SPE-171233-MS

Field Study of Temperature Simulator Applications for QuantitativeInterpretation of Transient Thermal Logging in a Multipay Well

R. Valiullin, A. Ramazanov, A. Sadretdinov, and R. Sharafutdinov, Bashkir State University;

V. Shako, and M. Sidorova, Schlumberger Moscow Research Center; D. Kryuchatov, Kogalymneftegeofizika

Copyright 2014, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Russian Oil and Gas Exploration and Production Technical Conference and Exhibition held in Moscow, Russia,14–16 October 2014.

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contentsof the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflectany position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the writtenconsent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations maynot be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract

Temperature wellbore studies are widely applied for well testing and logging at the stages of wellexploration and development. However, in nowadays practice the interpretation is still done mainlyqualitatively. In the last few years different approaches to joint interpretation of temperature, pressure andflow rates are discussed more and more widely. In particular, there is a special interest to determinationof flow profile and individual layer near-wellbore zone properties in multilayer reservoir from analysis oftransient temperature and pressure during different transitional regimes.

An application of mathematical modeling to quantitative interpretation of temperature field data frommultilayer wells is discussed in this work.

Different approaches to inverse problem solution are analyzed for assessing flow rates and hydrody-namic parameter of individual layers in a multilayer wells. It is shown that all available field data,including unsteady temperature logging data and stations, should be used in interpretation workflow toreach higher level of reliability. The reasons of field and simulation data mismatch are discussed.

The paper demonstrates how the complex approach to interpretation of thermal logging data, com-prising (1) application of sophisticated numerical temperature simulators to interpretation of the fullavailable field data set of temperature logs and fixed-depth measurements and (2) uncertainty analysis,allows increase reliability of qualitative interpretation results and provide quantitative analysis of flowprofiles and hydrodynamic properties of multilayer well.

IntroductionTemperature measurement was the first measurement in the well performed in 1906 at Apsheronpeninsula by professor D. V. Golubyatnikov.

Fast introduction of thermometry was started in the 30th of last century when the first wellboreelectronic thermometer was developed. Temperature logging applications included location of gas entries,detection of casing leaks and fluid movement behind casing, location of lost-circulation zones anddetermination of cement top location1,2.

First research studies of temperature fields of the fluid flowing in the porous media taking into accountthermodynamic effects were performed by B. B. Lapuk. In 1940 he published 3 articles devoted toresearch of thermodynamic effects during gas, crude oil and black oil filtration in the porous media. Heconsidered steady state flow in the horizontal reservoir as throttle (isoenthalpic) process, i.e. he showedthat the change of fluid temperature is caused by Joule-Thomson effect. B. B. Lapuk concluded that fluidand gas flow in the reservoir could be considered as isothermal due to small value of the temperaturechanges.

The new stage of wellbore thermometry started with development of high resolution temperaturesensors with resolution better than 0.1 degC in the early sixties. Development of wellbore measurementstechnology and methods of wells and reservoirs exploration allowed studying of temperature anomalieswith values 0.1 and even 0.01 degC. E. B. Chekalyuk 3 conducted a more sophisticated research of thetemperature phenomena in the oil reservoirs. He received the equations of non-isothermal single-phasefiltration in the porous media taking into account Joule-Thomson and adiabatic effects.

Mathematical models developed by E. B. Chekalyuk became a theoretical basis of wellbore temper-ature interpretation. Chekalyuk suggested the method, named thermal logging, for determination ofreservoir permeability distribution from the transient temperature of fluid inflowing into a well after wellstart-up with a constant flow rate. This method was applied to gas wells.

Field application of temperature measurements in the well, according to Chekalyuk, should be relatedto determination of reservoir pressure in each layer of multilayer reservoir and characterization ofnear-wellbore zone of reservoir. Chekalyuk developed method for quantitative interpretation of quasisteady-state temperature profiles in multilayer reservoir from calorimetric mixing in the wellbore. It wasassumed that the value of throttle temperature change in all layers is the same (i.e. coefficients ofJoule-Thomson are equal, pressure drawdowns in all layers are equal and formation temperature in alllayers has the same average geothermal gradient).

The method of reservoir thermal logging assumed:

– the well works with a constant flow rate;– transient temperature of a fluid inflowing from layer is measured;– transient bottomhole pressure is measured.

Data interpretation algorithm for determination of permeability distribution based on the linear anamor-phosis was developed. An analytical model was used for interpretation. All flow regimes, except flow witha constant rate, were characterized by Chekalyuk as unusable for thermal logging. The method of thermallogging didn’t find a broad application in the field. From our opinion, the reason of this was in the absenceof the wellbore tools for simultaneous measurements of temperature and pressure at that time. Anotherreason was in the lack of multilayer reservoirs thermo-hydrodynamic simulators taking into accountwellbore effects.

Further the main attention of wellbore thermometry was focused on identification of inflows, flowsbehind casing, leak detection in casing and pumping equipment. Wellbore pilot studies in the Romash-kinskoye field in Tataria and in the fields of Western Siberia have proved possibility of registration anduse of the temperature anomalies caused by thermodynamic effects during filtration of oil in the reservoir.In the 1970th research teams in Moscow (Vniineft, MINH and GP), in the Kazan state university and theBashkir state university developed the basics of the theory and methodology of wellbore thermometry inRussia. Since that time, the development of high-resolution thermometry started.

Nowadays the thermometry is the most informative method in production logging. The thermometryis used in development and appraisal of wells and reservoirs at all stages: drilling, completion andproduction4.

Recently, with new appearing opportunities of measurement of pressure with resolution better than0.01 psi and temperature better than 0.01 °C with multiple wellbore sensors, an interest to determination

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of hydrodynamic parameters of layers in multilayer reservoir from joint interpretation of transienttemperature and pressure data has increased.

Methods of damage zone parameters determination and models of thermo-hydrodynamic processes inwellbore and reservoir are described in a number of works by groups in Stanford University5-9, TexasUniversity10-12 and the Bashkir State University jointly with the Shlumberger Moscow Research Center13,14.

In the works of Stanford group one-dimensional radial model for pressure and temperature simulationsin one-layer reservoir and one-dimensional analytical model for simulation of wellbore temperature areused for single-phase and two-phase flows. Authors made a conclusion about possibility to obtaininformation additional to conventional pressure transient analysis, in particular, determination of porosity,permeability and saturation from transient temperature and flow rate. They also suggested using transienttemperature for inflow characterization.

Group in Texas University conducted a sensitivity study of transient wellbore temperatures, measuredabove each productive layer in multilayer reservoir, and bottomhole pressure to parameters of each layer.The mathematical model developed by authors, consists of coupled one-dimensional hydrodynamic andthermal models in the well and two-dimensional hydrodynamic and thermal models in the reservoir,describing a single-phase flow of slightly compressible fluid10 ,11 or gas12.

A number of works was devoted to characterization of reservoir and identification of oil, water or gasinflows13-15. Zonal allocation and quantitative characterization of inflow from individual layers in amultilayer reservoir with transient temperature and pressure data is of special interest now. The methodfor zonal allocation and determination of hydrodynamic parameters of productive zones in a horizontal,multi-zone, intelligent wells is described in papers16-19.

The methodology and case studies of interpretation of transient temperature data recorded along thewell, for characterization of flow in a gas well are presented in paper20. Authors used quasi steady-statesemi-analytical model for transient temperature profiles simulations. More sophisticated numerical modelfor simulation of transient temperature and pressure in the gas wells with coupling of wellbore, reservoirand impermeable formation, and field examples are presented in papers21,22.

Value of transient temperature-pressure analysisNon-isothermal well testing based on the coupled analysis of pressure (and rate) and temperaturesmeasurements will be referred to as PTRA (Pressure-Temperature-Rate transient Analysis). There are twotypes of PTRA in practice nowadays:

✓ measurements by well logging (moving of tools along the well);✓ transient measurements vs. time at a certain depth.

The first technology belongs to conventional geophysical well studies being a traditional part ofproduction logging. Measurement of temperature, pressure and flow rate profiles along a well is thetypical measurements during well exploration, appraisal and monitoring.

The second type of PTRA technology is used, for example, for appraisal of horizontal multilayer wellsin Russia (Tatneft, Surgutneftegaz). The main application of this technology is determination of hydro-dynamic parameters of each layer. Conventional isothermal well testing can be considered as approxi-mation partial of PTRA.

The temperature field in a well is determined by several thermal processes: Joule-Thomson effect inreservoir, adiabatic effect, effect of calorimetric mixing, convective heat transfer, degasing phenomenon(phase transitions) and heat conduction. The contribution of these effects to temperature distribution in awell depends on various factors: geothermal distribution of temperature, properties of reservoir and fluid,gas factor, water cut, fluid velocity in the wellbore, bubble-point pressure and well operational schedule.

SPE-171233-MS 3

Temperature transient profiles in the wellboredepend also on the direction and the speed of log-ging (so-called effect of non-instantaneous mea-surements). In this case the temperature profilealong the well, measured during log down or log up,represents the temperature change history for a pe-riod of time and in the range of depths logged by thetool.

The variety of these processes and measurementconditions make temperature distribution in eachwell quite unique. At the same time, temperaturedistribution across the productive layers in any wellis possible to divide into four typical zones. Anexample of temperature profile in an oil well isshown in Fig. 1.

Here the following zones are marked:

1 – the zone with undisturbed geothermal tem-perature in the rat hole

2 – the zone of disturbed geothermal temperature3 – intervals of fluid inflow4 – the zone of heat exchange between wellbore fluid and surrounding formation.

The field experience shows that the zone 1 with the geothermal distribution of temperature is identifiedeven after long production if the length of well rat hole exceeds 8-10 meters. This zone of temperatureprofile is very important for interpretation, as it is used for reconstruction of undisturbed formationtemperature along the whole zone of interest.

Above this zone, at the distance 5-10 meters below the bottom boundary of inflow, the geothermaldistribution could be disturbed due to heat exchange with the lowest productive layer. Temperature of thefluid flowing in the reservoir differs from undisturbed formation temperature due to “baro-thermal”effect. With time the surrounding formation below the reservoir is also warmed up due to heat conductionand in some cases due to natural thermal convection. Temperature distribution in zones 1 and 2 allowsidentification of fluid movement behind casing.

Fluid inflowing from productive layers mixes with wellbore fluid in the zones 3. Temperaturedistribution in this zone depends on rates of mixing flows. Temperature variations in time for transientregimes contains information on hydrodynamic parameters of layer.

Temperature distribution between productive layers and above the reservoir (zone 4) is determined byconvective heat exchange between wellbore fluid and surrounding formation. Quasi steady-state andtransient temperature profiles in this zone contain information about flow rate.

In presence of long enough zones 4 simulation of quasi steady-state temperature profiles after longproduction at constant rate is well known to allow characterization of flow rate from individual layers.Simulation of temperature transients additionally allows determination of hydrodynamic parameters oflayers.

Fig. 2 illustrates an example of pressure and temperature measurements at a fixed depth during gas liftoperations.

During well inflow initiation with the compressor the wellbore pressure increases and temperaturedecreases (Fig. 2). It is caused by liquid flow down along the wellbore. After opening the gas-lift valve,pressure in the annulus drops from 22.25 MPa to 13.04 MPa. Temperature decreases on 1.34 °C due toadiabatic effect and liquid redistribution between tubing and annulus. Further the slow monotonous

Figure 1—An example of temperature profile in naturally flowing oilwell with flow rate 250 m3/d

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pressure increase is observed as a result of fluidinflowing from the reservoir. Temperature increaseduring this period is observed due to Joule-Thomson effect. Change of temperature increaserate is related to changes in inflowing fluid phasecontent that is confirmed by resistivity and water cutmeasurements. Further in Chapter 4 these processesare considered in more details.

Simplified mathematical model ofsingle-phase reservoir flowLet us consider first the single-phase problem of thereservoir non-isothermal flow. Changes of temper-ature of fluid flowing in the productive layer are caused by convective and conductive heat transfer, andthermodynamic effects. Under assumption of absence of phase transitions the thermodynamic effectsinfluencing temperature variations are throttle and adiabatic effects. For a single-phase flow in ahorizontal porous layer we use the one-temperature energy equation3

(2.1)

or

(2.2)

Here

(2.3)

The first term on the right side of the equation (2.1) describes heat conductivity in the saturated porousmedia, the second term is fluid frictional heating, the third term is heating/cooling as a result of fluidcompression/expansion due to pressure transients, and the fourth term describes temperature coolingcaused by liquid expansion due to pressure drop in the layer.

In the equation (2.2) the 2nd and the 4th terms from equation (2.1) are combined in one term withJoule–Thomson coefficient �, and the 3rd term from equation (2.1) is written through adiabatic coefficient�.

Joule–Thomson coefficient � defined in this way is positive for liquids and normally negative for gases.It allows referring to the positive throttle effect for liquids and negative throttle effect for gases. It is alsoassumed above that the skeleton of the porous media is incompressible, i.e. ��/�t � 0.

It should be noticed that the maximum value of temperature variations caused by thermodynamiceffects in liquid flowing through porous media do not exceed 2-3 K. Therefore it is possible to neglecttemperature dependence of density and viscosity. Then the system of the equations of non-isothermalfiltration is linearized, we can define subsequently the pressure field p(r, t) and filtration velocity v(r, t)from the equation of the isothermal flow and then determine temperature field T(r, t) from the energyequation.

Figure 2—An example of pressure and temperature measurements at afixed depth above reservoir during gas lift inflow initiation.

SPE-171233-MS 5

In case of slightly compressible liquid pressure distribution could be defined from a diffusivityequation

(2.4)

with initial and boundary conditions determined by the transient process in coupled system wellbore/reservoir under study.

In general for the system comprising several layers, taking into account influence of the wellboreeffects, the problem should be solved numerically.

However, the approximate analytical solutions that could be used for the first estimations and obtainingof the initial approximations for the inverse problem solution are possible for simple flow geometries anda number of assumptions.

For example, as a first approximation, it is possible to neglect conductive (�) heat transfer in productivelayer in the energy equation for simulation of temperature field in a near- wellbore zone for producers andinjectors as its contribution is small in comparison with convective heat transfer.

Baro-thermal effectIf we neglect heat conduction in the porous media in equations (2.1) and (2.2), the fluid temperature willbe governed by the convective heat transfer in the productive layer and additional change of temperaturedue to Joule-Thomson and adiabatic effects.

The term “baro-thermal” effect is introduces to characterize the changes of temperature in thesaturated porous media, caused by pressure changes26. In case of a steady-state pressure field thebaro-thermal effect is equivalent to Joule–Thomson effect.

A simplified one-dimensional single-phase mathematical problem statement for simulation of temper-ature field in the reservoir caused by the baro-thermal effect is

(3.1)

(3.2)

Here u(r, t) is velocity of convective heat transfer3

(3.3)

c- is the ratio of volumetric heat capacity of fluid to volumetric heat capacity of fluid-saturated porousmedia

The solution of equations (3.1)-(3.2) for arbitrary transient pressure field p(r, t) along the characteristiclines r(t, r1), defined as solution of problem:

(3.4)

is26

(3.5)

The first term in (3.5) represents the convective heat flow of an initial temperature profile f(r) fromproductive layer. The second and the third terms in (3.5) represent temperature change caused bybaro-thermal effect. The second term is uniquely defined by the initial and current pressure values, and

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the third term characterizes dependence of baro-thermal effect on pressure changes on the way of fluidfiltrating in the layer.

Using (3.5), it is possible to calculate the temperature distribution in the layer for a given pressure fieldp(r, t). Further simplification of (3.5) depends on the type of p(r, t) and integrability of (3.5).

An analytical solution of baro-thermal effect problemAfter well start-up with constant boundary conditions, the time of the pressure field stabilization is definedby the reservoir piezoconductivity, and the temperature field depends on the speed of convective heattransfer that is proportional to the filtration velocity.

The characteristic times of these processes differ by 3-4 orders of magnitude: in a radial flow thecharacteristic time of pressure propagation is , and temperature propagation characteristic time is

.

Here q is the specific flow rate, m2/s; � is the reservoir piezoconductivity, m2/s; and R is characteristiclength, m.

Therefore, as a first approximation for studying the baro-thermal effect, it could be assumed thatpressure field stabilizes instantaneously in comparison with temperature field.

Instantaneously established pressure distribution after well start-up could be described with thefollowing function

where Pi is initial reservoir pressure; p(r) is stabilized pressure distribution.Then pressure derivative contains �–function

and integral in equation (3.5) will be

Change of temperature along the characteristic lines according to (3.5) could be expressed as

(4.1)

The physical meaning of the obtained solution is the following: the first term is initial temperature forthe characteristic line at point r1, the second term is the throttle warming up along the characteristic line,and the third term is cooling due to adiabatic effect caused by pressure decrease.

For a homogeneous reservoir layer we will use the well-known distribution of pressure

Here rw is the wellbore radius, re is the drainage radius, Pw is the wellbore pressure, Pe is the reservoirpressure at external boundary.

Convective heat transfer velocity according to (3.3) at t�0 is

and the solution of the problem (3.4), the characteristic lines is defined as

SPE-171233-MS 7

(4.2)

Temperature field in productive layer could be simulated using formulas (4.1) and (4.2). For thispurpose one needs to define a set of r1 values and increase time until the characteristic line (4.2) reachesthe wellbore wall rw. As a result the temperature variations in time for the control volume moving withthe speed of convective heat transfer from distance r1 in the reservoir to the well wall could be obtained.

Such approach for calculation along characteristic lines is not convenient as usually one is interestedin temperature variations with time at fixed points in the reservoir, usually at wellbore wall. Using Eulervariables, the temperature variations at well wall from (4.1) are given by the following expression

(4.3)

Here

(4.4)

is the radius (depth) of investigation, temperature signal with a velocity of convective heat transfercomes from this distance in the layer at time t.

Temperature change at the well wall, caused by the baro-thermal effect, for a steady-state flow rate Qis calculated with the formula

(4.5)

Time of temperature field stabilization tst is determined with the formulas

Fig. 3 shows a schematic of temperature change during fluid flow from a heterogeneous layer.Approximation parameters for linear curves in Fig. 3 are the following:

Slope coefficients for linear zones of a temperature curve depend on permeability of the zone ofinvestigation at current time moment (4.4).

Thus, in principal, it is possible to determine radial permeability distribution in the near-wellbore zonefrom sandface temperature interpretation. The small speed of thermal disturbance propagation in com-parison with the pressure field propagation gives opportunity to PTRA methods to become a goodcomplimentary source of information to the conventional pressure transient analysis.

It should be noticed that the analytical solution does not take into account effects related to thenon-radial flow in near-wellbore zone that could significantly affect the temperature trends, especiallyshort-term (like perforated zone flows or partial completion). Nevertheless, it can be used as an initialguess for inverse problem solution to determine hydrodynamic parameters of reservoir with PTRA afterstart-up of the production well of flow rate changes, with further more accurate inverse problem solution

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with a numerical model35. Approximate analyticalmodels for the specified transient bottomhole pres-sure37 and transient flow rate38 were developed.Numerical model was used in this paper for inter-pretation of PTRA data.

Two-dimensional two-phasenumerical modelWe consider the two-phase non-isothermal model ofoil and water flow in the coupled system well-reservoir-impermeable formation for analysis offield data. It is assumed in the model that the flowin the reservoir is axial symmetric in the near-wellbore zone of vertical or slightly deviated wells.

The mathematical model for permeable layers consists of black oil hydrodynamic model and theequation of the energy accounting for the thermodynamic effects, Joule-Thomson and adiabatic. Themathematical model of impermeable formation is the conductive heat transfer model simulating heattransfer between wellbore fluid and surrounding formation. Wellbore model assumes homogeneous flowof oil and water mixture, taking into account the adiabatic effect and heat exchange with surroundingformation. Model for the permeable reservoir layers and impermeable formation is two-dimensional (r-z),wellbore model is one-dimensional.

The equations of all model components, boundary conditions and coupling of models are given below.

Reservoir model

Mass balance equations

(5.1)

(5.2)

Energy balance equations

(5.3)

Linear Darcy’s equations:

(5.4)

Boundary condition at the top and the bottom boundaries of impermeable layer is no-flow, boundarycondition at the well wall and external reservoir boundary is the given pressure:

(5.5)

(5.6)

Figure 3—Temperature transients due to throttling heating in the flowthrough a heterogeneous layer

SPE-171233-MS 9

Temperature at external reservoir boundary is fixed, boundary conditions at the top and bottomboundaries of the permeable layers are zero heat flux, inner boundary condition at the well wall isrepresented by the condition of equality of the total heat flux between wellbore and formation

(5.7)

(5.8)

(5.9)

Impermeable formation thermal model

(5.10)

(5.11)

The first boundary condition in (5.11) provides coupling with wellbore model. The second condition in(5.11) is the boundary condition at the external reservoir boundary. Radius Rext could differ from externalreservoir boundary Rk in hydrodynamic problem but should be chosen in such a way that thermaldisturbance due to conductive heat transfer does not reach external boundary during the simulation time.

Wellbore flow modelThe momentum and mass balance equations:

(5.12)

(5.13)

Source terms in continuity equations are:

(5.14)

Energy balance equation:

(5.15)

Temperature measurements with the moving tool are described by a separate simple model. Additionalparameters (initial position of the tool Z0, time of the start of measurements t0, velocity of logging u) aredefined for description of tool movement. The current position of the tool is described with the followingexpression:

(5.16)

The model of the measured temperature accounts for the tool time constant . Measured temperatureis simulated with the following equation

(5.17)

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The solution of mathematical model (5.1)-(5.15) is found numerically with the method of controlvolumes. Coupling is realized iteratively with the source terms for the wellbore flow and boundaryconditions for reservoir and impermeable formation. IMPES method is used for solution of equations ofthe reservoir flow.

Example of PTRA during compressor well testingLet us revisit the problem of an artificial-lift well testing operations and analysis of pressure-temperaturetransients briefly discussed in chapter 1. Artificial-lift well testing is a complex of works in the cased-holewell after drilling or workover, connected with initiation of a short-term inflow from the reservoir.

Currently in Russia compressors, swabbing and jet pumps are used for testing of the low ratenon-flowing naturally wells. Most wells are tested with compressors.

The schematic of the well testing with compressor and corresponding changes of bottomhole pressureare given in Fig. 4.

There are few short-term transitional regimes during gas lift testing that take several hours.Well is shut-in during Period I. Static level of liquid in the wellbore is below the wellhead. Bottomhole

pressure is equal to reservoir pressure. In practice some fluid flow from or into the reservoir is observedduring this period. As a result the bottomhole pressure could differ from the reservoir pressure.

During Period II air or inert gas is pumped into the annulus with the compressor. Liquid level inannulus moves down, and liquid level in tubing moves up and, respectively, bottomhole pressureincreases. Wellbore fluid flows from the wellbore into the reservoir during this period.

At Period III the level of liquid decreases to the first gas lift valve and gas from annulus penetrates intotubing and decreases average wellbore fluid density. As a result the bottomhole pressure decreases andliquid starts to inflow from the reservoir. Usually at the end of this period the compressor is turned offand the gas pressure in the annulus goes down. As a result the bottomhole pressure decreases fast andreaches the minimum value.

During Period IV the fluid flow from the reservoir continues, it accumulates in the wellbore. Liquidlevel in the wellbore increases, the bottomhole pressure increases, and inflow from the reservoir graduallydecreases.

During Period V, when fluid level increases to the static level, the well and reservoir revert to theoriginal state with the bottomhole pressure equal to reservoir pressure.

Non-isothermal well testing (PTRA) could be easily utilized in frame of this testing scenario. There isa possibility of measurements at various short-term transitional well operation regimes. It providesadditional information on the processes happening in a well and reservoir. These pressure and temperaturetransients could be used for quantitative interpretation.

The main features of temperature processes during well exploration with the compressor are thefollowing:

Figure 4—The scheme of well testing with compressor and change of bottomhole pressure

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✓ bottomhole pressure is minimum and the flow rate is maximum at the time when compressor isturned off. Further, bottomhole pressure increases with time, gradually reaching the reservoirpressure, and flow rate is decreasing to zero;

✓ in the absence of phase transitions (wellbore pressure is higher than saturation pressure or water cutis more than 60% even at pressure below saturation pressure) fluid warming up due to thebaro-thermal effect after compressor turning off can continue growing for some time. Afterwardsthe baro-thermal effect decreases to some residual value and remains noticeable for a long timeafter inflow from reservoir is stopped;

✓ after the inflow from the reservoir is stopped, temperature in the reservoir changes due to verticaland horizontal heat transfer. This process is very slow and depending on thickness of layer can takethe considerable period of time;

✓ temperature change due to the baro-thermal effect depends on thermodynamic properties ofinflowing liquid, reservoir properties, pressure drawdown, and bottomhole pressure change in time;

✓ in the intervals of water and oil inflow the inversion of temperature anomaly from calorimetricmixing can be observed, warming up of more mobile water at the initial stage of inflow can exceedthe warming up of oil flow34;

✓ temperature change along the wellbore depends on the wellbore fluid velocity and can be used forflow rate characterization;

✓ when bottomhole pressure decreases below saturation pressure during fluid flow from the reservoir,wellbore temperature could decrease below initial reservoir temperature;

✓ temperature variations in the rat hole is caused by adiabatic effect and much less than temperaturevariations due to Joule-Thomson effect of fluid inflowing from reservoir;

✓ non-monotonous temperature changes in the rat hole and long transition zone in disturbedgeothermal temperature zone (see Fig. 1) in the rat hole can be caused by fluid flow behind casing;

✓ tool time constant impacts on temperature profiles measured during logging due to transitionalnature of thermal fields. The history of temperature changes in some time and length intervaldepending on the speed and the direction of tool movement is reflected in the measured temperatureprofiles. For this reason the additional model accounting for tool movement in the wellbore isimplemented.

Example of PTRA applicationAn example of quantitative interpretation of non-isothermal well testing data recorded in the oil well inone of Western Siberia oil fields is described below.

Information about well and well testWell design

Total depth 2491 M

Artificial bottom plug 2481.6 M

Tubing shoe 1307 M

Wellbore diameter 216 MM

Casing outer diameter 146 MM

Tubing outer diameter 73 MM

Casing inner diameter 132 MM

Tubing inner diameter 62 MM

Perforation intervals 2385-23882393-2398

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Average fluid density in the wellbore is 940 kg/m3 and compressibility is 0.00025 1/bar.Well was shut for a long time. Rimming and circulation were conducted before the well test with gas

lift. Detailed information on the performed works is not available.Logging methods used:

● Temperature● Gamma● SP● Flow rate● Pressure● Water cut● Resistivity● Heat indicator of inflow (STI)

Measurements:

● Measurements before well test● Logging while compression and after (39 measurements, including measurements before the test)● Measurements at fixed point during compression and after (31 measurements at a depth 2365 m)

Field data are presented in the plots below. Measurements of mechanical flow meter are not presentedin the plots as it was run separately and there is no pressure and temperature data available during this run.

Black curve in Fig. 5 is the temperature profile before the test.

Figure 5—Temperature profiles along the well (left plot) and in front of reservoir (right plot)

SPE-171233-MS 13

Blue curves are the temperature profiles during gas pumping in the annulus (correspond to Period IIin Fig. 4).

Red curves are the temperature profiles during fluid inflow into well (Period III), and green curves aretemperature profiles during shut-in after compressor was turned off (Period IV).

Blue filling represents perforation intervals.Temperature (red points) and pressure (blue points) measurement at the depth 2365 m, recorded with

the same logging tool between runs along the well, are presented in Fig. 6. Gray, blue, red and greenfillings represent time intervals when logging along the wellbore was done. Color coding in Fig. 6corresponds the one in Fig. 5.

Field data analysisThe results of interpretation by Kogalymneftegeofizika are presented in the tables below.

The table of dynamic fluid level (DFL) and dependence of flow rate on time from DFL:

Maximum flow rate calculated from pressure change after start of compression 74.63 m3/d

Maximum flow rate calculated from fluid level dynamics in 1 hour and 50 minutes 13.7 m3/d

Flow rate measure with mechanical flow meter 10 m3/d

Zonal contribution measured with mechanical flow meter 60% top layer40% bottom layer

Figure 6—Pressure and temperature transients at depth 2365 m during well test

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Inflowing fluid phase content is oil and water with 60% water cut. The water cut was estimated fromvariations of the phase interface depth.

Analysis of the liquid level build-up data gave the following results:

It was assumed during interpretation that reservoir porosity is 20%, formation rock compressibility is0.74·10-5 bar-1, and drainage radius is 250 m.

It should be noticed that these parameters are averages for 2 perforation intervals, and they wereobtained from analysis of the liquid level data during build-up and not from pressure transient analysis.

Transient pressure during well test is plotted in Fig. 7.It is seen from the plot in Fig. 7 that the bottomhole pressure decreased before the start of the test. Then

bottomhole pressure increased by more than 20 atm when gas was pumped into the annulus. After gas liftvalves were accessed, pressure decreased from 139 atm to 124 atm before the pump was switched off.After the pump stop the gas annulus pressure fast decreased by another 10 atm and reached 104 atm. Afterit the bottomhole pressure started to increase due to inflow from reservoir. Bottomhole pressure increasedby 33 atm within 10 hours during liquid level build-up.

Reservoir parameters were obtained from analysis of liquid level build-up highlighted in Fig. 7 usingmethod of non-linear regression. Within the area highlighted in Fig. 7 the simulated pressure build-upcurve (inflow curve) almost fits the actual pressure build-up curve with root mean square (RMS) � 0.02atm for the following parameters:

hydroconductivity � 12 D*cm/cP;skin s � -1.The flow rate accounting for the whole bottomhole pressure history and transient PI is used during

inflow curve simulations.

Unsteady productivity Index (PI) is calculated with the formula

where is the reservoir hydroconductivity, s is the skin, and � is reservoir piezoconductivity,

calculated with formula .

Time (hh:mm) Fluid level (m) Q (m3/d)

Start 699.5

0:42 988.5

1:50 937.5 13.65

4:53 829.5 10.59

7:15 793.0 4.63

7:57 787.0 2.58

Reservoir pressure 139.76 atm

PI 1.85 m3/d/m

Skin 2.28

Hydroconductivity (permeability*net pay/viscosity) 18.83 D*cm/cP

Piezoconductivity (permeability/viscosity/total compressibility/porosity) 350 sm2/s

Maximum flow rate 74.63 m3/d

SPE-171233-MS 15

The reservoir pressure 139.3 atm, the reservoir net pay 9.2 m, and formation compressibility �* �5.7·10-5 1/atm were used for simulation of inflow curves.

The set of values of hydroconductivity (from 1 to 100 D·cm/cP with step 1 D·cm/cP) and skin factor(from -5 to 50 with step 0.5) were used in simulations.

20 best-watch pairs of the skin and the hydroconductivity values are presented in Fig. 8. For thesevalues the simulated build-up pressure differs from the field data less than 0.1 atm. The best hydrocon-ductivity – skin pair ( � 12 D·cm/cP; s � -1) gives solution pressure RMS 0.02 atm. Reservoirparameters estimated by Kogalymneftegeofizika are in the same set � 18.8 D·sm/cP; s � 2.3.

As it is seen from Fig. 8 the best-match pairs of the skin and the hydroconductivity are almost linearlydependent

Figure 7—Transient pressure during well test with gas lift

Figure 8—Twenty best-match pairs of skin and hydroconductivity values for liquid level build-up pressure

16 SPE-171233-MS

As reservoir PI accounting for the skin factor is

it could be noticed that any pair of skin and hydroconductivity values from Fig. 8 gives approximatelythe same PI

One can draw a conclusion that build-up pressure recorded after pumping was stopped gives the inflowcurve, from which the values of skin and hydroconductivity could not be uniquely determined. Log-logplot with pressure change and pressure derivative vs. time is presented in Fig. 9. It confirms the abovestatement, the infinite acting radial flow was not reached and the whole pressure curve is influenced bywellbore storage effect (WBS). The wellhead was open and the fluid inflow from reservoir into wellboreand liquid level increase in tubing and annulus were observed during the whole well test.

The total flow rate from reservoir estimated from pressure data at the maximum drawdown after thecompression is 65 m3/d and in 10 hours decreases to 0.5 m3/d.

Thus, determination of parameters of individual layers only from wellbore pressure data is impossible.Further we discuss the possibility of determination of the quantitative parameters characterizing perfo-ration intervals separately from coupled analysis of wellbore pressure and temperature transients. Thesimulator with mathematical model described in Chapter 5 is used for this purpose.

Quantitative interpretation of pressure-temperature dataField data in the interval from 2365 m to 2400 m were used for quantitative interpretation. The start ofsimulation time corresponds to the time moment when recording of pressure and temperature at the depth2365 m was started.

The main problem in data analysis is absence of quantitative information about history of technologicaloperations in the well before the test. We have to build model using the available qualitative informationabout the initial well condition and technological operations carried in the well before the test.

Figure 9—Log-log plot with pressure change and pressure derivative vs. time

SPE-171233-MS 17

The well history is taken from general understanding of operations in the wells. The initial wellcondition after shut-in before the test is matched manually. After that the prehistory is fixed and thesolution of inverse problem to get the best match between simulated and field data for the fixed historyis conducted.

Figure 10—Comparison of field and simulated data obtained during inverse problem solution with the second objective function for the data fromthe first group (left plot) and second group (right plot)

Figure 11—Comparison of simulated and field transient temperature data at a fixed depth (second objective function)

18 SPE-171233-MS

Available field data are divided into two groups.One group is used for inverse problem solution, another group is used for verification of the obtained

solution. We used half of the wellbore temperature logs in the first group. The rest of temperature logsin the well and temperature transients at a fixed point were used in the second group.

Flow rate data was not included in calculation of objective function due to big uncertainty in the ratevalues obtained by different methods.

Objective function is described below in more details. It is a combination of simulation and field datain the interval from bottom of reservoir to the depth 15 m above the top of reservoir.

Inverse problem solution with perforation intervals provided by the operator did not allow to matchsimulated data with field data in the vicinity of inflow zones. Further we used the depth of perforationintervals obtained from qualitative interpretation.

Three zones of inflow with various parameters were allocated from preliminary data analysis:

1. 20.5 – 24.5 m.2. 25 – 29 m.3. 29.5 – 33.5 m.

Here we used the relative depths starting from the depth of measurements at the point 2365 m.At the first stage the prehistory of operations in the well is selected from available qualitative

information. The prehistory is defined by the following stages: injection, circulation, injection, build-up.The first three stages describe wellbore clean-up during which liquid is circulated in tubing and annulus,and injection of some circulation fluid into the reservoir. The build-up stage approximately describes ashut-in period between circulation stage and the well test.

At the following stage the inverse problem is solved to obtain the best match between simulated andfield data for the fixed prehistory. The stochastic method SCE38 is used for search of a global minimumof the objective function. Several types of the function are used for minimization:

1. Standard deviation of simulated and field data. The weight of points in front of inflow zones areequal 1, in the other interval is 1/3.

2. Standard deviation of simulated and field data. The weight of points in front of inflow zones are

Figure 12—Comparison of simulated flow rate with flow rate estimated from field data

SPE-171233-MS 19

equal 1, in the other interval is 1/10.3. Deviation of recorded temperature data and rate of temperature changes are minimized. Field data

is smoothed using a method of moving average with a window width 1 m. Objective function isthe following:

Temperature profiles logged at times 3.35, 6.17, 7.55 and 14.43 h were used in the first group forinverse problem solution. Temperature profiles logged at times 2.96, 4.17, 6.82, 9.38 h and temperaturetransients at fixed depth were used in the second group for solution verification.

Permeability of layers, skin factors of layers and initial saturation of layers were fitted during inverseproblem solution.

The analysis showed the following:

✓ The results of inverse problem solution on data from the first group are reasonable for all threeresidual functions.

✓ The general behavior of simulated wellbore temperature profiles have reasonable match with fielddata.

✓ Temperature anomalies in front of productive layers are reproduced for three measurements.Temperature measurement at 3.35 h (injection stage) is an exception. It is better described with theinverse solution obtained with the second form of the objective function.

✓ Temperature profiles from the second group (solution verification) are matched with approximatelythe same residuals in inverse problem solutions with the first and the second objective functions.

✓ Inverse problem solution with the third objective functions reproduces temperature profiles alongthe wellbore slightly worse than inverse problem solution with the first and the second residualfunctions.

✓ Inverse problem solution with the second residual functions was chosen as the best way for coupledanalysis of temperature profiles along the wellbore and transient temperature data at fixed depth.

Temperature profiles obtained with different methods for minimum residuals and reservoir parametersfor the best matches are shown below.

Inverse problem solution with the second objective function:Reservoir parameters that provided the best match with field data for three types of objective functions:

Reservoir parameters F �1 F �2 F �3

Permeability of the top layer, mD 7.51 3.45 3.57

Permeability of the middle layer, mD 8.64 9.4 8.39

Permeability of the bottom layer, mD 10.97 10.25 18.30

Skin of the top layer 9.68 9.75 9.31

Skin of the middle layer 0.01 0.04 0.40

Skin of the bottom layer 9.26 9.71 8.91

Saturation of the top layer 0.12 0.11 0.32

Saturation of the middle layer 0.33 0.39 0.1

Saturation of the bottom layer 0.14 0.10 0.1

20 SPE-171233-MS

Figure 13—Temperature profiles sensitivity study to parameters of middle layer (left plot) and prehistory of technological operations in the well (rightplot)

Figure 14—Sensitivity study of transient temperature data measured at fixed point to the parameters of middle layer (left plot) and prehistory oftechnological operations in the well (right plot)

SPE-171233-MS 21

Other parameters:

Sensitivity study of forward solution provides the understanding of influence of reservoir parametersand prehistory on the temperature data. Parameters of the middle layer, the initial temperature profile,duration and temperature of circulation, shut-in time before the test and volume of injected fluid duringcirculation were varied. Middle layer parameters were changed about 20% from the values obtained fromthe inverse problem solution. Variations of initial and circulation temperatures were 2 °C, variation of theshut-in time was 4 h, variation of circulation duration was 1 h, and variation of volume of injected fluidis 4 m3 .

Sensitivity study showed that prehistory influence plays the dominant role. The influence of reservoirpermeability has approximately the same value, all other parameters impact on temperature distributionless. This conclusion is very important for well testing with gas lift. It is necessary to record the full anddetailed prehistory of technological operations in the well before the test to get reliable hydrodynamicparameters of reservoir from coupled interpretation of pressure and temperature transients.

It is possible to get additional information from the coupled analysis of pressure and temperaturetransients for other well testing technologies, for example, for well testing with the jet pump. So, replacing

F �1 F �2 F �3

Total injected volume, m3 2.86 2.40 3.33

Total inflow volume, m3 4.48 3.84 8.56

Average flow rate, m3/d 9.62 8.24 18.38

Contribution of the top layer, % 24 13 7

Contribution of the middle layer, % 43 51 50

Contribution of the bottom layer, % 33 36 43

Figure 15—Comparison of sensitivity of temperature at fixed depth to some system parameters for field example (left plots) and optimized test (rightplot)

22 SPE-171233-MS

the stage of injection on several transient stages “flow - shut-in”, it is possible to obtain better sensitivityto layer parameters, and, as a result, it is possible to make more reliable quantitative interpretation.

Conclusions

1. The numerical model of transient thermo-hydrodynamic processes in the coupled system wellbore- reservoir is developed for two-phase flow of water and oil in a multilayer reservoir. The modelcomprises convective and conductive heat transfer in the reservoir and baro-thermal effect.

2. The numerical simulator for analysis of transient pressure, temperature and phase content wasdeveloped.

3. Pressure and temperature distributions along the wellbore and at a station, measured withconventional technologies during testing of low rate well with compressors, could be used forquantitative interpretation with the developed simulator to obtain the hydrodynamic properties ofindividual layers and flow rate estimations.

4. The knowledge of full and detailed prehistory of technological operations in the well before thetest is necessary to get reliable hydrodynamic parameters of reservoir from coupled interpretationof pressure and temperature transients.

AcknowledmentsThe authors extend their appreciation to Schlumberger for financial and technical support of this work andKogalymneftegeofizika for the help in preparation and interpretation of field data.

Nomenclature

c � specific heat capacity, J/kg/Kc � ratio of volumetric heat capacity of fluid to volumetric heat capacity of fluid saturated porous

mediaC � volumetric heat capacity, J/m3/Kp � pressure, PaPr � reservoir pressure, PaPw � downhole pressure, Par � radial coordinate, mR � inner radius of casing, mrw � wellbore radius, mrD � radius of damaged zone, mS � Holdups � saturation, skin-factorT� � formation temperature at wellbore� � filtration velocity, m/s � thermal expansion coefficient, 1/K � heat transfer coefficient, W/m2/K� � geothermal gradient, K/m� � Joule – Thomson coefficient, K/Pa� � adiabatic coefficient, K/Pa� � density, kg/m3

� � thermal conductivity, W/mKQ � flow rate, m3/sq � specific flow rate, m3/s/m

SPE-171233-MS 23

Subscripts

f � Formationfl � fluids � skeletonr � reservoir, formationo � oilW � waterw � well, wellbored, D � damage zone1 � oil2 � water

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