non-Darcy Seepage Research Articles

Nonlinear evolution characteristics and seepage mechanical model of fluids in broken rock mass based on the bifurcation theory

With the deep extension of coal mining in China, fault water inrush has become one of the major disasters threatening the safety production of coal mine. Based on the control equations of steady state and non-Darcy seepage in fractured rock mass, the multi-parameter nonlinear dynamic seepage equations of fractured rock mass are established in this paper. Based on the nonlinear dynamics theory, the function of the state variable in the system is derived, and the influence of the gradual change of non-Darcy flow factors on the structural stability of seepage system is studied. The research achievements show that there are three branches in the equilibrium state of the seepage system. Specifically, the stability of the equilibrium state changes abruptly near the limit parameter. The seepage dynamic system of fractured rock mass has the delayed bifurcation, and the coal mine disaster such as fault water inrush occurs easily at the bifurcation point. The research results are of great significance to enrich the theory of fault water inrush in coal mine, and to reveal the disastrous mechanism of fault water inrush and guide its prevention and control technology in coal mine, which can provide the theoretical reference for predicting the water seepage stability in fractured rock mass.

Coupling model of gas-water two-phase productivity calculation for fractured horizontal wells in tight gas reservoirs

Tight gas reservoirs are exploited mostly by using horizontal wells in combination with multi-stage hydraulic fracturing. Up to now, there have been plenty of studies conducted to establish a series of productivity models that consider various factors. Despite this, insufficient consideration has been given to the ramifications resulting from secondary fractures, permeability anisotropy, wellbore pressure drop, and the actual distribution of fractures. To accurately evaluate productivity, the superposition principle of potential and the conformal transformation theory are adopted to define the two-phase generalized pseudo-pressure, with consideration given to the characteristics of non-Darcy seepage in tight gas reservoirs and the effects of secondary fractures and permeability anisotropy on productivity. On this basis, the homogeneous flow model of the horizontal wellbore is introduced to establish the coupling model of gas-water two-phase productivity calculation for fractured horizontal wells. The results obtained from the empirical calculation and analysis of various factors affecting productivity are as follows. Firstly, the relative error is 5.35% between the theoretical open flow capability obtained from the proposed model and the open flow capability determined through the deliverability test, which attests to the proposed model's accuracy. Secondly, in response to an increase in gas slippage factor, fracture conductivity, the number of fractures, and the degree of permeability anisotropy, there is an improvement in the productivity of the gas well. However, with a rise in the water-gas volume ratio, threshold pressure gradient, reservoir stress sensitivity index, and the angle formed between the horizontal wellbore and the main permeability, the productivity declines. Lastly, the pressure drop along the horizontal wellbore is relatively insignificant, which has no substantial impact on fracture production. However, when the effect of secondary fractures is ignored, the predicted productivity is relatively low. This study offers theoretical direction for the application of productivity prediction and fracturing optimization design.

Seepage stability analysis of a deep-buried tunnel in fractured rocks based on a non-Darcy hydro-mechanical coupled method

Non-Darcy seepage commonly occurs in fractured rocks with an adequate groundwater system around deep-buried tunnels. This study develops a numerical method for capturing the non-linear hydro-mechanical coupled processes in a rock mass system comprised of a tunnel lining, grouting circle, and surrounding rocks with discrete fracture networks. The proposed model is validated by comparing it with analytical solutions and field data obtained from a diversion tunnel of the Jinping II Hydropower Station on the Yalong River. The investigation of the non-Darcy flow regions around the diversion tunnel indicates that fractures with relatively higher conductivities dominate the non-Darcy effect, and the highest Forchheimer number reaches 0.5 during the period of torrential rain. The seepage stability of the diversion tunnel lining under three different cases (failure of the seepage control capacity of the grouting circle, potential karst pipeline penetrating the diversion tunnel, and torrential rain attacking the area) is further studied. It is found that the pore pressure at the external edge of the lining increases with the enhancement of the grouting circle permeability, and reaches 2,472.4 kPa when the karst pipeline is connected. Additionally, it is observed that the pore pressures obtained based on the Darcian and non-Darcian assumptions differ by up to 468.8 kPa in Case I. However, the non-Darcy effect gradually decreases as the lining permeability decreases. The results of this study provide practically useful suggestions for the design and operation of diversion tunnels subjected to high water pressure.

A numerical pressure transient model of fractured well with complex fractures of tight gas reservoirs considering gas-water two phase by EDFM

Tight gas reservoirs considering gas-water two phase is common and show obvious permeability stress sensitivity and non-Darcy seepage characteristic. However, conventional analytical and semi-analytical solving method cannot simulate wellbore pressure response of tight gas reservoirs considering gas-water two phase. Numerical model can overcome the disadvantage of analytical and semi-analytical method and achieve wellbore pressure response simulation of fractured well with complex fracture. Therefore, a gas-water two phase mathematical model of the fractured well is established and solved by embedded discrete-fracture method (EDFM) and finite volume method. Transmissibility corrections is improved on the basis of the framework of Matlab Reservoir Simulation Toolbox (MRST). Result shows that the seven flow regimes can be found according to the pressure drop derivative curves characteristic. When permeability stress sensitivity and non-Darcy is considered, pressure and derivative curve of middle-late stage shows obvious ‘upwarp’ and non-Darcy make reservoirs pressure uneven distribution. Induced fractures parameters have obvious influence on pressure and derivative curves during early and middle flow regime. In addition, fracture intersection and neighbor well production style has obvious influence on pressure curve.

Three-dimensional analytical solutions of non-Darcy seepage in front of a grouted subsea tunnel face

In this paper, the impact of non-Darcy seepage on the steady seepage field in front of a grouted subsea tunnel face is analysed. Analytical solutions are proposed by combining the three-dimensional (3D) seepage continuity equation, which is expressed in spherical coordinates, with the Hansbo seepage model. These solutions, which are also suitable for calculating the hydraulic head distribution in front of the tunnel face without a grouted zone, are verified by a series of numerical simulations. Comparisons show that this study enabled the augmentation of the classical analytical model. Finally, the influence of relevant parameters, including the nonlinear parameters in the Hansbo seepage model, the relative permeability (i.e., kr/kg), and the grouted zone thickness, are investigated. The results show that the exponential parameter m and critical hydraulic gradient i1 in the Hansbo seepage model have a more significant impact on the water inflow than the water pressure on the grouted zone. Based on the non-Darcy seepage model, the water inflow is lower and the water pressure on the grouted zone is larger than that obtained by Darcy's model.

Mass-loss effects on the non-Darcy seepage characteristics of broken rock mass with different clay contents

A seepage testing system was designed and a series of seepage experiments on broken rock was conducted using different original porosity conditions and clay contents. The mass-loss process of the broken rock and the change in water flow velocity were investigated. After the mass-loss test, the non-Darcy seepage characteristics of the broken rock were tested through a step-by-step pressure-reduced seepage test. The experimental results show that the mass-loss and water velocity evolution during the water inrush could be divided into four stages: acceleration, stable with slight fluctuations, reacceleration, and stable. The lost-mass and change in water velocity were positively correlated with the clay contents and negatively correlated with the original porosity. By introducing the evolution equation of the Kozeny-Carman equation and the liquid limit index which characterises the effective particle size, the prediction model of the permeability coefficient was built. Six prediction models of the non-Darcy coefficient were verified against the testing results. The prediction model of the critical flow velocity from a Darcy flow to a non-Darcy flow using the Forchheimer number was also established. The results could provide an important reference for understanding water inrush mechanisms, adopting effective control measures for water inrush events, and calculating the water influx of tunnels.

One-Dimensional Nonlinear Consolidation for Soft Clays with Continuous Drainage Boundary Considering Non-Darcy Flow

Adopting the non-Darcy flow presented by Hansbo and considering the nonlinear compression and permeability characteristics of soils, the one-dimensional nonlinear consolidation problem of soft clays is investigated by means of a continuous drainage boundary. The numerical solutions of average consolidation degrees defined by settlement and excess pore water pressure are derived by using the finite difference method, and the correctness of these solutions is verified by comparing them with existing analytical and numerical solutions. Based on the proposed solutions, a parametric study is conducted to study the influence of interface parameter, non-Darcy flow parameter and soil nonlinearity on the consolidation behavior of soft clays. The results show that the solutions based on the continuous drainage boundary can be degenerated into the solutions based on the Terzaghi drainage boundary if the interface parameter is taken as a reasonable value. The soil consolidation behavior considering both non-Darcy seepage and nonlinear characteristics of soil is very complex.

Progress of Seepage Law and Development Technologies for Shale Condensate Gas Reservoirs

With the continuous development of conventional oil and gas resources, the strategic transformation of energy structure is imminent. Shale condensate gas reservoir has high development value because of its abundant reserves. However, due to the multi-scale flow of shale gas, adsorption and desorption, the strong stress sensitivity of matrix and fractures, the abnormal condensation phase transition mechanism, high-speed non-Darcy seepage in artificial fractures, and heterogeneity of reservoir and multiphase flows, the multi-scale nonlinear seepage mechanisms are extremely complicated in shale condensate gas reservoirs. A certain theoretical basis for the engineering development can be provided by mastering the percolation law of shale condensate gas reservoirs, such as improvement of productivity prediction and recovery efficiency. The productivity evaluation method of shale condensate gas wells based on empirical method is simple in calculation but poor in reliability. The characteristic curve analysis method has strong reliability but a great dependence on the selection of the seepage model. The artificial intelligence method can deal with complex data and has a high prediction accuracy. Establishing an efficient shale condensate gas reservoir development simulation technology and accurately predicting the production performance of production wells will help to rationally formulate a stable and high-yield mining scheme, so as to obtain better economic benefits.

A Study of the Non-Linear Seepage Problem in Porous Media via the Homotopy Analysis Method

A non-Darcy flow with moving boundary conditions in a low-permeability reservoir was solved using the homotopy analysis method (HAM), which was converted into a fixed-boundary mathematical model via similarity transformation. Approximate analytical solutions based on the HAM are guaranteed to be more accurate than exact analytical solutions, with relative errors between 0.0089% and 2.64%. When λ = 0, the pressure drop of the Darcy seepage model could be instantaneously transmitted to infinity. When λ > 0, the pressure drop curve of the non-Darcy seepage model exhibited the characteristics of tight support, which was clearly different from the Darcy seepage model’s formation pressure distribution curve. According to the results of the HAM, a moving boundary is more influenced by threshold pressure gradients with a longer time. When the threshold pressure gradients were smaller, the moving boundaries move more quickly and are more sensitive to external influences. One-dimensional, low-permeability porous media with a non-Darcy flow with moving boundary conditions can be reduced to a Darcy seepage model if the threshold pressure gradient values tend to zero.

Integrated flow model for evaluating maximum fracture spacing in horizontal wells

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

Analytical solutions on non-Darcy seepage of grouted and lined subsea tunnels under dynamic water levels

For the grouted and lined subsea tunnels, the fluid in the low permeability media follows non-Darcy's law rather than Darcy's law. To address this, we provide analytical solutions for the grouted and lined subsea tunnel with dynamic water levels. Our method uses complex variable functions and Hansbo non-Darcy seepage model. The analytical solutions divide into three cases according to the critical water inflow and can degenerate into the traditional method following Darcy's law. The numerical simulation, experimental data, in-situ data, and other analytical solutions are adopted to verify our proposed method. The calculated results subjected to non-Darcy seepage are close to the in-situ data compared with other analytical solutions. When degenerating into the traditional method, the results agree well with those of previous methods following Darcy's law. It reveals that our method is suitable and universal. The parametric analyses of the non-linear flow index m, the critical hydraulic gradient il, the dynamic water level ξci, the thickness of lining zone tl, and the relative permeability coefficient ratio of the grouting zone and lining zone kg/kl are carried out. The results show that non-linear seepage parameters and dynamic water levels significantly affect the water inflow and water head. The denser grouting material can apportion more water heads and make the lining apportion fewer water heads to protect the lining from fluctuation influencing of dynamic water levels.

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

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

Mechanical Constitutive and Seepage Theoretical Model of Water Storage Media Based on Fractional Derivative

Using an abandoned underground goaf in coal mines as water reservoirs has been successfully applied for protecting mine water resources in western China. The water storage media are composed of broken rock masses and the voids between the rock masses. It is critical for reservoir capacity calculation that the deformation characteristics and seepage evolution of the water storage media under triaxial stress are theoretically described. In this study, broken rock masses and the space among the rock masses are simplified as two springs in a series. The mechanical behavior of the broken rock mass is described by Hooke’s law of linear elasticity, and the deformation characteristics of the space among the rock masses are represented by a nonlinear elastic constitutive model. The nonlinear stress-strain constitutive model of the water storage media is established by combining Hooke’s law and the fractional derivative stress-strain model. Similarly, a non-Darcy seepage model of the water storage media is obtained. The nonlinear stress-strain model is verified by mechanical experiments, physical simulation tests, and field measured data, and parameter sensitivity analysis is performed. The non-Darcy seepage equation is fitted and analyzed by using the seepage experimental data of the broken rock mass under triaxial compression conditions. The fractional non-Darcy model rather than the Forchheimer equation can more accurately describe the nonlinear seepage process in water storage media.

Coupling Mechanism of Coal Body Stress–Seepage around a Water Injection Borehole

In China, coal seam water injection technology is widely used in the prevention and control of mine gas and dust disasters. The evolution law of stress–seepage field in the range of hydraulic influence in the process of coal seam water injection is the most basic theoretical problem in coal seam water injection technology. Therefore, this paper first uses the radial seepage experimental system to test the permeability of raw coal samples, and then numerically inverts the stress–seepage evolution law of the coal body around the water injection hole. The results show that the coal seam permeability decreases exponentially with the increase in the experimental overburden pressure. The dynamic water intrusion causes the pressure relief area and the stress concentration area to transfer to the deep part of the coal body, and abnormal stress occurs near the water injection hole wall. The composite stress area formed by the superposition of stress and pore water pressure reduces the permeability of the hole wall. During the radial development of water injection, the pore water pressure dropped sharply and eventually tended to zero, while Darcy’s velocity decreased significantly (within the pressure relief zone, it rapidly decreased from 7.26 × 10−3 m s−1 to 2.5 × 10−3 m s−1 (by 65.56%)). Near the initiation point, the maximum step-down height of Darcy’s velocity decreased from 6.73 × 10−3 m s−1 to 5.27 × 10−3 m s−1 (by 22.73%). This can make it clear that the seepage velocity presents a non-Darcy seepage phenomenon under the influence of pore pressure.

Calibration of Water Inrush Channel and Numerical Simulation of Water Inrush Process in Coal Mine Roof

The surrounding rock of a coal mine roof fractures with the excavation of the underground working face in coal mining. These mining-induced fractures are connected and extended upward to form water inrush channels. A water inrush accident may occur when there is a sufficiently large water source. To elucidate the formation mechanisms of the roof water inrush channel and the characteristics of water inrush in goafs, we performed a case study of No.18401 Panel of Xiqu Coal Mine in China and determined whether the roof water inrush channel is connected by theoretical calculation and microseismic monitoring. The modified mechanical parameters of rock masses were embedded into the numerical model based on microseismic data. Microseismic monitoring and numerical simulation were organically combined to analyze the connection process of the water inrush channel, after which the roof water inrush channel is calibrated on the No.18401 Panel. We established a non-Darcy flow model for water inrush in the water-conducting fractured zone of mines by coupling the Darcy, Forchheimer, and Navier–Stokes equations. Finite element language and its compiler (FELAC) was used to study the water inrush mechanism of non-Darcy seepage. The results show that the pressure, velocity, and porosity in the water-conducting fractured zone are non-uniform in water inrush occurrence and development, and the mixed fluid mainly passes through the “dominant channel.” The development of water inrush is accompanied by the release of the hydrostatic pressure in the aquifer, the sudden increase of the velocity at the water inrush position, and the increase in fluid concentration. Hence, the underground water inrush can be predicted and prevented by monitoring the aforementioned indicators. This research is of great significance for the calibration of the water inrush channel of the roof and the prediction of water inrush disasters.

A unified model including non-Darcy flow and viscoelastic mechanisms in tight rocks

Tight porous media have extremely low porosities and permeabilities, which can induce low-fluid mobility and a significant non-Darcy flow phenomenon. We build a unified mathematical model considering the non-Darcy seepage and the viscoelasticity mechanism, which includes two new parameters related to the non-Darcy flow mechanism. We call this the non-Darcy viscoelastic (NDV) model. Numerical experiments reveal that the two parameters have significant effects on the dispersion and attenuation of the P and S waves. We find that the dispersion and attenuation of the waves depend on the initial amplitude. A comparison is conducted between experimental data for tight porous media in the low (seismic) frequency band and the numerical results of the NDV model. The comparison demonstrates that the NDV model can predict the velocity dispersion in tight oil and gas reservoirs more accurately.

Production performance of the low-permeability reservoirs: Impact of contamination at the wellbore vicinity

Hydrocarbon reservoirs with low porosity and permeability are usually vulnerable to be contaminated owing to the drilling and completion fluids-induced formation damage, which significantly impedes the efficient production of hydrocarbon. However, most current productivity prediction models introduce the concept of skin factor to capture the near-wellbore contamination, which only considers the change of permeability before and after contamination, whereas neglecting the change of threshold pressure gradient (TPG) affected by the permeability reduction. Also, few works have been done to focus on the influences of contamination, TPG, sensitivity of rock porosity to internal stress, and fluid compressibility simultaneously in the actual production process of the low-permeability reservoir. Therefore, to bridge the knowledge gap, this study presents a new comprehensive productivity model encompassing those multiple influencing factors. Firstly, the seepage area is divided into contaminated zone and uncontaminated zone. Thereafter, the proposed model is established based on non-Darcy seepage theory and continuity principle, comprehensively taking account of the effects of different permeability and TPG between contaminated zone and uncontaminated zone, stress sensitivity based on primary stress in porous media, and the fluid compressibility. Subsequently, the newly proposed model is proved to be reliable through the consistency between the simplified model and the existing models in the literatures. Finally, the sensitivity analysis is employed to investigate the effects of key parameters. Main results show that a) the decreased permeability affected by contamination would result in increased TPG, further lowering the well productivity. b) Contamination, TPG and stress sensitivity will cause additional pressure drop and production loss, which will overestimate the production rate by neglecting them. While the fluid compressibility could boost the well productivity, which will underestimate the production rate by ignoring it. c) The order of productivity influencing factors is contamination > TPG > the fluid compressibility > stress sensitivity. d) Contamination and the TPG are significant influencing factors at stage of high bottomhole pressure. Moreover, the influence degree of contamination, the fluid compressibility and stress sensitivity gradually increase, while the influence degree of the TPG greatly decreases with lower pressure.

Calculation Method of Productivity for Shale Oil in Volumetric Fractured Horizontal Wells.

In order to evaluate quantitatively the productivity of shale oil in fractured horizontal wells and in view of the influence of non-Darcy seepage and pressure sensitivity on shale oil productivity, the methods of conformal transformation and superposition theory of complex potential are used to deduce the productivity formula in volumetric fractured horizontal wells with start-up pressure gradient and pressure sensitivity and to calculate quantitatively the productivity of each fracture. In addition, the influence factors on productivity, such as the fracture length, number of fractures, pressure sensitivity coefficient, starting pressure gradient, and other factors, have been analyzed as well. The study results indicate that the productivity of horizontal wells decreases with the increase of the start-up pressure gradient and pressure sensitivity coefficient. With the increase of the half-length and number of fractures, the productivity of fractured horizontal wells will increase, but the amplitude of increase decreases gradually. This method can be used for productivity calculation and optimization design of the fracturing parameter in volumetric fractured horizontal wells.

Effects of seepage behaviors on coal spontaneous combustion in longwall gobs: an investigation between Darcy and non-Darcy seepage

Seepage behaviors of air leakage directly affect the coal spontaneous combustion (CSC) in longwall gobs, which is usually regarded as Darcy flow, but in fact, non-linear seepage should better characterize the properties of air flow in the gobs. To address this issue, steady-state percolation tests were conducted on broken rocks with five particle sizes, and the non-Darcy permeability parameters versus porosity and particle size were obtained by binary fitting the experimental data, and then applied to an actual longwall gob. The multi-physics coupling model of CSC in longwall gobs was modified accordingly, which was then validated by comparing with on-site observation data. Numerical solving procedures based on Darcy seepage and non-Darcy seepage were developed separately to quantify the effects of these two types of seepage on the CSC. The results show that (i) the seepage tests confirm that the seepage behavior in the broken rocks is more consistent with the non-Darcy seepage; (ii) the distributions of oxygen concentration and solid temperature under the two seepage flows are relatively similar, but the high-temperature zone in the gob under non-Darcy seepage is larger and hotter; (iii) the non-Darcy permeability coefficient has a positive effect on the coal temperature rise in the gob, while the non-Darcy flow factor has almost no effect, so taking plugging measures at the working face can effectively prevent the spontaneous combustion in gobs. This work contributes to the understanding of the air seepage behavior in longwall gobs and its impact on the CSC.

Fracture evolution and nonlinear seepage characteristic of gas-bearing coal using X-ray computed tomography and the lattice Boltzmann method

Fracture is the primary channel of gas seepage in coal seams and controls the drainage efficiency of coal bed methane (CBM). However, the seepage characteristics and dynamic evolution law in the fractures of gas-bearing coal under external loads are yet to be clearly revealed. In this study, an industrial computer tomography (CT) scanner equipped with a triaxial loading system was used to conduct gas-seepage and CT scanning experiments under triaxial compression conditions. The results showed that the fracture volume and permeability decreased first and then increased during the complete stress–strain process of gas-bearing coal, displaying an approximate U-shaped variation trend involving a decrease stage, an increase stage, and a jump-increase stage. The lattice Boltzmann method (LBM) was applied to make the seepage processes of gas-bearing coal reappear, and a modified nonlinear permeability model was developed to represent non-Darcy seepage inside fractured coal. The LBM simulation results mirrored the dynamic evolution of the gas seepage field and gas permeability controlled by fracture propagation. The proposed modified nonlinear permeability model effectively reflected the nonlinear variation behaviour of gas permeability and was superior to the traditional Darcy's model in describing nonlinear seepage of gas-bearing coal.