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EMT simulation and effect of TTI anisotropic media in EMT signal

EMT simulation and effect of TTI anisotropic media in EMT signal An axisymmetric finite difference method is employed for the simulations of electromagnetic telemetry in the homogeneous and layered underground formation. In this method, we defined the anisotropy property using extensive 2D conductivity tensor and solved it in the transverse magnetic mode. Significant simplification arises in the decoupling of the anisotropic parameter. The developed method is cost-efficient, more straightforward in modeling anisotropic media, and easy to be implemented. In addition, we solved the integral operation in the estimation of measured surface voltage using Gaussian quadrature technique. We performed a series of numerical modeling of EM telemetry signals in both isotropic and anisotropic models. Experiment with 2D tilt transverse isotropic media characterized by the tilt axis and anisotropy parameters shows an increase in the EMT signal with an increase in the angle of tilt of the principal axis for a moderate coefficient of anisotropy. We show that the effect of the tilt of the subsurface medium can be observed with sufficient accuracy and that it is an order of magnitude of 5 over the tilt of 90 degrees. Lastly, consistent results with existing field data were obtained by employing the Gaussian quadrature rule for the computation of surface measured signal. Keywords Electromagnetic telemetry · Finite difference method · Anisotropy · Gaussian quadrature 1 Introduction and the return of instructions from the rig to the borehole. Borehole data transmission could be useful in geothermal In borehole drilling, effective geosteering and accurate well exploration, mineral exploration but are mostly used by the landing require quick access to point scale data about for- oil and gas industry. Borehole telemetry has been an essen- mation being drilled, bit trajectory, well condition and lots tial part of industrial drilling for oil and gas resources in more. Borehole telemetry is defined as the two-way transfer recent decades, especially in an unconventional reservoir of such data from the bottom hole assembly (BHA) to the rig and during horizontal drilling. It provides considerable time savings and minimizes the risks associated with geosteering and accurate well landing. Edited by Jie Hao This technique has been successfully applied in the drilling of several oil wells (Schnitger and Macpherson 2009; Baker * Olalekan Fayemi Hughes 2014). Telemetry methods can be classified based fayemiolalekan@mail.iggcas.ac.cn on the physical property considered in relation to the data Qing-Yun Di propagation, such as mud pulse telemetry, electromagnetic qydi@mail.iggcas.ac.cn telemetry and acoustic telemetry or combination of different Key Laboratory of Shale Gas and Geoengineering, Institute techniques, e.g., acoustic–electromagnetic technique. of Geology and Geophysics, Chinese Academy of Sciences, Electromagnetic telemetry (EMT) uses transmitted elec- Beijing 100029, China tromagnetic waves from bottom hole assembly (BHA) to Frontier Technology and Equipment Development Center the surface through the drill string, well and adjacent forma- for Deep Resources Exploration, Institute of Geology tions as a means of data transfer. Borehole telemetry aids in and Geophysics, Chinese Academy of Sciences, Beijing 100029, China accurate core point access to formation properties acquired close to the drill bit by logging while drilling (LWD) in Innovation Academy for Earth Science, Chinese Academy of Sciences, Beijing 100029, China real time, to fully assist in timely identification of zones of interest. EMT measurements can be acquired continuously College of Earth and Planetary Sciences, University of Chinese Academy of Sciences, Beijing 100049, China Vol:.(1234567890) 1 3 Petroleum Science (2021) 18:106–122 107 without the risk of data unavailability or insufficiency due casing on the current distribution at specified distances from to challenges with drilling fluid conditions as observed in the source and measured voltage at the surface. For mode- mud pulse telemetry. However, electromagnetic telemetry ling of an electromagnetic telemetry system in an anisotropic is profoundly affected by formation properties. Therefore, environment, and considering the accuracy requirement for effective simulation of the EMT response requires the con- practical application and the implemented complexity of the sideration of the effect of formation anisotropy on the cur - schemes, we intend to apply FDM to model EM propagation rent distribution pattern and, in turn, the voltage measured tools in axisymmetric formation with anisotropy. FDM can at the surface with respect to the source location. take all the features in an electromagnetic telemetry system, In EM study, formation anisotropy causes currents to such as drill pipes, drilling u fl id in the borehole and multiple depart from the direction of an exciting electric field within layers earth formation into consideration. the formation of interest (Lu et al. 2002). Therefore, accurate In this paper, we developed a scheme to model the EMT estimation of EMT signals at the rig floor can be marred response in a 2D underground formation by employing a by the effect of anisotropy on acquired data. Anisotropy is FD algorithm in the cylindrical coordinate system, to make a phenomenon that describes the variation in earth physi- use of the 2D structure to reduce the number of unknowns. cal properties as it occurs on various scales in a variety of geophysical applications. The measured earth properties could differ along the different measurement axes. Previous 2 Methodology applications of electromagnetic telemetry include marine controlled-source electromagnetic (CSEM) studies, reservoir 2.1 Isotropic medium characterization, SeaBed Logging (SBL), etc. (Brown et al. 2012; Cuevas 2019; He et al. 2019; Constable et al. 2019; Consider a homogenous and isotropic medium which is fully Li et al. 2019). For several decades, studies have been car- characterized by the constant parameters for magnetic per- ried out on electrical anisotropy as part of induction logging meability μ and electric permittivity ɛ. Next, assume that in efforts to better quantify hydrocarbon reservoir potential the source function and all the field variables exhibit the (Moran and Gianzero 1979; Anderson et al. 1998; Wilson j(t−kr) common time dependence e , where ω is the angular 2015). In reservoir evaluation, electrical anisotropy measure- frequency and k is the wave vector. Then, in terms of the ments may improve the estimate of hydrocarbons in place. fundamental field vectors, the electric intensity E and the The conductivity tensor that relates the induced current to magnetic intensity H are continuous and possess continuous the applied electric field has three principal values, each derivatives at ordinary points, such that corresponding to a direction of maximal conductivity. The resistivity of shale mostly controls the horizontal resistivity ∇× E =−iH − J (1) of the laminae, whereas the resistivity of hydrocarbon- bearing sands dictates the vertical resistivity  (Klein ∇× H =+ik E + J (2) 1993; Tabanou et al. 1999). Electrical anisotropy is caused by many different mechanisms such as conductive fluids fill- where k = i  − i ,  is the dielectric permittivity, and ing in fractured formations with voids, and laminated depo- s s J and J are the electric and magnetic source terms, respec- e m sition of substances of different electrical properties (e.g., tively.  in Eq. (2) is the electrical conductivity tensor, and sands and clays). In general, electrical anisotropy arises from it is defined as follows: the presence of sedimentary structures whose length scale is far smaller than the resolving power of measuring equip- ⎛  00 ⎞ ⎜ ⎟ ment, for example, shales and intercalation of thin layer of = 0  0 , (3) ⎜ ⎟ shale with sandstone are common phenomena resulting in ⎝ ⎠ formation anisotropy (Klein et al. 1995). The effect of anisotropy on EMT data can be investigated where  ,  and  are principle conductivities. Furthermore, considering a cylindrical coordinate system, via forward modeling using the finite difference method (FDM), the finite element method, etc. In this study, we con- an electromagnetic telemetry system is characterized by EM signals propagated using an insulating gap source on the drill sidered how formation anisotropy affects the current distri- bution pattern within the drill string and, in turn, the voltage string surrounded by a uniform vertical column of either conductive or resistive drilling fluid, and an outer column measured at the surface with respect to the source location during EMT. Emphasis was placed on investigating the of either layered formation and/or borehole casing (Fig. 1). The system, as defined radially outward from the center of effect of change in the orientation of the axis of symmetry of formation property in an anisotropic medium. We pay no the borehole, is solved as an axisymmetric and transverse magnetic problem. With the input voltage at the gap given attention to fluid invasion, but we consider the effect of the 1 3 108 Petroleum Science (2021) 18:106–122 2.2 2D Anisotropy Considering an electromagnetic telemetry system as Mud described above and displayed in Fig.  1, the governing Adjacent bed equation for EM propagation in full anisotropic media with conductivity tensor  in the cylindrical coordinate system is given as h h m D E − D E =−iH − J z  r z r h h m D E − D E =−iH − J r z z r  (6) h h m D E − D E + =−iH − J r z r  z r r h h e D H − D H = k E − k E − k E − J z  rr r r  rz z z r h h e D H − D H = k E − k E − k E − J r z r r   z z z r  (7) h h e D H − D H + = k E − k E − k E − J . r zr r z  zz z r  z r r Source Even though 3D forward modeling (Wang and Fang 2001; Weiss and Newman 2002; Davydycheva et al. 2003), combining high-performance computing, is available both in academic researches and in commercial software, full 3D methods are still prohibitively expensive for fast for- ward modeling. An intermediate solution between the 1D Fig. 1 Diagram of a borehole with a telemetry system (modified after Ribeiro and Carrasquilla 2014). (Jiang et al. 2017; Anderson 2001) model and fully 3D model is the 2D model which has uniform property along the strike axis and is less compu- tationally intensive. and the magnetic current circulating the drill string specie fi d In this study, a representative formation with anticlinal at the source position, the governing equation for transmis- structure, which shows uniformity in the phi-direction but sion of the radial magnetic field H is given as layering in the “r” and “z” plane, is considered. Generally, the transmitter of the EM tool consists of a very thin insu- 1   1 2 s H + + k  H =−J , lating gap along the drill string, generating controlled volt- 0 m k  z k z age difference along the conductive drill string. The source (4) transmission of the EM tool consists of radially transmit- where J = iM , and M is the magnetic current den- ted magnetic current. So the electric current transmission sity. The isotropic medium is characterized by a “radial is majorly in the vertical direction along the drill string conductivity”  which has the same value as the “vertical (which serves like air duct in guided waves). Even consid- conductivity”  at any point within the given layer, such z ering geological deformation, it is reasonable to assume that  =  . Relative permittivity value of 1 was used in z that within the propagation range of a typical EMT system this study. in a vertical or pseudo-vertical well (centimeters to tens The current flowing along the drill string is given as of meters), the geological formation is an asymmetrical J = 2H (Chen et al. 2017); therefore, Eq. (4) becomes 2D structure, which exhibits uniform EM properties in phi direction. A depth slice of formation with anticlinal 1  1   1 z 0 structure, which shows uniformity in the phi-direction but J + +  J =−J . z r z m 2   k  z k z inhomogeneity in the r-z plane, is shown in Fig. 2 using a (5) coordinate system image. Therefore, consider an axisymmetric problem (that is, the physical properties are radially isotropic) for an ani- sotropic media, whose conductivity tensor is expressed as in a cylindrical coordinate system (r, , z) as 1 3 Invaded zone Transition zone Uninvaded zone Petroleum Science (2021) 18:106–122 109 ⎛  0  ⎞ ϕ, ° rr rz ⎜ ⎟ = 0  0 . (8) r, m ⎜ ⎟ ⎝ ⎠ zr zz The medium is characterized by a “radial conductivity” which is constant in the horizontal directions and var- rr Z, m ies in the vertical direction for “vertical conductivity”  , zz such that  ≠  . The values of the subsurface conductivity rr zz tensor used in this study were defined using the anisotropy factor λ , longitudinal conductivity  , transverse conductiv- -8 -6 -4 -2 0 24 ity  and dip angle θ. (See “Appendix 1” for details.) The principal conductivity  is defined by the dip and resultant of the longitudinal conductivity and transverse conductivity. For simplicity, the uniform dielectric constant value of 1 was used in this study. Therefore, Eqs. (6) and (7) are reduced to h m −D E =−iH − J z r -8 -6 -4 -2 h h m 4 6 8 D E − D E =−iH − J r z z r (9) h m D E + =−iH − J r z 450 and h e −D H = k E − k E − J rr r rz z z r -8 -6 -4 -2 0 h h e 4 D H − D H =−k E − J 6 r z 8 z r (10) h e D H + = k E − k E − J . zr r zz z r z Thus, as for 2D (TM) with 2D anisotropic problems, the curl–curl operator cannot simply be replaced by the Lapla- cian operator. However, the simplified off-diagonal elements -8 -6 -4 -2 of the conductivity tensor make it possible to couple these equations into pure TM modes as in the 2D isotropic case. Therefore, solving the transverse magnetic (TM) problem, the reduced 2D equation is given as 1050 h −D H = k E − k E rr r rz z h h m D E − D E =−iH − J r z z r -8 (11) -6 -4 -2 6 8 D H + = k E − k E . zr r zz z 50 Ω·m 300 Ω·m 10 Ω·m By solving for E and E in the first and last terms in r z Fig. 2 Diagram of depth slice from an axisymmetric anticlinal struc- Eq. (11), respectively, and substituting the obtained values ture. Each depth slice shows different layers within a given radial dis- in the second term, the following is obtained: tance r = 10 m from the center (0). The radial distance is projected on the axis for  equal to 0°and 180°. The change in sign marks opposite sides of the radial length from negative to positive −k k UH − k k UJ =−D k H − D k H rr zz  rr zz z zz  r rr (12) e e e +D k H + D k k E − D k k E rr  rr zr r zz rz z r r z where U = i. Eliminating k E and k E from the equa- rz z zz z tions for E and E , respectively, and substituting the result- r z ing equations into Eq. (12), one can derive a system for esti- mating the radial magnetic field 1 3 110 Petroleum Science (2021) 18:106–122 D k U + D k U− D aup + D bup − D arup+ ⎛ z zz r rr ⎞ ⎛ ⎞ z r ⎜ ⎟ ⎜ ⎟ e e h m D ⋅ (AFZ ∗ crup)+ D k U − U + D D k U+ ⎜ ⎟ ⎜ zr ⎟ rr rz H = UJ r z r m (13) H = UJ (15) ⎜ ⎟ ⎜ ⎟ D ⋅ (AFZ ∗ cup)− ⎜ ⎟ ⎜ rz ⎟ e h e D D k U − D k U ⎝ zr rz ⎠ r z z ⎜ ⎟ D ⋅ (AFZ ∗ crup) − U ⎝ ⎠ where U = k k − k . rr zz 1 1 rz where aup = k U, bup = k U, arup = k U, crup = k U, zz rr rr rz We note in Eq. (13) that the upper term is identical to and r r and cup = k U. rz represents the 2D equation for isotropic medium, while the The results of this investigation were assessed by compar- lower terms are indicative of the anisotropic effect. Consider - ing the plots of the current distribution along the drill string ing Eq. (13), the upper terms can be considered as the primary and the measured EMT signal on the surface. To obtain the term, while the lower term can be considered for anisotropy, nodal values, the radial magnetic field value was interpolated the secondary term which can be updated for varying anisot- as the dot product of the face calculated radial magnetic field ropy conditions and effect. with the AFZ transpose matrix. Furthermore, we approximate the ordinary differential equations in terms of a series of numerical operators represent- ing higher-order forward and backward difference approxima- tions and averaging operators; the higher-order central differ - 3 Modeling ence approximation can be obtained by combining forward and backward difference operators. The above equation can be dis- In this study, we used these standard model parameters cretized and solved by the conventional first-order finite differ - unless specified otherwise: frequency range of 5–10 Hz, drill ence method. However, for better efficiency in this paper, we string inner radius of 0.25 m, while the outer radius is 0.3 m, used third-order forward and backward difference approxima- and the casing inner and outer radius of 0.4 m and 0.45 m, tions, and fifth-order central difference approximation for the respectively. Uniform conductivity value of 5 × 10 S/m was finite difference method. The discrete approximation of Eqs. used for the casing and drill string, while the drilling fluid (5) to (13) is formulated on a staggered 2D grid. In this for- conductivity was set as 1 S/m. The relative magnetic per- mulation, the electric components were located on cell faces, meability of the casing, borehole fluid and the formation is while the magnetic components were located on cell nodes. assumed to be unity. For the isotropic part, the above implementation is direct and We first consider a vertical well in a homogeneous under - requires N faces and N + 2 nodes for every vertical discrete ground formation with a resistivity value of 10 Ω.m. Solid layer/discretization. However, for the anisotropic term, a slight drill string with a radius of 0.25 m and casing with an inner complication arises from implementing the update parameters and outer radius of 0.35 m and 0.4 m, respectively, were used from the conductivity functions. In our implementation of this in this model. The total length of the drill string and the cas- term, we averaged the update parameters onto the nodes/cell ing is 1200 m and 800 m, respectively. The voltage source faces using a four-plane average matrix (Guo et al. 2018); with a value of 1 v and propagating frequency of 10 Hz were set 15 m away from the base of the drill string. To assess the ⎡ 1∕41∕4 m 1∕41∕40 0 ⎤ accuracy of the FD code, we compared the current distribu- ⎢ ⎥ 01∕41∕4 m 1∕4 m 0 ⎢ ⎥ tion along the drill string obtained from the FD method with AFZ = 0 01∕41∕4 mm 0 ⎢ ⎥ (14) the one obtained from the numerical simulation software ⎢ ⎥ n nn 0 nn n (COMSOL vs. 5.3a 2017), which is designed based on the ⎢ ⎥ 0 00 m 1∕4 m 1∕4 ⎣ ⎦ finite element method (Fig.  3). A conventional decrease in the amplitude of current along to ensure continuity. the drill string away from the source is recorded. Also, a With mapping operators and mapped model parameters, decrease in amplitude is recorded across the 800 m depth discretizing Eqs. (13) and (14) term by term, the complete mark due to the presence of casing in the vertical well. We matrix form for the magnetic phi-components on a staggered can see that current flow along the drill pipe calculated by grid can then be represented as both the FD method and the FEM method from COMSOL suite agrees relatively well with the exception of the depths below 1000 m where the FEM data show no definite trend and are less fitting. We further consider a vertical well in a homogeneous underground formation with a resistivity of 10 Ω.m, and a two-layered earth model with resistivity 1 Ω.m and 10 Ω.m, 1 3 Petroleum Science (2021) 18:106–122 111 Current along drill string respectively. The length of the drill string is constant, while the length of the borehole casing varied between 800 and 920 m depth, for both the homogeneous formation and two- layer earth model, respectively. The diameters of the drill pipe and the borehole are 0.3 m and 0.45 m, respectively. The resistivity of the drilling fluid is set as 1 Ω.m, and the conductivity of both drilling pipe and borehole casing is assumed to be 5 × 10 S/m. The downhole transmitter with 1v voltage source is set at 50 m behind the drill bit. Figure 4a shows the simulated magnitude of the current flowing through the drill string with a transmitting frequency of 5 Hz. The magnitude of the current flowing through the drill string gradually decreases with an increase in dis- tance from the source; this reflects the change in current density with loss of current into the adjacent formations. Comparison of current distribution result for cased borehole (black) and non-cased borehole (red) shows a decrease in the magnitude of the current flowing along the drill string from the depth of 800 m due to the presence of steel casing. FD 1200 m In the case of layered medium (Fig. 4b), with transmitting Comsol frequency of 10 Hz, slight deflection (reduction in current −6 −4 −2 0 2 magnitude) due to borehole casing was recorded at a depth of 920 m. However, the most notable perturbation is the Current log(A) current loss in the adjacent conductive layer at a depth of 550 m in the cased well. Fig. 3 Comparison of amplitude of current flow along the drill string in a homogeneous formation obtained from the FD method and FEM The general feature one might observe from these figures method in COMSOL Multiphysics software is that the current distribution along the drill string is much stronger without casing than with casing. The disparity in Current along drill string−comparison Current along drill string−comparison 0 0 (a) (b) 500 500 1000 1000 Uncased Uncased Cased Cased 1500 1500 −8 −6 −4 −2 0 2 −8 −6 −4 −2 02 10 10 Current Log (A) Current Log (A) Fig. 4 Amplitude of current flow along the drill string in; a a homogeneous formation and b 2-layered earth model. The current flow, as indi- cated above, is attenuated in the presence of both conductive adjacent formation and well casing. These form the major challenges encountered in EM telemetry 1 3 Depth, m Depth, m Depth, m 112 Petroleum Science (2021) 18:106–122 the magnitude of the current distribution might be explained 4 Recorded voltage estimation by the fact that a single metal cylinder in a given model can support a guided wave theory (Stratton 1941). In other The recorded voltage at the surface can be obtained by cal- words, the current induced by the current/voltage source culating the integral of the electric field calculated from the at the insulating gap along the drill string flows majorly obtained current distribution. (See “Appendix 2”) In prac- vertically along the drill string. However, with the second tice, the integrand is seldom smooth. However, this issue highly conductive metal cylinder included, more current gets is addressed in Gaussian quadrature by using weighting attenuated by the adjacent steel, leading to a reduction in the function, which results in removal of integrable singulari- surface measured EMT signal. ties (Mahesh and Sucharitha 2018; Piqueras et al., 2019; As mentioned above, the current density will produce a Hassan et al. 2020). Therefore, for higher accuracy calcula- pattern of vertical distribution within the drill string and tion, we used the explicit Legendre–Gauss–Lobatto (LGL), near-vertical distribution along the inner annulus. When Legendre–Gauss–Radau (LGR), Gauss–Lobatto (GLo) and compared to the current flow in a cased well, a similar flow Gauss–Legendre (GLe) quadrature techniques, which con- pattern is expected for current within the drill string, while verge accurately to estimate the potential across two measur- the magnitude of current within the annulus is reduced ing points (−1,1). The integral form of the quadrature tech- because there is a current loss to the conductive casing. niques is given in “Appendix 2.” When using the Gaussian These flow patterns are shown in Fig.  5, representing the quadrature techniques, selecting an appropriate bandwidth current distributions in the vertical plane (r-z plane) for cases for a kernel density estimator is important. Several tech- with and without a steel casing. The difference between the niques have been used to select the smoothing parameter current distributions of these two cases can be seen clearly for reasonable density estimate. These are categorized as with the current being weak along the borehole direction either plug-in bandwidth estimators, which tend to select within the cased region. larger bandwidths and could produce over-smoothed results, classical estimators which on the other hand could produce under smoothed results when the smoothing problem is severe (Loader 1999), or nonparametric estimation (Mar- ron and Chung 2001; Chan et al. 2010; Golyandina et al. 2012), which considers a family of smooths with a broad 10 10 Current distribution log Al Current distribution og A 0 0 (a) (b) 0 0 200 200 −2 400 400 600 600 −4 800 800 −6 1000 1000 1200 1200 8 −8 1400 1400 −10 −10 1600 1600 −12 −12 1800 1800 2000 2000 −14 −14 01020304050 01020304050 Radial distance, m Radial distance, m Fig. 5 Current flow distribution in a homogeneous formation with; a non-cased well and b cased well 1 3 Depth, m Depth, m Petroleum Science (2021) 18:106–122 113 range of bandwidths, instead of a single estimated function. reduces as the rate of attenuation of the current flowing Noting that the goal is to decide on selecting the smoothing through the drill string increases within the conductive parameter purely from the data, we used the nonparametric layer. technique. Furthermore, we used an example of a field telemetry Figure 6 shows an example of voltage responses gener- data acquired in Shandong Province, China. The resistiv- ated from a mixture of a Gaussian variable using the homog- ity log and constructed earth resistivity model are shown in enous earth model and 10 Hz propagating frequency. The Fig. 9. The drilling fluid resistivity value of 1 Ω.m was used kernel density was estimated using n (number of data points in this case. The working frequency and output/input current observed from a realization of the random variables) that of the downhole source were 10 Hz and 1.4 A, respectively. varies from 2 to 150, thereby varying the bandwidths. The The diameter of the drill pipe was set as 0.35 m, while the wide range of smoothing considered allows a contrast of borehole diameter was set as 0.5 m. In the case of the cased estimated features and possible point of convergence. In well simulation, the casing diameter was set as 0.6 m. *Note comparison, the accuracy of the estimated values increases that the resistivity of the first layer was set as 30 Ω.m con- with an increase in the value of n and at a different rate for sidering the inconsistency in the resistivity log and the high most of the techniques. However, with n equal to 150, the value of the measured field data within this depth. Also, the different techniques converge to given stable values. coefficients of anisotropy values of 1.8, 0.6, 0.8, 2 and 1.5 Using n equal to 150, the comparison of voltage meas- were used to define the longitudinal conductivity values at ured at the surface using both homogeneous earth and depths [320 366] m, [367 440] m, [441 480] m, [481 520] the three-layer earth model, with background resistivity m and [667 760] m, respectively. The measured EMT signal of 10 Ω.m, middle layer resistivity value of 2 Ω.m, and was simulated at depths ranging from 150 to 1080 m. source location varying from 200 to 1350 m depth at an Figure 10 shows the comparison of the simulated sur- interval of 45 m, is shown in Figs. 7 and 8. The confined face recorded telemetry signal using the different integral conductive layer was set between the depths of 440  m techniques with the field-measured data. The obtained plots and 580 m. A small difference in the value of the meas- show that the simulated responses follow the same trend as ured voltage at the surface is obtained by comparing the the field measurements over a large depth range. The simu- results from Gauss quadrature algorithms with the direct lated EMT signal in a cased well (Fig. 10b) shows a more method between the depth of 200–300 m for transmitting pronounced representation of the low-magnitude field data frequency of 10 Hz, and 300 m to 700 m for transmitting with the exception of a significant difference between the frequency of 5 Hz. Below these depths, the results from depths of 300 m and 480 m, as observed in the result for the different techniques are similar. In general, an increase the non-cased well signal as well (Fig. 10a). In general, the in measured voltage at the surface is obtained between magnitude of the measured EMT signal simulated without the depths of 440 m and 600 m, as the source is located borehole casing is higher with the given model. The simu- within the conductive layer. As the source depth increases, lated EMT signal is a close match with high-magnitude field the magnitude of the EMT signal recorded at the surface data within the first 300 m and at subsequent depths. EMT signal N = 2 EMT signal N = 50 EMT signal N = 150 0 0 0 (a) (b) (c) 200 200 200 400 400 400 600 600 600 800 800 800 1000 1000 1000 10 Hz direct 10 Hz direct 10 Hz direct 10 Hz LGL 10 Hz LGL 10 Hz LGL 10 Hz LGR 10 Hz LGR 10 Hz LGR 1200 1200 1200 10 Hz GLe 10 Hz GLe 10 Hz GLe 10 Hz GLo 10 Hz GLo 10 Hz GLo 1400 1400 1400 0 0.05 0.10 0.15 0 0.05 0.10 0.15 0 0.05 0.10 0.15 Voltage, V Voltage, V Voltage, V Fig. 6 Estimated voltage responses with several bandwidths initiated by changing the number of data point value from 2 to 150 1 3 Depth, m Depth, m Depth, m 114 Petroleum Science (2021) 18:106–122 EMT comparison EMT comparison 0 0 (a) (b) 200 200 400 400 600 600 800 800 1000 1000 10 Hz direct 10 Hz direct 1200 10 Hz LGL 1200 10 Hz LGL 10 Hz LGR 10 Hz LGR 10 Hz GLe 10 Hz GLe 10 Hz GLo 10 Hz GLo 1400 1400 00.050.10 0.15 00.050.100.15 Voltage, V Voltage, V Fig. 7 Simulated measured EMT signal for BHA transmitted signal at 10 Hz frequency set at depths ranging from 200 to 1370 m in a homoge- neous formation and b layered earth model EMT comparison EMT comparison 0 0 (a) (b) 200 200 400 400 600 600 800 800 1000 1000 5 Hz directout 5 Hz directout 1200 1200 5 Hz LGL 5 Hz LGL 5 Hz LGR 5 Hz LGR 5 Hz GLe 5 Hz GLe 5 Hz GLo 5 Hz GLo 1400 1400 00.020.040.060.080.10 00.020.040.060.08 0.10 Voltage, V Voltage, V Fig. 8 Simulated measured EMT signal for BHA transmitted signal at 5 Hz frequency set at depths ranging from 200 to 1440 m in a homogene- ous formation and b layered earth model Furthermore, we compared the EMT response from the layers results in better representation of the recorded the isotropic model with the final response of transverse voltage at the different source depths. The simulated EMT anisotropy (Fig. 11). Introducing anisotropy into some of signal with anisotropic model gave a better representation 1 3 Depth, m Depth, m Depth, m Depth, m Petroleum Science (2021) 18:106–122 115 Earth resistivity distribution Ohm m Well Well data.txt (a) (b) Rho, Ohm m −1 0 1 2 10 10 10 10 100 200 1000 1000 r25 r4 −50 050 Radial−direction, m Fig. 9 Resistivity well logging data (a) and respective earth resistivity model (b) used in EMT signal simulation (units in Ω.m). Established baselines with deterministic linear trends along the resistivity log were used to represent average vertical resistivity within each of the repre- sentative units EMT signal in non−cased well EMT signal in cased well ID resistivity 0 0 (a) (b) (c) 200 200 200 400 400 400 600 600 600 800 800 Mag low Mag low Mag high Mag high 10 Hz LGL 10 Hz LGL 1000 1000 1000 10 Hz LGR 10 Hz LGR 10 Hz GLe 10 Hz GLe 10 Hz GLo 10 Hz GLo 1200 1200 1200 0 1 2 0 0.005 0.010 0.015 0.020 0 0.005 0.010 0.015 0.020 10 10 10 Voltage, V Voltage, V Resistivity, Ω·m Fig. 10 Comparison of measured EMT signal (Mag high—maximum value in black, and Mag low—minimum value in red) with simulated EMT signal for BHA transmitted signal at 10 Hz frequency set at depths ranging from 150 to 1080 m in; a non-cased well, b cased well, and c 1D representation of the transverse resistivity model. LGL, LGR, GLe and GLo are the integral results obtained from using Legendre–Gauss– Lobatto, Legendre–Gauss–Radau, Gauss–Lobatto and Gauss–Legendre quadrature techniques of the field data between the depths of 150–280  m and is required, especially when the order of noise magnitude is 400–1080 m. The similarity between the measured voltage close to that of the expected signal. and the simulated voltage plots shows that these techniques However, the uncertainties between the earth model used can be used as an effective method for the calculation of and the resistivity log, as well as other cultural and drilling surface recorded voltage in cases where high-order accuracy conditions, will lead to discrepancies between numerical 1 3 Depth, m Depth, m Depth, m z−direction, m Depth, m 116 Petroleum Science (2021) 18:106–122 EMT signal in cased well EMT signal in cased well 1D Resistivity 0 0 0 (a) (b) (c) 200 200 400 400 600 600 800 800 800 Mag low Mag low Mag high Mag high 10 Hz LGL 10 Hz LGL 1000 1000 10 Hz LGR 10 Hz LGR 10 Hz GLe 10 Hz GLe 10 Hz GLo 10 Hz GLo 1200 1200 1200 0 1 2 00.005 0.010 0.015 0.020 00.005 0.010 0.015 0.020 10 10 10 Voltage, V Voltage, V Resistivity, Ω.m Fig. 11 Comparison of simulated EMT signal from the isotropic model with the result from the anisotropic model. a The result from the iso- tropic model, b result from the anisotropic model and c 1D representation of the isotropic resistivity model simulation results and actual measurements. For example, surface measured voltage of 0.11 mV, with a dipole distance the resistivity distribution near the surface is subject to of 100 m. Therefore, the development of hybrid drill string larger variation due to moisture; and the mud resistivity in material with low conductivity will be favorable in the appli- borehole may change with depth. Although the formation cation of electromagnetic telemetry in general. resistivity is considered as anisotropic due to successive Lastly, we considered a two-layer earth model where part intercalation of resistive and conductive earth material in of the first layer has anisotropic property. Given that (1) a each purported layer, the value of the anisotropic factor is at broad range of value of anisotropy factor from 0.5 to about best an estimate. Therefore, later in the study, we considered 30 could exist in a sedimentary environment, though the the effect of change in the angle of orientation of the axis of values are usually within 1 and 5, and (2) an increase in symmetry of the conductivity principal axis on the surface anisotropy factor represents either decrease in the transverse measured EMT signal. conductivity or increase in longitudinal conductivity, and In addition, we considered the effect of change in conduc- assuming that the model is radially isotropic, changes in tivity of the drill string on both the current distribution along surface recorded voltage with change in the dip of the prin- the string. We measured the EMT signal, using a three-layer cipal axis of subsurface material property (conductivity), earth model, with background resistivity of 10 Ω.m and with respect to a given anisotropy value, are considered. middle layer resistivity value of 2 Ω.m. The conductivity Here, we consider an axisymmetric model in which the resis- of the casing was set as 5 × 10 S/m. The source location tivity parameters do not change in the phi-direction. The was set at a depth of 1480 m, 280 m ahead of the casing and model properties may be described with reference to either 20 m behind the drill bit. The confined conductive layer was a cylindrical (measurement) coordinate frame involving the set between the depths of 440 m and 580 m. The obtained tensor elements ρ , ρ , ρ or a principal axis frame involv- rr rz zz results are shown in Fig. 12. The result shows an initial non- ing the components ρ , ρ and θ . Here, ρ  is the longitudi- h t h appreciable increase in the magnitude of the current flowing nal resistivity, ρ  is the transverse resistivity, and θ° is the along the drill string with a decrease in the conductivity angle of the symmetry axis relative to the vertical, which is of the drill string. However, the current flow via the drill physically meaningful since eigenvectors are aligned with string increases rapidly as the conductivity value reduces to the natural rock frame. The angle of the symmetry axis is 1 × 10 S/m. A plot of voltage recorded at the surface shows illustrated schematically in Fig. 13a. an initial relatively constant voltage value with a decrease By rearranging the orthogonal anisotropic resistivity in the conductivity of the drill string. However, when the model parameters (ρ  and ρ ), we may introduce an alterna- h t conductivity value of the drill string is reduced from 1 × 10 tive form of description for the media, namely the mean to 1 × 10 S/m, the simulated EMT signal increases expo- resistivity ρ  and the coefficient of anisotropy λ, given by: nentially. The drill string conductivity values of 1 × 10 S/m √ � give the highest voltage of 2.53 mV at the surface. In con-  =   and  = . (16) m t h trast, the conductivity value of 1 × 10 S/m gives the lowest 1 3 Depth, m Depth, m Depth, m Axis of symmetry Petroleum Science (2021) 18:106–122 117 Drill string conductivity comparison EMT comparison −2 (a) (b) 1e4, S/m 10 Hz LGL 1e5, S/m 10 Hz LGR 1e6, S/m 10 Hz GLe 1e7, S/m 10 Hz GLo −3 −4 −8 −6 −4 −4 −2 0 10 10 10 Current, log (A) Drill string conductivity, S/m Fig. 12 Comparison of simulated EMT signal with a change in the conductivity of the drill string; a current density along the drill string and b measured voltage at the surface 1D Resistivity Representation (a) (b) 0 1 2 10 10 10 Resistivity, Ohm m Fig. 13 a Simplified diagram of anisotropic 2-D TTI media, showing the axis of symmetry and principal resistivities. b 1D representation of the resistivity model. The geographic coordinate frame is r, z, while the principal axis frame (or natural rock frame) is tilted away from this direc- tion. The directions of the principal axes are the corresponding eigenvectors The quantity ρ  is the geometric mean of the longitudi- parameters and the orientation of the axis of symmetry and nal and transverse resistivities. It is equal to surface meas- considering its effect on electromagnetic telemetry signal. ured apparent resistivity. In this section, we focused on As an instructive preliminary investigation, we computed incorporating the coefficient of anisotropy with the model current distribution along the drill string and the surface 1 3 Depth, m Depth, m Voltage, V 118 Petroleum Science (2021) 18:106–122 measured signal, at a dipole length of 100 m, for the dif- To illustrate the differences with the change in the dip of ferent axes of symmetry, θ (°), and fixed value of ρ , λ. The the principal axis, the currents have been plotted. Since the current distribution along the drill string was calculated for horizontal conductivity is fixed in this study, plots of the cur - a two-layered anisotropic model with an anisotropic value rent distribution along the drill string exhibit the same pat- of λ = 5, ρ of 1 Ω.m and 10 Ω.m for the top and lower lay- terns as those of surface measured voltage. The plot of the ers, respectively. The anisotropic column was between the current distribution along the drill string (Fig. 14b) shows depths of 220 m and 440 m, and θ (°) varies from 2° to 90°. an increase in the magnitude of the current flowing along The simulated surface measurement related to θ = 90° is the drill string with an increase in the angle of orientation five orders of magnitude higher than  θ = 2° (Table 1). The of the axis of symmetry. At low angle (θ = 30° and 15°), existence of resistive anisotropic medium confined by con- the magnitude of the current distribution along the drill ductive isotropic materials (great simplifications arise in string decreases very fast within the anisotropic layer, and the decoupling of the anisotropy parameters) results in a the z-plane becomes almost entirely vertical (Figs. 13 and significant reduction in the magnitude of surface recorded 14b). When θ is between 45° and 75°, slight variation in the EMT signal. The rose diagram in Fig. 13a shows that the magnitude of the current distribution along the drill string sensitivities to change in θ angle increase with a decrease in was observed, and at θ > 75° no significant change in current θ and very rapidly for angles below 60°. The source of the density was recorded. In comparison with the result from amplitude difference in the sensitivity stems from the dif- the isotropic medium, the value of current flowing along the ference in the model parameters θ (°), leading to an increase drill string with θ = 45° has almost the same value as that of in the value of ρ   (Ω m) as θ increases, and its value tends the isotropic medium, while for θ > 45°, the current density toward the value of the major principal axis ρ  (Ω m). is higher. (The horizontal component is dominant.) Table 1 Surface measure voltage for electromagnetic telemetry Dip (°) 2 15 30 45 60 75 85 90 Voltage (V) 6.0E-08 0.99E-05 0.000068 0.00017 0.00029 0.00039 0.00043 0.00044 Current along drill string (a) (b) 210 330 15 deg 30 deg 45 deg 240 60 deg 75 deg 90 deg Isotropic −6 −4 −2 0 Current log(A) Fig. 14 Comparison of simulated EMT signal with the change in the angle of the principal axis of symmetry of the subsurface property; a rose diagram showing the change in EMT signal magnitude with a change in the angle of the principal axis of symmetry and b current density along the drill string at different angles of principal axis of symmetry 1 3 Depth, m Petroleum Science (2021) 18:106–122 119 otherwise in a credit line to the material. If material is not included in Although EM telemetry is used in subsurface wireless the article’s Creative Commons licence and your intended use is not communication, it is not as a quantitative well logging tool, permitted by statutory regulation or exceeds the permitted use, you will a substantial level of accuracy should be of priority for mod- need to obtain permission directly from the copyright holder. To view a eling of EM telemetry. This study shows the complexity in copy of this licence, visit http://creativ ecommons .or g/licenses/b y/4.0/. an accurate simulation of EMT measured signal and the extensive effect of anisotropy which could be of significant influence even if for a short section. Hence, a combination Appendix 1 of mud-pulse and EM telemetry could be useful in ensuring a continuous transmission of measurable signal with high In this section, the anisotropic parameters defining a magnitude. homogeneous TTI medium  ,  or  , λ , azimut h  and h t m the dip angle θ were used in deriving the conductivity values for TTI medium in a cylindrical coordinate sys- 5 Conclusion tem, where λ is the anisotropy factor,  represents the longitudinal conductivity,  represents the transverse con- In this paper, we presented an efficient FDM scheme for ductivity and  is the geometric mean of the principal simulations of electromagnetic telemetry in the axisym- conductivities and is defined by the dip and resultant of metric model. In terms of application, homogeneous and the longitudinal conductivity and transverse conductivity. layered earth models were used as examples. The current Given the diagonalized conductivity matrix in Cartesian density along the drill string and surface measured voltages coordinate which represents the three eigenvalues σ , σ , 1 2 were solved to understand the effect of change in drill string σ as conductivity and angle of axisymmetry of the earth physical ⎛  00 ⎞ ⎛  00 ⎞ property on electromagnetic telemetry. The algorithm and xx 1 ⎜ ⎟ ⎜ ⎟ = 0  0 = 0  0 , (17) model used straightforwardly assimilates the 2D anisotropy T yy 2 ⎜ ⎟ ⎜ ⎟ 00  00 ⎝ ⎠ ⎝ ⎠ and allows the computation of the anisotropy effect as an zz 3 update field. Also, we considered the Gaussian quadrature we define the relationship between the geometric mean, techniques in the effective estimation of measured EMT sig- the principal conductivities and coefficient of anisotropy nal. A field case was also given showing that the calculated as  =  =  ,  =  ,  =   and  =   for xx yy h zz t m h t h t results by the proposed method agree with the measured transverse anisotropic (VTI) medium. Compared to the data. Lastly, the effect of change in axis of symmetry of VTI medium, the conductivity tensor of TTI medium in earth layering and occurrence of confined resistive layers the Cartesian coordinate system can be expressed as should be taken into account in considering the effectiveness of EM telemetry. In general, we found that the accuracy of ⎛  00 ⎞ the 2D FDM scheme presented in this study is suitable for −1 ⎜ ⎟ = R 0  0 R, (18) T 2 ⎜ ⎟ practical applications and that the computational efficiency ⎝ ⎠ is much higher than for 3D modeling. It is believed that simulations in this paper will facilitate the feasibility study where of an electromagnetic system for real field jobs. cos  cos  sin  cos  − sin ⎛ ⎞ Acknowledgments This project was supported by the Strategic ⎜ ⎟ R = − sin  cos  0 . ⎜ ⎟ Priority Research Program of Chinese Academy of Sciences (No. cos  sin  sin  sin  cos ⎝ ⎠ XDA140500001). The anisotropic medium can thus be described by six Compliance with ethical standards independent components writing in Cartesian coordinate or recording frame as: Conflict of interest On behalf of all the authors, the corresponding au- thor states that there is no conflict of interest. 2 2 2 2 2 cos  cos  +  sin  +  sin  cos ⎛ ⎞ ⎛ ⎞ xx h h t � � ⎜ ⎟ ⎜ ⎟ 0.5 − +  sin  sin 2 xy Open Access This article is licensed under a Creative Commons Attri- h t � � ⎜ ⎟ ⎜ ⎟ bution 4.0 International License, which permits use, sharing, adapta- 0.5 − +  cos  sin 2 xz h t ⎜ ⎟ ⎜ ⎟ � � 2 2 tion, distribution and reproduction in any medium or format, as long ⎜  ⎟ ⎜ − +  sin  sin  +  ⎟ yy h t h � � as you give appropriate credit to the original author(s) and the source, ⎜ ⎟ ⎜ ⎟ 0.5 − +  sin  sin 2 yz h t provide a link to the Creative Commons licence, and indicate if changes ⎜ ⎟ ⎜ ⎟ sin  +  cos ⎝ ⎠ ⎝ ⎠ were made. The images or other third party material in this article are zz h t included in the article’s Creative Commons licence, unless indicated (19) where  =  =  and  =  . 1 2 h 3 t 1 3 120 Petroleum Science (2021) 18:106–122 z z However, for the general 2/2.5-D case where  = 0 , t hat J V = E z = z. (26) received z is, there is no dependency on the azimuth, we obtain: z z 0 0 cos  +  sin ⎛ xx ⎞ ⎛ ⎞ h t However, in this study, we considered only the conven- � � ⎜  ⎟ ⎜ ⎟ 0.5 − +  sin 2 xz h t tional surface measurement of EMT signal, where r is the = . 0 (20) ⎜ ⎟ ⎜ ⎟ yy h position on the blowout preventer and r represents second ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ sin  +  cos zz h t position tens of meters to over a hundred meters from r . To solve the integrand, we adopted Gaussian quadrature To translate these results into a cylindrical coordinate techniques using weighing function through which integrable system with situations where the axis of symmetry is tilted singularities are removed. Conventionally, the domain of inte- along the dip, we used the following expression: gration is taken as [a,b], whereas in Gaussian quadrature, the � � 2 2 domain is reduced to [−1,1]. The general Gaussian quadrature ⎛  cos  +  sin  0 0.5 − +  sin 2 ⎞ h t h t −1 ⎜ ⎟ can be represented as = Rc 0  0 Tc h � � ⎜ ⎟ 2 2 0.5 − +  sin 2 0  sin  +  cos ⎝ ⎠ h t h t 1 1 n−1 (21) f (x)dx = W(x)g(x)dx ≈ w g x . (27) i i where  is the conductivity tensor in the cylindrical coor- Tc i=1 −1 −1 −1 dinate system and Rc is the conversion matrix. The matrix In this study, we used the explicit Legendre–Gauss–Lobatto expression for the conversion matrix is given as (LGL), Gauss–Lobatto (GLo), Legendre–Gauss–Radau (LGR) ⎛ cos  sin  0 ⎞ and Gauss–Legendre (GLe) quadrature techniques, which con- −1 ⎜ ⎟ Rc = − sin  cos  0 . (22) verge accurately to estimate the potential across two measuring ⎜ ⎟ 0 01 ⎝ ⎠ points (−1,1). The integral form of the quadrature techniques is given as Finally, the TTI conductivity tensor is given as � � n−1 2 2 cos   cos  +  sin ⎛ rr ⎞ ⎛ h t ⎞ � � � � GLo - f (x)dx ≈ (f (−1)+f (1))+ w g x , i i ⎜  ⎟ ⎜ cos  0.5 − +  sin 2 ⎟ n(n − 1) rz h t i=2 = . (23) −1 ⎜ ⎟ ⎜ ⎟ cos ⎜ ⎟ ⎜ ⎟ (28) ⎝  ⎠ ⎝  sin  +  cos  ⎠ zz h t Note that there is no dependence on the axial direction LGL - f (x)dx ≈ w g x , (29) i i with  = 0 . 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EMT simulation and effect of TTI anisotropic media in EMT signal

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Springer Journals
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1672-5107
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1995-8226
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10.1007/s12182-020-00523-0
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Abstract

An axisymmetric finite difference method is employed for the simulations of electromagnetic telemetry in the homogeneous and layered underground formation. In this method, we defined the anisotropy property using extensive 2D conductivity tensor and solved it in the transverse magnetic mode. Significant simplification arises in the decoupling of the anisotropic parameter. The developed method is cost-efficient, more straightforward in modeling anisotropic media, and easy to be implemented. In addition, we solved the integral operation in the estimation of measured surface voltage using Gaussian quadrature technique. We performed a series of numerical modeling of EM telemetry signals in both isotropic and anisotropic models. Experiment with 2D tilt transverse isotropic media characterized by the tilt axis and anisotropy parameters shows an increase in the EMT signal with an increase in the angle of tilt of the principal axis for a moderate coefficient of anisotropy. We show that the effect of the tilt of the subsurface medium can be observed with sufficient accuracy and that it is an order of magnitude of 5 over the tilt of 90 degrees. Lastly, consistent results with existing field data were obtained by employing the Gaussian quadrature rule for the computation of surface measured signal. Keywords Electromagnetic telemetry · Finite difference method · Anisotropy · Gaussian quadrature 1 Introduction and the return of instructions from the rig to the borehole. Borehole data transmission could be useful in geothermal In borehole drilling, effective geosteering and accurate well exploration, mineral exploration but are mostly used by the landing require quick access to point scale data about for- oil and gas industry. Borehole telemetry has been an essen- mation being drilled, bit trajectory, well condition and lots tial part of industrial drilling for oil and gas resources in more. Borehole telemetry is defined as the two-way transfer recent decades, especially in an unconventional reservoir of such data from the bottom hole assembly (BHA) to the rig and during horizontal drilling. It provides considerable time savings and minimizes the risks associated with geosteering and accurate well landing. Edited by Jie Hao This technique has been successfully applied in the drilling of several oil wells (Schnitger and Macpherson 2009; Baker * Olalekan Fayemi Hughes 2014). Telemetry methods can be classified based fayemiolalekan@mail.iggcas.ac.cn on the physical property considered in relation to the data Qing-Yun Di propagation, such as mud pulse telemetry, electromagnetic qydi@mail.iggcas.ac.cn telemetry and acoustic telemetry or combination of different Key Laboratory of Shale Gas and Geoengineering, Institute techniques, e.g., acoustic–electromagnetic technique. of Geology and Geophysics, Chinese Academy of Sciences, Electromagnetic telemetry (EMT) uses transmitted elec- Beijing 100029, China tromagnetic waves from bottom hole assembly (BHA) to Frontier Technology and Equipment Development Center the surface through the drill string, well and adjacent forma- for Deep Resources Exploration, Institute of Geology tions as a means of data transfer. Borehole telemetry aids in and Geophysics, Chinese Academy of Sciences, Beijing 100029, China accurate core point access to formation properties acquired close to the drill bit by logging while drilling (LWD) in Innovation Academy for Earth Science, Chinese Academy of Sciences, Beijing 100029, China real time, to fully assist in timely identification of zones of interest. EMT measurements can be acquired continuously College of Earth and Planetary Sciences, University of Chinese Academy of Sciences, Beijing 100049, China Vol:.(1234567890) 1 3 Petroleum Science (2021) 18:106–122 107 without the risk of data unavailability or insufficiency due casing on the current distribution at specified distances from to challenges with drilling fluid conditions as observed in the source and measured voltage at the surface. For mode- mud pulse telemetry. However, electromagnetic telemetry ling of an electromagnetic telemetry system in an anisotropic is profoundly affected by formation properties. Therefore, environment, and considering the accuracy requirement for effective simulation of the EMT response requires the con- practical application and the implemented complexity of the sideration of the effect of formation anisotropy on the cur - schemes, we intend to apply FDM to model EM propagation rent distribution pattern and, in turn, the voltage measured tools in axisymmetric formation with anisotropy. FDM can at the surface with respect to the source location. take all the features in an electromagnetic telemetry system, In EM study, formation anisotropy causes currents to such as drill pipes, drilling u fl id in the borehole and multiple depart from the direction of an exciting electric field within layers earth formation into consideration. the formation of interest (Lu et al. 2002). Therefore, accurate In this paper, we developed a scheme to model the EMT estimation of EMT signals at the rig floor can be marred response in a 2D underground formation by employing a by the effect of anisotropy on acquired data. Anisotropy is FD algorithm in the cylindrical coordinate system, to make a phenomenon that describes the variation in earth physi- use of the 2D structure to reduce the number of unknowns. cal properties as it occurs on various scales in a variety of geophysical applications. The measured earth properties could differ along the different measurement axes. Previous 2 Methodology applications of electromagnetic telemetry include marine controlled-source electromagnetic (CSEM) studies, reservoir 2.1 Isotropic medium characterization, SeaBed Logging (SBL), etc. (Brown et al. 2012; Cuevas 2019; He et al. 2019; Constable et al. 2019; Consider a homogenous and isotropic medium which is fully Li et al. 2019). For several decades, studies have been car- characterized by the constant parameters for magnetic per- ried out on electrical anisotropy as part of induction logging meability μ and electric permittivity ɛ. Next, assume that in efforts to better quantify hydrocarbon reservoir potential the source function and all the field variables exhibit the (Moran and Gianzero 1979; Anderson et al. 1998; Wilson j(t−kr) common time dependence e , where ω is the angular 2015). In reservoir evaluation, electrical anisotropy measure- frequency and k is the wave vector. Then, in terms of the ments may improve the estimate of hydrocarbons in place. fundamental field vectors, the electric intensity E and the The conductivity tensor that relates the induced current to magnetic intensity H are continuous and possess continuous the applied electric field has three principal values, each derivatives at ordinary points, such that corresponding to a direction of maximal conductivity. The resistivity of shale mostly controls the horizontal resistivity ∇× E =−iH − J (1) of the laminae, whereas the resistivity of hydrocarbon- bearing sands dictates the vertical resistivity  (Klein ∇× H =+ik E + J (2) 1993; Tabanou et al. 1999). Electrical anisotropy is caused by many different mechanisms such as conductive fluids fill- where k = i  − i ,  is the dielectric permittivity, and ing in fractured formations with voids, and laminated depo- s s J and J are the electric and magnetic source terms, respec- e m sition of substances of different electrical properties (e.g., tively.  in Eq. (2) is the electrical conductivity tensor, and sands and clays). In general, electrical anisotropy arises from it is defined as follows: the presence of sedimentary structures whose length scale is far smaller than the resolving power of measuring equip- ⎛  00 ⎞ ⎜ ⎟ ment, for example, shales and intercalation of thin layer of = 0  0 , (3) ⎜ ⎟ shale with sandstone are common phenomena resulting in ⎝ ⎠ formation anisotropy (Klein et al. 1995). The effect of anisotropy on EMT data can be investigated where  ,  and  are principle conductivities. Furthermore, considering a cylindrical coordinate system, via forward modeling using the finite difference method (FDM), the finite element method, etc. In this study, we con- an electromagnetic telemetry system is characterized by EM signals propagated using an insulating gap source on the drill sidered how formation anisotropy affects the current distri- bution pattern within the drill string and, in turn, the voltage string surrounded by a uniform vertical column of either conductive or resistive drilling fluid, and an outer column measured at the surface with respect to the source location during EMT. Emphasis was placed on investigating the of either layered formation and/or borehole casing (Fig. 1). The system, as defined radially outward from the center of effect of change in the orientation of the axis of symmetry of formation property in an anisotropic medium. We pay no the borehole, is solved as an axisymmetric and transverse magnetic problem. With the input voltage at the gap given attention to fluid invasion, but we consider the effect of the 1 3 108 Petroleum Science (2021) 18:106–122 2.2 2D Anisotropy Considering an electromagnetic telemetry system as Mud described above and displayed in Fig.  1, the governing Adjacent bed equation for EM propagation in full anisotropic media with conductivity tensor  in the cylindrical coordinate system is given as h h m D E − D E =−iH − J z  r z r h h m D E − D E =−iH − J r z z r  (6) h h m D E − D E + =−iH − J r z r  z r r h h e D H − D H = k E − k E − k E − J z  rr r r  rz z z r h h e D H − D H = k E − k E − k E − J r z r r   z z z r  (7) h h e D H − D H + = k E − k E − k E − J . r zr r z  zz z r  z r r Source Even though 3D forward modeling (Wang and Fang 2001; Weiss and Newman 2002; Davydycheva et al. 2003), combining high-performance computing, is available both in academic researches and in commercial software, full 3D methods are still prohibitively expensive for fast for- ward modeling. An intermediate solution between the 1D Fig. 1 Diagram of a borehole with a telemetry system (modified after Ribeiro and Carrasquilla 2014). (Jiang et al. 2017; Anderson 2001) model and fully 3D model is the 2D model which has uniform property along the strike axis and is less compu- tationally intensive. and the magnetic current circulating the drill string specie fi d In this study, a representative formation with anticlinal at the source position, the governing equation for transmis- structure, which shows uniformity in the phi-direction but sion of the radial magnetic field H is given as layering in the “r” and “z” plane, is considered. Generally, the transmitter of the EM tool consists of a very thin insu- 1   1 2 s H + + k  H =−J , lating gap along the drill string, generating controlled volt- 0 m k  z k z age difference along the conductive drill string. The source (4) transmission of the EM tool consists of radially transmit- where J = iM , and M is the magnetic current den- ted magnetic current. So the electric current transmission sity. The isotropic medium is characterized by a “radial is majorly in the vertical direction along the drill string conductivity”  which has the same value as the “vertical (which serves like air duct in guided waves). Even consid- conductivity”  at any point within the given layer, such z ering geological deformation, it is reasonable to assume that  =  . Relative permittivity value of 1 was used in z that within the propagation range of a typical EMT system this study. in a vertical or pseudo-vertical well (centimeters to tens The current flowing along the drill string is given as of meters), the geological formation is an asymmetrical J = 2H (Chen et al. 2017); therefore, Eq. (4) becomes 2D structure, which exhibits uniform EM properties in phi direction. A depth slice of formation with anticlinal 1  1   1 z 0 structure, which shows uniformity in the phi-direction but J + +  J =−J . z r z m 2   k  z k z inhomogeneity in the r-z plane, is shown in Fig. 2 using a (5) coordinate system image. Therefore, consider an axisymmetric problem (that is, the physical properties are radially isotropic) for an ani- sotropic media, whose conductivity tensor is expressed as in a cylindrical coordinate system (r, , z) as 1 3 Invaded zone Transition zone Uninvaded zone Petroleum Science (2021) 18:106–122 109 ⎛  0  ⎞ ϕ, ° rr rz ⎜ ⎟ = 0  0 . (8) r, m ⎜ ⎟ ⎝ ⎠ zr zz The medium is characterized by a “radial conductivity” which is constant in the horizontal directions and var- rr Z, m ies in the vertical direction for “vertical conductivity”  , zz such that  ≠  . The values of the subsurface conductivity rr zz tensor used in this study were defined using the anisotropy factor λ , longitudinal conductivity  , transverse conductiv- -8 -6 -4 -2 0 24 ity  and dip angle θ. (See “Appendix 1” for details.) The principal conductivity  is defined by the dip and resultant of the longitudinal conductivity and transverse conductivity. For simplicity, the uniform dielectric constant value of 1 was used in this study. Therefore, Eqs. (6) and (7) are reduced to h m −D E =−iH − J z r -8 -6 -4 -2 h h m 4 6 8 D E − D E =−iH − J r z z r (9) h m D E + =−iH − J r z 450 and h e −D H = k E − k E − J rr r rz z z r -8 -6 -4 -2 0 h h e 4 D H − D H =−k E − J 6 r z 8 z r (10) h e D H + = k E − k E − J . zr r zz z r z Thus, as for 2D (TM) with 2D anisotropic problems, the curl–curl operator cannot simply be replaced by the Lapla- cian operator. However, the simplified off-diagonal elements -8 -6 -4 -2 of the conductivity tensor make it possible to couple these equations into pure TM modes as in the 2D isotropic case. Therefore, solving the transverse magnetic (TM) problem, the reduced 2D equation is given as 1050 h −D H = k E − k E rr r rz z h h m D E − D E =−iH − J r z z r -8 (11) -6 -4 -2 6 8 D H + = k E − k E . zr r zz z 50 Ω·m 300 Ω·m 10 Ω·m By solving for E and E in the first and last terms in r z Fig. 2 Diagram of depth slice from an axisymmetric anticlinal struc- Eq. (11), respectively, and substituting the obtained values ture. Each depth slice shows different layers within a given radial dis- in the second term, the following is obtained: tance r = 10 m from the center (0). The radial distance is projected on the axis for  equal to 0°and 180°. The change in sign marks opposite sides of the radial length from negative to positive −k k UH − k k UJ =−D k H − D k H rr zz  rr zz z zz  r rr (12) e e e +D k H + D k k E − D k k E rr  rr zr r zz rz z r r z where U = i. Eliminating k E and k E from the equa- rz z zz z tions for E and E , respectively, and substituting the result- r z ing equations into Eq. (12), one can derive a system for esti- mating the radial magnetic field 1 3 110 Petroleum Science (2021) 18:106–122 D k U + D k U− D aup + D bup − D arup+ ⎛ z zz r rr ⎞ ⎛ ⎞ z r ⎜ ⎟ ⎜ ⎟ e e h m D ⋅ (AFZ ∗ crup)+ D k U − U + D D k U+ ⎜ ⎟ ⎜ zr ⎟ rr rz H = UJ r z r m (13) H = UJ (15) ⎜ ⎟ ⎜ ⎟ D ⋅ (AFZ ∗ cup)− ⎜ ⎟ ⎜ rz ⎟ e h e D D k U − D k U ⎝ zr rz ⎠ r z z ⎜ ⎟ D ⋅ (AFZ ∗ crup) − U ⎝ ⎠ where U = k k − k . rr zz 1 1 rz where aup = k U, bup = k U, arup = k U, crup = k U, zz rr rr rz We note in Eq. (13) that the upper term is identical to and r r and cup = k U. rz represents the 2D equation for isotropic medium, while the The results of this investigation were assessed by compar- lower terms are indicative of the anisotropic effect. Consider - ing the plots of the current distribution along the drill string ing Eq. (13), the upper terms can be considered as the primary and the measured EMT signal on the surface. To obtain the term, while the lower term can be considered for anisotropy, nodal values, the radial magnetic field value was interpolated the secondary term which can be updated for varying anisot- as the dot product of the face calculated radial magnetic field ropy conditions and effect. with the AFZ transpose matrix. Furthermore, we approximate the ordinary differential equations in terms of a series of numerical operators represent- ing higher-order forward and backward difference approxima- tions and averaging operators; the higher-order central differ - 3 Modeling ence approximation can be obtained by combining forward and backward difference operators. The above equation can be dis- In this study, we used these standard model parameters cretized and solved by the conventional first-order finite differ - unless specified otherwise: frequency range of 5–10 Hz, drill ence method. However, for better efficiency in this paper, we string inner radius of 0.25 m, while the outer radius is 0.3 m, used third-order forward and backward difference approxima- and the casing inner and outer radius of 0.4 m and 0.45 m, tions, and fifth-order central difference approximation for the respectively. Uniform conductivity value of 5 × 10 S/m was finite difference method. The discrete approximation of Eqs. used for the casing and drill string, while the drilling fluid (5) to (13) is formulated on a staggered 2D grid. In this for- conductivity was set as 1 S/m. The relative magnetic per- mulation, the electric components were located on cell faces, meability of the casing, borehole fluid and the formation is while the magnetic components were located on cell nodes. assumed to be unity. For the isotropic part, the above implementation is direct and We first consider a vertical well in a homogeneous under - requires N faces and N + 2 nodes for every vertical discrete ground formation with a resistivity value of 10 Ω.m. Solid layer/discretization. However, for the anisotropic term, a slight drill string with a radius of 0.25 m and casing with an inner complication arises from implementing the update parameters and outer radius of 0.35 m and 0.4 m, respectively, were used from the conductivity functions. In our implementation of this in this model. The total length of the drill string and the cas- term, we averaged the update parameters onto the nodes/cell ing is 1200 m and 800 m, respectively. The voltage source faces using a four-plane average matrix (Guo et al. 2018); with a value of 1 v and propagating frequency of 10 Hz were set 15 m away from the base of the drill string. To assess the ⎡ 1∕41∕4 m 1∕41∕40 0 ⎤ accuracy of the FD code, we compared the current distribu- ⎢ ⎥ 01∕41∕4 m 1∕4 m 0 ⎢ ⎥ tion along the drill string obtained from the FD method with AFZ = 0 01∕41∕4 mm 0 ⎢ ⎥ (14) the one obtained from the numerical simulation software ⎢ ⎥ n nn 0 nn n (COMSOL vs. 5.3a 2017), which is designed based on the ⎢ ⎥ 0 00 m 1∕4 m 1∕4 ⎣ ⎦ finite element method (Fig.  3). A conventional decrease in the amplitude of current along to ensure continuity. the drill string away from the source is recorded. Also, a With mapping operators and mapped model parameters, decrease in amplitude is recorded across the 800 m depth discretizing Eqs. (13) and (14) term by term, the complete mark due to the presence of casing in the vertical well. We matrix form for the magnetic phi-components on a staggered can see that current flow along the drill pipe calculated by grid can then be represented as both the FD method and the FEM method from COMSOL suite agrees relatively well with the exception of the depths below 1000 m where the FEM data show no definite trend and are less fitting. We further consider a vertical well in a homogeneous underground formation with a resistivity of 10 Ω.m, and a two-layered earth model with resistivity 1 Ω.m and 10 Ω.m, 1 3 Petroleum Science (2021) 18:106–122 111 Current along drill string respectively. The length of the drill string is constant, while the length of the borehole casing varied between 800 and 920 m depth, for both the homogeneous formation and two- layer earth model, respectively. The diameters of the drill pipe and the borehole are 0.3 m and 0.45 m, respectively. The resistivity of the drilling fluid is set as 1 Ω.m, and the conductivity of both drilling pipe and borehole casing is assumed to be 5 × 10 S/m. The downhole transmitter with 1v voltage source is set at 50 m behind the drill bit. Figure 4a shows the simulated magnitude of the current flowing through the drill string with a transmitting frequency of 5 Hz. The magnitude of the current flowing through the drill string gradually decreases with an increase in dis- tance from the source; this reflects the change in current density with loss of current into the adjacent formations. Comparison of current distribution result for cased borehole (black) and non-cased borehole (red) shows a decrease in the magnitude of the current flowing along the drill string from the depth of 800 m due to the presence of steel casing. FD 1200 m In the case of layered medium (Fig. 4b), with transmitting Comsol frequency of 10 Hz, slight deflection (reduction in current −6 −4 −2 0 2 magnitude) due to borehole casing was recorded at a depth of 920 m. However, the most notable perturbation is the Current log(A) current loss in the adjacent conductive layer at a depth of 550 m in the cased well. Fig. 3 Comparison of amplitude of current flow along the drill string in a homogeneous formation obtained from the FD method and FEM The general feature one might observe from these figures method in COMSOL Multiphysics software is that the current distribution along the drill string is much stronger without casing than with casing. The disparity in Current along drill string−comparison Current along drill string−comparison 0 0 (a) (b) 500 500 1000 1000 Uncased Uncased Cased Cased 1500 1500 −8 −6 −4 −2 0 2 −8 −6 −4 −2 02 10 10 Current Log (A) Current Log (A) Fig. 4 Amplitude of current flow along the drill string in; a a homogeneous formation and b 2-layered earth model. The current flow, as indi- cated above, is attenuated in the presence of both conductive adjacent formation and well casing. These form the major challenges encountered in EM telemetry 1 3 Depth, m Depth, m Depth, m 112 Petroleum Science (2021) 18:106–122 the magnitude of the current distribution might be explained 4 Recorded voltage estimation by the fact that a single metal cylinder in a given model can support a guided wave theory (Stratton 1941). In other The recorded voltage at the surface can be obtained by cal- words, the current induced by the current/voltage source culating the integral of the electric field calculated from the at the insulating gap along the drill string flows majorly obtained current distribution. (See “Appendix 2”) In prac- vertically along the drill string. However, with the second tice, the integrand is seldom smooth. However, this issue highly conductive metal cylinder included, more current gets is addressed in Gaussian quadrature by using weighting attenuated by the adjacent steel, leading to a reduction in the function, which results in removal of integrable singulari- surface measured EMT signal. ties (Mahesh and Sucharitha 2018; Piqueras et al., 2019; As mentioned above, the current density will produce a Hassan et al. 2020). Therefore, for higher accuracy calcula- pattern of vertical distribution within the drill string and tion, we used the explicit Legendre–Gauss–Lobatto (LGL), near-vertical distribution along the inner annulus. When Legendre–Gauss–Radau (LGR), Gauss–Lobatto (GLo) and compared to the current flow in a cased well, a similar flow Gauss–Legendre (GLe) quadrature techniques, which con- pattern is expected for current within the drill string, while verge accurately to estimate the potential across two measur- the magnitude of current within the annulus is reduced ing points (−1,1). The integral form of the quadrature tech- because there is a current loss to the conductive casing. niques is given in “Appendix 2.” When using the Gaussian These flow patterns are shown in Fig.  5, representing the quadrature techniques, selecting an appropriate bandwidth current distributions in the vertical plane (r-z plane) for cases for a kernel density estimator is important. Several tech- with and without a steel casing. The difference between the niques have been used to select the smoothing parameter current distributions of these two cases can be seen clearly for reasonable density estimate. These are categorized as with the current being weak along the borehole direction either plug-in bandwidth estimators, which tend to select within the cased region. larger bandwidths and could produce over-smoothed results, classical estimators which on the other hand could produce under smoothed results when the smoothing problem is severe (Loader 1999), or nonparametric estimation (Mar- ron and Chung 2001; Chan et al. 2010; Golyandina et al. 2012), which considers a family of smooths with a broad 10 10 Current distribution log Al Current distribution og A 0 0 (a) (b) 0 0 200 200 −2 400 400 600 600 −4 800 800 −6 1000 1000 1200 1200 8 −8 1400 1400 −10 −10 1600 1600 −12 −12 1800 1800 2000 2000 −14 −14 01020304050 01020304050 Radial distance, m Radial distance, m Fig. 5 Current flow distribution in a homogeneous formation with; a non-cased well and b cased well 1 3 Depth, m Depth, m Petroleum Science (2021) 18:106–122 113 range of bandwidths, instead of a single estimated function. reduces as the rate of attenuation of the current flowing Noting that the goal is to decide on selecting the smoothing through the drill string increases within the conductive parameter purely from the data, we used the nonparametric layer. technique. Furthermore, we used an example of a field telemetry Figure 6 shows an example of voltage responses gener- data acquired in Shandong Province, China. The resistiv- ated from a mixture of a Gaussian variable using the homog- ity log and constructed earth resistivity model are shown in enous earth model and 10 Hz propagating frequency. The Fig. 9. The drilling fluid resistivity value of 1 Ω.m was used kernel density was estimated using n (number of data points in this case. The working frequency and output/input current observed from a realization of the random variables) that of the downhole source were 10 Hz and 1.4 A, respectively. varies from 2 to 150, thereby varying the bandwidths. The The diameter of the drill pipe was set as 0.35 m, while the wide range of smoothing considered allows a contrast of borehole diameter was set as 0.5 m. In the case of the cased estimated features and possible point of convergence. In well simulation, the casing diameter was set as 0.6 m. *Note comparison, the accuracy of the estimated values increases that the resistivity of the first layer was set as 30 Ω.m con- with an increase in the value of n and at a different rate for sidering the inconsistency in the resistivity log and the high most of the techniques. However, with n equal to 150, the value of the measured field data within this depth. Also, the different techniques converge to given stable values. coefficients of anisotropy values of 1.8, 0.6, 0.8, 2 and 1.5 Using n equal to 150, the comparison of voltage meas- were used to define the longitudinal conductivity values at ured at the surface using both homogeneous earth and depths [320 366] m, [367 440] m, [441 480] m, [481 520] the three-layer earth model, with background resistivity m and [667 760] m, respectively. The measured EMT signal of 10 Ω.m, middle layer resistivity value of 2 Ω.m, and was simulated at depths ranging from 150 to 1080 m. source location varying from 200 to 1350 m depth at an Figure 10 shows the comparison of the simulated sur- interval of 45 m, is shown in Figs. 7 and 8. The confined face recorded telemetry signal using the different integral conductive layer was set between the depths of 440  m techniques with the field-measured data. The obtained plots and 580 m. A small difference in the value of the meas- show that the simulated responses follow the same trend as ured voltage at the surface is obtained by comparing the the field measurements over a large depth range. The simu- results from Gauss quadrature algorithms with the direct lated EMT signal in a cased well (Fig. 10b) shows a more method between the depth of 200–300 m for transmitting pronounced representation of the low-magnitude field data frequency of 10 Hz, and 300 m to 700 m for transmitting with the exception of a significant difference between the frequency of 5 Hz. Below these depths, the results from depths of 300 m and 480 m, as observed in the result for the different techniques are similar. In general, an increase the non-cased well signal as well (Fig. 10a). In general, the in measured voltage at the surface is obtained between magnitude of the measured EMT signal simulated without the depths of 440 m and 600 m, as the source is located borehole casing is higher with the given model. The simu- within the conductive layer. As the source depth increases, lated EMT signal is a close match with high-magnitude field the magnitude of the EMT signal recorded at the surface data within the first 300 m and at subsequent depths. EMT signal N = 2 EMT signal N = 50 EMT signal N = 150 0 0 0 (a) (b) (c) 200 200 200 400 400 400 600 600 600 800 800 800 1000 1000 1000 10 Hz direct 10 Hz direct 10 Hz direct 10 Hz LGL 10 Hz LGL 10 Hz LGL 10 Hz LGR 10 Hz LGR 10 Hz LGR 1200 1200 1200 10 Hz GLe 10 Hz GLe 10 Hz GLe 10 Hz GLo 10 Hz GLo 10 Hz GLo 1400 1400 1400 0 0.05 0.10 0.15 0 0.05 0.10 0.15 0 0.05 0.10 0.15 Voltage, V Voltage, V Voltage, V Fig. 6 Estimated voltage responses with several bandwidths initiated by changing the number of data point value from 2 to 150 1 3 Depth, m Depth, m Depth, m 114 Petroleum Science (2021) 18:106–122 EMT comparison EMT comparison 0 0 (a) (b) 200 200 400 400 600 600 800 800 1000 1000 10 Hz direct 10 Hz direct 1200 10 Hz LGL 1200 10 Hz LGL 10 Hz LGR 10 Hz LGR 10 Hz GLe 10 Hz GLe 10 Hz GLo 10 Hz GLo 1400 1400 00.050.10 0.15 00.050.100.15 Voltage, V Voltage, V Fig. 7 Simulated measured EMT signal for BHA transmitted signal at 10 Hz frequency set at depths ranging from 200 to 1370 m in a homoge- neous formation and b layered earth model EMT comparison EMT comparison 0 0 (a) (b) 200 200 400 400 600 600 800 800 1000 1000 5 Hz directout 5 Hz directout 1200 1200 5 Hz LGL 5 Hz LGL 5 Hz LGR 5 Hz LGR 5 Hz GLe 5 Hz GLe 5 Hz GLo 5 Hz GLo 1400 1400 00.020.040.060.080.10 00.020.040.060.08 0.10 Voltage, V Voltage, V Fig. 8 Simulated measured EMT signal for BHA transmitted signal at 5 Hz frequency set at depths ranging from 200 to 1440 m in a homogene- ous formation and b layered earth model Furthermore, we compared the EMT response from the layers results in better representation of the recorded the isotropic model with the final response of transverse voltage at the different source depths. The simulated EMT anisotropy (Fig. 11). Introducing anisotropy into some of signal with anisotropic model gave a better representation 1 3 Depth, m Depth, m Depth, m Depth, m Petroleum Science (2021) 18:106–122 115 Earth resistivity distribution Ohm m Well Well data.txt (a) (b) Rho, Ohm m −1 0 1 2 10 10 10 10 100 200 1000 1000 r25 r4 −50 050 Radial−direction, m Fig. 9 Resistivity well logging data (a) and respective earth resistivity model (b) used in EMT signal simulation (units in Ω.m). Established baselines with deterministic linear trends along the resistivity log were used to represent average vertical resistivity within each of the repre- sentative units EMT signal in non−cased well EMT signal in cased well ID resistivity 0 0 (a) (b) (c) 200 200 200 400 400 400 600 600 600 800 800 Mag low Mag low Mag high Mag high 10 Hz LGL 10 Hz LGL 1000 1000 1000 10 Hz LGR 10 Hz LGR 10 Hz GLe 10 Hz GLe 10 Hz GLo 10 Hz GLo 1200 1200 1200 0 1 2 0 0.005 0.010 0.015 0.020 0 0.005 0.010 0.015 0.020 10 10 10 Voltage, V Voltage, V Resistivity, Ω·m Fig. 10 Comparison of measured EMT signal (Mag high—maximum value in black, and Mag low—minimum value in red) with simulated EMT signal for BHA transmitted signal at 10 Hz frequency set at depths ranging from 150 to 1080 m in; a non-cased well, b cased well, and c 1D representation of the transverse resistivity model. LGL, LGR, GLe and GLo are the integral results obtained from using Legendre–Gauss– Lobatto, Legendre–Gauss–Radau, Gauss–Lobatto and Gauss–Legendre quadrature techniques of the field data between the depths of 150–280  m and is required, especially when the order of noise magnitude is 400–1080 m. The similarity between the measured voltage close to that of the expected signal. and the simulated voltage plots shows that these techniques However, the uncertainties between the earth model used can be used as an effective method for the calculation of and the resistivity log, as well as other cultural and drilling surface recorded voltage in cases where high-order accuracy conditions, will lead to discrepancies between numerical 1 3 Depth, m Depth, m Depth, m z−direction, m Depth, m 116 Petroleum Science (2021) 18:106–122 EMT signal in cased well EMT signal in cased well 1D Resistivity 0 0 0 (a) (b) (c) 200 200 400 400 600 600 800 800 800 Mag low Mag low Mag high Mag high 10 Hz LGL 10 Hz LGL 1000 1000 10 Hz LGR 10 Hz LGR 10 Hz GLe 10 Hz GLe 10 Hz GLo 10 Hz GLo 1200 1200 1200 0 1 2 00.005 0.010 0.015 0.020 00.005 0.010 0.015 0.020 10 10 10 Voltage, V Voltage, V Resistivity, Ω.m Fig. 11 Comparison of simulated EMT signal from the isotropic model with the result from the anisotropic model. a The result from the iso- tropic model, b result from the anisotropic model and c 1D representation of the isotropic resistivity model simulation results and actual measurements. For example, surface measured voltage of 0.11 mV, with a dipole distance the resistivity distribution near the surface is subject to of 100 m. Therefore, the development of hybrid drill string larger variation due to moisture; and the mud resistivity in material with low conductivity will be favorable in the appli- borehole may change with depth. Although the formation cation of electromagnetic telemetry in general. resistivity is considered as anisotropic due to successive Lastly, we considered a two-layer earth model where part intercalation of resistive and conductive earth material in of the first layer has anisotropic property. Given that (1) a each purported layer, the value of the anisotropic factor is at broad range of value of anisotropy factor from 0.5 to about best an estimate. Therefore, later in the study, we considered 30 could exist in a sedimentary environment, though the the effect of change in the angle of orientation of the axis of values are usually within 1 and 5, and (2) an increase in symmetry of the conductivity principal axis on the surface anisotropy factor represents either decrease in the transverse measured EMT signal. conductivity or increase in longitudinal conductivity, and In addition, we considered the effect of change in conduc- assuming that the model is radially isotropic, changes in tivity of the drill string on both the current distribution along surface recorded voltage with change in the dip of the prin- the string. We measured the EMT signal, using a three-layer cipal axis of subsurface material property (conductivity), earth model, with background resistivity of 10 Ω.m and with respect to a given anisotropy value, are considered. middle layer resistivity value of 2 Ω.m. The conductivity Here, we consider an axisymmetric model in which the resis- of the casing was set as 5 × 10 S/m. The source location tivity parameters do not change in the phi-direction. The was set at a depth of 1480 m, 280 m ahead of the casing and model properties may be described with reference to either 20 m behind the drill bit. The confined conductive layer was a cylindrical (measurement) coordinate frame involving the set between the depths of 440 m and 580 m. The obtained tensor elements ρ , ρ , ρ or a principal axis frame involv- rr rz zz results are shown in Fig. 12. The result shows an initial non- ing the components ρ , ρ and θ . Here, ρ  is the longitudi- h t h appreciable increase in the magnitude of the current flowing nal resistivity, ρ  is the transverse resistivity, and θ° is the along the drill string with a decrease in the conductivity angle of the symmetry axis relative to the vertical, which is of the drill string. However, the current flow via the drill physically meaningful since eigenvectors are aligned with string increases rapidly as the conductivity value reduces to the natural rock frame. The angle of the symmetry axis is 1 × 10 S/m. A plot of voltage recorded at the surface shows illustrated schematically in Fig. 13a. an initial relatively constant voltage value with a decrease By rearranging the orthogonal anisotropic resistivity in the conductivity of the drill string. However, when the model parameters (ρ  and ρ ), we may introduce an alterna- h t conductivity value of the drill string is reduced from 1 × 10 tive form of description for the media, namely the mean to 1 × 10 S/m, the simulated EMT signal increases expo- resistivity ρ  and the coefficient of anisotropy λ, given by: nentially. The drill string conductivity values of 1 × 10 S/m √ � give the highest voltage of 2.53 mV at the surface. In con-  =   and  = . (16) m t h trast, the conductivity value of 1 × 10 S/m gives the lowest 1 3 Depth, m Depth, m Depth, m Axis of symmetry Petroleum Science (2021) 18:106–122 117 Drill string conductivity comparison EMT comparison −2 (a) (b) 1e4, S/m 10 Hz LGL 1e5, S/m 10 Hz LGR 1e6, S/m 10 Hz GLe 1e7, S/m 10 Hz GLo −3 −4 −8 −6 −4 −4 −2 0 10 10 10 Current, log (A) Drill string conductivity, S/m Fig. 12 Comparison of simulated EMT signal with a change in the conductivity of the drill string; a current density along the drill string and b measured voltage at the surface 1D Resistivity Representation (a) (b) 0 1 2 10 10 10 Resistivity, Ohm m Fig. 13 a Simplified diagram of anisotropic 2-D TTI media, showing the axis of symmetry and principal resistivities. b 1D representation of the resistivity model. The geographic coordinate frame is r, z, while the principal axis frame (or natural rock frame) is tilted away from this direc- tion. The directions of the principal axes are the corresponding eigenvectors The quantity ρ  is the geometric mean of the longitudi- parameters and the orientation of the axis of symmetry and nal and transverse resistivities. It is equal to surface meas- considering its effect on electromagnetic telemetry signal. ured apparent resistivity. In this section, we focused on As an instructive preliminary investigation, we computed incorporating the coefficient of anisotropy with the model current distribution along the drill string and the surface 1 3 Depth, m Depth, m Voltage, V 118 Petroleum Science (2021) 18:106–122 measured signal, at a dipole length of 100 m, for the dif- To illustrate the differences with the change in the dip of ferent axes of symmetry, θ (°), and fixed value of ρ , λ. The the principal axis, the currents have been plotted. Since the current distribution along the drill string was calculated for horizontal conductivity is fixed in this study, plots of the cur - a two-layered anisotropic model with an anisotropic value rent distribution along the drill string exhibit the same pat- of λ = 5, ρ of 1 Ω.m and 10 Ω.m for the top and lower lay- terns as those of surface measured voltage. The plot of the ers, respectively. The anisotropic column was between the current distribution along the drill string (Fig. 14b) shows depths of 220 m and 440 m, and θ (°) varies from 2° to 90°. an increase in the magnitude of the current flowing along The simulated surface measurement related to θ = 90° is the drill string with an increase in the angle of orientation five orders of magnitude higher than  θ = 2° (Table 1). The of the axis of symmetry. At low angle (θ = 30° and 15°), existence of resistive anisotropic medium confined by con- the magnitude of the current distribution along the drill ductive isotropic materials (great simplifications arise in string decreases very fast within the anisotropic layer, and the decoupling of the anisotropy parameters) results in a the z-plane becomes almost entirely vertical (Figs. 13 and significant reduction in the magnitude of surface recorded 14b). When θ is between 45° and 75°, slight variation in the EMT signal. The rose diagram in Fig. 13a shows that the magnitude of the current distribution along the drill string sensitivities to change in θ angle increase with a decrease in was observed, and at θ > 75° no significant change in current θ and very rapidly for angles below 60°. The source of the density was recorded. In comparison with the result from amplitude difference in the sensitivity stems from the dif- the isotropic medium, the value of current flowing along the ference in the model parameters θ (°), leading to an increase drill string with θ = 45° has almost the same value as that of in the value of ρ   (Ω m) as θ increases, and its value tends the isotropic medium, while for θ > 45°, the current density toward the value of the major principal axis ρ  (Ω m). is higher. (The horizontal component is dominant.) Table 1 Surface measure voltage for electromagnetic telemetry Dip (°) 2 15 30 45 60 75 85 90 Voltage (V) 6.0E-08 0.99E-05 0.000068 0.00017 0.00029 0.00039 0.00043 0.00044 Current along drill string (a) (b) 210 330 15 deg 30 deg 45 deg 240 60 deg 75 deg 90 deg Isotropic −6 −4 −2 0 Current log(A) Fig. 14 Comparison of simulated EMT signal with the change in the angle of the principal axis of symmetry of the subsurface property; a rose diagram showing the change in EMT signal magnitude with a change in the angle of the principal axis of symmetry and b current density along the drill string at different angles of principal axis of symmetry 1 3 Depth, m Petroleum Science (2021) 18:106–122 119 otherwise in a credit line to the material. If material is not included in Although EM telemetry is used in subsurface wireless the article’s Creative Commons licence and your intended use is not communication, it is not as a quantitative well logging tool, permitted by statutory regulation or exceeds the permitted use, you will a substantial level of accuracy should be of priority for mod- need to obtain permission directly from the copyright holder. To view a eling of EM telemetry. This study shows the complexity in copy of this licence, visit http://creativ ecommons .or g/licenses/b y/4.0/. an accurate simulation of EMT measured signal and the extensive effect of anisotropy which could be of significant influence even if for a short section. Hence, a combination Appendix 1 of mud-pulse and EM telemetry could be useful in ensuring a continuous transmission of measurable signal with high In this section, the anisotropic parameters defining a magnitude. homogeneous TTI medium  ,  or  , λ , azimut h  and h t m the dip angle θ were used in deriving the conductivity values for TTI medium in a cylindrical coordinate sys- 5 Conclusion tem, where λ is the anisotropy factor,  represents the longitudinal conductivity,  represents the transverse con- In this paper, we presented an efficient FDM scheme for ductivity and  is the geometric mean of the principal simulations of electromagnetic telemetry in the axisym- conductivities and is defined by the dip and resultant of metric model. In terms of application, homogeneous and the longitudinal conductivity and transverse conductivity. layered earth models were used as examples. The current Given the diagonalized conductivity matrix in Cartesian density along the drill string and surface measured voltages coordinate which represents the three eigenvalues σ , σ , 1 2 were solved to understand the effect of change in drill string σ as conductivity and angle of axisymmetry of the earth physical ⎛  00 ⎞ ⎛  00 ⎞ property on electromagnetic telemetry. The algorithm and xx 1 ⎜ ⎟ ⎜ ⎟ = 0  0 = 0  0 , (17) model used straightforwardly assimilates the 2D anisotropy T yy 2 ⎜ ⎟ ⎜ ⎟ 00  00 ⎝ ⎠ ⎝ ⎠ and allows the computation of the anisotropy effect as an zz 3 update field. Also, we considered the Gaussian quadrature we define the relationship between the geometric mean, techniques in the effective estimation of measured EMT sig- the principal conductivities and coefficient of anisotropy nal. A field case was also given showing that the calculated as  =  =  ,  =  ,  =   and  =   for xx yy h zz t m h t h t results by the proposed method agree with the measured transverse anisotropic (VTI) medium. Compared to the data. Lastly, the effect of change in axis of symmetry of VTI medium, the conductivity tensor of TTI medium in earth layering and occurrence of confined resistive layers the Cartesian coordinate system can be expressed as should be taken into account in considering the effectiveness of EM telemetry. In general, we found that the accuracy of ⎛  00 ⎞ the 2D FDM scheme presented in this study is suitable for −1 ⎜ ⎟ = R 0  0 R, (18) T 2 ⎜ ⎟ practical applications and that the computational efficiency ⎝ ⎠ is much higher than for 3D modeling. It is believed that simulations in this paper will facilitate the feasibility study where of an electromagnetic system for real field jobs. cos  cos  sin  cos  − sin ⎛ ⎞ Acknowledgments This project was supported by the Strategic ⎜ ⎟ R = − sin  cos  0 . ⎜ ⎟ Priority Research Program of Chinese Academy of Sciences (No. cos  sin  sin  sin  cos ⎝ ⎠ XDA140500001). The anisotropic medium can thus be described by six Compliance with ethical standards independent components writing in Cartesian coordinate or recording frame as: Conflict of interest On behalf of all the authors, the corresponding au- thor states that there is no conflict of interest. 2 2 2 2 2 cos  cos  +  sin  +  sin  cos ⎛ ⎞ ⎛ ⎞ xx h h t � � ⎜ ⎟ ⎜ ⎟ 0.5 − +  sin  sin 2 xy Open Access This article is licensed under a Creative Commons Attri- h t � � ⎜ ⎟ ⎜ ⎟ bution 4.0 International License, which permits use, sharing, adapta- 0.5 − +  cos  sin 2 xz h t ⎜ ⎟ ⎜ ⎟ � � 2 2 tion, distribution and reproduction in any medium or format, as long ⎜  ⎟ ⎜ − +  sin  sin  +  ⎟ yy h t h � � as you give appropriate credit to the original author(s) and the source, ⎜ ⎟ ⎜ ⎟ 0.5 − +  sin  sin 2 yz h t provide a link to the Creative Commons licence, and indicate if changes ⎜ ⎟ ⎜ ⎟ sin  +  cos ⎝ ⎠ ⎝ ⎠ were made. The images or other third party material in this article are zz h t included in the article’s Creative Commons licence, unless indicated (19) where  =  =  and  =  . 1 2 h 3 t 1 3 120 Petroleum Science (2021) 18:106–122 z z However, for the general 2/2.5-D case where  = 0 , t hat J V = E z = z. (26) received z is, there is no dependency on the azimuth, we obtain: z z 0 0 cos  +  sin ⎛ xx ⎞ ⎛ ⎞ h t However, in this study, we considered only the conven- � � ⎜  ⎟ ⎜ ⎟ 0.5 − +  sin 2 xz h t tional surface measurement of EMT signal, where r is the = . 0 (20) ⎜ ⎟ ⎜ ⎟ yy h position on the blowout preventer and r represents second ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ sin  +  cos zz h t position tens of meters to over a hundred meters from r . To solve the integrand, we adopted Gaussian quadrature To translate these results into a cylindrical coordinate techniques using weighing function through which integrable system with situations where the axis of symmetry is tilted singularities are removed. Conventionally, the domain of inte- along the dip, we used the following expression: gration is taken as [a,b], whereas in Gaussian quadrature, the � � 2 2 domain is reduced to [−1,1]. The general Gaussian quadrature ⎛  cos  +  sin  0 0.5 − +  sin 2 ⎞ h t h t −1 ⎜ ⎟ can be represented as = Rc 0  0 Tc h � � ⎜ ⎟ 2 2 0.5 − +  sin 2 0  sin  +  cos ⎝ ⎠ h t h t 1 1 n−1 (21) f (x)dx = W(x)g(x)dx ≈ w g x . (27) i i where  is the conductivity tensor in the cylindrical coor- Tc i=1 −1 −1 −1 dinate system and Rc is the conversion matrix. The matrix In this study, we used the explicit Legendre–Gauss–Lobatto expression for the conversion matrix is given as (LGL), Gauss–Lobatto (GLo), Legendre–Gauss–Radau (LGR) ⎛ cos  sin  0 ⎞ and Gauss–Legendre (GLe) quadrature techniques, which con- −1 ⎜ ⎟ Rc = − sin  cos  0 . (22) verge accurately to estimate the potential across two measuring ⎜ ⎟ 0 01 ⎝ ⎠ points (−1,1). The integral form of the quadrature techniques is given as Finally, the TTI conductivity tensor is given as � � n−1 2 2 cos   cos  +  sin ⎛ rr ⎞ ⎛ h t ⎞ � � � � GLo - f (x)dx ≈ (f (−1)+f (1))+ w g x , i i ⎜  ⎟ ⎜ cos  0.5 − +  sin 2 ⎟ n(n − 1) rz h t i=2 = . (23) −1 ⎜ ⎟ ⎜ ⎟ cos ⎜ ⎟ ⎜ ⎟ (28) ⎝  ⎠ ⎝  sin  +  cos  ⎠ zz h t Note that there is no dependence on the axial direction LGL - f (x)dx ≈ w g x , (29) i i with  = 0 . 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