Seepage Equation Research Articles

Coupling effects of porosity and particle size on seepage properties of broken sandstone based on fractional flow equation

Studying the seepage properties of broken rock is important for understanding the behavior of engineering projects and preventing seepage disasters from occurring. Therefore, a test system was developed to test the seepage properties of broken rock under different porosities and particle sizes. A non-linear seepage equation of broken rock was developed based on the Forchheimer equation and the theories of fraction calculus. The influence of the coupling mechanism of the porosity and particle size on the seepage properties of broken rock was analyzed. The results show that the non-linear seepage equation can describe the non-linear seepage properties of broken rock well. The relations between the permeability and the porosity and particles size can all be represented through an exponential function. It is thought that watercourses are developed in broken rock with high porosity and large particle size, which shows a stronger hydraulic conductivity capability. However, the inertial potential energy of a non-Darcy flow is relatively small.

A new deliverability evaluation method of gas condensate wells in gas–liquid two-phase state

Gas well deliverability evaluation and analysis are challenging due to the frequent abnormalities of deliverability test data of gas condensate wells caused by seepage of oil and gas phases in the reservoirs. To this end, based upon the pseudo-single-phase seepage equation and the oil–gas two-phase seepage equation, a new deliverability evaluation method was established which is applicable to the following two cases when the flow of a gas well reaches the quasi-steady stage, i.e., the pseudo-single-phase stable point deliverability evaluation for the case when the formation pressure is above the dew pressure; the gas–liquid two-phase stable point deliverability evaluation for the case when the formation pressure is below the dew pressure. Using this established deliverability evaluation method, based on the basic parameters of the Yaha gas field, Tarim Basin, the IPR curves were first obtained of gas wells do not get this at the same production gas–oil ratio and at the formation pressure above and below the dew point pressure; then, according to the four condensate gas fields, such as Yaha, Tazhong I, Qianmiqiao and Dina 2, the absolute open flow (AOF) potentials of condensate gas wells under different gas–oil production ratios were calculated. Finally, through statistical analysis of the calculation results from typical wells, the following findings were obtained. This new deliverability evaluation method under the two states of condensate gas wells with quasi-single-phase and gas–liquid two-phase stable points can be used to avoid cases due to the oil–gas flow in a condensate gas well which has remained unresolved by the classical deliverability evaluation methods. Also, with the increase of gas–oil ratios in gas condensate wells, a variable discrepancy is gradually reduced in AOF potentials calculated respectively by the quasi-single-phase and gas–liquid two-phase stable point deliverability evaluation equations. For the condensate gas wells with high condensate content and low condensate gas production rates, the AOF potentials calculated by the gas–liquid two-phase stable point deliverability equation is more appropriate and reliable compared with that obtained by classical methods.

A novel analytical well test model for fractured vuggy carbonate reservoirs considering the coupling between oil flow and wave propagation

Fractured vuggy carbonate reservoirs consist of matrix, fractures, and vugs. The existence of vugs in fractured vuggy carbonate reservoirs has long been observed. The volume of a vug is an important parameter owing to the large contribution of vugs to oil reserves. However, there are no reliable well test methods to estimate the volume of a vug yet. In this study, a new analytical well test model considering the coupling between oil flow and wave propagation is proposed. First, the mathematical model is established. Then, combined with the seepage equations in the outer formation, the analytical solutions of transient pressure in the wellbore are obtained through Laplace transformation and inversion. Second, based on the above analytical solutions, the standard log-log type curves are plotted to recognize the flow characteristics. It was found that, the flow in the wellbore and large vug can be divided into six distinct flow regimes. The volume of the large vug can be obtained by pressure transient analysis. Considering the coupling between oil flow and wave propagation, there will be one more V-shaped segment than typical triple-continuum pressure derivative curve. Third, a comparison with the triple-continuum model is performed to verify the validity and reliability of the proposed model in this study as well as a reservoir study. Finally, sensitivity analysis of some key parameters on the pressure derivative are carried out. Through the sensitivity analysis of different parameters for the well test type curves, it is found that the change in these well test curves is in accordance with the findings of the theoretical analysis. The curve behaviors are influenced by the friction coefficient, fluctuation coefficient, damping coefficient, storage ratios, and inter-porosity flow coefficients. The depth of the first V-shape segment is mainly affected by the friction, fluctuation, and damping coefficients. The depths and locations of the second and third V-shape segments are mainly influenced by the storage ratios of the fracture and vug, and inter-porosity flow coefficients within the matrix, fracture, and vug.

Transient modeling of borehole breakouts: A coupled thermo-hydro-mechanical approach

Wellbore breakout is critical in drilling operations, which can lead to a significantly enlarged borehole size. Wellbore breakout is a time-dependent process due to fluid seepage and temperature diffusion between the wellbore and formation. This paper presents a transient thermal-hydro-mechanical model for quantifying this time-dependent process. The model is developed based on a finite difference method that can solve the governing equations of unsteady-state fluid seepage and heat transfer. So it captures time-dependent diffusions of pore pressure and temperature around the wellbore. The model is validated with existing analytical solutions. The simulation results show that fluid seepage from the wellbore to the formation induces additional stresses in the near-wellbore region, which increase wellbore breakouts and makes it a time-dependent process. Cooling the formation around the wellbore has an opposite effect to fluid seepage on breakouts, i.e., mitigating breakouts. The proposed model provides a useful tool towards evaluating time-dependent breakouts.

Application of AutoBank Software in Earth-rock Dam Seepage Flow Computation

Earth-rock dam is one of the most widely used dams in the world, and has huge comprehensive benefits in flood controlling, irrigation, power generation, water supply, shipping, tourism and so on. This paper elaborate the theory and method of earth-rock dam seepage analysis, including the basic equation of seepage and boundary condition, and also introduce a hydraulic analysis software, Autobank seepage module, including the analysis steps and boundary conditions. Then two-dimension seepage analysis of earth-rock dam is carried out by using Autobank. The seepage line of the dam under three working conditions is calculated. This result can be used as reference for the design.

A three-dimensional two-level gradient smoothing meshfree method for rainfall induced landslide simulations

A three-dimensional two-level gradient smoothing meshfree method is presented for rainfall induced landslide simulations. The two-level gradient smoothing for meshfree shape function is elaborated in the three-dimensional Lagrangian setting with detailed implementation procedure. It is shown that due to the successive gradient smoothing operation without the requirement of derivative computation in the present formulation, the two-level smoothed gradient of meshfree shape function is capable of achieving a given influence domain more efficiently than the standard gradient of meshfree shape function. Subsequently, the two-level smoothed gradient of meshfree shape function is employed to discretize the weak form of coupled rainfall seepage and soil motion equations in a nodal integration format, as provides an efficient three-dimensional regularized meshfree formulation for large deformation rainfall induced landslide simulations. The exponential damage and pressure dependent plasticity relationships are utilized to describe the failure evolution in landslides. The plastic response of soil is characterized by the true effective stress measure, which is updated according to the rotationally neutralized objective integration algorithm. The effectiveness of the present three-dimensional two-level gradient smoothing meshfree method is demonstrated through numerical examples.

A numerical investigation on the effect of the temperature in the seepage calculation

Temperature difference in the soil and its environments is a common phenomenon. The permeability of the soil changes with the temperature mostly because of the variation of the viscosity of water in different temperatures. More realistic estimation of the seepage value through and beneath hydraulic structures leads to more efficient design of them. In this paper, the heat conduction equation is solved by a least squares based meshfree method to calculate the distribution of the temperature in a soil. The distribution of the permeability coefficients can be varied irregularly that may make some difficulties in the mesh-based methods. In these methods the permeability changes in each mesh and finer mesh or some kinds of interpolation are required in the solution procedure. Since there is no need to form elements or grids in a meshfree method, it can handle this irregular variation simply. Here, the seepage equation is solved by the same least squares based meshfree method. The method is integral free, simple and efficient in calculation thanks to its sparse and positive definite matrices. The scheme is validated by solving a simplified version of the governing equations. More complicated problems are dealt with to investigate the phenomenon numerically.

Numerical simulation on the coupling law of stress and gas pressure in the uncovering tectonic coal by cross-cut

During the process of cross-cut coal uncovering, the stress and gas pressure of the coal mass change with time and even lead to coal and gas outburst. This paper establishes a coupled model that includes the equation of coal deformation, gas diffusion, gas seepage and permeability evolution based on the dual poroelastic theory. This model is applied into COMSOL Multiphysics numerical software. The distribution and evolution of the stress and gas pressure in front of the working face can be obtained from the numerical results. The stepped increase mechanism of the combined potential energy in the tunneling process is proposed, which is used to analyze the evolution of elastic potential energy and gas potential energy. The results can be summarized as follows: under natural conditions, an obvious abnormal stress area occurs near the fault, and the stress of the coal seam is greater than that of the rock stratum. Under tunneling conditions, a original stress (OS), stress concentration (SC) and stress reduction (SR) are formed successively in the front of the working face. And correspondingly, the gas accumulates in SC due to the closure of cracks. The distributions of gas pressure pf and pm are approximately the same during tunneling, and they can reach an equilibrium state after 300 days. The coupled effects of stress and gas pressure gradually intensify, so the elastic potential energy and the gas potential energy of the coal continuously accumulate. The combined potential energy of coal increases with tunneling length, and an outburst is very likely to occur when the combined energy exceeds the surface energy of the coal body at the “Key point”. This research is helpful to understand the evolution mechanism of stress and gas pressure of coal seam in the process of cross-cut coal uncovering, and to know the effects of it to the dynamic disaster.

Seepage Flow Model and Deformation Properties of Coastal Deep Foundation Pit under Tidal Influence

As the coastal region is the most developed region in China, an increasing number of engineering projects are under construction in it in recent years. However, the quality of these projects is significantly affected by groundwater, which is influenced by tidal variations. Therefore, the regional groundwater dynamic characteristics under tidal impact and the spatiotemporal evolution of the seepage field must be considered in the construction of the projects. Then, Boussinesq function was introduced into the research to deduce the seepage equation under tidal influence for the coastal area. To determine the spatiotemporal evolution of the deep foundation pit seepage field and the coastal seepage field evolution model, numerical calculations based on changes in the tidal water level and seepage equation were performed using MATLAB. According to the developed model, the influence of the seepage field on the foundation pit supporting structure in the excavation process was analyzed through numerical simulations. The results of this research could be considered in design and engineering practice.

Propagation of elastic waves in saturated porous medium containing a small amount of bubbly fluid

It is very important to understand the acoustical properties of porous medium. To study the relationship between acoustical and other physical properties of porous medium will help us to use acoustical tools for determining the physical properties of porous medium. Many researchers have paid much attention to the properties of acoustic wave propagation in the gassy marine sediments based on the Biot model which is popularly used to predict the dispersion and attenuation of sound in saturated porous medium. The patchy model which contains gas inside the spherical water predicts that the existence of gas just has little effect on the propagation of acoustic wave in porous medium when the gas content is very small. However, the presence of a small number of bubbles in a fluid saturated sediment will lead to different acoustic responses. As is well known, the bubble vibration theory proposed by Keller and Miksis shows that a small number of bubbles existing in the liquid will have a great influence on sound velocity and attenuation. Therefore, in order to study the effect of a small amount of gas existing in fluid saturated porous medium on the property of acoustic wave propagation, we investigate a bubbly liquid saturated porous medium and consider the case of the bubbles vibrating linearly under the action of sound waves. First, we derive the continuity equation of the seepage according to the mass conservation of the pore fluid and the relationship between porosity differentiation and pore fluid pressure differentiation. Then, the bubble linear vibration theory given by Commander is used to deal with the time derivative of gas volume fraction in the continuity equation of the seepage, The bubble linear vibration theory gives the relationship between instantaneous bubble radius and background pressure of the medium. Through this relationship, we obtain the equation of time derivative of gas volume fraction and time derivative of pore fluid pressure. Then, we combine the obtained equation with the continuity equation of seepage, and obtain the modified continuity equation of seepage whose form is similar to that of Biot model. Finally, the modified Biot's equations for fluid saturated porous medium containing a small amount of bubbly fluid is obtained. As is well known, an effective density fluid model for acoustic propagation in sediments, derived from Biot theory, just can predict the acoustic properties of the fast compressional waves. However, the present model can predict the acoustic properties of fast, slow compressional waves and shear waves propagating in sediments. Through numerically calculating the dispersion, attenuation, amplitude ratios of pore fluid displacement to solid displacement for fast and slow compressional waves, it is found that the existence of a small number of bubbles has an influence on the acoustic properties of both the fast compressional waves and the slow compressional waves, especially the velocity of the fast compressional wave. In addition, the low-frequency speed approximation formula for the fast compressional wave is also presented. The approximate formula directly indicates the relationship between the velocity of fast compressional wave and the parameters of porous medium such as the gas volume fraction and the bubble radius. This study shows that the influence of a small number of bubbles in fluid saturated on acoustic wave propagation is noticeable. The modified Biot model presented in this paper provides one model to study the properties of acoustic waves in fluid saturated porous medium with a small number of bubbles.

A Fractal Model for Oil Transport in Tight Porous Media

Tight porous media are mainly composed of micro/nano-pores and throats, which leads to obvious microscale effect and nonlinear seepage characteristics. Based on the capillary bundle model and the fractal theory, a new nonlinear seepage equation was deduced, and a further fractal permeability model was obtained for oil transport in tight porous media by considering the effect of the boundary layer. The predictions of the model were then compared with experimental data to demonstrate that the model is valid. This model clarifies the oil transport mechanisms in tight porous media: the effective permeability is no longer a constant value and is governed by properties of tight porous media and oil. Furthermore, parameters influencing effective permeability were analyzed. The model can accurately present the seepage characteristics of the oil in tight porous media and provide a reliable basis for the development of unconventional reservoirs.

One‐Dimensional Seepage in Unsaturated, Expansive Soils

Core Ideas We present a new equation to describe seepage through unsaturated, swelling soils. This equation describes flow in saturated, unsaturated, rigid, and nonrigid soils. Rigid, unsaturated seepage equations overestimate dissipation in swelling soils. This study presents a new nonlinear partial differential equation to describe the dissipation of suction within unsaturated, swelling soils by vertical moisture flow using a material coordinate and soil parameters defined by constitutive surfaces from volume change theory. The equation takes a form similar to two classic equations that describe the flow of water within a soil matrix but accounts for the soil's interactions: the soil structure, water phase, suction, and stress. This new equation may simplify to Richards' equation for unsaturated flow through rigid media and to Terzaghi's consolidation equation under appropriate conditions. Although intended for unsaturated, swelling soils, this new equation may be applied to describe the vertical dissipation of pore pressures for all soils: saturated or unsaturated and rigid or nonrigid. Lastly, several simulations showed that the use of Richards' equation overestimates the rate of suction dissipation in swelling soils.

Experimental study of seawater seepage and heat transfer in a laboratory vertical beach well

Vertical beach well intake (VBWI) is considered as an ideal approach to provide seawater for seawater source heat pump (SWHP) in coastal areas due to its attractive advantages of seawater with high quality, lower temperature in summer and higher temperature in winter compared to raw surface seawater. A laboratory VBWI system was built and experimental strategy was proposed in this paper. First, pumping tests were conducted to obtain hydraulic parameters of the aquifer and estimate the state of water flow within aquifer region. As results of these tests, permeability coefficient of the aquifer was calculated to be 0.0209 m/s, and the water seepage in aquifer could be recognized as laminar flow. Then the influence of water-level variations on temperature in VBWI system was explored. Results showed that water temperature variation of the pumping well influenced by wave-induced water level fluctuations can be negligible in this laboratory VBWI. These testing results were adopted to provide information of hydraulic parameter and simplify the governing equation of seepage and heat transfer of VBWI model. And then numerical models describing seepage and heat transfer process of VBWI system for a SWHP were established. Finally, verification tests for the validity of the numerical model were conducted. Compared experimental results with numerical simulation of pumping temperature under cooling and heating mode respectively, simulation results showed great agreement with experimental data. The results revealed the numerical model can be well validated by laboratory experiments on VBWI system.

Control for well-posedness about a class of non-Newtonian incompressible porous medium fluid equations

Considering the non-Newtonian fluid equation of incompressible porous media, using the properties of operator semigroup and measure space and the principle of squeezed image, Fourier analysis and a priori estimate in the measurement space are used to discuss the non-compressible porous media, the properness of the solution of the equation, its gradual behavior and its topological properties. Through the diffusion regularization method and the compressed limit compact method, we study the overall decay rate of the solution of the equation in a certain space when the initial value is sufficient. The decay estimation of the solution of the incompressible seepage equation is obtained, and the asymptotic behavior of the solution is obtained by using the double regularization model and the Duhamel principle.

Experimental Investigation of Flow Domain Division in Beds Packed with Different Sized Particles

Seepage in a medium packed with particles of variable granular size can be seen in many fields of engineering applications. Due to the relative complex spatial aspect of pore geometry, there are notable differences in the critical parameters of flow transition (Reynolds number and Forchheimer number) between different structures. It is difficult to distinguish the available range of seepage equations and predict the water flux accurately. This work aims to establish the relationship between particle size and flow transition. This is conducted according to the results of flow region division, which obtains the application range for seepage equations. Experiments were carried out in sand columns with nine different particle sizes of sand with mean diameters of 0.0375, 0.1125, 0.225, 0.45, 0.8, 1.5, 2.18, 3.555 and 7.125 mm. Four flow regimes were identified (pre-Darcy regime, Darcy regime, Forchheimer regime and turbulent regime). The experimental data indicate that the permeability increases exponentially and the inertia factor reduces exponentially with an increase in particle diameter. The inertial effect becomes more significant in the medium with larger particles than with finer particles when the flow transition occurs.

Heuristic derivation of Brinkman's seepage equation

Brinkman’s law is describing the seepage of viscous fluid through a porous medium and is more acurate than the classical Darcy’s law. Namely, Brinkman’s law permits to conform the flow through a porous medium to the free Stokes’ flow. However, Brinkman’s law, similarly as Schro¨dinger’s equation was only devined. Fluid in its motion through a porous solid is interacting at every point with the walls of pores, but the interactions of the fluid particles inside pores are different than the interactions at the walls, and are described by Stokes’ equation. Here, we arrive at Brinkman’s law from Stokes’ flow equation making use of successive iterations, in type of Born’s approximation method, and using Darcy’s law as a zero-th approximation.

Fluid-solid coupling dynamic equations considering gas desorption con-traction and coal motion deformations

The coals are assumed as isotropic homogeneous saturated porous media, dynamic characteristics of porosity and permeability of coals and coal deformation caused by gas desorption are all considered, gas seepage equations about porosity, permeability and desorption contraction deformation are built based on Darcy’s law, and fluid-solid coupling equations for gas seepage are finally built, which are composed of the above-mentioned dynamic balance differential equations and gas seepage equations. In the end, one horizontal gas drainage hole is taken as a computational example, the gas pressures calculated by the new model built in this paper are compared with those by the normal model that does not include gas adsorption and motion deformation of coals, the new model is proved to be reasonable based on the mechanism of desorption contraction and seepage, and the characteristics of permeability, gas pressure and radial displacements around the hole are analyzed based on the new model.DOI: http://dx.doi.org/10.5755/j01.mech.23.3.18480

Interface condition for the Darcy velocity at the water-oil flood front in the porous medium.

Flood front is the jump interface where fluids distribute discontinuously, whose interface condition is the theoretical basis of a mathematical model of the multiphase flow in porous medium. The conventional interface condition at the jump interface is expressed as the continuous Darcy velocity and fluid pressure (named CVCM). Our study has inspected this conclusion. First, it is revealed that the principle of mass conservation has no direct relation to the velocity conservation, and the former is not the true foundation of the later, because the former only reflects the kinetic characteristic of the fluid particles at one position(the interface), but not the different two parts of fluid on the different side of the interface which required by the interface conditions. Then the reasonableness of CVCM is queried from the following three aspects:(1)Using Mukat’s two phase seepage equation and the mathematical method of apagoge, we have disproved the continuity of each fluid velocity;(2)Since the analytical solution of the equation of Buckley-Leveret equations is acquirable, its velocity jumps at the flood front presents an appropriate example to disprove the CVCM;(3) The numerical simulation model gives impractical result that flood front would stop moving if CVCM were used to calculate the velocities at the interface between two gridcells. Subsequently, a new one, termed as Jump Velocity Condition Model (JVCM), is deduced from Muskat’s two phase seepage equations and Darcy’s law without taking account of the capillary force and compressibility of rocks and fluids. Finally, several cases are presented. And the comparisons of the velocity, pressure difference and the front position, which are given by JVCM, CVCM and SPU, have shown that the result of JVCM is the closest to the exact solution.

Modeling interporosity flow functions and shape factors in low-permeability naturally fractured reservoir

A new method is proposed to derive interporosity flow functions (also called transfer functions) and shape factors between the fracture and low-permeability matrix in naturally fractured reservoirs with dual porosity model. The differential form of single-phase seepage equation and the analytical solution for pressure diffusion equation are adopted to deduce the analytical solution for shape factors in transient pressure diffusion equation. By employing this approach, the limitations resulting from the process of deducing interporosity flow function and shape factors under many assumptions are avoided, and the no-linear pressure gradients in the matrix can be well explained. The defects of the interporosity flow function and shape factors mostly are generated under the pseudo-steady state hypothesis, in which no-linear pressure gradients in the matrix cannot be accurately captured. The interporosity flow function, constant shape factors and time-varying shape factors considering threshold pressure in one, two and three dimensional fractures are derived for low permeability fractured reservoirs under no pseudo-steady state hypothesis. The newly developed analytical solution is useful in compressible fluid productivity prediction and transient well testing application. The field case study shows that those shape factors are able to provide more accurate parameters for compressible fluid productivity prediction and more reasonable theoretical basis for the correct interpretation of well testing.

Reconstruction of different scales of pore-fractures network of coal reservoir and its permeability prediction with Monte Carlo method

There are millimeter, micron and nanometer scales of pores and fractures in coal to describe different scales of coal pores and fissures communicating path and to quantitatively characterize their permeability. Such information provides an important basis for studying coalbed methane output mechanism. The pores and fissures in a large number of coal samples were observed and counted by scanning electron microscopy and optical microscopy. The probability distribution models of pore-fissure network were then established. Different scales of pore-fissures 2D network models were reconstructed by Monte Carlo method. The 2D seepage models were obtained through assignment zero method and using Matlab software. The effect of permeability on different scale pore-fractures network was obtained by two-dimensional seepage equation. Predicted permeability is compared with the measured ones. The results showed that the dominant order of different scale pore-fractures connected path from high to low is millimeter-sized fractures, seepage pores and micron-size fractures. The contribution of coal reservoir permeability from large to small is millimeter-size fractures, micron-size fractures and seepage pores. Different parameters in different scale pore-fractures are of different influence permeability. Reconstruction of different scale pore-fractures network can clearly display the connectivity of pore-fractures, which can provide a basis for selecting migration path and studying gas flow pattern.