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Journal of Geophysics and Engineering
ACCEPTED MANUSCRIPT
Studies of electrical properties of low resistivity sandstones based on
digital rock technology
To cite this article before publication: Weichao Yan et al 2017 J. Geophys. Eng. in press https://doi.org/10.1088/1742-2140/aa8715
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Page 1 of 24
Studies of electrical properties of low resistivity sandstones based on
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digital rock technology
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Weichao Yan*, , Jianmeng Sun , Jinyan Zhang , Weiguo Yuan , Li Zhang , Likai Cui
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and Huaimin Dong
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School of Geosciences, China University of Petroleum , Qingdao 266580, China
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Shengli Well logging Company, SINOPEC, Dongying, 257015, China
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Sinopec Exploration Company, Chengdu 610041, China
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d
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Technology, Tai’an 271019, China
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Department of Resources and Civil Engineering, Shandong University of Science and
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Abstract
The electrical properties are important parameters to quantitatively calculate water
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saturations in oil and gas reservoirs by well logging interpretation. It is usual that oil layers show
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high resistivity responses, while water layers show low resistivity responses. However, there are
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low-resistivity oil zones that exist in many oilfields around the world, leading to difficulties for
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reservoir evaluation. In our research, we used digital rock technology to study different internal
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and external factors to account for low rock resistivity responses in oil layers. We first
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constructed 3D digital rock models with five components based on micro-computed
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tomography (micro-CT) technology and X-ray diffraction (XRD) experimental results, and then
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oil and water distributions in pores were determined by the pore morphology method. When
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the resistivity of each component was assigned, rock resistivities were calculated by using
finite element method (FEM). We collected 20 sandstone samples to prove the effectiveness
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AUTHOR SUBMITTED MANUSCRIPT - JGE-101453.R1
*
Corresponding author.
Email addresses: yanweichaoqz@163.com
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AUTHOR SUBMITTED MANUSCRIPT - JGE-101453.R1
of our numerical simulation methods. Based on the control variate method, we studied the
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effects of different factors on the resistivity indexes and rock resistivities. After sensitivity
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analyses, we found the main factors which caused low rock resistivities in oil layers. For
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unfractured rocks, influential factors arranged in descending order of importance were
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porosity, clay content, temperature, water salinity, heavy mineral, clay type and wettability. In
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addition, we found that the resistivity index could not provide enough information to identify
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a low-resistivity oil zone by using laboratory rock-electric experimental results. These results
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can not only expand our understandings of the electrical properties of low resistivity rocks
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from oil layers, but also help identify low-resistivity oil zones better.
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Keywords: electrical properties, low-resistivity oil zone, digital rock technology
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1. Introduction
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Low-resistivity oil zones with high oil productions have been found throughout the world,
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and the electrical properties of low resistivity sandstones from these layers receive much
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interest because the remaining oil in pores is difficult to identify by resistivity log analysis. A
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low-resistivity oil zone is defined as an oil zone which is not significantly different from the
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water zone by using resistivity logging responses (Liu et al 2006). This phenomenon causes
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incorrect fluid identifications, which calculates higher water saturations for oil layers.
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This problem has been acknowledged for over 40 years, and the low-resistivity oil zones
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exist in many oilfields all over the world (Worthington 2000). The factors causing unfractured
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low-resistivity rocks from oil layers are complicated, which can be separated into two categories,
including internal factors and external factors. Most researchers attributed low resistivity contrast
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to one or combination of factors, including irreducible water, shoulder bed effects, clay type and
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content, high water salinity, heavy conductive minerals, and deep invasion by conductive muds
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(Palar et al 1997; Lubis et al 2016; Hamada et al 2000; Worthington 1997; Saha, 2003). In
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addition, wettability may be another factor that causes low resistivities because wettability
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influences rock resistivity (Montaron 2007).
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Due to the complexity of different factors, quantitative studies of different factors on rock
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resistivities are not easy to be performed by laboratory rock-electric experiments. As an advanced
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numerical simulation method, the digital rock simulation has been widely used in petrophysical
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properties analyses (Blunt 2001; Peng et al 2012; Arns and Melean 2009). Traditional digital rock
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models contain two parts, including solid matrix and pores. This method can be well-used in fluid
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flow numerical simulations because the pores dominate flow patterns, while it is not suitable for
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electrical properties simulations because of the existence of high conductive minerals in rocks.
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Therefore, it is necessary to construct 3D digital rock models with different solid components.
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However, due to complexity of rock components, it is difficult to identify different minerals only
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by using micro-CT technology. Another technique is needed to provide more information to
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calibrate detailed descriptions of rock components. Mutina et al (2012) combined both micro-CT
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and micro-X-Ray Fluorescence system to obtain information on spatial distribution of chemical
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elements in a rock. QemSCAN images determine the mineral phase at each individual voxel,
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which can be extended to the full 3D digital rock space (Golab et al 2013). Although several
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researchers studied the effects of some factors on rock electrical properties by using digital rock
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technology, they did not further investigate how these factors caused low rock resistivity responses
in oil layers.
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In this study, we first clarified a more specific concept of the “low-resistivity oil zone” by
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experimental results. We combined micro-CT, X-ray diffraction (XRD) and routine core lab
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experiments to construct 3D digital rocks with different components. Then we used pore
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morphology method and finite element method (FEM) to simulate fluids distributions and rock
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electrical properties respectively. The effectiveness of our methods was verified by 20 sandstone
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samples comparing both experimental rock-electric experimental results and simulation results.
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Seven influential factors were studied, including temperature, wettability, water salinity, heavy
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mineral, clay type, clay content and rock porosity. Finally, through sensitivity analyses, we
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found main factors causing low-resistivity oil zones.
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2. Numerical simulation method and theory
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2.1 3D digital rock model generation
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In order to employ the digital rock technology to study the electrical properties of low
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resistivity sandstones, constructing a 3D digital rock is the first step. The traditional digital rock
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generation method separates rock samples in two parts, solid matrix and pores. This method is
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useful for dealing with fluid flow simulations because solid matrix has little influence on flow
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properties. However, in order to study electrical properties, it is vital to identify different mineral
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components in a rock.
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Among 3D digital rock reconstruction techniques, micro-CT technology is widely used in the
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porous media analysis to investigate the real micro-structures of the rock samples. Although
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numerical simulations could extract the structural information from two dimensional (2D) slices,
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or from grain size distributions, micro-CT reconstruction method is much more accurate in
characterizing different components and pores. In our research, we combined CT images and XRD
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experimental results to generate 3D digital rock models of different rock samples. The principle of
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multiple component segmentation is based on the density differences of the mineral phases in a
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grayscale image of the rock sample (Halisch et al 2010). If one type of mineral has high density,
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such as pyrite, the gray value in the CT images will be high. It should be noted that due to the
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complex composition of different matters in rock matrix, gray values of mineral phases are not
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specific values, leading to difficulties in performing accurate segmentations. However, it is not
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necessary to separate different components accurately because the main purpose of our research is
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to study influential factors of the electrical properties of rocks.
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Based on routine porosity results and XRD experimental results, we divided a rock sample
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into five components, including pores, quartz, feldspar, clay minerals and heavy minerals. The
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content of each component can be easily acquired by using image segmentation. Figure 1 shows
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the analyzed results of three sandstones and one mudstone, and the porosities are 13.39 %, 6.94 %,
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7.20 % and 2.89 % respectively. In addition, these rocks have different mineral contents and pore
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size distributions. The first row color marks are the explanations of different colors. In this figure,
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red component refers to pores, dark blue refers to quartz, light blue refers to feldspar, brown refers
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to clay minerals and green refers to heavy minerals. 3D digital rocks after segmentations are
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shown in the second row. The third row images are thin sections, which are selected in the middle
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of 3D digital rocks. Size information is illustrated below, and the resolution of each voxel is 6.55
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μm.
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Pores
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AUTHOR SUBMITTED MANUSCRIPT - JGE-101453.R1
Quartz
Feldspar
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Clay
Heavy minerals
492×492×922 voxels
474×474×801 voxels
450×450×939 voxels
457×457 voxels
492×492 voxels
474×474 voxels
450×450 voxels
(a)
(b)
(c)
(d)
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457×457×758 voxels
Figure 1. Constructed 3D digital rock models based on X-ray CT images and XRD results; (a) -(c)
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are images of three sandstones, while image (d) belongs to a mudstone.
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2.2 Electrical simulation methods
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Low resistivity rocks may contain different types of fluids and contents, therefore, it is
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important to determine fluid distributions in pores before electrical simulations. In our research,
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we used pore morphology method to simulate oil and water distributions in pores under different
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water saturations in a digital rock which was constructed from X-ray CT images. The main
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algorithm of this method is an open morphology algorithm, which is expressed by:
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A B   A B  B
(1)
where both A and B are data sets. For set A, the open operation is an iteration of erosion by set B
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AUTHOR SUBMITTED MANUSCRIPT - JGE-101453.R1
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firstly and then follows a dilation calculation by set B. Erosion and dilation are two basic
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operations in mathematical morphology (Haralick et al 1987). This process is repeated until the
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program reaches to the maximum iteration. The water saturation value after each iteration can be
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calculated by counting voxels of different phases.
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After determining different fluid distributions, we used the electrical simulation methods to
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acquire rock resistivities under different water saturations. As an advanced electrical simulation
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method, FEM is often used to simulate rock resistivity. Data file of a digital rock model is a huge
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matrix, which consists of lots of voxels. We applied an electrical field between two opposite faces
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of the digital rock model, and the voltage distribution of each voxel determines the whole model
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energy. The effective resistivity of this model is determined by solving the problem of minimizing
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the system energy. In our research, the periodic boundary conditions were set as the restrictive
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condition of the calculation. Ideally, the energy gradient with respect to the variables of the
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voltage is zero. We defined a stopping criterion of C=1×10-7, and the simulation stopped when the
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norm squared of the gradient vector was less than C. The detailed descriptions of the pore
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morphology method and FEM can be found in Liu et al (2009) and Garboczi (1998).
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When we performed this simulation method on a digital rock, resistivity of each component
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was manually assigned as an input parameter. In our research, six components should be
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considered, including oil, water, quartz, feldspar, clay minerals and heavy minerals. We assumed
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that the resistivities of quartz, feldspar and oil were infinite and resistivities of other components
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were calculated based on temperature or cation exchange capacity (CEC, mol / 100g). The
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resistivity of clay was calculated by the total resistivity of each clay mineral. The percentages of
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clay minerals were measured by XRD experiments, and clay minerals’ resistivities (Rclay, ohm m)
were calculated by combining the following table and equation (2):
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AUTHOR SUBMITTED MANUSCRIPT - JGE-101453.R1
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Table 1. CEC of each clay mineral
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CEC range (mol / 100g)
Mean CEC (mol / 100g)
Chlorite
10 - 40
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Illite
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Kaolinite
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Smectite
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Rclay 
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Bt
CEC (1  t ) G
(2)
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Clay mineral
where ϕt is the total porosity of the rock, decimal; ρG is the average grain density, g / cm3; B is the
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equivalent ionic conductance of clay exchange cations, which can be obtained by the methods
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described in Dacy et al (2006). This parameter is related to the temperature, which means if the
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temperature changes, clay resistivity will be changed.
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Heavy minerals in sandstone samples have a wide range of resistivities, such as zircon (44
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ohm m), tourmaline (109 ohm m), garnet (107 ohm m), leucosphenite (103 ohm m), magnetite (10-3
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ohm m) and pyrite (10-3 ohm m). Although it is hard to separate different heavy minerals, mineral
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contents can be obtained by XRD experimental results.
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In order to study effects of the water salinity and the temperature on rock resistivity, we set
the input water resistivity value by using the following equations:



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81.77
Rw1  
 0.0123  

4
0.955
2.74

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
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Rw2  Rw1 
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1.8T1  39
1.8T2  39
(3)
(4)
where Cw is the water salinity, mg / L; T1 is the first temperature value, ℃; T2 is the second
temperature value, ℃; Rw1 the first water resistivity value, ohm m; Rw2 is the second water
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resistivity value, ohm m.
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3. Results and discussion
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3.1 A more specific concept of low-resistivity oil zone
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Some researchers have tried to identify the low-resistivity oil zone quantitatively, and they
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believe that the resistivity oil zone has a resistivity index (I) smaller than two (Liu et al 2006).
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However, we regard the “low-resistivity oil zone” as a relative concept, and it’s unwise to use a
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resistivity index value to characterize an oil zone. Although a small resistivity index means low
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resistivity theoretically, small resistivity indexes calculated by either well log interpretations or
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laboratory rock-electric experimental results cannot determine fluid types in rocks.
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In order to prove our concept, we conducted laboratory rock-electric experiments on two
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sandstones, which were No.1 and No.2 respectively. The water resistivity was 0.48 ohm m.
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Resistivity index was calculated by Archie’s equation, revealing the relationship between rock
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resistivity and water saturation (Archie 1942).
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RI 
Rt
b
 n
R0 Sw
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(5)
where Rt is the true resistivity of the rock sample; R0 is the resistivity of the water saturated rock; b
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is a lithology coefficient; Sw is the water saturation and n is the saturation exponent. Figure 2
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shows the experimental results of two rocks. It is indicated that the resistivity index of No. 1 (1.79)
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is higher than No. 2 (1.62), however, the rock resistivity of No.1 is nearly a half of No. 2 under the
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same water saturation. If No.2 is obtained from a water layer, then it will be difficult to determine
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fluid types of No.1 by resistivity logging responses. Therefore, laboratory rock-electric
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experiments of at least two sandstone samples are necessary for judging a low-resistivity oil zone,
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and resistivity index cannot provide accurate information.
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1
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(a)
(b)
Figure 2. Laboratory rock-electric experimental results of two sandstone samples; (a)
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relationships between water saturations and resistivity indexes; (b) relationships between water
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saturations and rock resistivities.
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3.2 Validations of our electrical properties simulation method
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Although some researchers had proved the effectiveness of using FEM to calculate electrical
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properties (e.g., Jiang et al 2011; Zhao et al 2013; Nie et al 2016), they divided a digital rock
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model into three parts, ignoring the influence of clay and heavy minerals. Based on routine core
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analyses and XRD experimental results, percentage of each phase can be obtained. We collected
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20 sandstone samples from Xihu depression and performed both laboratory experiments and
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numerical simulations. Following representative elementary volume (REV) analysis, a
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sub-volume of digital rock with a size of 300×300×300 voxels was segmented for each sandstone
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sample. These sub-volume digital rocks were selected in the middle of 3D digital rocks. Measured
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porosities of these rocks were ranging from 5.85% to 15.47%, and the average porosity was
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10.39%.
The input resistivity parameters for simulations were tried to be consistent with actual values.
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AUTHOR SUBMITTED MANUSCRIPT - JGE-101453.R1
We compared the saturation exponent values (n) in equation (5) of both laboratory rock-electric
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experiments results and simulation results, shown in figure 3. It is indicated that the saturation
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exponents calculated by numerical simulations match experimental results well, and the relative
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error is only 3.79%, which proves the validity of our simulation method.
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Figure 3. Saturation exponent values calculated by both experiments and simulations.
3.3 Effects of different factors on rock electrical properties
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In order to study different influential factors which cause low rock resistivities in oil layers,
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the control variate method was used. There are two main types of factors accounting for a
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low-resistivity oil zone, including internal and external causes. We studied the effect of injecting
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water salinity (external factor) on rock resistivity previously (Yan et al 2017), and found that rock
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resistivity dropped when high salinity fluids (low resistivity) penetrated to the oil layers. In our
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research, we focused on studying internal factors, such as clay, wettability, heavy minerals,
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formation water salinity and rock porosity. Among external factors, we tried to study effect of
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temperature on rock resistivity. Through numerical simulations of 20 different sandstone digital
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rocks, we found that the trends of the electrical properties were similar. Therefore, we used one
sandstone digital rock model (No. 3) to show the different effects of factors on resistivity index
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and rock resistivity. The resistivity index of this rock was 1.66. Table 2 shows the percentage of
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each component of this rock.
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Table 2. Percentage of each component of sandstone No.3.
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Heavy
Component
Pore
Illite-smectite
Illite
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minerals
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Firstly, we changed the rock temperature. The influence of temperature is different for each
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rock component. Due to high pressures and good sealing conditions in real oil reservoirs, we
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assumed that no water could evaporate in high temperatures. The resistivities of formation water
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and water film around grain surface are calculated by equation (4), and the resistivities of different
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clay minerals are changing with the parameter B in the equation (2) in different temperatures.
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Figure 4 shows the relationships between water saturations and the rock’s electrical properties. It
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is indicated that with the increase of temperature, the resistivity index decreases very slightly, but
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rock resistivity decreases obviously under the same water saturation. Three reasons account for the
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rock resistivity drop, including the low resistivities of free water, clay and water film. Although
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there is a big temperature difference between surface and underground, the parameter n in the
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Archie’s equation calculated by laboratory rock-electric experiments results can be used in a
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downhole situation. If the temperature of an oil zone is much higher than the water zone, it will be
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difficult to interpret fluid types by resistivity logging data.
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Figure 4. Effects of temperatures on rock electrical properties; (a) relationships between water
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saturations and resistivity indexes, (b) relationships between water saturations and rock
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resistivities.
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Secondly, we changed clay types and clay contents to study their effects on rock electrical
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properties. We assumed all clay components in sandstone No. 3 were pure kaolinite, illite,
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illite-smectite mixed-layer and chlorite respectively. Resistivity of each clay mineral was
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calculated by the equation (2), and the average CEC of illite-smectite mixed-layer was 70 mol /
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100g. We also studied “no clay” situation, which means the resistivity of clay was infinite.
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Numerical simulation results of rock electrical properties are shown in figure 5. It is obvious that
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clay has great effect on rock electrical properties. On one hand, the existence of clay decreases
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both resistivity index and rock resistivity. On the other hand, illite-smectite mixed-layer has the
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greatest influence on resistivity. Clay with a higher CEC has a lower parameter n and a lower rock
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resistivity value. Then, we studied the effects of clay contents on rock electrical properties. We
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assumed all clays were pure illite, and constructed clays around pores by random generations,
replacing solid voxels. The generated clay contents were 0 %, 3.95 %, 7.26 % and 16.86 %
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respectively. Numerical simulation results of rock electrical properties are shown in figure 6. We
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can see that with the increase of clay content, both resistivity index and rock resistivity decrease
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greatly. Therefore, high clay contents can cause low resistivity in oil layers. If an oil layer contains
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both high percentage of clay and high CEC, the resistivity contrast between an oil layer and a
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water layer will be low.
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(a)
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(b)
Figure 5. Effects of clay minerals on rock electrical properties; (a) relationships between water
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saturations and resistivity indexes, (b) relationships between water saturations and rock
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resistivities.
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(a)
(b)
Figure 6. Effects of clay contents on rock electrical properties; (a) relationships between water
saturations and resistivity indexes, (b) relationships between water saturations and rock
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resistivities.
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Then the effects of heavy minerals on electrical properties were analyzed. Due to the large
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scale of resistivities of different heavy minerals, we regarded pure garnet (high resistivity), zircon
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(medium resistivity) and pyrite (low resistivity) as heavy minerals respectively to perform
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numerical simulations. The simulation results are shown in figure 7, which indicates that only
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heavy minerals with low resistivity have great influence on rock electrical properties. Both
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resistivity index and rock resistivity decrease with the increase of heavy mineral conductivity.
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Low content of pyrite can lead to low rock resistivity, therefore, we believe high percentage of
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pyrite decreases rock resistivity dramatically. For interpreting oil or water layers, heavy minerals
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are quite useful clues. When low resistivity heavy minerals in reservoirs are abundant, they may
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cause a low-resistivity oil zone.
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(a)
(b)
Figure 7. Effects of heavy minerals on rock electrical properties; (a) relationships between water
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saturations and resistivity indexes, (b) relationships between water saturations and rock
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resistivities.
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Formation original water salinity was also regarded as an important factor that causes low
rock resistivity. We set salinity of water as 1000 mg / L, 5000 mg / L, 10000 mg / L, 50000 mg / L
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and 100000 mg / L respectively. By using equation (3), input water resistivities can be calculated.
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Figure 8 shows the relationships between water saturations and rock electrical properties. We can
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see that with the increase of water salinity, resistivity index increases, while rock resistivity
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decreases. High salinity irreducible or free formation water strengthens current field, leading to
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low resistivity responses in well logging data.
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Figure 8. Effects of water salinity on rock electrical properties; (a) relationships between water
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saturations and resistivity indexes, (b) relationships between water saturations and rock
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resistivities.
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Most sandstone reservoirs tend to be water wet or intermediate-wet, and water forms a film
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between pores and solid matrix. Although several researchers proved that rock wettability
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influenced resistivity by using digital rock technology (Liu et al 2009; Jiang et al 2011), their
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models were constructed by using electro-insulating solid matrix and fluid phases. In addition, it is
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necessary to study if wettability plays a significant role in causing low resistivity. Based on the
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descriptions by Liu et al (2009), five water film conductivities were chosen as 0 S/m (oil wet),
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0.02 S/m, 0.04 S/m, 0.08 S/m and 0.16 S/m respectively in our research. Figure 9 shows the
results of different wettabilities on electrical properties. Although water wet rocks have lower
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resistivity indexes and rock resistivities than oil wet rocks, it is difficult to form low-resistivity oil
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zones.
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Figure 9. Effects of wettability on rock electrical properties; (a) relationships between water
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saturations and resistivity indexes, (b) relationships between water saturations and rock
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resistivities.
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Core experiments of rocks in a low-resistivity oil zone usually show low porosity results, and
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then some researchers may think low porosity is a key factor for identifying oil zone which cannot
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be interpreted by resistivity well logging techniques. We used our methods to prove if low
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porosity caused low rock resistivity. 20 sandstone digital rock samples were not used because of
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their small range of porosities and different clay contents. We used a process-based method
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(Bryant et al 1995) to construct rock models with different porosities. Figure 10 shows the 3D
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images of these rocks, including 5.0 %, 10.0 %, 15.0 %, 20.0 %, 25.0% and 30.0% respectively.
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Red represents pores and blue represents solid matrix. Figure 11 shows electrical properties of
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these rock models. With the increase of porosity, water saturated rock resistivity (R0) is decreasing
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following the Archie’s equation. However, rock resistivity is also decreasing, which is inconsistent
with the traditional concept. From our simulation results, we can conclude that the resistivities of
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oil layers with high porosities may be much lower than the low porosity water layers. In order to
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explain this difference between laboratory observations and our simulation conclusions, we
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studied the relationships between experimental measured porosities and shale contents of an actual
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oil well, shown in figure 12. It is obvious that with the increase of porosity, clay content decreases.
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In order to study the effect of rock porosity on electrical properties, we ignored clay minerals in
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digital rock models. Our results change the traditional concept about the effect of porosity on
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resistivity. We have proved that with the increase of clay content, rock resistivity decreases.
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Therefore, it is not low porosity but clay content that causes a low-resistivity oil zone.
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(e)
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Figure 10. Constructed 3D digital rock models with different porosities; (a) 5.0 %; (b) 10.0 %; (c)
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15.0 %; (d) 20.0 %; (e) 25.0%; (f) 30.0%.
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(a)
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Figure 11. Resistivity properties of constructed rocks; (a) relationships between rock porosities
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and R0; (b) relationships between water saturations and rock resistivities.
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Figure 12. Relationships between rock porosities and shale contents.
After studies of seven possible factors which may cause low-resistivity oil zones, it is
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necessary to find out the main factors. Sensitivity analyses can be used to select main factors of
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altering rock resistivities. In this paper, the Morris screening method (Lenhart et al 2011) was
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used to determine main factors. It used sensitivity index, which was expressed by:
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n 1
s
i 0
(Yi 1  Yi ) / Y0
/ (n  1)
( Pi 1  Pi ) /100
(6)
where s is the sensitivity index; Yi is the ith output value; Yi+1 is the (i+1)th output value; Y0 is
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the initial value; Pi is the percentage change between ith output value and initial value; Pi+1 is
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the percentage change between (i+1)th output value and initial value; n is the number of
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calculations.
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Higher s means higher sensitivity, and sensitivity degrees of different factors are
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classified as follows: |s| ≥1 means very high sensitivity parameter; 0.2≤ |s| <1 means high
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sensitivity parameter; 0.05≤ |s| <0.2 means medium sensitivity parameter; 0≤ |s| <0.05 means
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small to negligible sensitivity parameter. Figure 13 shows the results of sensitivities of
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different factors that cause a low-resistivity oil zone. The following influential factors are
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arranged in the descending order of importance: porosity, clay content, temperature, water
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salinity, heavy mineral, clay type and wettability. In order to identify low resistivity oil layers,
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comparisons of these factors between different layers should be taken into consideration.
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Figure 13. Sensitivity analyses of different factors that cause a low-resistivity oil zone.
4. Conclusions
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The effects of internal and external factors on rock resistivities are deeply explored by
using digital rock technology. The 3D digital rock models are constructed with five
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components, including pores, quartz, feldspar, clay minerals and heavy minerals. The electrical
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properties of rocks are studied using the pore morphology method and FEM. According to the
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experimental and numerical simulation results, the results show that the low-resistivity oil
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zone is a relative concept, and it is necessary to perform laboratory experiments on at least
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two rock samples from different layers to judge an oil layer. Different factors have different
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effects on rock electrical properties. Through sensitivity analyses, we believe the influential
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factors arranged in the descending order of importance are porosity, clay content, temperature,
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water salinity, heavy mineral, clay type and wettability. Our results explain the possible
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reasons of low rock resistivity responses in oil layers by quantitative analyses, which help to
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improve the low-resistivity oil zone identification and evaluation.
10
Acknowledgments
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This work was supported in part by the National Science and Technology Major Project
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(No. 2016ZX05006002-004) and Fundamental Research Funds for the Central Universities
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(No.16CX06049A).
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