Skip to main content
SpringerLink
Log in

Laboratory sandbox validation of pollutant source location methods

  • Original Paper
  • Published:
Stochastic Environmental Research and Risk Assessment Aims and scope Submit manuscript

Abstract

Inverse methods can be used to recover the pollutant source location from concentration data. In this paper, the relative effectiveness of two proposed methods, simultaneous release function and source location identification (SRSI) and backward probability model based on adjoint state method (BPM-ASM) are evaluated using real data collected by using experimental equipment. The device is a sandbox that reproduces an unconfined aquifer in which all the variables are controlled. A numerical model was calibrated using experimental observations. The SRSI is a stochastic procedure which finds the source location and the release history by means of a Bayesian geostatistical approach (GA). The BPM-ASM provides the backward probability location of the pollutant detected at a monitoring point by means of a reverse transport simulation. The results show that both methods perform well. While the simultaneous release function and SRSI method requires a preliminary delineation of a probable source area and some weak hypotheses about the statistical structure of the unknown release function, the backward probability model requires some hypothesis about the contaminant release time. A case study was performed using two observation points only, and despite the scarcity of data, both methodologies were able to accurately reconstruct the true source location. The GA has the advantage to recover the release history function too, whilst the backward probability model works well with fewer data. If there are many observations, both methodologies may be computationally heavy. A transfer function approach has been adopted for the numerical definition of the sensitivity matrix in the SRSI method. The reliability of the experimental equipment was tested in previous laboratory works, conducted under several different conditions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Abbreviations

b p × 1:

Unknown coefficients

C(x ,t):

Concentration at point x and time t

C 0 :

Solution concentration

\( \hat{C}_{i} \) :

Observed concentration

C w :

Initial condition in backward model

D :

Dispersion tensor

F (t):

Concentration of the water injected at the source as function of time t

F 0 :

Constant and known mass rate input function

\( f_{{\hat{C}_{i} }} \) :

Measured concentrations PDF

f(x ,t):

Transfer function at position x and time t

f X (x,t):

Backward location PDF

H m × n :

Sensitivity matrix

h(s) m × 1:

Vector that describes the transport process

H L (x,t):

Load term

H SF (t):

Heaviside step function

h D :

Head level downstream

h U :

Head level upstream

J :

Generic sub-areas

K :

Hydraulic conductivity

m :

Number of observations

m o :

Release source mass

M :

Random source mass

M p × n :

Multipliers

n :

Number of unknowns

n :

Normal versor

p :

Number of unknown coefficients

Q(θ) n × n :

Matrix, covariance of the unknown process

Q in :

Injected flow rate

Q w :

Initial condition in backward model

q 0 :

Injected solution discharge

q I :

Source inflow rate per unit volume

R m × m :

Error covariance matrix

s n × 1:

Unknowns

s(t):

Unknown release function

\( \tilde{s} \) :

Transformed unknown function

\( {\hat{\mathbf{s}}} \) n × 1 :

Vector of estimated release function

t :

Time

\( \bar{t} \) :

Sampling time

t start :

Starting time

t end :

Ending time

u :

Velocity tensor

v m × 1:

Measurement errors

V n × n :

Matrix, covariance of the estimate of the errors

x :

Position in the domain

x 0 :

Source location

x inj :

Longitudinal coordinate of injector

X 0 :

Random source location

X n × p :

Matrix, mean of the unknown process

x w :

Observation location

z m × 1:

Observations

z inj :

Vertical coordinate of injector

α :

Positive number

α L :

Longitudinal dispersivity

α T :

Transversal dispersivity

β x :

Normalization factor

Γi :

Boundaries

δ:

Dirac delta function

Δt :

Numerical model time step

Δx:

Longitudinal size of numerical cell grid

Δy:

Transversal size of numerical cell grid

Δz:

Vertical size of numerical cell grid

ε i :

ith measurement error

η :

Time

θ :

Structural parameters of the covariance function

λ s :

Correlation time length of the unknown release function s(t)

Λ n × m :

Kriging coefficients

Ξ m × m :

Dummy matrix

σ 2 R :

Variance of the measurement error

σ 2 S :

Variance of the unknown release function s(t)

Σ m × m :

Dummy matrix

τ:

Backward time

τ w :

Backward sampling time

φ :

Porosity

ψ* :

Adjoint state

\( \nabla \) :

Nabla operator

References

  • Bear J, Verruijt A (1987) Modeling groundwater flow and pollution. Springer, Dordrecht, Holland

    Book  Google Scholar 

  • Box GEP, Cox DR (1964) An analysis of transformations. J R Stat Soc Ser B 26(2):211–252

    Google Scholar 

  • Butera I, Tanda MG (2003) A geostatistical approach to recover the release history of groundwater pollutants. Water Resour Res 39(12):1372. doi:10.1029/2003WR002314

    Article  Google Scholar 

  • Butera I, Tanda MG, Zanini A (2004) La ricostruzione della storia del rilascio di inquinanti in acquiferi sede di moto non uniforme mediante approccio geostatistico. In: XXIX Convegno di Idraulica e Costruzioni Idrauliche, Trento, 7–10 Settembre 2004. Bios, Cosenza

  • Butera I, Tanda MG, Zanini A (2006) Use of numerical modeling to identify the transfer function and application to the geostatistical procedure in the solution of inverse problems in groundwater. J Inverse Ill-Posed Probl 14(6):547–572. doi:10.1163/156939406778474532

    Article  Google Scholar 

  • Butera I, Tanda MG, Zanini A (2013) Simultaneous identification of the pollutant release history and the source location in groundwater by means of a geostatistical approach. Stoch Environ Res Risk Assess 27(5):1269–1280. doi:10.1007/s00477-012-0662-1

    Article  Google Scholar 

  • Citarella D, Tanda MG, Zanini A (2010) Setup and calibration of an experimental device aimed at the validation of geostatistical procedures. In: IAHR International Groundwater Symposium, Valencia (E), 22–24 September 2010

  • Gzyl G, Zanini A, Frączekc R, Kura K (2014) Contaminant source and release history identification in groundwater: a multi step approach. J Contam Hydrol 157:59–72. doi:10.1016/j.jconhyd.2013.11.006 ISSN 0169-7722

    Article  CAS  Google Scholar 

  • Harbaugh AW, Banta EW, Hill MC, McDonald MG (2000) MODFLOW-2000, the U.S. Geological Survey Modular Ground-Water Model–User Guide to Modularization Concepts and the Ground-Water Flow Process. United States Geological Survey: Open File Report 00-92

  • Jury WA, Roth K (1990) Transfer functions and solute movement through soil: theory and applications. Birkhäuser Verlag, Boston

    Google Scholar 

  • Kitanidis PK (1995) Quasi-linear geostatistical theory for inversing. Water Resour Res 31(10):2411–2419. doi:10.1029/95WR01945

    Article  Google Scholar 

  • Kitanidis PK (1996) On the geostatistical approach to the inverse problem. Adv Water Resour 19(6):333–342. doi:10.1016/0309-1708(96)00005-X

    Article  Google Scholar 

  • Mathworks (2010) MATLAB 7.10.0. 7.10.0 edn. The MathWorks Inc., Natick, MA

  • Michalak AM, Kitanidis PK (2002) Application of Bayesian inference methods to inverse modeling for contaminant source identification at Gloucester Landfill, Canada. In: Hassanizadeh SM, Schotting RJ, Gray WG, Pinder GF (eds) Computational methods in water resources XIV, vol 2. Elsevier, Amsterdam, pp 1259–1266

    Google Scholar 

  • Michalak AM, Kitanidis PK (2003) A method for enforcing parameter nonnegativity in Bayesian inverse problems with an application to contaminant source identification. Water Resour Res 39(2):1033. doi:10.1029/2002WR001480

    Article  Google Scholar 

  • Michalak AM, Kitanidis PK (2004a) Estimation of historical groundwater contaminant distribution using the adjoint state method applied to geostatistical inverse modeling. Water Resour Res 40(8):W08302. doi:10.1029/2004wr003214

    Article  Google Scholar 

  • Michalak AM, Kitanidis PK (2004b) Application of geostatistical inverse modeling to contaminant source identification at Dover AFB, Delaware. J Hydraul Res 42(EXTRA ISSUE):9–18

    Article  Google Scholar 

  • Neupauer RM, Lin R (2006) Identifying sources of a conservative groundwater contaminant using backward probabilities conditioned on measured concentrations. Water Resour Res 42:W03424. doi:10.1029/2005WR004115

    Google Scholar 

  • Neupauer RM, Wilson JL (1999) Adjoint method for obtaining backward-in-time location and travel time probabilities of a conservative groundwater contaminant. Water Resour Res 35(11):3389–3398

    Article  Google Scholar 

  • Neupauer RM, Wilson JL (2001) Adjoint-derived location and travel time probabilities for a multidimensional groundwater system. Water Resour Res 37(6):1657–1668. doi:10.1029/2000wr900388

    Article  Google Scholar 

  • Neupauer RM, Wilson JL (2002) Backward probabilistic model of groundwater contamination in non-uniform and transient flow. Adv Water Resour 25(7):733–746. doi:10.1016/S0309-1708(02)00073-8

    Article  CAS  Google Scholar 

  • Neupauer RM, Wilson JL (2003) Backward location and travel time probabilities for a decaying contaminant in an aquifer. J Contam Hydrol 66(1):39–58. doi:10.1016/S0169-7722(03)00024-X

    Article  CAS  Google Scholar 

  • Neupauer RM, Wilson JL (2004a) Forward and backward location probabilities for sorbing solutes in groundwater. Adv Water Resour 27(7):689–705. doi:10.1016/j.advwatres.2004.05.003

    Article  CAS  Google Scholar 

  • Neupauer RM, Wilson JL (2004b) Numerical implementation of a backward probabilistic model of ground water contamination. Ground Water 42(2):175–189. doi:10.1111/j.1745-6584.2004.tb02666.x

    Article  CAS  Google Scholar 

  • Neupauer RM, Wilson JL (2005) Backward probability model using multiple observations of contamination to identify groundwater contamination sources at the Massachusetts Military Reservation. Water Resour Res 41:W02015. doi:10.1029/2003WR002974

    Article  Google Scholar 

  • Neupauer RM, Lin R, O’Shea H (2007) Conditioned backward probability modeling to identify sources of reactive groundwater contaminants. Water Resour Res 43:W11403. doi:10.1029/2006WR005580

    Article  Google Scholar 

  • Neupauer RM, Wilson JL, Bhaskar A (2009) Forward and backward temporal probability distributions of sorbing solutes in groundwater. Water Resour Res 45(1):W01420. doi:10.1029/2008wr007058

    Article  Google Scholar 

  • Snodgrass MF, Kitanidis PK (1997) A geostatistical approach to contaminant source identification. Water Resour Res 33(4):537–546

    Article  CAS  Google Scholar 

  • Zheng C, Wang PP (1999) MT3DMS: a modular three-dimensional multispecies transport model for simulation of advection, dispersion, and chemical reactions of contaminants in groundwater systems; Documentation and user’s guide. U.S. Army Engineer Research and Development Center No. SERDP-99-1, Vicksburg, MS

Download references

Acknowledgments

We are grateful to the reviewers for their valuable comments.

Author information

Authors and Affiliations

  1. DICATeA, Università degli Studi di Parma, Parco Area delle Scienze, 181/A, 43124, Parma, Italy

    Fausto Cupola, Maria Giovanna Tanda & Andrea Zanini

Authors
  1. Fausto Cupola
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maria Giovanna Tanda
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andrea Zanini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrea Zanini.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cupola, F., Tanda, M.G. & Zanini, A. Laboratory sandbox validation of pollutant source location methods. Stoch Environ Res Risk Assess 29, 169–182 (2015). https://doi.org/10.1007/s00477-014-0869-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00477-014-0869-4

Keywords

Search

Navigation