Radial Derivative and Radial Inversion for Interpreting 4D Gravity Anomaly Due to Fluids Injection Around Reservoir

TELKOMNIKA, Vol.16, No.6, December 2018, pp.2855~2863
ISSN: 1693-6930, accredited First Grade by Kemenristekdikti, Decree No: 21/E/KPT/2018
DOI: 10.12928/TELKOMNIKA.v16i6.9468  2855
Received April 2, 2018; Revised September 26, 2018; Accepted October 30, 2018
Radial Derivative and Radial Inversion for Interpreting
4D Gravity Anomaly Due to Fluids Injection Around
Reservoir
Muhammad Zuhdi*
1
, Sismanto
2
, Ari Setiawan
3
, Jarot Setyowiyoto
4
, Adi Susilo
5
,
Muhammad Sarkowi
6
1
Physics Education Department, Universitas Mataram, Indonesia.
2,3
Physics Department, Universitas Gadjah Mada, Indonesia
4
Geology Department, Universitas Gadjah Mada, Indonesia
5
Geophysics Laboratory, Physics Department, Universitas Brawijaya, Indonesia
6
Geophysics Department, Universitas Lampung, Indonesia
*Corresponding author, e-mail: mzuhdi@mail.unram.ac.id
Abstract
The 4D gravity or time lapse gravity has been used many reseracher to identify fluid injection in
oil reservoir. The objective of this study is to find the better way in interpreting 4D gravity anomaly due to
fluid injection around the reservoir. Radial Derivatives are derivative values of gravity anomalies against
horizontal distances in the radial direction. Radial inversion is a two-dimensional inversion of lines with
radial directions resulting in a 3-dimension model. Time lapse microgravity research have been performed
in “X Oil Field” with amount of 604 data point covering area of 4000 m x 5000 m. This Radial derivative and
Radial inversion have been aplied at an injection well of the X oil field. The yield show that 4D gravity
anomaly value due to injection is 0.02 mGal to 0.36 mGal. Radial derivative value in the area is
0 micro Gal/cm to 0,012 mGal/meter. Radial inversion shows radius of fluid front movement is 304 meters
to 1120 meters. Radial derivative and Radial inversion have been proven fairly good to identify injected
fluid movement in the reservoir.
Keywords: radial, derivative, inversion, 4D gravity, reservoir
Copyright © 2018 Universitas Ahmad Dahlan. All rights reserved.
1. Introduction
The 4D micro-gravity method or well known as time lapse microgravity is the
development of gravity method with time as the fourth dimension. This method is characterized
by repeatable measurements either daily, weekly, monthly or yearly using very high-accuracy
gravity measurements supported by positioning and height measurements with high
accuracy [1]. The 4D gravity method has been widely used for the identification of subsurface
changes. Eiken et al, using intermittent micro gravity, to monitor gas production in subsea
reservoirs with a gross gravimeter of up to 4 micro Gals [2]. Gettings et al, measured the value
of 4D micro gravity around geothermal geysers sources to detect subsidence [3].
Akasaka and Nakanishi separated the gravity anomaly of geothermal sources from the
effect of groundwater change [4]. Rahman et al. succeeded in monitoring fluid injection in
reservoirs using the 4D gravity method [5]. Dafis et al., measured the 4D gravity anomaly to
monitor the rate of water injection in artificial aquifer storage and recovery (ASR) aquifers in
Leyden Colorado [6]. Sarkowi examines the relationship between ground water changes with
vertical gravity gradient changes in the city of Semarang and the surrounding areas [7].
Tsuboi dan Kato, examine the properties of the second derivative of gravity data based
on the Laplace equation [8]. Reilly, studies formulations for differentiation, smoothing and
interpolation of gravity data [9]. Abdelrahman dkk. [10], using derivatives analytically to detect
fault zones [10]. Oruc, detect boundaries of anomaly source and estimate depth using horizontal
gravity gradient data. This method is performed in the Kozakli-Central area, Anatolia,
Turkey [11]. Ansari, proposed a new method for detecting anomaly edges with Analitic Signal of
Tilt Angle (ASTA). This method has been tested on magnetic and gravity data in southwestern
England [12]. Aydogan, extracting lines of gravity anomalies and performing horizontal gradient
calculations for interpretation of geological structure boundaries. This method has been tested

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TELKOMNIKA Vol. 16, No. 6, December 2018: 2855-2863
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in the Tuzgolu basin, Anatolia, Turkey [13]. Tatchtum, make geological tilt models in various
shapes, the anomaly respon and differentiated responses horizontally. This method was tested
in the Foumbon fault zone in Cameroon [14]. Chijun, measure vertical derivatives directly to get
a more accurate geoid value up to the order in centimeter level [15]. Aku, using vertical
derivative analysis to determine lithological vertical boundaries to sharpen the analysis of
gravitational interpretation in Gusau, Nigeria [16]. Askari, using the slope angle of the total
normalized derivative of the gravity anomaly to detect geological structural boundaries. This
method has been performed in the Sea of Oman, southeast of Iran [17]. Wahyudi, use vertical
derivatives from 3D gravity data for interpretation of geological structures. This method has
been tested in a very wide area, namely Central Java, East Java and Madura [18]. Pasteka
dkk., using derivative transformations from Bouguer anomalies for geological interpretation. This
method test was carried out in a very large area in Slovakia [19].
The world's oil demand is always increasing over time, while oil availability is running
low. Efforts to fulfill oil demand are performed by exploring new oil resources and optimizing on
producing oil reservoirs wich also well known as Enhancement Oil Recovery (EOR). One way to
increase production is injecting fluids into the reservoir. This fluid will move away from the
injection well and push the oil toward the production well. In the fluid injection process, the
density change of the pore fluid mass will result in a 4D gravity anomaly. Time lapse gravity
anomaly due to injection is the gravity value difference of post-injection with pre- injection. The
reservoir model and the 4D gravity response due to the fluids injection is shown in Figure 1. The
4D anomaly is positive because the density of injected fluid (water) is graeter than the oil filling
the pore.
Figure 1. Reservoir model and the 4D gravity response due to fluids injection in reservoir
The identification of fluid injection in reservoir using gravity method has been
conducted by many researchers. The use of this method is divided into two major parts of the
method of gravity on the surface and gravity of the well hole. Vasilevskiy et al used the gravity
data of the well-hole by interpretation through inversion [20]. Sarkowi made the interpretation of
surface gravity with the forward model [21], while Zuhdi et al made the same way by performing
radial derivative as shown in Figure 2 (a) [22]. The main problem of 4D gravity in identification of
fluid injection in reservoir is the difficulties to identify contact of injected fluid and pre-injection

TELKOMNIKA ISSN: 1693-6930 
Radial Derivative And Radial Inversion for Interpreting 4D Gravity... (Muhammad Zuhdi)
2857
fluid. To solve this main problem authors propose radial derivatives and radial inversion. The
objective of this research is to construct computer program of radial derivative and radial
inversion of 4D gravity anomaly and show the abilty to show boundary of density contrast in
injected reservoir. The data was taken with Lacoste Romberg type 504 and Scintrex C 63,
consist of 604 measurement point covering area of 4000 mx5000 m, in 3 periods ie. January,
May and September in 2003. Radial derivative and radial inversion have been performed in the
part of the area covering 1000m x 1000m with an injection well as the center.
2. Research Method
2.1. Time Lapse Micro Gravity Method
According to Kadir, a 3-dimensional object with a mass density distribution  (α,β,γ),
gives the effect of gravity at point P(x,y,z) on the surface within a given time interval (t) given
by [23]:
(1)
In (1) if the change of gravity anomaly only depens on density change (no geometrical
change), then it can be approximated by:
g(x,y,z,t) κ.(x,y,z,t) (2)
where κ is a constant corresponding to the geometry and volume of anomaly source objects,
whereas:
g(x,y,z,t)=g(x,y,z,t’)-g(x,y,z,t) (3)
g(x,y,z,t) is a gravity change related to density changes due to fluid injection in reservoir. If
pore volume is filled with oil at time t, and there is a replacement of oil with injected fluid at time
t, then the density change is given by the (4) [24]:
Δ = ’ - = Φ (f - o) (So-1) (4)
where f is density of injected fluid and o is density of oil. Oil saturation So is defined as the
ratio of volume Vm to the overall pore volume of Vp, and Φ is the total porosity.
Δg = k Φ (f - o) (So-1) (5)
by replacing So with Vo / Vp, (5) becomes:
Δg = Φ (f - o) {(Vo/Vp) - 1} (6)
the anomaly g will be positive because the oil (low density) is replaced with water (higher
density). If the injected fluid has bigger density, its time lapse anomaly will be positive.
2.2. Radial Derivative
Radial derivative is a derivative of the gravity anomaly value to the horizontal distance
with the radial direction of a particular point. The horizontal derivative of a gravitational anomaly
is the derivative of the gravity value in the direction of a particular straight line. If the straightline
cuts perpendicular to a contrast density boundary, the derivative value will be of great value. For
these purposes radial derivatives are made. The radial derivative center point is chosen based
on the center of anomaly.
Radial derivatives will be easier to perform in vertical cylindrical coordinates. The gravity
anomaly value Δgz (x, y, z) caused by the density anomaly Δρ (α, β, γ) is written as:
 
  
     
3
2 2 2 20
, , ,
, , ,
t z
g x y z t K d d d
x y z
    
  
  
  

  
  
     
 
  

 ISSN: 1693-6930
2858
(7)
can be converted into vertical cylindrical coordinates, so that Δgz (x, y, z) becomes Δgz (R, z, λ)
and Δρ (R ', z', λ '), with relation [25]: x = R cos λ, y = R sin λ, z = z, α = R cos λ’, β = R sin λ’
and γ = z’.The mass element is Δρ dr dz rdλ , so that the gravity value of the axis z can be
written as:
(8)
The radial derivative of the gravity anomaly in vertical cylindrical coordinates Δgz (R, z,
λ) can simply be written as:
(9)
Figure 2 shows the field diagram used to calculate the radial derivative. ABCD, EFGH is
a cube with length of ribs along AB which is also the same as AJ. Points A, B, C and D are point
2 of measurement gravitation. A to B is the line towards the x-axis while A to D is towards the
y-axis. If the difference in gravity at point B with A is dx whose vector length is proportional to
BR, whereas the difference in the value of gravity of point A with D is the length of its vector
along the DS, then the APNQ is a plane with the same gravity value. If dT is the directional
value of the derivative then:
(a) (b)
Figure 2. (a) Plane diagram of radial derivative approach
(b) Radial inversion lines consist of 8 lines to make a radial inversion model
dT = JN= IP +KQ; (10)
with dT having a vector length equal to JN whose value is equal to the value of IP and KQ.It was
easy to understand, the total derivatif become:
dT = dx cos θ + dy sin θ (11)
  
        
 


 


0 2
3
222
,,
),,( 


ddd
zyx
z
Kzyxgz
  
        
 
 


0
2
0 2
3
222
'',','
),,(




 rdrdzd
zyx
zzzR
KzRgz
   
      
  
 
 





0
2
0 2
3
222
'',','



rdrdzd
zyx
zzzR
K
R
FRD

2859
2.3. Radial Inversion
The 4D gravity anomaly of the reservoir contaminated with shallow effect anomaly is
then separated by filtering. The residual (local effect) of the filtering is an undesirable shallow
anomaly, while the regional one is the anomaly part of the reservoir which is then be inverted by
inversion program. Figure 2 (b) shows some 2D inversion lines that produce a radial inversion.
The radial inversion performed in this case is assumed to be an anomaly derived only from the
reservoir. This inversion is based on the inversion of a cylindrical-like reservoir shape with
thickness corresponding to the reservoir thickness. The inversion result is expected to illustrate
the actual reservoir geometry shape, which includes density contrast and the fluid movement
radius in the reservoir.
Inversion theory can be regarded as an art to obtain various physical parameters with
sufficient knowledge about forward modeling [26]. Relation of observed data and model can be
written as:
d = g (m) (12)
where d is the observed data, m is the model parameter and g is forward function which is the
prediction function of the model known as Kernel function. If the observed data are N and the
model parameter of M can be related to the Kernel’s matrix by the (13):
d = G m (13)
where d is a Nx1 matrix, G is a NxM Kernel matrix and m is a model parameter of Mx1 size. The
inversion relationship of model parameters with observed data can be written as:
m = [G
T
G]
-l
G
T
d (14)
The measured gravity value actually has a non-linear relationship to the parameters of
the model in search of density contrast, radii and position, but this relationship can be solved by
linear approximation. The principle of this method is the LSQR with special development. The
formulation of non-linear inversion method with approach linear principle using the first Taylor
series and ignoring the higher rates of the Taylor series around x + Δx as follows:
f(x+∆x) = f(x)+f’(x) ∆x (15)
where f(x+∆x) is the measurement data written with d, f(x) is the result of the model response ie.
g(m0) and f’(x) is first derivative of the model parameter forming the Jacobi matrix J0, so:
d = g(m0) + J0Δm0 (16)
The Jacob's matrix consists of:
𝐽 𝐴 = [
𝜕𝑔 𝑖
𝜕𝑚 𝐴
] , 𝐽 𝑑 = [
𝜕𝑔 𝑖
𝜕𝑚 𝑑
] and 𝐽𝑟 = [
𝜕𝑔 𝑖
𝜕𝑚 𝑟
]
where A is the Amplitude factor d is the depth and r is the radius of the fluid injection front, so
the Jacobi matrix of the cylinder model can be written as:
J0 = [ JA Jd Jr ]
The Kernel model of this radial derivative function is based on Finite horizontal slab
wich can be written as:
𝑔 = 2𝐺𝜌𝑡 {𝜋 + 𝑡𝑎𝑛−1
(
𝑥+0.5𝐿
𝑧
) + 𝑡𝑎𝑛−1
(
0.5𝐿−𝑥
𝑧
)} (17)
from (16), we can obtain:
d - g(m0) = J0Δm (18)
so,
Δm = [J0
T
J0]
-1
J0
T
(d - g(m0)) (19)

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2860
with Jo
T
is the transpose of matrix J0, since
d - g(m0) = Δd (20)
then Δd has a linear relationship with Δm so that it can be solved by LSQR method.
The principle of linear inversion solution is a recurring count that will yield the value
m=m0+Δm so that the minimum Δd value is obtained. With the RMS value of Δd small enough
to some extent, iteration is then stopped. With the very small value of Δd then the value of m
m0 so that the model parameters very close to the real value [26]. The flowchart of the process
is shown in Figure 3.
Figure 3. Flow chart in the radial inversion programming
3. Results and Analysis
3.1. Radial Derivatives for Injection Fluid Identification
An anomaly due to injection that has been removed from shallow effect is then treated
with radial derivative. This treatment is intended to find the front of fluid injection. The front of
fluid injection made a density contrast wich result measurable anomaly of gravity on the surface.
Figure 4 shows the radial derivative of 4D gravity anomaly due to injection from January to May.
The middle of the blue zone on the image can be interpreted as the contact boundary between
the injection fluid and the previous pore fluid. The yield show maximum radial derivative in the
area is 0,012 mGal/meter and the minimum value of 0 micro Gal/m.
(a) (b)
Figure 4. Radial derivative of 4D gravity anomaly due to injection:
(a) January to May period, (b) January to September period


2861
3.2. Radial Inversion For identification of Injection Fluid
The input of this Radial Inversion program is a file of 4D gravity anomaly containing
position (x, y) and gravity values in regular grid. The data is then extracted by the number of
specific lines to get the inversion lines. Each inversion line is then performed inversion process
with kernel function as finite horizontal slab (17). The yields of the inversion parameter are the
amplitude factor and the radius of fluid injection front. The result of this inversion data is then
unified and re-gridded to obtain 3-dimensional model of reservoir injection fluid.
Figure 5 shows the fluid injection volume model in the reservoir based on the radial
inversion method. The flat blue zone is interpreted as uncontaminated reservoir and the red
zone is volume of the injection fluid infiltrated the reservoir. The volume of the injection fluid has
a shape tends to be straight to the radii because the kernel matrix of the program only allows us
to obtain a uniform volumes thickness value toward the radius. Reservoir model due to fluid
injection as a result of radial inversion has maximum value of 1120 meter and minimum value of
304 meter.
(a)
(b)
Figure 5. Injection fluid volume model on the reservoir based on radial inversion results:
(a) January to May period, (b) January to September period

 ISSN: 1693-6930
2862
4. Conclusion
Radial derivatives and Radial Inversion has been proven fairly good for identification of
fluid injection in the reservoir. This method is useful to help more accurate analysis of 4D gravity
around the reservoir. Radial derivatives are not only applicable to 4D anomalies due to fluid
injection in the reservoir, but also effective for the analysis of ordinary gravity anomalies with
horizontal slab shape. Radial Derivative Methods can also be applied to ordinary gravity
anomalies and become tools for further interpretation of existing gravity data.
Radial inversions can be applied to time lapse gravity data around injection wells for
identification of additional mass (positive density contrast) or mass reduction (negative density
contrast). Radial inversion can also be used in an ordinary gravity anomaly (3D) to detect
boundary of horizontal slab. Radial Inversion is fairly good to provide a physical reservoir
parameters. This program is useful for general interpretation of gravity and can be a reliable
additional software as a new way of interpretation.
References
[1] Reynolds, J. M., An Introduction to Applied and Environmental Geophysics. John Willey and Sons,
England. 1997.
[2] Eiken, O., Zumberge, M., Stenvold, T., Gravimetric monitoring of gas production from the Troll field,
SEG Int'l Exposition and 74
th
Annual Meeting, Denver, Colorado. 2004.
[3] Gettings, P., Harris, R.N., Allis, R.G., Chapman, D.S., Gravity Signals at The Geysers Geothermal
System, Proceedings, Twenty-Seventh Workshop on Geothermal Reservoir Engineering, Stanford
University. 2002.
[4] Akasaka, C., Nakanishi, S. Evaluation of Microgravity Background at the Undisturbed Oguni
Geothermal Field, Japan. Proceedings, Twenty-Fifth Workshop on Geothermal Reservoir
Engineering Stanford University, 2000.
[5] Rahman, A., Mashud, M. I., Rahmawati, A. D., Susilo, A., Sarkowi, M., Hydrocarbon Reservoir
Monitoring Using Gravity 4d Method, In “X” Field in Southern Sumatra Area. Proceedings Joint
Convention Bali 2007The 32
nd
HAGI And The 36
th
IAGI Annual Convention and Exhibition, 2007.
[6] Dafis, K., Li, Y., Batzle, M. Time-lapse gravity monitoring: A systematic 4D approach with application
to aquifer storage and recovery. GEOPHYSICS. 73(6). 2008. A61–WA69, 13 FIGS.
10.1190/1.2987376
[7] Sarkowi, M., Vertical Gradient of Micro Gravity Between Times and Its Relationship with Groundwater
Dynamics (in Indonesia: Gradient Vertikal Gravitasi Mikro Antar Waktu dan Hubungannya Dengan
Dinamika Air Tanah). Prosiding Seminar Nasional Sains dan Teknologi-II 2008 Universitas Lampung,
17-18 November 2008.
[8] Tsuboi and Kato. The First and Second Vertical Derivative of Gravity. Journal of Physics of The
Earth. 1952; 1(2): 95-96.
[9] Reilly, W. I. Interpolation, smoothing, and differentiation of gravity anomalies. New Zealand Journal of
Geology and Geophysics. 1968; 12(4): 609-627. ISSN: 0028-8306 (Print) 1175-8791 (Online) Journal
homepage: http://www.tandfonline.com/loi/tnzg20
[10] Abdelrahman, E. M., El-Araby, H. M., El-Araby, T. M., and Abbo-Ezz, E. R. A least-squares
derivatives analysis of gravity anomalies due to faulted thin slabs. Geophysics. 2003; 68(2): 535–543.
[11] Oruc, B. Edge Detection and Depth Estimation Using a Tilt Angle Map from Gravity Gradien
horizontalt Data of the Kozaklı-Central Anatolia Region. Pure and Applied Geophysics. 168(10):1-12.
[12] Ansari, A.H. and Alamdar, K. A new edge detection method based on the analytic signal of tilt angle
(ASTA) for magnetic and gravity anomalies. Iranian Journal of Science & Technology (IJST). 2011;
A2: 81-88
[13] Aydogan, D. Extraction of lineaments from gravity anomali maps using the gradien horizontalt
calculation: Application to Central Anatolia. Earth Planets Space. 2011; 63: 903–913.
[14] Tatchum, C.N., Tabod, T. C., Koumetio, F., and Manguelle-Dicoum, E. A Gravity Model Study for
Differentiating Vertical and Dipping Geological Contacts with Application to a Bouguer Gravity
Anomali Over the Foumban Shear Zone, Cameroon, Geophysica. 2011; 47(1–2): 43–55.
[15] Chijun, Z., Shaofeng, B., Zhounm, Y., Lingtao, L. and Jian, F. Refining geoid and vertical gradien
horizontalt of gravity anomaly. Geodesy and Geodynamics. 2011; 2(4): 1 – 9.
[16] Aku, M. O. Application Of Second Vertical Derivative Analytical Method To Bouguer Data For The
Purpose Of Delineation Of Lithological Boundaries. Science World Journal. 2014; 9(3): 27-32.
[17] Askari, A. Edge detection of gravity anomali sources via the tilt angle, total horizontal derivative, total
horizontal derivative of the tilt angle and new normalized total horizontal derivative. Scholars Journal
of Engineering and Technology (SJET.) ISSN 2321-435X (Online) Sch. J. Eng. Tech. 2014; 2(6B):
842-846

2863
[18] Wahyudi. E. J., Kynantoro, Y., and Alawiyah, S. Second Vertical Derivative Using 3-D Gravity Data
for Fault Structure Interpretation. International Conference on Energy Sciences (ICES 2016) IOP
Publishing IOP Conf. Series: Journal of Physics: Conf. Series 877 (2017) 012039 doi :10.1088/
1742-6596/877/1/012039.
[19] Pasteka, R., Zahorec, P., Kusnirak, D., Bosansky, M., Papco, J., Szalaiova, V., Krajnak, M.,
Marusiak, I., Mikuska, J., and Bielik, M. High resolution Slovak Bouguer gravity anomali map and its
enhanced derivative transformations: new possibilities for interpretation of anomalous gravity fields,
Contributions to Geophysics and Geodesy. 2017; 47(2): 81–94.
[20] Vasilevskiy A.N., dan Dashevsky, Y.A. Gravity monitoring at oil and gas fields: data inversion and
errors. Russian Geology and Geophysics. 2015; 56: 762–772.
[21] Sarkowi, M., Kadir, W.G.A., Santoso. Dj. Strategy of 4D Microgravity Survey for the Monitoring of
Fluid Dynamics in the Subsurface. Proceedings World Geothermal Congress, Antalya, Turkey 24-29.
2005: 1-5.
[22] Zuhdi, M., Sismanto, Setiawan. A, Setyowiyoto.J. Identifiction Of Oil-Gas Contact Using Radial
Derivative Of 4D Gravity Anomaly. IOSR Journal of Applied Geology and Geophysics (IOSR-JAGG).
5 Ver. II (Sep.–Oct. 2017): 8-13.
[23] Kadir, W. G. A. 4D Gravity Survey and Surface Source dynamics (in Indonesia: Survey Gravitasi 4D
dan dinamika Sumber Bawah Permukaan). Proceeding Himpunan Ahli Geofisika Indonesia, 1999,
94-99.
[24] Schon, J. H. Physical Properties of Rocks: Fundamental and Principles of Petrophysics. Oxford. UK,
Elsevier Science. 1998; 24: 73.
[25] Jacoby, W.,Smilde, P.L. Gravity Interpretation Fundamentals and Application of Gravity Inversion and
Geological Interpretation. Springer-Verlag Heidelberg, Berlin. 2009: 25-26.
[26] Grandis. H. Introduction to Geophysical Inversion modeling (in Indonesia: Pengantar pemodelan
Inversi Geofisika). Institute Teknologi Bandung, Himpunan Ahli Geofisika Indonesia, Bandung. 2009:
105-113.

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