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1.
Karst spring responses examined by process-based modeling 总被引：8，自引：0，他引：8
Ground water in karst terrains is highly vulnerable to contamination due to the rapid transport of contaminants through the highly conductive conduit system. For contamination risk assessment purposes, information about hydraulic and geometric characteristics of the conduits and their hydraulic interaction with the fissured porous rock is an important prerequisite. The relationship between aquifer characteristics and short-term responses to recharge events of both spring discharge and physicochemical parameters of the discharged water was examined using a process-based flow and transport model. In the respective software, a pipe-network model, representing fast conduit flow, is coupled to MODFLOW, which simulates flow in the fissured porous rock. This hybrid flow model was extended to include modules simulating heat and reactive solute transport in conduits. The application of this modeling tool demonstrates that variations of physicochemical parameters, such as solute concentration and water temperature, depend to a large extent on the intensity and duration of recharge events and provide information about the structure and geometry of the conduit system as well as about the interaction between conduits and fissured porous rock. Moreover, the responses of solute concentration and temperature of spring discharge appear to reflect different processes, thus complementing each other in the aquifer characterization. 相似文献
2.
Flow and transport simulation in karst aquifers remains a significant challenge for the ground water modeling community. Darcy's law–based models cannot simulate the inertial flows characteristic of many karst aquifers. Eddies in these flows can strongly affect solute transport. The simple two-region conduit/matrix paradigm is inadequate for many purposes because it considers only a capacitance rather than a physical domain. Relatively new lattice Boltzmann methods (LBMs) are capable of solving inertial flows and associated solute transport in geometrically complex domains involving karst conduits and heterogeneous matrix rock. LBMs for flow and transport in heterogeneous porous media, which are needed to make the models applicable to large-scale problems, are still under development. Here we explore aspects of these future LBMs, present simple examples illustrating some of the processes that can be simulated, and compare the results with available analytical solutions. Simulations are contrived to mimic simple capacitance-based two-region models involving conduit (mobile) and matrix (immobile) regions and are compared against the analytical solution. There is a high correlation between LBM simulations and the analytical solution for two different mobile region fractions. In more realistic conduit/matrix simulation, the breakthrough curve showed classic features and the two-region model fit slightly better than the advection-dispersion equation (ADE). An LBM-based anisotropic dispersion solver is applied to simulate breakthrough curves from a heterogeneous porous medium, which fit the ADE solution. Finally, breakthrough from a karst-like system consisting of a conduit with inertial regime flow in a heterogeneous aquifer is compared with the advection-dispersion and two-region analytical solutions. 相似文献
3.
We discuss techniques to represent groundwater flow in carbonate aquifers using the three existing modeling approaches: equivalent porous medium, conduit network, and discrete fracture network. Fractures in faulted stratigraphic successions are characterized by dominant sets of sub-vertical joints. Grid rotation is recommended using the equivalent porous medium to match higher hydraulic conductivity with the dominant orientation of the joints. Modeling carbonate faults with throws greater than approximately 100 m is more challenging. Such faults are characterized by combined conduit-barrier behavior. The barrier behavior can be modeled using the Horizontal Flow Barrier Package with a low-permeability vertical barrier inserted to represent the impediment of horizontal flow in faults characterized by sharp drops of the piezometric surface. Cavities can occur parallel to the strike of normal faults generating channels for the groundwater. In this case, flow models need to account for turbulence using a conduit network approach. Channels need to be embedded in an equivalent porous medium due to cavities a few centimeters large, which are present in carbonate aquifers even in areas characterized by low hydraulic gradients. Discrete fracture network modeling enables representation of individual rock discontinuities in three dimensions. This approach is used in non-heavily karstified aquifers at industrial sites and was recently combined with the equivalent porous medium to simulate diffusivity in the matrix. Following this review, we recommend that the future research combines three practiced modeling approaches: equivalent porous medium, discrete fracture network, and conduit network, in order to capture structural and flow aspects in the modeling of groundwater in carbonate rocks. 相似文献
4.
Analytical solutions are presented for solute transport in rivers including the effects of transient storage and first order decay. The solute transport model considers an advection–dispersion equation for transport in the main channel linked to a first order mass exchange between the main channel and the transient storage zones. In case of a conservative tracer, it is shown that different analytical solutions presented in the literature are mathematically identical. For non-conservative solutes, first order decay reactions are considered with different reaction rate coefficients in the main river channel and in the dead zones. New analytical solutions are presented for different boundary conditions, i.e. instantaneous injection in an infinite river reach, and variable concentration time series input in a semi-infinite river reach. The correctness and accuracy of the analytical solutions is verified by comparison with the OTIS numerical model. The results of analytical and numerical approaches compare favourably and small differences can be attributed to the influence of boundary conditions. It is concluded that the presented analytical solutions for solute transport in rivers with transient storage and solute decay are accurate and correct, and can be usefully applied for analyses of tracer experiments and transport characteristics in rivers with mass exchange in dead zones. 相似文献
5.
Diffusive mass exchange of non‐reactive substances in dual‐porosity porous systems – column experiments under saturated conditions 
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下载免费PDF全文 Diffusive mass exchange into immobile water regions within heterogeneous porous aquifers influences the fate of solutes. The percentage of immobile water is often unidentified in natural aquifers though. Hence, the mathematical prediction of solute transport in such heterogeneous aquifers remains challenging. The objective of this study was to find a simple analytical model approach that allows quantifying properties of mobile and immobile water regions and the portion of immobile water in a porous system. Therefore, the Single Fissure Dispersion Model (SFDM), which takes into account diffusive mass exchange between mobile and immobile water zones, was applied to model transport in well‐defined saturated dual‐porosity column experiments. Direct and indirect model validation was performed by running experiments at different flow velocities and using conservative tracer with different molecular diffusion coefficients. In another column setup, immobile water regions were randomly distributed to test the model applicability and to determine the portion of immobile water. In all setups, the tracer concentration curves showed differences in normalized maximum peak concentration, tailing and mass recovery according to their diffusion coefficients. These findings were more pronounced at lower flow rates (larger flow times) indicating the dependency of diffusive mass exchange into immobile water regions on tracers' molecular diffusion coefficients. The SFDM simulated all data with high model efficiency. Successful model validation supported the physical meaning of fitted model parameters. This study showed that the SFDM, developed for fissured aquifers, is applicable in porous media and can be used to determine porosity and volume of regions with immobile water. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
6.
We present an analytical expression for the shear dispersion during solute transport in a coupled fracture–matrix system. The dispersion coefficient is obtained in a fracture with porous walls by taking into account an accurate boundary condition at the interface between the matrix and fracture, and the results were compared with those in a non-coupled system. The analysis presented identifies three regimes: diffusion-dominated, transition, and advection-dominated. The results showed that it is important to consider the exchange of solute between the fracture and matrix in development of the shear dispersion coefficient for the transition and advection-dominated regimes. The new dispersion coefficient is obtained by imposing the continuity of concentrations and mass fluxes along the porous walls. The resulting equivalent transport equation revealed that the effective velocity in a fracture increases while the dispersion coefficient decreases due to mass transfer between the matrix and fracture. A larger effective advection term leads to greater storage of mass in the matrix as compared with the classical double-porosity model with a non-coupled dispersion coefficient. The findings of this study can be used for modeling of tracer tests as well as fate, transport, and remediation of groundwater contaminants in fractured rocks. 相似文献
7.
Analytical solutions for the water flow and solute transport equations in the unsaturated zone are presented. We use the Broadbridge and White nonlinear model to solve the Richards’ equation for vertical flow under a constant infiltration rate. Then we extend the water flow solution and develop an exact parametric solution for the advection-dispersion equation. The method of characteristics is adopted to determine the location of a solute front in the unsaturated zone. The dispersion component is incorporated into the final solution using a singular perturbation method. The formulation of the analytical solutions is simple, and a complete solution is generated without resorting to computationally demanding numerical schemes. Indeed, the simple analytical solutions can be used as tools to verify the accuracy of numerical models of water flow and solute transport. Comparison with a finite-element numerical solution indicates that a good match for the predicted water content is achieved when the mesh grid is one-fourth the capillary length scale of the porous medium. However, when numerically solving the solute transport equation at this level of discretization, numerical dispersion and spatial oscillations were significant. 相似文献
8.
A method to estimate reactive transport parameters as well as geometric conduit parameters from a multitracer test in a karst aquifer is provided. For this purpose, a calibration strategy was developed applying the two-region nonequilibrium model CXTFIT. The ambiguity of the model calibration was reduced by first calibrating the model with respect to conservative tracer breakthrough and later transferring conservative transport parameters to the reactive model calibration. The reactive transport parameters were only allowed to be within a defined sensible range to get reasonable calibration values. This calibration strategy was applied to breakthrough curves obtained from a large-scale multitracer test, which was performed in a karst aquifer of the Swabian Alb, Germany. The multitracer test was conducted by the simultaneous injection of uranine, sulforhodamine G, and tinopal CBS-X. The model succeeds to represent the tracer breakthrough curves (TBCs) of uranine and sulforhodamine G and verifies that tracer-rock interactions preferably occur in the immobile fluid region, although the fraction of this region amounts to only 3.5% of the total water. However, the model failed to account for the long tailing observed in the TBC of tinopal CBS-X. Sensitivity analyses reveal that model results for the conservative tracer transport are most sensitive to average velocity and volume fraction of the mobile fluid region, while dispersion and mass transfer coefficients are least influential. Consequently, reactive tracer calibration allows the determination of sorption sites in the mobile and immobile fluid region at small retardation coefficients. 相似文献
9.
Variable-density groundwater flow and solute transport in porous media containing nonuniform discrete fractures 总被引：1，自引：0，他引：1
Variations in fluid density can greatly affect fluid flow and solute transport in the subsurface. Heterogeneities such as fractures play a major role for the migration of variable-density fluids. Earlier modeling studies of density effects in fractured media were restricted to orthogonal fracture networks, consisting of only vertical and horizontal fractures. The present study addresses the phenomenon of 3D variable-density flow and transport in fractured porous media, where fractures of an arbitrary incline can occur. A general formulation of the body force vector is derived, which accounts for variable-density flow and transport in fractures of any orientation. Simulation results are presented that show the verification of the new model formulation, for the porous matrix and for inclined fractures. Simulations of variable-density flow and solute transport are then conducted for a single fracture, embedded in a porous matrix. The simulations show that density-driven flow in the fracture causes convective flow within the porous matrix and that the high-permeability fracture acts as a barrier for convection. Other simulations were run to investigate the influence of fracture incline on plume migration. Finally, tabular data of the tracer breakthrough curve in the inclined fracture is given to facilitate the verification of other codes. 相似文献
10.
《Advances in water resources》2001,24(6):607-619
Solute discharge moments (mean and variance) are computed using numerical modeling of flow and advective transport in two-dimensional heterogeneous aquifers and are compared to theoretical results. The solute discharge quantifies the temporal evolution of the total contaminant mass crossing a certain compliance boundary. In addition to analyzing the solute discharge moments within a classical absolute dispersion framework, we also analyze relative dispersion formulation, whereby plume meandering (deviation from mean flow path caused by velocity variations at scales larger than plume size) is removed. This study addresses some important issues related to the computation of solute discharge moments from random walk particle tracking experiments, and highlights some of the important differences between absolute and relative dispersion frameworks. Relative dispersion formulation produces maximum uncertainty that coincides with the peak mean discharge. Absolute dispersion, however, results in earlier arrival of the uncertainty peak as compared to the first moment peak. Simulations show that the standard deviation of solute discharge in a relative dispersion framework requires increasingly large temporal sampling windows to smooth out some of the large fluctuations in breakthrough curves associated with advective transport. Using smoothing techniques in particle tracking to distribute the particle mass over a volume rather than at a point significantly reduces the noise in the numerical simulations and removes the need to use large temporal windows. Same effect can be obtained by adding a local dispersion process to the particle tracking experiments used to model advective transport. The effect of the temporal sampling window bears some relevance and important consequences for evaluating risk-related parameters. The expected value of peak solute discharge and its standard deviation are very sensitive to this sampling window and so will be the risk distribution relying on such numerical models. 相似文献
11.
Modelling of the groundwater flow and of tracer movement in the porous and fissured media: Chalk Aquifer (Northern part of Paris Basin,France) 下载免费PDF全文
下载免费PDF全文 Adel Zghibi Lahcen Zouhri Ismail Chenini Amira Merzougui Jamila Tarhouni 《水文研究》2016,30(12):1916-1928
A groundwater flow model has been developed in order to study the chalk aquifer of Paris Basin, based on most of the geological and hydrological available data. The numerical processes are intended to modelling the groundwater flow in the Senonian (Late Cretaceous) formations and to visualize the tracer movement in groundwater resources in the experimental site of LaSalle Beauvais (northern part Paris Basin). Both objectives were achieved as follows: (i) the comprehension of the spatial distribution of the hydraulic conductivity in the chalk aquifer taking into account the characteristics of the hydrogeological system and (ii) the use of the analytical solution for describing one‐dimensional to two‐dimensional solute transport in a unidirectional steady‐state flow tracer with scale‐dependent dispersion. Advection and diffusion mechanisms are taken into account. Comparison between the breakthrough curves of the analytical and the numerical solutions provided an excellent agreement for various ranges of scale‐related transport parameters of interest. The developed power series solution facilitates fast prediction of the breakthrough curves at each observation point. Thus, the derived new solutions are widely applicable and are very useful for the validation of numerical transport. The numerical approach is carried out by MT3DMS, a Modular 3‐D Multi‐Species Transport Model for Simulation of Advection, Dispersion, and Chemical Reactions of Contaminants in Groundwater Systems, and based on total variation‐diminishing method using the ULTIMATE algorithm. The estimation of the infected surface could constitute an approach in water management and allows to prevent the risks of pollution and to manage the groundwater resource from a durable development perspective. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
12.
AbstractThe impact of pollution incidents on rivers and streams may be predicted using mathematical models of solute transport. Practical applications require an analytical or numerical solution to a governing solute mass balance equation together with appropriate values of relevant transport coefficients under the flow conditions of interest. This paper considers two such models, namely those proposed by Fischer and by Singh and Beck, and compares their performances using tracer data from a small stream in Edinburgh, UK. In calibrating the models, information on the magnitudes and the flow rate dependencies of the velocity and the dispersion coefficients was generated. The dispersion coefficient in the stream ranged between 0.1 and 0.9 m2/s for a flow rate range of 13–437 L/s. During calibration it was found that the Singh and Beck model fitted the tracer data a little better than the Fischer model in the majority of cases. In a validation exercise, however, both models gave similarly good predictions of solute transport at three different flow rates. 相似文献
13.
J.A. Barker 《Advances in water resources》1982,5(2):98-104
The main processes affecting the migration of a solute in a fissured aquifer will be advection and dispersion in the fissures, diffusion into the porous matrix; and adsorption. This paper considers solute transport in an idealized fissured aquifer consisting of slabs of saturated rock-matrix separated by equally spaced, planar fissures. The solution of the transport equations is developed as far as Laplace transforms of the solute concentrations in the fissure and matrix water. Numerical inversion of the transforms is used to investigate characteristic behaviour of the model for a number of special cases. 相似文献
14.
The success of transient storage (TS) modeling for natural streams depends, in part, on the ability to describe the dispersion process accurately. Evidence based on stream tracer data shows that solute transport processes often do not follow the classical second-order dispersion model (e.g., early breakthrough and faster than Fickian travel times were observed). While models based on space-fractional dispersion are a promising alternative, different definitions of fractional derivatives exist in the literature. Unlike integer-order derivatives, fractional derivatives represent convolutions of concentration with long-range spatial correlation and numerical approximations can produce dense matrices. Therefore issues of both accuracy and computational efficiency need to be examined to successfully identify model parameters for natural streams. In this paper, we first compare the performance of several numerical approaches for solving the space-fractional dispersion equation. We examine three different numerical approaches to approximate the space-fractional derivatives including: (a) a fully-implicit scheme based on the shifted Grünwald–Letnikov (GL) approximation (b) a three-point implicit representation based on the GL formula and (c) a three-point implicit scheme based on mass conservation and the Caputo definition of the fractional derivative. We then use an operator-splitting technique to evaluate a TS model based on space-fractional dispersion (the FSTS model) and test the model against analytical solutions and stream tracer data. A sequence acceleration method (Richardson extrapolation) significantly improves the performance of all schemes examined. Results indicate that the fully-implicit GL method with Richardson extrapolation produces the most accurate solutions while the three-point implicit GL scheme has a stringent time-step restriction to produce acceptable solutions. The three-point implicit scheme based on the Caputo derivative produces accurate solutions in a fraction of the time taken by the fully-implicit GL method and represents the best trade-off between accuracy and computational efficiency for practical applications. The scheme is suitable for parameter estimation and is used to successfully describe tracer data in a natural stream. 相似文献
15.
Li G 《Ground water》2011,49(4):584-592
Often the water flowing in a karst conduit is a combination of contaminated water entering at a sinkhole and cleaner water released from the limestone matrix. Transport processes in the conduit are controlled by advection, mixing (dilution and dispersion), and retention-release. In this article, a karst transport model considering advection, spatially varying dispersion, and dilution (from matrix seepage) is developed. Two approximate Green's functions are obtained using transformation of variables, respectively, for the initial-value problem and for the boundary-value problem. A numerical example illustrates that mixing associated with strong spatially varying conduit dispersion can cause strong skewness and long tailing in spring breakthrough curves. Comparison of the predicted breakthrough curve against that measured from a dye-tracing experiment between Ames Sink and Indian Spring, Northwest Florida, shows that the conduit dispersivity can be as large as 400 m. Such a large number is believed to imply strong solute interaction between the conduit and the matrix and/or multiple flow paths in a conduit network. It is concluded that Taylor dispersion is not dominant in transport in a karst conduit, and the complicated retention-release process between mobile- and immobile waters may be described by strong spatially varying conduit dispersion. 相似文献
16.
Hydraulic/partitioning tracer tomography (HPTT) was recently developed by Yeh and Zhu [Yeh T-CJ, Zhu J. Hydraulic/partitioning tracer tomography for characterization of dense nonaqueous phase liquid source zones, Water Resour Res 2007;43:W06435. doi:10.1029/2006WR004877.] for estimating spatial distribution of dense nonaqueous phase liquids (DNAPLs) in the subsurface. Since discrete tracer concentration data are directly utilized for the estimation of DNAPLs, this approach solves the hyperbolic convection–dispersion equation. Solution to the convection–dispersion equation however demands fine temporal and spatial discretization, resulting in high computational cost for an HPTT analysis. In this work, we use temporal moments of tracer breakthrough curves instead of discrete concentration data to estimate DNAPL distribution. This approach solves time independent partial differential equations of the temporal moments, and therefore avoids solving the convection–dispersion equation using a time marching scheme, resulting in a dramatic reduction of computational cost. To reduce numerical oscillations associated with convection dominated transport problems such as in inter-well tracer tests, the approach uses a finite element solver adopting the streamline upwind Petrov–Galerkin method to calculate moments and sensitivities. We test the temporal moment approach through numerical simulations. Comparing the computational costs between utilizing moments and discrete concentrations, we find that temporal moments significantly reduce the computation time. We also find that tracer moment data collected through a tomographic survey alone are able to yield reasonable estimates of hydraulic conductivity, as indicated by a correlation of 0.588 between estimated and true hydraulic conductivity fields in the synthetic case study. 相似文献
17.
致密碳酸盐岩在成岩和后成岩过程中形成了复杂的孔隙结构特征,其速度等地震弹性参数不仅与孔隙度有关,而且还与孔隙结构特征密切相关.为了进一步研究致密碳酸盐岩内部流体相关的速度频散特征,针对致密碳酸盐岩进行实验室的频散测量与频散理论分析尤为重要.本研究选用了一块典型的致密型碳酸盐岩样品,在对样品进行了精细的包括CT扫描与镜下薄片的孔隙结构描述基础上,进行了实验室跨频段(从地震频段-超声频段)的频散测量与频散响应分析.比较实验室跨频段岩石物理频散测量可以获得如下认识:1)较之于典型的"喷射流"机制,改进的"喷射流"模型可以半定量地解释频散测量的结果,这大大提高了对致密碳酸盐岩频散响应的理解与认识;2)改进的"喷射流"模型还不能完全精确地匹配实验室频散测量结果,这说明除了微观尺度下的"喷射流"机制,还存在着其他控制频散与衰减的机制;3)本项工作对研究致密碳酸盐岩储层中不同频段地震波响应以及对储层预测与流体识别提供了理论依据. 相似文献
18.
Bioremediation in Fractured Rock: 1. Modeling to Inform Design,Monitoring, and Expectations 下载免费PDF全文
下载免费PDF全文 Claire R. Tiedeman Allen M. Shapiro Paul A. Hsieh Thomas E. Imbrigiotta Daniel J. Goode Pierre J. Lacombe Mary F. DeFlaun Scott R. Drew Carole D. Johnson John H. Williams Gary P. Curtis 《Ground water》2018,56(2):300-316
Field characterization of a trichloroethene (TCE) source area in fractured mudstones produced a detailed understanding of the geology, contaminant distribution in fractures and the rock matrix, and hydraulic and transport properties. Groundwater flow and chemical transport modeling that synthesized the field characterization information proved critical for designing bioremediation of the source area. The planned bioremediation involved injecting emulsified vegetable oil and bacteria to enhance the naturally occurring biodegradation of TCE. The flow and transport modeling showed that injection will spread amendments widely over a zone of lower‐permeability fractures, with long residence times expected because of small velocities after injection and sorption of emulsified vegetable oil onto solids. Amendments transported out of this zone will be diluted by groundwater flux from other areas, limiting bioremediation effectiveness downgradient. At nearby pumping wells, further dilution is expected to make bioremediation effects undetectable in the pumped water. The results emphasize that in fracture‐dominated flow regimes, the extent of injected amendments cannot be conceptualized using simple homogeneous models of groundwater flow commonly adopted to design injections in unconsolidated porous media (e.g., radial diverging or dipole flow regimes). Instead, it is important to synthesize site characterization information using a groundwater flow model that includes discrete features representing high‐ and low‐permeability fractures. This type of model accounts for the highly heterogeneous hydraulic conductivity and groundwater fluxes in fractured‐rock aquifers, and facilitates designing injection strategies that target specific volumes of the aquifer and maximize the distribution of amendments over these volumes. 相似文献
19.
It has been known for many years that dispersivity increases with solute travel distance in a subsurface environment. The increase of dispersivity with solute travel distance results from the significant variation of hydraulic properties of heterogeneous media and was identified in the literature as scale-dependent dispersion. This study presents an analytical solution for describing two-dimensional non-axisymmetrical solute transport in a radially convergent flow tracer test with scale-dependent dispersion. The power series technique coupling with the Laplace and finite Fourier cosine transform has been applied to yield the analytical solution to the two-dimensional, scale-dependent advection–dispersion equation in cylindrical coordinates with variable-dependent coefficients. Comparison between the breakthrough curves of the power series solution and the numerical solutions shows excellent agreement at different observation points and for various ranges of scale-related transport parameters of interest. The developed power series solution facilitates fast prediction of the breakthrough curves at any observation point. 相似文献
20.
Understanding the transport of chemical components in porous media is fundamentally important to many reservoir processes such as contaminant transport and reactive flows involved in CO2 sequestration. Carbonate rocks in particular present difficulties for pore-scale simulations because they contain large amounts of sub-micron porosity. In this work, we introduce a new hybrid simulation model to calculate hydrodynamic dispersion in pore-scale images of real porous media and use this to elucidate the origins and behaviour of stagnant zones arising in transport simulations using micro-CT images of carbonates. For this purpose a stochastic particle model for simulating the transport of a solute is coupled to a Lattice-Boltzmann algorithm to calculate the flow field. The particle method incorporates second order spatial and temporal resolution to resolve finer features of the domain. We demonstrate how dispersion coefficients can be accurately obtained in capillaries, where corresponding analytical solutions are available, even when these are resolved to just a few lattice units. Then we compute molecular displacement distributions for pore-spaces of varying complexity: a pack of beads; a Bentheimer sandstone; and a Portland carbonate. Our calculated propagator distributions are compared directly with recent experimental PFG-NMR propagator distributions (Scheven et al., 2005; Mitchell et al., 2008), the latter excluding spin relaxation mechanisms. We observe that the calculated transport propagators can be quantitatively compared with the experimental distribution, provided that spin relaxations in the experiment are excluded, and good agreement is found for both the sandstone and the carbonate. However, due to the absence of explicit micro-porosity from the carbonate pore space image used for flow field simulations we note that there are fundamental differences in the physical origins of the stagnant zones for micro-porous rocks between simulation and experiment. We show that for a given micro-CT image of a carbonate, small variations in the parameters chosen for the segmentation process lead to different amounts of stagnancy which diffuse away at different rates. Finally, we use a filtering method to show that this is due to the presence of spurious isolated pores which arise from the segmentation process and suggest an approach to overcome this limitation. 相似文献