Modelling Wellbore Transient FluidTemperature and Pressure During

Diagnostic Fracture-Injection Testing inUnconventional Reservoirs

B. Nojabaei, SPE, Pennsylvania State University; A.R. Hasan, SPE, Texas A&M University; andC.S. Kabir, SPE, Hess Corporation

Summary

Diagnostic fracture-injection testing (DFIT) has gained wide-spread usage in the evaluation of unconventional reservoirs. DFITentails injection of water above the formation-parting pressure,followed by a long-duration pressure-falloff test. This test is apragmatic method of gaining critical reservoir information (e.g.,the formation-parting pressure, fracture-closure pressure, and ini-tial-reservoir pressure), leading to fracture-completion design andreservoir-engineering calculations.

In typical field operations, pressure is measured at the well-head, not at the bottom of the hole, because of cost considerations.The bottomhole pressure (BHP) is obtained by simply adding aconstant hydrostatic head of the water column to the wellheadpressure (WHP) at each timestep. Questions arise whether thispractice is sound because of significant changes in temperaturethat occur in the wellbore, leading to changes in density and com-pressibility throughout the fluid column. This paper explores thisquestion and offers an analytical model for estimating the tran-sient temperature at a given depth and timestep for computing theBHP. Furthermore, on the basis of the premise of a line-sourcewell, we have shown that the early-time data can be representedby the square-root of time formulation, leading to the new modi-fied Hall relation for the injection period.

Introduction

Historically, many studies have explored various interpretationaspects of DFIT in unconventional shale reservoirs, encompassingmicro- to nanodarcy formations. These studies include those ofMayerhofer et al. (1995), Abousleiman et al. (1994), Solimanet al. (2010, 2011), Craig et al. (2005), Barree et al. (2009), Soli-man and Kabir (2012), and Nojabaei and Kabir (2012), amongothers. DFIT entails inducing a hydraulic fracture by injecting asmall volume of fluid into the formation and shutting the well infor a long-duration falloff. Typically, this type of test allows esti-mation of pfb, pfc, initial reservoir pressure (pi), the leakoff type,and some measure of formation conductivity. The injection periodleads to the determination of pfb, whereas the falloff analysisyields the remaining parameters.

The use of BHP is implicit in all interpretation methods. How-ever, the economic reality in field operations suggests the use ofWHPs in most settings. Questions arise whether the WHP datalend themselves for transient interpretation without rigorous well-bore modelling, given significant changes in water density andcompressibility as a function of shut-in time. Although compressi-bility of water is significantly lower than that of hydrocarbons,relevant papers (Kabir and Hasan 1998; Izgec et al. 2009) pointout that gauge-placement issues in conventional gas and oil reser-voirs suggest that this question merits thorough vetting.

Accordingly, this paper expands upon the previous study ofNojabaei and Kabir (2012) for translating WHP into BHP with atransient wellbore-heat-transfer model. In other words, we ex-plored the question of whether a constant-hydrostatic-head correc-tion to the WHP suffices during the falloff period. To this end,this study presents an analytical model for temperature transientsthat allows for the evaluation of water density, compressibility,and thermal expansion at each depth step for evaluating BHP. Afluid-temperature model during injection also allows rigorousdetermination of BHP to allow analysis of injection data (e.g.,with the modified Hall method). To that end, a new semianalyticalformulation of the modified Hall method allows rigorous treat-ment of injection data involving linear flow.

Temperature Model During Pressure-Falloff Test

After a well is shut in at the surface, afterflow at the sandface isnegligible because of the low compressibility of the injected water.Upon cessation of injection, the cold injection water begins to gainheat from the surroundings. In other words, heat flow from the for-mation into the wellbore will result in increased internal energy ofthe wellbore fluid and of the composite tubing/casing/cement mate-rial. The general energy-balance equation can be written as

Q ¼ dðmEÞcv

dtþ dðm0E0Þw

dt: ð1Þ

The heat received from the formation, Q, is written as

Q � cPðTei � Tf ÞL0R: ð2Þ

The alternative relaxation distance parameter, L0R (which omitsflow rate, w) is given by

L0R ¼2pcP

rUke

ke þ ðrUTDÞ

� �; ð3Þ

where L0R is equivalent to wL0Rfor injection.Eq. 1 is rewritten by replacing the internal energy of the fluid

in the control volume with its enthalpy, H, and its pressure andvolume, and then noting that fluid mass per unit depth, m, equalsAq. As Hasan and Kabir (2002) noted, the temperature rise of thecement/tubular material at any time may be taken to be a multipleof the rise in the fluid temperature. In that case, we may write thefirst term of the right side of Eq. 1 as

d

dtðmEþ m0E0Þ ¼ d

dt½mcPTf ð1þ CTÞ�: ð4Þ

Combining Eqs. 1 through 4, we obtain the following first-order,linear differential equation for fluid temperature in time and space

dTf

dt¼ L0R

mð1þ CTÞðTei � Tf Þ: ð5Þ

Eq. 5 is solved easily if L0R is assumed constant. However, asEq. 3 shows, L0R depends on TD, which depends on time. For

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. . . . . . . . . . . . . . . . . . .Copyright VC 2014 Society of Petroleum Engineers

This paper (SPE 166120) was accepted for presentation at the SPE Annual TechnicalConference and Exhibition, New Orleans, 30 September–2 October 2013, and revised forpublication. Original manuscript received for review 13 October 2013. Revised manuscriptreceived for review 1 February 2014. Paper peer approved 3 April 2014.

May 2014 Journal of Canadian Petroleum Technology 161

dimensionless shut-in periods of tD< 1.5, Hasan and Kabir (2002)have shown that the following expression for TD applies:

TD ¼ 1:1282ffiffiffiffiffitD

pð1� 0:3

ffiffiffiffiffitDpÞ � 1:1282

ffiffiffiffiffitD

p; ð6Þ

where tD ¼ at=r2wb. When ke is small, one may approximate L0R as

L0R ¼2pke

cp1:1282ffiffiffiffiffitDp ¼ 5:57 ke

cpffiffiffiffiffitDp ¼ 5:57 kerwb

cp

ffiffiffiffiatp ¼ a00=

ffiffitp:

� � � � � � � � � � � � � � � � � � � ð7Þ

With these assumptions, Eq. 5 has the following solution:

Tf ¼ Tei þ ðTfo � TeiÞ e�a0ffitp; ð8Þ

where the parameter a0 ¼ 2a00/(mþmCT) combines various con-stants embedded in Eqs. 5 through 7 and Tfo is the fluid-tempera-ture profile just before shut-in. When Tfo data are unavailable, asis often the case, the injection-fluid temperature estimates can beused. Appendix A presents the transient injection-fluid-tempera-ture model and also its steady-state counterpart.

Field Example

We discuss two field examples from different settings. Well 1 isin the Bakken tight oil play, whereas Well 2 comes from the gas/condensate setting in Eagle Ford. Nominally, a Bakken well hasapproximately 10,000-ft vertical trajectory and 10,000-ft lateraltrajectory. The Bakken DFIT entails dropping a ball in a well’svertical section, which is then pushed by the injected fluid at alow rate filling in approximately 10,000 ft in the lateral sectionuntil it seals the packer near the well’s toe. Thereafter, the injec-tion rate is stepped up to induce hydraulic fracturing in the forma-tion. Note that the ball isolates the float collar ahead of thepacker. However, the DFIT operation in Eagle Ford and other set-

tings is fundamentally different in that no wellbore fill-up isrequired before initiating the fracture.

Modelling Temperature Transient During Falloff Test. Thepressure-computation algorithm entails two steps. First, we evalu-ate the temperature as a function of time at various depthsthroughout the wellbore. As expected, the bottomhole temperature(BHT) will exhibit the largest excursion. Second, the fluid proper-ties of density and compressibility are evaluated at each depthstep corresponding to the temperature profile, leading to the BHPestimation at a given timestep. Allow us to illustrate the computa-tional approach with a field example.

Fig. 1 presents the falloff data measured during a test, dis-cussed earlier by Nojabaei and Kabir (2012). The rapid rise intemperature during the first 10 hours suggests potentially largechanges in compressibility and density of water. Therefore, if thefracture closure occurs within this time period, uncertainty inBHP estimation may affect analysis. Fig. 2 shows the quality ofmatch obtained for temperature data with the square-root-of-timemodel presented by Eq. 8.

Establishing time-dependent temperature profiles with Eq. 8 isa first step toward computing BHP from WHP. Thereafter, the flu-id’s expansivity and compressibility are estimated as outlined inAppendix B. Fig. 3 displays both the temperature and density pro-files. Note that the error between the corrected and the uncor-rected pressures begins to diminish with time because the changein density or expansivity is counteracted by the fluid’s compressi-bility, as Fig. 4 suggests.

Analysis of Pressure and Temperature Transients During

Falloff Test. Because the detailed interpretation of the falloffpressure response was discussed previously, we only explore theadditional information that can be learned from this test. Fig. 5exhibits the pre- and post-closure responses, where a fracture-

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10,000 230

220

210

200

190

180

170

160

150

8,000

6,000

4,000

2,000

00 50 100

BHP

WHP

BHT

Shut-In Time, hours

BH

T, °

F

WH

P, B

HP,

psi

g

Fig. 1—BHP, BHT, and WHP data gathered during a falloff test.

230

220

210

200

190

180

170

160

1500 25 50 75

DataModel

100Time, hours

BH

T, °

F

Fig. 2—Good agreement between the model and temperaturedata.

250 67

66

65

64

200

150

100

50

00 3,000 6,000 9,000

ρ

Well Depth, ft True Vertical Depth

12,000

@24 hours

ρ @0.5 hours ρ, Ib

m/ft

3

T, °

F

Initial ρ

Initial T

T @24 hours

T @0.5 hours

Fig. 3—Time-dependent temperature gradient affects wellbore-fluid density.

65

50

35

20

5

–100 5 10 15

Shut-In Time, hours

Err

or, p

si

20 25

Fig. 4—Nonlinear correction of pressure data with increasingshut-in time.

162 May 2014 Journal of Canadian Petroleum Technology

closure time of 13 hours is indicated. This plot comparing theBHP with WHP responses suggests that the pressure-conversionissue becomes moot for the problem at hand because the twocurves converge well before the closure time.

The analysis of temperature transients also suggests the domi-nance of linear flow because of the slow thermal diffusion process,as shown in Fig. 6. A fracture closure time of 12 hours is estimatedfrom the diagnostic plot, which is in good agreement with its pres-sure counterpart. We surmise that the upward shift in the tempera-ture derivative at approximately 0.8 hours is a manifestation ofnonidealized fracture geometry. However, the smooth transitionfrom the higher elevation appears to be a reflection of the fractureclosure. The relatively long fracture-closure transition is a manifes-tation of low-leakoff rate. Recent studies (Wallace et al. 2014) havebegun to explore the underlying physics involving full-wellboretransients and storage; hydraulic behaviour through induced frac-

tures; and complex interactions between rock, fluids, and naturalfractures. We expect that the use of thermal considerations as shownto correct BHP is prudent, and to identify how thermal effects couldconfound the early falloff pressure transients of DFITs. For instance,a large number of DFIT preclosure periods show a slope that isbetween bilinear- and linear-flow regimes. Whether thermal transi-ents contribute to this signature or poroelasticity triggers it, is yetunknown. Also, many of our closure times (for the primary closureevent) occur in 6 to 18 hours, even in shale DFIT.

Analysis of Injection Pressure With Modified Hall Integral

Method. Analysis of injection data has been fraught with uncer-tainty because of complex mechanisms of fracture initiation, fracturepropagation, variable fluid loss, and fluid efficiency that are all inplay during a short period of time. Nolte’s seminal studies (Nolte1979, 1986a, b, 1991) paved the way for understanding this complexprocess. Nolte (1991) proposed log-log diagnosis (log Dp vs. logtime or cumulative injection volume), and he suggested 1/8 to 1/4slope at early times to reflect restricted height and unrestricted exten-sion, followed by a plateau period indicating stable fracture growth.Finally, the unit-slope response suggests restricted fracture extensionof two active wings. However, field experiences suggested a depar-ture from the idealized behaviour postulated originally. The estima-tion of formation permeability was attempted by many over theyears (Mayerhofer et al. 1995) in moderate-permeability systems.Subsequently, Valko and Economides (1999) offered modificationof the Mayerhofer et al. model to handle variable leakoff from vari-ous segments of the fracture. Others have reported analysis of injec-tion data. For example, Mayerhofer and Economides (1997) shedlight on various possible formulations involving superposition ofinjection history and filter-cake resistance at the fracture/formationinterface. The use of log-log diagnosis was strongly recommendedbefore embarking on any analysis with specialized plots.

This subsection explores the use of the modified Hall approach(Izgec and Kabir 2009) to establish the formation-breakdownpressure with injection data. Nojabaei and Kabir (2012) showedthat the numerical derivative is a good method to arrive at thebreakdown pressure. Appendix C presents development of the an-alytical derivative involving linear flow in unconventional forma-tions at early times. As Fig. 7 shows, the break-over point agreesclosely with that of the numerical derivative. As expected, neitherthe radial-flow model nor the log-time derivative shows any pointof inflection. Despite the new semianalytical formulation with lin-ear flow, we expect that the numerical derivative provides aclearer picture of the breakdown pressure.

Another interesting observation emerges when the same modi-fied Hall data are graphed on the log-log plot. The features of alog-log plot finesse subjectivity, such as that in the basic pumppressure vs. time Cartesian plot used in field operations. As theexpanded version of Fig. 7 data, Fig. 8, indicates, the expectedhalf-slope response emerges after the fracture breakdown occurs,but over a short time span. The earlier unit-slope line suggests

1,000 1/2-slopePreclosure Linear Flow

–1/2-slopeAfter-Closure Linear Flow

–1-slopeRadial Flow

Fracture-Closure Time = 13 hoursClosure Pressure = 8,485 psig

100

10

10.1 1.0 10.0 100.0

WHP

BHP

Shut-In Time, hours

(a)

(b)

(ti+

Δt)dp ws/d

Δt, p

si

1,000.0

1,000

100

100.1 1.0

WHP

BHP

Shut-In Time, hours

(ti+

Δt)dp ws/d

Δt, p

si

10.0

Fig. 5—Estimating fracture-closure pressure and time: (a) totalresponse, (b) early-time response. The expanded version ofFig. 5a suggests that the two curves meet in approximately 5hours.

100

–1/2 slope

Fracture closure(12 hours)

–1/2 slope10

1

0.10.01 0.1 1 10

Δt, hours

Δt×

dT/d

Δt, °

F

100 1,000

Fig. 6—Temperature diagnosis suggests linear flow and thesuspected fracture-closure transition. HI 5 Hall integral. DHI 5derivative of Hall integral.

200,000Modified Hall IntegralNumerical derivativeSquare-Root-of-Time derivativeLog-time derivative

160,000

120,000

80,000

40,000

00 50 100 150

Cumulative Injection, STB

HI o

r D

HI,

psi-D

200 250

Fig. 7—The modified Hall formulation justifies the square-root-of-time model, Well 1.

May 2014 Journal of Canadian Petroleum Technology 163

that the wellbore was being loaded with the injection water, whichmay be construed as wellbore storage, used in the context of tran-sient-pressure testing. Fig. 9 exhibits a similar signature for Well2, where only approximately 5 bbl of water accounts for the shorttime span, compared with approximately 33 bbl in Well 1. Notethat the short-duration fracture response is hard to ascertain fromthe traditional log-log plot, as shown in Fig. 10. Normally, thestate of the fracture is inferred from the shut-in response despiteconstant-rate injection. For instance, Fig. 11 showing the G-func-tion plot for Well 2 suggests that a diffused fracture had devel-oped, as indicated by the analysis of the falloff data.

We note that the degree of separation of the derivative of theHall-integral curve from the Hall-integral curve is a measure offracture conductivity in conventional formations, as shown byIzgec and Kabir (2011). However, such separation cannot provideclues about the pressure behaviour during the shut-in period. Infact, analysis of injection data of several wells from reservoirs ofdiverse geomechanical and fluid-efficiency considerations sug-gested that the short-duration linear flow, followed by the unit-slope response, is the norm as observed here. We speculate thatthe dominance of the late-time unit-slope response, preceded bythe half-slope signature, suggests that the fluid has not had time todiffuse into the formation given the high-rate injection over ashort time span in a tight formation.

Discussion

The transient-temperature model and the computational approachpresented here are intended primarily for the falloff test run duringany DFIT in an unconventional setting to account for large changesin fluid temperature at early times. As expected, the early-timeinjection data for the modified Hall formulation also require the lin-ear-flow treatment in micro- and nanodarcy formations.

Questions arise when downhole pressure measurementsbecome a necessity to avoid any uncertainty arising from thepotential transient-temperature issue. Using Fig. 5 as a guide, one

can surmise that if the fracture closure occurs within the first logcycle (1 to 10 hours) or earlier, one needs to consider running adownhole gauge. Otherwise, wellhead measurements suffice. Tothat end, this study provides some clues about pressure measure-ments in a setting where considerable fluid injection occurs (70þbbl) either before initiation of the fracture (as in Well 1) or duringits propagation (as in Well 2).

Although the conventional modified Hall plot diagnoses thefracture-breakdown pressure quite well, graphing the same dataon a log-log plot is even more illuminating in that it clearly delin-eates both the early-time unit-slope and the late-time half-sloperesponses. The half-slope response is in accordance with linearflow, as shown in Appendix C. But, the derivative of the modifiedHall formulation on a log-log plot suggests that the half-slope pe-riod is short-lived. The subsequent development of the unit-sloperesponse is speculative in that it may be associated with fluid stor-age within the fracture.

We also note that the underlying assumptions in DFIT inter-pretation (e.g., a planar biwing fracture) may be too simplistic forall situations. Interception of the induced fracture by a swarm ofnatural fractures and its termination on one side can conceivablyaccount for the development of an ineffective fracture. In fact,understanding mechanisms of complex hydraulic fractures in dif-ferent rock-property and geologic environments has been a sub-ject of many recent studies, including those of Cipolla et al.(2010), Chuprakov et al. (2011), and Gu et al. (2012). Wallaceet al. (2014) also sheds light on complexity in preclosure eventscaused by the presence of fissures.

Conclusions

1. An analytical solution is obtained for estimating the transientfluid temperature at any point in the wellbore during a pres-sure-falloff test. The resultant computational algorithm requiresconverting WHP into BHP by accounting for changes in thecompressibility, thermal expansion, and density of the fluid col-umn. A field example demonstrates the approach presentedhere. Generally speaking, if the fracture closure occurs within10 hours in a setting where considerable injection occurs, thentemperature modelling is needed for pressure correction; other-wise, hydrostatic-head correction suffices.

2. Analysis of temperature transients during the post-closure pe-riod supports the fracture-closure time estimated from its pres-sure counterpart.

3. A new modified Hall formulation is developed on the premiseof linear flow for the short-duration injection period. The plotsuggests that the linear flow is short-lived, and the dominantmechanism appears to be fluid storage within the fracture,given high-rate injection over a short time span.

Nomenclature

B ¼ formation volume factor, RB/STBc ¼ fluid compressibility, 1/psi

Ceff ¼ leakoff coefficient, ft/min0.5

60,000 100,000

10,00010 100

50,000

40,000

30,000

20,000

10,000

00 10 20 30 40

Cumulative Injection, STB Cumulative Injection, STB

1/2-Slope Line

Unit-Slope Line

Fracture Breakdown Pressure = 8,409 psig

HI o

r D

HI,

psi-D

HI o

r D

HI,

psi-D

Inje

ctio

n R

ate,

ST

B/m

in

50 60 70 800

1

2

3

4

5

6

7

Fig. 9—The log-log signature on the right captures the short-duration fracture linear-flow response, Well 2.

Cumulative Injection, STB

HI o

r D

HI,

psi-D

50,000100

1/2-Slope Line

Unit-Slope Line

Fig. 8—The log-log graph delineates the short-duration fracturelinear-flow response, Well 1.

164 May 2014 Journal of Canadian Petroleum Technology

cp ¼ heat capacity, Btu/(lbm- �F)ct ¼ total system compressibility, 1/psi

cw ¼ water compressibility, 1/psiD ¼ wellbore diameter, ftE ¼ Young’s modulus, psiE0 ¼ plane-strain modulus, psig ¼ gravitational acceleration, ft/sec2

gc ¼ conversion factor, 32.17 (lbm-ft)/(lbf-sec2)gG ¼ geothermal gradient, �F/ftG ¼ fracture G-function, dimensionlessh ¼ net-pay thickness, fthf ¼ fracture height, fthp ¼ leakoff height, ft

i ¼ water-injection rate, STB/DJ ¼ ft-lbf to Btu conversion factor, dimensionlessk ¼ formation permeability, md

LR ¼ relaxation distance parameter, 1/ftp ¼ pressure, psi

pe ¼ pressure at the injectant/reservoir-fluid interface, psipi ¼ initial reservoir pressure, psire ¼ outer reservoir radius, ftrw ¼ wellbore radiuss* ¼ pseudoskin, dimensionless

t ¼ total test time, hourstinj ¼ injection time, hoursto ¼ reference time, hours

Tei ¼ formation temperature at initial condition at any depth, �FTeiwh ¼ formation temperature at wellhead, �F

Tf ¼ fluid temperature, �FTfis ¼ stabilized fluid temperature during injection, �F

Tfwh ¼ fluid temperature at wellhead, �FTs ¼ fluid shut-in temperature, �Fv ¼ average fluid velocity during injection, ft/sec

Wi ¼ cumulative water injection, STBxf ¼ fracture half-length, ftz ¼ well depth, fta ¼ wellbore inclination with horizontal, degreesb ¼ volumetric expansion coefficient, 1/�F

Dt ¼ shut-in time, hoursl ¼ fluid viscosity, cpq ¼ fluid density, lbm/ft3

r ¼ Poisson’s ratio/ ¼ lumped parameter defined by Eq. A-4

Subscripts

bh ¼ bottomholec ¼ closure

fb ¼ fracture breakdown

fc ¼ fracture closurewh ¼ wellheadDt ¼ shut-in time, hr

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Fig. 10—The traditional log-log plot does not provide revealingclues, Well 2.

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Fig. 11—The G-function plot suggests poorly defined fracture,Well 2.

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Wallace, J., Kabir, C.S., and Cipolla, C. 2014. Multiphysics Investigation

of Diagnostic Fracture Injection Tests in Unconventional Reservoirs.

Presented at the SPE Hydraulic Fracturing Technology Conference,

The Woodlands, Texas, USA, 4–6 February. SPE-168620-MS. http://

dx.doi.org/10.2118/168620-MS.

Appendix A—Transient-Temperature ModelDuring Injection

Setting up a general energy balance for a control volume at depthz from surface with a wellbore differential length of dz, we obtain

Q ¼ dðmEÞcv

dtþ dðm0EÞw

dtþ d

dzw H þ 1

2v2 þ gz

� �� �:

� � � � � � � � � � � � � � � � � � � ðA-1Þ

In terms of temperature and pressure, we obtain

dTf

dz¼ aðTei � Tf Þ þ ae�zLRðTfwh � TeiwhÞ �

awLRð1� e�zLRÞ

� � � � � � � � � � � � � � � � � � � ðA-2Þwhere

a ¼ wcpLR

mcpð1þ CTÞ; ðA-3Þ

/ ¼ � v

cpJgc

dv

dzþ CJ

dp

dz; ðA-4Þ

and

w ¼ gGsina þ / � gsinacpJgc

: ðA-5Þ

The solution for Eq. A-2, with a constant LR and initial condi-tion of Tf¼Tei at t¼ 0, is

Tf ¼ Tei �wLRð1� e�zLRÞð1� e�atÞ

þ e�zLRðTfwh � TeiwhÞð1� e�atÞ: � � � � � � � � � � ðA-6Þ

Eq. A-6 can be used for estimating fluid-temperature profile atany time during injection. The steady-state injection temperatureprofile, Tfo, can be obtained by using t¼1 in Eq. A-6 as

Tfo ¼ Tei �wLRð1� e�zLRÞ þ ðTfwh � TeiwhÞ e�zLR : ðA-7Þ

However, if the injection period is short and the variation ofLR with t needs to be accounted for, we can follow the approachoutlined in the body of the paper and use Eq. 7, L0R ¼ ½2pke=ðTDcpÞ� ¼ a00=ð

ffiffitpÞ. With those assumptions, Eq. A-2, which is

similar in form to Eq. 5, has the following solution:

Tf ¼ Tei þ Tfo � Tei �b

a02

� �e�2a0

ffitpþ b

2a02 �bffiffitp

a0 :

� � � � � � � � � � � � � � � � � � � ðA-8Þ

Appendix B—BHP Computational Algorithm

Fluid Temperature. For this vertical well, gG¼ 0.01526 �F/ft;hence, Tei¼ 70þ 0.01526 z.

For initial fluid temperature before shut-in, we use Eq. A-7,the steady-state injection temperature (with LR/w¼ 76.1977,LR¼ 0.0001834),

Tfo ¼ Tei � 76:1977 ð1� e�0:0001834zÞþ ð40� 70Þ e�0:0001834z:

ðB-1Þ

Fluid temperature as a function of depth and time after wellshut-in is obtained from Eq. 8 (with a¼ 0.04, rwb¼ 0.5ft)

Tf ¼ ðTfo � TeiÞe�1:91915ffiffiffitDpþ Tei : ðB-2Þ

Fluid Pressure. To obtain the BHP from the WHP, only statichead qgh needs to be calculated. Density of a liquid, however,depends on both pressure and temperature and is usually com-puted with the following expression:

q ¼ qoecwDP�b�DT ; ðB-3Þ

where qo is the fluid density at standard conditions; cw is the fluidcompressibility, given by �(1/q)(@q/@p)T, and b is the expansiv-ity, given by (1/q)(@q/@T)p. These fluid properties themselves areweak functions of pressure and temperature. We used the follow-ing expressions for fluid compressibility and expansivity (with pin psi and Tf in �F):

cw ¼ ðaþ bTf þ cTf2Þ10�6; ðB-4Þ

where

a ¼ 3:8546� 0:000134pb ¼ �0:01052þ 4:77� 10�7pc ¼ 3:9267� 10�5 � 8:8� 10�10p;

ðB-5Þ

and

b ¼ ð2:0� 10�9T2f þ 1:9� 10�6Tf � 7:64� 10�5Þ:

� � � � � � � � � � � � � � � � � � � ðB-6ÞTo accomplish the pressure-traverse computation at any timestep,the wellbore is divided into a number of discrete depth steps for com-putation of fluid temperature at each depth location. Thereafter, cw

and b are calculated, allowing estimation of q. Pressure at the nextdepth location is then calculated with Eq. B-3. The process is thenrepeated until the bottom of the hole is reached for a given timestep.

Appendix C—Derivation of Modified Hall Integralin Unconventional Formations

The derivative of the Hall integral in given by (Izgec and Kabir2009):

DHI ¼d

ðDpdt

dlnW¼ a1WpwD: ðC-1Þ

At early times, the exponential-integral function reduces to thefollowing form (Hasan and Kabir 2002):

pwD ¼ 1:128ffiffiffiffiffitDpð1� 0:3

ffiffiffiffiffitD

pÞ: ðC-2Þ

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166 May 2014 Journal of Canadian Petroleum Technology

Combining Eqs. C-1 and C-2, we obtain the following expres-sion for the derivative of the Hall integral for early-time (tD< 1.5)transient flow:

DHI ¼ a1W½1:128ffiffiffiffiffitDpð1� 0:3

ffiffiffiffiffitD

pÞ�; ðC-3Þ

where

a1 ¼141:2Bl

kh: ðC-4Þ

tD ¼2:64� 10�4kt

ulctr2w

: ðC-5Þ

Fluid flow in nanodarcy formations at early times may beapproximated by

tD << 1,ffiffiffiffiffitD

p>> tD: ðC-6Þ

Therefore, combining Eqs. C-1, C-3, and C-6, one obtains

DHI ¼ 1:128a1WffiffiffiffiffitD

p: ðC-7Þ

Bahareh Nojabaei is a PhD degree candidate in petroleumand natural gas engineering at the Pennsylvania State Univer-sity. Her research areas include DFIT analysis, reservoir simula-tion and well testing in unconventional reservoirs, and phasebehaviour in tight rocks. Nojabaei holds an MS degree in me-chanical engineering from Tehran Polytechnic University anda BS degree from Iran University of Science and Technology.She is currently a technical editor for SPE Reservoir Evaluation& Engineering and SPE Journal. Nojabaei received the SPE

Nico van Wingen Fellowship in 2014. She can be reached atbahareh@psu.edu.

A. Rashid Hasan is a professor of petroleum engineering atTexas A&M University. He has more than 35 years of teaching,consulting, and research experience in many areas, includingfluid and heat flows in wellbores and pressure-transient testing.Hasan is an expert in the areas of production engineering; hefocuses on modelling complex transport processes in variouscomponents of petroleum production systems. He has alsoworked with the National Aeronautics and Space Administra-tion on various aspects of multiphase flow and thermohy-draulic transients. Hasan has published extensively, and is acoauthor of the SPE textbook Fluid Flow and Heat Transfer inWellbores. He has served on various SPE committees, includingthe Editorial Review Committee for SPE Production & Facilitiesand SPE Journal, and was the recipient of the 2011 SPEProduction and Operations Award. Hasan holds Masters andPhD degrees in chemical engineering from the University ofWaterloo.

C. Shah Kabir is a global reservoir engineering adviser at HessCorporation in Houston. His experience spans more than 30years in the areas of transient-pressure testing, fluid- and heat-flow modelling in wellbores, and reservoir engineering. Besidescoauthoring more than 100 papers, Kabir coauthored the2002 SPE textbook Fluid Flow and Heat Transfer in Wellboresand contributed to the 2009 SPE monograph Transient WellTesting. He has served on various SPE committees, includingthe Editorial Review Committees for SPE Production & Facilities,SPE Reservoir Evaluation & Engineering, and SPE Journal. Cur-rently, Kabir serves as an associate editor for SPE ReservoirEvaluation & Engineering. He has chaired SPE Applied Tech-nology Workshop and SPE Forum Series meetings. Kabir was anSPE Distinguished Lecturer during the 2006–07 season andbecame an SPE Distinguished Member in 2007. He receivedthe 2010 SPE Reservoir Description and Dynamics Award.

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May 2014 Journal of Canadian Petroleum Technology 167