Angermann L. (ed.). Numerical Simulations - Applications, Examples and Theory

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Numerical Modeling of Reflector Antennas

231

From fig.16, 19 and 20 it follows that when the antenna receives a spherical wave from a

point on the focal plane, the isolation coefficient between the feed-horn in the focusing point

(the feed-horn 1) and the adjacent (feed-horn 2) is not less than 16-17 dB. The isolation

coefficient at the feed-horn 1 with the rest feed-horns is not less than 20dB. If the feed-horns

location in the focusing plane provides crossing the antenna diagrams at the level of -3dB

then the level of the side lobes in antenna with the dimensions of the feed-horn corrected in

the focusing plane and along the focal axis is distinctly lower than in an antenna without the

dimension correction and positions of the feed-horns. The isolation of the feed horns in

these two cases doesn’t differ much.

0 2 4 6 8 10121416

P1/Pn

0246810121416

P1/Pn

a) with dimension correction b) without dimension correction

Fig. 19. The dependence of isolation coefficient on feed-horn number.

Angle Teta, deg.

302826242220181614121086420-2-4-6-8

-10

-20

-30

-40

-50

-60

-70

Fig. 19. The antenna diagram of multi-beam antenna without correction of feed-horns

dimensions and its location.

When the antenna receives a spherical wave from the point located at Y

≠ 0 some

peculiarities occur. It is connected with the fact that the distribution of field amplitude in the

focusing plane is asymmetrical regarding the maximum. Therefore the isolation coefficient

between the feed-horn concerned and adjacent feed-horn located to the left and to the right

from it is different.

Numerical Simulations - Applications, Examples and Theory

232

Monopulse double-reflector Cassegrain antenna.

Monopulse antennas are designed for forming two differential diagrams and one sum

diagram. The schemes for construction of monopulse double-reflector antennas (MDA) are

described in literature. The simplified scheme of a Cassegrain antenna is shown in fig. 22.

Designations: 1- is paraboloid (reflector); 2 is a hyperboloid (sub-reflector); 3 – is monopulse

horn. The antenna receives a spherical electromagnetic wave coming from the point P (R,0,f)

of the far-field zone with angular coordinates 0, f. On the fig. 22 one beam in the receiving

mode is depicted in fig. 22 with dotted lines.

The feed-horn must generate three signals: two differential and one summary. Further the

numerical modeling results are given and the main principles are described for two types of

MDA: a) with a feed-horn in the form of 4-horns executed by TE01 wave mode; b) with a

multimode feed-horn. The numerical modeling is produced for antennas with paraboloid

diameters the

D=30λ and the ratio

D/F =0,4. The diameter of the hyperboloid depends

on the eccentricity “e” and on the far focal distance Fg1.

A multimode feed-horn is a pyramidal horns with waves

. The

scheme of simultaneous excitation of these wave modes though the isolated inputs is

described in literature. These waves form necessary antenna diagrams: the wave

form

summary antenna diagram, the wave

forms a differential antenna diagram in the

magnetic plane (H plane); the sum of waves

forms the differential diagrams

in the electrical plane (E plane). The parameters of the antenna diagram of the sum and

differential channels depend on the dimensions of the feed-horn cross-section

and

the eccentricity E.

Further the dependencies of antenna parameters from

A ,

B are analyzed. The numerical

modeling is produced for antennas with diameters of the paraboloid

D=30λ and the ratio

D/F =0,4.

The diameter of the hyperboloid depends on the eccentricity E and the distance between

paraboloid apex and the feed-horn aperture. The hyperboloid must be inscribed in the

aperture angle of paraboloid, as it is shown in fig. 6. The diameter of the hyperboloid

must not be larger than 0.25

D to reduce the shadow effect. Further results of modeling are

given for the case of H=3λ and

D =0,22

(the eccentricity e=2). For these parameters the

level of side lobes in the sum channel in E and H planes is equal and it doesn’t exceed -25dB;

if the condition

=3,33λ;

=2,33λ (

=1,428) is fulfilled. The antenna diagrams

of the sum channel for the mentioned parameters are shown in fig. 22; in the plane E with a

thick line and for H plane with a thin line. The

values, the diameters of

hyperboloid

D and antenna parameters in the summary and differential channels depend

on the hyperboloid eccentricity.

The side lobes level and the width of the main lobe of antenna diagram for the summary

and differential channels depend on

/λ and

/λ ratios. With growth of the

eccentricity the hyperboloid diameter decreases if the condition of equality of the paraboloid

aperture angles and the hyperboloid aperture angle from the focus of paraboloid (the near

focus of hyperboloid) is fulfilled. The dependence of the ratio

D on the eccentricity E

with

=3,33λ and

=2,33λ is shown in fig. 23. This figure also depicts the dependence

of the antenna efficiency factor (Kef) in the summary channel from the hyperboloid

eccentricity. The antenna efficiency factor is the value that connects the antenna gain (G), the

Numerical Modeling of Reflector Antennas

233

area of paraboloid aperture (S) and the wavelength. This relation is determined from well-

known formula:

λ/KefSπ4G ⋅⋅= , the 4/DπS

= . (28)

Fig. 22. An Cassegrain antenna scheme.

Angle Teta, deg.

1050-5-10

-10

-20

-30

-40

-50

Fig. 23. The antenna diagram for the sum

1.0 1.5 2.0 2.5 3.0 3.5

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

Dg/Dp

Kef

Fig. 24. The

D and Kef versus eccentricity and differential channels.

P(R,θ,φ)

Numerical Simulations - Applications, Examples and Theory

234

Depending on the eccentricity the side lobe of summary and differential channels in the E-

and H- planes for

A =3,33λ,

B =2,33λ change as it is shown in fig. 24. In our calculation

we used a lens in the horn to achieve equal phase distribution in the aperture.

With the growth of the eccentricity the hyperboloid diameter increases. Therefore if the

values

A ,

B remain the same the horn field level decreases at the edge of hyperboloid

surface. It leads to reducing the side lobes level and to increasing the main lobe width. At

the expense of the side lobes reducing the antenna gain increases. At the expense of the

main lobe extension the antenna gain decreases.

These two factors lead to the situation when at a certain level of the horn field at the edge of

the hyprboloid the antenna gain reaches its maximum. In this case the antenna gain is

maximal. Numerical analysis indicates that the maximum of antenna gain in the summary

channel is observed when the field level at the edge of the hyperboloid in comparison with

the center is

≈Δ 0,3.

1.0 1.5 2.0 2.5 3.0 3.5

-29

-28

-27

-26

-25

-24

-23

-22

-21

-20

-19

E-plane

H-plane

1.0 1.5 2.0 2.5 3.0 3.5

-20

-19

-18

-17

-16

-15

-14

-13

E-plane

H-plane

a) sum channel b) differential channels

Fig. 24. Side lobe levels versus hyperboloid eccentricity

The dependence of the efficiency factor and the antenna gain in differential channels on the

eccentricity is shown on the fig. 24. The maximum of Kef is observed when the level on the

edge of the aperture in the plane E is

≈Δ

0,4; in the plane H is

≈Δ

0,3.

From fig. 22 it follows that for reducing of side lobes level in the summary channel it’s

necessary to increase eccentricity in comparison with e=2 besides side lobes in the summary

channel in the E- and H-planes are equal. The efficiency factor increases, if E<2.5.

But then the width of main lobe in the summary channel is increased and the antenna

efficiency coefficient for the summary channel is decreased.

4. Conclusion

In the chapter the mathematical model of a reflector antenna is described with using the

physical optical method and the waveguide excitation theory. New results of research of

regularity and parameters of reflector antennas are:

• The field distribution in a near-field zone in a tangent plane and along a focal axis in

radiation mode and receiving mode at different wave dimensions of antenna elements;

in the receiving mode the antenna is illuminated by a spherical wave from the point

with known coordinates, located in any space zone;

Numerical Modeling of Reflector Antennas

235

• The isolation between channels in a multi-beam antenna in receiving mode of spherical

wave, radiated from near-field zone point depending on geometrical parameters of

antenna elements.

• Changing diagram parameters of a reflector antenna during scanning and focusing in to

given distance.

The results obtained can be used with designing a different purpose antenna.

5. References

Charles, M. An Extension of Rushh’s Asymptotic Physical Optics Diffraction Theory of a

Paraboloid Antenna. IEEE Trans. Antennas Propagat., vol.23, pp.741-743, Sept.,

1975.

Chen, J & Xu, Y. Analysis and Calculation of Radiation Patterns of Cassegrain Antennas.

IEEE Trans. Antennas Propagat., vol. 38, pp. 823 – 830.

Fitzgerald, W. The Efficiency of Near-Field Cassegrainian Antennas. IEEE Trans. Antennas

Propagat., AP-14, no.9, pp.648-650, Sept. 1972.

Hannan, P.W.

Optimum feeds for all three modes of a monopulse antenna I: Theory, IRE Trans.

Antennas Propagat., vol. 9, pp. 444 - 454, September 1961.

Houshmand, D; Lee, S-W; Rahmat-Samii, Y & Lam, P. Analysis of Near-Field Cassegrain

Refletctor: Plane Wave Versus Element-by-Element Approach. 1988 IEEE Int.

Antennas Propagat. Symp. Dig. vol. 26, pp. 124 - 127, June 1988.

Khayatian, B; Rahmat-Samii, Y;

Characteristics of dual reflector antennas with gaps placed on the

subreflector: MoM and PO analysis

, 1999 IEEE Int. Antennas Propagat. Symp. Dig.

vol. 37, pp. 2332 - 2335, June 1999.

Laybros, S; Combes, P.F.; &Mametsa, H.J. The «Very-Near-Field» Region of Equiphase

Radiating Apertures. IEEE Antennas & Propagation Magazine. 2005, vol. 47, No.4,

pp.50-66.

Narasimhan,M; Ramanujam,P; & Raghavan, K.

GTD analysis of a hyperboloidal subreflector

with conical flange attachment, IEEE Trans. Antennas Propagat., vol. 29, pp. 865 - 871,

November 1981.

Narasimhan, N and Christopher S. A New Method of Analysis of the Near and Far Fields of

Paraboloidal Reflectors. IEEE Trans. Antennas Propagat., AP-32, no.1, pp.13-19,

January 1984.

Narasimhan M & Govind, K. Front-to-back ratio of paraboloidal reflectors, IEEE Trans.

Antennas Propagat., vol. 39, pp. 877 - 882, July 1991.

Rahmat-Samii, Y.

Subreflector extension for improved efficiencies in Cassegrain antennas--

GTD/PO analysis

, IEEE Trans. Antennas Propagat., vol. 34, pp. 1266 - 1269, October

1986.

Rahmat-Samii, Y;

Jacobi-Bessel analysis of reflector antennas with elliptical apertures, IEEE Trans.

Antennas Propagat., vol. 35, pp. 1070 - 1074, September 1987.

Radar Handbook. Editor-In-Chif Skolnik M.I. McGraw_Hill Book Company, 1970.

Rusch, W. Physical-optics diffraction coefficients for a paraboloid. Electronic Lett., vol.10,

pp.358-360, Aug.22, 1974.

Valentino, A & Toulios,P. Fields in the Focal Ragion of Offset Parabolic Antennas. IEEE

Trans. Antennas Propagat., vol. 24, pp. 859 - 865, November 1976.

Numerical Simulations - Applications, Examples and Theory

236

Watson, W. The Field Distribution in Focal Plane of a Paraboloidal Reflector. IEEE Trans.

Antennas Propagat., vol. 12, pp. 561 - 569, September 1964.

Modeling of Microwave Heating

and Oil Filtration in Stratum

Serge Sysoev

and Anatoli Kislitsyn

Surgut State University,

Tyumen State University

Russia

1. Introduction

Extraction of high-viscosity oil and bitumen is an important practical problem, because the

reserves of such deposits are significant, and their role in common stocks of organic raw

materials is constantly increasing. However, on account of the high viscosity of oil, and but

also because of frequent blockage bottomhole zone due to sediments of colloidal surface-

active components of oil extraction is becomes possible only after preliminary heat

treatment of the stratum. Traditional methods of thermal treatment - hot steam or hot liquid

- are in this case ineffective. Moreover, their widespread use may lead to severe

environmental consequences in the form of violations of the hydrogeological environment.

One of the promising methods of thermal treatment is an electromagnetic heating of the

productive layers. Due to deep penetration and the volumetric heat release, and absence of

coolant, electromagnetic radiation can provide (compared to traditional methods) high

speed and uniform heating, the possibility of optimal control and automation of

technological processes, virtually eliminate the harmful effects on the environment. The

results of laboratory and field trials in Russia (Sayakhov et al., 1970, Makogon et al., 1989,

Sayakhov et al., 2002) and practical experience of using this technology on an industrial

scale in the U.S. and Canada (Da Mata et al., 1997, Vermeulen & McGee, 2000, Sahni et al.,

2000, Chhetri & Islam, 2008) show perspective utility of this trend. However, the effective

realization of these opportunities is hindered by the lack of reliable data on the study of heat

and mass transfer processes in multiphase media, typical for the oil and gas technologies,

when subjected to these media microwave electromagnetic radiation. The main objective is

to determine optimal modes of stimulation, namely: the frequency and power of a source of

microwave radiation, the parameters of the antenna, the possibility of using nonlinear

properties of the medium to enhance impact on the models as close as possible to real

conditions.

In Russia, work on the effects of high frequency electromagnetic radiation on the oil

reservoir was started in the late 60-ies by a team from Bashkir State University under the

leadership of F.L. Sayakhov (Sayakhov et al., 1970, Sayakhov et al., 1975). Their industrial-

scale plant was successful tested at Sushuglinsk' and Mordovo-Karmalsk' oilfields. At this

industrial-scale plant, the electromagnetic energy from the high-frequency generator for the

feeder (two coaxial tubes) is inserted into the well. The outer sheath of coaxial cable joins the

casing and the central thread cable - to the tubing at a depth of about 5 meters, so that the

Numerical Simulations - Applications, Examples and Theory

238

upper part of the column and tubing formed the short-circuited line, equal to 1/4

wavelength. Inside the tubing is submerged rod pump. Casing and tubing are used as a

coaxial transmission line for supplying a high frequency electromagnetic energy to the

radiator. The radiator consists of the lower casing and the bottom of the tubing, which

stands below the casing with 1/4 wavelength (quarter-wave linear radiator). 2.5-inch

duralumin tubes were used as tubing. A diameter of casing is 9 inches. Insulating washers

(plastic ring 15 mm thick) placed every 8-10 m along the tubing, were used for insulation of

tubing from the casing. Generator with an operating frequency of 13.56 MHz and output

power (under optimal conditions) 63 kW was used as a source of high frequency

electromagnetic radiation. The average yield before the start of heating was 0.1 m3/day;

water cut was 30%. As a result of heating at the output power of 20-30 kW steady-state

temperature set at 110 C after 7 days; yield increased to 0.25 m3/day, i.e. 2,5 times; water cut

was reduced to 7-8%, i.e. more than 3 times. A well operated with a high yield flow rate for

17 days after the end of the electromagnetic effects.

In the U.S.A., research of electromagnetic effects on the oil wells was started in the late 70's.

These studies culminated in a series of successful tests on the oil fields of the United States

and Canada, and current technology of high-frequency electromagnetic radiation is reduced

to a cost-effective and competitive level that allowed to move to its practical use in

industrial scale. The effect of electromagnetic heating of the near-well zone can be illustrated

by the tests carried out on the field in Alberta. To assess the effectiveness of the heating for

this field computer simulations were carried out. The modeling predicted approximately

twofold increase of well production by electromagnetic effects. The well was drilled in

January 1986, and in March 1986 started production of oil. Up to the impact well gave about

6 barrels of oil per day. A month after the start of operation was launched electromagnetic

heating. A few days later oil production increased and set at about 20 barrels per day, i.e.

even higher than had been predicted by numerical simulation.

Successful tests were conducted in several other fields of the United States and Canada (in

Oklahoma, Utah, Texas, California). Currently in the U.S.A. (New Jersey) the company

Global Resource Corp. has been successfully working in this area. This company is a

developer of a patented microwave technology and machinery that extracts oil and

petroleum products from shale deposits, tar sands, capped oil wells, coals and processed

materials such as tires and plastics as well as dredged soil from harbors and river bottoms.

This process produces significantly greater yields and lower costs than are available using

existing technologies.

Over the past 10-15 years several reviews of methods of electromagnetic heating for

enhanced oil recovery (Da Mata et al., 1997, Vermeulen & McGee, 2000, Sahni et al., 2000,

Chhetri & Islam, 2008) have been published.

Theoretical studies of heat and mass transfer in the oil stratum under the influence of high

frequency electromagnetic radiation were carried out by teams of specialists under the

leadership of R.I. Nigmatulin and F.P. Sayakhov (Zyunk Ngok Khai et al., 1987, Kislitsyn &

Nigmatulin, 1990, Sayakhov et al., 1998, Sayakhov et al., 2002, Kovaleva et al., 2004).

The process of heating and filtration of bitumen in porous medium volume heat source,

arising due to absorption of energy of electromagnetic waves was studied (Zyunk Ngok

Khai et al., 1987). One-dimensional problem was solved taking into account the phase

transition. They found a stationary solution for spherically symmetric source of

electromagnetic waves and self-similar solution for a cylindrically symmetric source.

Numerical simulation of the space heating and filtration of oil in the presence of a moving

Modeling of Microwave Heating and Oil Filtration in Stratum

239

front of melting was performed in the one-dimensional model. Quantitative estimates of the

size of the heat zone were obtained. It was also pointed out to the danger that the source of

too much power will cause overheating near wellbore zone. The negative consequences of

such overheating are the decomposition of oil near the well, the deformation of the skeleton

of porous rock, the destruction of wells, etc.

The theoretical study of heat and mass transfer in the oil stratum when it is heated by high-

frequency electromagnetic radiation was performed on one- and two-dimensional models in

research (Sayakhov et al., 1998, Sayakhov et al., 2002, Kovaleva et al., 2004, Kislitsyn, 1993,

Kislitsyn, 1996). In these studies, considerable attention was paid to the propagation of

electromagnetic waves in the oil reservoir and the distribution of the density of volumetric

volume heat sources. The valuation of efficiency and cost effectiveness of the method in

terms of energy balance has been made.

The filtration processes in porous media filled with a solid gas hydrate or liquid, with

depression and thermal effects (including the electromagnetic heating), which leads to phase

transitions (gas hydrate decomposition, boiling liquid) were studied in research (Shagapov

& Syrtlanov, 1994). Optimal regimes of the heating stratum by using high-frequency

electromagnetic radiation were determined to gas hydration control in the near wellbore

zone.

Summarizing this brief review, it should be noted that there are a number of studies which

examined the processes of heating and filtration of oil in stratums when exposed to high-

frequency electromagnetic field. In these studies important results that may be used to

estimate the depth and duration of heating and to select the optimal modes of exposure were

obtained. Overall, however, the problem can not be well studied. In all the works cited above

the equation for the electromagnetic field with the type of radiating antenna, temperature and

frequency dependences of dielectric loss tangent of the medium wasn't used. Neglect of these

circumstances can cause significant inaccuracies, and even erroneous results.

In this paper we propose a mathematical model closer to the actual conditions that includes

two-dimensional system of interrelated equations of heat transfer and piezoconductivity,

supplemented by the equation for the electromagnetic field with the type of radiating

antenna, temperature and frequency dependence of dielectric loss tangent of the medium.

2. Model and equations

Numerical studies were performed with a two-dimensional axisymmetric model, a diagram

of which is shown in Fig.1. The petroleum stratum 2 is contained between planes

perpendicular to the z-axis (1 - cap rock, 3 - underlying bedrock). The plate is bounded

above and below by an infinite medium, the physical characteristics of which (thermal

conductivity, density, heat capacity) differ from those of the plate. An electromagnetic

radiation source 4 with an antenna is placed in the well. In this model, the antenna consists

of a coaxial cable with a ring-shaped slot 7 cut on the outer conductor 6 from the short-

circuited tip (5 - the central conductor of coaxial cable). The isolines of magnetic strength 8

in the coaxial cable, the petroleum stratum and the adjacent rock are shown in Fig.1.

Electromagnetic waves propagate in a radial direction about the well; they are absorbed and

volume heating of the plate and adjacent rock occurs. Because of the heating the viscosity of

the oil decreases and its flow into the well increases.

For fixed source power the size of the heated zone depends on the physical parameters of

the medium and the electromagnetic wave penetration depth. This depth, in turn, depends

Numerical Simulations - Applications, Examples and Theory

240

on the frequency of the radiation and can thus be controlled. For too great penetration

depths (too low a frequency) the source energy is dissipated in a large region and leaks into

the adjacent rock without producing the required heating. For too small a penetration depth

(too high a frequency) intense heating of a small region surrounding the source occurs, a

high temperature gradient develops, and heat is lost intensely upward and downward

without providing the required radial heating. In both cases the heated zone is small and

heating is ineffective. Consequently, there must exist some optimum frequency at which (for

fixed source power) the most effective heating can be produced. As for source power, within

the framework of the model used, the higher that power, the higher the well yield, but also

the higher the heat loss. Therefore the efficiency of heating (ratio of the increase in

petroleum yield to energy expended) can prove low for too high power level. Moreover, the

radiated power is limited by the fact that it is undesirable to heat the oil above the

temperature at which it decomposes. Determination of optimum values for radiation

frequency and power is the basic task of our numerical modeling.

z,m

r,m

Fig. 1. A diagram of the model: 1 - cap rock; 2 - petroleum stratum; 3 - underlying bedrock; 4

- an electromagnetic radiation source with an antenna; 5 and 6 - a central conductor and an

outer conductor of coaxial cable, respectively; 7 - a ring-shaped slot; 8 - isolines of magnetic

strength

The model takes advantage of the problem’s rotational symmetry, which allows modeling in

2D using cylindrical coordinates as indicated in Fig.1. When modeling in 2D, we can select a

fine mesh and achieve excellent accuracy. The model uses a frequency-domain problem

formulation with the complex-valued azimuthal component of the magnetic field as the

unknown.

The radial and axial extent of the computational domain is in reality larger than indicated in

Fig.1. This problem does not model the interior of the metallic conductors, and it models

metallic parts using boundary conditions, setting the tangential component of the electric

field to zero.

An electromagnetic wave propagating in a coaxial cable is characterized by transverse

electromagnetic fields (TEM). Assuming time-harmonic fields with complex amplitudes

containing the phase information, the appropriate equations are