Adjoint Modeling to Quantify Steam Flow Changes Due to Aquife Pumping Roseanna M. Neupaue, ph. D Univesity of Coloado Novembe 2013 Completion Repot No. 225
Acknowledgements The autho thanks Stanley Leake of the U.S. Geological Suvey fo poviding his goundwate flow model of the Uppe San Pedo Basin, Aizona. This epot was financed in pat by the U.S. Depatment of the Inteio, Geological Suvey, though the Coloado Wate Institute. The views and conclusions contained in this document ae those of the authos and should not be intepeted as necessaily epesenting the official policies, eithe expessed o implied, of the U.S. Govenment. Additional copies of this epot can be obtained fom the Coloado Wate Institute, E102 Engineeing Building, Coloado State Univesity, Fot Collins, CO 80523-1033 970-491-6308 o email: cwi@colostate.edu, o downloaded as a PDF file fom http://www.cwi.colostate.edu. Coloado State Univesity is an equal oppotunity/affimative action employe and complies with all fedeal and Coloado laws, egulations, and executive odes egading affimative action equiements in all pogams. The Office of Equal Oppotunity and Divesity is located in 101 Student Sevices. To assist Coloado State Univesity in meeting its affimative action esponsibilities, ethnic minoities, women and othe potected class membes ae encouaged to apply and to so identify themselves.
ADJOINT MODELING TO QUANTIFY STREAM FLOW CHANGES DUE TO AQUIFER PUMPING By Roseanna M. Neupaue Depatment of Civil, Envionmental, and Achitectual Engineeing Univesity of Coloado Boulde COMPLETION REPORT i
Acknowledgements: The autho thanks Stanley Leake of the U.S. Geological Suvey fo poviding his goundwate flow model of the Uppe San Pedo Basin, Aizona. Disclaime: This epot was financed in pat by the U.S. Depatment of the Inteio, Geological Suvey, though the Coloado Wate Institute. The views and conclusions contained in this document ae those of the authos and should not be intepeted as necessaily epesenting the official policies, eithe expessed o implied, of the U.S. Govenment. Additional copies of this epot can be obtained fom the Coloado Wate Institute, Coloado State Univesity, E102 Engineeing Building, 1033 Campus Delivey, Fot Collins, CO 80523-1033, 970-491- 6308 o email: cwi@colostate.edu, o downloaded as a PDF file fom http://www.cwi.colostate.edu. Coloado State Univesity is an equal oppotunity/affimative action employe and complies with all fedeal and Coloado laws, egulations, and executive odes egading affimative action equiements in all pogams. The Office of Equal Oppotunity and Divesity is located in 101 Student Sevices. To assist Coloado State Univesity in meeting its affimative action esponsibilities, ethnic minoities, women and othe potected class membes ae encouaged to apply and to so identify themselves. ii
Abstact As populations gow and demand fo wate inceases, new souces of wate must be found. If goundwate esouces ae developed to meet these gowing demands, the inceased pumping of aquifes should not educe flows in ives to levels that would limit the availability of wate fo dinking wate supply, iigation, and ipaian habitat. Steam depletion is the tem fo the change in the ive flow ate due to pumping in an aquife that is hydaulically connected to the ive. In many egions of the U.S., a new well cannot be sited until it is shown that pumping the new well will not cause substantial steam depletion. Numeical simulations ae typically used to quantify steam depletion. In the standad appoach, two numeical simulations ae un one without pumping and one with pumping in a well at the poposed location. In both simulations, the wate flux between the ive and aquife is calculated, and the diffeence between these fluxes is the steam depletion due to pumping at the poposed well location. If multiple well locations ae consideed, one addition simulation must be un fo each additional potential well location; thus, this appoach can be inefficient fo siting new wells. The goal this eseach was to develop an adjoint-based modeling appoach to efficiently quantify steam depletion due to aquife pumping. In a single simulation of an adjoint model, steam depletion is calculated fo a well at any location in the aquife; thus, it is computationally efficient when the numbe of well locations o possible well locations is lage. The adjoint appoach was developed to be used with standad goundwate flow simulatos, and theefoe can be applied in pactice. The eseach included igoous development of the adjoint equation fo calculating steam depletion in confined and unconfined aquifes with vaious models of goundwate/suface wate inteaction, along with numeical simulations to veify the adjoint equation. In addition, we used the adjoint method to investigate the sensitivity of steam depletion to the hydaulic conductivity of the steam channel, a paamete which is known to be uncetain. Keywods: Steam depletion, steamflow, adjoint method, steambed conductance, non-tibutay goundwate iii
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Table of Contents Justification of Wok Pefomed... 1 Review of Methods Used... 2 Fowad Flow Equations and Steam Depletion Calculations... 2 Adjoint Equations and Steam Depletion Calculations... 3 Discussion of Results and thei Significance... 4 Adjoint Steam Depletion Calculations... 4 Efficiency of Adjoint Simulations... 7 Souces of Eo in Adjoint Simulations... 8 Limitations... 9 Investigation of the Effects of Steambed Hydaulic Conductivity on Steam Depletion... 10 Pincipal Findings, Conclusions, and Recommendations... 13 Pincipal Findings... 13 Recommendations... 14 Summay... 14 Refeences... 15 v
List of Figues FIGURE 1. Model domain. Contous epesent head (m) in the unconfined aquife (thick line) and in the confined aquife (thin line). Thick dashed lines epesent tibutaies and ives. Bounday conditions fo the fowad and adjoint models ae shown along the boundaies (the same bounday conditions ae used at x = 0 km and at x = 160 km). Gay shaded egion epesents the aea whee evapotanspiation is occuing. The two filled cicles epesent the two well locations used in Figue 4.... 5 FIGURE 2. Steam depletion (m 3 /d) in the ive due to pumping in the unconfined (left column) and confined (ight column) aquifes at a ate of 2.5 x 10 4 m 3 /d, calculated using one adjoint simulation (top ow) and multiple fowad simulations (bottom ow). Thick dashed lines epesent tibutaies and ives.... 6 FIGURE 3. Pecent diffeence in steam depletion values calculated using the adjoint method and the standad method in the (a) unconfined aquife and (b) confined aquife. Wheeve steam depletion is less than 4000 m 3 /d, the value is set to zeo. Thick dashed lines epesent tibutaies and ives.... 7 FIGURE 4. Effects of vaious pumping ates on steam depletion fo the two well locations shown in Figue 1. Thin solid lines epesent the adjoint-based steam depletion estimates.... 9 FIGURE 5. Steam depletion calculated as a faction of pumping ate (dq /dq p ) at each potential well location fo (a) a heteogeneous steam channel with high K nea ive bends and low K in staight ive sections, (b) a heteogeneous steam channel with low K nea ive bends and high K in staight ive sections, (c) a homogeneous steam channel with K equal to the mean K fom the heteogeneous scenaios (1 x 10-7 m/s), and (d) a homogeneous steam channel with K equal to the maximum K fom the heteogeneous scenaios (2 x 10-7 m/s). The white S-shaped cuve is the ive, which flows in the y diection.... 11 FIGURE 6. Steam depletion calculated as a faction of pumping ate (dq /dq p ) due to pumping at a well at (x,y) = (77.5 km, 152.5 km) fo K values anging fom 10-9 m/s to 10-2 m/s at incements of 0.25 log units. Steam depletion inceases as K inceases, i.e. the uppemost cuve coesponds to K = 10-2 m/s; the cuve immediately below it coesponds to K = 10-2.25 m/s; etc. The thick black lines epesent the bounds of the sensitive ange: K = 1 x 10-8 m/s fo the lowe bound and K = 1 x 10-6 m/s fo the uppe bound.... 12 FIGURE 7. Steam depletion calculated as a faction of pumping ate (dq /dq p ) at each potential well location fo a heteogeneous steam channel with tempoally-vaying heteogeneity pattens. Fo half of each yea, the patte is high K nea ive bends and low K in staight ive sections, while fo the othe half of the yea, the patten is low K nea ive bends and high K in staight ive sections. The spatial mean value at any time and the tempoal mean value at any location is K = 1 x 10-7 m/s.... 13 vi
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Justification of Wok Pefomed In many egions of the U.S. and the wold, population gowth is inceasing the demand fo wate, while climate change may lead to a eduction in the available of wate. Fo example, ove the next thity yeas, population gowth of 46% and 29% is foecasted fo the westen and southen U.S. espectively (Hansen 2012), while climate models pedict that doughts in the 21 st centuy will last longe and be moe intense (MacDonald 2010). Consideing that goundwate compises 33% of public and 99% of domestic dinking wate, the incease in wate demands may be met though futhe development of goundwate esouces (Kenny et al., 2009; Voosmaty et al., 2000). Howeve, the inceased pumping of aquifes should not educe flows in ives to levels that would limit the availability of wate fo dinking wate supply, iigation, and ipaian habitat. Steam depletion is the eduction in the flow ate in a ive as a esult of pumping in an aquife that is hydaulically connected to the ive. Steam depletion has many negative consequences, such as eduction in wate supply fo municipal, agicultual, and domestic uses; failue to satisfy existing wate ights; and destuction of the ecosystems that depend on steams and ives. Quantifying steam depletion is theefoe cucial fo potecting wate supplies, suface wate ights, and envionments that depend on the steams and ives. The standad appoach fo using numeical models to calculate steam depletion is to fist un one goundwate flow simulation without pumping to detemine the exchange of wate between the ive and the aquife, and then to un an additional simulation with pumping at one location to detemine the change in the flow ate of wate between the ive and the steam. If the location of a new well is to be chosen, many possible well locations may be unde consideation. It may be necessay to choose a location that limits depletion in a neaby steam; theefoe, steam depletion must be calculated fo many diffeent well locations. Assuming that the aquife is sufficiently complex to equie numeical models to simulate steam depletion, the standad appoach must be epeated fo each potential well location, and can become computationally inefficient if many potential well locations ae consideed. In ou wok, we developed the adjoint method fo calculation steam depletion in a ive due to pumping in an adjacent aquife. With the adjoint method, only one simulation is needed to calculate steam depletion fo a well at any location in the aquife; thus it is moe efficient than the standad appoach when multiple potential well locations ae consideed. The accuacy of steam depletion pedictions depends on the how well the paametes used in the model match the physical popeties of the system. Many studies have shown that steam depletion is sensitive to the hydaulic conductivity of the steambed, K (e.g., Spalding and Khaleel, 1991; Sophocleous et al., 1995; Hunt 1999; Zlotnik and Huang 1999; Butle et al., 2001). Futhemoe, the steambed conductance, which includes K, along with steambed thickness and the channel width, was identified as the most uncetain paamete in steam depletion estimations (Chistensen, 2000). In addition, studies that have chaacteized K in natual channels have shown that it vaies both in time (Geneeux et al., 2008) and in space (Spinge et al., 1999; Cadenas and Zlotnik, 2003; Chen, 2004; Chen, 2005; Ryan and Boufadel, 2006; Chen et al., 2008; Cheng et al., 2011). Unfotunately, steambed hydaulic conductivity is typically not measued in steam depletion studies, and a single assumed value of K is assigned 1
to the entie steambed. In ou wok, we investigated the implications of using a single value of K, in tems of how it impacts estimations of steam depletion and the selection of feasible well locations. Review of Methods Used Fowad Flow Equations and Steam Depletion Calculations The main activity in this eseach was the development and veification of adjoint equations to calculate steam depletion. The adjoint equation is deived fom a fowad govening equation of flow in a ive and aquife system. Fo illustative puposes in this epot, we conside a twodimensional, unconfined aquife with a pumping well and a ive. Fo this system, the fowad govening equations of flow ae h K S y = [ K( h ζ ) h] Qp δ ( x x w ) + ( h h) B( x) (1) t b Q K = ( h h)w (2) s b whee h and h ae head in the ive and aquife espectively, t is time, x = (x,y) is the position vecto, S y is specific yield, K is the hydaulic conductivity tenso fo the aquife, z is the elevation of the aquife bottom, Q p is the pumping ate, x w is the location of the pumping well, K and b ae the hydaulic conductivity and thickness of the ive sediment, w is the ive channel width, Q is the flow ate in the ive, s is the spatial coodinate along the ive channel in the diection of flow, and B(x) is an indicato function that is unity at the ive and zeo elsewhee. Hee we assume that the head in the ive changes quickly elative to the change in head in the aquife, theefoe we neglect tansient changes in the ive volume. Fo illustation puposes, we use the following bounday conditions (although othe boundaies conditions can be used) h x,0 = h x h ( ) o ( ) ( x, t) = g1( x, t) h n = g 2( x, t) * ( s = 0, t) = Q ( t) on on Γ Γ 1 2 Q whee h o (x) is the initial head in the aquife, g 1 (x,t), g 2 (x,t), and Q * (t) ae known functions, and G 1 and G 2 ae the aquife boundaies. Steam depletion, DQ (s=l,t=t c ), is the decease in the ive flow ate at a compliance point at s=l at a compliance time t=t c as a esult of pumping. Fo small pumping ates, steam depletion is popotional to the pumping ate and can be expessed mathematically as dq ( L, t c ) Δ Q ( L, t c ) Q p ( x w ). (4) dq x p ( ) w Using the standad method to calculate steam depletion, (1) and (2) ae solved once with Q p = 0 and the flow ate in the ive is ecoded at s=l fo t=t c. Then (1) and (2) ae solved again with pumping at x = x w, and the new flow ate in the ive is also ecoded at s=l fo t=t c. The diffeence between these flow ates is steam depletion. To calculate steam depletion due to pumping at a diffeent well location, (1) and (2) must be solved again with pumping at a diffeent 2 (3)
location. Note that the only component of ive flow that is affected by pumping is the exchange of wate between the aquife and the ive, which is epesented by the last tem in (1) and the tem on the ight hand side of (2). Thus, altenatively, one can calculate steam depletion as the change in the ate at which wate is exchanged between the aquife and the ive. Adjoint Equations and Steam Depletion Calculations The deivative on the ight hand side of (4) is the sensitivity of ive flow ate to pumping, o equivalently, the steam depletion pe unit pumping ate. The adjoint deivation is based on sensitivity analysis using this sensitivity as the focus of the deivation. Details of the adjoint deivation ae povided in Neupaue and Giebling (2012), Giebling (2012), and Giebling and Neupaue (2013). Hee we pesent the final adjoint equations and show how they ae used to calculate steam depletion. The adjoint of (1) (3) is given by (Giebling and Neupaue, 2013) ψ K S y = [ K( ho ζ ) ψ ] + ( ψ ψ ) B( x) τ b (5) Q ψ K = ( ψ ψ )w h s b (6) ψ ψ ( x, τ = 0) = B( x) ( x, t) = 0 on ψ n = 0 on ψ K b S ( s = L, τ ) = 0 y Γ Γ 1 2 (7) whee t = t t c is backwad time and y and y ae the adjoint states of h and h. The steam depletion in (4) can be ewitten in tems of these adjoint states as (Giebling and Neupaue, 2013) ΔQ t ( ) c L t Q ψ (, τ ) dτ c p, x, (8) 0 which implies that the sensitivity of the ive flow ate to pumping ate is given by dq ( L, t dq p ( x ) w c ) tc = ψ, 0 ( x τ ) dτ. (9) Note that in (4), steam depletion is calculated fo pumping at a well at a paticula location x w, while in (8), steam depletion is calculated fo pumping at a well at any location x in the domain. To calculate steam depletion using (8), the adjoint state y (x,t) is obtained by solving (5) and (6) with the bounday conditions in (7). At each location x, the adjoint state is integated ove the time domain (as shown in (8)). Multiplying this esult by the pumping ate poduces steam depletion due to pumping at that location x in the aquife. Thus, only one simulation of the adjoint equation is needed to obtain steam depletion fo pumping at a well at any location in the aquife. 3
The adjoint equations, (5) and (6), have simila foms as the fowad equations, (1) and (2), so they can be solved using any goundwate flow simulato that solves (1) and (2). In this wok, we use MODFLOW-2000 (Habaugh et al., 2000) to solve both the fowad and the adjoint equations. The left hand side of (6) has a diffeent fom than the left hand side of (2), so we modified the souce code of the MODFLOW-2000 Steam package (Pudic, 1989) to solve (6). See Giebling (2012) fo moe details of the code modifications. Othe diffeences between the adjoint equations and the fowad equations ae: The state vaiables in the fowad equations ae head, which have units of length; the state vaiables in the adjoint equations ae adjoint states, which have units of ecipocal time. The time vaiable in the fowad equation is fowad time, while the time vaiable in the adjoint equation is backwad time; thus, in the adjoint simulation, infomation is popagated backwad in time. The pumping tem in the fowad equation does not appea in the adjoint equation. The bounday conditions on the adjoint goundwate equations ae homogeneous. The initial conditions ae homogeneous, except whee the aquife is adjacent to the ive. The goundwate flow tem in the fowad equation fo the unconfined aquife is nonlinea in head h; theefoe the elated tem in the adjoint is linea in the adjoint state but contains the fowad state vaiable, h. As an appoximation, we assume that the satuated thickness of the unconfined aquife can be appoximated as constant, so we eplace the time-dependent satuated thickness h-z with h o -z. We teat the unconfined aquife as if it wee a confined aquife with a tansmissivity of K(h o -z). In the fowad ive equation, the sign on the flow tem is positive, while in the adjoint equation, it is negative. The bounday condition on the fowad ive equation is a specified flow at the upsteam bounday, while the bounday condition fo the adjoint ive equation is a specified adjoint state at the downsteam bounday of the ive. These diffeences equie that the adjoint simulations with MODFLOW use somewhat diffeent inputs than a fowad simulation, and also equie a somewhat diffeent intepetation of the outputs. See Giebling and Neupaue (2013) fo infomation on these diffeences. Discussion of Results and thei Significance Adjoint Steam Depletion Calculations The adjoint equations fo calculating steam depletion wee developed and tested fo vaious aquife and ive systems, and wee shown to be accuate and efficient. As an example, conside the aquife and ive system shown in Figue 1. This system is compised of an unconfined aquife, an undelying confining unit, and a lowe confined aquitad. Two tibutaies and a ive ae hydaulically connected to the unconfined aquife and flow in the southwad diection. The aquife expeiences natual echage and evapotanspiation. We assume that the ive and tibutaies can be appoximated as wide ectangula channels, an assumption that is used in the MODFLOW STR package (Pudic, 1989). Fo details on the model paamete values, see 4
Giebling and Neupaue (2013). Figue 1 shows the head distibution in the confined and unconfined aquifes in the absence of pumping. FIGURE 1. Model domain. Contous epesent head (m) in the unconfined aquife (thick line) and in the confined aquife (thin line). Thick dashed lines epesent tibutaies and ives. Bounday conditions fo the fowad and adjoint models ae shown along the boundaies (the same bounday conditions ae used at x = 0 km and at x = 160 km). Gay shaded egion epesents the aea whee evapotanspiation is occuing. The two filled cicles epesent the two well locations used in Figue 4. To calculate steam depletion using the adjoint appoach, we solved (5) and (6) using MODFLOW-2000 and the adjoint STR package to obtain the adjoint states, which wee then used in (8) to calculate steam depletion. The esults ae shown in Figue 2a,b fo the unconfined and confined aquifes, espectively. Fo a given location in the model domain, these plots show the amount of steam depletion in the downsteam teminus of the ive afte 50 yeas of pumping at a ate of 2.5 x 10 4 m 3 /d at the given location. The esults show that steam depletion is highest fo wells nea the ive and tibutaies, and deceases as the distance between the well and the ive o tibutay inceases. All of this infomation was obtained with just one simulation of the adjoint model. 5
FIGURE 2. Steam depletion (m 3 /d) in the ive due to pumping in the unconfined (left column) and confined (ight column) aquifes at a ate of 2.5 x 10 4 m 3 /d, calculated using one adjoint simulation (top ow) and multiple fowad simulations (bottom ow). Thick dashed lines epesent tibutaies and ives. Fo compaison, we also calculated steam depletion using the standad fowad appoach. The domain is discetized into 40,960 1.25 km by 1.25 km gid blocks, and each one could be consideed a potential well location. We consideed the potential well locations to be the cells at the intesection of evey fouth ow and evey fouth column, fo a total of 2,560 potential well locations. We an 2,560 additional fowad simulations to estimate steam depletion fo each potential well. The esults ae shown in Figue 2c,d. Compaison with Figue 2a,b shows that the adjoint simulation poduces vey simila esults as the standad appoach. Fo both aquifes, the pattens ae vey simila between the esults of the two methods, with some slight vaiations nea the tibutaies and ives. Figue 3 shows the pecent diffeence between the steam depletion values calculated fo the adjoint and fowad simulations. The diffeences ae less than 4% thoughout most of the domain, except whee the steam depletion is low. In these aeas, the absolute eo is low. 6
FIGURE 3. Pecent diffeence in steam depletion values calculated using the adjoint method and the standad method in the (a) unconfined aquife and (b) confined aquife. Wheeve steam depletion is less than 4000 m 3 /d, the value is set to zeo. Thick dashed lines epesent tibutaies and ives. Efficiency of Adjoint Simulations The esults in Figue 2a,b wee obtained with one simulation of the adjoint model, which an in 140 seconds on a Dell Latitude E6530 with an Intel Coe i7-3720qm pocesso at 2.60 GHz. The adjoint simulation poduced steam depletion estimates fo a well at any of the 40,960 cells in the model domain, i.e., at 1.25 km by 1.25 km esolution. The esults of the standad appoach, shown in Figue 2c,d, wee obtained at 5-km by 5-km esolution, poducing 1/16th of the amount of infomation as obtained fom the adjoint simulations. The 2560 fowad simulations an in 582 minutes, o appoximately 250 times longe than the single adjoint simulation. The adjoint simulation was 250 times faste than the fowad appoach and poduced 16 times moe infomation. Pio to unning adjoint simulations, it is still necessay to develop and calibate a fowad model of the ive and aquife system. The paameteization of the adjoint model is developed based on the fowad model, and the adjoint model equies as input the steady-state aquife head and ive head that ae obtained fom a fowad simulation in the absence of pumping. Once the model is developed, the adjoint simulation is moe efficient than the standad appoach fo calculating steam depletion when the numbe of possible well locations is lage. Typically, it is not necessay to obtain steam depletion infomation fo well locations thoughout the entie model domain, so the efficiency of the adjoint simulation may be lowe than the 250- fold decease in simulation time seen hee. Howeve, fo lage models with simulation times on the ode of multiple hous, the time savings of a single adjoint simulation compaed to just 10 o so fowad simulations may be substantial. In addition, in pefoming a sensitivity analysis to 7
calculate the sensitivity of steam depletion to vaious model paametes, a modele may want to un simulations with seveal diffeent paamete sets. Even if the sensitivity is desied fo only a small subset of well locations, the computational time fo unning fowad simulations fo all combinations well locations and paamete sets may become pohibitive, and the adjoint appoach may be moe efficient. Souces of Eo in Adjoint Simulations The adjoint states ae elated to maginal sensitivities of head to the pumping ate; thus each tem in the adjoint equation epesents the sensitivity of vaious pocesses to the pumping ate. The adjoint deivation equies diffeentiation of each tem of the fowad equation with espect to pumping ate. Fo the non-linea tems (e.g., the fist tem on the ight hand side of (1)), these deivatives ae lineaized aound the pe-pumping conditions. A souce of eo in the adjoint model as compaed to the esults of the standad appoach is due to this lineaization. The tansmissivity of an unconfined aquife depends on the satuated thickness of the aquife, which can vay ove time as the head in the aquife vaies. In a pumping scenaio, the head, satuated thickness, and tansmissivity would decease ove time. The adjoint appoach ignoes the time vaiation of the satuated thickness; thus, the tansmissivity used in the adjoint simulation can be highe than the tansmissivity used in equivalent fowad simulations. Anothe potential souce of eo is in the assumption that steam depletion vaies linealy with pumping ate, as shown in (4). We an fowad simulations fo a ange of pumping ates (Q p = 0 to 5 x 10 6 m 3 /d) fo two diffeent well locations (Wells 1 and 2 in Figue 1) in the unconfined aquife. Steam depletion caused by this ange of pumping ates at each of the wells is shown in Figue 4. Well 1 is close to the tibutay, so dawdown is highe fo pumping at Well 1 than fo pumping at Well 2, fo any given pumping ate. Fo pumping at eithe well, steam depletion vaies appoximately linealy until the simulated pumping ate exceeds a theshold pumping ate above which the model cell containing the well goes dy. If the cell goes dy, pumping ceases in the simulation causing steam depletion to dop and appoach zeo. This theshold pumping ate is highe fo Well 1 because the satuated thickness of the aquife is geate whee the head is highe. Fo compaison, the adjoint-deived steam depletion is also shown in Figue 4. In the adjoint appoach, we calculate a single value of dq /dq p fom (9), which is the slope of the adjoint steam depletion cuves in Figue 4. Thus, non-zeo steam depletion is calculated even fo pumping ates that exceed the well yield. Futhemoe, the adjoint steam depletion slightly exceeds the steam depletion calculated fom the fowad simulations fo almost all pumping ates. This discepancy is likely caused by the assumption in the adjoint appoach that the satuated thickness is unchanged duing pumping. 8
FIGURE 4. Effects of vaious pumping ates on steam depletion fo the two well locations shown in Figue 1. Thin solid lines epesent the adjoint-based steam depletion estimates. In the STR package of MODFLOW, if the aquife head is above the ive channel bottom, the ate of flow acoss the steambed is popotional to the head diffeence between the ive and the aquife. Howeve, if the aquife head dops below the ive channel bottom, the ate of flow acoss the steam bed is popotional to the head diffeence between the ive and the bottom of the bed sediment, whee the pessue head is assumed to be zeo. Thus, in this case, the flow ate between the ive and aquife is independent of the head in the aquife, and the ive is no longe hydaulically connected to the aquife. Since the adjoint model only solves fo the adjoint state and not fo the aquife head, it is not possible to detemine when the ive is no longe hydaulically connected to the aquife. Thus, the adjoint model esults ae only accuate fo pumping scenaios fo which the ive emains hydaulically connected to the aquife. If the adjoint model is used fo situations in which steam depletion is not linealy popotional to the pumping ate, the adjoint model esults would oveestimate steam depletion. Thus, although the esults would be inaccuate, they would be consevative. Limitations The elationship between flow ate in the ive, Q, and ive head, h, is non-linea; thus the adjoint of the ive flow equation (6) does not have the same fom as the fowad ive flow equation (2). In addition, the code that solves the fowad ive equation must be modified to solve the adjoint ive equation. We have developed the adjoint appoach fo two specific ive models: The head in the ive is known. With this assumption, h in (1) is known, so (2) is not needed. This assumption is equivalent to the assumptions of the Rive package in MODFLOW. The adjoint equations fo this case ae deived in Neupaue and Giebling (2012). The head in the ive vaies as a esult of pumping, and the ive channel is appoximated as having a wide, ectangula coss section. This assumption is equivalent to the 9
assumptions of the Steam (STR) package in MODFLOW (Pudic, 1989). The adjoint equations fo this case ae deived in Giebling and Neupaue (2013). Many steam depletion models use MODFLOW with eithe the Rive package o the Steam package, so although the adjoint method has only been developed fo two ive models, it can be applied widely. The othe MODFLOW package that is commonly used to simulate steam and aquife inteaction in steam depletion simulations is the Steamflow Routing (SFR) package (Niswonge and Pudic, 2005), which allows fo moe complicated ive channel geometies and fo unsatuated flow beneath the ive bottom. Developing an adjoint model fo the SFR package is the subject of futue wok. Investigation of the effects of steambed hydaulic conductivity on steam depletion In the final pat of the poject, we investigated the effects of steambed hydaulic conductivity on steam depletion. We used the adjoint appoach to calculate steam depletion in a ive due to pumping at a well at any location in an unconfined aquife. Details of the model geomety and paamete values can be found in Lackey (2013). Figue 5c,d show dq /dq p afte 200 yeas of pumping with K = 1 x 10-7 m/s and K = 2 x 10-5 m/s, espectively. Steam depletion is lowe fo a low steambed hydaulic conductivity (Figue 5c) than fo a high steambed hydaulic conductivity (Figue 5d). With a low steambed hydaulic conductivity, less pumped wate is taken fom the steam because wate cannot flow as easily acoss the steambed. As stated ealie, many numeical investigations of steambed hydaulic conductivity use an assumed value of K ; these esults demonstate that choosing an incoect value of K can lead to incoect estimates of steam depletion. 10
FIGURE 5. Steam depletion calculated as a faction of pumping ate (dq /dq p ) at each potential well location fo (a) a heteogeneous steam channel with high K nea ive bends and low K in staight ive sections, (b) a heteogeneous steam channel with low K nea ive bends and high K in staight ive sections, (c) a homogeneous steam channel with K equal to the mean K fom the heteogeneous scenaios (1 x 10-7 m/s), and (d) a homogeneous steam channel with K equal to the maximum K fom the heteogeneous scenaios (2 x 10-7 m/s). The white S- shaped cuve is the ive, which flows in the y diection. Although these esults show that steam depletion is sensitive to K, we found that fo a given aquife hydaulic conductivity, a finite ange of K exists fo which steam depletion is sensitive to K. Fo example, Figue 6 shows steam depletion at a well at (x,y) = (77.5 km, 152.5 km) fo K values anging fom 1 x 10-9 m/s to 1 x 10-2 m/s. This ange epesents values of steam depletion that have been used in numeous field and numeical studies (Calve, 2001). Fo low values of K (less than 1 x 10-8 m/s), steam depletion is vey low and does not vay with K. Fo high values of K (geate than 1 x 10-6 m/s), steam depletion is high and again does not vay with K. Fo an intemediate ange (fom less than 1 x 10-8 m/s to 1 x 10-6 m/s), steam depletion is sensitive to K. We call this ange the sensitive ange. We found that the sensitive ange of K vaies with the aquife hydaulic conductivity (Lackey, 2013). 11
FIGURE 6. Steam depletion calculated as a faction of pumping ate (dq /dq p ) due to pumping at a well at (x,y) = (77.5 km, 152.5 km) fo K values anging fom 10-9 m/s to 10-2 m/s at incements of 0.25 log units. Steam depletion inceases as K inceases, i.e. the uppemost cuve coesponds to K = 10-2 m/s; the cuve immediately below it coesponds to K = 10-2.25 m/s; etc. The thick black lines epesent the bounds of the sensitive ange: K = 1 x 10-8 m/s fo the lowe bound and K = 1 x 10-6 m/s fo the uppe bound. We also investigated the effects of steambed heteogeneity on steam depletion. Using a mean steambed hydaulic conductivity of 1 x 10-7 m/s, we geneated a spatial distibution of K with high values nea the bends in the ive and low values in the staight sections, which is a epesentative patten unde high flow conditions (Andews, 1979; Sea, 1996; Clayton and Pitlick, 2007). Figue 5a shows dq /dq p fo this scenaio. In addition, we geneated a spatial distibution of K with low values nea the bends in the ive and high values in the staight sections, which is a epesentative patten unde low flow conditions (Andews, 1979; Sea, 1996; Clayton and Pitlick, 2007). Figue 5b shows dq /dq p fo this scenaio. In both cases, steam depletion is highe fo wells hea the high K stetches and lowe fo wells nea the low K stetches. Moe noticeably, the spatial patten of steam depletion (plotted as a function of well location) is diffeent fo heteogeneous steambeds (Figue 6a,b) than fo homogeneous steambed (Figue 5c,d). Thus, assuming a single unifom value of K in simulations of steam depletion may poduce incoect estimates, unless K does not fall in the sensitive ange (Lackey, 2013). We also investigated the effects of tempoal changes in steambed heteogeneity. We assumed that high flow conditions exist fo half of each yea and low flow conditions exist fo the othe half of each yea. Using the adjoint method, we calculated steam depletion in the ive due to pumping at a well at any location in the unconfined aquife afte 200 yeas of pumping. Results ae shown in Figue 7. Although the steambed is heteogeneous, the tempoal mean value of K at any location along the steambed is 1 x 10-7 m/s, the same value used fo the homogenous steambed in Figue 5c. These esults show that the cyclic tempoal vaiations in K lead to steam depletion patten that ae simila to steam depletion patten that would be obtained if the tempoal mean values of K ae used. 12
FIGURE 7. Steam depletion calculated as a faction of pumping ate (dq /dq p ) at each potential well location fo a heteogeneous steam channel with tempoally-vaying heteogeneity pattens. Fo half of each yea, the patte is high K nea ive bends and low K in staight ive sections, while fo the othe half of the yea, the patten is low K nea ive bends and high K in staight ive sections. The spatial mean value at any time and the tempoal mean value at any location is K = 1 x 10-7 m/s. Pincipal Findings, Conclusions, and Recommendations Pincipal Findings This wok developed an adjoint method to calculate steam depletion caused by pumping in an aquife that is hydaulically connected to a ive. We developed the adjoint appoach fo two diffeent ive models, consistent with the assumptions in the MODFLOW Rive package and MODFLOW Steam package, which ae widely used in pactice to calculate steam depletion. We tested the adjoint appoach fo seveal diffeent scenaios using hypothetical aquifes. The advantage of the adjoint method ove the standad method fo calculating steam depletion is that only one adjoint simulation is needed to calculate steam depletion due to pumping at a well at any location in the aquifes. In the standad fowad method, one fowad simulation must be un fo each possible will location, which can be computationally pohibitive. The esults show that the adjoint solution accuately appoximates steam depletion when the assumptions of small changes in ive and aquife head ae satisfied, and that the adjoint appoach is computationally moe efficient than the standad fowad method. Thus, the adjoint 13
method is a useful appoach fo identifying optimal locations fo new wells that minimize steam depletion o fo identifying aeas along a ive section that ae most sensitive to pumping. Steam depletion is sensitive to the steambed hydaulic conductivity fo a ange of K values that depends on the aquife hydaulic conductivity. Steam depletion is also sensitive to steambed heteogeneity and tempoal vaiations. Thus, numeical modeles who ae using simulations to estimate steam depletion should use measued steambed hydaulic conductivity values if available, unless it can be shown that the steambed hydaulic conductivity is outside of the sensitive ange. Recommendations The adjoint appoach has been tested using synthetic aquifes. To fully evaluate the adjoint appoach, it must be tested on an actual aquife. We ae cuently woking on a case study fo the San Pedo Basin in Aizona using a goundwate flow model that was developed fo estimation of steam depletion using the standad fowad modeling appoach (Leake et al., 2008). Seveal pactitiones have expessed inteest in using the adjoint method fo steam depletion calculations. In ode to make the method feasible fo widespead application in pactice, a code should be witten that ceates the adjoint MODFLOW input files fom MODFLOW input and output files fom a fowad simulation in the absence of pumping. Wok on the code development is cuently undeway. The code uses MODFLOW suboutines fo eading existing MODFLOW input and out files and fo witing adjoint MODFLOW input files. In many steam depletion studies, the MODFLOW Rive o Steam package is used to simulate the flow between the aquife and the suface wate body, fo which the adjoint method has been developed. The Steamflow Routing (SFR) package is also used in steam depletion studies; theefoe, the adjoint method should be developed fo the ive models that ae used in the SFR package. Summay This wok has developed and tested a new adjoint appoach fo calculating steam depletion. With one adjoint simulation, we obtain estimates of steam depletion fo pumping at a well at any location in the model domain. The adjoint appoach is efficient if multiple well locations ae being consideed and steam depletion must be estimated fo each one. We developed a pocedue fo using MODFLOW to solve the adjoint equations. The method has been developed and tested fo two ive models: (1) head is the ive is known and is not impacted by pumping (equivalent to the assumptions in the MODFLOW Rive package), and (2) a wide ectangula ive channel geomety, and negligible changes in ive stoage (equivalent to the MODFLOW Steam package). 14
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