I. INTRODUCTION DIRECTIONAL drilling technology involves directing

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1 144 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 65, NO. 1, JANUARY 2016 Continuous Real-Time Measurement of Drilling Trajectory With New State-Space Models of Kalman Filter Qilong Xue, Henry Leung, Fellow, IEEE, Ruihe Wang, Baolin Liu, and Yunsheng Wu Abstract Rotary-steerable drilling provides unique features such as an extended reach and accurate well trajectory control. These characteristics can notably increase drilling efficiency and safety. One of the main technical difficulties of rotarysteerable drilling is dynamically measuring the spatial attitude at the bottom of the rotating drillstring as the drillstring rotates. We developed a strap-down measurement system, with a triaxial accelerometer and triaxial magnetometers installed near the bit. The inclination and azimuth can be measured in real time even as the drillstring rotates. Although the magnetic system is the norm, we can use this system to achieve continuous measurement while drilling; to achieve this, a novel state-space model is proposed here and the Kalman filter is used to estimate the states. Simulation and experiments show that a continuous-survey system with a Kalman filter approach can improve measurement precision and reduce errors produced by drillstring vibration. Index Terms Continuous measurement while drilling (MWD), directional drilling, Kalman filter, state-space model, strap down. I. INTRODUCTION DIRECTIONAL drilling technology involves directing a wellbore along a predefined trajectory, which dramatically reduces costs and saves time during drilling operations [1 [3. During the last several years, more and more attention had been paid to the development of directional well-drilling technologies. Technology for directionaldrilling navigation is currently based on an integrated magnetometer and accelerometer triad [4. To compute the bottom-hole assembly (BHA) position, data of the earth s magnetic field and the force of gravity are measured. The surveying system is executed along the well trajectory at stationary survey stations. In drilling engineering, the bottom Manuscript received April 15, 2015; revised August 3, 2015; accepted August 4, Date of publication October 6, 2015; date of current version December 7, This work was supported in part by the Fundamental Research Funds for the Central Universities under Grant and in part by the Public Welfare Fund Project through the Ministry of Land and Resources under Grant The Associate Editor coordinating the review process was Dr. George Xiao. Q. Xue and B. Liu are with the Key Laboratory on Deep GeoDrilling Technology, Ministry of Land and Resources, School of Engineering and Technology, China University of Geosciences, Beijing , China ( xql@cugb.edu.cn). H. Leung is with the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB T2N 1N4, Canada. R. Wang is with the College of Petroleum Engineering, China University of Petroleum, Qingdao , China. Y. Wu is with State Grid Haixing Power Supply Company, Cangzhou , China. Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TIM drillstring attitude (inclination and azimuth) measurement is usually carried out when the drillstring is not rotating. However, as drilling technology improves, continuous measurement of the well trajectory becomes increasingly important. It is also essential in a rotary-steerable system (RSS). An RSS [5 [7 is a mechatronics tool developed for directional drilling. It can drill a more economical and smoother borehole. Since the introduction of RSS, rotary-steerable technology has achieved notable progress in reliability and has become a standard drilling tool in many worldwide applications. The application of RSS is restricted to high-cost offshore sites and is becoming more common in cost-sensitive land work, especially in shale-gas and shale-oil drilling [8. Despite popular use of the RSS, field-trial results of continuous measurement of inclination and azimuth have not been well documented in the literature. However, measuring the posture of downhole tools as the drillstring rotates is essential because of its closed-loop control structure [9, [10. A bottom-drilling tool shows complex dynamics while rotating owing to the combined effects of nonlinear vibrations, such as vertical vibration, horizontal vibration, eddy, and sticky slip [11, [12. The effects of such vibrations cause the measurement sensors to generate large errors. This is a huge challenge for signal processing [9, which is completely different from the measurement-while-drilling (MWD) survey systems in current oil and gas industry. Continuous MWD is studied under laboratory conditions using a gyroscope-based system [13 [15. ElGizawy et al. [13, Elgizawy et al. [14, and Jurkov et al. [15 proposed an advanced inclination and direction sensor package based on an inertial navigation system (INS). They verified the reliability of the algorithm through simulation, which used INS to achieve continuous MWD with high accuracy. The influences of vibration and temperature on MWD were also analyzed [16 [18. Yanshun et al. [19 and Chen et al. [20 conducted a similar study by developing an MWD instrument based on a predigested inertial measurement unit. However, they did not consider the downhole complex situations, severe vibration, and high temperature. These are great challenges for measurement of accuracy and sensors lifetime. Drillstring vibration can greatly affect the life of the gyroscope, and an increasing temperature can cause drift error in the gyroscope. The approaches mentioned above can increase the accuracy of measurements to a certain extent. However, effort still needs to be made in order to improve the performance of the MWD IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 XUE et al.: CONTINUOUS REAL-TIME MEASUREMENT OF DRILLING TRAJECTORY 145 Fig. 1. (a) Multimodel measurement system at the drill bit. (b) Construction of measurement system. TABLE I CHARACTERISTICS OF SENSORS instrument. Qilong Xue et al. [9 developed a strap-down multimode surveying system. They gave more consideration to the actual situation of the drilling process and used field test data to study the measurement algorithm. This system is more conducive to field applications. Moreover, improvement in accuracy is crucial, especially for the continuous MWD of an RSS, for precise and efficient measurement of wellbores drilled for oil and gas exploration. Within the signal-processing community, Kalman filter still remains a very active topic. This paper is concerned with improving the accuracy and stability of inclination and azimuth measurements of the accelerometer and magnetometer. We developed new state-space models that were applied to the Kalman filter model. The algorithm greatly improved the accuracy of well-trajectory measurements and is expected to be applied to ordinary magnetic surveying systems, which are more widely used in drilling engineering. II. MEASUREMENT SYSTEM We develop a strap-down MWD system here that incorporates three-axis magnetometers and three-axis accelerometers arranged in three mutually orthogonal directions [9, [10, as shown in Fig. 1(a). The magnetic measuring tools are installed in the interiors of nonmagnetic drill collars. These special drill collars are usually designed from monel metal to avoid external interference with measurements taken by magnetic MWD surveying tools [3, [9. Fig. 1(b) shows the structure of the downhole measurement system, and Table I shows the characteristics of the sensors. a x, a y,anda z are defined as survey signals of the triaxial accelerometers on the x, y, andz axes, respectively. Moreover, m x, m y,andm z are defined as survey signals of the triaxial magnetometers on the x, y,andz axes, respectively. Assume that the earth s magnetic field strength is M. Obviously, M = (m 2 x + m2 y + m2 z )1/2. Under a certain sample frequency (100 Hz), the measuring signal can be considered as a time series. Assume that the acceleration of gravity is G, and g x, g y, and g z are defined as survey signals of gravity acceleration on the x, y, and z axes, respectively. Then, G = (gx 2 + g2 y + g2 z )1/2. The survey signals (a x, a y, a z ) of the triaxial accelerometers include not only the acceleration of gravity but also the acceleration of the drillstring vibration. Electronic instruments consisting of three-axis accelerometers and three-axis magnetometers are shown to relate to the measured inclination and azimuth. The drillstring posture is determined via the following known equations: gx 2 Inc = arctan + g2 y (1) g z [ G (g x m y g y m x ) Azi = arctan ( m z g 2 x + g 2. (2) y) gz (g x m x g y m y ) The three-axis inclination and six-axis azimuth equations [22 are used in the static MWD surveys as industrystandard survey methods. The azimuth direction is determined in a stationary mode by using three-axis magnetometers, while the inclination and the tool face angle are determined using three-axis accelerometers. Drilling has to pause frequently at surveying stations in order to allow the inclination and azimuth to be surveyed. The well trajectory is then computed between the two surveying stations based on some mathematical assumptions; It is assumed that the drilled distance is a smooth arc. In (1) and (2), only the acceleration of gravity is concerned, and the drillstring vibration signals are considered as noise. The determination of azimuth while drilling (while the sensors are rotating) was not demonstrated owing to limitations in the capabilities of downhole processing. The survey signals of triaxial accelerometers include not only the acceleration of gravity but also the acceleration of drillstring vibration. Unlike the conventional MWD, the inclination and azimuth should be dynamically solved when the

3 146 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 65, NO. 1, JANUARY 2016 Fig. 2. Energy spectrum of lateral vibration signals (left) and longitudinal vibration signals (right). drillstring rotates. Accelerometers are hypersensitive to vibrations, so the violent vibration of a BHA leads the measured signals to be submerged in noise signals. The continuous MWD in this paper is based on a strap-down multimodel surveying system [9. According to the data of field test, we completed a frequency analysis of the vibration signals to evaluate if the white noise hypothesis. Frequency analysis is indispensable when discussing the most advanced approach that involves dynamic spectral analysis and analyzing how the spectrum of a signal evolves in time. This analysis is performed with Gabor transforms [25, which are close relatives of wavelet transforms. Gabor transforms are particularly useful in analyzing drillstring vibration signals. The transform is employed here to determine the sinusoidal frequency and phase content of local sections of a signal as it varies with time. The function is first multiplied by a Gaussian window function. The resulting function is then transformed by the Fourier transform. We use a time frequency box to express the energy distribution of the Gabor function. The time frequency spectrum of the vibration signal is plotted in Fig. 2. The Gabor transform can help us discriminate the rotational movement pattern. Define G u,ξ (t) as the time window function, where u is center point, σ is the width, and the Fourier transform of G u,ξ (t) can be expressed as Ĝ u,ξ (ω) = Ĝ(ω ξ)e iu(ω ξ). (3) As shown in Fig. 2, from the energy spectrum of lateral vibration signals (left) and longitudinal vibration signals (right), we find that the spectrum does not particularly have the energy concentration point, and the signals can be considered as random. Therefore, the hypothesis of white noise in the new Kalman filter is satisfied. III. NEW KALMAN FILTER APPROACH The application of KF [23, [24 requires that both the system and the measurement models of the underlying process be linear. A discrete-time linear state-space system is described by x k = k,k 1 x k 1 + G k 1 w k 1 (4) where k,k 1 is the state transition matrix, x k is the state vector, G k 1 is the noise distribution matrix, w k 1 is the process noise vector, and k is the measurement epoch. Fig. 3. Solving process for the system. The measurement equation of the system is given by z k = H k x k + η k (5) where z k is the measurement vector of the system output, H k is the observation or design, and η k is the measurement noise. The system noise w k and the measurement noise η k are unassociated zero-mean white-noise processes with determined autocovariance functions. We developed an algorithm with new state-space models by analyzing drillstring dynamics. In the solution system, we define the input vector of KF-1 as [ ax a X = y a z m x m y m z which is measured by the three-axis magnetometers and three-axis accelerometers. The input vector [ of KF-2 is the inclination and azimuth, defined as M = IA,whereI is the inclination of the borehole and A is the azimuth of the borehole. Using the solving process as shown in Fig. 3, we develop two KFs for the entire drilling process. After KF-1, we can obtain the more precise signals of gravity acceleration g x, g y,andg z, that defined as the output of KF-1. Using the gravity accelerations, we can obtain the inclination and azimuth by the equation developed when the drillstring rotates. KF-2 is then used to further smooth the drilling trajectory. The output of KF-2 is defined as M =[I A T,whichismore precise. A. State-Space Model for KF-1 The sensors installed at the center of the drillstring, and the measurement signals of the x and y axes, will exhibit a sine wave during rotation. Theoretically, accelerometer and magnetometer signals have the same rule. In the actual drilling process, the vibration of the drillstring affects less

4 XUE et al.: CONTINUOUS REAL-TIME MEASUREMENT OF DRILLING TRAJECTORY 147 on the magnetometer signals. That is, the magnetometer signals are used to calibrate the accelerometer signals. From the laboratory testing, we can conclude that the changes of signals of fluxgate in line with signals of gravity acceleration, as shown in Fig. 6, will be provided in the following. Assuming that the angular velocity is ω x,y,z and the sampling interval is t, then m x,y (k) m x,y (k 1) = = M sin ω x,y t M sin ω x,y (t t) G sin ω x,y t G sin ω x,y (t t) = g x,y(k) g x,y (k 1). (6) [ gx (k) In the KF-1, we define the state vector as x g (k) = g y (k), g z (k) as shown in Fig. 2, from the vibration signals of a x, a y,anda z, we can obtain the signals of gravity acceleration x g (k). Thus [ ax the measurement vector of the system output as z(k) = a y, a z when the drillstring rotates, the measurement signals of the x- andy-axis will exhibit a sine wave except the signal of the z-axis. So the transformation matrix is defined as m x (k) 0 0 m x (k 1) H k = m y (k) 0 0 m y (k 1) The system noise w k and the measurement noise η k are uncorrelated zero-mean white-noise processes. Therefore, we obtain the state-space model of the KF-1 as follows: m x (k) 0 0 m x (k 1) x g (k) = m y (k) 0 0 m y (k 1) x g(k 1) w x(k 1) w y (k 1) (7) w z (k 1) z(k) = H k x g (k) + η(k). (8) B. Calculating the Inclination and Azimuth When the drillstring rotates, the above equations are not applicable. The sensors installed at the center of the drillstring, and the measurement signals of the x- andy-axis, will exhibit a sine wave during rotation. Through KF-1, we obtain g x, g y,andg z which are defined as survey signals of gravity acceleration on the x, y, andz axes, respectively. [ Then define the input vector of the system as X = gx g y g z m x m y m. z Define the rotational speed as R. If R = 0, (1) and (2) are used to calculate the inclination and azimuth. R = 0 indicates that the drillstring is rotating, and computational formulas of inclination I and azimuth A are given as follows: ( ) 2Np 2 ( ) I = tg 1 T M =0 g 2Np 2 xdt M + T M =0 g ydt M 2Np T M =0 g (9) zdt M A = tg 1 ( 2Np T M =0 G(m ) yg x m x g y )dt M 2Np T M =0 m z( g 2 x +g 2 ) y dtm 2Np T M =0 g z(m x g x + m y g y )dt M (10) where T M is the magnetic toolface angle: T M = tg 1 ( m y /m x ). Toolface angle used for near-vertical wells. Magnetic toolface is the angle, or azimuth, of the borehole survey instrument within the wellbore measured clockwise relative to magnetic north and in the plane perpendicular to the wellbore axis; the north, east, south, and west directions have magnetic toolface angles of 0, 90, 180, and 270, respectively. Magnetic toolface may be corrected to reference either grid north or true north. Although the drillstring rotational speed is a way to determine whether the string is rotating or not, it reliability of this approach is not high. Instead, using the standard-deviation statistical methods to determine the drillstring movement is more effective because it reflects the degree of dispersion among the individuals within the group. Using 50 data points as a time window, assumed them to be x 1, x 2,...,x 49, x 50,we get the standard deviation σ = ((1/N) N i=1 (x i x) 2 ) 1/2. The drillstring can be considered static when the standard deviation σ is close to zero. C. State-Space Model for KF-2 In the drilling process, the movement states that σ = 0or σ = 0 will appear alternately. When σ = 0, the results are more accurate because the vibration is slight when the drillstring is not rotating. We develop another Kalman filter (KF-2), as shown in Fig. 2, to smooth the trajectory of the drilling. (KF-2 was divided into KF-2.1 and KF-2.2, respectively, as showninfig.5.) In a normal drilling process, the drilling operation has to frequently stop at measurement stations in order to measure the inclination and azimuth. The well trajectory is then computed between the two surveying stations based on mathematical assumptions. For instance, it may be assumed that the drilled distance is a straight line, smooth arc, or polygonal line; each requires a different calculating method. Assuming that the 3-D coordinates of the measuring Nth point of the actual drilling trajectory are (x N, y N, z N ), the measuring (N + 1)th point is (x N+1, y N+1, z N+1 ), and the well depth, vertical depth, inclination, and azimuth are L N, H N, θ N,andψ N, and L N+1, H N+1, θ N+1,andψ N+1, respectively. The well trajectory between the two points can then be defined as follows: L = L N+1 L N H = H N+1 H N (11) θ = (θ N + θ N+1 )/2 ψ = (ψ N + ψ N+1 )/2.

5 148 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 65, NO. 1, JANUARY 2016 Fig. 4. Trajectory prediction using method of inclined plane cirque. If the vertical depth H is a known, the 3-D coordinates of the measuring (N + 1)th point can be defined as x N+1 = x N + H tan θ cos ψ y N+1 = y N + H tan θ cos ψ (12) z N+1 = z N + H. We can obtain the spatial coordinates of each point by recursive calculation, thereby obtaining the entire drilling trajectory. As shown in Fig. 4, we can use the wellbore trajectory extrapolation method to establish a recursive relationship between two adjacent measurement points. Assume L is the depth of the drilling, γ is the angle of the drilling trajectory and its tangent. Then L(k) = L(k 1) + L(k) (13) γ = arccos [cos(i k 2 ) cos(i k 1 ) + sin(i k 2 ) sin(i k 1 ) cos(a k 1 A k 2 ) (14) γ γ(k) = L(k). (15) L(k 1) From (13) to (15), we can use two points to estimate the next point, so the well trajectory can be smoothed. As shown in Fig. 5, we can calibrate the drilling trajectory using Kalman filter 2.2, wherein the kth measurement point is defined as P(k) = [I k A k. The system inputs are P(k-2) and P(k-1), KF-2.2 estimated P(K) combined with measured values and the theoretical calculation values. KF-2.2 can smooth the well trajectory using inclination and azimuth [ as inputs. Assume that the state vector is x(k) = Ik A and the measurement vector of the system k [ Ik A k m output is y(k) =, which is calculated from (8) and (9). [ The transformation matrix is defined as H (k) = According to (12) (14), the state-space model for KF-2.2 can be obtained in (16) and (17), as shown at the bottom of this page. As shown above, we should determine L as the KF-2.2 input (the distance that the drill bit moves forward Fig. 5. Solving process for KF-2. during t). As shown in Fig. 4, we can use the measurement of acceleration on the z-axis to calculate the displacement. a z is the signal of the triaxial accelerometers on the z-axis, which is combined with the gravitational acceleration and vibration acceleration. The measurement of acceleration on the z-axis time series is defined as a z (k). So before calculating the displacement L, we should exclude the impact of gravity as follows: f gz (k) = a z (k) G cos(i k 1 ) (18) where f gz (k), the acceleration time series function by removing the acceleration of g z, can be calculated from a z (k) and inclination I k 1 corresponding to the same time. Then, we can calculate the depth ( L) of drilling using acceleration on the z-axis time series f gz (k). Define the state [ L(k) vector as L(k) = L(k), the measurement vector of L(k) the system output as z(k) = f gz (k), we develop a state-space model for KF-2.1 as follows: 1 L(k) 1 t L(k) = 2 t2 L(k 1) ε(k) 0 1 t L(k 1) + ε(k) (19) L(k) L(k 1) ε(k) f gz (k) = [ L(k) L(k) + η(k). (20) L(k) By preprocessing the measured signals, a dynamic azimuthand inclination-solving algorithm is established based on dynamic analysis of the bottom rotating drilling tool. Based on the theoretical model, we develop a Kalman filter to improve the solver accuracy as well; the state-space equations used for the Kalman filter have been established based on [ sin(γ + γ(k)) arccos cos(i k 2 ) cos(γ + γ(k)) (cos(i k 2 ) cos γ cos(i k 1 )) sin(γ [ ) cos(γ (k)) cos(ik 1 ) cos(i k ) A k 1 + (A k 1 A k 2 ) arccos sin(i k 1 ) sin(i k ) y(k) = H (k)x(k) + v(k) [ Ik A k = + [ wi (k) w A (k) (16) (17)

6 XUE et al.: CONTINUOUS REAL-TIME MEASUREMENT OF DRILLING TRAJECTORY 149 Fig. 6. (a) Laboratory bench. (b) 3-D simulation model of the test bench. (c) Experimental equipment at inclinations of 5, 2, 0, 2, and 5 (shown by the red line). the drilling trajectory predicted. The dynamic measurement algorithm with the Kalman filter is a new model that can greatly reduce the solution errors. IV. EXPERIMENTAL RESULTS Dynamic measurement algorithms developed in this paper were tested through laboratory bench and field measurement data, respectively. A. Laboratory Testing The combined measurement system was tested first in a laboratory environment, as shown in Fig. 6. Measurement data were obtained under different inclination and rotating speed conditions. In the experimental system, we used the encoder to measure the drillstring rotational speed and positioned the inclination mechanically. As shown in Fig. 6(b), the power simulation motor (with digital 1 representation), will work to provide electricity, although this time the drillstring not continuously rotates, the power simulation motor will generate vibration making the static solver results as above are not very accurate. The platform rotating motor (with digital 7 representation) working makes the drillstring continuously rotates, we could use (1) and (2) to dynamically solve the inclination and azimuth. In the experiment, we set the laboratory test at different inclinations and different rotational speeds, as shown in Fig. 6(c). When the drillstring continuously rotates with a constant speed and is located at a particular borehole inclination and azimuth, accelerometer (x, y, z axes) and fluxgate (x, y, z axes) measurement data are obtained as shown in Fig. 7(b) and (c), respectively. We put the entire system on a laboratory bench [Fig. 6(a) to conduct an accuracy test of the measurement system. Fig. 7(b) and (c) shows the measurement data for the magnetometers (x, y, z axes) and accelerometers (x, y, z axes). Here, we can calculate the signals of the magnetometers and accelerometers if we know the inclination according to (9) and (10). Assumed that the reference of the measurements data is the reverse theoretical calculation results, the relative errors of measurement data as shown in Fig. 8, it can be seen that the accelerometer noises are relatively much larger than the magnetometer signal noises, the peaks in Fig. 8 show that the measurement error increases when drillstring appear stick-slip vibration. The main reason is that accelerometers are hypersensitive to drillstring vibrations. In the field test, the vibration of the drillstring is more violent. Using the components of gravity acceleration on the x, y, andz axes can help us calculate the inclination and then combine the fluxgate measurement signals to obtain the azimuth [Fig. 7(d). In engineering applications, the inclination error of 0.1 will suffice. As shown in Fig. 6(c), when the experimental equipment is at inclinations of 5, 2, 0, 2, and 5, the test results are shown in Table II. Through laboratory tests,

7 150 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 65, NO. 1, JANUARY 2016 Fig. 7. Measurement system. (a) Construction of measurement system. (b) Laboratory survey signals of triaxial magnetometers on the x, y, and z axes. (c) Laboratory survey signals of triaxial accelerometers on the x, y, and z axes. (d) Continuous survey method, g x, g y, and g z are the gravity accelerations on the x, y, and z axes, in (8) and (9), we are only concerned about the acceleration of gravity. Fig. 8. data. Relative errors of accelerometers and magnetometers measurement TABLE II ERROR OF INCLINATION the theoretical models can be verified as entirely feasible when the drillstring rotates. Experiments show that the dynamic solving methods in this paper meet engineering requirements. B. Field-Drilling Testing In the field, a few methods such as a low-pass filter and moving average filter are considered, but the results are not Fig. 9. Schematic of field test. The electronic circuit is installed on a compressive cylinder and placed in the axis of the drill collar, the sampling frequency of measurement and the control system is 100 Hz, and the measurement data can be stored in real time. satisfactory when the drillstring rotates. The accelerometer signals of the field tests are completely different from those obtained via laboratory survey. The azimuth is determined using three-axis magnetometers, while the inclination is determined using three-axis accelerometers. Therefore, the drillstring vibration dramatically amplifies the error of the continuous survey when the drillstring rotates. The algorithms developed in this paper will solve this problem by improving the accuracy of continuous MWD with the proposed statespace models. Fig. 9 shows the schematic of the field test. From the measurement data stored in real time, we can obtain the vibration signals measured by the accelerometer (x, y, andz axes), which contain the gravity acceleration on the x, y, andz axes. We use the real field-test data to demonstrate the feasibility of our algorithm. The results are shown in Fig. 10. During the first step of the solving process for the system (Fig. 3),

8 XUE et al.: CONTINUOUS REAL-TIME MEASUREMENT OF DRILLING TRAJECTORY 151 Fig. 10. Signals of triaxial accelerometers on the x, y, and z axes, as shown in subparts (a), (b), and (c) respectively, before KF-1 and after KF-1. Fig. 11. Solving results of inclination and azimuth at XuanYe 1. the signals of the triaxial accelerometers (a x, a y, a z ) going though KF-1 get the gravity acceleration (g x, g y, g z ). The results of KF-1 are shown in Fig. 10. The red line is the output of KF-1, that is, the gravity acceleration. Because the accelerometer on the z-axis is not influenced by the rotation of the drillstring, the curve tends to be a straight line when the inclination does not change [Fig. 10(c). We cannot estimate the error of this process, because the data came from the field test, which impossible to know the exact measurements. In the actual drilling process, we need to find a way to measure how good the measurement accuracy is. For a continuous drilling trajectory, there is no method that could be used to accurately measure the wellbore trajectory. So we could only use the MWD static measuring points for comparison. It should be feasible to evaluate the advantages of the dynamic measurement algorithm. Tests of more than ten wells were conducted throughout China. The dynamic Fig. 12. Inclination error of solving results. measurement system was deployed at multiple formations. The field tests were carried out on wells using automatic vertical or rotary steerable drilling. Three- and four-rib actuators were adopted. A lot of raw measurement data

9 152 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 65, NO. 1, JANUARY 2016 TABLE III BASIC PARAMETERS OF FIELD TEST Fig. 13. Solving result of inclination at Anshun1. were accumulated. These data were input into our algorithm for authentication. Conclusions on bit movement rules were drawn by analyzing and summarizing large amounts of data. Two wells in China (Xuanye1 and Anshun1) are selected here to evaluate the algorithm. The results for inclination and azimuth are shown in Figs. 11 and 12. The basic parameters of field tests are shown in Table III. The proposed Kalman filter method can significantly improve the precision of the survey and reduce vibration interference in the solution results. We can also use the periodic static measurements to improve the dynamic solver accuracy. Dynamic solution approach to the bottom of the rotating drillstring attitude is proposed in [9. With nonrotating string, filtered real time signals on three axes are all used for calculation, and at the same time, filtered signals on the x and y axes were stored; on condition of rotation string, real time filtered signals on the z-axis and stored signals of the x and y axes with nonrotating string are adopted. In addition, stick-slip state of the down-hole drilling tool is considered as a nonrotating stationary state. Inclination and azimuth show great fluctuations when the drilling string rotates. The results obtained using the method we developed are shown again in Figs. 11 and 12. Comparing static measurement points, it can be seen that the dynamic measurement inclination error is less than 1, and the azimuth error is relatively large, between 5 and 20, because it is calculated under small inclination angles. When the angle of inclination is small, accelerometer signals of G x and G y are

10 XUE et al.: CONTINUOUS REAL-TIME MEASUREMENT OF DRILLING TRAJECTORY 153 small as well; this leads to an increased impact of noise. Moreover, the error accumulated in formula (2) causes the azimuth error to become larger. 1) Drilling at Xuanye1: This drilling field test was conducted on May 19, We use the static measurement data to evaluate the continuous solving results. From Figs. 11 and 12, we can see that the inclination error is less than 1, additionally, the error is gradually reduced. Unfortunately, we did not get the static field measurement data of azimuth. 2) Drilling at Anshun1: This drilling field test was conducted on January 23, We use the static measurement data to evaluate the continuous solving results. From Fig. 13, we can see that the inclination error is less than 1, and that the azimuth error is between 5 and 20. V. CONCLUSION In the directional drilling and rotary-steerable drilling technology, it is a challenge to measure the spatial attitude (inclination, azimuth, and tool face) of the bottom-drilling tool accurately in real time while the drillstring is rotating. The drillstring vibration seriously affects the accuracy of the dynamic solution. In drilling engineering, the bottom drillstring attitude is usually measured when the drillstring does not rotate. However, with recent progress in drilling technology, continuous measurement of the well trajectory has become increasingly important. It also becomes necessary in the RSS and automatic vertical drilling system. In recent years, the requirement for a continuous survey that captures the actual trajectory between stationary surveying stations has become an urgent need. This allows for a better estimation of the casing pipe and cementing of the borehole. In addition, this provides an actual estimation of the curvature along the well trajectory. Therefore, in this paper, the trajectory between the two surveying stations is continuously surveyed using a triad of accelerometers and a triad of magnetometers. The calculation algorithm is based on a strap-down computation mechanism and Kalman filtering. We developed downhole measurement signal-processing methods and an algorithm to dynamically solve the spatial attitude of the bottom rotating drillstring. Initially, we established a theoretical system of dynamic solving and laid a theoretical and technical foundation for continuous MWD system and RSS development. As the core of an RSS, MWD urgently needs to improve its accuracy and reliability. However, due to the complexity of BHA vibration and dynamic real-time measurement, precise solution has become a key technology affecting the rapid development of MWD. For the downhole signal filtering of sensors, a specific algorithm needs to be proposed by analyzing the motion state of the bottom-hole drillstring. After filtering, the measurement signals are input into a space attitude dynamic measurement algorithm model that is based on motion analysis. The use of stick-slip vibration phenomena can improve the accuracy of the solution. Using the dynamic measurement theory model and a trajectory prediction, a new space-state equation can be established. A more effective Kalman filter model was developed. This further reduced the errors in the results of a dynamic solution. The dynamic measuring algorithm proposed is analyzed through downhole measurement instance data. The results show that the algorithm can effectively reduce inclinationand azimuth-solving errors. Field measurement data analysis shows that the algorithm can effectively solve the dynamic inclination and azimuth when the inclination is large. 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11 154 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 65, NO. 1, JANUARY 2016 [18 E. Pecht and M. P. Mintchev, Observability analysis for INS alignment in horizontal drilling, IEEE Trans. Instrum. Meas., vol. 56, no. 5, pp , Oct [19 Z. Yanshun, W. Shuwei, and F. Jiancheng, Measurement-while-drilling instrument based on predigested inertial measurement unit, IEEE Trans. Instrum. Meas., vol. 61, no. 12, pp , Dec [20 C. Chen, Z. Yanshun, and L. Chunyu, Surveying method of measurement while drilling based on the inertial sensor, in Proc. 1st Int. Conf. PCSPA, Harbin, China, Sep. 2010, pp [21 P. H. Walters, Method of determining the orientation of a surveying instrument in a borehole, U.S. Patent , Dec. 1, [22 M. K. Russell and A. W. Russell, Surveying of boreholes, U.S. Patent , Aug. 7, [23 A. Noureldin, T. B. Karamat, and J. Georgy, Fundamentals of Inertial Navigation, Satellite-Based Positioning and Their Integration. Berlin, Germany: Springer-Verlag, [24 G. Minkler and J. Minkler, Theory and Application of Kalman Filtering. Palm Bay, FL, USA: Magellan Book Co., [25 J.-J. Ding, Time frequency analysis and wavelet transform class note, Dept. Elect. Eng., Nat. Taiwan Univ., Taipei, Taiwan, Tech. Rep., Ruihe Wang was born in He received the M.Sc. degree in petroleum engineering and the Ph.D. degree in drilling engineering from the China University of Petroleum, Qingdao, China, in 1989 and 1995, respectively. He is currently a Professor and the Vice President with the China University of Petroleum, and directing the Oil and Gas Engineering Research Institute. As a Cross-Century Academic and Technical Leader with China National Petroleum Corporation, Beijing, China, an expert who enjoys the State Council Special Allowance, he focuses on oil and gas engineering fluid mechanics, rock mechanics, and new technology and methods on rock breaking of drilling. Qilong Xue was born in He received the bachelor s degree in electrical engineering and the Ph.D. degree in drilling engineering from the China University of Petroleum, Beijing, China, in 2006 and 2014, respectively. He was with PetroChina, Beijing, from 2006 to He was a Joint Researcher with the University of Calgary, Calgary, AB, Canada, from 2012 to 2013, where he was involved in real time measurement of bottom drilling tool s attitude and rotary steerable drilling system. He is currently an Assistant Professor with the China University of Geosciences, Beijing. His current research interests include measurement and control technology of drilling engineering. Baolin Liu was born in He received the Ph.D. degree in engineering from the China University of Geosciences, Beijing, China. He is currently a Professor and the Director of the Key Laboratory on Deep GeoDrilling Technology of the Ministry of Land and Resources, China University of Geosciences. His current research interests include scientific drilling, environmental science drilling, computer controlled drilling, and new technology drilling engineering. Dr. Liu is a member of the Professional Committee of Mineral Exploration Engineering. Henry Leung (F 15) received the Ph.D. degree in electrical and computer engineering from McMaster University, Hamilton, ON, Canada. He was with Defence Research Establishment Ottawa, Ottawa, ON, Canada, where he was involved in the design of automated systems for air and maritime multisensor surveillance. He is currently a Professor with the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, Canada. His current research interests include chaos, computational intelligence, information fusion, data mining, robotics, sensor networks, and wireless communications. Yunsheng Wu was born in He received the bachelor s degree in electric system and automatism from Hebei Radio and Television University, Hebei, China, in He has been with State Grid Haixing Power Supply Company, Cangzhou, China, since 2000.