# Application Guide

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The Application Guide is organized into several sections that group similar classical groundwater problems and presents their solution using the STOMP simulator. The examples in this guide were selected to demonstrate the application of the simulator to a variety of thermal and hydrogeologic flow and transport problems while illustrating a range of features available in the simulator. Simultaneously, the application examples serve as verification and benchmark cases wherever possible through comparison to analytic solutions or results reported elsewhere in the literature for similar problems solved using other computer codes. The following outline provides a synopsis of the problems addressed in the Application Guide.

# Flow in Saturated Media

**Theis Problem:**This problem examines the classical transient radial flow problem in which water is extracted by a well that fully penetrates a single aquifer. Comparisons against Theis' analytical solution are given. [Theis, C.V. 1935.*"The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using groundwater storage."***Transaction American Geophysical Union 2:519-524.**]

**Two-Aquifer Problem:**This problem illustrates flow to a well in a confined aquifer multiple-layered system, where two identical aquifers are separated by an aquitard. The well produces only from the lower aquifer, where it is fully penetrating. This problem is known in the literature as the leaky aquifer problem. Comparisons against analytical solutions are given.**Flow to Two Wells in a nonhomogeneous Domain:**This problem illustrates the effects of pumping in a complex confined aquifer made up of seven anisotropic soil/rock types. Two pumping wells produce from the bottom quarter of the aquifer. Comparisons against the published numerical solutions of Morris and Reddell are given [Morris, G.L. and D.L. Reddell. 1991.*"The Laplace transform finite difference method for simulation of flow through porous media."*.**Water Resources Research**27(8):1873-1884.].

# Transport in Saturated Media

**One-Dimensional Transport in a Uniform Steady Flow Field:**This problem illustrates the transport of a solute within a steady-state, uniform flow field. An initial square pulse of solute mass is instantaneously introduced into the flow field and transported downstream. The pulse undergoes advection, dispersion, and molecular diffusion. An analytical solution by van Genuchten and Alves is used for comparison. [van Genuchten, M.T., and W.J. Alves. 1982.**Analytical Solutions of the One-Dimensional Convective-Dispersive Solute Transport Equation.**ARS Technical Bulletin 1661, USDA, Washington, D.C.]**Patch Source:**The patch source problem was used by Daus and Frind as a test problem for a finite element Galerkin transport code, while an analytical solution was given by Cleary and Ungs. In this problem a fixed concentration boundary condition is used as a source in a steady, uniform, two-dimensional flow field. Results are compared against the analytical solution of Cleary and Ungs. [Cleary, R.W., and M.J. Ungs. 1978.**Groundwater Pollution and Hydrology.**Report 78-WR-15, Department of Civil Engineering, Princeton University, Princeton, New Jersey.].

# Salt-Water Intrusion & Density-Driven Flow

**Henry's Problem:**Henry's problem addresses the steady-state solution of a diffused salt water wedge within a confined aquifer balanced against a flowing fresh-water field. Fresh water enters the confined aquifer at a constant rate from a hypothetical inland boundary and discharges into a hypothetical coastal boundary. Salt water from the costal boundary advances and mixes against the discharging fresh water. Because both the inland and costal boundary conditions are invariant, a steady-state condition is reached, which balances the intruding sea-water wedge against the fresh-water flow field. Henry published an analytical solution to this problem in a U.S. Geological Survey publication, and the problem has henceforth become a classical test for numerical simulators with solute-dependent density capabilities. Unfortunately, no other numerical method has been able to successfully duplicate Henry's solution. This accordingly resulted in some doubt about its validity. Ségol revisited Henry's solution and noted several discrepancies in the published solution. Ségol's revisited solution to this classical problem shows close agreement with the numerical solution of Voss and Souza. Comparisons for this application problem will be made against the revisited solution of Ségol. [Ségol, G. 1994.**Classical Groundwater Simulations: Proving and Improving Numerical Models,**Prentice-Hall, Englewood Cliffs, New Jersey.]**Elder's Problem:**The original problem described by Elder addresses transient thermal convection in porous media. Elder considered a two-dimensional rectangular enclosure filled with porous media. The bottom surface was heated over a segment of its horizontal extent and the remaining walls were held at constant temperature conditions. All surfaces of the enclosure were considered to be impermeable to fluid flow and thermally conducting. Isothermal and hydrostatic initial conditions are assumed. Elder showed generally close agreement between laboratory observations and the numerical solutions. This application problem involves the solution of an analogous problem to Elder's problem, which was first formulated by Voss and Souza as a verification exercise for numerical simulators to represent bulk fluid flow driven only by density differences. Density-driven advection in Elder's problem occurs thermally, whereas the Voss and Souza formulation involves coupled flow and solute transport, where aqueous phase density is dependent on the solute concentration. Either Elder's original thermally driven density problem or Voss and Souza's solute-driven density problem could have been selected for inclusion in this application guide. The latter was chosen as an additional coupled flow and solute transport problem. [Voss, C.I. and W.R. Souza. 1987.*"Variable density flow and solute transport simulation of regional aquifers containing a narrow freshwater-saltwater transition zone."***Water Resources Research.**23(10):1851-1866.].

# Flow and Transport in Unsaturated Porous Media

**Haverkamp's Infiltration Experiments:**This problem addresses the infiltration of water into a uniform laboratory-scale soil column following the experiments of Haverkamp et al. for very dry soils. STOMP simulation results are compared against the Haverkamp's experimental results. [Haverkamp, R., M. Vauclin, J. Touma, P.J. Wierenga and G. Vachaud. 1977.*"Comparison of numerical simulation models for one-dimensional infiltration."***Soil Science Society of America Journal,**41:285-294.]**Tank Leak Simulation:**This problem illustrates a hypothetical but practical case. The bottom of a liquid storage tank sits at ground level above a sandy soil. The tank is a repository for liquid waste and has been leaking for a long time. At simulation-time zero a new waste/solute is introduced to the tank, at which point it begins to leak into the underlying soil. The sandy soil contains isolated lenses of either clay or gravel, and the water table below has a gradient from west to east. The development of the resulting contaminant plume is simulated here.**Two-Phase, One-Dimensional Infiltration:**Touma and Vauclin demonstrated, both experimentally and numerically, the effects of air flow on water infiltration in a 93.5-cm column of soil. Touma and Vauclin applied three types of boundary conditions to the top of the soil column: 1) positive time-constant liquid head, 2) positive time-constant aqueous-phase liquid flux less than the saturated soil hydraulic conductivity, and 3) positive time-constant aqueous- phase liquid flux greater than the saturated hydraulic conductivity. In this section, the positive time-constant aqueous-phase liquid head (referred to as ponded infiltration) simulations of Touma and Vauclin are repeated using STOMP to demonstrate the impact of air flow on water infiltration rates and compare solutions with the previously published experimental and numerical results. Two cases are described: a column open to air flow at the base, and a column closed at the base so that air must flow out the top of the column as it is displaced by infiltrating water. [Touma, J. and M. Vauclin. 1986.*"Experimental and numerical analysis of two-phase infiltration in a partially saturated soil."***Transport in Porous Media,**27-55.]**Radon Vapor Transport into Dwellings:**Investigators have shown that subsurface air advection patterns can significantly affect radon entry into buildings. Nero reported that when structures are located on soils with large intrinsic permeabilities, extraordinarily high indoor radon concentrations can occur even when the radium content of the soil is low. Other researchers have documented high radon activity in groundwater. The combination of these two factors poses an interesting question of whether degassing of radon from ground water may contribute to indoor radon levels, especially when the structures are slightly under pressurized with respect to atmospheric conditions. In previous modeling efforts, the source of radon is assumed to be evenly distributed radon-producing radium in the soil; no consideration was given to groundwater as a potential source of radon. [Nero, A. 1989.*"Earth, air, radon and home."***Physics Today,**42(4):32-39.]

# Nonisothermal Flow

**Single-Node Evaporation:**The single-node evaporation problem involves the evaporation of aqueous phase water within a closed adiabatic volume of partially saturated porous medium. The evaporation process occurs through the addition of heat into the control volume. Sufficient heat is added to the system to desaturate the porous medium. Because the control volume surfaces are closed and adiabatic, the initial quantities of air and water mass will remain constant throughout the simulation. Upon desaturation of a node, STOMP switches the primary variable for the water mass conservation equation from the liquid pressure to the water-vapor mass fraction. This problem demonstrates this numerical transition from two-phase conditions to gas-phase-only conditions. The problem requires the coupled solution of the water mass, air mass, and energy conservation equations.**Single-Node Condensation:**The single-node condensation problem involves the condensation of water vapor within a closed adiabatic volume of porous medium initially desaturated. The condensation process occurs through the removal of heat from the control volume. Sufficient heat is removed from the system to cause condensation and partially saturate the porous medium. Because the control volume surfaces are closed and adiabatic, the initial quantities of air and water mass should remain constant throughout the simulation. Upon partial saturation of a desaturated node, STOMP switches the primary variable for the water mass conservation equation from the the water-vapor mass fraction to the aqueous pressure. This problem demonstrates this numerical transition from gas-phase-only conditions to two-phase conditions. The problem requires the coupled solution of the water mass, air mass, and energy conservation equations.**Single-Node Thawing:**The single-node thawing problem involves the melting of ice within a closed adiabatic volume of porous medium initially desaturated. The thawing process occurs through the removal of heat from the control volume. Sufficient heat is removed from the system to cause melting and partially saturate the porous medium. Because the control volume surfaces are closed and adiabatic, the initial quantities of air and water mass should remain constant throughout the simulation. This problem demonstrates this numerical transition from frozen to unfrozen conditions. The problem requires the coupled solution of the water mass, air mass, and energy conservation equations under freezing soil conditions.**Flow from Hot Two-Phase Conditions:**This problem is concerned with two-phase flow between two adjacent cubes of porous medium with sharply differing initial conditions. The problem starts with both nodes in two-phase conditions; however, one node is nearly saturated with relatively cool water and the other is nearly desaturated at an elevated temperature. The problem involves simulating imbibition and vaporization of the liquid water from relatively cool saturated conditions to hot dry conditions. The control volume surfaces surrounding the two nodes are adiabatic and impermeable to fluid flow. The problem proceeds until equilibrium conditions are reached. This simulation demonstrates the conservation of water mass, air mass, and thermal energy by the STOMP simulator for two-phase flow conditions. An energy balance error analysis is presented.

# Heat Pipe Flow and Transport

**Evaporation/Condensation Heat Pipe:**Hydrogeologic heat pipes have been shown to occur in partially saturated soils subjected to thermal gradients. For example, radial heat pipes have been created around the nuclear waste packages emplaced in variably saturated and fractured rock during the Engineering Barrier Design Test at the Yucca Mountain Exploratory Shaft Test Site. The general requirements for creating countercurrent hydrothermal (i.e., heat pipe) flow in geologic media are a heat source and heat sink separated by partially saturated porous media. Because of the importance of heat pipe flow to the overall heat transfer of engineered geologic systems, the ability of the numerical simulator to accurately and efficiently predict these complex and multiple-phase flow structures is imperative. The heat pipe problem chosen for solution is a modified version of the problem posed and solved by Udell and Fitch. [Udell, K.S., and J. S. Fitch. 1985.*"Heat and mass transfer in capillary porous media considering evaporation, condensation, and noncondensible gas effects."*In**Proceedings of 23rd ASME/AIChE National Heat Transfer Conference,**Denver, Colorado.]**Freezing/Thawing Heat Pipe:**Frozen soil barriers, referred to as cryogenic barriers, have been proposed for temporarily containing plumes of radioactive and/or organic contamination within the subsurface environment. Predicting the effectiveness of cryogenic barriers and near-surface barriers in temperate or arctic climates requires capabilities for numerically modeling subsurface flow and transport for freezing soil conditions. Field-scale experiments of frozen soil barriers require significant investments in refrigeration and monitoring equipment and time for planning and executing. Numerical modeling of cryogenic barrier systems with proven, physically based simulators can cut the requirements for field-scale testing, by providing mechanisms for appropriately scaling laboratory experiments to field applications. Critical components of a physically based simulator for freezing conditions are the constitutive relations for predicting liquid water and ice saturations and aqueous relative permeabilities as a function of temperature, interfacial pressure differences, and osmotic potential. This problem demonstrates the application of the STOMP simulator to a problem involving hydrothermal flow in a horizontal cylinder where one end of the cylinder is held below the freezing point, following the experiments of Jame. [ Jame, Y.W., and D.I. Norum. 1980.*"Heat and mass transfer in a freezing unsaturated porous medium."***Water Resources Research,**16(4):811- 819.]

# NAPL Flow and Transport

**Infiltration and Redistribution of Oil in a Hypothetical, Two-Dimensional Aquifer:**The objective of this example is to investigate the effects of fluid density and viscosity on the movement of NAPLs after a spill in a partly saturated, hypothetical, aquifer. Infiltration and redistribution of a finite quantity of oil in a vertical section are considered. STOMP generated results are compared with simulations conducted with the MOFAT code of Kaluarachchi and Parker. [Kaluarachchi, J.J, and J.C. Parker. 1989.*"An efficient finite element method for modeling multiphase flow."***Water Resources Research,**25(1):43-54].**Infiltration and Redistribution of Dense and Light NAPLs in Partially Saturated Sand Columns:**In this problem STOMP generated data are compared with experimentally determined fluid saturations during the infiltration and redistribution of an LNAPL (Soltrol¨) and a DNAPL (carbon tetrachloride) in a partly saturated one-dimensional column. The main objective is to evaluate the performance of the Brooks-Corey and the van Genuchten pressure-saturation relations in combination with either the Burdine or Mualem pore-size distribution model. The experimentally determined fluid saturations are compared with simulated results from four relative permeability-saturation-pressure (k-S-p) models. The four models are the Brooks-Corey-Burdine (BCB), Brooks-Corey-Mualem (BCM), van Genuchten- Burdine (VGB), and van Genuchten-Mualem (VGM) models. Simulations are compared against the experimental data of Oostrom. [Oostrom, M., R.J. Lenhard, and M.D. White. 1995.*"Infiltration and redistribution of dense and light nonaqueous phase liquids in partly saturated sand columns."*In**Proceedings of Fifteenth Annual American Geophysical Union Hydrology Days,**Fort Collins, Colorado, 215-226.]**Density-Dependent Gas Advection of Trichloroethylene (TCE):**In this problem results obtained with the STOMP simulator are compared to experimentally determined TCE gaseous concentrations. The experimental investigation was conducted to evaluate whether apor-density effects are important in moving contaminant vapors through the subsurface. TCE was studied because it represents a volatile organic compound (VOC) that has a high vapor pressure and molecular weight and it is a ubiquitous contaminant in the subsurface. VOCs, such as solvents and hydrocarbon fuels, are commonly found in the subsurface at many sites. Typically, industrial VOCs have entered the subsurface as nonaqueous-phase liquids (NAPLs) via chemical spills, leaks in storage or transmission structures, and direct disposal to waste sites. VOCs have characteristically a high vapor pressure at normal temperatures and pressures near the earth's surface; therefore, a substantial mass of VOCs will likely be present in the gaseous phase of the subsurface. Simulations are compared against the experimental data of Lenhard and Oostrom. [Lenhard, R.J., M. Oostrom and M.D. White. 1995.*"Investigation of density-dependent gas advection of trichloroethylene: experiment and a model validation exercise."***Journal of Contaminant Hydrology**19(1):47-67.]

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