Over few previous decades, numerous methods have been developed to predict transport and the fate of contaminants in the soil. However, these methods are mostly limited to homogenized one-, two- or, rarely, three-dimensional models and tend to require a high level professionals in order to prepare input data and analyze the results. Typical estimates for exposure and environmental transport currently use the multimedia transport models that utilize compartmental models.
These models account for the presence of and substance transport through various media in gaseous, liquid or solid phases but tend to obscure the physics of the transport processes both within individual phases and at interface boundaries. Hence, the results of compartmental models combined with uncertainties about the transport properties of specific substances provided, as well as of multiscale morphology of underground soil layers at the very best, an order of magnitude estimations.
A hierarchical physics VAT modeling approach to the solution of soil and groundwater contamination transport alleviates the added degrees of uncertainty by including, instead of marginalizing, the physics of the Heterogeneous Multiscale transport processes.
Theoretical developments of HSP-VAT in the area of porous-media-transport-processes-modeling, and in groundwater modeling allow one to develop more precise methods of tracing the fate of chemicals released into the soil. It is clear that the coefficient of permeability, K in the Darcy law notation is at least the very simplified (and combined) characteristic and can not be used by itself as the influence of many physical mechanisms occurring in porous media fluid flow, even for one-structured morphology, is not contained in its variant models.
Homogeneous equations for momentum, heat and mass transport in use at this time are only appropriate for small scale laboratory experiments with a simple single structure morphology porous medium (plus conventional approximation of effective constant coefficients). Effective transport coefficient modeling would be satisfactory if provided on the basis of a specific soil type, the corresponding heterogeneous transport HSP-VAT equations and scaled closure modeling developments.
In the soil and groundwater hierarchical method based on HSP-VAT developed, has usually important features and characteristics that are typically omitted from the modeling. Among the many possible extensions and improvements of existing Homogeneous models we provided consideration of specific soil types and specific morphology (which is usual as for one scale models, but include when HSP-VAT base used the precise morphology description and direct involvement of it), the lower level scale fluctuations of variables, cross-effects of different variable fluctuations, phase and interface variable fluctuations effects.
The new integro-differential HSP-VAT transport equations for a heterogeneous soil and groundwater media and applications of these non-classical types of equations is the topical issue. As it is now silently (at least) agreed upon, in most of the cases, quasi-homogeneous and quasi-stochastic methods are not sufficient for description of the physical process featured in the heterogeneous media.
The results of our middle 90th studies, conceptualization of the modeling of transport phenomena in a multiscale soil and contaminated groundwater system were reported to DOD ARPA in 1994-1995. The smallest scales for continuum phenomena and their modeling for soil and groundwater contamination systems are the cornerstone areas of interest in groundwater modeling. The primary elements of a physical models and new mathematical models for nonlinear transport in different morphological pore systems were developed based on the HSP-VAT.
New mathematical models for nonlinear transport in different morphological pore systems were developed and are described. Analysis of the impact of nonlinear and non-Newtonian viscosity models on flow in nano- and micron scale random pore distribution was made. Morphological analysis of the soil and mining layers and an uncertainty analysis for the specific needs of HSP-VAT modeling were carried out and results were analyzed.
Software development commencement resulted in several computer based nonlinear transport codes. New results were obtained for the nonlinear transport equations. These numerical simulations show the importance of nonlinear and non-Newtonian transport at the lowest morphology level; pore surface, single pore, primary pore network, random pore size distribution morphology. A Top-Down software design and its analysis were reported.
The theory of Hierarchical Scaled Physics and Volume Averaging (HSP-VAT) has been applied directly to develop the five scale transport equation sets
for soil and groundwater contamination systems (SGCS).
The methodology developed dealt with flow and solute modeling in a five scale soil systems. The procedure includes: 1- solid-liquid surface scale modeling; 2 - pore scale; 3 - third (small) scale homogeneous porous medium two phase and two-concentration modeling; 4 - fourth (large) scale heterogeneous porous layer modeling when the Heterogeneity of the that scale morphology of media are taken for modeling, then the very Upper meter and tens of meters scale (5th) underground environment media (mostly of layered structure ) used and modeled. The modeling procedures suggested are believed to give more accurate results while considering the transport of constituents in each of the phases at each level of this hierarchy.
The HSP-VAT based study brought in the range of results confirming in the middle of other that the morphometrical scaled soil characteristics should be an ingredient in local site assessment and used in the multiscaled soil and groundwater modeling methodology.
Among the main results of our scaled SGCS studies should be cited that:
1) Basic subsurface soil types and their physico-chemical characteristics should be incorporated into a database to be used as a supplemental modeling tool. Conceptual decision making procedures are being designed for each level of the model hierarchy. We will assume for our purposes that soil characteristics such as saturated hydraulic conductivity, clay content, bulk density, available soil water content be admitted as classification criteria. These and other parameters of the mining layer will serve as a characterization vector in model selection and for model tuning procedures.
2) As long as sorption from contaminated groundwater to solid surfaces of soil and minerals plays a major role in the distribution and fate of organic pollutants in the SGCS, work must be devoted to developing reasonable phenomenological descriptions of adsorption using the specifics of the current scaled HSP-VAT approach. The specifics of the mineral and pollutant composition prescribe the possibility for attaining a sorption equilibrium.
3) The type of sorption of organic compounds depends greatly on the soil (mineral) organic matter content and ion-exchange for some mineral compositions. Computation of equilibrium compositions for simple aqueous solutions clearly demonstrated that describing solid phase interfaces should be the goal of a separate software development program, as the known codes (for example, MINEQL, MINTEQ etc.) are found to be not applicable.
4) The modified structure of contaminated water affects the filtration process in the real underground environment and leads to inapplicability of the Darcy law. Concerning the applicability of the different forms of the governing equations for irregular curved pores, the first stage of the analysis revealed that for pores with curvature, the governing equations have two parameters which characterize the flow. The influence of curvature on flow is evaluated in terms of these two parameters and in most cases this curvature effect could be neglected. Further, the flow in a curved pore is much more stable than in a straight pore
5) In fine pores, it was found that due to polymorphism of the surface fluid layers the modeling should use the structured near surface film of liquids because it is different from that of the bulk liquid. Calculations of the flow rates on fine scale in models in our HSP-VAT models must take into account the dependence of viscosity on distance from the pore wall. As long as water boundary layers at nanoscale remoteness from the solid phase are characterized by elevated viscosity, and do not display a noticeable critical shear stress, they are important. Measurements of water flow velocity in capillaries at a number of radii ( ranging from 5x10-2 :m to 1 :m ) have been used to determine the Newton viscosity of water in proximity of a solid hydrophobic surface. The average velocity for variable viscosity was calculated and compared with the average velocity for the Poiseuille flow. For pores with R = 10-1 :m the effect was about 20 %. The modified structure of contaminated water was shown to affect the filtration process and leads to inapplicability of the Darcy law even for lowest spatial scales.
6) Further, much of the previous research results were based on substitution of porous medium properties and functions using artificial network simulation capabilities without satisfying the initial equations. The concern here is that the physical problem, which is simulated with the help of network modeling, must still be modeled in such a way that its initial physical and mathematical statements are properly treated. The development of models of irregular and random networks of pores in the subsurface REV, with consequent substitution of closed morpho-convective and morpho-diffusive terms into the transport equations, was a part of the research.
7) The key elements in the design of groundwater network models are correct modeling of pore momentum, mass and heat transport and at the same time incorporating the present level of Heterogeneous (HSP-VAT based) fluid mechanics knowledge about channel junction hydrodynamic resistance. In the Upper the network model, the junction resistance calculation models are also included as ingredients in the overall hydrodynamic resistance presentation. The goal of network simulation was to close the morpho-convective and morpho-diffusive terms in averaged upper scale transport equations.
8) The development of a prototype network momentum computer code has been completed and preliminary results have been obtained. This was the initial step in modeling of a groundwater network at the 3d level (scale). The aim was to use these results in our implementation of the next (fourth) level of the averaged transport equations. Several network morphologies at millimeter scale were studied by computer simulation. Two computer programs were produced. One simulates very low momentum transport while the other focuses on the non-linear analysis of a capillary network. This effort resulted in a different approach for flow resistance calculations.
9) A random pore diameter morphology model was developed and implemented in a computer code. The objective of this scaling (two lower neighboring scales) modeling was to determine the impact of a random pore distribution on averaged and fluctuation values of momentum and heat transfer.
10) The quantitative results of non-Newtonian fluid modeling in porous media differentiate greatly from those for a Newtonian fluid (water) obtained in the groundwater contamination study. An analysis of statistical data modeled on exact solutions for assigned capillary morphology was provided. It showed the tremendous importance of the lowest size pore distribution phenomena. It was found that a negligible part of the overall momentum transport goes through the smallest pores, yet the heat and mass transport through these pores is a large fraction of the bulk transport due to their significant interface transport and enhanced transport peculiarities in fine pores.
11) A non-Newtonian fluid model has been introduced for use with the specific capillary morphology and incorporated into multiscale SGCS for use with the random capillary morphology and that was coded. Simulation results show the importance of non-Newtonian viscosity model in the modeling of bulk and local transport characteristics.
12) Analysis of closed forms of the porous medium transport HSP-VAT averaged equations developed in the work was accomplished for lower three scales and some analytically. These equations usually incorporate two features: nonlinearities of different kinds and nonuniformity of porosity. Analytical solutions and evaluation of some simplified transport equations yielded the tools for analysis of the influence of nonlinearity and nonuniformity of porosity. The closed form of the steady state averaged momentum equation in a porous channel was solved using a Weirstrass function with two invariants. The equation for fluctuating energy was reduced to Abel's equation and expressed in terms of elementary functions.
13) For relatively small nonlinearities, expansion of the VAT averaged fluid momentum equation solution into a power series allowed one to calculate the mutual influence of nonuniform porosity and nonlinearity. The first terms in the expansion show the influence of nonuniformity alone. For a regular porous structure of spherical packing, the velocity function is expressed in terms of modified Bessel functions. Analytical solutions and evaluation of some simplified transport equations yielded the tools for analysis of the influence of nonlinearity and nonuniformity of porosity.
14) The following important cases have been analyzed: the existence of simultaneous strong nonlinearity and slowly varying nonuniformity of porosity. If the change in porosity is characterized by a function f(x) with an assumed slow change on the scale of a typical pore radius a, the rescaling of the coordinate z is possible in a way that a new so-called slow variable Z= Lz was used.
15) Using the nonlinear solutions obtained in the work, a Ginsburg-Landau equation for slowly varying modulation of the equation solution was obtained. This equation is very well studied and its solutions are expressed in terms of Hermitian polynomials allowing one to get analytical solutions for studying many different morphologies.
16) The theory of scaled volume averaging (HSP-VAT) has been applied directly to the development of second, third level (preliminary results for 4-th and 5-th scale systems were obtained) transport multiphysics governing equations. Spatial scales for the channel's hydraulic diameter must be in the range from 10-9 m to 10-3 m if the flow regime studied is to be in the creeping or laminar regimes.
17) One of the research goals was to develop a dual-porosity "near" pore model and computer code featuring a random and irregular interface surface (random surface morphology), permeable solid phase model for transport of momentum, mass and heat with the correct boundary conditions on the pore (crack) surface.
18) A computer code for rough crack (or large pore) momentum and heat (mass) transport has been produced. This code is capable of simulating highly intensive even turbulent transport regimes in a pore or crack of 1 mm to 1 cm hydraulic diameter scale. The results obtained to date are very encouraging. Transport processes in a rough pore with different morphological types of roughness were simulated.
19) The effects of different pore level blocking using the regular pore wall roughness morphology models were simulated. Numerical experiment results were obtained with pore wall roughness of various structure.
20) A second code for rough pore modeling of the laminar momentum transport regime has been developed. The results of a numerical simulation were obtained and analyzed for both regimes and gave new insights. For example, for Turbulent flow regime the main impact on flow characteristics is by height of the obstacle. In Laminar flow the three regimes of the flow exist, depending on the distance between obstacles and their height. All these features should be incorporated into the scaled SGCS modeling and simulation.
21) Exact modeling in each of the large numbers of straight pores was combined using the HSP-VAT closure analysis for a given morphology. Two kinds of pore diameter distributions were used: 1) uniformly distributed diameters in the range 0 to a few millimeters or few centimeters and 2) two-mode regularly distributed pore diameters. These two distributions resulted in dramatically different distributions of velocity, dispersion coefficient and other characteristics.
22) An analysis of statistical data modeled with exact solutions for assigned capillary morphology was provided. It confirmed the known fact while showed the tremendous importance of the lowest size pore distribution phenomena.
23) The most sensitive parameters in the equations of slow flow of contaminated groundwater in the soil were identified. A sensitivity analysis allowed us to determine the parameters with the highest impact on the final results.
As it is clear from the available above texts that the many if not all of the SGCS properties concerned above while being in combined, collective implementation can not be found in the groundwater literature, studies, because of the few reasons:
1) The basis for governing scaled equations were the HSP-VAT techniques, methods. Having the peculiarities through the involvement of nonlinear phenomena treatment.
2) The closure and bi-directional interscale simulation were available due to using the subject of HSP-VAT advancements not only for theoretical mathematical reasoning and equations development, but mainly because the theory was supported by the means and ways of scaled numerical simulation. That path from the theory to reality of numerical simulation was and is still not understood in the community.
3) In spite that in the most of our multiscale modeling and simulation problems and codes were used the known specified shape, pore, network morphologies, that was only to demonstrate the power and capabilities of HSP-VAT applied solutions. The real underground morphologies of any level (scale) can be used and incorporated into the specific site simulation. That, of coarse, requests the more specific and applied geological field's scanning and morphology specification. That is the unavoidable and the must do part of any scaled, hierarchical HSP-VT application, and of groundwater contamination as well.