Heterogeneous Porous Formations Research Articles

Numerical solution of non-linear advection-dispersion equation in a finite porous domain

This study presents a numerical simulation of the one-dimensional concentration-dependent convection-dispersion equation in a finite heterogeneous porous formation. The solution of the convection-diffusion equation with variable coefficients is obtained with the help of MATLAB pdepe solver. The groundwater flow velocity depends on the pollutant concentration, and the dispersion coefficient is proportional to the groundwater flow velocity. The effects of the zero-order production term and the first-order decay are also considered. The aquifer is assumed to be heterogeneous and finite, with sources concentrated in the flow direction. It is assumed that the porous media and the pollutant are chemically non-reactive. Initially porous domain is considered not solute free. The model assumes a uniform continuous input point source and a variable input point source released from the left end of the aquifer domain. The obtained results graphically describe the importance of the dispersion coefficient and other relevant parameters for solute transport in porous media. The developed numerical solution is verified with an analytical solution, and it is found that they are in good agreement.

Theoretical investigation of nonergodic solute dispersion in natural porous formations characterized by persistent and antipersistent power-law log-conductivity correlations

Tracer dispersion within log-conductivity fields represented by power-law semivariograms is investigated by an analytical first-order Lagrangian approach that, in treating subsurface flow and transport, resorts to the superposition principle of an infinite double hierarchy of mutually independent scales of heterogeneity. The results of the investigation, which are corroborated by a preliminary field validation, and also interpreted in terms of probabilistic risk analysis, say that transport anomaly is intrinsically associated with evolving-scale heterogeneous porous formations, regardless of their semivariogram scaling exponent b. In contrast with what was found by previous studies that dealt with this subject in a Lagrangian framework, it is demonstrated that: (1) the magnitude of nonergodic dispersion is nonmonotonically related to b; (2) consistently assuming a characteristic advective-heterogeneity length-scale leads to a universal (and quadratic) dependence of the dimensionless macrodispersion coefficients on the dimensionless time. Additionally, it is demonstrated that, in the presence of fractal heterogeneity, and unlike what happens for short-range correlations, diffusion acts as an antagonist mechanism in terms of Fickian dispersion achievement. Finally, the reinterpretation of antipersistent and persistent correlations as a double hierarchy of oscillatory nonperiodic and periodic fields, respectively, besides allowing for a technical explanation of all the detected trends, envisions a possible alternative methodology for their numerical generation.

Nearly perfectly matched layer implementation for time domain spectral element modelling of wave propagation in 3D heterogeneous and anisotropic porous media

Numerical modelling of wave propagation in the heterogeneous and anisotropic porous media is important for seismic exploration. When modelling time-domain poroelastic waves in an unbounded region, the implementation of perfectly matched layer (PML) is a common technique to absorb the outgoing waves. The PML in terms of first-order equations has been well proposed. However, the first-order PML needs fundamental and nontrivial reconstructions when applied to the second-order systems in displacements, such as finite element method (FEM) and spectral element method (SEM). It is well known SEM is an efficient algorithm for simulating seismic waves for its high accuracy and geometrical flexibility. However, at present, few literature studies the implementation of PML into time-domain SEM for simulating waves in the 3D unbounded heterogeneous porous media. In this work, combining with auxiliary ordinary differential equations (ODEs), we first systematically extend nearly PML (NPML) into time-domain SEM in terms of the second-order poroelastic system. The scheme preserves the original form of wave equations, leading to a simple and flexible incorporation into the existing algorithm. It avoids the temporary convolution through solving a set of first-order ODEs in time domain, resulting in high computational efficiency. Furthermore, to solve the coupling poroelastic/acoustic/elastic waves, we employ domain decomposition technique to handle the different regions independently, and apply the corresponding boundary conditions at the region interfaces to guarantee the coupling. Numerical examples demonstrate the accuracy and stability of the proposed method by comparing the numerical results with the analytical solutions and other independent numerical solutions. Finally, we apply the algorithm to solve wave propagation in a large-scale model involved a fluid-filled borehole embedded in heterogeneous porous formations.

Solute Transport in a Doublet‐Type Flow Configuration Through a Weakly Heterogeneous Porous Formation

AbstractSteady flow generated by an injecting and a pumping well (doublet) takes place in a porous formation where the spatially variable hydraulic conductivity K is modeled as a stationary, lognormal, random field with anisotropic two‐point autocorrelation. The latter is characterized by a vertical integral scale, that is, Iv, smaller than the horizontal one, that is, I. A solute, either passive or reactive, is injected in the medium, and we aim at computing the breakthrough curve (BTC) and its moments not only at the recovery (pumping) well, but also at any location between the two wells. The strong coupling between K and the nonuniformity of the flow renders the problem very difficult. Nevertheless, a simple (analytical) solution is obtained by adopting a few assumptions: (a) wells are replaced by lines of singularity, (b) a perturbation solution which regards the variance of the log‐conductivity Y = ln K as a perturbation parameter is employed, (c) the study is limited to strongly anisotropic heterogeneous formations (for which the anisotropy ratio λ = Iv/I is much smaller than one), and (d) the impact of pore‐scale dispersion is neglected. Central for the computation of the BTC is the statistics of the travel time of a fluid particle released at the injecting well and reaching a control plane located at any position x1 along the distance connecting the two wells. It is shown that the spatial variability of Y acts de facto like a dispersion mechanism: it enhances spreading, especially in the early arrivals. Useful closed form expressions for moments of the travel time along the central trajectory are also obtained. Finally, the theoretical framework presented in this study is applied to two transport experiments in order to compute the second‐order (temporal) moment as function of x1, and therefore to quantify dispersion occurring in the zone delimited by the two wells.

Well‐Type Steady Flow in Strongly Heterogeneous Porous Media: An Experimental Study

AbstractSteady well‐type flow was monitored in an aquifer that was artificially packed in order to reproduce a given, highly heterogeneous, statistical distribution of the log‐conductivity Y. In particular, we focus on pumping tests carried out at 10 volumetric flow rates. The experimental arrangement was composed by a pumping well and several surrounding observation piezometers. The unique feature of this experimental study is that the high heterogeneity structure of Y is known fairly well. Thus, the study lends itself as a valuable tool to corroborate theoretical findings about flows driven by sources through porous formations, where the variance (in the present study equal to 3.79) of Y is large. Besides discussing experimental findings, we tackle the crucial issue of upscaling the hydraulic conductivity in a well‐flow configuration. In particular, we deal with the equivalent conductivity (EC) as that pertaining to a homogeneous (fictitious) medium which conveys the same volumetric flow rate of the real one, under the same boundary conditions. Hence, the EC can be identified straightforwardly by means of head measurements. Even if we show that the EC is a proper parameter to reproduce measurements, it is experimentally shown (in line with the theoretical results) to be position‐dependent, and therefore, it cannot be regarded (unlike groundwater‐type flow) as a formation’s property. This implies that the EC applies only to the configuration at stake. Then, we show that the EC, combined with a recent model of effective conductivity in well‐flows through highly heterogeneous porous formation, leads to a reasonably reliable estimate of , some limitations and approximations, notwithstanding. It is hoped that the present experimental study will be useful for other researchers who are engaged with similar research‐topics.

Experimental investigation to characterize simple versus multi scaling analysis of hydraulic conductivity at a mesoscale

The spatial variability of the aquifers' hydraulic properties can be satisfactorily described by means of scaling laws. The latter enable one to relate the small (typically laboratory) scale to the larger (typically formation/regional) ones, therefore leading de facto to an upscaling procedure. In the present study, we are concerned with the spatial variability of the hydraulic conductivity K into a strongly heterogeneous porous formation. A strategy, allowing one to identify correctly the single/multiple scaling of K, is applied for the first time to a large caisson, where the medium was packed. In particular, we show how to identify the various scaling ranges with special emphasis on the determination of the related cut-off limits. Finally, we illustrate how the heterogeneity enhances with the increasing scale of observation, by identifying the proper law accounting for the transition from the laboratory to the field scale. Results of the present study are of paramount utility for the proper design of pumping tests in formations where the degree of spatial variability of the hydraulic conductivity does not allow regarding them as “weakly heterogeneous”, as well as for the study of dispersion mechanisms.

Large-Scale Model for the Dissolution of Heterogeneous Porous Formations: Theory and Numerical Validation

Large-Scale Model for the Dissolution of Heterogeneous Porous Formations: Theory and Numerical Validation

Improving the computational efficiency of first arrival time uncertainty estimation using a connectivity-based ranking Monte Carlo method

The first arrival time of a solute plume migrating from its source to an environmentally sensitive target is one of the key quantities of interest when assessing the risks of groundwater contamination. First arrival times are correlated with the hydraulic connectivity properties of spatially heterogeneous porous formations. Hydraulic connectivity leads to the presence of preferential flow paths which in turn control the transport dynamics of the leading edge of the solute plume and therefore first arrival times. In applications, these arrival times are subject to uncertainty given the lack of a detailed site characterization. The Monte Carlo method is commonly adopted to estimate the uncertainty of solute arrival times, however it leads to a computational burden. In this work, we build upon the existing knowledge regarding the correlation between connectivity and first arrival times to propose an innovative connectivity-based ranking Monte Carlo approach to quantify the uncertainty of first arrival times. The proposed method is tailored to predict first arrival times and allows to alleviate the computational costs when compared to the traditional Monte Carlo method. Our method consists of ranking the randomly generated spatially heterogeneous hydraulic conductivity fields according to their connectivity. The connectivity metric adopted is based on the concept of the minimum hydraulic resistance and can be obtained at a very low computational cost through the use of graph theory. We illustrate the methodology by analyzing the convergence rate of the first arrival time means and standard deviations. We compare the convergence rate of the first arrival times statistics obtained through the proposed methodology with those computed through the traditional Monte Carlo method. Overall, our results indicate the that the proposed methodology ensures a faster convergence of the considered quantities, thus reducing the time required for their estimation and the associated computational burden.

Average steady flow toward a drain through a randomly heterogeneous porous formation

We consider the problem of steady pumping of water from a line drain on the surface of a wet ground. Unlike the classical formulation, which regards the conductivity parameter K as uniformly distributed in the domain, the problem here is solved within a stochastic framework in order to account for the irregular (random), and more realistic, spatial variability of K. Due to the linearity of the problem at stake, we focus on the derivation of the mean Green function G. This is computed by means of an asymptotic expansion.The fundamental result is an analytical (closed form) expression of G which generalizes the classical solution. Based on this, we develop an equivalent conductivity Keq which enables one to tackle the problem similarly to the classical one. In particular, it is shown that the equivalent conductivity grows monotonically with the radial distance r from the drain, and it lies within the range Keq(0) ≤ Keq(r) ≤ Keq(∞) < ∞.

Efficient Image‐Based Simulation of Flow and Transport in Heterogeneous Porous Media: Application of Curvelet Transforms

AbstractFlow, transport, reaction, adsorption, and deformation (FTRAD) are phenomena that occur in a wide variety of heterogeneous porous media and have been studied for decades. Models of heterogeneous porous formations involve approximations, simplifications, and even unjustified assumptions. With advances in instrumentation, obtaining highly resolved images of such media has become feasible, opening the way to direct image‐based simulation of the FTRAD. Such simulations require, however, intensive computations, limiting their utility. We propose a novel methodology whereby (i) the image is taken to the curvelet space, (ii) all the curvelet coefficients that are smaller than a threshold are set to 0, and (iii) the image is reconstructed. By carrying out image‐based simulations of diffusion and fluid flow in two distinct types of heterogeneous porous media, we demonstrate that simulations in the reconstructed images yield as accurate results as those in the original ones while speeding up the computation by 2 orders of magnitude or better. Thus, the methodology represents a significant step toward the ultimate goal of making direct image‐based simulation of the FTRAD in heterogeneous porous media a standard practice.

How does resolution of sedimentary architecture data affect plume dispersion in multiscale and hierarchical systems?

The potential impacts of the spatial resolution of sedimentary structure data on solute dispersion in heterogeneous porous formations are investigated with Lagrangian-based transport models. The models rely on a covariance function that represent a hierarchical organization of sedimentary facies types by using facies physical properties, such as volume proportions and mean length as well as auto- and cross-transition probabilities and their log-permeability covariance. A detailed sedimentary architecture data (i.e. indicator data) provides better representation of the spatial correlation structures of the global covariances through capturing its underlying structure defined by transition probabilities more accurately. However, the extent to which such data affect time-dependent transport parameters (i.e. dispersivity) is unclear. In this study, we parameterize transport models using detailed collocated sedimentary architecture and permeability data from an outcrop in Española Basin, NM. In addition, we perform global sensitivity analysis based on Polynomials Chaos Expansion to understand the significance of parameters in the transport models. The results show that dispersivity and particle displacement variance are under-estimated if less resolved facies (i.e. indicator) data are used even if the global covariance structure is well captured. Dispersivity is sensitive to the correlation scale that is directly calculated from sedimentary architecture data, and the mean log-permeability. However, anisotropy ratio and mean log-permeability are the most sensitive parameters for the transverse dispersivity.

Experimental evidence of the stochastic behavior of the conductivity in radial flow configurations

We deal with the spatial distribution of the hydraulic conductivity K within heterogeneous porous formations where a radial flow (typical of pumping and slug tests) is taking place. In particular, the study provides a wide data-set which is instrumental to corroborate theoretical findings about the stochastic behavior of K in the above flow configuration. Here, K-data pertain to a series of slug tests conducted within a large caisson which was densely instrumented in order to capture the transitional behavior of the conductivity from the near field (close to the pumping well) to the far field (away from the pumping well). For the experiments at stake, it is shown that the apparent conductivity $$K_{\mathrm{app}}$$ is a very robust property. In fact, with the exception of a very tiny annulus surrounding the pumping well, $$K_{\mathrm{app}}$$ can be used to solve flow (and transport) problems in close analogy to the effective theory approach adopted for a groundwater-type flow. It is hoped that the data-set exploited in the present study will be useful for other researchers who are engaged with similar studies.

Minimum Hydraulic Resistance Uncertainty and the Development of a Connectivity‐Based Iterative Sampling Strategy

AbstractThe presence of well‐connected paths is commonly observed in spatially heterogeneous porous formations. Channels consisting of high hydraulic conductivity (K) values strongly affect fate and transport of dissolved species in the subsurface environment. Several studies have established a correlation between connectivity properties of the spatially variable K‐field and solute first arrival times. However, due to limited knowledge of the spatial structure of the K‐field, connectivity metrics are subject to uncertainty. In this work, we utilize the concept of the minimum hydraulic resistance and least resistance path to evaluate the connectivity of a K‐field in a stochastic framework. We employ a fast graph theory‐based algorithm to alleviate the computational burden associated with stochastic computations in order to investigate both the impact of the hydrogeological structural conceptualization and domain dimensionality (2‐D vs. 3‐D) on the uncertainty of the minimum hydraulic resistance. Finally, we propose an iterative data acquisition strategy that can be utilized to identify the least resistance path (which is linked to preferential flow channels) in real sites. A synthetic benchmark test is presented, showing the advantages of the proposed sampling strategy when compared to a regular sampling strategy. By using the iterative data sampling strategy, we were able to reduce first arrival time uncertainty by 47% (when compared to the regular sampling strategy), while maintaining site characterization efforts constant.

An alternative approach to prepare alginate/acemannan 3D architectures

Current research has explored biomedical applications using polymers and compounds derived from plant sources. In the present work, the interactions between acemannan, the main polysaccharide of aloe vera leaves, and alginate, a marine origin polymer were investigated to design biomaterials. As a proof-of-concept, alginate/acemannan beads were produced through an extrusion dipping method followed by freeze-drying technique to form porous beads. Characterization data confirmed that the Ca2+ presence in acemannan composition resulted in a stronger interaction between alginate and acemannan promoting gel formation and further the development of beads with a micro/nano roughness on their surface and heterogeneous porous formation in their interior. Results obtained until now suggest that the present approach represent a feasible and straightforward platform to design 3D biomaterial architectures for biomedical applications.

On a Class of Integrals Useful to Solve Well‐Type Flows in Heterogeneous Porous Formations

AbstractWe provide analytical (closed form) expression for a class of integrals that frequently arise into modeling well‐type flows through heterogeneous porous formations. In particular, these integrals are encountered when ensemble (second‐order) moments of the flow variables are computed by means of a perturbation approach that regards the variance of the random field (being K the hydraulic conductivity) as a small parameter.

Effective conductivity in steady well-type flows through porous formations

A steady flow generated by a well of given strength takes place in a two-dimensional heterogeneous porous formation where the conductivity K is modeled as a random space function (RSF). As a consequence, the flow-variables become RSF s, and we wish to compute the effective conductivity (EC) by means of the self-consistent approximation. Toward this aim, the porous formation is sought as a collection of circular, non overlapping, inclusions with different (and statistically independent) conductivities. We compute the EC by adapting a procedure which was originally developed for mean uniform flows. Overall, the EC is found to be position-dependent, and therefore it can not be regarded as a medium’s property (unless one is dealing with large distances from the well). Then, it is shown how results of the present study can be used for practical applications.

Revisitation of the dipole tracer test for heterogeneous porous formations

Abstract In this paper, a new analytical solution for interpreting dipole tests in heterogeneous media is derived by associating the shape of the tracer breakthrough curve with the log-conductivity variance. It is presented how the solution can be used for interpretation of dipole field test in view of geostatistical aquifer characterization on three illustrative examples. The analytical solution for the tracer breakthrough curve at the pumping well in a dipole tracer test is developed by considering a perfectly stratified formation. The analysis is carried out making use of the travel time of a generic solute particle, from the injection to the pumping well. Injection conditions are adapted to different possible field setting. Solutions are presented for resident and flux proportional injection mode as well as for an instantaneous pulse of solute and continuous solute injections. The analytical form of the solution allows a detailed investigation on the impact of heterogeneity, the tracer input conditions and ergodicity conditions at the well. The impact of heterogeneity manifests in a significant spreading of solute particles that increases the natural tendency to spreading induced by the dipole setup. Furthermore, with increasing heterogeneity the number of layers needed to reach ergodic conditions become larger. Thus, dipole test in highly heterogeneous aquifers might take place under non-ergodic conditions giving that the log-conductivity variance is underestimated. The method is a promising geostatistical analyzing tool being the first analytical solution for dipole tracer test analysis taking heterogeneity of hydraulic conductivity into account.

On the Longitudinal Dispersion in Conservative Transport Through Heterogeneous Porous Formations at Finite Peclet Numbers

AbstractWe consider transport of a conservative solute through an aquifer as determined: (i) by the advective velocity, which depends upon the hydraulic conductivity K and (ii) by the local spreading due to the pore‐scale dispersion (PSD). The flow is steady, and it takes place in a porous formation where, owing to its erratic spatial variations, the hydraulic log conductivity is modeled as a stationary Gaussian random field. The relative effect of the above mechanisms (i)–(ii) is quantified by the Peclet number which, in most of the previous studies, was considered infinite (i.e., no PSD) due to the overtake of advective heterogeneities upon the PSD. Here we aim at generalizing such studies by accounting for the impact of finite Pe on conservative transport. Previous studies on the topic required extensive numerical computations. In the present note, we remove the computational burden by adopting the rational approximate expression of Dagan and Cvetkovic (1993) for the covariance of the velocity field. This allows one to obtain closed form expressions for the quantities characterizing the longitudinal plume's dispersion. Transport can be straightforwardly investigated by dealing with a modified Peclet number incorporating both the PSD and the aquifer's anisotropy. The satisfactory match to Cape Cod field data suggests that the present theoretical results lend themselves as a useful tool to assess the impact of the PSD upon conservative transport through heterogeneous porous formations.

Physicochemical properties, antimicrobial activity and oil release of fish gelatin films incorporated with cinnamon essential oil

Fish skin gelatin films incorporated with various concentrations of cinnamon essential oil (CEO) were prepared and characterized. The results showed that tensile strength (TS), elongation at break (EAB), and water content (WC) of the gelatin based film decreased with the increasing concentrations of CEO, but water vapor permeability (WVP) increased. Addition of CEO improved light barrier property of the film. The Scanning electron microscope (SEM) showed that the heterogeneous surface and porous formation appeared in gelatin-CEO films. Fourier transform infrared spectroscopy analyses (FTIR-ATR) spectra indicated the interactions existed between gelatin and CEO. The gelatin-CEO films exhibited good inhibitory effects against the tested microorganisms (Escherichia coli, Staphylococcus aureus, Aspergillus niger, Rhizopus oryzae, and Paecilomyces varioti) and their antifungal activity seemed to be more effective than the resistance to bacterial growth. In vitro release studies showed an initial burst effect of CEO release and that subsequently slowed down at 40 °C, but the initial burst release was not obvious at 4 °C. The obtained results suggested that incorporation of CEO as a natural antimicrobial agent into gelatin film has potential for developing as active food packaging.

Solution of one-dimensional space- and time-fractional advection–dispersion equation by homotopy perturbation method

This study develops solution of one-dimensional space–time fractional advection–dispersion equation (FADE). Various forms of dispersion and velocity profiles (i.e. space dependent and both space–time dependent) are considered throughout the study. Homotopy perturbation method (HPM) is used to solve the problem semi-analytically. The advantage of HPM is that it does not require much information about the boundary of the aquifer. The initial condition may be measured for an aquifer, but sometimes it is very difficult to specify the boundary conditions. The FADE is employed for modeling the fate of contaminants in both homogeneous and heterogeneous porous formations subject to an increasing spatially dependent source condition. It is found that the contaminant concentration changes with the order of FADE as fractional-order derivative contains the memory of the system, i.e. how the system changes from one integer order to another integer order. FADEs are used to model the non-local system, hence this study helps understand the physical meaning of parameters involved in the velocity and dispersion.