desorption isotherms, including scanning loops. Additionally, the coal exhibited a two-porosity matrix that was incorporated numerically.     

TRANSIENT WELL TEST MODEL FOR THE WELL WITH GAS AND WATER DISTRIBUTED IN COALBED

煤层气是一种高效清洁的非常规天然气资源,其开采过程是一个排水降压采气的过程. 由于煤层气主要是以吸附态的形式存在于煤层中,当煤层压力降低到临界解吸压力以下时煤层气从煤层中解吸出来并与水一起采出,因此煤层中流体是气水两相分布的. 本文根据煤层气藏排采过程中的解吸特征,通过考虑气水两相分布的渗透率关系,提出了一种与解吸区域大小相关的煤层气井不稳定试井模型. 该模型较好地描述了煤层气排采过程中煤层内气水的流动状态,采用分区模式对气水两相进行描述. 通过有限体积方法求解了所建立的试井模型,计算得到了煤层气井气水两相分布不稳定试井理论曲线,分析了煤层气解吸系数、解吸复合半径、气水饱和度分布等对试井理论曲线的影响.     

Monte Carlo Simulations of Observation Time-Dependent Self-Diffusion in Porous Media Models

Observation time-dependent self-diffusion coefficients can be used to obtain microstructural information of porous media. This paper presents two different kinds of Monte Carlo simulations of the self diffusion process of fluids like water in porous systems, a lattice-free method and a lattice-based method. The results for simple porous media model geometries agree well with each other and with published analytical as well as semi-analytical equations. The use of these equations, which are important for the interpretation of Pulsed Field Gradient-Nuclear Magnetic Resonance (PFG-NMR) time-dependent diffusion data with respect to properties of porous media, is discussed.     

Riemann problem for one-dimensional binary gas enhanced coalbed methane process

With an extended Langmuir isotherm, a Riemann problem for one-dimensional binary gas enhanced coalbed methane (ECBM) process is investigated. A new analytical solution to the Riemann problem, based on the method of characteristics, is developed by introducing a gas selectivity ratio representing the gas relative sorption affinity. The influence of gas selectivity ratio on the enhanced coalbed methane processes is identified.     

基于变边界分段模型的页岩损失气量和解吸气量评价方法

储层含气量的准确评估是目前制约非常规天然气高效开发的重要因素, 直接法采用损失气估算模型结合解吸曲线估算储层含气量, 但现有损失气估算模型均基于煤层气的常压边界条件和球形颗粒假设, 如美国矿业局提出的USBM方法, 为埋藏深、柱状岩心的页岩气藏含气量的估算带来较大误差. 本文基于扩散理论, 采用时变压力边界条件和柱坐标系求解一维扩散方程获得解析解, 从而提出了新的损失气估算模型, 即变边界分段模型, 该模型能够反演出提钻和解吸两个阶段气体逸散的不同特征. 结果表明: 在提钻阶段, 环境压力不断降低, 岩心内外压差增大, 气体逸散速率加快, 从而是下凸函数; 在解吸阶段, 环境压力恒定, 岩心内压力随气体逸散而下降, 内外压差减小, 气体逸散速率减慢, 因而是上凸函数. 进一步为证明模型的准确性, 基于相似原理在实验室搭建了损失气?解吸气复原实验系统, 采用圆柱状页岩岩心复现提钻过程和解吸过程的气体逸散情况, 得到的实验结果与变边界分段模型吻合, 而已有的USBM方法不能进行准确预测, 验证了本文提出的变边界分段模型正确性. 根据川南地区Y151井现场测试数据, 采用变边界分段模型进行拟合预测, 所得结果良好, 验证了变边界分段模型的适用性.      

Analytical Solutions to Gas Flow Problems in Low Permeability Porous Media

In most of conventional porous media the flow of gas is basically controlled by the permeability and the contribution of gas flow due to gas diffusion is ignored. The diffusion effect may have significant impact on gas flow behavior, especially in low permeability porous media. In this study, a dual mechanism based on Darcy flow as well as diffusion is presented for the gas flow in homogeneous porous media. Then, a novel form of pseudo pressure function was defined. This study presents a set of novel analytical solutions developed for analyzing steady-state and transient gas flow through porous media including effective diffusion. The analytical solutions are obtained using the real gas pseudo pressure function that incorporates the effective diffusion. Furthermore, the conventional assumption was used for linearizing the gas flow equation. As application examples, the new analytical solutions have been used to design new laboratory and field testing method to determine the porous media parameters. The proposed laboratory analysis method is also used to analyze data from steady-state flow tests of three core plugs. Then, permeability (k) and effective diffusion coefficient (D _e) was determined; however, the new method allows one to analyze data from both transient and steady-state tests in various flow geometries.     

Conditional model for sorption in porous media with fractal properties

In this study, the conditional moment closure approach, which is proven to be very useful for modelling of reactions in turbulent flows, is extended to characterise adsorbing, desorbing or reacting flows in porous media. A complete specification of the porous distance conditioned moment closure model, which is formulated in terms of single-conditioned expectations, is presented. The closure of the model equations is obtained assuming the diffusion approximation for fluxes of the reactive species. The model simulates complex multi-cascade processes of convective and diffusive transport of species between pores in a continuous and consistent manner and is a generalisation of dual (or triple) porosity concept. The model addresses the major difficulty of describing transport, entrapment and sorption processes in porous media with fractal properties, where distant transport occurs in the largest pores or fractures, while the adsorbing or desorbing surface is mainly allocated in small pores. The model is able to simulate various regimes of methane replacement by CO₂ in a coal sample, which makes it useful for optimising the design and parameters of enhanced coal bed methane recovery operations. It is demonstrated that the power-low decrease in downstream methane concentration, which has been observed experimentally, can be accurately reproduced by the model.     

Transient gas flow around boreholes

We present numerical solutions and analytical approximate solutions to problems of gas flow in porous media arising in the modelling of outbursts in coal mines and the efficient recovery of methane from coal seams.     

Time-Dependent Diffusion and Surface-Enhanced Relaxation in Stochastic Replicas of Porous Rock

Understanding the connection between pore structure and NMR behavior of fluid-saturated porous rock is essential in interpreting the results of NMR measurements in the field or laboratory and in establishing correlations between NMR parameters and petrophysical properties. In this paper we use random-walk simulation to study NMR relaxation and time-dependent diffusion in 3D stochastic replicas of real porous media. The microstructures are generated using low-order statistical information (porosity, void–void autocorrelation function) obtained from 2D images of thepore space. Pore size distributions obtained directly by a 3D pore space partitioning method and indirectly by inversion of NMR relaxation data are compared for the first time. For surface relaxation conditions typical of reservoir rock, diffusional coupling between pores of different size is observed to cause considerable deviations between the two distributions. Nevertheless, the pore space correlation length and the size of surface asperity are mirrored in the NMR relaxation data for the media studied. This observation is used to explain the performance of NMR-based permeability correlations. Additionally, the early time behavior of the time-dependent diffusion coefficient is shown to reflect the average pore surface-to-volume ratio. For sufficiently high values of the self-diffusion coefficient, the tortuosity of the pore space is also recovered from the long-time behavior of the time-dependent diffusion coefficient, even in the presence of surface relaxation. Finally, the simulations expose key limitations of the stochastic reconstruction method, and allow suggestions for future development to be made.     

Simulation of Pressure Transient Behavior for Asymmetrically Finite-Conductivity Fractured Wells in Coal Reservoirs

Based on Fick’s law in matrix and Darcy flow in cleats and hydraulic fractures, a new semi-analytical model considering the effects of boundary conditions was presented to investigate pressure transient behavior for asymmetrically fractured wells in coal reservoirs. The new model is more accurate than previous model proposed by Anbarci and Ertekin, SPE annual technical conference and exhibition, New Orleans, 27–30 Sept 1998 because new model is expressed in the form of integral expressions and is validated well through numerical simulation. (1) In this paper, the effects of parameters including fracture conductivity, coal reservoir porosity and permeability, fracture asymmetry factor, sorption time constant, fracture half-length, and coalbed methane (CBM) viscosity on bottomhole pressure behavior were discussed in detail. (2) Type curves were established to analyze both transient pressure behavior and flow characteristics in CBM reservoir. According to the characteristics of dimensionless pseudo pressure derivative curves, the process of the flow for fractured CBM wells was divided into six sub-stages. (3) This paper showed the comparison of transient steady state and pseudo steady state models. (4) The effects of parameters including transfer coefficient, wellbore storage coefficient, storage coefficient of cleat, fracture conductivity, fracture asymmetry factor, and rate coefficient on the shape of type curves were also discussed in detail, indicating that it is necessary to keep a bigger fracture conductivity and fracture symmetry for enhancing well production and reducing pressure depletion during the hydraulic fracturing design.     

Modeling of Coal Fine Migration During CBM Production in High-Rank Coal

During CBM (coalbed methane) production, the interaction of coal fracture surface with water flow commonly generates and starts coal fine flow. Part of flowing coal fines deposit in coal fracture system due to water production reduction and methane production increase. The fine sedimentation results in the reduction of coal permeability and well productivity. Despite the increasing awareness of the importance of fine migration, limited research has been carried out on the flow model of coal fine coupled with water and gas. In this paper, a flow model of coal fine is established coupled with water and gas flow, taking coal fine generation, migration and sedimentation process into consideration. Then, case simulations are conducted to illustrate effects of water production schedule, permeability performance and gas content on production performance in flow model. The simulation results indicate that methane rate with the lowest initial water rate is observed to have the highest production in late production period. This is mainly due to the reason that the low water flow cannot generate and start the flow of coal fine. Further, the case with high initial water production has faster gas and water flow rate, thus higher coal fine generation rates, which can improve well productivity at earlier production period. As water production declines quickly, both permeability and production performance decrease, which leads to the loss of well productivity. Meanwhile, higher gas content will lead to a faster water production decline at late production period. This indicates that a portion of coal fines plugged in the fracture as water production deceases and the CBM reservoir with high gas content should not adopt a high initial water production schedule.