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Proceedings of the 3rd Applied Science for Technology Innovation, ASTECHNOVA 2014 International Energy Conference
Yogyakarta, Indonesia, 13-14 August 2014
238
Thermal Analysis in Geothermal Prospect Pasema Air Keruh- District Empat Lawang, South Sumatera
J. R. Kelibulin
Department of Physics, Pattimura University
Jln. Ir. Putuhena, Ambon
Desi Kiswiranti, Wahyudi
Department of Physics, Gadjah Mada University
Jln. Bulak Sumur, Yogyakarta
ABSTRACT
Temperature measurements have been done in the prospect area of geothermal Pasema Air Keruh,
District Empat Lawang, South Sumatra, to determine the temperature distribution on subsurface.
Temperature measurement in monitoring and mapping is done on an area of 800 m x 800 m with two
different locations, Penantian and Air Keliansar.
Retrieval data of temperature measurement using a temperature sticks and electronic thermometer
incorporating a thermistor as a sensor such as IC LM35. The working principle of this instrument is that
the physical phenomena detected temperature sensor is converted into electrical signals in the potential
difference, then converted in degrees Celsius.
The results of measurements of temperature monitoring at the Penantian obtained average surface
temperatures between 26,5 oC – 29,1 oC while for Air Keliansar obtained average surface temperature
average between 27,3 oC – 30,5 oC. Distribution of heat on subsurfaces is calculated by solving the heat
conduction equation (Laplace) with a finite difference method that has been constructed (known as the
Liebmann method). The modeling assumes that two-dimensional objects shaped the plate. The results of
modeling heat source on subsurface with finite difference methods estimate the temperature reservoir in
Penantian is 400 oC at a depth of 1100 m, while for Air Keliansar the temperature reservoir is 400 oC at a
depth of 1300 m.
.
KEYWORDS: geothermal, monitoring, mapping of surface temperature, finite difference methods,
modelling of heat source
1 INTRODUCTION
Energy needs in human life at this time has increased to a primary requirement. Along with the
increase in human population, energy requirements will also increase. Dependence of human life on fossil
energy (oil , gas and coal) has shifted the priority of energy requirements commensurate with the need for
food, clothing and shelter. Therefore, a wide range of efforts made to meet the energy needs.
Utilization of geothermal resources has been known for a long time, but is still limited to cooking,
bathing and tourist attractions, while the use of geothermal energy is still very little. Geothermal energy is
an alternative energy to reduce dependence on fossil fuels. Geothermal energy has a very minimal risk,
239
because of the relatively safe for the environment and the availability of highly prospective land of
Indonesia.
Indonesia was dominated by the energy sector such as oil, gas and coal. Petroleum holds 57 % of
the energy needs in Indonesia, gas 23 %, coal 18 %, while 2 % is held by a carbon emission (such as
hydroelectric and geothermal). Data obtained from MEMR (the Ministry of Energy and Mineral
Resources), indicates that the supply of geothermal energy in Indonesia amounted to 20,000 MW.
Indonesia has 20 years developing the geothermal energy, but was only able to develop 787 MW, it means
that only 4 % of the total potential in Indonesia (Wulandari , 2004).
Potential area of geothermal in Indonesia is South Sumatra. It is encouraging research in the areas of
thermal methods Pasema Air Keruh, District Empat Lawang, South Sumatra. Research areas include
Penantian and Keliansar. Geographically, the study area is located between the coordinates (3°45'-3°65')
LS and (102°37'5"- 102°52'5") BT as in Figure 1.
Figure 1: Map manifestation research sites (Virgo, et.al, 2013)
2 THEORY
2.1 Geological Research Area
Regional geology of Sumatra is one of the magmatic belt in Indonesia. Sumatra lies along the shores
of Southwest Asian Plate is colliding with the plate so as to experience the subduction of the Indian
Ocean. This form of subduction has been released periodically through the transform fault is parallel to the
plate edges and centered along the Sumatra fault system that stretches along the island of Sumatra
(Kusuma et al., 2005).
Research areas include Penantian and Air Keliansar, PasemaAir Keruh, District Empat Lawang.
Geological mapping is a mapping surface through field observations using a certain trajectory.
Observations made in the field include field orientation, morphology, and rock outcrop observations,
240
qx
Profil
temperatur
T
x
measurements, and rock sampling. Geological map of the study area image shown in Figure 2, with a
scale of 1: 50000.
Figure 2: Geological map of the study area (Virgo, et.al, 2013)
2.2 Basic Law of Heat Transfer by Conduction
The basic relationship for heat transfer by conduction method proposed by French scientists JBJ
Fourier in 1882 (Lienhard et al., 2003). This relationship states that the rate of heat flow by conduction in
a material way equal to the product of the amount of three ingredients namely thermal conductivity (k),
cross-sectional area (A) and temperature gradient in the cross section or the rate of change of the
temperature T of the distance in the direction of flow heat (∂ T / ∂ x). The law of thermodynamics states
that heat will flow automatically from point to point towards the high temperature low temperature, the
heat conduction flow qx is positive if the temperature gradient is negative. Besides the increase in distance
x toward a positive direction of heat flow (Figure 3).
Figure 3: Direction of heat flow
241
To obtain the equation of heat flow rate plus the rate of heat flow entering the heat generation
sources will result in the rate of heat flow out and the rate of change in energy (Wati, 2013). Here used the
z-axis (depth states) are assumed to be positive.
(𝑞𝑥 + 𝑞𝑧) + 𝑞(𝑑𝑥 𝑑𝑧) = (𝑞𝑥+𝑑𝑥 + 𝑞𝑧+𝑑𝑧) + 𝑐𝜌(𝑑𝑥𝑑𝑧)𝜕𝑇
𝜕𝜃 (1)
where,
𝑞𝑥= heat flow in x-direction (𝑊),
𝑞𝑧 = heat flow in z-direction (𝑊),
c = specific heat (𝑊𝑠
𝑘𝑔, ℃),
𝜌 = density (𝑘𝑔/𝑚), 𝑇 = temperature (℃ ),
𝜃 = time (s ).
𝜕
𝜕𝑥(𝑘
𝜕𝑇
𝜕𝑥) +
𝜕
𝜕𝑧(𝑘
𝜕𝑇
𝜕𝑧) + 𝑞 = 𝑐𝜌
𝜕𝑇
𝜕𝜃 (2)
If the system is homogeneous, specific heat c and density 𝜌 is independent of temperature and is
also considered to be uniform, then the Eq. (2) can be written as:
𝜕2𝑇
𝜕𝑥2 +𝜕2𝑇
𝜕𝑧2 +𝑞
𝑘=
1
𝛼
𝜕𝑇
𝜕𝜃 (3)
where the constant α= k/c 𝜌 called thermal diffusivity. Equation (3) is known as the heat transfer equation
in general. These equations govern the flow temperature and conduction in a solid is a homogeneous
physical properties. If a system does not contain a heat source, then Eq. (3) becomes:
𝜕2𝑇
𝜕𝑥2 +𝜕2𝑇
𝜕𝑧2 =1
𝛼
𝜕𝑇
𝜕𝜃 (4)
If there is a heat source and the system is in steady state, then the equation would be:
𝜕2𝑇
𝜕𝑥2 +𝜕2𝑇
𝜕𝑧2 +𝑞
𝑘= 0 (5)
If the flow of body temperature in the steady state and without a heat source, it will be:
𝜕2𝑇
𝜕𝑥2 +𝜕2𝑇
𝜕𝑧2 = 0 (6)
A system is in a steady state rate of heat flow when a system does not change with time, is the rate
constant, the temperature at any point does not change (Kreith, 1985).
2.3 Numerical Methods
The method is a numerical solution technique is mathematically formulated in a way operation
count/arithmetic and done repeatedly with computer assistance or manually (hand calculation). Basic use
of numerical methods to solve partial differential equation is any partial derivative of a differential
equation that is used is replaced by a finite difference approach. When the finite difference approach is
242
applied throughout the dots variables contained in the concept model, then the solution of a series of
simultaneous equations that are used directly or can be determined using the method of iteration
(Setiawan, 2006).
Taylor series is the basis for solving problems in numerical methods, especially the completion of
partial differential equations. If a function T (x) is known at the point xi and all the derivatives of T with
respect to x is known at that point, then the Taylor series can be stated value of T at the point xi +1 is
located at a distance Δx from the point xi (Triatmojo, 2002). General form of the Taylor series are as
follows:
𝑇(𝑥𝑖+1) = 𝑇(𝑥𝑖) + 𝑇 ′(𝑥𝑖)∆𝑥
1!+ 𝑇"(𝑖)
∆𝑥2
2!+ ⋯ + 𝑇(𝑛)(𝑥𝑖)
∆𝑥𝑛
𝑛!+ 𝑅𝑛 (7)
where,
T(xi) : function at xi,
T(xi+1) : function at xi+1,
T, T’, T”, T”’, ..., T(n) : The first derivative, second, third, ..., n-th of function,
∆x : distance between xi and xi+1
Rn : truncation error
! : factorial operator
If we let y = z (z is assumed to express the depth), then the first derivative of the variables x and z in a row
can be written in differential form developed as follows:
𝜕𝑇
𝜕𝑥=
𝑇(𝑥𝑖+1, 𝑧𝑗) − 𝑇(𝑥𝑖 , 𝑧𝑗)
∆𝑥 (8)
𝜕𝑇
𝜕𝑧=
𝑇(𝑥𝑖+1, 𝑧𝑗) − 𝑇(𝑥𝑖 , 𝑧𝑗)
∆𝑧 (9)
To simplify the writing, the form T (xi, zj) is written into Ti, j, with i and j indicate the components in the
direction of the x-axis and z-axis (Triatmojo, 2002), so that Eqs. (8) and (9) can be written as:
𝜕𝑇
𝜕𝑥=
𝑇𝑖+1,𝑗 − 𝑇𝑖,𝑗
∆𝑥 (10)
𝜕𝑇
𝜕𝑧=
𝑇𝑖+1,𝑗 − 𝑇𝑖,𝑗
∆𝑧 (11)
Laplace equation with a heat source is,
𝑇𝑖+1,𝑗−2𝑇𝑖,𝑗+𝑇𝑖−1,𝑗
∆𝑥2 +𝑇𝑖+1,𝑗−2𝑇𝑖,𝑗+𝑇𝑖−1,𝑗
∆𝑧2 +𝑞
𝑘= 0 (12)
If ∆x = ∆z, Eq. 12 becomes,
𝑇𝑖+1,𝑗 + 𝑇𝑖−1,𝑗 + 𝑇𝑖,𝑗+1 + 𝑇𝑖,𝑗−1 − 4𝑇𝑖,𝑗 +𝑞
𝑘= 0 (13)
If ∆x ≠ ∆z, Eq. 12 becomes,
243
(𝑇𝑖+1,𝑗)(∆𝑧2) + (𝑇𝑖−1,𝑗)(∆𝑧2) + (𝑇𝑖,𝑗+1)(∆𝑥2) + (𝑇𝑖,𝑗−1)(∆𝑥2) − 𝑇𝑖,𝑗(2∆𝑥2 + 2∆𝑧2) +
𝑞(∆𝑥2)(∆𝑧2)
𝑘= 0 (14)
Liebmann equation is constructed of Eq. 14,
𝑇𝑖,𝑗 = (𝑇𝑖+1,𝑗)(∆𝑧2) + (𝑇𝑖−1,𝑗)(∆𝑧2) + (𝑇𝑖,𝑗+1)(∆𝑥2) + (𝑇𝑖,𝑗−1)(∆𝑥2) + 𝑞(∆𝑥2)(∆𝑧2)
𝑘/ (2∆𝑥2 +
2∆𝑧2) (15)
Laplace equation without heat sources indicate that there is no heat source in the two-dimensional model
of heat flow (q = 0). Therefore, the Eq. 12 becomes,
(𝑇𝑖+1,𝑗)(∆𝑧2) + (𝑇𝑖−1,𝑗)(∆𝑧2) + (𝑇𝑖,𝑗+1)(∆𝑥2) + (𝑇𝑖,𝑗−1)(∆𝑥2) − 𝑇𝑖,𝑗(2∆𝑥2 + 2∆𝑧2)
= 0 (16)
Liebmann subsequent equations can be constructed from Eq. (16) becomes,
𝑇𝑖,𝑗 = (𝑇𝑖+1,𝑗)(∆𝑧2) + (𝑇𝑖−1,𝑗)(∆𝑧2) + (𝑇𝑖,𝑗+1)(∆𝑥2)
+ (𝑇𝑖,𝑗−1)(∆𝑥2) / (2∆𝑥2 + 2∆𝑧2) (17)
To be able to solve Eqs. (15) and (17), boundary conditions must be determined so that
dapatdiperoleh unique solution. For simple cases, the temperature at the edges of the plate determined a
fixed value (Figure 4). Such cases are referred to the Dirichlet boundary conditions (Setiawan, 2006). For
this case the x-axis is used as a surface and a depth z.
1.j+1
1.j-1
I,j I,j+1I,j-1
x
z
Figure 4: Grid for solution of finite difference
Subsequently formed equation solved iteratively for i = 1 to n and j = 1 to m with over-relaxation
equation.
𝑇𝑖,𝑗𝑏𝑎𝑟𝑢 = 𝜆𝑇𝑖,𝑗
𝑏𝑎𝑟𝑢 + (1 − 𝜆)𝑇𝑖,𝑗𝑙𝑎𝑚𝑎 (18)
with 𝑇𝑖,𝑗𝑛𝑒𝑤 and 𝑇𝑖,𝑗
𝑜𝑙𝑑is present and previous iteration values, while λ is the relaxation coefficient magnitude
can be taken between 1 and 2. λ can be found using the equation
𝜆 =2
1 + √1 − 𝜔2 (19)
244
with,
ω = the magnitude of the relaxation parameters
𝜔 =1
1 + (∆𝑥
∆𝑦)
2 [cos𝜋
𝑚+ (
∆𝑥
∆𝑦)
2
cos𝜋
𝑛] (20)
Over-relaxation is used to speed of stability by using the formula in Eq. (18) is followed in each
iteration (Chapra and Canale, 2002). As the initial value of the state at the point of the interior is taken
equal to zero. Iteration can be terminated if the relative error has reached the limit. The magnitude of the
relative error or an error value is defined as (Setiawan, 2006).
|(𝜀𝑎)𝑖,𝑗| = |𝑇𝑖,𝑗
𝑏𝑎𝑟𝑢−𝑇𝑖,𝑗𝑙𝑎𝑚𝑎
𝑇𝑖,𝑗𝑏𝑎𝑟𝑢 | 𝑥 100% (21)
For steady state with no heat source, can be applied to the Laplace equation assuming a constant value of
thermal conductivity. The solution of these equations can be solved by analytic method. Form of the final
solution can be written as follows (Holman, 2010):
𝑇−𝑇1
𝑇2−𝑇1=
2
𝜋∑
(−1)𝑛+1+1
𝑛
∞
𝑛=1𝑠𝑖𝑛
𝑛𝜋𝑥
𝑊
sinh (𝑛𝜋𝑧/𝑊)
sinh (𝑛𝜋𝐻
𝑊)
(22)
3 RESEARCH RESULTS AND DISCUSSIONS
Based on the tools used, how the data retrieval and data processing thermal methods in PasemaAir
Keruh, District Empat Lawang, which is in the Penantian, Air Keliansar and surrounding areas, this
research can be divided into two parts, namely the temperature monitoring and mapping temperature
thermal conductivity of rocks . The outline of the study is depicted in Figure 5.
Desain survei
Informasi Geologi
Monitoring Suhu Pemetaan Suhu (Mapping)
Kurva Data
Kurva Data
Cetak Model
Distribusi Suhu
Selesai
Mulai
Pemodelan
Kurva Model
Pencocokan
Kurva Model dan
Pengukuran
tidak
ya
Interpretasi
Kesimpulan
Figure 5: Flowchart of the thermal methods
245
3.1 Temperature monitoring
Measurement of the temperature monitoring has been done, namely the first phase 26 to 28 June
2013 and the second phase of 3 to 5 July 2013, the duration of monitoring for 48 hours with intervals one
hour. The first stage is done at 3 monitoring stations, where each station measurements are given names
for easy identification matches the name of the local village. Stations in the first phase is in Penantian and
has three measurement stations namely: Penantian 1.1, Penantian 1.2 and Penantian 1.3. The number of
measurement stations for the second phase is in the Air Keliansar village also has three measurement
stations are named, Air Keliansar 1.1, Air Keliansar 1.2 and Air Keliansar1.3. The topographic map can
be seen in Figures 6 and 7.
Figure 6: Topographic map Penantian
Figure 7: Topographic map Air Keliansar
246
3.2 Temperature Mapping
Temperature mapping results showed the presence of several concentrations of heat that leads to
the north. Some heat concentration showed a pattern of heat flow controlled by the presence of the fault
with a northwest-southeast direction. Temperature measurements were carried out in the area penenlitian,
shows temperature information spread evenly. The closure temperatures were in some places it has not
shown the limits of the horizontal flow pattern. This pattern is most likely still in the area above the weak
zone fault plane so that the spread of the temperature evenly. The concentration of high temperatures in
some areas where the fault that caused the heat flow to the surface. In order to determine the concentration
and distribution of temperature measurement points in the study area can be seen in Figure 8 and Figure 9.
Figure 8: Concentration temperature in Penantian
.
Figure 9: Concentration temperature in Air Keliansar
3.3 Quantitative Interpretation
Quantitative interpretation of temperature data to create cross-sectional model of the subsurface in
the study area. Subsurface modeling as a plate-shaped fluid reservoir or a place to move. Selection of
247
numerical methods with finite difference techniques to the completion of differential equations of heat
conduction in a two-dimensional framework. This settlement will result in a linear system. For the
distribution grid is large, then the solution will be more difficult, and result in sizable errors. To resolve
this usually Liebmann method. In the equation expressing Liebmann method of two-dimensional finite
difference and solving iteratively for i = 1 to n and j = 1 to m (Setiawan, 2006).
Utilization of algebraic equations at each grid coupled with other supporting information
(temperature, conductivity and temperature gradient), it can be presumed depth of the source and reservoir
temperature below the earth's surface. Discrete models are constructed with Liebmann method further
solved using a Matlab 2012b. The results are displayed in the GUI (modified from Wati, 2013) as follows:
Figure 10: Display program of model heat flow 2D
Figure 11: Selection of temperature data incision at Penantian (A-A')
248
Temperature contour measurement data in Figures 11 and 12 chosen the most symmetrical incisions
and then matched with the modeling of heat distribution. In the measurement data matching and data
model will be seen patterns or trends of each test value. Basis for making an incision was cut geothermal
manifestations are located along the Sumatra fault and fault Musi.
Figure 12: Selection of temperature data incision at Air Keliansar (B-B')
3.4 Analysis Models
The results of geochemical analysis indicates that the reservoir temperature at the location of
Penantian and Air Keliansar of 300 0C and 320 0C (Virgo et al., 2012). From this information, it is made
of two-dimensional heat flow model. The test results by entering the temperature variation attached.
Information about the test by numerical methods in conduction heat distribution in a two-dimensional
framework very suitable as a first step in the survey design for three-dimensional problems. Discussed
here only two dimensions and the data obtained from the second matching pattern subsurface temperature
distribution has a similar pattern in the study area.
3.4.1 Penantian 2D Models (at depth 1100 m)
In this section will be given the results of the simulation and visualization of heat flow in steady
state with no sources. Given number of grid 145 and the value of error of 0.01%. The results of the
simulation and visualization can be seen in Figure 13. Next will be given the results of the simulation and
visualization of heat flow in steady state the heat source. Given the thermal conductivity of 2.8 W / MOC,
the number of grid 145, the value of error of 0.01%, and the heat rate of 0.0615 W. The results of the
simulation and visualization can be seen in Figure 14.
249
Figure 13: Graph of heat flow (surface) at Penantian with an error value of 0.01% at a depth of 1100 m
(without a heat source)
Figure 14: Graph of heat flow (surface) at Penantian with an error value of 0.01% , heat flow 0,0615 W at
a depth of 1100 m (with a heat source)
Heat flow graph above explains that by inserting the values of parameters such as image
visualization produced Figures 13 and 14. To obtain equilibrium with 0.01% error value at each point Ti, j,
carried 654 times for iteration and 619 times without a source with the source. At the temperature of the
flow picture looks for red areas are areas that have a high temperature is the temperature above 350 ° C,
for the orange area is an area with a temperature between 300 ° C to 350 ° C, for the yellow area is an area
with a temperature between 250 oC to 300 oC. For the green area is an area with a temperature between
200 ° C to 250 ° C, for the aqua-colored areas are regions with a temperature between 150 ° C to 200 ° C,
for the blue area is an area with a temperature between 100 ° C to 150 ° C, for the area the light blue area
temperature is between 50 ° C to 100 ° C, for the dark blue area is an area with a temperature below 50 °
C, which means that the lower the temperature on the color image is getting older (dark blue). Results
matching measurement data and model data at a depth of 1,100 meters at Penantian can be seen in Figures
15(a) and 15(b).
250
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55
60
65S
uh
u (
oC
)
Jarak lintasan (m)
Data Pengukuran
Data Model
0 50 100 150 200 250 300
20
25
30
35
40
45
50
55
60
65
Su
hu
(0C
)
Jarak Lintasan (m)
Data Pengukuran
Data Model
(a) (b)
Figure 15: The results of the measurement data matching and data model depth of 1100 m, (a) without a
heat source), and (b) with a heat source
3.4.2 Air Keliansar 2D Models (at depth 1300 m)
In this section will be given the results of the simulation and visualization of heat flow in steady
state with no sources. Given number of grid 134 and the value of error of 0.01%. The results of the
simulation and visualization can be seen in Figure 17. Next will be given the results of the simulation and
visualization of heat flow in steady state the heat source. Given the thermal conductivity of 2,8 W / MOC,
the number of grid 134, the value of error of 0.01%, and the heat rate of 0.0615 W. The results of the
simulation and visualization can be seen in Figure 18.
Figure 17: Graph of heat flow (surface) at Air Keliansar with an error value of 0.01% at a depth of 1300 m
(without a heat source)
251
Figure 18: Graph of heat flow (surface) at Air Keliansar with an error value of 0.01%, heat flow 0,0615 W
at a depth of 1300 m (with a heat source)
Heat flow graph above explains that by inserting the values of parameters such as image
visualization produced 17 and 18. To obtain equilibrium with 0.01% error value at each point Ti, j, carried
475 times for iteration and 463 times without a source with the source. At the temperature of the flow
picture looks for red areas are areas that have a high temperature is the temperature above 350 ° C, for the
orange area is an area with a temperature between 300 ° C to 350 ° C, for the yellow area is an area with a
temperature between 250 oC to 300 oC. For the green area is an area with a temperature between 200 ° C
to 250 ° C, for the aqua-colored areas are regions with a temperature between 150 ° C to 200 ° C, for the
blue area is an area with a temperature between 100 ° C to 150 ° C, for the area the light blue area
temperature is between 50 ° C to 100 ° C, for the dark blue area is an area with a temperature below 50 °
C, which means that the lower the temperature on the color image is getting older (dark blue). Results
matching measurement data and model data at a depth of 1.300 meters at Air Keliansar can be seen in
Figures 19 and 20.
0 50 100 150 200
30
35
40
45
50
55
60
Su
hu
(oC
)
Jarak lintasan (m)
Data Pengukuran
Data Model
0 50 100 150 200
30
35
40
45
50
55
60
Su
hu
(oC
)
Jarak lintasan (m)
Data Pengukuran
Data Model
(a) (b)
Figure 19: The results of the measurement data matching and data model depth of 1300 m (a) without a
heat source), and (b) with a heat source
252
3.5 2D Model Verification Using Mathematical Analysis
For steady state with no heat source, can be applied to the Laplace equation assuming a constant
value of thermal conductivity. The solution of these equations can be solved by analytic method. Shape of
the eventual solution to the equation as 22.
Suppose that at the surface (x = 0 m, y = 0 m) with H = 1300 m and L = 200 m, then the
mathematical solution:
𝑇−𝑇1
𝑇2−𝑇1=
2
𝜋∑
(−1)𝑛+1+1
𝑛
∞
𝑛=1𝑠𝑖𝑛
𝑛𝜋𝑥
𝐿
sinh(𝑛𝜋𝑧
𝐿)
sinh(𝑛𝜋𝐻
𝐿)
𝑇−32
30−32=
2
𝜋{2 sin 0
sinh 0
sinh(1300𝜋
200)
+2
3sin 0
sinh 0
sinh(3900𝜋
200)
+ ⋯ }
𝑇−32
30−32=
2
𝜋{0}
𝑇 − 32 = 0
T=32𝑜C
If taken any position, x = 10 m and y = 10 m, then the solution is the math:
𝑇−𝑇1
𝑇2−𝑇1=
2
𝜋∑
(−1)𝑛+1+1
𝑛
∞
𝑛=1𝑠𝑖𝑛
𝑛𝜋𝑥
𝐿
sinh(𝑛𝜋𝑧
𝐿)
sinh(𝑛𝜋𝐻
𝐿)
𝑇−32
32−30=
2
𝜋{2 sin
10𝜋
200
sinh(10𝜋 200⁄ )
sinh(1300𝜋
200)
+2
3sin
30𝜋
200
sinh(30𝜋 200⁄ )
sinh(3900𝜋
200)
+ ⋯ }
𝑇−32
2=
2
𝜋{0,313 𝑥
0,158
3,69 𝑥 108 + 0,303 𝑥 0,489
20,2 𝑥 1025}
𝑇 − 32 = 4
𝜋
T= 33,27𝑜C
The verification of the results shows that the results of analytical and numerical calculations on the
surface has the same temperature value that is equal to 32 °C, whereas at an arbitrary point (x = 10 m and
y = 10 m) temperature values analytically and numerically in a row is 33,27 °C and 32.35 ° C with an
error value of 2.8%.
4 CONCLUSION
Based on a map of the spread of surface temperature, concentration and temperature in Penantian
and Air Keliansar have two different temperatures closure. The temperature closure spread evenly but has
not shown the limits of heat flow pattern horizontally. This means that temperature closures is still above
the weak zone fault plane.
The results of modeling the heat source below the surface with a 2D finite difference method can
estimate the order of the source reservoir temperature of 400 oC Waiting locations at a depth of 1100 m,
whereas the Air Keliansar order for the location of the source reservoir temperature of 400 0C at a depth of
1300 m.
253
5 ACKNOWLEDGEMENTS
In this paper the authors have completed a lot of guidance, assistance, and support from various
parties. Therefore, the authors would like to thank Dr. Wahyudi, MS, Dr. rer. nat. Wiwit Suryanto, M.Si
and friends who always gives motivation.
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