Research ArticleReal-Time Interpretation Model of Reservoir CharacteristicsWhile Underbalanced Drilling Based on UKF

Miao He ,1,2 Yihang Zhang ,1 Mingbiao Xu,1,2 Jun Li,3 and Jianjian Song 1

1School of Petroleum Engineering, Yangtze University, Wuhan 430100, China2Hubei Cooperative Innovation Center of Unconventional Oil and Gas, Yangtze University, Wuhan 430100, China3China University of Petroleum Institute of Petroleum Engineering, Beijing 102249, China

Correspondence should be addressed to Miao He; hemiao@yangtzeu.edu.cn and Yihang Zhang; luckyzhangyh@foxmail.com

Received 22 October 2019; Revised 15 March 2020; Accepted 6 May 2020; Published 19 May 2020

Academic Editor: Shengnan Nancy Chen

Copyright © 2020 Miao He et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This study presents a novel interpretation model for reservoir characteristics while underbalanced drilling (UBD), byincorporating an unscented Kalman filter (UKF) algorithm in a three-phase variable mass flow model of oil, gas, andliquid. In the model, the measurement parameters are simplified to bottomhole pressure and liquid outlet flow, fordecreasing the amount of the computation and time. By taking into account real-time measurements, the permeability andreservoir pressure along the well can be continuously updated. Three cases including single-parameter and double-parameter estimations have been simulated, and the performance is tested against the extended Kalman filter (EKF). Theresults show that single-parameter estimation of reservoir permeability or pressure achieves superior performance. Thefiltered values of bottomhole pressure and outlet flow trace the measured values in real time. When a new section of areservoir is opened, the estimated reservoir permeability or pressure can always be quickly and accurately returned to itstrue value. However, it is not possible for the double-parameter estimation to obtain good results; its interpretationaccuracy is low. UKF is superior to EKF in both estimation accuracy and convergence speed, which further illustrates thesuperiority and accuracy of the novel interpretation model based on UKF. Benefits from this model are seen in accuratebottomhole pressure and reservoir characteristic predictions, which are of major importance for safety and economicreasons during UBD and follow-up completion operations.

1. Introduction

Underbalanced drilling (UBD) is a technical means of dril-ling under the condition of negative pressure in a wellbore[1, 2]. UBD began in the 1930s and developed rapidly inthe recent years [3], because of its great advantages [4] inimproving rate of penetration (ROP), discovering reservoirs,and reducing reservoir damage. When a reservoir is opened,formation fluid will enter the wellbore under the action ofnegative pressure difference, which will lead to changes inannulus pressure, flow rate, and other parameters [5, 6].Because of the coupling effect between the wellbore and for-mation, reservoir characteristics such as reservoir pressureand permeability may be obtained by using measured dataacquired during drilling operation, such as bottomhole pres-sure, flow rate, and injection rate.

Estimation of the near wellbore characteristics of a reser-voir gives important information in the drilling and comple-tion process, helping the technical crew make betterdecisions. It is of great significance for early identifying a res-ervoir, reducing drilling risk, and improving drilling time.On this basis, Kardolus and Kruijsdijk [7] put forward theidea of formation characteristic interpretation while under-balanced drilling. Differently from the traditional approach,it needed not to go down the test string; the permeability pro-file near the wellbore was estimated while drilling based on asimplified analytical model. Actually, the joint hydraulicsmodel and estimation algorithm is the basic theory of param-eter interpretation. Several researches have addressed thebasic theory works, respectively, for a two-phase flow model-ing or estimation algorithm of UBD. Rommetveit et al. [8]presented a transient UBD simulator, DynaFloDrill, and the

HindawiGeofluidsVolume 2020, Article ID 8967961, 15 pageshttps://doi.org/10.1155/2020/8967961

predictions were validated by the full-scale experiments per-formed with either parasite string or drill string gas injection.Lage et al. [9] proposed a composite discrete scheme fordynamic two-phase flow modelling, combining the first-order Lax-Friedrichs and the second-orderMacCormack for-mat, to describe the transient behavior in UBD. Fjelde et al.[10] established a new multiphase flow model of UBD, whichused the MUSCL format to modify the classical upwind for-mat, and could better describe the pressure fluctuation lawduring pipe connection. Perez-Tellez et al. [11] proposed amechanical model for predicting the pressure of two-phaseflow in an annulus and a drill pipe, in which a drift flux modelis coupled. Khezrian et al. [12] developed a two-fluid model inthe Eulerian frame of reference for simulation of gas-liquidtwo-phase flow in the UBD operation. On this basis, Vefringet al. [13] evaluated the performance of the nonlinear leastsquare methodology. The Levenberg-Marquardt optimizationalgorithm was used to estimate reservoir pressure and perme-ability in the UBD process. Tang [14] applied the damped leastsquare method to study the interpretation of reservoir charac-teristics in underbalanced drilling. However, those approachesare both nonrecursive, which need to use all historical data, sothey are best suited for postanalysis of data.

Kalman filter (KF) estimation methods have been widelyused in oil industry in recent years, because of their better per-formance in real time. Vefring et al. [15] are the representativeresearchers in reservoir characteristic interpretation whileunderbalanced drilling. The measurements were set up aspump pressure, bottomhole pressure, and gas and liquid outletflow. Based on the Levenberg-Marquardt and extended Kal-man filter (EKF), interpretation models of reservoir pressureand permeability were established, in which EKF was a real-time recursive algorithm. Nazari et al. [16] introduced theunscented Kalman filter (UKF) to estimate the gas-liquid mix-ing velocity in the drill string and annulus. Several pressuresensors were installed in the annulus to improve the accuracyand robustness. Lorentzen et al. [17] designed an EKF methodbased on a two-phase flow model to regulate the key parame-ters of a drift flux model in UBD operation. Nikoofard et al.[18, 19] introduced a kind of nonlinear horizontal movingobserver, which was used to estimate the liquid quality and liq-uid production coefficient of annular air in an underequili-brium state. Nygaard et al. [20] established a wellborepressure control method based on the UKF algorithm toaddress the problem of pressure fluctuation caused by singleconnection and flow change when a gas reservoir was openedand conducted inversion interpretation for reservoir perme-ability. Gravdal et al. [21] established a pressure interpretationmodel by using the UKF algorithm, but only the friction coef-ficient was used as the estimated parameter.

Based on the previous researches, most of the Kalman fil-ter studies on UBD use pump pressure, bottomhole pressure,and gas and liquid outlet flow parameters as the measure-ments. However, with more measurement parameters, theamount of the computation and time will be increasedaccordingly. The measurement parameters in the model needto be further optimized and evaluated due to their internalrelations. Furthermore, currently there is still less researchof using UKF in the UBD process.

In the present paper, a novel interpretation model for res-ervoir characteristics while underbalanced drilling is devel-oped, by incorporating the UKF algorithm in a three-phasevariable mass flowmodel of oil, gas, and liquid. The measure-ment parameters are simplified to bottomhole pressure andliquid outlet flow, to continuously update the permeabilityand reservoir pressure along the well. Based on this interpre-tation, the future state of the reservoir system near the well-bore can be predicted, helping the management of thewellbore during UBD and follow-up completion operations.

The outline of the paper is as follows. First, the three-phase variable mass flow model of oil, gas, and liquid, includ-ing a drift flux model and frictional pressure loss model, ispresented. Then, the real-time estimation method of UKF isdescribed. Finally, results from simulations performed withthree cases, like reservoir permeability and pressure single-parameter estimation and double-parameter estimation, areconducted. The performance of UKF is also evaluated againstEKF in permeability single-parameter estimation.

2. Three-Phase Variable Mass Flow Model ofOil, Gas, and Liquid

Underbalanced drilling technology is realized by gas injec-tion into a drill string. When underbalanced drilling encoun-ters reservoirs, the oil and gas in the formation will continueto flow into the wellbore. With the elapse of time, reservoirgas and oil rise upward in the wellbore and reservoir openinglength prolongs, resulting in a gradual increase in reservoirproduction. Therefore, the wellbore is actually a variablemass flow system consisting of injected gas, drilling fluid,produced gas, produced oil, and cutting multiphasecomponents.

A mathematical model of three-phase variable massflow of oil, gas, and liquid is established based on the the-ory of a wellbore multiphase flow and dynamic reservoirmodel [22, 23]. The basic assumptions are as follows:

(1) The wellbore fluid flows in one dimension, ignoringthe radial flow change

(2) Drilling fluid is water-based mud (WBM), set as theHerschel-Bulkley model, without considering themass transfer between oil, gas, and liquid phases

(3) Ignoring the influence of heat transfer between thewellbore and formation, one can calculate the tem-perature in the wellbore by a linear geothermalgradient

(4) The effect of cuttings on wellbore flow is small, so it isnot considered

2.1. Multiphase Flow Equation. The mass conservation equa-tion of injected gas is

∂∂t

xigρgαgA� �

+ ∂∂z

xigρgαgvgA� �

= 0: ð1Þ

2 Geofluids

The mass conservation equation of drilling fluid is

∂∂t

ρlαlAð Þ + ∂∂z

ρlαlvlAð Þ = 0: ð2Þ

The mass conservation equation of produced gas is

∂∂t

xfgρgαgA� �

+ ∂∂z

xfgρgαgvgA� �

= qfg: ð3Þ

The mass conservation equation of produced oil is

∂∂t

ρoαoAð Þ + ∂∂z

ρoαovoAð Þ = qo: ð4Þ

The three-phase momentum conservation equation ofoil, gas, and water is

∂∂t

ρgαgvgA + ρlαlvlA + ρoαovoA� �+ ∂∂z

ρgαgv2gA + ρlαlv

2l A + ρoαov

2oA

� �+ ρgαg + ρlαl + ρoαo� �

g sin θA + ∂ pAð Þ∂z

+ A∂pf∂z

= 0:

ð5Þ

The P-V-T equation is

ρg =Mgp

ZRT: ð6Þ

The gas-liquid drift flux model is calculated as

vg = c0vm + vgr: ð7Þ

The model of frictional pressure loss along the well is cal-culated as

∂pf∂z

= 2f ρmvmD

vmj j + ∂pac∂z

, ð8Þ

where A is the annular area. ρg, ρl, and ρo are the density ofgas, drilling fluid, and oil, respectively. αg, αl, and αo are thevolume fraction of gas, drilling fluid, and oil, respectively.vg, vl, and vo are the actual flow rates of gas, drilling fluid,and oil, respectively. xig and xfg are the mass fraction ofinjected gas and produced gas, respectively. qfg and qo arethe influx rate of gas and oil phases. g is the gravitationalacceleration. θ is the angle between the wellbore and horizon-tal direction. pf is the pressure drop. pac is the accelerationpressure drop. Mg is the molar mass of the gas. p is the well-bore pressure. Z is the deviation factor, solved by the PR-EOSmodel [24]. R is a general gas constant. c0 is the gas phase dis-tribution coefficient. vgr is the gas slip velocity. f is the fan-ning friction coefficient. ρm is the density of gas-liquidmixture. vm is the velocity of gas-liquid mixture. D is theequivalent diameter.

Distribution coefficients, slip velocity parameters, andpressure drop model will change with the change of a gas-liquid two-phase flow pattern. Therefore, accurate flow pat-tern identification is an important prerequisite for establish-ing a comprehensive multiphase flow mathematical model.The flow pattern transition of gas-liquid two-phase flow is acomplex physical process. With the change of volume frac-tion, velocity, pressure, and relative position of the two-phase medium, the shape of the interface changes, whichleads to the change of the flow pattern. At present, there isno mature theoretical support for the physical mechanismof flow pattern transition. For different flow pattern changes,empirical formulas of relevant parameters are often fitted bymeans of experiments, and then, critical conditions of flowpattern transition are determined. According to previousresearch results [25–27], two-phase flow patterns in the ver-tical wellbore are divided into five categories: bubble flow,dispersed bubble flow, slug flow, churn flow, and annularflow. For different flow patterns, the distribution coefficients,slip velocity, and pressure drop along the well are deter-mined. The multiphase flowmodel is solved by a simple fronttracking technique and finite difference numerical method.The specific algorithm and formula are not described here.A previous paper by He et al. [28] has introduced the algo-rithms and formulas in detail.

2.2. Dynamic Reservoir Model. When reservoir oil and gasinflux occurs in the drilling process, the influx mode is nega-tive pressure influx, which can be regarded as plane radialflow in isotropic homogeneous elastic porous media. TheDake model [29] with an analytic solution is used to describethe flow process. In the process of underbalanced drilling, theopened zones of the reservoir are all involved in the coupledflow with the wellbore, thus forming the whole variable massflow process. The opened reservoir is divided into n unitsalong the axis of the wellbore, as shown in Figure 1. Thedynamic reservoir model can be expressed as follows:

qi tið Þ = 4πkiΔh pr,i − pa,i� �

μ 2S + ln 4kiti/eγϕμcrw2ð Þ½ � , ð9Þ

where pa is the annular pressure. pr is the reservoir pressure. qis the influx rate. k is the reservoir permeability. S is the skinfactor. Δh is the length of the open reservoir interval per unittime. t is the duration of the open reservoir interval. φ is thereservoir porosity. μ is the viscosity of reservoir fluid. c is thecompressibility coefficient of reservoir fluid. rw is the bore-hole radius. γ is the Euler constant.

qi-1 qi+1qi

Figure 1: Diagram of wellbore-reservoir coupling variable massflow.

3Geofluids

3. Interpretation Model of ReservoirCharacteristics Based on UKF

3.1. Determination of Interpretation Model Parameters. Thedescription of the gas injection in the drill string of underba-lanced drilling technology is as follows: gas passes throughthe gas injection pipeline, mixes with the drilling fluidpumped by a mud pump, and jointly injects into the drillstring. With the opening of the reservoir section during dril-ling, reservoir oil and gas fluids gradually influx into the well-bore and return to the wellhead together with the injected gasand drilling fluids. After passing through the gas-liquid sep-arator on the ground, the gas phase and liquid phase arefinally separated. The schematic diagram is shown inFigure 2.

In the previous literature, pump pressure, bottomholepressure, and gas and liquid outlet flow rate were often regu-lated as measurement parameters. The bottomhole pressureis measured by pressure while drilling (PWD) instruments.However, according to the principle of the U-tube, the pres-sure transfer relation can be constructed in the drill stringbetween bottomhole pressure and pump pressure. In addi-tion, there is a mathematical relationship between the gas-liquid outlet flow rate and wellbore-reservoir coupled flowsystem. Because the mass transfer of gas and liquid is notconsidered, according to the superposition principle, thesum of gas-liquid outlet flow is equal to the sum of inlet flow,

reservoir influx rate, and gas expansion rate. The formulasare as follows:

pb = ps + ph − Δpd − Δpt,qout,l + qout,g = qin,l + qin,g + qinf + qexp:

ð10Þ

Therefore, the measurement parameters can be simpli-fied into two categories: bottomhole pressure and liquid out-let flow. The expressions are as follows:

pb = pb t1ð Þ, pb t2ð Þ,⋯,pb tNð Þ½ �,qout,l = qout,l t1ð Þ, qout,l t2ð Þ,⋯,qout,l tNð Þ� �

,ð11Þ

where pb is the bottomhole pressure. ps is the pump pressure.ph is the hydrostatic column pressure in the drill string. Δpd isthe circulating pressure loss in the drill string. Δpt is the pres-sure loss of the drill bit. qout,l and qout,g represent liquid andgas outlet flow rates, respectively. qin,l and qin,g indicate liquidand gas inlet flow, respectively. qinf is the reservoir influx flowrate. qexp is the gas expansion rate. t1, t2,⋯, tN is theN pointsof time corresponding to the measured data.

Generally, reservoir lithology is complex, and the internalpore and fracture distribution is random and nonuniform.Therefore, the reservoir to be drilled can be subdivided intoseveral units. Reservoir permeability k and reservoir pressurepr of each unit may be different, as shown in Figure 2. Reser-voir permeability and pressure are determined as interpreta-tion parameters of the model, and the expressions are asfollows:

k = k t1ð Þ, k t2ð Þ,⋯,k tNð Þ½ �,pr = pr t1ð Þ, pr t2ð Þ,⋯,pr tNð Þ½ �:

ð12Þ

3.2. UKF Algorithm. The initial Kalman filtering technique isonly suitable for linear systems [30], but most of the realproblems are essentially nonlinear. Then, an extended Kal-man filter and unscented Kalman filter are developed succes-sively for nonlinear systems [31–33]. The main differencebetween EKF and UKF is the way Gaussian random variablesare represented for propagating through system dynamics.UKF adopts a deterministic sampling method with a minimalset of carefully chosen sample points, instead of the local lin-earization in EKF. These sample points are propagatedthrough the nonlinear system and capture the posteriormean and covariance accurately to the third order for allnonlinearities. In contrast to EKF, UKF appears later andrequires no calculation of the complex Jacobian matrix andis a more advanced nonlinear filtering technology.

Using UKF, it is possible to combine the informationobtained from the measurements with the model to get animproved real-time estimate of the state vector of the system.The state vector is the interpretation parameter. Combinedwith the discussion in Section 3.1, the state vector x and themeasurement z in the model are determined as follows,

Reservoir …

i

……

Mud pumpStandpipe pressure gaugePWDCasing pressure gaugeChoke valve

Gas-liquid separatorLiquid flow meterGas flow meterGas injection pipeline

1 k=k1

k=ki

pr=pr,1

qin,g

qin,lqout,l

qout,g

pr=pr,i

①

①

⑤

⑥

⑦

⑧

⑨

②

③

④

⑨

②

③

④ ⑤

⑦

⑥

⑧

Figure 2: Diagram of underbalanced drilling of gas injection in thedrill string.

4 Geofluids

where the subscript k denotes the time k.

x = k tkð Þ, pr tkð Þ½ �T ,z = pb tkð Þ, pout,l tkð Þ� �T

:ð13Þ

The state space form of the nonlinear system based onstate estimation is as follows:

xk = f xk−1ð Þ +wk−1,zk = h xkð Þ + vk,

ð14Þ

where xk is a state vector. fð⋅Þ is a nonlinear system statefunction. zk is a measurement vector. hð⋅Þ is a nonlinear mea-surement function, which represents the variable mass flowmodel in this work. wk and vk are, respectively, the processnoise and measurement noise of the system, which satisfythe zero mean white noise distribution, wk~Nf0,Qkg, andvk~Nf0, Rkg, and they are not related to each other.

The structure of UKF is basically the same as that of EKF,which is mainly divided into two processes: state update andmeasurement update. The process of state update means thatthe estimated value at the previous time is known, the currentstate variables and the estimated error covariance matrix arepredicted by using the state model, and a prior estimate isconstructed for the next time state. The measurement updateprocess is responsible for feedback, and the posterior esti-mate of the current state is corrected by combining the priorestimate with a new measurement data. Therefore, this algo-rithm can also be called a predictor-corrector algorithm.

Firstly, the prior estimation mean of the state vector atthe initial time is set to x0, and the estimation error covari-ance matrix P0 is set.

According to the state vector of timestep k − 1, the esti-mation of timestep k can be obtained.

xk = xk−1,Pk = Pk−1 +Qk−1:

(ð15Þ

2L + 1 sigma points xi are constructed, where L is thedimension of the state vector, and the set of sigma pointscan be expressed as

xi,k−1 =

xk−1, i = 0,

xk−1 +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL + λð ÞPk−1

p� �i, i = 1,⋯, L,

xk−1 −ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL + λð ÞPk−1

p� �i, i = n + 1,⋯, 2L,

8>>>><>>>>:

ð16Þ

where ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðL + λÞPk−1p Þi is the ith column of the matrix square

root, which is calculated by the Cholesky decomposition. αrepresents the spread of the sigma points around the currentstate vector and is usually set as a small positive value, rang-ing from ½0, 1�. λ = α2ðL + κÞ − L is a scaling parameter. κ is asecondary scaling parameter. If L ≤ 3, κ = 3‐L; if L > 3, κ = 0.

For each sigma point, the transformed result is obtainedby nonlinear transformation fð⋅Þ.

xi,k/k−1 = f xi,k−1ð Þ, i = 0, 1,⋯, 2L: ð17Þ

xi,k/k−1 is weighted to predict the prior estimates of meanxk/k−1 and covariance matrix Pk/k−1.

xk/k−1 = 〠2L

i=0W mð Þ

i xi,k/k−1,

Pk/k−1 = 〠2L

i=0W cð Þ

i xi,k/k−1 − xk/k−1½ � xi,k/k−1 − x∧k/k−1½ �T +Qk−1,

W mð Þi =

λ

L + λ, i = 0,

12 L + λð Þ , i ≠ 0,

8>><>>:

W cð Þi =

λ

L + λ+ 1 + β − α2, i = 0,

12 L + λð Þ , i ≠ 0,

8>><>>:

ð18Þ

whereWðmÞi andWðcÞ

i are weighted by mean and covariance,respectively. β is a scaling parameter, used to include theinformation about the distribution. For the Gaussian distri-bution, β = 2 is optimal.

Similarly, the nonlinear measurement function hð⋅Þ isused to transfer the sigma point to zi,k/k−1 and to predict themeasurement value zk/k−1, the autocovariance matrix Pzk ,and the crosscovariance matrix Pxkzk .

zi,k/k−1 = h xi,k−1ð Þ, i = 0, 1,⋯, 2L,

zk/k−1 = 〠2L

i=0W mð Þ

i zi,k/k−1,

Pzk= 〠

2L

i=0W cð Þ

i zi,k/k−1 − zk/k−1½ � zi,k/k−1 − z∧k/k−1½ �T + Rk,

Pxkzk= 〠

2L

i=0W cð Þ

i xi,k/k−1 − xk/k−1½ � zi,k/k−1 − z∧k/k−1½ �T :

ð19Þ

Finally, calculate the gain matrix Kk , and update the statemean xk and covariance matrix Pk at timestep k based on anew measured value zk .

Kk = PxkzkP−1zk,

xk = xk/k−1 + Kk zk − zk/k−1ð Þ,Pk = Pk/k−1 − KkPzk

KTk :

ð20Þ

5Geofluids

4. Numerical Simulation Analysis

Based on the interpretation model while underbalanced dril-ling established above, real-time estimation of reservoir char-acteristics is carried out. Because the interpretationparameters of the model are reservoir pressure and perme-ability, they are difficult to be obtained under the current lab-oratory and field test conditions. Thus, this paper uses thesynthetic data instead of the experimental data as measure-ments. Calculation steps are as follows: Firstly, a set of reser-voir characteristic parameters (pressure and permeability)are preset as the estimates of interpretation parameters ofthe open reservoir section at the initial time. Then, the bot-tomhole pressure and outlet flow are simulated by using theestablished hydraulics model as measurements. Finally, theinterpretation parameters are online estimated by using theUKF algorithm.

Well X is drilled in underbalanced condition by gas injec-tion in the drill string. The injected gas phase is nitrogen. Thereservoir is drilled at 3000m, which is a pure oil reservoir.The reservoir is a typical medium-permeability sandstonereservoir with permeability ranging from 50mD to 500mD.Shale is the upper caprock, and low-permeability mudstoneis the interlayer. Vertical distribution of sand and mud isinteractive. The basic data needed for the well calculationare shown in Table 1.

The drilling process of the 3000-3100m reservoir sectionis simulated. By dividing the reservoir into five units, eachunit is 20m in length. The reservoir pressure and permeabil-ity of each unit are constant. In order to analyze the influenceof interpretation parameter dimension on the results ofmodel estimation, we have carried out three cases, each ofwhich has different interpretation parameters. Case 1 andcase 2 are single-parameter estimates. The single parameterof reservoir permeability or formation pressure is estimatedas the interpretation parameter, respectively. Case 3 is adouble-parameter estimate, and the reservoir permeabilityand pressure are both estimated as interpretation parameters.The true value of reservoir permeability and pressure of fiveunits is given in each case: case 1: reservoir permeability is[250, 500, 350, 150, 50] mD and reservoir pressure has a fixedvalue (38MPa); case 2: reservoir pressure is [38, 44, 40, 36,

42] MPa and reservoir permeability has a fixed value(400mD); and case 3: reservoir permeability is [250, 500,350, 150, 50] mD and reservoir pressure is [38, 44, 40, 36,42] MPa.

Note that the proper specification of the covariancematrix for the modeling error is crucial to get better perfor-mance of the filter. The covariance matrix for state parametererror Q and measurement error R are almost diagonal, whichdepends on the specific case of the research object, while theuncertainty parameters may be related to each other.Although the relationship between those parameters andthe effect on the interpretation performance are not the focusof this paper, they can be a topic for further research. Weassume that the errors in the measurements and state param-eter are statistically independent and are set as a suitablevalue with personal experience. The standard deviations ofreservoir pressure and permeability process noise are0.15MPa and 0.5mD, respectively, while the accuracy of bot-tomhole pressure and outlet flow measurement is 0.05% and0.1%, respectively; i.e., the standard deviations of measure-ment noise are 0.02MPa and 0.00003m3/s, respectively.Thus, the covariance matrix for parameter and measurementerror in three cases is as follows:

Case 1. Q = diag ½0:52�, R = diag ½0:022,0:000032�

Case 2. Q = diag ½0:152�, R = diag ½0:022,0:000032�

Case 3. Q = diag ½0:52,0:152�, R = diag ½0:022,0:000032�

Cases 1, 2, and 3 show that gas is injected from the drillstring and then returned from the annulus wellhead, reach-ing to a steady state (0-75min). The gas injection process isthe common part of three cases, which are described in detailhere. 0-15.5min is the process of injected gas from the drillstring to the drill bit. At this stage, the bottomhole pressureincreases slightly and basically keeps unchanged, and theoutlet flow increases with the gas injection at the initial time,then gradually decreases. 15.5-51.1min is the process of gasmigration from the bottomhole to the wellhead annulus. Inthis stage, the bottomhole pressure continues to decrease,

Table 1: Basic parameters for well X.

Parameters Value Parameters Value

Well depth (m) 3000 Yield point (Pa) 3.84

Casing shoe depth (m) 2200 Surface temperature (°C) 20

Casing ID (mm) 228.47 Geothermal gradient (°C/m) 0.0266

Bit diameter (mm) 215.9 Casing pressure (MPa) 1

Drill pipe OD (mm) 127 Gas injection rate (Nm3/s) 0.5

Drill pipe ID (mm) 108.6 ROP (m/h) 16

Bit nozzle area (mm2) 660 Oil density (kg/m3) 850

Pump rate (L/s) 30 Oil viscosity (mPa·s) 30

Mud density (kg/m3) 1180 Compressibility of liquid (Pa-1) 1:0 × 10−9

Consistency index (Pa·sn) 0.37 Reservoir porosity 0.15

Flow behavior index 0.68 Skin factor 0.013

6 Geofluids

while the outlet flow gradually increases due to the gas migra-tion. 51.1-75min is the process of gas front reaching wellheadannulus to wellbore stable-state flow. Bottomhole pressurecontinues to decrease at first, then tends to be constant, andthe outlet flow quickly falls back to the inlet pump rate of0.03m3/s.

For case 1, the reservoir pressure is known to be 38MPaand the reservoir permeability is estimated. The calculationresults are shown in Figures 3–5. Figures 3 and 4 show thecomparison between the estimated data and measurementsof bottomhole pressure and outlet flow rate, respectively. Itcan be seen that 75min is the initial point of drilling in thereservoir section and real-time interpretation starts synchro-nously at this time. 450 minutes is the end point, when thereservoir is just opened 100m in length. The estimated valueof bottomhole pressure and outlet flow basically coincideswith the measured value. Because of the influence of mea-surement noise, the measured value is somewhat burr, whilethe filtered curve (i.e., the blue line) is smoother. However,the calculated values of bottomhole pressure and outlet flowrate without filter processing (i.e., the green line) deviatefrom the measurements. The setting condition of the nonfil-ter is that the reservoir permeability is a constant value.

Figure 5 shows the comparison between the true and esti-mated permeability values. The simulation results show thatthe true values of permeability of each reservoir unit are [250,500, 350, 150, 50] mD and the initial value is set to 100mD.75-150min is the time period of opening reservoir unit 1.With the elapse of time, the estimated value of reservoir per-meability gradually rises from 100mD to its true value near150mD and then basically fluctuates slightly around the true

value. When unit 2 is opened (150-225min), the reservoirpermeability gradually approaches its true value of 500mDfrom 150mD, and then, analogies are made until the whole100m reservoir section is opened. Therefore, the estimatedvalue of reservoir permeability is in good agreement withthe true value. When a new section of the reservoir is opened,the estimated value can always be accurately and quicklyreturned to its true value.

For case 2, the reservoir permeability is preset as 400mDand the reservoir pressure is estimated. The calculationresults are shown in Figures 6–8. Figures 6 and 7 show thecomparison between the estimated data and measurementsof bottomhole pressure and outlet flow rate, respectively.The estimated values of bottomhole pressure and outlet floware basically in agreement with the measurements, while thenonfiltered data deviate greatly from the measurements.Figure 8 shows the comparison between the true and esti-mated reservoir pressure values. It can be seen that the truevalues of reservoir pressure in each unit are [38, 44, 40, 36,42] MPa and the initial value is set to 34MPa. The estimationof reservoir pressure is in good agreement with the real value.When a new section of the reservoir is opened, the estimationof reservoir pressure can quickly approach its true value.Compared with example 1, it takes shorter time and con-verges faster.

The double parameters of reservoir permeability andpressure are estimated simultaneously in case 3. The simula-tion results are shown in Figures 9–12. Figures 9 and 10,respectively, show the comparison between the estimateddata and measurements of bottomhole pressure and outletflow. Similarly, the estimated values of bottomhole pressure

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Figure 3: Bottomhole pressure variation (case 1).

7Geofluids

0 50 100 150 200 250 300 350 400 4500.025

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id o

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Figure 4: Liquid outlet rate variation (case 1).

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Figure 5: True and estimated reservoir permeability (case 1).

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Figure 6: Bottomhole pressure variation (case 2).

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Figure 7: Liquid outlet rate variation (case 2).

9Geofluids

and outlet flow rate are highly consistent with the measuredvalues. Bottomhole pressure gradually decreases with theincrease in crude oil production during 150-225min. In

225-300min, the effects of the increase in crude oil produc-tion and the prolongation of the reservoir section on the bot-tomhole pressure offset each other, thus basically remaining

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Figure 8: True and estimated reservoir pressure (case 2).

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Figure 9: Bottomhole pressure variation (case 3).

10 Geofluids

unchanged. In 300-450min, the prolongation of the reservoirsection becomes the main controlling factor, which results inthe gradual increase in bottomhole pressure. It is obvious that

there are five different stages of slope change in the outletflow rate, which matches with five different reservoir unitsdrilled.

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Figure 10: Liquid outlet rate variation (case 3).

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Figure 11: True and estimated reservoir permeability (case 3).

11Geofluids

Figures 11 and 12 show the contrasting results of true andestimated reservoir permeability and pressure, respectively. Itcan be seen that the true values of reservoir permeability andpressure of each unit are [250, 500, 350, 150, 50] mD and [38,44, 40, 36, 42] MPa, respectively, and the initial values are setto 100mD and 34MPa, respectively. The estimated values ofreservoir permeability and pressure are quite different fromthe true values, which basically do not coincide with eachother. When the permeability estimates are lower than thetrue values, the reservoir pressure estimates are higher andvice versa.

In summary, the single-parameter estimation of reservoirpermeability or pressure in case 1 and case 2 can achieve real-time and accurate interpretation. However, the double-parameter estimation of reservoir permeability and pressurein case 3 cannot be realized and the interpretation accuracyis low. The reason is that for the two-parameter estimation,the reservoir permeability and pressure are only related tothe crude output according to the dynamic reservoir model.Two measurements in this model are too few to obtainenough effective information. However, it is also difficult toobtain effective information for the four measurements inthe previous researches. It is not feasible to simply increasethe number of measurement parameters, which are bothrelated to pressure or flow rate. Therefore, by increasing mea-surement dimensions (e.g., dielectric constant or resistivity)and digging useful measurements in depth, synchronous esti-mation of two or even multiple parameters of reservoir char-acteristics may be realized.

Taking the single-parameter estimation of reservoir per-meability in case 1 as an example, the UKF algorithm and

EKF algorithm are compared and analyzed, as shown inFigure 13. The simulation results are shown in Table 2. Theaverage reservoir permeability while drilling through eachreservoir unit is used as the estimated value. Compared withEKF, the estimated values of UKF in units 1-5 are both closerto the true value. Moreover, it takes less time for UKF estima-tion error to converge to 10% than it is for EKF. Only whenreservoir unit 5 is used, time of convergence to 10% is higherthan that for EKF, which may be related to the randomness ofmodel noise. The numerical results show that UKF is supe-rior to EKF in both estimation accuracy and convergencespeed, which further indicates the superiority and accuracyof this model.

5. Conclusion

(1) A novel interpretation model while underbalanceddrilling based on the UKF and three-phase variablemass flow model of oil, gas, and liquid is established.The measurement parameters are simplified to bot-tomhole pressure and liquid outlet flow, to continu-ously update the permeability and reservoirpressure along the well, helping the management ofthe wellbore during the UBD and follow-up comple-tion operations

(2) Three cases including reservoir permeability andpressure single-parameter estimation and double-parameter estimation are simulated. The results showthat the interpretation accuracy of single-parameterestimation is high and the filtered values of

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Figure 12: True and estimated reservoir pressure (case 3).

12 Geofluids

bottomhole pressure and outlet flow trace the mea-sured values in real time. When a new section of thereservoir is opened, the estimated reservoir perme-ability or pressure can always be quickly and accu-rately returned to its true value. However, thedouble-parameter estimation cannot be realized,and the interpretation accuracy is low

(3) The comparison between the UKF and the EKF iscarried out. The results show that UKF is superiorto EKF in both estimation accuracy and convergencespeed, which further illustrates the superiority andaccuracy of the new model for interpretation whileunderbalanced drilling based on UKF

(4) Although the presented study is promising, foraddressing the problem of joint state-and-parameter estimation of a multiphase flow during

drilling, further development of the methodologyis needed. Future works will include how toachieve the double-parameter or even multiple-parameter estimation for reservoir characteristicsand the application of the methodology to actualoilfield experiments

Data Availability

The data used to support the findings of this study are avail-able from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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UKF estimated value

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Figure 13: Comparison between EKF and UKF of the true and estimated value (case 1).

Table 2: Comparative analysis of EKF and UKF estimated results.

Reservoir unitReservoir permeability (mD)

Time of convergence to10% (min)

True value UKF EKF UKF EKF

1 250 218 213 21 26

2 500 450 439 21 26

3 350 379 386 18 25

4 150 189 198 25 29

5 50 69 71 27 25

13Geofluids

Authors’ Contributions

Miao He conceived and designed the study and performedthe modelling. Yihang Zhang wrote the manuscript. Min-gbiao Xu reviewed and edited the manuscript. Jun Li per-formed the data analyses. Jianjian Song helped perform theanalysis with constructive discussions.

Acknowledgments

The authors would like to thank the National Natural ScienceFoundation of China (Grant Nos. 51904034 and 51734010)for the financial support and the National Major Scienceand Technology Projects of China (Grant numbers2016ZX05060 and 2017ZX05032004-004).

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