state–space mixing cell model of unsteady solute transport in unsaturated soil

Environmental Modelling & Software 14 (1999) 161–169

State–space mixing cell model of unsteady solute transport inunsaturated soil

V.J. Bidwell *

Lincoln Environmental, PO Box 84, Lincoln University, Canterbury, New Zealand

Received 10 August 1997; received in revised form 7 April 1998; accepted 15 June 1998

Abstract

This state–space model enables application of stochastic linear systems theory to solute transport in soil. A series of mixing cellsmodels one-dimensional, vertical, advective–dispersive transport. Cumulative soil water drainage is the index variable for applicationto unsteady flow in unsaturated soil. For each cell, solute transfer between mobile and immobile soil water, as well as equilibriumand non-equilibrium linear adsorption, are represented as lumped processes by two fractions linked by rate-limited transfer. Plantuptake of solutes is treated as an unknown disturbance. For a uniform soil, the model has four parameters and can be describedin MATLAB with about 10 lines of code. This software library is used to produce the discrete form of the model, which isunconditionally stable for any drainage interval, as well as to implement state estimation and control algorithms. The model isdemonstrated with breakthrough curve data of15N, Br−, and 35S leached from an undisturbed field soil under pasture receivingrainfall and irrigation. Process knowledge is combined with estimates of measurement and process uncertainty, by means of theKalman filter, to forecast the response of35S to the solute pulse application, beyond the available data record. 1999 ElsevierScience Ltd. All rights reserved.

Keywords:Solute transport; Mixing cell; State–space; Monitoring; Control; Kalman filter

Software availabilityName of software: SSMC (State–Space

Mixing Cell)Developer: Vince Bidwell, Lincoln

Environmental, PO Box84, Lincoln University,Canterbury, New Zealand.E-mail:[email protected];Phone:1 64-3-325 3704;Fax: 1 64-3-325 3725.

Year first available: 1998Hardware required: specified by Mathworks

Inc. for the selectedversion ofMATLAB

Software required: MATLAB with ControlSystem Toolbox

Program language: text file for compilation as

* Tel.: 1 64-3-325-3704; fax: 1 64-3-325-3725; e-mail:[email protected]

1364-8152/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved.PII: S1364-8152 (98)00067-X

an M-function file withinMATLAB

Program size: source code is 10–20 linesof text

Availability and cost: available as a text file,free to researchers

1. Introduction

Application of fertilisers or wastes to the land surfacecan degrade the quality of underlying groundwater ifthese substances are in excess of crop requirements orthe assimilative capacity of the soil. The modelpresented in this paper is designed to support manage-ment of the amount and timing of substance applicationsto the land so that groundwater quality is protected. Theproposed management process involves monitoringleachate quality by means of lysimeters and using thesedata for operational decision making.

Each lysimeter is an undisturbed column of soil, sur-

162 V.J. Bidwell /Environmental Modelling & Software 14 (1999) 161–169

rounded by a sealed vertical casing, from which freelydraining soil water is collected for analysis (Cameron etal., 1992). The collected leachate is a 100% sample ofdrainage from the horizontal area of the lysimeter andtherefore substance mass balances are measured for thesoil volume. A reliable estimate of water and solute massflux in the field is obtained with appropriate replicationof lysimeter units.

For reasons explained in the following section, cumu-lative drainage is the index variable rather than time. Forpractical monitoring, leachate data are collected aftersignificant rainfall or irrigation events, and the drainageinterval since the previous collection is not predeter-mined. Therefore, the leachate concentration data rep-resent the mean values during these irregular intervalsof cumulative drainage.

2. Model

2.1. State–space for monitoring and control

Transport and transformation of substances in soilinvolves processes which are highly variable and haveproperties which are difficult to measure accurately.Management under these circumstances is facilitated bythe application of control theory for stochastic linear sys-tems (e.g. Bertsekas, 1976; Davis and Vinter, 1985),which enables formulation of rules for monitoring andcontrol with optimal performance. This theory requiresthe processes of interest be described by a mathematicalmodel with the ‘state–space’ structure, which relates themeasured outputsY(t) of a system to the internal statesX(t) in response to inputsU(t) in the form:

dX(t)dt

5 AX (t) 1 BU(t)

Y(t) 5 CX(t) 1 DU(t) (1)

The coefficient matricesA, B, C, andD for the presentpaper are assumed to be constant with respect to theindex variablet. A model in the form of Eq. (1) caneasily be coded inMATLAB software for access to alibrary of algorithms for linear system control (Grace etal., 1992).

2.2. Unsteady, unsaturated solute transport

For one-dimensional, vertical, steady, advective–dis-persive transport of solute through soil, the probabilitydensity functionf(z,t) of the distancez travelled by soluteparticles in timet is:

f (z,t) 51

√2p(2Dt)expH 2

(z 2 Vt)2

2(2Dt) J (2)

whereV is the mean pore water velocity andD is thedispersion coefficient. The distribution (Eq. (2)) is Gaus-sian, centred on the mean travel distanceVt and withvariance 2Dt. There is experimental and analytical evi-dence that this process is also applicable to non-uniformwater content (De Smedt and Wierenga, 1978) and tran-sient water flow (Wierenga, 1977; Jury et al., 1990). Thisbehaviour depends on dispersion being linearly relatedto velocity by the dispersivityl. Substitution ofD 5lV into Eq. (2) results in dispersion which is dependentonly on the mean distance travelled by the solute par-ticles. If I is the total amount of vertical drainage whichhas transported the solute pulse down from the groundsurface, then the mean depth of travel isI/u whereu isthe fraction of mobile soil water in the profile after drain-age has ceased. With the additional substitutionVt 5I/u, Eq. (2) specifies the Gaussian distributionf(z,I) withmeanI/u and variance 2lI/u. This Lagrangian descrip-tion f(z,I) can be transformed (Bartlett, 1978) to the Eul-erian descriptionf(I,L) at a particular observation depthz 5 L:

f(I,L) 5LI

1√2p(2lI/u)

expH 2(L 2 I/u)2

2(2lI/u) J (3)

mean5 Lu variance5 2lLu2

The ratio of variance to mean for the distribution (Eq.(3)) is 2lu, which is independent of the travel distance.This information is used in the next section to developa model to simulate Eq. (3) with a structure suitable forEq. (1).

2.3. Mixing cell simulation of advection–dispersion

The mixing cell concept has a long history in disper-sion modelling (Bear, 1969; Bajracharya and Barry,1994) but is generally presented as an approximation tothe finite difference solution of the advection–dispersionequation, with optimal results when the spatial step isequal to twice the dispersivity. Numerical dispersion andstability restrict the size of the time step. Bajracharyaand Barry (1992) developed an ‘improved’ mixing cellmodel which overcomes these problems but it does nothave the state–space structure desired for the presentapplication.

A mixing cell is a volumeW of water (per area trans-verse to the flow) through which flows a water fluxq(t).Solute of concentrationcin(t) in the inflow is instan-taneously mixed with the contents of the cell, and theoutflow concentrationc(t) is the same as that within thecell. Solute mass balance for the cell is:

Wdc(t)

dt5 q(t)cin(t) 2 q(t)c(t) (4)

By substituting incremental drainagedI 5 q(t)dt into

163V.J. Bidwell /Environmental Modelling & Software 14 (1999) 161–169

Eq. (4), and omitting the index variableI from c andcin,for convenience:

WdcdI

1 c 5 cin (5)

which is a first order linear differential equation. Theimpulse response solution to Eq. (5) gives the distri-bution of solute transit through the cell in terms ofI as:

f (I) 51W

expH 21WJ (6)

which is the exponential distribution with meanW andvarianceW2. The ratio of variance to mean isW, and byputting W 5 2lu a series of mixing cells becomes asimple mathematical concept for simulating the first twomoments of Eq. (3). The number of cells required is themean of Eq. (3),Lu, divided by the mean for one mixingcell, 2lu, which is n 5 L/2l. The resulting impulseresponse forn cells in series is the gamma distribution:

f (I) 5In 2 1

(n 2 1)!(2lu)n expH 21

2luJ (7)

The shape of Eq. (7) becomes closer to Eq. (3) asn(or L/2l) increases, and the mixing cell series is a goodpractical representation of advection–dispersion. Thevertical space step is restricted to a size of 2l but thisis not usually a practical difficulty given the precisionof soil profile data.

The differential equations for the series ofn mixingcells can be expressed in terms of Eq. (5), witha 5 1/W5 1/2lu:

dc1

dI5 2 ac1 1 acin

dc2

dI5 2 ac2 1 ac1

$$$ (8)

dcn

dI5 2 acn 1 acn 2 1

cout 5 cn

wherecj is the solute concentration in thejth cell, andcout is the outflow concentration from the last cell. Eq.(8) has the state–space structure of Eq. (1) when writ-ten as:

ddI

c1

c2

c3

$

cn 2 1

cn

=

2a 0 $ $ $ 0

a 2a 0 $ $ 0

0 a 2a 0 $ 0

$ $ $ $ $ $

0 $ 0 a 2a 0

0 $ $ 0 a 2a

A

c1

c2

c3

$

cn 2 1

cn

+

a

0

0

$

0

0

B

cin

cout 5 |1|cn 1 |0|cin

C D(9)

The continuous drainage model (Eq. (9)) is the basisfor obtaining the discrete state–space models for anydrainage interval by means ofMATLAB software.These models are unconditionally stable, and can includefiltering to account for leachate concentration being themean value over the drainage interval rather than theinstantaneous value at the end of the interval.

2.4. Solute transformations

With advective–dispersive transport represented bythe system of mixing cells, solute transformations maybe considered as lumped processes within each mixingcell subsystem (Sardin et al., 1991). The following dis-cussion considers only linear, equilibrium and non-equi-librium solute transformations. These are described froma process point of view in this section and will then bereparameterised in terms of dynamic behaviour in Sec-tion 2.5.

2.4.1. Mobile–immobile soil waterEach mixing cell has a mobile water fractionum with

solute concentrationc, which is part of the advective–dispersive transport through all the cells, and an immo-bile fractionuim with concentrations, which is connectedonly to the mobile fraction of the same cell (Fig. 1). Thedifferential equations for thejth cell are, for mass bal-ance:

2lum

dcj

dI1 2luim

dsj

dI5 cj 2 1 2 cj (10)

and rate-limited diffusion:

dsj

dI5 r(cj 2 sj ) (11)

in which r is a transfer coefficient with dimension(drainage units)−1 rather than (time)−1, because the non-equilibrium processes are with reference to cumulativedrainage.


Fig. 1. Conceptual structure of the state–space mixing cell (SSMC)model.

2.4.2. Linear adsorption–desorptionLinear equilibrium adsorption within either the mobile

or immobile fraction can be quantified by a ‘retardationcoefficient’R, specific to each fraction, which is the ratioof total resident mass in equilibrium, per volume of soil,to the solute mass in the soil water fraction. Inclusionof equilibrium adsorption in Eq. (10) gives:

2lumRm

dcj

dI1 2luimRim

dsj

dI5 cj 2 1 2 cj (12)

The only form of non-equilibrium adsorption–desorp-tion considered in the present conceptual discussion isthat due to the rate-limited diffusion described by Eq.(11). Therefore, any equilibrium adsorption within theimmobile fraction is observed externally as being non-equilibrium.

2.5. The complete dynamic model

By making the substitutions:

E 5 2lumRm N 5 2luimRim (13)

in Eq. (12), and substituting Eq. (11) into Eq. (12), thedifferential equations for thejth cell become:

dcj

dI5

1E

cj 2 1 2(1 1 rN)

Ecj 1

rNE

sj (14)

dsj

dI5 rcj 2 rsj

The four parametersE, N, r, and the number of cellsn, are the maximum set that can be identified from a

series of leachate concentration data. However, theequivalences (Eq. (13)) can be useful for suggestinginitial values from other process knowledge, or for relat-ing calibrated parameter values to physical reality. Anyform of adsorption–desorption with first order dynamics(relative to drainage) will be indistinguishable from theconcepts used in the model development.

The state–space structure (Eq. (9)) is extended toinclude the additional statessj (j 5 1, n) and the termsin Eq. (14):

ddI

c1

$

cn

s1

$

sn

5CC CS

SC SS

c1

$

cn

s1

$

sn

11E

1

0

$

$

$

0

cin (15)

cout 5 cn

CC 51E

2 1 0 $ $ $ 0

1 2 1 0 $ $ 0

0 1 2 1 0 $ 0

$ $ $ $ $ $

0 $ 0 1 2 1 0

0 $ $ 0 1 2 1

2 CS

CS 5rNE

I (n) SC 5 rI (n) SS5 2 rI (n)

where I (n) is the identity matrix of sizen.The model (Eq. (15)) can be described inMATLAB

with about 10 lines of code as an M-function file, whichwill be referred to as SSMC (state–space mixing cell).

3. Demonstration

The emphasis in the following demonstration is on therole of the model in describing prior knowledge aboutthe dynamic behaviour of solute transport, which canthen be incorporated into a Bayesian approach to predic-tion and control with uncertain information.

3.1. Experimental data

The data are from an experiment (McLaren et al.,1993; Fraser et al., 1994) on five replicated lysimeters(800 mm diameter, 1100 mm depth) of undisturbed soil(a free-draining silt loam) under pasture, which receivedrainfall and irrigation over a period of one year. Several


solutes were applied as a surface pulse to the soil (atfield capacity) with 4 mm of water to represent animalurine, and immediately followed by 10 mm of simulatedrainfall. Leachate drainage, in this research environment,was sampled at regular intervals of 30 mm.

Three sets of solute breakthrough curve (BTC) datawere selected for estimation of model parameters:15N,of which only about 8% of the pulse application wasrecovered in the leachate; Br−, 47% recovered; and35S-labelled sulphate, 13% recovered. Each data set wasscaled for unit area under the breakthrough curve for theperiod of record (17–20 values at 30 mm intervals). Thiswas for the purpose of estimating the parameters of theSSMC models which were similarly scaled.

3.2. Parameter estimation

The solute breakthrough curves do not represent thepulse response of transport by vertical soil water move-ment through the soil, because of plant uptake of soluteand water in the root zone, as demonstrated by the lowvalues of solute recovery in the leachate. Solute, whichdoes arrive at the measurement depth, has been trans-ported by vertical water movement which is larger thanthe measured cumulative drainage. The result is that thesolute input history, in terms of measured cumulativedrainage, is unknown and therefore the usual systemidentification techniques cannot be applied.

The parametersn, E, N, and r of the SSMC model(Eq. (15)) were fitted by visually comparing the BTCdata with the response of the model to a unit mass ofsolute initially resident in the top cell. This modelresponse takes account of the sampling interval and thatthe data represent average solute concentrations over thatinterval. For ease of visual comparison, the data andmodel response were both scaled to unit area for thelength of record. Automatic optimisation techniqueswere not used.

Fig. 2 shows that the15N data were fitted very wellby just the advective–dispersive portion of the model,with n 5 20, E 5 10.3 mm. This result suggests that thesmall amount of recovered15N (mainly as nitrate) wasleached through the near-surface zone during a periodof drainage without much intervening plant uptake ofwater.

The Br− data were modelled by retaining the previousvalues of n andE, and varyingr andN to simulate thedynamics of the ‘tail’ of the response curve which iscontrolled by rate-limited diffusion from the immobilewater fraction. The results in Fig. 3 were obtained withn 5 20,E 5 10.3 mm,N 5 4 mm, andr 5 0.005 mm−1.The model does not fit the data well over the remainderof the curve near the peak because the higher recoveryof Br− now includes drainage events which do not reflectwater movement in the near-surface zone, and the BTCis not a pulse response.

The 35S data (Fig. 4) exhibit a much slower dynamicresponse which is modelled by increasing the value ofNto simulate adsorption–desorption in the immobile waterfraction, whilst retaining the previously derived valuesof n, E, andr. The visual fitting objective was to simu-late the slope of the rising trend of the BTC which hadnot reached a peak during the period of record. Theresulting value ofN 5 28 mm, in comparison to thevalue for the Br− data, corresponds to a retardation coef-ficient of Rim 5 7, and the non-equilibrium behaviourdepends on the value ofr for diffusion between mobileand immobile water.

For the measurement depthL 5 1100 mm, the valuesof n 5 20, E 5 10.3 mm correspond to dispersivityl5 27.5 mm, and mobile water fractionum 5 0.19, fromthe relationshipsn 5 L/2l andE 5 2lum. The value ofN 5 4 mm obtained for Br− (no adsorption) implies thatthe immobile water fraction isuim 5 0.07 (from N 52luim).

The 35S data were selected for the following demon-stration of estimating the state of the solute transport pro-cesses for the purpose of forecasting future behaviour ona real-time basis.

3.3. State estimation with the Kalman filter

Application of substances to the land surface inresponse to feedback information from monitoring ofleachate concentration may be considered as a controlledsystem which tracks the unknown consumption of sol-utes by plant uptake, decay, or other biochemical pro-cesses. The best information for feedback to operationalaction is based on estimation of the current states of thesystem. For the model (Eq. (15)) the states are the valuesof cj andsj, the resident concentrations of the cells.

The Kalman filter (e.g. Davis and Vinter, 1985, pp.117–127) is a Bayesian statistical procedure for combin-ing prior knowledge about a dynamic system togetherwith information from measured data to produce an opti-mal estimate of the states of the system, given assump-tions about the noise properties which representmeasurement error and process uncertainties.

Uncertainty about process knowledge, and imperfectmeasurement, can be accounted for by appending errorcomponents to Eq. (15). This is more concisely illus-trated by the discrete form of the general state–spacesystem (Eq. (1)), which includes process noisew(k) andmeasurement noisev(k):

X(k 1 1) 5 AdX(k) 1 BdU(k) 1 Gw(k) (16)

Y(k) 5 CX(k) 1 DU(k) 1 v(k)

In this particular formulationw(k) andv(k) are uncor-related with each other and have covariance matricesQand R, respectively. Evaluation of the state estimator


Fig. 2. Solute pulse response of the advective–dispersive component of the SSMC model (n 5 20, E 5 10.3 mm), fitted to the15N data (*).

Fig. 3. Solute pulse response of the SSMC model of Fig. 2 with the addition of the mobile–immobile soil water component (n 5 4.0 mm,r 50.005 mm−1), to represent the Br− data (*).

depends on the relative magnitudes ofQ and R, andtherefore the variance of the measurement error was setasR 5 1. The matrixQ was set with only diagonal non-zero elements of magnitudes based on subjective judge-

ment about processes in the soil profile in relation tomeasurements of the variance of the lysimeter data.

The values of these variances of process uncertainty,for the 20 cells of the example, are shown in Fig. 5.


Fig. 4. Solute pulse response of the SSMC model of Fig. 3 modified (n 5 28 mm) to simulate adsorption–desorption of35S-labelled sulphate (*).

Fig. 5. Uncertainty about process knowledge for each cell expressed as error variance relative to measurement error variance.

The most significant feature is the high relative varianceascribed to the root zone (cells 1–5), especially near theground surface, in comparison to the lower values forthe deeper part of the profile where there is insignificantplant uptake.

The matrix G, which relates the uncertainty of thestatesrj, sj to the uncertainty of processes within each

cell was of size 403 20 comprising two identity matr-ices of size 203 20. This configuration implies that thestate uncertainties are correlated within cells but uncor-related between cells.

In the present demonstration, model (Eq. (15)) wascombined with the35S BTC data and the uncertainty pro-file of Fig. 5, by means of theMATLAB Control Sys-


tem procedures ‘dlqe’ and ‘destim’ (Grace et al., 1992),to form the state estimator which is itself in state–space form.

The state estimator was tested for its ability to forecastthe system states (cj, sj), and hence the outputcout, atone drainage interval beyond each observed BTC datavalue, using the procedure ‘dlsim’ with the data seriesas input. The initial state values at the beginning of theseries were set to zero except for the top cell which wasdetermined from the value of the solute pulse applied atthe ground surface. The water depth applied with thesolute pulse (14 mm) was approximately the combinedwater fraction for one cell, and therefore the solute pulsedose of 740 Mbq m−2 35S was allocated to an equilibriumconcentration in the total water storage equivalent (E 1N 5 38.3 mm) of the top cell, so thatc1 5 s1 519.32 MBq l−1. Fig. 6 shows the results of the one-step-ahead forecasts.

Forecasts of leachate concentration can, of course, bemade for any number of drainage intervals beyond ameasured value, but then the knowledge basis is themodel (Eq. (15)) with initial state values as estimatedfrom this last value and with no further correction. Fig.7 shows the full leachate response to the solute pulse asforecasted (using ‘dinitial’) from the last measuredvalue. Approximately 13% of the35S dose had beenleached at the end of the measured portion of the BTC,whereas the complete forecasted response accounts for66% at 3000 mm of drainage. This is a conservativeforecast because the possibility of further plant uptakehas not been taken into account.

Fig. 6. One-step-ahead forecast of the35S data (*).

4. Discussion and conclusions

The purpose of the model development was to enableaccess to control theory and supporting software so thatthis technology could be applied to managing the impacton groundwater of applying wastes and fertilisers to theland. The model is designed for application to leachatedata from undisturbed field soils where uncertainty ofprocess parameters and measurement is high. Processesof advective–dispersive transport, mobile–immobile soilwater, and linear adsorption (equilibrium and non-equilibrium) were explicitly included in the model devel-opment, but have been shown to be represented by asmaller set of dynamic characteristics. Sources andsinks, such as plant uptake, are implicitly included in theconcept of this model being part of a controlled systemwhich is tracking these assimilative processes.

Demonstration of the model was centred on its roleas a component in an application of the Kalman filter,a pivotal theory in estimation and control under uncer-tainty. The results (Fig. 6) display satisfactory dynamictracking in response to new data but the error betweenforecast and measurement shows some persistence,which indicates imperfections in specification of themodel and uncertainty profile (Fig. 5). The forecast ofthe complete response, at 1100 mm depth, to a soluteapplication at the land surface (Fig. 7) demonstratesagain how the model can be used to incorporate availabledata and subjective judgements about uncertainty intomaking an optimal estimate of dynamic behaviour.


Fig. 7. Forecast of the complete breakthrough curve for35S beyond the existing data record (*) and forecast of Fig. 6.

Acknowledgements

The experimental data used for the model develop-ment and demonstration were contributed by RonMcLaren (staff) and Trish Fraser (former postgraduatestudent), Soil Science Department, Lincoln University.The reported research was part of the Groundwater Qual-ity Protection programme funded by the Foundation forResearch, Science and Technology, New Zealand.

References

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Bajracharya, K., Barry, D.A., 1994. Note on common mixing cell mod-els. Journal of Hydrology 153, 189–214.

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Bertsekas, D.P., 1976. Dynamic Programming and Stochastic Control.Academic Press, New York.

Cameron, K.C., Smith, N.P., McLay, C.D.A., Fraser, P.M., McPher-son, R.J., Harrison, D.F., Harbottle, P., 1992. Lysimeters withoutedge flow: an improved design and sampling procedure. SoilScience Society of America Journal 56, 1625–1628.

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Jury, W.A., Dyson, J.S., Butters, G.L., 1990. Transfer function modelof field-scale solute transport under transient water flow. SoilScience Society of America Journal 54, 327–332.

McLaren, R.G., Cameron, K.C., Fraser, P.M., 1993. A comparison ofthe effects of subsoiling on plant uptake and leaching losses ofsulphur and nitrogen from a simulated urine patch. Plant and Soil155–156, 375–378.

Sardin, M., Schweich, D., Leij, F.J., Van Genuchten, M.Th., 1991.Modeling the nonequilibrium transport of linearly interacting sol-utes in porous media: a review. Water Resources Research 27 (9),2287–2307.

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state–space mixing cell model of unsteady solute transport in unsaturated soil

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