ANALYSIS OF GAS PRODUCTION FROM HYDRAULICALLY FRACTURED WELLS IN THEHAYNESVILLE SHALE USING SCALING METHODS

Frank Male, Akand W. Islam, Tad W. Patzek, Svetlana Ikonnikova, John Browning and Michael P. Marder

University of Texas at Austin

ABSTRACT

The Haynesville Shale is one of the largest unconventionalgas plays in the US. It is also one of the deepest, with wellsreaching more than 10,000 feet below ground. This uncom-mon depth and overpressure lead to initial gas pressures ofup to 12,000 psi. The reservoir temperature is also high, upto 3000F. These pressures are uniquely high among shale gasreservoirs, and require special attention when modeling wellsin the Haynesville Shale. We show that the method developedby Patzek, et al. (PNAS, 110, 19731-19736) scales cumu-lative gas production histories of individual wells such thatthey all collapse onto two universal curves. The first curve isfor the wells in which consecutive hydrofractures interfere,and the second one is for the wells in pure transient flow.Haynesville wells can take months or years for flowing tub-ing pressure to stabilize, so we modified the universal curvesto take this delay into account. Furthermore, we introduce aPVT solver in order to calculate gas properties up to the highinitial reservoir pressure and temperature. When we applythe Patzek et al. scaling theory to 2,199 individual well pro-ductions in the Haynesville Shale, we find 1,580 wells whichhave entered exponential decline due to pressure interference.We use a simple physical model to determine the time to inter-ference for wells with geologic parameters typical of the Hay-nesville Shale, and use this time to interference to determinea field-wide stimulated permeability. Using this permeabil-ity, we arrive at the times to interference for the remainder ofHaynesville wells, and production forecasts for all individualwells.

Index Terms— Reservoir Engineering, EUR, UniversalCurve

1. INTRODUCTION

Before the advent of hydraulic fracturing technology and hor-izontal drilling, the Haynesville Shale could not have existedas a play. The Haynesville Formation lies at depths from10,300 ft to 14,000 ft (Parker et al., 2009), and has a payzonethickness of only 150 ft. Since laboratory estimates for per-meability obtained from shale core samples are on the order

Thanks to Sloan Foundation for funding

of nanodarcies (Nunn, 2012), it would be impossible to ex-tract economic quantities of natural gas from the HaynesvilleShale without horizontal wells and massive hydrofractures.

In addition to being deeper than other major shale gasplays, the Haynesville is hotter, with temperatures that reachabove 300◦F. It is abnormally pressured, with gradients ofapproximately 0.95 psi/ft, leading to reservoir pressures thatcan reach beyond 13,000 psi (Hammes et al., 2011). TheHaynesville play lies in East Texas and North Louisiana, andwas deposited during the Jurassic period, with the prehistoricSabine Island at the southern boundary. There has been essen-tially no condensate production, and adsorption is negligibledue to the high reservoir temperature.

This paper applies the methodology of Patzek et al.(2013), developed to study the Barnett Shale, to determinewell decline curves and productivity for the wells drilled inthe Haynesville Shale. We start by describing the underlyingtheory of our type curves using scaling arguments. We mod-ify the conditions to include early-time throttling of wellsperformed by operators using chokes, and then introduce aPVT solver to calculate gas properties in well conditions. Weperform fits of well production to the resulting curves and usethe fits to estimate the field’s average stimulated permeabil-ity. Finally, we provide estimates of time to interference andultimate recovery for producing wells.

We performed this work as part of a project to forecastfield-wide production for the Haynesville Shale; the. Theworkflow is similar to that in Browning et al. (2013), cov-ering predictions for the Barnett Shale, and Browning et al.(2014), forecasting production in the Fayetteville Shale. Weuse production histories and geologic data from Smye et al.(in preparation) to forecast production, which is then used totier well acreage and build well inventories. These are then“mined” by an activity-based economic model to arrive at fullfield production forecasts.

2. STATEMENT OF THEORY AND DEFINITIONS

In order to build a minimal model for natural gas flow in a hy-draulically fractured well, we must start with the basic equa-tions that govern flow. The first is Darcy’s law that relates

flow in a porous medium to the gradient of pressure:

ug =k

µg

∂p

∂x. (1)

The second equation is mass balance of the gas:

−∂(ρgug)∂x

=∂

∂t[φSgρg + (1− φ)ρa] . (2)

These equations can be generalized to three dimensions, butthe simplest picture of flow near a hydrofractured well hap-pens to be that of a cuboid volume of stimulated rock with auniform set of planar hydrofractures draining it. This mentalmodel lends itself to one-dimensional analysis. Combiningthe two equations leads to the following partial differentialequation:

− ∂

∂x

(kρgµg

∂p

∂x

)≈ φSg

∂ρg∂p

∂p

∂t+(1−φ)∂ρa

∂ρg

∂ρg∂p

∂p

∂t. (3)

Here we introduce the analysis of Patzek et al. (2013), whoapply the Kirchoff integral transformation of gas pressure pi-oneered by Al-Hussainy et al. (1966), leading to an equationbased on the real gas pseudopressure, then introduce a set ofscaling factors to build a nondimensional version of the re-sulting equations. The scaling factors are:

M = (N + 1)4ρiLHdφSg

and

τ = d2/αi.

The scaled transport equation is:

−∂m∂t

=α

αi

∂2m

∂x2,

m(x, t = 0) = mi(x),

m(x = 0, t) = f(t/ts),

∂m/∂x|x=1 = 0,

(4)

where the function of scaled real gas pseudopressure at x = 0slowly varies with time to correspond to the flowing wellborepressures experienced in a typical Haynesville well. Afterthis system is solved numerically, the mass flow rate into thefracture can be calculated via

m =Mτ

∂m

∂x

∣∣∣∣0

(5)

Integrating this equation with respect to dimensionless timeprovides the cumulative production, which can be written as

m

M = RF(t), where RF(t) ≡∫ t

0

dt′∂m

∂x

∣∣∣0(t′) (6)

The solution for the fractional recovery factor (RF) followsthree distinct phases. In the initial period, the well is throttled,and the pressure at the fracture face slowly varies. In thisperiod, production rate declines slowly. After the pressureat the fracture face stabilizes, production rate declines as oneover the square root of time, so that

m ∝ 1√t− t0

where t0 varies as a function of initial pressure, pi, and thecharacteristic time for choking, ts, but is usually between 0and 4 months. After t = τ , flow begins to be boundary-dominated and the rate follows exponential decline.

3. DESCRIPTION AND APPLICATION OFPROCESSES

The initial boundary value problem (4) is solved at varyinginitial reservoir pressures using a fully implicit numericalsolver, implemented in Python.

Reservoir pressures in the Haynesville average 10,000 psi.This leads to bottomhole flowing pressures that take up to twoyears to stabilize as the flowing tubing pressure is slowly re-duced through choke changes to the prevailing pressure of fa-cilities. In order to account for this slow bottomhole pres-sure decline, we allow the fracture pressure to vary with time.State well tests for Haynesville wells reported that flowingtubing pressure was initially 7,000 psi, dropped to 2,500 psiafter one year, and stabilized at 1,000 psi starting at yeartwo. We converted that timescale into nondimensional timethrough a scaling factor of 0.3, used for all Haynesville wells.We selected the scaling factor to match the universal declinecurve with real well productions. The scaling factor is arbi-trary, and will only hold for future wells if 1) wells are throt-tled in the same manner by operators, and 2) wells continueto exhibit pressure interference at approximately 1-2 years.

A further adjustment is necessary in light of the highreservoir pressures. We developed a PVT package consist-ing of the appropriate equation-of-state (EOS) and modelsfor calculations of thermophysical properties of natural gasmixtures. This package calculates the density (ρ), the com-pressibility factor (Z), the viscosity (µ), and the isothermalcompressibility (c) of a natural gas mixture for given temper-ature, pressure, and molar fractions of gas components. Asmentioned, temperatures and pressures can be very high in theHaynesville. Therefore, PVT computations cover very wideranges of 100–340 F and 100–15000 psi, respectively. It isobserved that with appropriate mixing rules for critical prop-erties (Sutton et al., 1985), Peng-Robinson computes densitybetter than other models, such as that by Redlich and Kwong(1949), the Dranchuk et al. (1975) model, etc. Figure 1shows calculations of typical Haynesville gas mixtures (CH4

95%, C2H6 1%, CO2 4.8%, and N2 0.1%) extracted froma technical report by Bullin (2008). The results shown are

validated against the closest compositions data (CH4 86.3%,C2H6 6.8%, C3H8 2.4%, n-C4H10 0.48%, iso-C4H10 0.43,C5H12 0.22%, C6H14 0.1%, C7+ 0.04%, and CO2 3.2%)available in the API monograph by Gonzalez et al. (1970).While selecting the sample mixture, we gave more weightto CO2 concentration because PVT properties of acid gaseschange widely, especially at high temperatures and pressures.Our package shows good agreement with experimental data.Small deviations occur at higher pressures because the com-positions were not identical. Figure 2 shows validations ofviscosity data. Lee’s formulae were used in the gas viscositycalculations (Lee et al., 1966). The formula from Matterand Brar (1975) was used for calculations of the isothermalcompressibility of gas.

We have obtained production data and directional surveysfor 2,696 horizontal wells in the Haynesville from the IHSCERA database. Pay zone thickness, clay-corrected porosity,and pressure data were extracted for each individual well fromthe geologic model built by Smye et al. (in preparation) fromwell-log analysis. The RF curves were calculated for initialreservoir pressures ranging from 7,000 psi to 14,000 psi, andinterpolated in pressure for individual well pressures using theSciPy Interpolate package. We performed least-squares fit-ting with the SciPy Optimize package, rejecting fits that leadto time to interference greater than 0.64 the age of the well toassure that all fitted wells had shown signs of pressure inter-ference. We then estimate a field-wide average permeabilityby inverting the equation for the characteristic interferencetime, τ , matched from the individual wells.

In order to estimate τ for wells that have not shown signsof interference, we applied a field-wide average permeabilityof 230 nanodarcy. With fluid properties at the initial reservoirpressure, a distance between hydrofracture wings of 1/9 thewell horizontal length, and the log-derived average porositynear each well, we were able to arrive at an estimated τ foreach well. 1,515 wells had estimated times to interferencethat implied that they had already began interference. For allof these wells, production forecasts were done from the fitwith a set τ , and only M allowed to vary. Forecasts weremade for each well to determine the expected ultimate recov-ery (EUR), with production cut off at either a 25-year limit or0.3 MMCF/M, whichever came first.

4. PRESENTATION OF DATA AND RESULTS

We were able to apply the model to 2,199 wells that had atleast 18 months of production data. Of these wells, 66 hadjumps in production consistent with recompletion, and theywere removed from consideration. Of the remaining wells,we found that three-quarters, 1,515 wells, had begun expo-nential decline, due to pressure interference between adjoin-ing hydrofractures. We show the dimensionless recovery fac-tor, RF, for the average initial reservoir pressure in Figure 3.The Recovery Factor over time is compared directly to the set

of wells that have experienced pressure interference. Thesewells have been offset one month in time, and fitted for cumu-lative production from month 4 of production until the monthof July, 2013. The set had a median τ of 1.13 years, and amedian 25-year EUR of 3.26 Bcf.

In order to forecast production for wells that have not yetexhibited interference, we had to estimate either the M orτ for the fit. Because calculating well original gas in place(OGIP) from first principles is unreliable, we decided to esti-mate τ , which is only dependent upon the hydrofracture spac-ing and hydraulic diffusivity. We assumed that all wells in thefield were hydrofractured in 9 equally-spaced stages, and thatthese treatments created an effective permeability that wasrelatively constant throughout the field. Applying those as-sumptions to wells that have already exhibited interference,given each well’s unique pressure and porosity, we found anaverage field-wide permeability of 240 nanodarcy with a stan-dard deviation of 150 nanodarcy. We then calculated the ex-pected time-to-interference for each well that had not yet ex-hibited interference to fit for M. From that, we made esti-mates of 25-year EUR’s. The median EUR from this set ofwells was 3.97 Bcf, and the average time to interference was1.14 years.

To test the fits for wells that have not yet exhibited inter-ference, we looked at the correlation between the cumulativeproduction in the first 12 months for each well and its 25-yearEUR. With an incorrect estimation of time-to-interference,the populations of already-interfering and not-yet-interferingwells diverge. Figure 4 shows that the two are highly corre-lated, and that the two populations completely overlap. TheEUR values seem low when compared to industry expecta-tions, but are above those predicted by Kaiser and Yu (2011),who expect the middle 40% of wells to yield EURs from 2 to3 Bcf.

5. CONCLUSIONS

We have built the minimal model for fitting and predictingproduction from gas wells in the Haynesville Shale. Themodel uses a cuboid volume of hydraulically stimulated reser-voir with a uniform array of hydrofractures that act as adsorb-ing boundaries. We assume constant permeability of the stim-ulated rock. Adsorption can be neglected, but the adsorbingboundaries must have pressures that decrease over time to thefinal values. The timescale for pressure at the boundary to de-crease was set so as to match historical production. We tookcare to treat the gas equation of state realistically and used acustom PVT solver to calculate the required properties natu-ral gas mixtures, given the molar composition, pressures, andtemperature found in the field. The scaling curve we usedprovides excellent agreement with wells in the HaynesvilleShale.

For 1,515 wells in the Haynesville, pressure interferencehas set in, and we were able to determine interference time

τ and gas in place M. The typical time to interference wasaround 1 year. One can deduce the stimulated permeabilityfrom these wells, and we arrived at values for permeability kof roughly 240 nanodarcy. These effective permeabilities aremuch larger than the values found in core experiments (Nunn,2012). EUR was estimated to be 3.26 Bcf for the median wellin this set.

For 618 wells in the Haynesville where interference hasnot been observed, we used the field-wide permeability ob-tained from the interfering wells to estimate interference timeand then fit to determine gas in place. Typical values for timeto interference were 1-2 years, and EUR values were centeredabout 4 Bcf for this set of wells. While there is uncertaintydue to the process through which we arrived at interferencetime for these wells, the EURs were found to correlate withinitial production in the same manner as for the wells withobserved interference.

We have shown that the theoretical tools provided hereincan be used to provide production forecasts. As more databecome available, we will be able to make increasingly accu-rate forecasts, which can be used in many applications. Theresults of this model are used in a detailed economic analysis,which will be published elsewhere and follows the approachof Gulen et al. (2013). This work should assist policymakersin making rational decisions about how the future develop-ment of shale gas resources will proceed.

6. ACKNOWLEGDEMENTS

The production data and flowing pressure data were ex-tracted from the IHS Cambridge Energy Research Associatesdatabase, licensed to the Bureau of Economic Geology. Thispaper was supported by the Shell Oil Company/Universityof Texas at Austin project “Physics of Hydrocarbon Recov-ery,” with T.W.P. and M.M. as coprincipal investigators, andthe Bureau of Economic Geologys Sloan Foundation-fundedproject “The Role of Shale Gas in the U.S. Energy Transition:Recoverable Resources, Production Rates, and Implications.”The work on the PVT solver was conducted with the supportof the Reservoir Simulation Joint Industry Project, a consor-tium of operating and service companies for Petroleum andGeosystems Engineering at the University of Texas at Austin.M.M. acknowledges partial support from the National Sci-ence Foundation Condensed Matter and Materials Theoryprogram.

7. NOMENCLATURE

c = Isothermal compressibility, 1/psi [1/Pa]d = hydrofracture spacing, ft [m]k = permeability, mdH = height of hydrofractures, ft [m]L = length of hydrofractures, ft [m]m = real gas pseudopressure,

M = mass of gas in place, lb [kg]N = number of fracturesp = pressure, psia [kPa]RF = recovery factor functionS = saturationt = time, s [s]u = superficial flow rate ft/s [m/s]x = distance, ft [m]α = hydraulic diffusivity, ft2 /s [m2/s]φ = porosityρ = density, lb/ft3 [kg/m3]µ = viscosity, cp [MPa.s]τ = time to interference, d [s]

7.1. Subscripts

g = gasi = initial conditionss = scaling for throttling

7.2. Diacritics

∼= scaled version of quantity

8. REFERENCES

ReferencesR Al-Hussainy, HJJ Ramey, and Crawford PB. The flow of

real gases through porous media. AIME Petrol Transac-tions, 237:624–636, 1966.

John Browning, Scott W Tinker, Svetlana Ikonnikova, GurcanGulen, Eric Potter, Qilong Fu, Susan Hovarth, Tad WPatzek, Frank Male, William Fisher, Forrest Roberts, andKen Medlock III. BARNETT SHALE MODEL-1: Bar-nett study determines full-field reserves, production fore-cast. Oil and Gas Journal, 111:62, 2013.

John Browning, Scott W Tinker, Svetlana Ikonnikova, GurcanGulen, Eric Potter, Qilong Fu, Susan Hovarth, Tad WPatzek, Frank Male, William Fisher, and Forrest Roberts.Study develops Fayetteville shale reserves, productionforecast. Oil and Gas Journal, 112, 2014.

K Bullin. Composition variety complicates processing plansfor us shale gas. In Annual Forum, Gas Processing Associ-ation, Houston, TX, October 2008.

PM Dranchuk, H Abou-Kassem, et al. Calculation of z fac-tors for natural gases using equations of state. Journal ofCanadian Petroleum Technology, 14(03), 1975.

Mario Hugo Gonzalez, Bertram Eugene Eakin, and An-thony L Lee. Viscosity of natural gases: monograph onAPI Research Project 65. American Petroleum Institute,1970.

Gurcan Gulen, John Browning, Svetlana Ikonnikova, andScott W Tinker. Well economics across ten tiers in low andhigh Btu (british thermal unit) areas, Barnett Shale, Texas.Energy, 60:302–315, 2013.

Ursula Hammes, H Scott Hamlin, and Thomas E Ewing. Ge-ologic analysis of the Upper Jurassic Haynesville Shale ineast Texas and west Louisiana. AAPG bulletin, 95(10):1643–1666, 2011.

Mark J Kaiser and Yunke Yu. Haynesville shale well perfor-mance and development potential. Natural resources re-search, 20(4):217–229, 2011.

Anthony L Lee, Mario H Gonzalez, Bertram E Eakin, et al.The viscosity of natural gases. Journal of Petroleum Tech-nology, 18(8):997–1000, 1966.

L Matter and GS Brar. Compressibility of natural gases. JCan Petrol Technol., pages 77–80, August 1975.

Jeffrey A Nunn. Burial and thermal history of the HaynesvilleShale: Implications for overpressure, gas generation, andnatural hydrofracture. GCAGCS Journal, 1:81–96, 2012.

Mark A Parker, Dan Buller, James Erik Petre, Douglas TroyDreher, et al. Haynesville shale-petrophysical evaluation.In SPE Rocky Mountain Petroleum Technology Conference.Society of Petroleum Engineers, 2009.

Tad W Patzek, Frank Male, and Michael Marder. Gas pro-duction in the Barnett Shale obeys a simple scaling theory.Proceedings of the National Academy of Sciences, 110(49):19731–19736, 2013.

Ding-Yu Peng and Donald B Robinson. A new two-constantequation of state. Industrial & Engineering Chemistry Fun-damentals, 15(1):59–64, 1976.

Otto Redlich and JNS Kwong. On the thermodynamics ofsolutions. v. an equation of state. fugacities of gaseous so-lutions. Chemical Reviews, 44(1):233–244, 1949.

Katie M Smye, Eric Potter, Susan Hovarth, Forrest Roberts,and Scott W Tinker. Geological analysis of the HaynesvilleShale for ‘bottom up’ production forecasting and resourceestimation. in preparation.

RP Sutton et al. Compressibility factors for high-molecular-weight reservoir gases. In SPE Annual Technical Con-ference and Exhibition. Society of Petroleum Engineers,1985.

Fig. 1. Validation of density calculations (lit-literature fromGonzalez et al., (1970), cal - calculated.)

9. APPENDIX

Here we show calculations of density, viscosity, and isother-mal compressibility.

Density Density, ρ, in g/cc is calculated from original Peng-Robinson equation of state using following correlations ofcritical properties (Sutton et al., 1985).

Pc = 756.8− 131× SG− 3.6× SG2 (7)

Tc = 169.2 + 349.5× SG− 74× SG2 (8)

Here, Pc and Tc are in psi and ◦R, respectively. SG representsspecific gravity, and is given by

SG =1

28.964

n∑i=1

yi ×MWi (9)

where MWi is the molecular weight of component i, withmolar fraction yi. Tr and Pr are the reduced temperature andpressure, and are defined by the relations Tr = T/Tc andPr = P/Pc.

Viscosity Viscosity µ in cp (centipoise), is computed byLee’s formula (1966), which is presented as follows:

µ = 10−4K exp(XρY

)(10)

where

X = A1 +A2

T+A3

n∑i=1

yiMWi (11)

Fig. 2. Validation of viscosity calculations (lit-literature fromGonzalez et al., (1970), cal - calculated.)

K =

(A4 +A5

n∑i=1

yiMWi

)×

T 1.5

A6 +A7

∑ni=1 yiMWi + T

(12)

Y = A8 −A9X (13)

The coefficients are given by

A1 = 3.448

A2 = 986.4

A3 = 0.01009

A4 = 9.379

A5 = 0.01607

A6 = 209.2

A7 = 19.26

A8 = 2.447

A9 = 0.2224

Isothermal Compressibility First compressibility factor,Z, is determined from the Peng-Robinson Equation of State.The reduced compressibility cr is calculated from Equation(14) where cr = cPc. Here c is in psi−1.

cr =1

Pr−

0.27 ∂Z∂ρr

ZTr

(1 + ρr

∂Z∂ρr

/Z) , (14)

Where

∂Z

∂ρr= B1 +B2ρr + 5B3

ρ4rTr

+ 2ρr exp(−A8ρ2r)B4

T 3r

. (15)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Square root scaled time√t

0.0

0.2

0.4

0.6

0.8

1.0

Rec

over

yfa

ctor

RF

(t)

Simulationindividual well RFs

Fig. 3. Comparison of 1,515 wells with scaling function. Thewell histories are scaled so as it fit our scaling function, whichassumes an initial reservoir pressure of 10,200 psi and a wellflowing pressure of 1,000 psi. Each well’s dimensionless timereaches .64 or more.

The reduced density is given by ρr = ρ/ρc and ρc =0.27Pr/ZTr. The constants B come from the relations.

B1 = A1 +A2

Tr+A3

T 3r

(16)

B2 = A4 +A5

Tr(17)

B3 = A5 ×A6 (18)

B4 = A7

(1 +A8ρ

2r −A2

8ρ4r

)(19)

The necessary coefficients of Equations (16-19) are

A1 = 0.31506237

A2 = −1.0467099A3 = −0.57832729A4 = 0.53530771

A5 = −0.61232032A6 = −0.10488813A7 = 0.68157001

A8 = 0.68446549.

0 1 2 3 4 5

12 Month Cumulative (BCF)

0

2

4

6

8

10

12

25ye

arE

UR

(BC

F)

not interferinginterfering

Fig. 4. Direct comparison of initial year of production to es-timated ultimate recovery for all 2,133 wells with at least 18months of production history and have not been subjected toan apparent recompletion. The average well is predicted toproduce 3.47 Bcf over its first 25 years, and several wells areexpected to produce more than 10 Bcf.

0 1 2 3 4 5

Time to interference (years)

0

2

4

6

8

10

12

25ye

arE

UR

(BC

F)

not interferinginterfering

Fig. 5. Values for EUR and interference time τ for the 2,133wells in Figure 4. The median well is expected to have aninterference time of 1.14 years.