Notes on the GoldSim Plume Function

Transcription

1 NAC-0036_R1 Note on the GoldSim Plume Function Augut 014 Prepared by John auxe NEPUNE AND COMPANY, INC th St, Suite B, o Alamo, NM 87544

2 itle: Decription: hi document calculation detail of the GoldSim plume function, a interpreted from the ource code: PlumeFuncNode.cpp Printed Name Signature Date Originator John auxe 16 Sep 008 Reviewer Ian Miller 9 Sep 008 Remark: 4 Sep 08 Corrected a typo in the in-line equation in Section.0, and added the image ource illutration in Figure. J 8 Sep 08 Added Figure 1, and made clarifying edit to text. IM 13 Sep 10 Corrected the text in ection 3..3 and to clarify the lack of dilution in cae where the obervation point i within the plume. Aug 014 Revied for releae a a Neptune NAC document. - J Augut 014 ii

3 CONENS 1.0 Introduction Aumption and Boundary Condition Functionality of PlumeFuncNode.cpp Setting Up Initial alue Relative ength Adjutment ranvere Diperion Narrow (Point) Source Wide (General) Source Negligible ranvere Diperion ertical Diperion Fully Mixed Aquifer Ue of Image Source hin Source hick Source Negligible ertical Diperion Final Aembly Implication for Ue of the plume() Function at SRS... 9 FIGURES Figure 1. Plume function geometry (courtey of GoldSim echnology Group) Figure. Placement of image ource relative to the aquifer and actual ource Augut 014 iii

4 1.0 Introduction he plume() function in GoldSim produce a value between 0 and 1, which i to be ued a a multiplier applied to the concentration at the end of a Pipe Pathway element in order to account for lateral and vertical diperion when the Pipe i intended to repreent an aquifer, for example. hee are not allowed in the Pipe calculation, which ha a pecified cro-ectional area and circumference around impermeable wall more like a traw than an aquifer. A point of obervation i identified in three dimenion,,, and, correponding to longitudinal, tranvere (lateral), and vertical location in a coordinate ytem baed on the center of the uptream edge of the ource zone and aligned with the flow of water in the Pipe. he longitudinal location correpond to the end of the Pipe. Figure 1 indicate the geometry that i aumed. Figure 1. Plume function geometry (courtey of GoldSim echnology Group). It hould be noted that a of GoldSim verion 10, the plume() function wa upgraded to handle input that are GoldSim vector and matrice. hee calculation are imply term-by-term calculation, a if the plume() function had been applied to each term independently. hi i ueful in GoldSim programming, ince a ingle ue of the function can be ued to determine plume correction at a large number of independent location. GoldSim contaminant tranport (C) model developed in upport of Performance Aement (PA) at the Savannah River Site (SRS) have applied thi plume correction to concentration at the end of a column of Cell Pathway element, alo ued to repreent an aquifer. hi technique offer everal advantage to the Pipe, the dicuion of which i beyond the cope of thi document. For implicity, reference to Pipe in the remainder of thi document could be replaced with column of Cell, except where noted. Augut 014 1

5 he plume() function ha eleven argument, in the following order. he correponding item in Figure 1 are hown in bracket: length of the Pipe, or, in a Cell column, the longitudinal poition of the obervation point, which hould be made coincident with the center of the Cell nearet the deired column length [1] A cro-ectional area of the Pipe length of the ource parallel to the flow direction [] vertical poition of the obervation point [3] tranvere poition of the obervation point [4] D vertical depth to the top of the ource from the top of the aquifer [5] W width of the ource, tranvere to the aquifer flow [6] b thickne of the ource [7] b thickne of the aquifer [8] α diperivity in the tranvere direction diperivity in the vertical direction α All thee input except the area have dimenion of (length), with the area A in. GoldSim evaluate all argument to enure conitent dimenion and unit, o thi need not be addreed further in thi dicuion. he diperivity in the longitudinal direction doe not enter thi equation directly, ince it i included in the Pipe olution of concentration. For the column of Cell, longitudinal diperion i handled numerically. Other time- and chemical-related parameter are alo handled in the contaminant tranport (C) calculation, a dicued in the following ection..0 Aumption and Boundary Condition A clever apect of the plume() function i that it doe not include a full-blown plume calculation, which would necearily involve tranport-relevant parameter like retardation, decay and ingrowth, and uch. hee are all handled by the tranport Pathway (Pipe or Cell column), o all the plume() function need to do i modify the patial ditribution of contamination. It doe o by providing the calar by which the end-of-the-pipe concentration i modified, effectively dipering the traw-like Pipe calculation in pace. Boundary condition include the aumption of zero concentration at infinite ditance, and a pecial handling of the output of the GoldSim Pipe element. he Pipe element provide a contaminant concentration (in dimenion of M/ 3 ) at the end of the Pipe expreed a a ma ma fluxdicharge(m/) flux dicharge per flow rate, that i,. A dicued below (in Section 3 Pipe flow rate ( /) 3.1.1) the factor calculated by the plume function include the cro-ectional area of the Pipe, A, o that the calculated Pipe concentration i thereby a ma per volume (M/ 3 ): areal ma flux(m/ ). Darcyvelocity (/) Augut 014

6 3.0 Functionality of PlumeFuncNode.cpp PlumeFuncNode.cpp i the C++ ource code for the plume() function, kindly provided by GoldSim echnology Group for the purpoe of thi documentation. 3.1 Setting Up After running all argument through the GoldSim Eval() function, which aure that they all have proper numeric and dimenionally conitent value, and after determining whether the argument are calar, or vector or matrice of calar value, the plume() function begin contructing the anwer that it will return the value (or vector or matrix of uch value) between 0 and 1 that i to be multiplied by the end-of-the-pipe concentration to arrive at a concentration at the deignated obervation location(). hi reult i called dilute within the function Initial alue he value of dilute i calculated in part, with all part multiplied in the end, a decribed in Section 3.4. Initially, we et dilute init to the cro-ectional area of the Pipe, A. A the value of dilute i modified for diperion effect in the ubequent calculation, it i twice divided by a length, leaving it dimenionle. In effect, thi counteract the ue of cro-ectional area in the Pipe calculation. In the development of thi calculation, we hall keep track of the variou part of the equation for dilute that are proffered. So far, we have dilute init = A, (1) where A i the cro-ectional area of the Pipe or column of Cell Pathway element. Equation (1) convert the end-of-the-pipe concentration to the total ma flux leaving the Pipe divided by the total flow in the pipe, a hown in Section.0. In order to convert that to a concentration at an obervation point, it need to be converted to the ma flux rate acro a (mall) unit area at the monitored location, divided by the Darcy velocity. hi i done by multiplying by the -D patial denity at the obervation point, for the denity function in the plane perpendicular to the flow, of a unit ma input uniformly ditributed over the ource region, a dipered after traveling the pecified advective tranport ditance Relative ength Adjutment he length of the Pipe i aumed to be meaured from the uptream end of the ource. It i then modified in order to center the ource at the longitudinal origin. he following cae are conidered: Cae: if <= 0, then the point of obervation i uptream of the ource, and dilute = 0 () init At thi point, i alo et to = 1 (with unit of m implied), to preclude divide-by-zero error. Augut 014 3

7 Cae: if >, then the point of obervation i downtream of the ource, and i adjuted thu = o that the ource end of the Pipe i moved to the center of the ource. hi adjuted length i denoted. Cae: if neither of the above cae i true, then <=, meaning that the Pipe i horter than the ource. In thi cae, i halved: = hi, in effect, truncate the ource to be the length of the Pipe, and the uptream end i moved to the center of the ource. In the latter two cae, the value of dilute init remain et equal to A. 3. ranvere Diperion In accounting for tranvere diperion, variou cae are conidered: a narrow (point) ource, a wide ource, and the cae where the obervation point i too far afield to be conidered part of the plume. he cae are preented a follow: 3..1 Narrow (Point) Source A narrow ource trigger the point-ource olution for tranvere diperion. In comparing the ource width W to the tranvere tandard deviation of the plume α, a narrow ource olution i ued if W < α 100 where the value of 100 i hard coded into the function, baed on the judgment of the developer. In thi cae, the point-ource olution i ued, and the tranvere portion of dilute, hereafter called dilute, aume the form of the normal ditribution denity function:, (3) (4) (5) dilute exp = π α α exp 4α = 4πα (6) Augut 014 4

8 Keep in mind that dilute i to be multiplied by dilute init a defined in Section 3.1 in order to arrive at the final anwer. 3.. Wide (General) Source A wide ource, the general full olution, i indicated if the following i true: W < (7) 1,000,000 α hi applie in the cae where a ource i not negligibly wide, and diperion cannot be ignored. In thi cae, the point-ource olution i integrated over the width of the ource, and dilute i dilute W W + erf erf = 4 α 4α W (8) 3..3 Negligible ranvere Diperion If neither of the previou two ituation occur, then the tranvere diperion i negligible (i.e. the plume doe not pread at all), and the obervation point i teted to ee if it i inide or outide of the plume. he tet for being outide the plume i if the tranvere location of the obervation point i greater than half the ource width from the plume centerline: W > In effect, if the obervation location i further from the centerline than the edge of the ource, then it i out of the plume. If equation (9) hold true, then the point i outide the plume, and dilute (and therefore dilute, a decribed in Section 3.4) i et to zero. Otherwie dilute i et equal to the invere of the ource width, W. 3.3 ertical Diperion In the cae of diperion in the vertical direction, again perpendicular to flow, five cae are conidered: a fully mixed aquifer, a thin ource, a thick ource, no ource geometry adjutment, and the cae where the obervation point i too hallow or deep to be conidered part of the plume. Since the flow field i aumed to have natural boundarie on the top and bottom, but not on the ide, the calculation of concentration at vertical location within the aquifer take advantage of uperpoed image ource mathematic. Note that, unlike the tranvere cae, where the ource i alway centered horizontally, the vertical poition of the ource may be anywhere within the flow field. Similar to the tranvere cae, however, the vertical apect of the ource may be relatively thin or thick compared to vertical diperion. he cae are preented a follow: (9) Augut 014 5

9 3.3.1 Fully Mixed Aquifer If an aquifer (or, more generically, the flow field) i ufficiently thin in comparion to diperion in the vertical direction, then the flow field can be aumed to be fully mixed in the vertical direction, and no image ource are ued. hat i, if we compare the aquifer thickne b to the vertical tandard deviation of the plume α, and b < α (where again the value of 3 i choen by the developer) then dilute i merely divided by that thickne, o that the vertical term dilute become 3 1 dilute = b (10) (11) 3.3. Ue of Image Source If the aquifer i not ufficiently thin to be conidered fully mixed, it i examined for uitability in uing nine image ource to refine the vertical ditribution of contaminant, a illutrated in Figure. In effect, the plume from each image ource, identical to the original ource, i offet and it contribution to the concentration in the aquifer i uperimpoed. hi ha the ame effect a bouncing the original plume off the top and bottom of the aquifer. Image ource are invoked if, analogou to equation (7), b < α (1) 1,000,000 where the value of 1,000,000 i elected by the developer. If thi inequality i not true, then no image ource are ued, and no further change are made to dilute for vertical diperion (and the following ection teting for a thin or thick ource are alo precluded). ike the tranvere cae preented in equation (7), equation (1) i true only for very mall diperivitie. Augut 014 6

10 Figure. Placement of image ource relative to the aquifer and actual ource hin Source A thin ource trigger the point-ource olution for vertical diperion, expanded to include the additional image ource (the development of which i not covered here). In comparing the aquifer thickne b to the vertical tandard deviation of the plume olution i ued if b < α 100 α, a thin ource In thi cae, the point-ource olution i ued with i image at vertical location i relative to the ource vertical centerline, and dilute i the ummation of a number of normal ditribution denity term: (13) Augut 014 7

11 dilute = i exp i 4α 4πα Here dilute i to be multiplied by dilute init a defined in Section 3.1 and dilute from Section 3. in order to arrive at the final anwer. If the criterion for uing image i met, that i, equation (1) i true, and that for a thin ource i not met, that i, equation (13) i fale, then the ource i aumed to be relatively thick hick Source In the cae of a ource that i relatively thick in comparion to vertical diperion, the pointource olution i integrated over the thickne of the ource b and i ummed over i image ource, o that dilute i (14) dilute erf i b + erf i = 4α 4α b i b (15) Negligible ertical Diperion In the cae where we till do not aume vertical mixing and alo do not invoke the ue of image ource, that i equation (10) and (1) are fale, then a final tet i made to determine whether the obervation point i vertically outide the plume. If the vertical location of the obervation point relative to the vertical centerline of the ource i greater than half the ource thickne, expreed a b > then the obervation point i conidered to be out of the plume and dilute (and therefore dilute, a decribed in Section 3.4) i et to zero. Otherwie dilute i et equal to the invere of the ource thickne. 3.4 Final Aembly Having completed the variou multiplicative factor involved in the calculation of dilute, we aemble the final calculation a the product (16) dilute = dilute dilute dilute. (17) init Augut 014 8

12 In ome cae, one or more of thee term may be equal to unity, in effect removing it influence, or zero, in effect making the entire calculation equal to zero. he entire proce i repeated for each item in the vector or matrix, if the plume() function i fed them rather than a collection of calar argument. 4.0 Implication for Ue of the plume() Function at SRS At the SRS, GoldSim PA model ue the plume() function in the calculation of aturated zone contaminant tranport from beneath the variou ource (trenche in E-Area, high-level wate tank in F- and H-Area, and Salttone vault in Z-Area) to a number of potential location for receptor groundwater well, generally located 100 m from the collection of ource. Each plume i modeled with a ingle horizontal column of Cell Pathway element, and the plume() function i ued to derive a PlumeCorrectionFactor for each ource. Each well location i tranpoed into coordinate relative to each ource, and the plume() function i applied to that longitudinal and tranvere and. he aquifer i aumed to be fully mixed vertically, reulting from the aumption of a fully penetrating water well. he concentration at each well i the uperpoition of the contribution from each ource. Since each ource originate in the unaturated zone above the aquifer, contamination i introduced at the top of the aquifer, uing the ame areal footprint a the ource (i.e. trench, tank, or vault). So, the aumption in invoking the plume() function in the SRS PA model are that there i no explicit vertical diperion (ince complete vertical mixing i aumed), and that horizontal tranvere diperion i controlled only by the upplied diperivity α. Parameter upplied to the plume() function are therefore length of the Cell column from the uptream end of the ource to the receptor well A cro-ectional area of the Cell column (ource width Cell column thickne) length of the ource parallel to the flow direction vertical poition of the obervation point (et to 0 m) tranvere poition of the obervation point, perpendicular to the flow centerline for a given ource, at D vertical depth to the top of the ource from the top of the aquifer (et to 0 m) W width of the ource, tranvere to the aquifer flow b thickne of the ource (et to 0 m) b thickne of the aquifer α diperivity in the tranvere direction diperivity in the vertical direction (et to an arbitrarily large value) α For thi pecific et of input argument, the plume() function will perform the following calculation: Firt, dilute init i et to the cro-ectional area in equation (1): Augut 014 9

13 dilute init = A. (18) In all cae, the receptor well are at leat 100 m from the downtream edge of the ource, o > and equation (3) adjut the aumed length of the column to be =. he value of varie with the dipoal facility, and can be a large a 00 m, reducing the column length from, for an extreme example, 300 m to = (300 00/) m = 00 m, a reduction of 33%. At the other extreme, the tank are circular with a diameter of about 3 m, and the ditance to the well may be a large a 310 m, o the column length would be reduced from 333 m to = (333 3/) m = 31.5 m, a reduction of only about 6%. In conidering tranvere diperion, we conider the width of the ource (again ranging from about 3 m to 00 m), and the tranvere diperivity. Currently, the tranvere diperivity α i expreed a about 1/3 of the longitudinal diperivity α, which in turn i a ratio of the column length, about / 7.5, or α / 8.5. o tet the poibility of a narrow ource, examine the narrowet ource and the longet path length, or 3 m and 310 m, repectively. In thi cae, plume() would be upplied the argument = 310 m, W = 3 m, and α 3.8 m. Since the value for i adjuted internally by plume() (in thi cae to about 300 m), the tandard deviation of the tranvere diperion in the plume i about = 3.8m 300m = 48m. tet for a narrow ource come from equation (5): W < (19) α? 48 m (0) or 3m < Clearly, 3 m > 0.48 m, o the narrow ource tet fail. Even when the SRS model are modified to accommodate contaminant tranport to groundwater eep a mile or more away, the narrow ource tet will fail. We may alo reconider the etimation of diperivitie, baing them on groundwater model calibration rather than rough literature review. For now, however, uffice it to ay that the ource are not claified a narrow. Neither would they be expected to be particularly wide, though the widet apect ratio of ource to receptor location could be at the Salttone facility, where the ype I and I vault are 00 m wide, and may be a cloe a 100 m from the receptor well. In thi cae, α = 1. m 100 m = 16 m. However, if we conider the definition of a wide ource uing equation (7), we ee that W < 1,000,000 or α? 00m < m. (1) Augut

14 Augut Certainly, thi i true, and would be for all cae. It eem that, for the purpoe of calculation, the plume() function conider all the ource a wide ource. If that i the cae, then equation (8) would be ued to calculate the tranvere diperion, dilute. he vertical diperivity α in the SRS model i upplied to plume() a an arbitrarily large number. herefore the tandard deviation of the vertical diperivity, α i alo very large. he tet for a fully mixed aquifer in equation (10) therefore pae, and equation (11) i invoked: b dilute 1 =. In the SRS application, therefore, a long a the argument, D, and b are all 0 m and the vertical diperivity α i very large, the final expreion for dilute (ued a the PlumeCorrectionFactor in the SRS model) i baed on equation (8), including the column croectional area A and dividing by the aquifer thickne b, and with the ubtitution of the modified for column length, jut to keep the equation in term of the original input bw W W A 4 erf 4 erf + α α () Since through careful definition of the Cell column, the area A normal to flow equal the ource width W time the aquifer thickne b, the equation for PlumeCorrectionFactor implifie to. 4 erf 4 erf + W W α α (3)