Designation: D5719 − 13Standard Guide forSimulation of Subsurface Airflow Using Groundwater FlowModeling Codes1This standard is issued under the fixed designation D5719; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (´) indicates an editorial change since the last revision or reapproval.1. Scope1.1 This guide covers the use of a groundwater flowmodeling code to simulate the movement of air in the subsur-face. This approximation is possible because the form of thegroundwater flow equations are similar in form to airflowequations. Approximate methods are presented that allow thevariables in the airflow equations to be replaced with equiva-lent terms in the groundwater flow equations. The modeloutput is then transformed back to airflow terms.1.2 This guide illustrates the major steps to take in devel-oping an airflow model using an existing groundwater flowmodeling code. This guide does not recommend the use of aparticular model code. Most groundwater flow modeling codescan be utilized, because the techniques described in this guiderequire modification to model input and not to the code.1.3 This guide is not intended to be all inclusive. Othersimilar techniques may be applicable to airflow modeling, aswell as more complex variably saturated groundwater flowmodeling codes. This guide does not preclude the use of othertechniques, but presents techniques that can be easily appliedusing existing groundwater flow modeling codes.1.4 This guide is one of a series of standards on groundwa-ter model applications, including Guides D5447 and D5490.This guide should be used in conjunction with Guide D5447.Other standards have been prepared on environmentalmodeling, such as Practice E978.1.5 The values stated in SI units are to be regarded asstandard. No other units of measurement are included in thisstandard.1.6 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.1.7 This guide offers an organized collection of informationor a series of options and does not recommend a specificcourse of action. This document cannot replace education orexperience and should be used in conjunction with professionaljudgment. Not all aspects of this guide may be applicable in allcircumstances. This ASTM standard is not intended to repre-sent or replace the standard of care by which the adequacy ofa given professional service must be judged, nor should thisdocument be applied without consideration of a project’s manyunique aspects. The word “Standard” in the title of thisdocument means only that the document has been approvedthrough the ASTM consensus process.2. Referenced Documents2.1 ASTM Standards:2D653 Terminology Relating to Soil, Rock, and ContainedFluidsD5447 Guide forApplication of a Groundwater Flow Modelto a Site-Specific ProblemD5490 Guide for Comparing Groundwater Flow ModelSimulations to Site-Specific InformationE978 Practice for Evaluating Mathematical Models for theEnvironmental Fate of Chemicals (Withdrawn 2002)33. Terminology3.1 Definitions:3.1.1 For definitions of general technical terms used withinthis guide, refer to Terminology D653.3.2 Symbols:3.2.1 A—cross-sectional area of cell [cm2].3.2.2 g—acceleration due to gravity [cm/s2].3.2.3 h—air-phase or water phase head [cm].3.2.4 k—air phase permeability [cm2].3.2.5 K—hydraulic conductivity [cm/s].3.2.6 P—air phase pressure [g/cm-s2].1This guide is under the jurisdiction ofASTM Committee D18 on Soil and Rockand is the direct responsibility of Subcommittee D18.21 on Groundwater andVadose Zone Investigations.Current edition approved April 1, 2013. Published April 2013. Originallyapproved in 1995. Last previous edition approved in 2006 as D5719 – 95 (2006).DOI: 10.1520/D5719-13.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at

[email protected] For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.3The last approved version of this historical standard is referenced onwww.astm.org.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States13.2.7 P0—reference air-phase pressure [g/cm-s2].3.2.8 qs—specific discharge vector for air [cm/s].3.2.9 q—volumetric flow of water through cell [cm3/s].3.2.10 q*—model-computed term related to airflow in unitsg2-cm/s4.3.2.11 qv—volumetric airflow [cm3/s].3.2.12 qm—mass airflow [g/s].3.2.13 R—universal gas constant = 8.314 × 107[g-cm2/s2-mol-K].3.2.14 Sa—air storage coefficient.3.2.15 Ss—specific storage of the porous material [cm−1].3.2.16 t—time [s].3.2.17 T—temperature [K].3.2.18 z—elevation head [cm].3.2.19 ∂h—hydraulic head difference [cm].3.2.20 ∂l—length of model cell [cm].3.2.21 ρ—density of air [g/cm3].3.2.22 θ—air-filled porosity [-].3.2.23 φ—pressure-squared (P2) [(g/cm-s2)2].3.2.24 ω—average molecular weight of air [g/mol].3.2.25 µ—dynamic viscosity of air [g/cm-s].4. Summary of Guide4.1 The flow of gas (air in this case) through unsaturatedporous media can be approximated using groundwater flowmodeling codes. This is accomplished through substitution ofair-phase parameters and variables into the groundwater flowequations. There are two substitution techniques discussed inthis guide, the pressure-squared technique (1),4and the pres-sure substitution technique (2). These substitutions are sum-marized as follows:4.1.1 The dependent variable, usually head, in the ground-water flow equation becomes pressure or pressure-squared;4.1.2 Saturated hydraulic conductivity (K), both horizontaland vertical components, becomes air permeability (k orintrinsic permeability) in the pressure-squared technique andan equivalent air hydraulic conductivity in the pressure substi-tution technique.4.1.3 Storage coefficient (S) becomes the air storage coeffi-cient (Sa);4.1.4 The Vadose zone is considered a confined aquifer;and,4.1.5 All boundary conditions are expressed in terms of airpressure-squared, although constant flux boundary conditionsmay be used in the pressure substitution technique.4.2 The groundwater modeling code is executed using theseparameter and variable substitutions. The model results mustthen be transformed to values representative of air. Thesecalculations are summarized as follows:4.2.1 If the problem is formulated in terms of air pressure-squared, the square root of the model-computed dependentvariable is computed at each cell;4.2.2 Flow rates computed by the pressure-squared ap-proach must be transformed into equivalent airflow terms forvolumetric flow rates (qv) or mass flow rates (qm).4.2.3 No transformation of the output is required by thepressure substitution technique, although the pressures may beconverted to more convenient units.5. Significance and Use5.1 The use of vapor extraction systems (VES), also calledsoil vapor extraction (SVE) or venting systems, is becoming acommon remedial technology applicable to sites contaminatedwith volatile compounds (3, 4). A vapor extraction system iscomposed of wells or trenches screened within the vadosezone. Air is extracted from these wells to remove organiccompounds that readily partition between solid or liquid phasesinto the gas phase. The volatile contaminants are removed inthe gas phase and treated or discharged to the atmosphere. Inmany cases, the vapor extraction system also incorporateswells open to the atmosphere that act as air injection wells.NOTE 1—Few model codes are available that allow simulation of themovement of air, water, and nonaqueous liquids through the subsurface.Those model codes that are available (5, 6), require inordinate computehardware, are complicated to use, and require collection of field data thatmay be difficult or expensive to obtain. In the future, as computercapabilities expand, this may not be a significant problem. Today,however, these complex models are not applied routinely to the design ofvapor extraction systems.5.2 This guide presents approximate methods to efficientlysimulate the movement of air through the vadose zone. Thesemethods neglect the presence of water and other liquids in thevadose zone; however, these techniques are much easier toapply and require significantly less computer hardware thanmore robust numerical models.5.3 This guide should be used by groundwater modelers toapproximately simulate the movement of air in the vadosezone.5.4 Use of this guide to simulate subsurface air movementdoes not guarantee that the airflow model is valid. This guidesimply describes mathematical techniques for simulating sub-surface air movement with groundwater modeling codes. Aswith any modeling study, the modeler must have a thoroughunderstanding of site conditions with supporting data in orderto properly apply the techniques presented in this guide.6. Pressure-Squared Substitution Procedure6.1 The pressure-squared substitution procedure is adaptedfrom Baehr and Joss (1). The technique allows simulation ofthe flow of gas (air in this case) through porous media usinggroundwater flow modeling codes. This is accomplishedthrough substitution of air-phase parameters and variables intothe groundwater flow equations. These substitutions are sum-marized as follows:6.2 Airflow Equation—The following presentation outlinesthe essential assumptions of the airflow equation. A moredetailed presentation providing justification of the variousassumptions is provided by Baehr and Hult (7).6.2.1 The conservation of mass equation for airflow in anunsaturated porous medium is given by the following:4The boldface numbers in parentheses refer to a list of references at the end ofthis standard.D5719 − 132]]t~ρθ!1π·~ρ;qs! 5 0 (1)6.2.2 Darcy’s Law for airflow is assumed as follows:;qs52ρgµ k πh (2)6.2.3 Hubbert (1940) defined the head for a compressiblefluid as follows:h 5 z11g*P0P1ρdP (3)6.2.4 The Ideal Gas Law is assumed to relate pressure anddensity and thus provides a model for air compressibility asfollows:ρ 5ωPRT(4)6.2.5 Substituting Eq 4 into Eq 3, assuming ω and T areconstant, neglecting the elevation component of head (that issmall for air compared to the pressure component) andsubstituting into Eq 2 gives the following expression forDarcy’s Law in terms of P:;qs521µ k πP (5)6.2.6 Substituting Eq 4 and Eq 5 into Eq 1, and then usingthe following linearizing change of variable suggested byMuskat and Botset (8) for airflow:φ 5 P2(6)yields the following three-dimensional airflow equation inCartesian coordinates that is analogous in form to the ground-water flow equation solved by many groundwater flow models(MODFLOW (9), for example):]]xSkxx]φ]xD1]]ySkyy]φ]yD1]]zSkzz]φ]zD5 Sa]φ]t(7)where x, y, and z are Cartesian coordinates aligned along themajor axes of the permeability tensor with diagonal compo-nents kxx, kyy, and kzz.6.2.7 Air-phase permeability is assumed to be independentof P, therefore, the Klinkenberg slip effect (10) can only bemodeled as constant with respect to P. The coefficient Sais thepneumatic equivalent of specific storage and if air-filledporosity is constant with respect to time (that is, watermovement is neglected) then:Sa5θµ=φ(8)6.2.8 The change of variable φ = P2results in a linearequation for steady-state airflow. The transient equation islinearized by assuming φ1/2= Patmin the definition of Sa,where Patmis the prevailing atmospheric pressure.6.2.8.1 Massmann (2) describes the errors involved with thepressure-squared substitution described above, as well assimply substituting pressure for head. The error in the pressure-squared substitution is less than 1 % when the pressuredifference between any two points in the flow field is less than0.2 atmospheres (atm) and less than 5 % when the pressuredifference is less than 0.8 atm. When substituting pressure(instead of pressure-squared) for head, the errors are similar forpressure differences less than 0.2 atm, but are quite large forpressure differences greater than 0.5 atm. In most cases, thepressure differences will be less than 0.2 atm; therefore, eithersubstitution may be used in environmental modeling (seeSection 7 for a description of the pressure substitution tech-nique).6.2.9 Eq 7 can be directly compared to the linear ground-water flow equation. The simplifying assumptions needed toarrive at this linear airflow equation are summarized asfollows:6.2.9.1 Darcy’s law is valid for airflow;6.2.9.2 The elevation component of pneumatic head isneglected;6.2.9.3 Temperature effects are neglected;6.2.9.4 The Ideal Gas law is a valid model for compress-ibility;6.2.9.5 The Klinkenberg slip effect is neglected;6.2.9.6 Water movement and consolidation are neglected,therefore porosity is constant with respect to time; and6.2.9.7 φ1/2= Patmin definition of storage coefficient Sa.6.2.10 Baehr and Hult (7) examined the consequences of theassumptions presented in 6.2.9. The authors found that thelinear airflow model given by Eq 7 is a good working model foressentially all environmental applications.6.3 Groundwater Flow Equation—The following ground-water flow equation is solved by many groundwater flowmodels:]]xSKxx]h]xD1]]ySKyy]h]yD1]]zSKzz]h]zD2 W 5 Ss]h]t(9)where: x, y, and z are Cartesian coordinates aligned along themajor axes of the hydraulic conductivity tensor with diagonalcomponents Kxx, Kyy,Kzz.6.3.1 The purpose of the procedure presented in this guide isto facilitate airflow simulations by matching Eq 7 and Eq 9 sothat the numerical solution coded in groundwater flow modelscan be used to solve the airflow equation. This is accomplishedwith the following parameter matches:h⇒φ (10)K⇒k (11)Ss⇒Sa(12)6.3.2 The parameter matching allows the hydraulic headand flow output from the groundwater model to be interpretedfor the airflow model in accordance with 6.3.6.4 Boundary Conditions—There are only two permissibletypes of boundary conditions when using the pressure-squaredsubstitution described above. These include constant pressureand no-flow boundaries.6.4.1 Constant pressure cells are actually constant pressure-squared cells. Constant pressure cells are used in two ways:6.4.1.1 Constant pressure cells are set around the perimeterof the model to allow air to flow into the model horizontally,andD5719 − 1336.4.1.2 Venting wells and trenches are defined as constantpressure cells where the pressure is the absolute pressure(squared) maintained in the venting well.6.4.2 An extra layer of constant pressure cells should beadded at the top of the model domain to simulate theconnection between the vadose zone and the atmosphere. Thecells in this top layer and the constant pressure cells around theoutside of the model are maintained at the preva