International Journal of Automation and Computing  2018, Vol. 15 Issue (5): 625-636   PDF    
Improvement of Wired Drill Pipe Data Quality via Data Validation and Reconciliation
Dan Sui, Olha Sukhoboka, Bernt Sigve Aadnøy     
Petroleum Engineering Department, University of Stavanger, Stavanger 4036, Norway
Abstract: Wired drill pipe (WDP) technology is one of the most promising data acquisition technologies in today's oil and gas industry. For the first time it allows sensors to be positioned along the drill string which enables collecting and transmitting valuable data not only from the bottom hole assembly (BHA), but also along the entire length of the wellbore to the drill floor. The technology has received industry acceptance as a viable alternative to the typical logging while drilling (LWD) method. Recently more and more WDP applications can be found in the challenging drilling environments around the world, leading to many innovations to the industry. Nevertheless most of the data acquired from WDP can be noisy and in some circumstances of very poor quality. Diverse factors contribute to the poor data quality. Most common sources include mis-calibrated sensors, sensor drifting, errors during data transmission, or some abnormal conditions in the well, etc. The challenge of improving the data quality has attracted more and more focus from many researchers during the past decade. This paper has proposed a promising solution to address such challenge by making corrections of the raw WDP data and estimating unmeasurable parameters to reveal downhole behaviors. An advanced data processing method, data validation and reconciliation (DVR) has been employed, which makes use of the redundant data from multiple WDP sensors to filter/remove the noise from the measurements and ensures the coherence of all sensors and models. Moreover it has the ability to distinguish the accurate measurements from the inaccurate ones. In addition, the data with improved quality can be used for estimating some crucial parameters in the drilling process which are unmeasurable in the first place, hence provide better model calibrations for integrated well planning and realtime operations.
Key words: Data quality     wired drill pipe (WDP)     data validation and reconciliation (DVR)     drilling, models    
1 Introduction

The oil and gas industry needs to reduce the operational cost, and to be able to operate safely and sustainably in remote, vulnerable and hazardous areas. Motivating factors include better planning of the well to be drilled, remote control of heavy machinery, removing personnel from hazardous areas, better understanding of the drilling process, and increasing safety for personnel and the environment. However the one important aspect of preventing developments in the drilling industry is that information from the downhole has been very limited. Limitations in the amount and quality of data from downhole adversely impact health, safety and the environment (HSE) and drilling efficiency. Today, information is available only from the near bit area of the drill string via mud-pulse telemetry[1], which by its nature has several disadvantages, i.e., transmission rate as low as $5$ to $20$ bits/s, signal delay depending on the length of the wellbore because the mud wave travels at about $1\, 200$ m/s, reduced signal strength and quality after traveling through the wellbore, unavailability during tripping/connection operations etc. The cost for drilling operations in such conditions has been dramatically increased.

Wired drill pipe (WDP) technology[2-4] has been described as a game-changer for the oil and gas industry. It has the ability to deliver high-resolution data to the surface which can substantially improve real-time decision making, enable precise control of the drilling process by closing the control loop between surface and downhole equipment, and provide visibility of wellbore conditions along the drill string for drilling operations including tripping. The development of WDP technology has been motivated by the need to have more and better data from the near bit area and along the drill string in real-time. WDP technology has allowed sensors including temperature sensors, pressure sensors and many more to be discretely positioned along the drill string, presenting new opportunities for the direct measurement of drilling parameters, that were previously only discernible by semi-real-time models, surface response characteristics, or were directly recorded to memory for subsequent post section analysis. The measurements provide a high frequency feedback to the real-time drilling models which help the drillers construct an overview of the downhole conditions. WDP has been field tested in more than $130$ wells and the number is still increasing. Results show that it has the ability and great potential to play a major roll in higher level control[5-8], and in improving safety and reducing nonproductive time.

Nevertheless, one challenge raised with WDP is the data quality. Because of the nature of the sophisticated conditions in the wellbore, the sensors are inevitably exposed to the environmental effects like noises, tear and worn, possibility of sensor drifting, requirement of re-calibration, transmission errors, etc. Data received from WDP are not necessarily always in good quality. It has been long recognized that poor data quality is hampering the attempts to make use of them in integrated planning, burden collaborative environments and the whole operation workflow. In light of this, many attempts have been made to reduce and eventually eliminate the effects of the poor quality data in order to achieve better decision making and more efficient drilling operations[9-13].

In this paper, an advanced technology, data validation and reconciliation (DVR)[14], has been applied to improve WDP data quality. In practice, the measurements (temperature, pressure, flow rate, stress, strain etc.) themselves are inevitably subjected to measurement errors, usually classified as random, systematic and gross. These errors often affect the results of the measurements. The data validation means improving the quality, achieved in particular by elimination of gross and compensation of systematic errors, and minimizing the influence of random errors. A fundamental method applied to the validation of data is their reconciliation to make them consistent to the mathematical models. Data reconciliation is usually performed by minimizing a least-squares objective function subject to model equations, which come from maximum likelihood formulations and the assumption of normal error distributions. Such model equations range from simple material, component, and energy balances to full models involving all system variables and parameters. A more detailed background review on the DVR technique and its variants can be found from the recent work[15, 16]. In this paper DVR has been applied to improve WDP data quality by using the data redundancy of the mathematical models (fluid density model, fluid temperature model and friction model) as the source of information to calibrate the measurement data (downhole pressure) and calculate the unmeasured drilling parameters (density, friction factor and heat transfer coefficient) simultaneously.

2 Wired drill pipe technology

Up to now and in a reasonably long period into the future, mud pulse telemetry (MPT) is and still will be the most commonly used technology in oil and gas industry. The advantages include low cost in design, production, commissioning and maintenance, and good adaptability to deep wells (up to $12\, 000$ m)[17]. However, the low transmission rate, limited bandwidth, data absence during connection and tripping, and decoding issues hamper the understanding of downhole behavior and in turn potentially increase non-production time. To improve MPT, electromagnetic telemetry has been developed where the transmission rate is around $100$ bits/s. Electromagnetic telemetry allows communication with downhole tools while drilling with aerated fluids such as air, mist and/or foam. Therefore it can be used for managed pressure drilling and underbalanced drilling. However, electromagnetic telemetry has limitations towards deepwater wells due to high electrical conductivity of water, meanwhile it cannot be used for ultra-deep or extended reach wells due to the signal attenuation effect[17].

WDP technology has the ability to provide immediate access to the downhole measurements at a transmission rate of $57\, 600$ bits/s regardless of the fluid environment (e.g., hydrocarbon, water, air or foam) or even total losses. The wider bandwidth delivers data rates approximately three orders of magnitude greater than MPT can do. Using WDP technology makes it possible to continuously provide the measurements (temperature, pressure, shock, vibration, borehole caliper, stress and strain etc.) near the bit, as well as along the drill string. The first commercial use of WDP technology was in Andaman sea 2006 as a solution to a complex problem including narrow drilling pressure margin, complicated geological environment and depleted reservoirs. WDP provides clearer and better vision of the borehole conditions than other telemetry methods.

The networked drill string data provided by WDP, together with advanced wellbore models can bring a lot of benefits to real-time drilling process[6, 10, 11, 18-21]. Most benefits are listed as below

1) Reinforce well control procedure (more information to detect kick volume, location, etc.)

2) Enhance managed pressure drilling operation

3) Strengthen equivalent circulation density management

4) Identify lost circulation or influx zone

5) Monitor annular pressure fluctuations

6) Monitor hole cleaning efficiency and hydraulics (for instance, cuttings loading distribution)

7) Detect drilling problems in early phase conditions, such as stuck pipe, pack off, washout, leakage, etc.

Three different placements of multiple WDP sensors during the deployments are summarized[20]: sensors concentrated in the open hole, sensors biased towards the open hole with some coverage inside casing, and sensors spaced evenly along the drill string. The third placement has high capacity to monitor cutting solids movement along the whole annulus, to evaluate hole cleaning performance, to detect variations from the normal and indicate the location of events quickly.

Compared to MPT the most criticized feature of WDP is its cost during the life cycle. However we shall also see that WDP enables the next generation of downhole measurements which can reduce potential non-productive time. In [8], $10$ distributed pressure sensors are placed along the drill string. The lowest pressure sensor at the drill bit while the distance between two sensors is $150$ m of vertical depth. In the case study it illustrates that multiple sensors have fast response and sensitive kick detection since the sensor close to the kick location could detect the earlier changes and with a higher accuracy. Rasmus et al.[21] describe a case study where WDP is used in aerated drilling fluid environment. Multiple pressure sensors are used to provide data for earlier drilling problem detection and quick decision making while two lost circulation events occurred. Shils et al.[11] shows that WDP is beneficial not just for pressure control but also for indication of state of cuttings transport, as it can give an evidence of solid movements in the well. Furthermore, WDP also helps to decrease drilling time by minimizing number of drill bits and bottom hole assembly trips to fix tool failure[11]. Papers[10, 11, 20] illustrate good field case studies of implementation of WDP systems. As can be seen, WDP with the placement of several sensors shows reliable results and provides the volume of data needed for accurate well control and for understanding of downhole processes, which makes drilling operations safer and less expensive.

The performance of WDP can be summarized in the following three points[11]:

1) Data transmission time. Compared with MPT, WDP has $72\%$ reduction in telemetry time per well.

2) Network maintenance. Compared with MPT, WDP has $77\%$ reduction in non-production time during the 1st year and has $295\%$ improvement in production time.

3) Higher drilling performance.

So even though WDP requires a high cost for implementation, cost savings can be achieved in terms of shorter telemetry time, less maintenance, higher drilling performance, earlier and easier fault detections, better visibility and enhancement of automated drilling. Furthermore, a second generation of WDP with more stable, robust and reliable network system is developed[10]. It is anticipated to further lower the investment cost and improve cost-effective deployment. More practical developments related to optimal position of distributed sensors, improve level of accuracy and investigate possible drilling scenarios are next steps for consideration.

3 Mathematical models 3.1 Drilling fluid density model

Drilling fluid serves as the first line of providing hydrostatic pressure to prevent fluids in the formation from flowing into the wellbore, removing cuttings from the wellbore, suspending and releasing cuttings, cooling, lubricating and supporting the bit and drilling assembly and ensuring adequate formation evaluation, etc. Close monitoring of the properties of drilling fluid can help rig-site personnel make decisions, for instance, provide early warnings of some potential well control problems.

The density of drilling fluid determines the hydrostatic pressure and is the basis for controlling formation pressure. A too high mud density can lead to lost circulation and differential sticking, while a too low mud density can result in well cleaning problems, kick and wellbore instability. Continuous evaluation of the mud density, where the mud flows in the drill string and out in the annulus, is thereby of critical importance. The more the information of the mud density along the drill string, the better the drillers understand the downhole situation.

When drilling operation starts, the drilling fluid is subjected to increasing pressure and temperature. While the higher pressure increases the drilling fluid density, the increased temperature results in the mud density reduction. The accurate knowledge of the behavior of the drilling fluid density is necessary as the wellbore pressure and fluid temperature change along drillpipes. From Fig. 1, it is easily known that pressure and temperature have a major impact on the density. Neglecting the effect of temperature and pressure on fluid density especially for high pressure and high temperature (HPHT) wells, can yield wrong estimation of density, further in turn the erroneous bottom hole pressure estimation. The fluid density model may be written as, see[22]

$ \begin{equation} \rho=\rho~(P, T) \end{equation} $ (1)
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Fig. 1. Density as a function of pressure and temperature where the water based mud is used in Section 5

where $P$ is the pressure, $\rho$ is the density, and $T$ is the temperature. In general the changes in mud density as a function of pressure and temperature are small for a liquid. It is common to use the linearized equation to approximate the fluid density

$ \begin{equation} \rho=\rho_0+\frac{\rho_0}{\beta}(P-P_0)-\rho_0\alpha(T-T_0)\label{density} \end{equation} $ (2)

where $\rho_0, P_0$ and $T_0$ are defined as the reference points for the linearization, $\beta$ is the isothermal bulk modulus of the liquid, and $\alpha$ is the cubical expansion coefficient of the liquid. Fig. 2 illustrates the linearized case of a liquid density model. From Fig. 2 it is understandable that the difference between the true measurement and the calculation from the linearized density model is insignificant. In this paper the linearized density model (2) is employed to estimate the fluid densities along drill string, where the best-fit coefficients ($\alpha$ and $\beta$) can be calculated by using linear least squares method[23, 24].

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Fig. 2. Illustrating the differences between the true density measurements and values calculated from the linearized density model

While drilling with aerated fluids such as air, mist and/or foam, the density model can be expressed as[25]

$ \begin{equation} \rho_f = \alpha_f\rho_g+(1-\alpha_f)\rho \label{density_f} \end{equation} $ (3)

where $\rho$ is the liquid density, $\rho_g$ is the density for gas phase and

$ \begin{equation} \rho_g = \frac{PM_g}{Z_gR_gT} \label{density_g} \end{equation} $ (4)

where $M_g$ is the molar weight of gas, $Z_g$ is the compressibility factor and $R_g$ is gas constant. Substitute (2) and (4) into (3), the density model while drilling with aerated fluids can be formulated as

$ \begin{align} \rho_f = &\frac{\alpha_fPM_g}{Z_gR_gT}+(1-\alpha_f)(\rho_0+\frac{\rho_0}{\beta}(P-P_0)- \nonumber\\&\rho_0\alpha(T-T_0)). \label{density_all} \end{align} $ (5)
3.2 Drilling fluid temperature model

During conventional drilling, heat is transported from the formation up to the surface. Drilling fluid temperature information is essential for modeling and is significantly related to mud density, mud viscosity and other fluid parameters influenced by temperature. Knowledge of fluid temperature is important for prediction of the bottom hole pressure, cuttings transport, drillstring deformation and interpretation of well kicks as mud ballooning is caused by temperature[26]. Hydrostatic pressure increases while cooling the circulating fluid[27] as in this case liquid expands due to the increase of amount of applied heat. Thermal expansion of drilling fluid can be significant in HPHT wells and also in deepwater wells. Circulating fluid temperature depends on numbers of factors related to fluid properties, geology and other, for instance, well depth, circulation rate, formation properties, thermal properties, temperature distribution of the surrounding rocks, mud properties, inlet mud temperature, circulation time and geometry of wellbore and drillpipe etc. Temperature modelling is important and essential for real time operating and monitoring as it can help to detect problems during well operations and for better prediction of drilling fluid and cement behavior.

Most of the investigation methods for fluid temperature modelling in the wellbore are based on Ramey$'$s work[28] where the author estimated the fluid temperature as a function of wellbore depth and circulation time. This analytical temperature model is an excellent approximation but the temperatures calculated in an early transient period are significantly overestimated[29]. An improved mechanistic model for prediction of temperature profiles of tubing and annulus was developed[30], where the formation dimensionless circulation time depending on the formation borehole heat transfer boundary conditions was presented and is widely used by many different temperature models. K${\rm\dot{a}}$rstad and Aadnøy[31] developed a more advanced temperature model to estimate the temperature distribution while drilling for forward and reverse circulation systems. One of the main advantages of the model is that the dominant parameters which influence temperature are considered and an analytical model is provided. Furthermore, such model can be applied to onshore and offshore wells as well as for water-based and oil-based drilling fluids and for completion fluids. The expression for the annular fluid temperature profiles is

$ \begin{equation} T_a(z, t)=\alpha_a{\rm e}^{\lambda_1z}+\beta_a{\rm e}^{\lambda_2z}+g_Gz-Bg_G+T_{sf}\label{temperature} \end{equation} $ (6)

where

$ \begin{align} \lambda_1&=\frac{1}{2A}(1-\sqrt{1+\frac{4A}B}) \end{align} $ (7)
$ \begin{align} \lambda_2&=\frac{1}{2A}(1+\sqrt{1+\frac{4A}B}) \end{align} $ (8)
$ \begin{align} A&=\frac{\omega C_{fl}}{2\pi r_cU_a}\left(1+\frac{r_cU_af(t_D)}{k_f}\right)\label{f_tD} \end{align} $ (9)
$ \begin{align} B&=\frac{\omega C_{fl}}{2\pi r_dU_a} \end{align} $ (10)
$\begin{align} \alpha_a&=\frac{(T_{in}+Bg_G-T_{sf})\lambda_2{\rm e}^{\lambda_2D}+g_G}{\lambda_1{\rm e}^{\lambda_1D}-\lambda_2{\rm e}^{\lambda_2D}} \end{align} $ (11)
$ \begin{align} \beta_a&=\frac{(T_{in}+Bg_G-T_{sf})\lambda_1{\rm e}^{\lambda_1D}+g_G}{\lambda_1{\rm e}^{\lambda_1D}-\lambda_2{\rm e}^{\lambda_2D}}. \end{align} $ (12)

In (9), $f(t_D)$ is a dimensionless time function, which describes how the transient heat flow from the formation to the wellbore changes with time. It is mathematically expressed (with correct units) as follows:

$ \begin{align} f(t_D)=&(1.1281\sqrt{t_D})\times(1-0.3\sqrt{t_D}), \nonumber\\&\text{if}~~10^{-10}\leq t_D \end{align} $ (13)
$ \begin{align} f(t_D)=&(0.406\, 3+0.5\ln{t_D})\times(1+\frac{0.6}{t_D}), \nonumber\\&\text{if}~~ t_D>1.5 \end{align} $ (14)

where the dimensionless time, $t_D$, is given by

$ \begin{equation} t_D=\frac{\alpha_ht}{r_c^2}\times 3\, 600 \end{equation} $ (15)

and the thermal diffusivity is defined as

$ \begin{equation} \alpha_h=\frac{k_f}{\rho_fC_f}. \end{equation} $ (16)

Table 1 summarizes all parameters introduced in Section 3.2. Most parameters involved in (6) can be measured or calculated. However, determination of overall heat transfer coefficients is appreciably difficult. Lots of research work has been done in determining overall heat transfer coefficients over the past decades. Semi-empirical works were published to estimate heat transfer coefficients in annuli in terms of different fluid regimes[32]. In general, the overall heat transfer coefficients depend on the resistances to heat flow through the flowing fluid, tubing metal, casing metal and the cement, etc. It can be expressed as

$ \begin{equation} \frac1{U_a}=\frac{1}{h_d}+\frac{r_d}{k_p}\ln(\frac{r_{do}}{r_d})+\frac{r_{d}}{r_{do}}\frac1{h_a}.\label{ua} \end{equation} $ (17)
Table 1
Drilling parameters used in Section 3.2

To calculate $U_a$, the coefficients of heat transfer of fluid, $h_d$ and $h_a$ should be known, which mostly rely on the Rayleigh number, $N_{REp}$, and Prandtl number $N_{Pr}$. In McAdams$'$s work[33], $h_d$ is determined with correct units by

$ \begin{equation} h_d=0.023\frac{k}{2r_d}(N_{REp})^{\frac{4}{5}}(N_{Pr})^{\frac{2}{5}} \label{ha} \end{equation} $ (18)

where the Rayleigh number is a dimensionless number associated with buoyancy driven flow which is affected by fluid viscosity, flow rate and density

$ \begin{equation} N_{REp}=\frac{2\omega}{r_d\mu}.\label{REp} \end{equation} $ (19)

Prandtl number is defined as the ratio of momentum diffusivity to thermal diffusivity. That is, the Prandtl number is given as

$ \begin{equation} N_{Pr}=\frac{C_{fl}\mu}k. \end{equation} $ (20)

From the above discussion, the overall heat transfer coefficients are influenced by many factors, such as flow regime, flow rate, fluid density, viscosity, thermal conductivity, geometry of wellbore and drillpipe, etc. Its calculation mainly relies upon empirical data and is with high uncertainty and inaccuracy. In this paper, the overall heat transfer coefficient can be extracted with the help of multiple WDP sensors and data reconciliation method. It is a crucial factor to determine the heat energy transmission in the wellbore, to calibrate fluid temperature, hydraulic, torque and drag and rate of penetration models, and in turn to evaluate solid transports and thermally induced stresses in the wellbore and to enhance drilling performance for real-time operation and improve well planning for design phase.

3.3 Drilling fluid friction model

During circulation of drilling fluids, the pressure in the wellbore consists of two components, the hydrostatic pressure and the dynamic fluid pressure loss. The hydrostatic pressure $P_{h}$ is calculated as

$ \begin{equation} P_{h}=\rho gL\cos(I)\label{hy} \end{equation} $ (21)

where all related parameters are given in Table 2. Frictional pressure loss is a function of several factors:

Table 2
Drilling parameters used in Section 3.3

1) Fluid rheological behavior and properties (e.g., viscosity, density, etc.),

2) Flow regime (laminar, transitional or turbulent flow),

3) Flow rate,

4) Wellbore geometry and drill string configuration.

It is generally regarded that frictional pressure loss is directly proportional to its length, the fluid density, the fluid velocity squared and inversely proportional to the conduct diameter. It is calculated from the Fanning equation[34]

$ \begin{equation} P_{Loss}=\frac{2 f\rho v^2L}{D}.\label{loss} \end{equation} $ (22)

In general, the friction factor $f$, called the Fanning friction factor, depends on Reynolds number, $Re$, and the surface conditions of the drill string defined by the roughness of the pipe $\dfrac{\large \epsilon}{D}$. ${ \epsilon}$ represents the average depth of the pipe wall irregularities. Normally the smoother the pipe, the lower the value $\dfrac{\large \epsilon}{D}$. According to (22), $f$ can be calculated by

$ \begin{equation} f=\frac{P_{Loss}D}{2\rho v^2 L}.\label{ff} \end{equation} $ (23)

The Reynolds number $Re$ gives a measurement of the ratio of inertial forces to viscous forces. It is a dimensionless, empirically-deduced parameter. Over the years, the Reynolds number is the most important parameter to define the regime of a drilling fluid flow (laminar, transient, or turbulent). With the correct units, $Re$ is defined as

$ \begin{equation} Re=\frac{Dv\rho}{\mu}. \end{equation} $ (24)

Generally, the flow regime of a liquid changes from laminar to turbulent at a fairly well-defined Reynolds number value[35]. For flow of a Newtonian fluid, the flow is considered laminar if the Reynolds number is less than approximately $2\, 000$, transitional from $2\, 000$ to $3\, 000$, and the turbulent for Reynolds numbers greater than $3\, 000$. In turbulent flow, the friction factor $f$ has to be estimated from experimental data. For instance, the pressure loss $P_{Loss}$ for a liquid is experimentally measured, and $f$ may then be calculated from (23). Several mathematical relationships among friction factor, Reynolds number and roughness ${\large \epsilon}$ have been developed. However, the Reynolds number does not always give an exact evaluation of the regime of flow and the Reynolds numbers calculated from different rheological models may not be consistent. Therefore, determination of the friction factor is with high uncertainty especially in the turbulent flow and in the transition phase. Due to multiple sensors of WDP technology, the pressure difference between one segment of the drill string is measured, the friction factor in the segment could be easily calculated from (23). More discussions will be given in Section 5.

3.4 Model framework

Various types of WDP sensors can be used and placed along the drill string, including pressure sensors, temperature sensors, inclinometers, bending/vibration and rotation sensors, flow-rate sensors and many more. Among them, pressure and temperature sensors that measure rapidly changing conditions may be particularly valuable, since it can easily and precisely monitor the change of downhole situation, for instance, wellbore/formation stability, equivalent circulation density measurements, wash out etc. Placing multiple sensors along the drill string establishes distributed measurements which vary with time and depth.

In this paper, according to the location of the sensors positioned along the drill string, we consider the downhole network system by dividing the annulus between the drill string and the wall of the borehole into a number of control volumes or segments ($n_s$), as shown in Fig. 3. In each annular segment ($j$) the drilling fluid is assumed to have a uniform pressure, $P_{(j)}$, a uniform temperature $T_{(j)}$ and a uniform density $\rho_{(j)}$. And each segment has its own wellbore diameter, $D_{W(j)}$, drill string diameter $D_{D(j)}$, inclination, $I_{(j)}$ and length $L_{(j)}$.

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Fig. 3. Illustration of annulus with $n_s=9$ control volumes/segments

Between one segment, the pressure difference $P_{(j+1)}-P_{(j)}$ consists of two components, the hydrostatic pressure $P_{h(j)}$ and the dynamic fluid pressure loss $P_{Loss(j)}$. The hydrostatic pressure $P_{h(j)}$ is then calculated from (21)

$ \begin{equation} P_{h(j)}=\rho_{(j)}gL_{(j)}{\rm cos}(I_{(j)}), \quad \forall j=1, \cdots, n_s-1. \end{equation} $ (25)

The pressure loss $P_{Loss(j)}$ is calculated according to (22):

$ \begin{equation} P_{Loss(j)}=\frac{2f_{(j)}\rho_{(j)}v_{(j)}^2}{D_{a(j)}} L_{(j)}, \quad \forall j=1, \cdots, n_s-1 \end{equation} $ (26)

where $f_{(j)}$ is friction factor in the $j$-th segment and

$ \begin{align} &v_{(j)}=\frac{Q}{A_{(j)}} ~~ \text{with}~~ A_{(j)}=\frac\pi4(D_{W(j)}^2-D_{D(j)}^2) \end{align} $ (27)
$ \begin{align} &D_{a(j)}=D_{W(j)}-D_{D(j)}. \end{align} $ (28)

Therefore, the pressure $P_{(j+1)}$ can be expressed as

$ \begin{align} P_{(j+1)}=P_{(j)}+P_{h(j)}+P_{Loss(j)}, \quad\forall j=1, \cdots, n_s-1. \label{pressure} \end{align} $ (29)
4 DVR method

To be able to understand the complicated downhole processes, a certain amount of data is necessary. Fast and accurate decision making is also built on availability of qualitative data. As is known that WDP has the ability to provide a big amount of data in real-time. However, it does not necessarily mean improved understanding of downhole processes or better decision making. When the volume of data is beyond some limit, it creates confusion for the observer and reduces quality of decision making dramatically; see the discussion in [7]. As a result, more efforts should be made to understand and interpret the received data such as data interpretation technique and some new tools to increase a variety of available sensors for accuracy, diversity and observability of WDP technique.

4.1 Mathematical formulation

DVR is an advanced technology which uses process information and mathematical methods in order to automatically correct raw measurements, estimate model parameters/unmeasured variables in industrial processes. The use of DVR allows for extracting accurate and reliable information from raw measurement data and produces a consistent set of data representing the most likely process operation. The interest of applying DVR techniques has started from 1980$'$s. Nowadays DVR has been widely used in various processing industries, such as refinery, chemical and petrochemical, etc.

In the general case, not all variables of the process are measured. The estimates of unmeasured variables as well as model parameters are also obtained as part of the DVR problem. Data reconciliation can be formulated by a constrained weighted least squares optimization problem, where the measurement errors are minimized with process model constraints. Given $n$ measurements, DVR can mathematically be expressed as an optimization problem of the following form

$ \begin{align} \min\limits_{y^*, x}J(x, y^*)&=\sum\limits_{i=1}^n(\frac{y_i^*-y_i}{\sigma_i})^2\label{cost} \end{align} $ (30)
$ \begin{align} &\text{subject to}\nonumber\\ &f_m(x, y^*)=0 \end{align} $ (31)
$ g_m(x, y^*)\leq0 $ (32)

where $y_i$ is the raw measurement value of the $i$-th measurement, $y^*=\{y_1^*, \cdots, y_n^*\}$, $y_i^*$ is the reconciled value of the $i$-th measurement, $x$ is a vector of estimates for unmeasured values of the process and $\sigma_i$ is the standard deviation of the $i$-th measurement. $f_m$ is a vector describing the functional form of model equality constraints and $g_m$ is a vector describing the functional form of model inequality constraints which include simple upper and lower bounds. Solving this optimization problem provides simultaneously the measurement error corrections and the estimates for unmeasured variables.

4.2 WDP data quality improvement

Pressure $P_{(n_s)}$ and temperature $T_{(n_s)}$ measurements from near the bit are combined with measurements $\{P_{(j)}, T_{(j)}\}$ that are measured at various locations along the networked drill string. The additional sensors provide an increased accuracy as well as redundancy that are inherent through the multiple sensors. That is, in case of sensor malfunction, wrong data transmit or other faults in the wellbore, the DVR technique is applied to the network system in order to correct the raw measurements $P_{(j)}$ by using the data redundancy of the system. By using the above mentioned mathematical models in Section 3, the unmeasured variables $\rho_{(j)}(j=1, \cdots, n_s-1)$, unknown drilling parameters $f_{(j)}(j=1, \cdots, n_s-1)$ and $U_a$ are estimated simultaneously as the part of the problem. Suppose $\{P_{(j)}, T_{(j)}, j=1, \cdots, n_s\}$ are measured by WDP. The DVR problem can be formulated as the following form

$ \begin{align} &\min\limits_{P^*_{(j)}, \rho_{(j)}, f_{(j)}, U_a}J(P^*_{(j)}, \rho_{(j)}, f_{(j)}, U_a)=\nonumber\\ &\qquad\qquad\qquad \sum\limits_{j=1}^{n_s}\left(\left(\frac{P^*_{(j)}-P_{(j)}}{\sigma_{P, j}}\right)^2 +\left(\frac{T^*_{(j)}-T_{(j)}}{\sigma_{T, j}}\right)^2\right)\label{cost1} \end{align} $ (33)

subject to

$ \quad P^*_{(j+1)}=P^*_{(j)}+P_{h(j)}+P_{Loss(j)}, ~~\forall j=1, \cdots, n_s-1 $ (34)
$ \quad P_{h(j)}=\rho_{(j)}gL_{(j)}\cos(I_{(j)}), ~~\forall j=1, \cdots, n_s-1 $ (35)
$ P_{Loss(j)}=\frac{f_{(j)}\times\rho_{(j)}\times v_{(j)}^2}{D_{a(j)}}\times L_{(j)}, ~~\forall j=1, \cdots, n_s-1 $ (36)
$ \begin{align} & \rho_{(j)}=\rho_0+\frac{\rho_0}{\beta}(P^*_{(j)}-P_0)-\rho_0\alpha(T_{(j)}-T_0), \nonumber\\ &~~~~~~~~~ \forall j=1, \cdots, n_s-1 \end{align} $ (37)
$ \begin{align} & T^*_{(j)}=\alpha_a{\rm e}^{\lambda_1z(j)}+\beta_a{\rm e}^{\lambda_2z(j)}+g_Gz(j)-Bg_G+T_{sf}, \nonumber\\ &~~~~~~~~~ \forall j=1, \cdots, n_s-1 \end{align} $ (38)
$ P^*_{(j)}>0, ~~\forall j=1, \cdots, n_s $ (39)
$ T^*_{(j)}>0, ~~\forall j=1, \cdots, n_s $ (40)
$ f_l<f_{(j)}<f_u, ~~\forall j=1, \cdots, n_s-1 $ (41)
$ U_{a, l}<U_a<U_{a, u} $ (42)
$ \rho_l<\rho_{(j)}<\rho_u, ~~\forall j=1, \cdots, n_s-1 $ (43)

where $\sigma_{P, j}$ and $\sigma_{T, j}$ are the standard deviation of the $j$-th measurement of pressure and temperature respectively. $z(j)$ is the depth where the sensor $(j)$ is located. $f_l$ and $f_u$ are given lower and upper limit of the friction factor. $U_{a, l}$ and $U_{a, u}$ are given lower and upper limit of the overall heat transfer coefficient. $\rho_l$ and $\rho_u$ are given lower and upper limit of the density. Solving this optimization problem, the corrected pressure measurements $P_{(j)}^* (j=1, \cdots, n_s)$, the estimates for unmeasured variables $\rho_{(j)}, f_{(j)} (j=1, \cdots, n_s-1)$ and $U_a$ are obtained, which is illustrated by a case study in Section 5. Note that in the problem (33)$-$(34) only the liquid is considered. When aerated fluids are used, density function (37) is replaced by (5).

In this paper, only the steady-state models (2), (6) and (29) are considered. The main consideration is to employ data reconciliation method on steady-state models to eliminate gross errors, to minimize random errors, to compensate systematic errors and calibrate mathematical steady-state models. Therefore, models calibration, data post-analysis and model verification are our main purpose. Recently, in papers [5-8, 36], a low order dynamic hydraulic model introduced by Stamnes et al.[37] is used for parameters estimation and controller design for managed pressure drilling operation using WDP technology. In their work, dynamic data reconciliation methods (moving horizon estimation or its combination with extended Kalman filter) are used to determine the dynamic parameters, such as fluid density, friction factor and gas influx flow rate etc. The optimal annulus friction factor and density are fed to the controller for real-time operations. To feed all the required information into the dynamic model for controller design to manipulate drilling operational parameters, the dynamic data reconciliation method[9] is more advantageous. More discussions about benefits and challenges of dynamic reconciliation methods are summarized in [9].

5 Simulations and discussions

In the case study we use the NPSOL sequential quadratic programming algorithm to solve the DVR problem (33)$-$(43). Matlab is used for simulation and computations, with the tomlab interface to NPSOL.

The simulated well is a simple vertical well with an initial depth of $1\, 980$ m. The well has one casing with length $1\, 800$ m, including $455$ m of riser. The remainder of the well is an open hole. At the start of simulation, the bit is inside the openhole at $1\, 980$ m. For simplicity, the mud is an incompressible fluid with specific gravity $1.1$ sg. Pressure and temperature are recorded every $90$ m along the drill string from the drill bit to the $720$ m above the bit. Therefore $n_s=9$. It is important to catch the dynamics and friction loss by the bottom hole assembly and drill collars, where the drill string diameter changes and the flow area is narrow. The bottom hole assembly (BHA) is a typical configuration with various components up to $270$ m above the bit.

The flow rate is shown in Fig. 4. The well and mud data used in the case study are shown in Table 3. Density measurement with respect to pressure and temperature is shown in Fig. 1. The disturbed pressure and temperature values along drill string are shown in Figs. 5 and 6. Color versions of Figs. 5 and 6 are available online. Some observations are made:

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Fig. 4. Flow rate

Table 3
Drilling parameters used in Section 5

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Fig. 5. Distributed temperature measurements

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Fig. 6. Distributed pressure measurements

1) Due to assumed incompressible mud, the pressure in the annulus increases with the depth and the flow rate during drilling. In this case an increase of $10$ bars is observed between each sensor. The situation is a bit complicated around the BHA. The pressure drops/increases (refer to $P_{(6)}, P_{(7)}, P_{(8)}, P_{(9)}$) depend on the BHA component outer diameters $D_{w(j)}$, flow rate and mud density with changes in terms of the temperature and friction factor $f_{(j)}$.

2) From Fig. 6, in the first $5$ mins, $P_{(6)}$ and $P_{(7)}$ overlap. After the first $5$ mins, the pressure difference between $P_{(6)}$ and $P_{(7)}$ is very small, while the pressure difference between $P_{(5)}$ and $P_{(6)}$ is very large. When the time is within $35$ mins and $40$ mins, the pressure difference ($P_{(6)}$ and $P_{(7)}$) is less than $5$ bars. Considering the surface fluid density is 1 100 kg/m$^3$, the hydrostatic pressure in each segment is around $9$ bars. When pumps are off ($t\leq 5$ mins), the pressure difference in each segment should be close to $9$ bars (friction pressure loss is zero due to no circulation in the wellbore). When pumps are on ($t\geq 5$ mins), most friction pressure loss is around the BHA due to the narrow flow area. It means that the pressure difference between $P_{(6)}$ and $P_{(7)}$ should be much greater than $9$ bars. Therefore, it indicates that the values of pressures $P_{(5)}$, $P_{(6)}$ and $P_{(7)}$ are incorrect.

In order to make $P_{(j)}$ and $T_{(j)}$ from simulator more closed to true measurement, Gaussian noises are intendedly added to $P_{(j)}$ and $T_{(j)}$ to emulate the measurement with random errors. Solving the DVR problem (33)$-$(43), the pressure measurement $P_{(j)}$ is corrected, which is represented by $P_{(j)}^*$, see Fig. 7. From Section 3.2, we understand that friction factor is severely affected by the wellbore geometry, drill string, BHA configuration and flow regime, etc. Since the diameters of drill string/BHA and annulus, and flow regime are the same in the drill string segments ($j=1, \cdots, 5$), as well as the BHA segment ($j=6, 7$), therefore, only three friction factors are considered to simplify the problem, which is friction factor around the drill bit ($f_{(8)}$), friction factor around the BHA where ($f_{(6)}=f_{(7)}$), friction factor around drill string where ($f_{(1)}=\cdots=f_{(5)}$).

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Fig. 7. Distributed pressure calculations via DVR

From Figs. 6 to 8, pressures $P_{(j)}$, especially pressures $P_{(5)}$ and $P_{(6)}$, are corrected. It is easy to see that the pressure difference between sensors in the BHA part ($P_{(j+1)}^*-P_{(j)}^*, j=6, \cdots, 8$) is much larger than the one above the BHA part($P_{(j+1)}^*-P_{(j)}^*, j=1, \cdots, 5$), which is coincident with the fact that the most frictional loss happens in BHA part.

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Fig. 8. Comparison of measurements before and after correction

As the part of the solution, the unmeasured density pressures $\rho_{(j)}$, $f_{(j)}$ and $U_a$ are obtained. The results are shown in Figs. 9 to 11.

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Fig. 9. Distributed density calculations via DVR

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Fig. 10. Distributed friction factor calculations via DVR

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Fig. 11. Overall heat transfer coefficient calculations via DVR

In Fig. 1, it shows that the density is increased by approximately 3 kg/m$^3$ when the pressure is increased from $172$ bar to $344$ bar under the same temperature; while the density is decreased by around 16 kg/m$^3$ when the temperature is dropped by $30^\circ$C under the same pressure. Therefore, the temperature seems the more dominant factor influencing the density. Fig. 9 shows that the densities $\rho_{(j)}, j=1, \cdots, 6 $ are significantly affected by the fluid temperatures. Fig. 5 shows that the temperatures at the nodes ($j=1, \cdots, 7$) increase by around $40^\circ$C during the simulation. From Fig. 1, it indicates that the density between two nodes $(j=1, \cdots, 6)$ might decrease by approximately 15$-$20 kg/m$^3$. It is coincident with Fig. 9. The change of the density $\rho_{(7)} $ is also significantly influenced by the temperature. Since the mud cools the drill bit and is warmed by the formation when it is circulated from the bottom, the highest temperature is measured at a few hundred meters above the bit, see Fig. 5. Therefore the density $\rho_{(7)} $ is a little bit higher than the density $\rho_{(4, 5, 6)}$ since the mud is warming by formation. It is also observed that density $\rho_{(8)} $ is obviously increasing when it starts drilling and keeps a slight variation when it is continuously drilling and circulating. Since the mud is cooling the drill bit, $T_{(9)}$ is dropping when starting drilling. After $5$ mins, $T_{(9)}$ becomes stable since the fluid is continuously circulated. The pressure becomes the dominant factor to affect the density. Then, the change of $\rho_{(8)} $ follows the change of $P_{(9)}$, which is also observed in Fig. 9. We know that the friction factor of the bit is the largest while the one around the drill string is smallest due to the different flow regime and fluid velocity, which is consistent to the observations. Also from Fig. 10 we can see that the change of the friction factor is mainly dominated by the change of the flow rate. In the laminar, the lower the flow rate, the larger the friction factor. Fig. 11 shows the calculated overall heat transfer coefficient $U_a$. From (11), (18) and (19), it is easy to know that the overall heat transfer coefficient is proportional to the flow rate. The larger flow rate, the larger the overall heat transfer coefficient, which is consistent to the curve of $U_a$ shown in Fig. 11.

6 Conclusions

This paper presents WDP data quality control issues. Some overall conclusions are summarized as below:

1) Through an optimization approach, model parameters are refined based upon additional data from sensors along the drill string, provided by the networked drill string.

2) Including all parameters essential to the well and drill string system results in accurately reproducing actual drilling parameters.

3) The framework is used with respect to the networked pressure and temperature measurements.

4) This approach allows for both improving measurements and monitoring the downhole conditions.

5) Conditions of sensor malfunction and miss-calibration of a sensor could be distinguished.

6) This proposed method has the capacity of distinguishing a sensor defect from a kick or other unwanted events.

7 Future work

The procedure used in this paper to calibrate pressure at a given sensor could extend to detect a malfunctioning sensor. A similar approach can be used to detect slowly developing problems in the wellbore, such as bad hole cleaning. Extending this to a full system for fault detection and event identification is a priority to the authors.

Acknowledgements

This work is supported by University of Stavanger, Norway. The authors wish to thank SINTEF, the Center for Integrated Operations in the Petroleum Industry and the management of National Oilwell Varco IntelliServ for their contribution and support in publishing this paper.

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