on the simulation of unsaturated oil flow in soils avner...

Scientific Basis for Water Resources Management (Proceedings of the Jerusalem Symposium, September 1985). IAHS Publ. no. 153.

O n the simulation of unsaturated oil flow in soils

AVNER KESSLER & HILLEL RUBIN Faculty of Civil Engineering, Technion - Israel Institute of Technology, Haifa 32000, Israel

ABSTRACT This study concerns the development of a numerical model being able to simulate unsaturated flow of oil spills in soils. Such a model should provide important information concerning variations of the groundwater quality and the remedial measures required against the hazardous contamination by oil spills. Previous study indicated that short-term negative environmental effects can be simulated by applying the Richards equation. It was also found that the parameters needed for the use of this equation can often be evaluated by applying available water infiltration data. Such an approach is applied in the present article. The numerical scheme developed here stems from the tjj-based equation leading to the determination of the characteristics of soil spill migration in the unsaturated zone.

Simulation d'écoulement d'hydrocarbures en milieu non saturée RESUME Cette étude traite de la mise au point d'un modèle numérique pour la simulation d'écoulements d'hydrocabures en milieu non saturé. Un tel modèle est en mesure de fournir des informations importantes sur les variations de la qualité des eaux souterraines et sur les mesures qu'il faudrait prendre pour prévenir leur contamination. Des etudes précédentes ont montré qu'il était possible de simuler les effets de cette pollution à court terme en utilisant l'équation de Richards. Les paramètres nécessaires à l'utilisation de cette équation peuvent être évalués, comme il a été montré, à partir de données sur l'infiltration d'eau. Une telle approche est appliquée dans cette étude. Le modèle numérique est mis au point à partir d'une équation basée sur le potentiel capillaire IJJ , dont la solution permettra de déterminer les caractéristiques de la migration d'hydrocarbures en milieu non saturé.

INTRODUCTION

The increasing demand for oil products by the industrial society is associated with increasing risks of oil spills. Accidental spills

*This study was partially supported by Technion V.P.R. Fund -M.SC. Papo Research Fund.

195

196 Avner Kessler S Hillel Rubin

from various sources are described in various publications (e.g. Schwille, 1975; King & Kruijer, 1974; Schellekens, 1974; Vanlooke et al., 1975). Remedial measures against the contamination of the soil and groundwater by such spills require information concerning the oil spill migration in the unsaturated zone. Such information can be provided by using the appropriate numerical model parameters representing the characteristics of the soil and the contaminant. In a previous study (Kessler & Rubin, 1985) the authors showed that in various cases, considering the short-term migration of oil spills, available data concerning water infiltration can be useful for the evaluation of the parameters determining the soil spill flow in the unsaturated zone. The present study applies such an approach. Here a finite difference numerical model is developed, water infiltration data are converted in order to predict the oil migration parameters. Later numerical simulations provide the characteristics representing this process.

BASIC EQUATIONS

Unsaturated flow is sometimes viewed as a two phase phenomenon in which liquid (water, oil etc.) represents the wetting phase, and gas (air, natural gas etc.) represents the non-wetting phase. During the simultaneous flow of liquid and gas in a porous medium, each point in the flow region is represented by a relative liquid saturation and by a relative gas saturation while their sum is equal to unity. Applying the principles of continuity and motion for the liquid and the gas, Richards equation (Richards, 1931) is obtained as follows

ff = V-(KVH) (1)

where 9 = volumetric liquid content; K = hydraulic conductivity; H = piezometric head. Representing the piezometric head as consisting of the elevation, Z, and the capillary pressure head, 4>, equation (1) yields the following equation:

3 fl

j ^ = V . (KVZ + KV4>) (2)

Equation (2) can be modified into several versions represented in the following paragraphs.

The ty-based equation can be derived by defining the specific moisture capacity, C, as follows

C(ifO = 36/3^ (3)

Introducing equation (3) into equation (2) we obtain

c|| = V-(KVZ + KVijj) (4)

Provided that the hydraulic conductivity is represented as a function

Simulation of unsaturated oil flow in soils 197

of tjj, equation (4) represents the transport of tjj in the porous medium. Therefore it is called the ij;-based equation for unsaturated flow.

The Q-based equation can be derived by defining the hydraulic diffusivity as follows:

D(9) = K3i|V86 (5)

Substituting this expression into equation (2) we obtain the 9-based equation as follows:

~ = V-(KVZ + DV9) (6) ot

The advantage of the ijj-based equation stems from its meaningful expressions for all possible ranges of saturation whereas equation (6) stops to be applicable in the saturated zones of the porous medium where D ->- °°. However, in the vicinity of a sharp wetting front, small changes in the value of 8 may often be associated with extremely large variations in the magnitude of tjj. Such conditions are especially common during infiltration into relatively dry soils. In such cases the use of the ip-based equation may lead to numerical models subject to problems of convergence and stability.

The F-based equation stems from the application of the Klrchoff integral transformation (Carslaw & Jaeger, 1959) being defined as follows :

F(i{0 = J* K(*)d(p (7)

where ty0 = lower bound for the pressure head in the porous medium. Introducing equation (7) into equation (2) we obtain the F-based equation as follows:

clj = |f VF-VZ + KV2F (8) dt dip

Basically equations (4), (6) and (8) are of the same type, and they lead to similar numerical models. The application of each model is directed towards different oil spill problems as well as different boundary conditions of the specific flow field. However, the majority of practical problems of oil spill migration may be simulated by models being based on the ijj-based equation.

THE NUMERICAL MODEL

As stated in the previous section there are three possible equations that can be utilized for the development of the numerical schemes simulating the oil spill migration in the unsaturated zone. Schemes resulting from all three equations are very similar. However special attention is paid to the ijj-based model due to its applicability to most of the possible practical cases of oil spill migration.

198 Avner Kessler & Hillel Rubin

Referring to the one-dimensional flow directed downwards, equation (4) yields:

9* 9 3i|) (9)

Using an implicit finite difference scheme, and following the approach outlined by Kroszynski (1977), this equation yields:

AZ , ^i-1 1-5

, AZ. ,n+l l

At

1 2

AZ. AZ. ,n+l

AZi+J 1 + 1 X Atn X i+l 1-9 (10)

where

AZ (AZ. x + AZ. 1)/2; 1+2 1-2

, n n+1 At = t (11)

In equations (10) and (11), subscript i represents a grid nodal point; superscript n represents the time level. The coefficients K and C should be calculated at time n + 1 . Their values are predicted by using the "two points" extrapolation pattern leading to the following formulae:

n+1 n , n n = 9 + a ( 6

i i i

n + 1 = c ( e n + 1 ) ; 1 1

- e-1) i

K n + 1 i±i

= K

_ n + l „ n + l

' l i + l

\ 2 (12)

where a = an extrapolation factor whose value is optimized through the simulation.

The numerical model is subject to two types of boundary conditions: (a) constant oil discharge penetrating the soil surface and (b) variable ponded oil level at the soil surface. The first type of boundary cpnditions is relevant to minor leakages, the latter is relevant to substantial abrupt spillage leading to oil ponding above the ground surface.

VERIFICATION OF THE MODEL APPLICABILITY

The reliability of the model is verified by comparing its water infiltration results with data reported in various publications. Water infiltration into initially dried sand is the first example used for the tp-based model verification. This example was previously treated by Rubin & Steinhardt (1963), using a S-based finite difference scheme. Attempts to solve this problem by t|j-based schemes failed as reported by Neuman (1972) and Kroszynski (1977). This failure is attributed to the ill conditions imposed


on the tridiagonal matrix which has to be solved during the simulation process. Rubin & Steinhardt (1964) provide data relevant to Rehovot sand for which the following parameters, determining unsaturated water flow, are defined:

4>(cm) = -1.02 11.3 + -~ 0.05exp(156) + exp(16.3 - 5759)

K(cm s x) = 8400/ JL 1.02

+ 14.45" (13)

According to Brooks & Corey (1964) the relationships between the unsaturated flow parameters are represented as follows:

S e = <i|;c/,|0

K r = OPc/IO

A

2+3A

S = K = 1.0 e r

for \ty\ > \ty

for 1̂1

1*1 S U

(14)

where ipc = the air entry pressure head; S e = effective saturation; À = a constant.

Figures 1 and 2 show how equations (14) are applicable for Rehovot sand. The experimental data represented in these figures were measured by Rubin & Steinhardt (1964). Introducing the relationships represented in Figs 1 and 2 into the numerical model and simulating constant water discharge we obtain the results represented in Fig.3. Our results are compared with those obtained by Rubin & Steinhardt (1963). The comparison generally indicates

o.o

— --

—

-

-

I

o \

1 11 I I I

\°

...1 1 1 I 1 1

* "I5(cm)

\ -1.355

9 r » 0 . 0 i

se "(T) For + ' l5cm

°\

\ o

\° I I I 1 I \ i 1 I 1

y 100

FIG.l Calculated and experimental relationships between the effective saturation and the capillary pressure for Rehovot sand.

200 Avner Kessler & Hillel Rubin

10'

id s

Id3

id-

id5

id*

id'

i n 9

-

-

1 1 I 1 | 1 1 1

° o \

Vo

1

o\

1

s°o

1 J

, . . .6 .065 K r*(*) For +> 15 cm

o \

o \

\ ° 1 1 111 1 \ 1 1 1 1 1 1

"Y(cm) 50 100 500 1000

FIG.2 Calculated and experimental relationships between the relative hydraulic conductivity and the capillary pressure for Rehovot sand.

TIME INTERVAL 3375.8 SEC 8.829 8.648 8.888 8.888 8.16 J I I I L

KOISTURE CONTENT 8.128 0.149 0.169 8.188 J I I , I

FIG.3 Comparison between water content profiles obtained by the ip-based implicit model (solid line) and by Rubin S Steinhardt model (circles) for infiltration in Rehovot sand with constant discharge of 3.528 x lO'^cm s_1 at the soil surface.


good agreement except for minor differences at the edge of the wetting front. In our calculations we also obtain a good cumulative mass balance, equal to 0.7%, using an extrapolation factor of a = 0.34.

The next stage is to verify the validity of the numerical model with information relevant to Yolo light clay being reported by Philip (1957). The soil-water characteristics of this soil are given as follows:

= 274.17/[739 + (In|^|) ] for 14» I > 1 cm

= 0.495 for | if) | < 1 cm

K(cm s_1) = 0.00153/(124.6 + | \p | 1 • 7 7 ) (15)

0.124; 0.495; Kc 1.23 x 10 5cm s"1

where 6r = residual moisture content; 6g, Ks = values of moisture content and hydraulic conductivity under complete saturation.

Figure 4 shows that equation (15) describes very well the experimental results obtained with Yolo light clay. In this particular case the relationships of Brooks & Corey (1964) failed probably due to swelling effects of this soil. Figure 5 represents the comparison between the results obtained by Philip's (1958) analytical solution, based on a power series expansion, and those obtained by the numerical model developed in the present study. The analytical and numerical results are in good agreement.

OIL SPILL MIGRATION CHARACTERISTICS

In the case of oil migration the original water-soil relationships are transferred into oil-soil relationships using the conversion

E o

\ \

\ : \

;

\

0.2 0.J 0.4 Oil

^tcmVcm")

E o

1 . , v

N > ft

t

\

102

t//(cm)

FIG.4 ty(Q) and K(\p) relationships for Yolo light clay. The crosses are the experimental results and the solid lines are the functional relationships given by equation (15).


HOISTURE CONTENT .8S8 0.168 B.ISB 8.269 B.2S9 0.388 B.350 0.408 0.458

a.

12-

18.

24.

38.

30.

42.

48_

54_

fifl

I I I -J

>~"S~^*

/ ^

fi

l 0*î î ï_———•

6 | 0 * J « _ _ _ -tr

Zio'sec^,^;

—r~"~~^

^**^* °

-^*

FIG.5 Comparison between water content profiles obtained by the ty-based implicit model (solid line) and by Philip's solution (circles) for infiltration in Yolo clay with a constant head of 25 cm at the soil surface.

factor developed in a previous report (Kessler & Rubin, 1985) The IJJ-0 relationships are transferred according to

Va<9> a P

F ~ ~ T ) ^ w > a < e )

w,a <

(16)

where a = the interfacial tension, p = density and the subscripts o, w,a refer to oil, water and air respectively.

The K-8 relationships are transferred by assuming that the permeability depends mainly on the geometrical properties of the pore space. The resulting ratio between Kw and K 0 is:

K0(9) = Kw(9)- (17)

where vw and vQ represent the kinematic viscosity of water and oil. Referring to the density, viscosity and surface tension of

kerosene (Schwille, 1981) we obtain the following conversion formulae:

*o,a = 0 . 4 4 7 ^ ; K0 = 0.2KW (18)

Considering unsaturated flow of kerosene in Rehovot sand, equations (13) and (18) yield:


\jj(cm) = -0.5460 [11.3 + 3' 1 9 - 0.05exp(156) + exp(16.3 - 5759)]

K(cm s •"•) = 1680/ 0.4560

+ 14.45' (19)

Applying these relationships we simulate unsaturated kerosene flow with constant discharge of 3.528 x 10" (cm s _ 1 ) . Figure 6 represents the differences between the characteristics of water infiltration and the oil spill migration. According to the numerical results shown in Fig.6 the moisture content limiting values for the oil and the water are 0.256 and 0.167, respectively. Rubin & Steinhardt (1964) showed that under a constant rate of infiltration the limiting water content, 6^, is given through the following relationship :

q = K(6A) (20)

where q = specific discharge. In our particular simulations equation (20) yields:

• « - 1 * 3.528 x 10 H = K o(0 £ j O) = Kw( wv "£,,„) <>w (21)

where subscripts o and w refer to oil and water respectively. Applying equations (19) and (21) we obtain limiting values of

n

7-

14.

2L

2f.

1

°42.

4Û.

sa.

e&

7 i

TDC INTERVAL 6758. S SEC 0,033 D.Ë30 Û.ES3 1.129

f i l l

/" ^^^"^

B.IEt i

K0ISTIRE CONTENT S. IM 0.210 0 .20 D.270

i i i i

-r 5623 sac ^s 1

/. ~r 12375 08C ^ ^ |

/ 1

1 /

- ^ 2 5 8 7 5 »8C __^-^

FIG.6 Comparison between water content profile (solid line) and oil content profile (dashed line) obtained by the Tp-based implicit model for infiltration in Rehovot sand with constant discharge of 3.528 x 10~ cm s at the soil surface.


0£ w = 0.1675 for water infiltration, and 6^ 0 = 0.260 for oil migration. These values are in good agreement with the numerical results obtained.

In the next stage we study the migration of oil in Yolo light clay. It is assumed that the oil spill is significant enough to lead to temporary ponding of the oil over the soil surface. The thickness of this ponded oil layer controls the oil spill migration process. Figure 7 describes the process of oil spill migration and water infiltration in Yolo light clay provided that the initial thickness of the ponded layer is 5 cm. The following expressions are assumed through the numerical simulation:

0.238 for t = 0 (22)

ft ^o = 5 ~ J °ldt for t à 0 at the soil surface

= 0.238 for t > 0 at the bottom of the flow field.

Through the simulation the effect of the swelling mechanism of the clay is completely ignored. Clay swelling is expected only in the presence of water, leading to decreasing advancement velocity of the water front.

Comparing the oil and the water content profiles obtained under the same initial and boundary conditions, we obtain the following differences :

(a) The oil advances more slowly than the water. In the case of

MOISTURE CONTENT B.3S8 8.486 B.4S8 ' ' L.

FIG.7 Comparison between water content profile (solid line) and oil content profile (dashed line) obtained by ip-based implicit model for infiltration in Yolo clay with initial head of 5 cm at the soil surface.


a constant oil leakage the advancement rate is completely controlled by the K(6) term. Under a positive head and during the redistribution process the advancement rate is controlled by both K(6) and t|K6) relationships.

(b) The oil wetting front has higher moisture gradient than that of the water. The moisture gradient of the wetting front is mainly determined by the capillary term which is smaller in the case of oil migration.

REFERENCES

Brooks, R.H. & Corey, A.T. (1964) Hydraulic properties of porous media. Hydrol. Pap. no.3, Colorado State University, Fort Collins, Colorado.

Carslaw, H.S. & Jaeger, J.C. (1959) Conduction of Heat in Solids. Oxford Univ. Press, London.

Kessler, A. & Rubin, H. (1985) Development of a practical method simulating oil spill migration (OSPIM) in soils. ASCE Hydraulics Div. Speciality Conference, Orlando, Florida.

King, E.M. & Kruijer, G.C. (1974) Spillage from oil industry cross-country pipelines in W.Europe. CONCAWE Report no.2/73, 1/74, 5/74.

Kroszynski, U.I. (1977) European hydrologie system. Report no.3, Danish Hydraulic Institute.

Neuman, S.P. (1972) Finite element computer programs for flow in saturated unsaturated porous media. Technion-Israel Inst, of Techn.

Philip, J.R. (1957) The theory of infiltration: 1. The infiltration equation and its solution. Soil Sci. 83, 345-357.

Philip, J.R. (1958) The theory of infiltration: 6. Effect of water depth over soil. Soil Sci. 85, 278-286.

Richards, L.A. (1931) Capillary conduction of liquids through porous medium. Physics 1, 318-333.

Rubin, J. & Steinhardt, R. (1963) Soil water relations during rain infiltration: I. Theory. Soil Sci. Am. Proc. 27, 246-251.

Rubin, J. & Steinhardt, R. (1964) Soil water relations during rain infiltration: III. Water uptake at incipient ponding. Soil Sci. Am. Proc. 28, 614-619.

Schellekens, G.A.P. (1974) Olieverontreininging pompstation Geulhen. H20 7, 140-143.

Schwille, F. (1975) Groundwater pollution by mineral oil products. In: Groundwater Pollution (Proc. Moscow Symp., August 1971), 226-240. IAHS Publ. no.103.

Schwille, F. (1981) Groundwater pollution in porous media by fluid immiscible with water. The Science of the Total Environment 21, 173-185.

Vanlooke, R., DeBorger, R., Voets, J.P. & Verstraete, W. (1975) Soil and groundwater contamination by oil spills: problems and remedies. Int. J. Environ. Studies 8, 99-111.

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