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Petroleum 1 (2015) 133e138

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Original article

Characterizing hydraulic fractures in shale gas reservoirs usingtransient pressure tests

Cong Wang*, Yushu WuColorado School of Mines, United States

a r t i c l e i n f o

Article history:Received 21 April 2015Accepted 29 May 2015

Keywords:Hydraulic fracturesShale gas reservoirsTransient pressure testsReservoir simulationFormation characterizationGas adsorption

* Corresponding author.E-mail addresses: cowang@mines.edu, congwang2

ywu@mines.edu (Y. Wu).Peer review under responsibility of Southwest Pe

Production and Hosting by Elsev

http://dx.doi.org/10.1016/j.petlm.2015.05.0022405-6561/Copyright © 2015, Southwest Petroleum Uaccess article under the CC BY-NC-ND license (http://

a b s t r a c t

Hydraulic fracturing combined with horizontal drilling has been the technology that makes itpossible to economically produce natural gas from unconventional shale gas or tight gas reservoirs.Hydraulic fracturing operations, in particular, multistage fracturing treatments along with hori-zontal wells in unconventional formations create complex fracture geometries or networks, whichare difficult to characterize. The traditional analysis using a single vertical or horizontal fractureconcept may be no longer applicable. Knowledge of these created fracture properties, such as theirspatial distribution, extension and fracture areas, is essential information to evaluate stimulationresults. However, there are currently few effective approaches available for quantifying hydraulicfractures in unconventional reservoirs.

This work presents an unconventional gas reservoir simulator and its application to quantifyhydraulic fractures in shale gas reservoirs using transient pressure data. The numerical model in-corporates most known physical processes for gas production from unconventional reservoirs,including two-phase flow of liquid and gas, Klinkenberg effect, non-Darcy flow, and nonlinearadsorption. In addition, the model is able to handle various types and scales of fractures or het-erogeneity using continuum, discrete or hybrid modeling approaches under different well pro-duction conditions of varying rate or pressure. Our modeling studies indicate that the mostsensitive parameter of hydraulic fractures to early transient gas flow through extremely lowpermeability rock is actually the fracture-matrix contacting area, generated by fracturing stimu-lation. Based on this observation, it is possible to use transient pressure testing data to estimate thearea of fractures generated from fracturing operations. We will conduct a series of modeling studiesand present a methodology using typical transient pressure responses, simulated by the numericalmodel, to estimate fracture areas created or to quantity hydraulic fractures with traditional welltesting technology. The type curves of pressure transients from this study can be used to quantifyhydraulic fractures in field application.

Copyright © 2015, Southwest Petroleum University. Production and hosting by Elsevier B.V. onbehalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND

license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

68@gmail.com (C. Wang),

troleum University.

ier on behalf of KeAi

niversity. Production and hostcreativecommons.org/licenses/b

1. Introduction

For the unconventional gas reservoirs, hydraulic fracturescharacterization is important in assuring the maximum stim-ulation efficiency [1e3]. A lot of researches have been carriedout on the flow behavior analysis of vertical wells with finite-conductivity or infinite-conductivity hydraulic fractures. Cinco-Ley and Samaniego summarized fluids flow in a hydraulicfractured well could be divided into four periods: fracturelinear flow, bilinear flow, formation linear flow and pseudo-radial flow [4]. Nobakht and Clarkson pointed out that thedominant flow regime observed in most fractured tight/shale

ing by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an openy-nc-nd/4.0/).

C. Wang, Y. Wu / Petroleum 1 (2015) 133e138134

gas wells is the third one, formation linear flow, which maycontinue for several years [5]. Transient pressure analysis ofthis linear flow behavior is able to provide plenty of usefulinformation, especially, the total contact area between hy-draulic fractures and tight matrix.

Pseudo-pressure, a mathematical pressure function thataccounts for the variable compressibility and viscosity of gaswith respect to pressure, is widely used for the transientpressure analysis in conventional gas reservoirs [6]. Comparedwith conventional reservoirs, gas flow in ultra-low perme-ability unconventional reservoirs is subject to more nonlinear,coupled processes, including nonlinear adsorption/desorption,non-Darcy flow (at both high flow rate and low flow rate),strong rockefluid interaction, and rock deformation withinnano-pores or micro-fractures, coexisting with complex flowgeometry and multi-scaled heterogeneity. Therefore, quanti-fying flow in unconventional gas reservoirs has been a signif-icant challenge and traditional REV-based Darcy law, forexample, may not be generally applicable. For gas flow in theseunconventional reservoirs, our previous work indicates thatgas-slippage effect and adsorption/desorption play an impor-tant role to describe the subsurface flow mechanisms, whichcannot be neglected [7]. To the authors' knowledge, these twofactors were not considered in the previous pseudo-pressurederivation. In this paper, a new derivation of pseudo-pressureis provided.

This paper presents our continual efforts in developing nu-merical models and tools for quantitative studies of uncon-ventional gas reservoirs [8,9]. Specifically, we explore thepossibility of performing well testing analysis using the devel-oped simulator. The numerical model is able to simulate real-istic processes of single-phase or two phase flow inunconventional reservoirs, which considers the Klinkenbergeffects and gas adsorption/desorption. We use the numericalmodel to verify our new derived pseudo-pressure formulation.We also apply it to generate type-curves of transient gas flow inunconventional reservoirs with horizontal well and multistagehydraulic fractures. The type curves of pressure transients fromthis study can be utilized to quantify hydraulic fractures in fieldapplication.

2. Derivation of new pseudo pressure

In 1965, Al-Hussainy and Ramer derived the pseudo pressurewhich has been successfully used to analyze the flow of real gasin the gas reservoirs.

mðPÞ ¼ 2ZP

P0

P0

mZdP0 (1)

where P0 is the reference pressure; P is gas pressure; m is the gasviscosity and Z is gas pressure Z factor.

The concept of the real gas pseudo-pressure promises aconsiderable simplification. It brings improvement in all pha-ses of gas well aWnalysis and gas reservoir calculations. Theseanalysis and calculations in terms of pseudo-pressure workvery well for the conventional reservoirs but meet someproblems when it is directly applied in the unconventionalreservoirs analysis. This is mainly because gas flow in ultra-low permeability unconventional reservoirs, different fromthe gas flow in conventional reservoirs, is subject to morenonlinear, coupled processes, including nonlinear adsorption/desorption, non-Darcy flow, and strong rockefluid interaction,

and rock deformation within nanopores or micro-fractures,coexisting with complex flow geometry and multi-scaledheterogeneity.

Considering the Klinkenberg effects and gas adsorption, theprinciple of conversation of mass for isothermal gas flow througha porous media is expressed by the expression:

V$

�rkðPÞmðPÞVP

�¼ v

vt

�frþmgðVÞ

V

�(2)

The pressure-dependent permeability for gas is expressed byKlinkenberg as:

kg ¼ k∞

�1þ b

Pg

�(3)

where k∞ is constant, absolute gas-phase permeability in highpressure (where the Klinkenberg effect is minimized); and b isthe Klinkenberg b-factor, accounting for gas-slippage effect.

The mass of adsorbed gas in formation volume, V, is describedby Refs. [10,11,7]:

mgðVÞ ¼ rKrgf ðPÞV (4)

where mg(V) is absorbed gas mass in a volume V, rK is rock bulkdensity; rg is gas density at standard condition; f(P) is theadsorption isotherm function. If the adsorbed gas terms can berepresented by the Langmuir isotherm (Langmuir, 1916), thedependency of adsorbed gas volume on pressure at constanttemperature is given below,

f ðPÞ ¼ VLP

P þ PL(5)

where VL is the gas content or Langmuir's volume in scf/ton (orstandard volume adsorbed per unit rock mass); P is reservoir gaspressure; and PL is Langmuir's pressure, the pressure at which50% of the gas is desorbed.

For real gas,

r ¼ MRT

�P

ZðPÞ�

(6)

Substitute Equations (3)e(6) into Equation (2),

V$

24P

�1þ b

P

�

mðPÞZðPÞVP35 ¼ f

k∞

v

vt

�P

zðPÞ þRTrKrgVLPfMðP þ PLÞ

�(7)

From the definition of the isothermal compressibility of gas:

cgðPÞ ¼ ZðPÞP

ddP

�P

ZðPÞ�

(8)

We also define the “compressibility” from the adsorption:

caðPÞ ¼ ZðPÞP

ddP

�RTrKrgVLPfMðP þ PLÞ

�¼ ZðPÞ

PRTrKrgVLPLfMðP þ PLÞ2

(9)

Let the total compressibility:

ctðPÞ ¼ caðPÞ þ cgðPÞ (10)

C. Wang, Y. Wu / Petroleum 1 (2015) 133e138 135

Equation (7) will be:

V2P2 �d�ln mðPÞZðPÞ

1þb=P

�

dP2

�VP2

�2 ¼ fmðPÞctðPÞkð1þ b=PÞ

vP2

vt(11)

Assume the viscosity and gas law deviation factors change slowlywith pressure changes; the second part of Equation (11) becomesnegligible. Equation (11) becomes:

V2P2 ¼ fmðPÞctðPÞkð1þ b=PÞ

vP2

vt(12)

We define the new pseudo-pressure m(P) as follows:

mðPÞ ¼ 2ZP

P0

P0ð1þ b=PÞmðPÞZðPÞ dP0 (13)

Equation (2) can be rewritten in terms of variable m(P) using thedefinition of ct(P) given by Equations (8)e(10) as

V2mðPÞ ¼ fmðPÞctðPÞkðPÞ

vmðPÞvt

(14)

Based on Equation (14), we could apply this form of the flowequation, quasi-linear flow equation, to the analysis of real gasflow behavior in unconventional reservoirs.

Table 1Data used in the calculation of two compressibility.

Parameters Value Units

Gas type MethanePorosity, f 0.05Temperature, T 122 FRock density, rK 2.7 g/ccLangmuir's volume, VL 218.57 Scf/tonLangmuir's pressure, PL 2285.7 psi

3. Linear gas flow

The early flow regime observed in fractured tight/shale gaswells is linear flow from formation, which may continue forseveral years. Sometimes decline curve may indicate outerboundary effects, but no pseudo-radial flow. Wattenbarger et al.gave the “short-term” approximations for this linear flow withconstant rate production conditions [12];

mDi �mDwf ¼ qBffiffiffiffiffiffiffipm

fct

r1ffiffiffik

pA

ffiffit

p(15)

In Equation (15), mD is the pseudo pressure; q is the gas pro-duction rate; B is the gas FVF; f is the formation porosity; ct is thetotal compressibility, k is the formation permeability; A is thehydraulic fracture area; t is the time; and subscript i refers toinitial condition and subscript wf refers to the wellborecondition.

Equation (15) indicates that for the constant-flowing-rateproduction condition, linear flow appears as a straight line onthe plot of normalized pressure vs. the square root of time. Theslope of this square-root-of-time plot can be used to estimate thetotal contact area between hydraulic fracture and the tight ma-trix. The estimation accuracy is influenced by initial pressure,formation average permeability and total compressibility [5].

This model is limited by the assumptions of only one infinite-conductivity hydraulic fracture and the neglect of gas adsorp-tion/desorption effect. For the gas flow analysis in unconven-tional reservoirs, these assumptions are generally unacceptable.Multi-stage hydraulic fracturing is the key technology in devel-oping the unconventional shale gas or tight gas reservoirs. Inaddition, our previous work indicates that adsorbed gas couldcontribute more than 30% to the total production in some un-conventional shale gas reservoirs [7]. For the above situations,estimations considering the effects of gas adsorption/desorption

will bemore accurate and it will avoid the overestimation of totalhydraulic fracture area.

One efficient approach to include the adsorption is by addingan “adsorption compressibility” term into the total compress-ibility. Based on the derived “adsorption compressibility”formulation in Equation (9), the value of adsorption compress-ibility is calculated below. Table 1 lists the data used in thiscalculation and Fig. 1 shows the gas corresponding Z factor valuewith pressure.

Fig. 2 is the calculated adsorption compressibility value andthis value is in the same magnitude with the rock and fluids totalcompressibility. These calculation results indicate that when thepressure increases from 2000 psi to 5000 psi, this compress-ibility drops significantly from 2.71�10�4/psi to 4.29� 10�5/psi.

4. Numerical model

The unconventional oil/gas reservoir simulator developedcoupled multiphase fluid flow with the effects of rock defor-mation, gas slippage, non-Darcy flow and chemical reaction ofadsorption and desorption processes in unconventional reser-voirs. In numerical formulation, the integral finite differencemethod [13] is used for space discretization of multidimensionalfluid and heat flow in porous and fractured reservoirs using anunstructured/structured grid. Time is discretized fully implicitlyas a first-order backward finite difference. Time and space dis-cretization of mass balance equations results in a set of couplednon-linear equations, solved fully implicitly using Newton iter-ation [7].

In our model, a hybrid-fracture modeling approach, definedas a combination of explicit-fracture (discrete fracture model),MINC (Multiple Interacting Continua) approach [13] on thestimulated zones, and single-porosity modeling approaches onunstimulated areas (Figs. 3 and 4), is used for modeling a shalegas reservoir with both hydraulic fractures and natural fractures[14e16]. This is because hydraulic fractures, which have to bedealt with for shale gas production, are better handled by theexplicit fracture method. On the other hand, natural fracturedreservoirs are better modeled by a dual-continuum approach,such as MINC for extremely low-permeability matrix in shale gasformations, which cannot be modeled by an explicit fracturemodel.

Fig. 4 illustrates the conceptual model including a horizontalwell, multistage hydraulic fractures and their correspondingstimulated reservoir volume (SRV). The stimulated reservoirvolume (SRV) is defined as the 3D volume of microseismic-eventcloud nearby the hydraulic fractures, inside which complexnatural fracture networks are stimulated. We apply the methodof MINC to model this the porous and fractured medium. Asingle-porosity model is applied in the region outside the SRV. Inthis method, hydraulic fractures are represented by gridsmatching the real fracture geometric data. Then the properties ofthe fractures, such as permeability and permeability, areassigned to the corresponding fracture grids.

0.94

0.96

0.98

1

1.02

1.04

1.06

2000 3000 4000 5000

Z fa

ctor

for M

etha

ne

Pressure (psi)

Fig. 1. Methane Z factor at 122 F from 2000 to 5000�psi.

0.00E+00

1.00E-04

2.00E-04

3.00E-04

4.00E-04

5.00E-04

6.00E-04

2000 2500 3000 3500 4000 4500 5000

Coom

pres

sibi

lity

(/ps

ia)

Pressure (psia)

Free GasCompressibility

Adsorp onCompressibility

Fig. 2. “Adsorption compressibility” vs. pressure.

C. Wang, Y. Wu / Petroleum 1 (2015) 133e138136

5. Model applications

In this section, we apply the numerical model to analyzetransient pressure behaviors vs. fracture areas. Herewe present aset of simulation cases to study the gas flow characteristics infractured wells.

In the first simulation case, the formation is homogeneouswith no natural fractures. A refining grid system is built for thesimulation. The simulation input parameters are summarized inTable 2. The fluids include gas and water, but water is at residualor immobile, so it is a single-phase gas flow problem. The

Fig. 3. Discrete fracture methods and refining

hydraulic fractures are represented by a discrete fracture withinfinite conductivity. With the input parameters as shown inTable 2, the dimensionless fracture conductivity is calculated byits definition, Cfd ¼ kfwf=kmwm. Its value is large enough thusthe hydraulic fracture could be treated as infinite conductivity.

Three different fracture models with the same total fracture-matrix contact area are built, as demonstrated in Figs. 5e7. Fig. 8illustrates the simulated wellbore bore pseudo-pressure vs. timefor these three cases. Their pressure change with time are almostsame, which indicates that fracture number and fracture geom-etry, as long as the total fracture area keep the same, have littleinfluence on the wellbore pressure change for gas productionfrom extremely low permeability rock.

Next we compare the transient pressure behavior in earlytime with different fracture-matrix total contact area. Two caseswith different area are run and Fig. 9 is the simulation result. Alarger surface area will lead to a slower increase of the dimen-sionless pseudo-pressure. Slope of this line is inversely propor-tional to the fracture area.

Gas adsorption influence is also analyzed. Two cases withdifferent initial pressures are run, one is 2350 psi and the other is3800 psi, respectively. The results are illustrated in Figs. 10 and11. Two conclusions could be drawn from these simulationworks:

1. Gas flow with adsorption will also behave straightly in thenormalized pressure vs. the square root of time plot for thelinear flow period. This is identical with our previous analysisthat adsorption could be treated by a compressibility factor ifthe pressure changes a little.

2. Adsorption will have different influences on the linear flowbehavior at different initial pressure, as shown in Fig. 2 that“adsorption compressibility” drops with pressure increases.

6. Conclusions

In this paper, we derive a new formulation of gas pseudo-pressure for the transient pressure analysis in unconventionalgas reservoirs. We prove its efficiency and accuracy by running aset of relevant simulation examples. We use the developed nu-merical model to simulate shale gas production with a contin-uum, discrete or hybrid modeling approach. Our modelingstudies indicate that the most sensitive parameter of hydraulicfractures to early transient gas flow responses through extremelylow permeability rock is actually the fracture-matrix contactingarea, generated by fracturing stimulation. We observed that gasflow with adsorption will also behave straightly in the normal-ized pressure vs. the square root of time plot for the linear flowperiod, if an adsorption term is included in the total gas

grids technology in our numerical model.

Fig. 4. The hybrid fracture model.

Table 2Data used for the case study and discussion.

Reservoir length, Dx, ft 5500 Natural fracture total compressibility, cnf, psi�1 2.5E�04Reservoir width, Dy, ft 2000 Hydraulic fracture total compressibility, chf, psi�1 2.5E�04Formation thickness, Dz, ft 250 Initial reservoir pressure, Pi, psi 3800Horizontal well length, Lh, ft 4800 Constant flowing bottomhole pressure, Pwf, psi 500Production rate, Q, Mscf/day 5.0Eþ03 Reservoir temperature, T, �F, 122Hydraulic fracture width, wf, ft 0.02 Klinkenberg coefficient, psi 200Hydraulic fracture half-length, Xf, ft 250 Non-Darcy flow constant, cb m3/2 3.2E�06Viscosity, m, cp 0.0184 Langmuir's pressure, PL, psi 2285.7Matrix permeability, km, md 1.0E�04 Langmuir's volume, VL, scf/ton 218.57Matrix porosity, Fm 0.05 Natural fracture total compressibility, cnf, psi�1 2.5E�04Matrix total compressibility, ctm, psi�1 2.5E�04 Hydraulic fracture permeability, khf, md 1E05Natural fracture porosity, Fnf, md 0.3 Hydraulic fracture porosity, Fhf 0.5Natural fracture permeability, knf, md 100

Fig. 5. Case 1, horizontal well with one fracture.

Fig. 6. Case 2, horizontal well with two fractures, symmetrical with the well.

Fig. 7. Case 3, horizontal well with two fractures, asymmetrical with the well.

0.0E+001.0E+182.0E+183.0E+184.0E+185.0E+186.0E+187.0E+188.0E+189.0E+18

0 20 40 60

Pseu

do P

ress

ure

(SI)

Time1/2 (days1/2)

Case 1Case 2Case 3

Fig. 8. Pseudo pressure vs. time square for those 3 cases.

0.0E+00

2.0E+18

4.0E+18

6.0E+18

8.0E+18

1.0E+19

1.2E+19

1.4E+19

0 20 40 60

Pseu

do P

ress

ure

(SI)

Time1/2 (days1/2)

Fracture Area:7500m^2

Fracture Area:16000m^2

Fig. 9. Pseudo pressure vs. square root of time for those 2 cases with differentfractures.

C. Wang, Y. Wu / Petroleum 1 (2015) 133e138 137

0.0E+001.0E+182.0E+183.0E+184.0E+185.0E+186.0E+187.0E+188.0E+18

0 10 20 30 40 50 60Time1/2 (days1/2)

Ini al Pressure is 2,350psi

Withou Adsorp on

With Adsorp on

Pseu

do P

ress

ure

(SI)

Fig. 10. Pseudo pressure vs. time square analysis about adsorption with initialpressure 2350�psi.

0.0E+001.0E+182.0E+183.0E+184.0E+185.0E+186.0E+187.0E+188.0E+189.0E+18

0 10 20 30 40 50 60Time1/2 (days1/2)

Ini al Pressure is 3,800psi

With Adsorp on

Without Adsorp on

Pseu

do P

ress

ure

(SI)

Fig. 11. Pseudo pressure vs. time square analysis about adsorption with initialpressure 3800�psi.

C. Wang, Y. Wu / Petroleum 1 (2015) 133e138138

compressibility. Based on this observation, we demonstrate thatit is possible to use transient pressure testing data to estimate thearea of fractures generated from fracturing operations. A meth-odology using typical transient pressure responses, simulated bythe numerical model, to estimate fracture areas created or toquantity hydraulic fractures with traditional well testing tech-nology is presented. The methodology as well as type curves ofpressure transients from this study can be used for quantifyhydraulic fractures in field application.

Acknowledgments

This work was supported in part by EMG Research Center andUNGI of Petroleum Engineering Department at Colorado Schoolof Mines, and by Foundation CMG.

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