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
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.
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
The heat received from the formation, Q, is written as
Q cPTei Tf L0R: 2
The alternative relaxation distance parameter, L0R (which omitsflow rate, w) is given by
where L0R is equivalent to wL0Rfor injection.
Eq. 1 is rewritten by replacing the internal energy of the fluidin 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
dtmE m0E0 d
dtmcPTf 1 CT: 4
Combining Eqs. 1 through 4, we obtain the following first-order,linear differential equation for fluid temperature in time and space
m1 CTTei Tf : 5
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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This paper (SPE 166120) was accepted for presentation at the SPE Annual TechnicalConference and Exhibition, New Orleans, 30 September2 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:
where tD at=r2wb. When ke is small, one may approximate L0R as
p 5:57 kecp
p 5:57 kerwbcp
With these assumptions, Eq. 5 has the following solution:
Tf Tei Tfo Tei ea0 ffitp ; 8
where the parameter a0 2a00/(mmCT) 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.
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 wellsvertical 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 wells 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-ids 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 fluids compressi-bility, as Fig. 4 suggests.
Analysis of Pressure and Temperature Transients DuringFalloff 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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00 50 100
Shut-In Time, hours
Fig. 1BHP, BHT, and WHP data gathered during a falloff test.
1500 25 50 75
Fig. 2Good agreement between the model and temperaturedata.
00 3,000 6,000 9,000
Well Depth, ft True Vertical Depth
@0.5 hours , Ib
T @24 hours
T @0.5 hours
Fig. 3Time-dependent temperature gradient affects wellbore-fluid density.
100 5 10 15
Shut-In Time, hours
Fig. 4Nonlinear 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 IntegralMethod. 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. Noltes 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
Fracture-Closure Time = 13 hoursClosure Pressure = 8,485 psig
10.1 1.0 10.0 100.0
Shut-In Time, hours
Shut-In Time, hours
Fig. 5Estimating fra...