Equivalent Continuum Model Research Articles

Upscaled Anisotropic Methane Hydrate Critical State Model for Turbidite Hydrate‐Bearing Sediments at East Nankai Trough

AbstractTurbidite formation is a common feature of natural hydrate‐bearing sediments that has been observed and reported at several hydrate exploration sites. It is therefore important to incorporate this anisotropic geological feature into the constitutive modeling when evaluating the geomechanical risks involved during hydrate‐based gas production. To date, a number of constitutive models have been proposed to capture the isotropic geomechanical behavior of homogeneous hydrate‐bearing sediments. Since the turbidite formation contains soil layers at a scale much smaller than the size of the numerical element used for reservoir scale simulations, it is necessary to upscale the geomechanical behavior of a layered system to an equivalent anisotropic continuum model by adopting some homogenization techniques. In this study, an anisotropic methane hydrate critical state model is developed by modifying the original isotropic version of Uchida et al. (2012, https://doi.org/10.1029/2011JB008661). The calibration methodology of the anisotropic model parameters for a given set of hydrate heterogeneity and the turbidite formation at the Eastern Nankai Trough is proposed and demonstrated. The upscaled parameters are calibrated by curve fitting the numerically simulated stress‐strain curves of the layered system with the original isotropic constitutive model at the layered scale. Forty‐two sets of model parameters are calibrated from different site element models of this site. They are used to develop empirical correlations between the model parameters and the site input properties within the turbidite formation. This paper presents the details of the new anisotropic constitutive model and the performance of the proposed upscaling procedure for the Eastern Nankai Trough case.

3D modelling of the Excavation Damaged Zone using a Marked Point Process technique

The Excavation Damage Zone (EDZ) is a key aspect in the assessment of engineering feasibility and long-term safety of deep geological repositories for radioactive waste. The EDZ represents a possible release path for radionuclides dissolved in the porewater and for gases from corrosion and degradation processes which needs to be addressed quantitatively in Safety Assessment (SA). The EDZ is often represented by a discrete fracture network, whose complexity precludes its implementation in conventional SA models. To overcome this problem, two-dimensional abstraction methodologies have been developed that replace the complex geometry of the discrete fracture network by a hydraulically equivalent continuum model. Although such simple 2D models are amenable to SA modelling, they cannot capture the inherent three-dimensional and stochastic nature of the phenomena associated with the EDZ. In this paper we present a methodology based on Marked Point Processes to infer 3D stochastic continuum models of the EDZ from 2D discrete characterizations of the fracture network. The methodology is illustrated with the calibration of a multi-rate injection experiment in Mont Terri (Switzerland).

Experimental and Numerical Investigations of Influence on Overland Flow and Water Infiltration by Fracture Networks in Soil

Preferential flow is common in clay or expansive clay soils, involving water bypassing a large portion of the soil matrix. Dye tracer experiment and numerical modeling are used to simulate the surface runoff and subsurface preferential flow patterns influenced by the soil fracture network of a relatively steep hillslope system (slope angle equals to 10 degrees). The result of the experiments indicates that part of the water is infiltrated through cracks, leading to the delay of the initial runoff-yielding time and reduction of the discharge of the surface runoff. The soil water flow is mainly in the matrix when the intensity of precipitation is low. With the increasing of precipitation, soil water movement may become in the form of preferential flow through cracks. In addition, the nonuniformity of soil water infiltration and the depth of the average water infiltration increase as the precipitation intensity increases. To this end, the complete coupling model was established by using the surface-matrix-crack (SMC) model to simulate water flow within discrete fracture as well as to simulate water flow in the soil matrix based on the concept of dual permeability using the traditional Richards’ equation. In this model, the “cubic law” of fluid motion in cracks within smooth parallel plates and the two-dimensional diffusion wave approximation to Saint-Venant equations with momentum term ignored (two-dimensional shallow water equations) were used. The model divides soil water infiltration into two forms and uses the overall method to calculate the exchange of water between the crack networks and matrix regions as well as the exchange water between surface runoff and infiltration water. Results indicate that the SMC model has better performance compared with the traditional equivalent continuum model when those models are used to simulate the surface runoff movement and the soil water movement in the presence of cracks.

Vibrational analysis of sandwich sectorial plates with functionally graded sheets reinforced by aggregated carbon nanotube

In the present work, by considering the agglomeration effect of single-walled carbon nanotubes, free vibration characteristics of functionally graded nanocomposite sandwich sectorial plates are presented. The volume fractions of randomly oriented agglomerated single-walled carbon nanotubes are assumed to be graded in the thickness direction. To determine the effect of carbon nanotube agglomeration on the elastic properties of carbon nanotube-reinforced composites, a two-parameter micromechanical model of agglomeration is employed. In this research work, an equivalent continuum model based on the Eshelby–Mori–Tanaka approach is considered to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented straight carbon nanotubes. The two-dimensional generalized differential quadrature method as an efficient and accurate numerical tool is used to discretize the equations of motion and to implement the various boundary conditions. The proposed sectorial plates are simply supported at radial edges, while all possible combinations of free, simply supported, and clamped boundary conditions are applied to the other two circular edges. The benefit of using the considered power-law distribution is to illustrate and present useful results arising from symmetric and asymmetric profiles. The effects of agglomeration, geometrical, and material parameters together with the boundary conditions on the frequency parameters of the sandwich functionally graded nanocomposite plates are investigated. It is shown that the natural frequencies of structure are seriously affected by the influence of carbon nanotubes agglomeration. This study serves as a benchmark for assessing the validity of numerical methods or two-dimensional theories used to analyze the sandwich sectorial plates.

Seepage simulation of high concrete-faced rockfill dams based on generalized equivalent continuum model

This research focused on the three-dimensional (3D) seepage field simulation of a high concrete-faced rockfill dam (CFRD) under complex hydraulic conditions. A generalized equivalent continuum model of fractured rock mass was used for equivalent continuous seepage field analysis based on the improved node virtual flow method. Using a high CFRD as an example, the generalized equivalent continuum range was determined, and a finite element model was established based on the terrain and geological conditions, as well as structural face characteristics of the dam area. The equivalent seepage coefficients of different material zones or positions in the dam foundation were calculated with the Snow model or inverse analysis. Then, the 3D seepage field in the dam area was calculated under the normal water storage conditions, and the corresponding water head distribution, seepage flow, seepage gradient, and seepage characteristics in the dam area were analyzed. The results show that the generalized equivalent continuum model can effectively simulate overall seepage patterns of the CFRD under complex hydraulic conditions and provide a reference for seepage analysis of similar CFRDs.

NURBS-based analyses of functionally graded carbon nanotube-reinforced composite shells

We in this paper present an effective computational approach for static and free vibration analyses of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shells. The governing equations are approximated based on the first-order shear deformation shell theory (FSDT) and non-uniform rational B-Spline (NURBS) basis functions. The advantage of NURBS in modeling exact geometries of shell structures is exploited. Material properties of FG-CNTRC shells are assumed to be graded through the thickness and estimated according to an equivalent continuum model based on the rule of mixture. The high accuracy and reliability of the present formulation is verified through several numerical examples in comparison with various reference solutions. In addition, some numerical results of FG-CNTRC panels and cylinders are provided. This may be useful for future references.

Application of ERT, Saline Tracer and Numerical Studies to Delineate Preferential Paths in Fractured Granites.

Accurate quantification of in situ heterogeneity and flow processes through fractured geologic media remains elusive for hydrogeologists due to the complexity in fracture characterization and its multiscale behavior. In this research, we demonstrated the efficacy of tracer-electrical resistivity tomography (ERT) experiments combined with numerical simulations to characterize heterogeneity and delineate preferential flow paths in a fractured granite aquifer. A series of natural gradient saline tracer experiments were conducted from a depth window of 18 to 22 m in an injection well (IW) located inside the Indian Institute of Technology Hyderabad campus. Tracer migration was monitored in a time-lapse mode using two cross-sectional surface ERT profiles placed in the direction of flow gradient. ERT data quality was improved by considering stacking, reciprocal measurements, resolution indicators, and geophysical logs. Dynamic changes in subsurface electrical properties inferred via resistivity anomalies were used to highlight preferential flow paths of the study area. Temporal changes in electrical resistivity and tracer concentration were monitored along the vertical in an observation well located at 48 m to the east of the IW. ERT-derived tracer breakthrough curves were in agreement with geochemical sample measurements. Fracture geometry and hydraulic properties derived from ERT and pumping tests were further used to evaluate two mathematical conceptualizations that are relevant to fractured aquifers. Results of numerical analysis conclude that dual continuum model that combines matrix and fracture systems through a flow exchange term has outperformed equivalent continuum model in reproducing tracer concentrations at the monitoring wells (evident by a decrease in RMSE from 199 to 65 mg/L). A sensitivity analysis on model simulations conclude that spatial variability in hydraulic conductivity, local-scale dispersion, and flow exchange at fracture-matrix interface have a profound effect on model simulations.

An Equivalent Continuum Approach to Efficiently Model the Response of Steel Wire Meshes to Rockfall Impacts

Steel wire meshes are a key component of rockfall protection barriers. The efficiency in reproducing the structure response with numerical methods relies upon the specific modelling technique employed to capture the wire mesh behaviour. The fabric of some rockfall meshes, such as chain-links is quite complex, which leads to sophisticated and costly numerical models, if modelled accurately. This paper presents an efficient approach to model the response of steel wire meshes to rockfall impacts by using shell elements to develop an equivalent continuum model. An elastoplastic behaviour is prescribed to the shell elements to reproduce the results of a set of experimental data, carried out on mesh portions under various load paths and boundary conditions. The idea is that simple laboratory tests can be used to calibrate an effective numerical model of the steel wire mesh with a significantly lower computational cost if compared to other effective solutions. The model’s ability in yielding consistent results when implemented at the structure scale is then assessed, based on the data of full-scale impact tests on a three-span low-energy rockfall barrier. The method can be extended to other wire mesh types and can find convenient application on exploring the response of a rockfall barrier with a cost-effective tool.

Evolution of permeability in sand injectite systems

Sand injectite complexes comprise kilometer-scale clastic intrusion networks that act as effective conduits for the migration, accumulation and then recovery of hydrocarbons and other fluids. An equivalent continuum model is constructed to represent a sand injectite reservoir, coupling stress and fluid flow in fractured rock using the continuum simulator TOUGHREACT coupled with FLAC3D to follow deformation and fluid flow. A permeability model, which uses staged percolation models, is proposed to improve permeability estimation of fracture networks by accommodating four different levels of fracture connectivity. This permeability model is confirmed against field and laboratory data, corresponding to the different connectivities of fracture networks. The new constitutive permeability model is incorporated into the coupled hydro-mechanical simulator framework and applied to sand injectites with the analysis of permeability evolution mechanisms and mechanical sensitivity. The results indicate that when the magnitudes of principal stresses increase in a constant ratio, normal closure is the dominant mechanism in reducing fracture aperture and thereby permeability. Conversely, the evolution of stress difference can accentuate aperture and permeability due to an increase in shear dilation for critically or near-critically oriented fractures. Also, the evolution of aperture and related permeability of fractured rock are more sensitive at lower stress states than at higher stress states due to the hyperbolic relationship between normal stress and normal closure of the fractures.

Bounded acceleration traffic flow models: A unified approach

Bounded acceleration plays a critical role in safety, fuel consumption, vehicle emissions, and capacity drop in traffic flow. We present here equivalent second-order continuum and car-following models with bounded accelerations that encompass existing models, including the LWR model with bounded acceleration, Lebacque’s two-phase model, and the bounded acceleration version of Newell’s car-following model.Analytical solutions are presented for (i) an accelerating rarefaction wave on a homogeneous road, where traffic is still in equilibrium in the acceleration zone, and (ii) an inhomogeneous road with varying speed limits, where an accelerating standing wave is continuous over an acceleration zone and traffic states are non-equilibrium but stationary. We find that traffic states with bounded acceleration can be equilibrium or not and stationary or not, and bounded acceleration does not lead to capacity drop with variable speed limits. These theoretical results are verified with numerical solutions of corresponding lead-vehicle problems. This study provides a theoretical foundation of applying variable speed limits to eliminate capacity drop at lane-drop, sag, tunnel, and other bottlenecks. In the future we are interested in extending such bounded acceleration models for other bottlenecks with inhomogeneous jam densities and time gaps.

Multiscale multiphysical analysis of photo-mechanical properties of interphase in light-responsive polymer nanocomposites

The design of the adaptive mechanical behavior of polymer nanocomposites driven by light irradiation has attracted considerable attention in the fields of physical chemistry and mechanical engineering. To assess the photo-mechanical properties of photo-responsive polymer (PRP) nanocomposites reinforced with gold nanoparticles via continuum mechanics, we first investigated interfacial properties of PRP nanocomposites by using a novel multiscale multiphysical model. Particularly, the interfacial interactions between the nanoparticle and polymer matrix under external loading were addressed by combining all-atom molecular dynamics (MD) simulations with finite element (FE) analysis. To explicitly characterize the role of the interfacial interactions between the nanoparticle and polymer matrix in strain energy density under photo-mechanical loading, the effective stiffness and the spatial range of the interphase layer were numerically identified by matching the deformation energy in the FE model with that in the MD model using the energy method and the homogenization theory. It is verified that the equivalent continuum model obtained by the multiscale method satisfactorily predicts not only the overall mechanical properties of nanocomposites, but also the local stress distribution at the interphase as well as the inherent nanoparticle size and photo-isomerization reaction effects of the nanocomposites. The present multiscale analysis reveals that the effective interfacial region around the nanoparticle is considerably weakened as the photo-isomerization reaction progresses, as well as the decrease of the thickness of the interfacial region.

Application of the Mohr-Coulomb Yield Criterion for Rocks with Multiple Joint Sets Using Fast Lagrangian Analysis of Continua 2D (FLAC2D) Software

The anisotropic behavior of a rock mass with persistent and planar joint sets is mainly governed by the geometrical and mechanical characteristics of the joints. The aim of the study is to develop a continuum-based approach for simulation of multi jointed geomaterials. Within the continuum methods, the discontinuities are regarded as smeared cracks in an implicit manner and all the joint parameters are incorporated into the equivalent constitutive equations. A new equivalent continuum model, called multi-joint model, is developed for jointed rock masses which may contain up to three arbitrary persistent joint sets. The Mohr-Coulomb yield criterion is used to check failure of the intact rock and the joints. The proposed model has solved the issue of multiple plasticity surfaces involved in this approach combined with multiple failure mechanisms. The multi-joint model was implemented into Fast Lagrangian Analysis of Continua software (FLAC) and is verified against the strength anisotropy behavior of jointed rock. A case study considering a circular excavation under uniform and non-uniform in-situ stresses is used to illustrate the practical application. The multi-joint model is compared with the ubiquitous joint model.

Comparison Between Equivalent Continuum and Discrete Crack Models of Transient Radon Transport at the Soil–Building Foundation Crack Interface Using TOUGH2/EOS7Rn

In the framework of radon risk management in France, it is necessary to enhance knowledge on radon transfer from its source to exposure areas (e.g., buildings) by developing simple, accurate, numerical models for transient radon transport in three-dimensional (3D) unsaturated porous materials. The equivalent continuum model (ECM) of flow and transport at the interface between the soil and cracks (fissures) in a building foundation (e.g., slab on grade, basement) is attractive, since equivalent (effective) continuum properties assigned to model cells can represent the combined effect of individual cracks and solid matrix of the cracked concrete of the foundation (slab and blocks walls). Although the ECM approach based on the volume averaging method has been used to model flow and transport through cracks at the soil–building interface, it has never been verified numerically. Thus, the goal of the present work is to develop an ECM using this averaging method and to quantify its uncertainties based on its comparison to an accurate numerical discrete crack model (DCM) for flow and transport in the crack. As a first step, the DCM implemented in the TOUGH2/EOS7Rn module has been verified numerically through a comparison to a reference 3D steady-state numerical solution for radon transport into a house with basement under constant negative pressure. Then, 3D results of the DCM and ECM approaches were compared, under time-dependent indoor–outdoor pressure differentials conditions, for two crack line configurations in the basement slab floor and two different soil configurations with different soil permeability and radium \(^{226}\)Ra mass content values. Results of this comparison show that, for a homogeneous soil configuration, discrepancies between ECM and DCM simulated indoor radon activity concentrations decrease with the increase in soil permeability, regardless crack line configuration in the slab floor and soil radium mass content. However, ECM uncertainties were not within the range of absolute errors on measured radon concentration for the higher soil permeability \((1\times 10^{-9}, 1\times 10 ^{-8} \hbox { m}^{2})\) and the higher \(^{226}\hbox {Ra}\) mass content values (4500 \(\hbox {Bq\;kg}^{-1})\), especially for high radon pics induced by sudden increase in indoor air pressure drop. Regardless soil \(^{226}\hbox {Ra}\) mass content and crack line configuration in the slab floor, the ECM showed to be conservative for the two-layered soil configuration with the presence of aggregates beneath the slab foundation, generally practiced in buildings constructions.

Hydro-mechanically coupled finite-element analysis of the stability of a fractured-rock slope using the equivalent continuum approach: a case study of planned reservoir banks in Blaubeuren, Germany

Construction works associated with the building of reservoirs in mountain areas can damage the stability of adjacent valley slopes. Seepage processes caused by the filling and drawdown operations of reservoirs also affect the stability of the reservoir banks over time. The presented study investigates the stability of a fractured-rock slope subjected to seepage forces in the lower basin of a planned pumped-storage hydropower (PSH) plant in Blaubeuren, Germany. The investigation uses a hydro-mechanically coupled finite-element analyses. For this purpose, an equivalent continuum model is developed by using a representative elementary volume (REV) approach. To determine the minimum required REV size, a large number of discrete fracture networks are generated using Monte Carlo simulations. These analyses give a REV size of 28 × 28 m, which is sufficient to represent the equivalent hydraulic and mechanical properties of the investigated fractured-rock mass. The hydro-mechanically coupled analyses performed using this REV size show that the reservoir operations in the examined PSH plant have negligible effect on the adjacent valley slope.

King post truss as a motif for internal structure of (meta)material with controlled elastic properties.

One of the most interesting challenges in the modern theory of materials consists in the determination of those microstructures which produce, at the macro-level, a class of metamaterials whose elastic range is many orders of magnitude wider than the one exhibited by ‘standard’ materials. In dell’Isola et al. (2015 Zeitschrift für angewandte Mathematik und Physik 66, 3473–3498. (doi:10.1007/s00033-015-0556-4)), it was proved that, with a pantographic microstructure constituted by ‘long’ micro-beams it is possible to obtain metamaterials whose elastic range spans up to an elongation exceeding 30%. In this paper, we demonstrate that the same behaviour can be obtained by means of an internal microstructure based on a king post motif. This solution shows many advantages: it involves only microbeams; all constituting beams are undergoing only extension or compression; all internal constraints are terminal pivots. While the elastic deformation energy can be determined as easily as in the case of long-beam microstructure, the proposed design seems to have obvious remarkable advantages: it seems to be more damage resistant and therefore to be able to have a wider elastic range; it can be realized with the same three-dimensional printing technology; it seems to be less subject to compression buckling. The analysis which we present here includes: (i) the determination of Hencky-type discrete models for king post trusses, (ii) the application of an effective integration scheme to a class of relevant deformation tests for the proposed metamaterial and (iii) the numerical determination of an equivalent second gradient continuum model. The numerical tools which we have developed and which are presented here can be readily used to develop an extensive measurement campaign for the proposed metamaterial.

A comparative study of discrete fracture network and equivalent continuum models for simulating flow and transport in the far field of a hypothetical nuclear waste repository in crystalline host rock

One of the major challenges of simulating flow and transport in the far field of a geologic repository in crystalline host rock is related to reproducing the properties of the fracture network over the large volume of rock with sparse fracture characterization data. Various approaches have been developed to simulate flow and transport through the fractured rock. The approaches can be broadly divided into Discrete Fracture Network (DFN) and Equivalent Continuum Model (ECM). The DFN explicitly represents individual fractures, while the ECM uses fracture properties to determine equivalent continuum parameters. We compare DFN and ECM in terms of upscaled observed transport properties through generic fracture networks. The major effort was directed on making the DFN and ECM approaches similar in their conceptual representations. This allows for separating differences related to the interpretation of the test conditions and parameters from the differences between the DFN and ECM approaches. The two models are compared using a benchmark test problem that is constructed to represent the far field (1×1×1km3) of a hypothetical repository in fractured crystalline rock. The test problem setting uses generic fracture properties that can be expected in crystalline rocks. The models are compared in terms of the: 1) effective permeability of the domain, and 2) nonreactive solute breakthrough curves through the domain. The principal differences between the models are mesh size, network connectivity, matrix diffusion and anisotropy. We demonstrate how these differences affect the flow and transport. We identify the factors that should be taken in consideration when selecting an approach most suitable for the site-specific conditions.

Buckling analyses of double-shell octagonal lattice truss composite structures

To reveal the compression failure modes of one-dimensional hierarchical double-shell octagonal lattice truss composite structures (DLTCSs), finite element modeling and equivalent continuum models were developed. DLTCS has three typical failure modes: (a) fracture of the strut, (b) global buckling, and (c) local buckling. Failure mode maps were constructed. It is found that column of long enough length will collapse at global buckling. When the column length decreases, the failure mode will turn to local buckling and strut fracture successively. Bay length greatly influences the buckling mode. Longer bay length could change the buckling mode from global buckling to local buckling. Compared with single-shell lattice truss composite structure, DLTCS has advantage in load carrying when the column fails at strut fracture or global buckling, while local buckling tolerance of DLTCS is smaller.

An enhanced equivalent continuum model for layered rock mass incorporating bedding structure and stress dependence

Layered rock masses exhibit significant anisotropy in both the deformability and strength. Properties of this rock mass are largely affected by its bedding structure and also by the stress state change. For the purpose of characterizing these salient features using the equivalent continuum concept, an enhanced model is proposed in this paper based on the ubiquitous-joint model. In the enhanced model, the intact bedded rock is assumed to behave as a transversely isotropic elastic body. A modified anisotropic strength criterion is adopted to describe the direction dependence of the strength of intact bedded rock. In addition, stress-sensitive parameters are utilized for both the intact rock and bedding plane, among which the dilation angle and strength parameters of intact rock are influenced by confining pressure and loading history, whereas the stiffnesses of bedding planes are closely related to the current normal stress. The effect of layer thickness on the mechanical behavior of rock mass is reflected by continuously updating the global stiffness matrix after failure of bedding plane, and by conditions for conducting bedding plane related calculations determined by the relative scales between element size and layer thickness. Preliminary validation of this model is conducted by comparing with the closed-form solutions and with laboratory tests. Generally good agreements can be achieved between numerical simulations and theoretical/experimental results, which indicate that applicability of the enhanced model on underground engineering issues related to layered rock mass is promising.

A plate model for multilayer graphene sheets and its finite element implementation via corotational formulation

An equivalent continuum model for multilayer graphene sheets (MLGSs) and its plate model are developed to analyze the deformation behavior of MLGSs. Hyperelastic material models are introduced for the MLGS continuum model by examining the atomistic structures of MLGSs and obtaining their mechanical properties by means of molecular statics simulations. The MLGS plate model, a structural model for MLGSs, is developed by applying kinematics assumptions to the MLGS continuum model subjected to infinitesimal deformation. Finite element methods (FEM) with the corotational formulation are adopted to analyze the mechanical behavior of MLGSs under small-strain deformation and large rotation conditions. The MLGS plate element passes several basic numerical tests, including patch tests, eigenvalue analyses, and geometrically nonlinear benchmark problems. Finally, the deflections of a plane-strain cantilever and spherical indentations are analyzed by the proposed MLGS plate element and molecular dynamics (MD) simulations. These results show that the MLGS plate element properly represents the deformation behaviors of MLGSs from the atomic scale to the macroscopic continuum scale.

An experimental and numerical study on radon transport from UMT-contaminated sand to a house basement under variable weather conditions

Numerical modelling is necessary in order to better understand transient phenomena of radon transport from radium-contaminated soils to a house basement where high radon levels were observed. The present work aims to develop a robust and simplified subsurface numerical model which can simulate radon transport at the soil-basement foundation interface under variable weather conditions around a house. The equivalent continuum modelling (ECM) approach has been used assuming an equivalent porosity and permeability of the basement foundation cracked concrete materials. This approach has been applied for simulating two-dimensional (2D) soil-gas and radon-entry into a house basement built on sand contaminated by uranium mill tailings (UMT) considering the presence of an aquifer with water-table, rainfall, evapotranspiration, atmospheric air pressure, and house under-pressurization.Sensitivity analysis of the model foundation parameters, performed at steady-state, showed that the numerical model predictions at transient conditions were in a good agreement with measurements of the radon-entry-rate through the basement foundation and radon concentration in the basement. The optimized parameters are physically sound and consistent with the observations and literature data.