A Stochastic Model for BOD and DO with Random Initial Conditions and Distance Dependent Velocity

A Stochastic Model for BOD and DO with Random Initial Conditions and Distance Dependent Velocity ALEX S. PAPADOPOULOS Department of Mathematics, University of North Carolina at Charlotte, Charlotte, North Carolina 28223 Received 2 7 September

1982; revised 4 May 1983

ABSTRACT A random

differential

equation

arises

in stream

pollution

models

when

the initial

conditions are stochastic and the stream velocity is distance dependent. The mean squared solution gives the biochemical oxygen demand (BOD) and the dissolved oxygen (DO) at any distance downstream from the pollution source. The probability density function and some of the moments of the BOD and DO are obtained, thus completely characterizing the BOD and DO process. ones.

1.

An example

illustrates

the results and compares

them with previous

INTRODUCTION

In the past, several mathematical models have simulated the pollution caused in streams by the discharge of organic waste materials. It is generally agreed that the two main indicators of water quality are the biochemical oxygen demand (BOD) and the dissolved oxygen (DO). The BOD is defined as the amount of oxygen required by the bacteria while organic matter is decomposed with the help of dissolved oxygen in the water. Dissolved oxygen is obtained directly from the air by aeration and indirectly through the photosynthetic process of aquatic plants. The concentration of DO has to be sufficiently high so as to support aquatic life, and various agencies, such as the Environmental Protection Agency, have specified minimum levels of DO in streams and rivers. The purpose of this study is to extend some of the previous models in a more realistic way so as to predict the amount of BOD and DO at any point on a stream. Beginning with the classical model of Streeter and Phelps [lo] in 1925 (the first equations to model the behavior of the DO and BOD), various mathematical models have been proposed. In 1964 Dobbins [2] derived the MATHEMATICAL

BIOSCIENCES

67:19-31

19

(1983)

OElsevier Science Publishing Co., Inc., 1983 52 Vanderbilt

Ave., New York, NY 10017

00255564/83/$03.00

20

ALEX S. PAPADOPOULOS

basic differential equations for BOD and DO which are the foundations of previous papers as well as of this one. Later Padgett [5], Padgett and Durham [6], Padgett, Schultz, and Tsokos [8], Padgett and Papadopoulos

[7], and Bell

and Papadopoulos [l] studied the model from a stochastic and hence a more realistic point of view. Loucks and Lynn [4] used a Markov model to predict the probability that the concentration of DO will be within specified limits at a point downstream from a source of pollution. Thayer and Krutchkoff [ll] used generating functions to obtain approximate (discrete) probability distributions for DO and/or BOD. Both of the models assume that the DO and BOD concentration exists in only finite number of possible states. Padgett [5] proposed that the deterministic differential equations of Dobbins [2] be considered as differential equations with random initial conditions and random inhomogeneous terms. Furthermore, Padgett and Durham [6], Padgett, Schultz, and Tsokos [8], and Padgett and Papadopoulos [7] studied the stochastic differential equations more extensively. Bell and Papadopoulos [l] considered the situation where the speed of the stream was random and thus the differential system had random initial conditions and random coefficients. The general assumptions that will be made here are the same ones mentioned by Dobbins [2] and Padgett [5]. It will be assumed that there are five major activities in the stream: (i) Bacterial action decreases the pollution BOD and DO. The rate of decrease is proportional to the amount of pollution present with proportionality constant

k,, in units of DO (ppm) per day, and there is always some

DO present. (ii) The reaeration process increases the DO at a rate proportional to the DO deficit with proportionality constant k, in units of DO (ppm) per day. (iii) The pollution only is decreased by sedimentation and absorption at a rate proportional to the amount of pollution present with proportionality constant k, in units of DO (ppm) per day. (iv) The pollution is increased from small sources along the stretch of the stream with rate I, in ppm per day, which is independent of the amount of pollution present. (v) The dissolved oxygen is decreased at a rate d, in ppm per day. The variable d, may have positive or negative values and represents the net change in DO due to Benthal demand and respiration and photosynthesis of plants. In Section 2 the general model in terms of two systems of random differential equations and their solutions will be presented. In Section 3 the probability distributions of DO and BOD will be derived. Finally, in Section 4 an example and computations will be given for a typical stream to illustrate how the results could be utilized to predict the BOD and DO at any point down the stream.

A STOCHASTIC MODEL FOR BOD AND DO

2.

DERIVATION

21

OF THE MODEL

In addition to the above assumptions it will be assumed that the conditions at every cross section of the stream are unchanged with time. Under these assumptions the classical equations derived by Dobbins [2] are: .i(t)-(k,+k,)Z(t)+I,=O, -zl~(t)+k2[C,-c(t)]-kklz(t)-dg=0

(1)

(’ = d/d), where t is the distance downstream from the source of pollution, f(t) is the BOD in ppm at distance t, c(t) is the concentration of DO in ppm at a distance t downstream, c, is the oxygen saturation concentration, and u is the velocity of the stream. The initial conditions for the system (1) are f(0) = I, and c(0) = cc,. In this study the trajectories of BOD and DO as well as their joint probability distribution at each downstream point will be considered under assumptions similar to those of Esen and Rathbun [3] and Padgett [5]. Thus it will be assumed that the effects of k,, I,, and d, are negligible and that the initial conditions I, and c,, are random variables. The corresponding random variables will be denoted by capital letters. In this paper it is assumed that L, and C,, are random variables. However, it also incorporates the notion that more realistically the velocity of the stream is distance dependent. It is realistic to assume that the velocity of the stream is described by different equations at certain segments, i.e. ut( 1) = beof, z+(t)=et+f, us(t)

t,dtdt,,l,

(2) (3)

tjGtGtj+l,

= u3,

t, d t< tk+l.

(4

Equation (2), depending on the coefficient a, describes the section of the stream where the velocity increases or decreases drastically. Equation (3) represents the section of the stream where the velocity is increasing or decreasing in a constant rate, and Equation (4) describes the section of a stream where the velocity is constant. Thus the differential equations (1) become random differential equations with random coefficients and inhomogeneous terms, -u(t)i(t)-k&(t)=O, (5)

u(t)&(t)+k,[c,-C(t)]-k,L(t)=O, with L(0) = L,, C(0) = C,,, and u(t) to be given by either Equation or (4). Equation (5) can be written in a random vector form, k(t)

= A(t)X(t)+Y(t),

t> 0,

(2), (3),

22

ALEX

S. PAPADOPOULOS

where

X(t) =

L(t) [ c(t) 1 ’

Y(t)

1’

= k%s [

u(t)

(6)

and

A(r) =

The differential equation (6) has only random initial conditions, and according to Soong [9] we know that a solution exists. Using variation of constants (Soong

[9], 1973),

the mean

squared

solution

of (6) can be found

and

presented as follows: Case 1.

If the velocity is given by Equation

(2),

x,(t) =

where C(t,) Case 2.

and L(t,)

are the random initial conditions at point t,

If the velocity is given by Equation

(3),

(8) where L(t,)

and C(t,)

are the random initial conditions at point t,

23

A STOCHASTIC MODEL FOR BOD AND DO

The equation for X(t) when the velocity is given by Equation (4)

Case 3. is

x,(t)=

I+[ L(t

k

L(t,)exp(

-

1

%)

)

Mexp(-y)-exp(-v)+C(t,)exp(-v)

y)]]’

cs[l-expl-

where again L( tk) and C(t,)

3.

PROBABILITY

(9)

are the random initial conditions

DISTRIBUTION

at point t,.

OF BOD AND DO

Using a Liouville type theorem we can obtain the joint probability of L(t)

and C(t)

for a given value of t. Equation

density

(6) can be written in the

general form

k(t)=h(X(t);t)

(10)

with random initial conditions C=X(O). Let the probability function of C be fa(C); then the joint density of X(t), g(C; t), is given by the Liouville theorem which states: THEOREM

3.1

Assume that Equation (10) has a mean squared solution for X(t). Then the joint probability function of X( t), f (X( t), t), satisfies the Liouville equation af(x;t>

at

+ 5

‘IfijI i=l -=ax,

o

7

f(x~o)=fo(Xo)

where h, and xj are the jth components of h and x respectively. Using the above theorem, a solution for f(x, t) can be derived as follows. Let the mean squared solution for Equation (7) be written as

-dy”v.h[x=g(c;

f(x,f)=fo(C)exp( c=g-‘(x;

t)

t); t] dt),

24

ALEX S. PAPADOPOULOS

where v. h is the divergence of the vector h. In this study

I

,

k, - -L(t)

u(t)

h(X(t); t> =

W(t) + -k&s ’ u(t) u(t) /

k

-‘L(+\

u(t)

v.h=(

c=

;

(11)

-&&).h

=- k, + k,

u(t) .

The mean squared solution of (11) depends on u(t) and will be as follows: Case 1.

If the velocity is given by (2), then

\ c, = )),

=

g-l@(t);

(12)

t),

and the joint probability function of (L(t),

C(t))

fl(~,c>=flo(~,,c,)exp-i:‘+‘v-hdt [

=fio(l,,c,)exp(

is given by

I

-v(e-u’-

If we proceed as in case 1, the joint density of (L(t), for cases 2 and 3. Thus:

eKa’~)]. C(t))

(13)

can be derived

Case 2. (14)

A STOCHASTIC MODEL FOR BOD AND DO

25

26

ALEX Case

S. PAPADOPOULOS

3.

f3(/,c) =AO(lky+)exp

(k, + k*N- t/c)

1.

U

i

(15)

Using equations (13), (14), (15), the marginal density of L(t) or C(t) can be derived as well as the moments. In the next section an example will illustrate the result of Sections 2 and 3.

4.

EXAMPLE

AND COMPUTATIONS

Some data on DO and BOD as well as values for k,, k,, and c, were obtained by monitoring a creek in Lexington County, South Carolina and

0

I 5

I

10

I

1'5 Distance

FIG.2.

Deterministic

I

1

2'0

25

(miles)

case.

I

3'0

.

A STOCHASTIC MODEL FOR BOD AND DO

21

28

ALEX

Distance FIG. 4.

S. PAPADOPOULOS

(miles) p = 0.0.

will be used to illustrate the usefulness of this study. A &i-squared goodness of fit test on the DO and BOD showed that Lo is approximately N(pL, = 5.16, a& = 2.6), Co is app roximately N( p,-, = 7.39, u& = 1.99), and k, = 0.23, k, = 0.5, c, = 12. Furthermore, the velocity is considered distance dependent, with a possible configuration shown in Figure 1. From t = 0 until t = 7 it is increasing,

from t = 7 to t = 18 it decreases exponentially,

and from t = 18 on

it stays constant. In addition it will be assumed that the random variables L, and Co are independent (p = 0) and have truncated normal distributions with respective means 5.17 and 7.39 and variances 2.6 and 1.99. Thus the joint probability density of Lo and Co is given by

g(kl,ccl)

=

d,d, 2n\/o(1.99)

_exp

i

1 2

(I, -5.16)’

(

+ (co -7.39)2

2.6

1.99

0 < I, < co, where d, and d, are truncation joint density of X(t)

factors. From Equations

is obtained as follows:

}I ’

0 < co < c,,

(2), (3), and (4) the

A STOCHASTIC

MODEL

FOR BOD AND

FIG. 5.

f,,(x; t) =

29

DO

p = 0.0.

d1d2

2aJo(1.99)

+ [ %(c(t),~(t))-7.39]2 1.99

3

30

ALEX

S. PAPADOPOULOS

when 7g tg18,

[4(l(4~(~))-5.16]~ 2.6

+ [4(t),+))-7.3912 1.99

andwhen18,(t
[r2(f(r),c(~))-5.16]2+ 2.6 xexp

[~~(l(t),c(t))-7.39]* 1.99

(kr + k2)(t-l8) i

U

17

and C,( c( t), I(t)) are the joint probability densities for the initial conditions of c(t) and f(t) in the intervals [0,7],

whereG(c(t), Qt)>, C,(c(t), 4r)),

[7,18] and [18, co]. Then the mean functions E( L( t)) and E(C( t)) and the variances a&,) and u&,) may be obtained by integrating the joint density of L(t) and C(t) by the usual methods. Figure 2 shows the solutions of the deterministic equations using the values of I, = 5.16, ca = 7.39, k, = 0.23, k, = 0.5, and c, = 12. These solutions will be used for comparison. Figure 3 shows simulated trajectories of the solution process using the assumed distributions for L, and C,. The mean and standard deviation of L(t) C(t) are shown in Figures 4 and 5.

and

REFERENCES K. M. Bell and A. S. Papadopoulos. A stochastic model for BOD and DO when the discharged pollutants and the flow of the stream are random qualities, Internut. J. Environ. Studies 11131-42 (1979). W. E. Dobbins, BOD and oxygen relationships in streams, J. Sanlt. En,erg. Dir). ASCE (X43) 90:53-78 (1964). I. I. Esen and R. E. Rathbun, A stochastic model for predicting the probability distribution of the dissolved oxygen deficit in streams, Geological Survey Professional Paper 913, U.S. Government Printing Office, Washington. D.C., 1976.

A STOCHASTIC 4

5

MODEL

FOR BOD AND

31

DO

D. P. Loucks and W. R. Lynn, Probabilistic models for predicting stream quality, Water Resources Res. 2:593-605 (1966). W. J. Padgett, A stochastic model for stream pollution, Math. Biosci. 25:309-317

6

(1975). W. J. Padgett and S. D. Durham, A random differential equation pollution, Trans. ASME Ser. G J. Dynamic Sysfems Measurement

I

(1976). W. J. Padgett and A. S. Papadopoulos, Stochastic models DO in streams, Ecological Modeling 6:289-303 (1979).

8

9 r10

11 12

arising in stream Control 98:32-36

for prediction

of BOD and

W. J. Padgett, G. Schultz, and C. P. Tsokos, A stochastic model for BOD and DO in streams when pollutants are discharged over a continuous stretch, Internat. J. Enuiron. S&dies 11:45-55 (1977). T. T. Soong, Random Differential Equations in Science and Engineering, Academic, New York, 1973. H. W. Streeter and E. B. Phelps, A study of the pollution and natural purification of the Ohio River, Public Health Bulletin 146, U.S. Public Health Service, Washington, D.C., 1925. R. P. Thayer and R. G. Krutchkoff, Stochastic model for BOD and DO in streams, J. Sanit. Engrg. Dia. ASCE (SA3) 93:59-72 (1967). C. P. Tsokos and W. J. Padgett, Random integral equations sciences and engineering, Academic, New York, 1974.

with applications

to life