Journal of Petroleum Science and Engineering 162 (2018) 617–632

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Journal of Petroleum Science and Engineering

journal homepage: www.elsevier.com/locate /petrol

Benchmarking EUR estimates for hydraulically fractured wells with andwithout fracture hits using various DCA methods

Yuqi Hu, Ruud Weijermars *, Lihua Zuo, Wei Yu

Harold Vance Department of Petroleum Engineering, Texas A&M University, 3116 TAMU College Station, TX 77843-3116, USA

A R T I C L E I N F O

Keywords:Production forecastingEUR estimatesModified Hyperbolic Decline ModelDuong modelLogistic Growth Analysis ModelPower Law Exponential

* Corresponding author.E-mail address: r.weijermars@tam.edu (R. Weijermars

https://doi.org/10.1016/j.petrol.2017.10.079Received 28 June 2017; Received in revised form 13 SepAvailable online 31 October 20170920-4105/© 2017 Elsevier B.V. All rights reserved.

A B S T R A C T

Various decline curve analysis (DCA) methods can be applied to forecast the production performance of hydro-carbon wells, including horizontal wells stimulated with hydraulic fracturing. Yet, which method is more pref-erable remains in doubt. The objective of this study is to evaluate various DCA methods by history matching andhindcasting both synthetic and field production data in order to assess for each method the reliability of pro-duction forecasts and the estimated ultimate recovery (EUR). Five DCA methods have been evaluated because oftheir computational simplicity and broad application. These methods are the Modified Hyperbolic Decline model(MHD), Duong model, Logistic Growth Analysis Model (LGM), and the Power Law Exponential (PLE, with D∞ ¼ 0and D∞ ¼ ∞) Decline models. Each method was evaluated using three data sets: synthetic shale oil productionrates based on a CMG reservoir model using Eagle Ford reservoir characteristics, data from producing Eagle Fordwells and from Austin Chalk wells. The hindcast method was applied to the Eagle Ford synthetic wells to establishthe EUR deviation between the synthetic reservoir model production forecast and the various DCA methods. Theweighted residuals of history matched production rates for Eagle Ford indicate that the MHD and Duong modelsgive the lowest matching errors (as low as 1.32%) and the highest EUR estimations (as high as 3,447,609 stb). ThePLE model (with D∞≠0) gives the most conservative EUR estimation, and also the least reliable history matchingmethod (on our Eagle Ford data sets), which therefore is the least preferable DCA method. For the Austin Chalkwells in our study, all methods generate fairly similar EUR predictions with similar matching errors (from 13% to15%) so that the DCA method preference becomes indifferent. This study also touches upon the relationshipbetween DCA results when well interference occurs due to various hydraulic fracture hits. Shale wells that pri-marily contain hydraulic fractures and few natural fractures (Eagle Ford) give the most reliable forecasts using theMHD and Duong DCA methods. However, wells in the Austin Chalk, a naturally fractured reservoir, show littledifference between the forecast accuracy from the various DCA methods.

1. Introduction

Many historically gas-focused companies have shifted their emphasisto oil production, when US natural gas prices started to decline afterpeaking a decade ago. Subsequently, Texas, possessing numerous reser-voirs with ultra-low matrix permeability (e.g., Eagle Ford formation,Wolfcamp formation, and Austin Chalk), has experienced an unprece-dented revival of its oil production output (Martin et al., 2011), princi-pally due to broad application of hydraulic fracturing and horizontaldrilling. After hydraulic fracturing, oil production of individual wellswith only limited production data can be forecasted with various DCAmethods. However, it is unclear which DCA method suits best for thesereservoirs. Hence, determining the appropriate DCA method to evaluate

).

tember 2017; Accepted 27 October 2

the estimated ultimate recovery (EUR) remains shrouded in uncertaintybut is vital for accurate reserves estimations to quantify the assetcollateral value essential for financing the future growth of shale com-panies (Cirilo Agostinho andWeijermars, 2017; Weijermars et al., 2017a,b). To help operators in deciding which DCA method to use, the presentstudy analyzes and compares both older and recently developed DCAmethods and uses these methods to benchmark historical and simulatedproduction data from a couple of unconventional fields. In brief, thispaper uniquely analyzes and compares several recent decline curvemethods (such as Duong Model, Logistic Growth Model, and Power LawExponential Decline Methods), including historical ones (TraditionalArps’DeclineModel, andModified Hyperbolic DeclineModel) in order tofind the most appropriate DCA method for EUR forecasting of the well

017

Y. Hu et al. Journal of Petroleum Science and Engineering 162 (2018) 617–632

productivity from specific formations with different types of natural andhydraulic fractures, using synthetic data from a calibrated reservoirmodel as a benchmark bottom line. Such an integrated approach has notbeen published before. Our results are timely and relevant given therapidly evolving industry focus on unconventional resources and theimminent need to provide reliable EUR estimations as a basis for thevalidated determination of economically viable reserves classes andcategories.

The traditional Arps' decline model (1945) has been successfullyused for production forecast in conventional reservoirs, but if appliedunmodified becomes unreliable for unconventional resources. Rob-ertson (1988) therefore modified the traditional Arps' decline model byimposing a predetermined minimum decline rate and thus proposed theModified Hyperbolic Decline (MHD) model. Separately, Ilk et al. (2008)developed the Power Law Exponential (PLE) decline model based onthe Arps’ decline curves and used the power law decline to approximatethe production rate decline, specifically for shale gas wells. Valko(2009) presented the Stretched Exponential Decline Model (SEDM),which attempts to avoid the arbitrariness of the MHD model whenestimating long-term reserves (Mishra, 2012). Clark et al. (2011) pro-posed the Logistic Growth Analysis Model (LGM) as an empiricalmethod to forecast oil and gas reservoirs with extremely low perme-ability. Likewise, Duong (2011) put forward a DCA model for shallowtight gas reservoirs and shale reservoirs, particularly applied to frac-tured shale gas reservoirs. More recently, Zuo et al. (2016) developedthe Fractional Decline Curve (FDC) model based on the anomalousdiffusion phenomena for shale gas reservoirs. Yu et al. (2016) proposeda probabilistic method to forecast the EUR based on this FDC model.

Seshadri and Mattar (2010) concluded that for tight gas wells, theMHD model is more preferable than the PLE model, because the MHDmodel is well understood and accepted while the PLE model is complexand non-intuitive. Meyet Me Ndong et al. (2013) showed that the MHDmodel generated the highest EUR estimations, and concluded that theLGM model was not reliable when using short-term production data.Yet, the PLE model and the Duong model (with zero production atinfinite times) showed the least variation when history matching pro-duction data for different time spans and across various plays, welltypes, and fluid types. Paryani et al. (2016) illustrated that for EagleFord shale oil production, Arps’ hyperbolic model provided overlyoptimistic reserves estimations, while the PLE model consistentlyyielded the most conservative production forecast. The Duong modelmatched reasonably well with longer and less noisy production history,but still needs improvement (Paryani et al., 2016). According to thelatter authors, the LGM model was the most reliable method and gavereasonable EUR estimates.

In our study, the MHD, Duong, LGM, and two PLE models have beenselected to analyze three sets of well production data. The five DCAmethods are relatively easy to apply and have been widely accepted bythe industry for unconventional oil and gas production forecasting andreserves estimations. The data sets investigated are historic productionrates from: (1) a synthetic reservoir model (CMG) with various fracturehits using typical Eagle Ford reservoir parameters, (2) four Eagle Fordoil wells in Brazos County, Texas, and (3) six Austin Chalk wells also inBrazos County, Texas. All three data sets contain horizontal wells witheither multiple hydraulic fractures (Eagle Ford) or natural fractures(Austin Chalk).

This study is organized as follows. First, the methodology of the fiveevaluated DCA models is briefly described. Then, history matching isapplied to generate production forecasts for each data set using theselected DCA models. Next, the EUR forecasts for each well series arecompared and subjected to an error analysis. In addition, the results arecombined with the hydraulic fracturing parameters to gain more insightin how hydraulic fracturing impacts unconventional oil production.

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2. Model description

This section briefly explains the five DCA methods evaluated in ourstudy: MHD, Duong, LGM, and the two PLE methods. Advantages andlimitations of each model are also highlighted.

2.1. Traditional Arps’ decline model

Arps (1945) proposed the simplest empirical DCA method, which hasbeen widely used for conventional reservoirs as well as unconventionalresources. The method is based on the observation that “first differencesof the loss ratios are approximately constant”. The general hyperbolicform of Arps’ decline is:

qðtÞ ¼ qið1þ bDitÞ�1=b; (1)

where t is time, q is production rate at time t, qi is initial production rate,b is decline exponent, and Di is initial nominal decline rate. When b ¼ 1,Eq. (1) becomes harmonic:

qðtÞ ¼ qið1þ DitÞ�1; (2)

When b ¼ 0, Eq. (1) exhibits exponential decline:

qðtÞ ¼ qie�Dit; (3)

The Arps model assumes that the bottomhole pressure, the drainagearea, along with the skin factor are constant and the flow regime isboundary dominated flow (Arps, 1945). This method is broadly used inconventional reservoirs with medium to high permeability and a rela-tively short transient flow phase followed by boundary dominated flow(Paryani et al., 2016). However, this method is not recommended forunconventional resources because unconventional resources have verylow permeability and boundary dominated flow phase is rarely reached,unless adjacent wells impose stagnation flow boundaries (Weijermarset al., 2017c, d).

2.2. Modified Hyperbolic Decline Model

Robertson (1988) proposed the MHD model, where hyperbolicdecline in the early life of a well is shifted to exponential decline in thelate life by imposing a predetermined minimum decline rate, Dmin. Thisminimum decline rate occurs at time t* in the production forecast and isdetermined purely empirical, and usually amounts to about 5% per year.Nevertheless, this method is commonly used. The MHD model uses:

qðtÞ ¼�qiðqþ bDitÞ�1=b;

�t< t*

�qie�Dit;

�t � t*

� �; (4)

In our study, the Solver tool build in Excel was used to compute qi, b andDi simultaneously. An initial guess was made first and, then the variableswere changed through multiple iterations in Solver to meet the objectivefunction of minimized residual values between the curve fitting andhistory matched production rates. Constraints of each variable were alsoinputted as follows. After obtaining reasonable regression results forhistorical production rate and cumulative production, a minimumdecline rate, Dmin is imposed to generate forecasts up to 360 months (30years). The DCA forecast allows calculation of the EUR.

2.3. Duong Model

Duong (2011) proposed this method for shallow tight gas reservoirsand fractured shale gas reservoirs. The production rate is computed in theDuong Model using:

Y. Hu et al. Journal of Petroleum Science and Engineering 162 (2018) 617–632

qðtÞ ¼ q1tða;mÞ þ q∞ (5)

where:

tða;mÞ ¼ t�mea

1�mðt1�m�1Þ; (6)

and the cumulative production is:

GpðtÞ ¼ qatm; (7)

The first step is to check the data andmake necessary corrections (such ascorrecting wet-gas to dry-gas for high condensate/gas ratio). The secondstep is to determine the parameters a and m using a log-log plot of q=Gpvs. time. In the third step, q1 can be determined using a semi-log plot of qvs. tða; mÞ. Finally, the production forecast can be calculated using Eq.(5), and the EUR can be determined using Eq. (7) for production atminimum economic rate (Duong, 2011). The four steps are illustrated inFig. 1, using data of Eagle Ford synthetic Case 4 (see later).

The typical ranges for the DCA parameters in the Duong model are1 � m � 2 and 0< a � 2 (Paryani et al., 2016). A limitation of the Duongmodel is that proper initialization per pressure is needed to get accuratevalues of a andmwhen the well is shut in for a longer time (Paryani et al.,2016). In addition, values of a and m will increase if there is a suddendecrease in decline rate due to water breakthrough (Paryani et al., 2016).The parameter q∞ normally is zero, but may attain non-zero values incertain well operating situations. Joshi and Lee (2013) suggest to forceq∞ to be 0 in order to minimize errors for estimating production

Fig. 1. Four steps of comput

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forecasts. In our study, q∞ was forced to be 0 to minimize errors.

2.4. Logistic Growth Model

Clark et al. (2011) developed this model as an empirical method toforecast oil and gas reservoirs with extremely low permeability. Thecumulative production in this model reaches a maximum carrying ca-pacity, when there will be no further growth. The cumulative productionfor forecast is calculated as follows:

QðtÞ ¼ Ktnba þ tn; (8)

where Q is cumulative production, K is carrying capacity, ba is a constant,n is hyperbolic exponent and t is time. K can be determined by volumetricmethods or curve fitting methods. n controls the curvature of the declinecurve with possible values between 0 and 1. However, an enigmaticupward inflection in the curve will occur if n>1 (Meyet Me Ndong et al.,2013). The constant a is similar to the initial decline parameter in theArps model and occurs when half of the carrying capacity is reached.

The production rate of this model can be obtained by differentiatingEq. (8), which is shown below:

qðtÞ ¼ dQdt

¼ Knbatn�1

ðba þ tnÞ2; (9)

The relevant parameters (K, n, and ba) can be obtained from either opti-mizing the parameters with a numerical scheme or linearizing the

ation for Duong model.

Table 1Basic reservoir and fracture properties used for the simulations (SPE-184825).

Parameter Value Unit

Initial reservoir pressure 8000 psiReservoir temperature 270

�F

Reservoir permeability 470 nDReservoir porosity 12% –

Initial water saturation 17% –

Total compressibility 3 � 10�6 psi�1

Fracture half-length 225 ftFracture conductivity 100 md-ftFracture height 100 ftFracture width 0.01 ftFracture spacing 200 ftWell spacing 700 ft

Y. Hu et al. Journal of Petroleum Science and Engineering 162 (2018) 617–632

equations and plotting the data. In our study, the former strategy isadopted. K, n, and ba were found simultaneously by using Solver in Excelwith an initial guess of these three parameters.

The main assumption in this model is that the entire reservoir can bedrained by a single well over a long period (Paryani et al., 2016). Themajor advantage is that the reserves estimate is constrained by theparameter K as well as the production rate, which terminates at infinitetime (Clark et al., 2011).

2.5. Power Law Exponential Decline Methods (D∞≠0 and D∞ ¼ 0)

Ilk et al. (2008) developed the Power Law Exponential (PLE) DeclineModel based on the Arps’ decline curves and used the power law declineto approximate the production rate decline. This model is developedspecifically for shale gas wells, but is also applied to unconventional oilwells. The production rate is shown below:

qðtÞ ¼ bqie½�D∞ t�bDitbn �; (10)

where bqi is the rate “intercept”, bDi is the initial decline constant, D∞ is thedecline constant at infinite time and bn is the time exponent. In addition,

the parameters D and bDi are defined as follows:

D ¼ D∞ þ D1t�ð1�bnÞ; (11)

bDi ¼ D1bn ; (12)

The above DCA variables account for both transient and boundarydominated flow. However, four unknowns, bqi, bDi, D∞ and bn in this modelcause too many degrees of freedom and sometimes can be cumbersome touse. Several methods were proposed to solve the model parameters andmultiple solutions are acceptable. In our study, the Solver tool built inExcel is used to solve for the parameters simultaneously. Then, the fittingcurve is extrapolated to 30 years to calculate the EUR.

The situation for D∞ ¼ 0 was also applied to all data sets consideredin this paper. For more optimistic estimations Eq. (11) reduces to thefollowing form:

qðtÞ ¼ bqie½�bDitn'�; (13)

However, Eq. (13) only holds true during transient flow (Mattar andMoghadam, 2009). Therefore, attention is needed to select appropriatedata when using Eq. (8).

3. History matching production rate and EUR

Two measures were used to quantify how the computed DCA valuesdeviate from the historic production rate of a well (Section 3.1) and howthe consequent EUR estimations differ from the prototype well (Sec-tion 3.2).

3.1. Weighted residual of production rate

The weighted residual of history matched production rates in ourpaper summarizes the fraction made up by the absolute difference be-tween the actual production rate and the computed production rate in thenominator, and in the denominator occurs the actual production rate ateach time step. The residual takes the arithmetic average of all valuesover the total history matched time period to observe how much eachcomputed DCA value deviates from the actual production rate form thephysical well. Our method accounts for the relative magnitude so that thelarger production rates in the early life will equally impact the error termas the smaller ones in the later stages of production. The formula used tocompute the weighted residual (WR) is:

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WR ¼ 1 XN abs�qactual;i � qcalculated;i

�� 100%; (14)

N i¼1 qactual;i

3.2. EUR deviation

The hindcast method was used to compute the EUR deviation. Thehistory matched EUR estimations using the DCA methods can becompared to the simulated 30-year production data and EUR for theEagle Ford synthetic data. The EUR deviation is calculated by taking theabsolute difference between the simulated EUR and the computed EURdivided by the simulated EUR. The ratio shows howmuch deviation thereis between the simulated EUR and the history match based EUR forecastin order to determine the reliability of each of the five DCA methods. Theequation used is:

DevEUR ¼ absðEURactual � EURcalculatedÞEURactual

� 100%; (15)

4. Production data analyzed

Three sets of oil production data are used to evaluate the five DCAmethods. The data sets are: (1) shale oil production rates generated witha CMG reservoir model based on Eagle Ford play characteristics, (2)actual Eagle Ford shale oil production rates from four wells in BrazosCounty, Texas, and (3) Austin Chalk oil production data, from 6 wells in anaturally fractured reservoir in Brazos County, Texas. For the three datasets, the regression and forecast periods cover a total time spam of 30years. For the Eagle Ford synthetic oil production data, production isextracted from 0 to 4994 days (which is slightly less than half of the totalproduction life for the hindcast history matching of the 30 years ofsimulated production from CMG). Importantly, Dmin of the MHDmodel isassumed to be 5% per year which is an industry convention (Joshi andLee, 2013; Meyet Me Ndong et al., 2013). Outliers of the data sets (fielddata only) are eliminated before history matching. Outliers in the fielddata (rates that are either zeros or too far from the general trend) wereexcluded from the history matching because they would amplifymatching errors unjustifiably. The rejection of outliers is based on aformal procedure which first calculates the average of five adjacentproduction rates. Next the average is multiplied by a ±20% error-tolerance rate. Historic production rates that fall within the tolerancewindow were included in the forecast, and any outliers were rejected.

4.1. Eagle Ford synthetic cases

4.1.1. Data descriptionThe Eagle Ford synthetic data was built based on publicly available

Eagle Ford data used in a previous study (Yu et al., 2017). Table 1 lists thebasic reservoir and fracture properties used in the CMG simulation. Eachhorizontal well has a fracture spacing of 200 ft and a total of 30 fractures,with an inter-well spacing of 700 ft, and horizontal well length of 5800 ft.

Fig. 2. a-d─Map views showing well architecture for four CMG cases with different intensity of well interference. Comparison of the production rates (e) and cumulative production (f) forproducing well only (data based on SPE-184825).

Y. Hu et al. Journal of Petroleum Science and Engineering 162 (2018) 617–632

Fig. 2a–d shows map views of the four cases of hydraulically fracturedwells with different well interference intensity. Case 1 shows a singlehorizontal well without well interference. Case 2 considers 2 wells withany well interference only possible through matrix permeability withoutany connecting hydraulic fractures. Cases 3 and 4 both have well inter-ference through both matrix permeability and hydraulic fracture hits.The two wells in Case 3 are interconnected through five hydraulic frac-ture hits, and the two wells in Case 4 are interconnected via 15 hydraulicfracture hits. The fracture conductivity of the fracture hits is 10 md-ft,while the regular hydraulic fractures have higher conductivities of 100md-ft. The production is simulated for 30 years. The upper horizontalwell produces under a constant bottom-hole pressure of 1000 psi and thelower horizontal well remains shut-in at all times for Cases 2 to 4 (Yuet al., 2017).

Fig. 2e plots the oil production rate for each case (in bbl/day) versustime (in days). Fig. 2f represents the cumulative production (in Mbbl)versus time (in days). Case 1 and Case 2 have very close productionforecasts. Case 3 has production rates and cumulative production higher

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than for Cases 1 and 2, while Case 4 has highest cumulative production.Using the synthetic data set is beneficial to understand the influence ofwell interference and fracture hits on the reliability of DCA forecasts. Forexample, we found that the accuracy of the history matched data isindifferrent to the presence of any fracture hits. Additionally, certainDCA methods are consitently more accurate than others (see below).

4.1.2. DCA fitting parameters and production forecastIn order to examine the inter-relationship of the fitting parameters,

the ranges of fitting parameters for the five DCA methods are summa-rized in Table 2. These parameters are the exponential terms in thecorresponding models, which most significantly impact the shape and fitof the history matched decline curves. Table 2a–e shows the fitting pa-rameters for the MHD, Duong, LGM, and PLE (D∞≠0 and D∞ ¼ 0)models, respectively. The forecasts are based on stock tank barrel of oilper day.

Based on the history matching with the MHD model, the b parameterranges between 1.45 and 1.70 (Table 2a). For the Duong model, the a

Table 2Fitting parameters of five DCA methods using Eagle Ford synthetic reservoir data.

(a) Fitting parameters for MHD model

Cases qi Di b Dmin

– Stb/day /year (Nominalrate)

– /year (Nominalrate)

Case1

3508 9.223 1.495 0.05

Case2

3806 11.010 1.518 0.05

Case3

4309 8.907 1.690 0.05

Case4

5180 6.869 1.594 0.05

(b) Fitting parameters for Duong model

Cases a m q1 q∞– (day�1) – (Stb/

day)(Stb/day)

Case1

1.0420 1.1114 4637.7 0

Case2

1.0242 1.1086 4763.04 0

Case3

1.1257 1.1084 4361.91 0

Case4

1.3366 1.1374 4124.78 0

(c) Fitting parameters for LGM model

Cases K n ba– Stb – dayn

Case1

2,651,315 0.6569 257.65

Case2

2,656,987 0.6569 258.31

Case3

4,312,545 0.6952 427.85

Case4

5,061,751 0.7045 400.46

(d) Fitting parameters for PLE model with D∞≠0

Cases bqi bDi D1 bn D∞

– (Stb/day) (day�1) (day�1) – (day�1)

Case1

3038.49 0.196 0.065 0.330 2.35E-04

Case2

3141.06 0.199 0.066 0.331 2.35E-04

Case3

3363.40 0.198 0.060 0.305 1.74E-04

Case4

5212.18 0.198 0.065 0.330 1.00E-04

(e) Fitting parameters for PLE model with D∞ ¼ 0

Cases bqi bDi D1 bn D∞

– (Stb/day) (day�1) (day�1) – (day�1)

Case1

10,756.87 0.839 0.173 0.206 0

Case2

10,937.58 0.852 0.174 0.204 0

Case3

11,302.70 0.792 0.157 0.198 0

Case4

12,334.46 0.654 0.143 0.218 0

Y. Hu et al. Journal of Petroleum Science and Engineering 162 (2018) 617–632

parameter ranges between 1.00 day�1 and 1.35 day�1 and the mparameter is within the range of 1.10–1.15 (Table 2b). For LGM model,the n parameter is within the range of 0.65–0.75 (Table 2c). For the PLEmodel with D∞≠0, the bn parameter lies between 0.30 and 0.35, the bDi

parameter is between 0.195 and 0.200, and the D∞ parameter is between1.00E-04 and 2.35E-4 (Table 2d). For the PLE model with D∞ ¼ 0, the bnparameter lies between 0.15 and 0.25, and the bDi parameter between

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0.65 and 0.90 (Table 2e). In addition, Table 2d and e indicate that forcingthe D∞ term to be zero changes the other fitting parameters significantlyfor the PLE model. The fitting parameters of Table 2 were used tocompare the decline curve fits when history matching the historic pro-duction rates of the synthetic reservoir model.

Fig. 3 plots for Cases 1 and 2 the historic oil production rates andcumulative production as well as the history matched curves using thefive DCA methods. Fig. 4 plots the corresponding forecasts for Cases 3and 4, with increasing levels of well interference. The oil production ratefor all cases decreases rapidly at early times and declines much slower inthe later stages of the well life. This trend is also reflected in the cumu-lative production graphs. Comparing the production plots for the Cases1–4, we conclude that Cases 3 and 4 have higher cumulative productionthan Cases 1 and 2, which reflects an increasing level of communicationwith the shut-in well.

The given production rate and the given cumulative productionsimulated in CMG for 30 years, are also plotted Figs. 3 and 4 for hindcastcomparison. In spite of the different degrees of inter-well communicationfor Cases 1–3, the best fit to the reservoir model for all cases is the Duongmodel, which also gives the most optimistic EUR forecast. Notably, thesimpler MHD model gives only slightly lower production rates andcorrespondingly slightly lower cumulative production. However, theLGM and PLE (D∞ ¼ 0) models both generate EUR forecasts that aresignificantly lower than the EUR forecasts of both the Duong and MHDmodels. The worst fit is provided by the PLE model with D∞≠0, whichyields the most conservative results for daily production rates and lowestcumulative production. The D∞ term imposes an extra decline effect onthe exponential term, which contributes sharply to the decline in the PLEmodel with D∞≠0.

4.1.3. History matching resultsIn order to quantify the accuracy and reliability of the forecast by the

various DCA methods (Section 3.1 and 3.2), the forecasted EUR werecompared to the EUR obtained from the actual CMG production simu-lation for the full 30-year reservoir life, as summarized in Table 3. TheEUR deviations and weighted residuals of production rates are depictedin Fig. 5 for each DCA method.

In Table 3, the highest EUR and the lowest EUR estimates are high-lighted in yellow and green, respectively. The MHD model has thehighest EUR forecast for Case 4 and the second highest EUR values forCases 1–3. The Duong model has the highest EUR forecasts for Cases 1–3and the second highest EUR value for Case 4. The PLE model with(D∞≠0) predicts the lowest EUR for all cases. The LGM and PLE (D∞ ¼ 0)models both give the medium EUR estimates for Cases 1–4 (Table 3).

Compared with the CMG numerical solution, Fig. 5a illustrates thatthe MHDmodel gives the lowest EUR deviation for Case 1 and the secondlowest deviations for Cases 2–4. The Duong model gives the least EURdeviations for Cases 2–4 and the second lowest for Case 1. The PLE model(with D∞≠0) yields the greatest EUR deviations and thus is the leastaccurate for all well scenarios (Cases 1–4, Fig. 5a).

We also computed the matching error of production rate at each timestep (Fig. 5b), which shows that the MHD model gives the lowestweighted residuals for production rates for Cases 1–3 and the secondlowest value for Case 4. The Duong gives the lowest production ratematching error (residual) for Case 4 and the second lowest values forCases 1–3. The PLE model with (D∞≠0) yields the greatest weightedresidual of production rates for all cases and thus is the least accurate.The LGM and PLE (D∞ ¼ 0) models both provided medium error valuesfor both the EUR deviation (Fig. 5a) and the weighted residual of pro-duction rate (Fig. 5b).

Based on the above analysis we conclude that the simple MHDmodel,while least cumbersome to use, provides the best and most accurateproduction rate forecasts for Cases 1–3 and the second most accurate forCase 4, while being nearly as accurate as the Duong method in EURforecasting for all cases, the latter method being slightly more convoluteto use (Fig. 1) than the MHD method as a DCA matching technique. We

Fig. 3. Map view of synthetic cases 1 and 2 demonstrating the different intensity of well interference (a)–(b) and related production forecast (c)–(f).

Y. Hu et al. Journal of Petroleum Science and Engineering 162 (2018) 617–632

also conclude that the degree of interwell communication does notworsen the matching result for any of the DCA methods used, but theMHD and Duong models give the least mismatch for both EUR andproduction rate forecasts (Fig. 5a and b).

4.1.4. Estimation of EUR per fractureSince the MHD model gives reasonably accurate history matching

results, the EUR from the MHD model is used to separately analyze theeffect of lateral length and fracture number (Table 4). The EUR pereffective lateral length and EUR per fracture increases from Case 1 to 4,because of the increasing interwell communication via the matrix (Case2) and the fracture hits (Cases 3 and 4). Table 4 confirms that fracture hits(Cases 3 and 4) have a much greater impact on the EUR produced fromeach fracture than has matrix permeability (Case 2). The presence of asecond well communicating with the first well via the matrix (Case 2) andvia an increasing number of fracture hits (Cases 3 and 4) generatesincreasingly larger EUR, which is adequately captured by the MHD

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model (Table 4).

4.2. Eagle Ford field cases

4.2.1. Data descriptionWe next test the MHD, Duong, LGM, and two PLE models to accu-

rately match historic oil production data from four 4 Eagle Ford wells.The field data were privately obtained, and the wells (Well R, Well O,Well H and Well M) were completed at the Brazos County in 2014. Theeffective lateral length, fracture spacing, fracture number and totalproppant injected are obtained for each well from the companycompletion reports and listed in Table 5. The actual production of thewells with the know fracture attributes and well details can be used tocompare the history matching accuracy of the five DCA methods. Thewell trajectories of Well R and Well O are sub-parallel to each other, withspacing varying between 750 and 1250 ft; the trajectories of Well H andWell M wells are nearly perfectly parallel with 1250 ft horizontal

Fig. 4. Map view of synthetic cases 3 and 4 demonstrating the different intensity of well interference (a)–(b) and related production forecast (c)–(f).

Table 3Forecasted EUR using five DCA methods for the Eagle Ford synthetic data.

Y. Hu et al. Journal of Petroleum Science and Engineering 162 (2018) 617–632

distance between the wells. The schematics of the four wells are shown inFig. 6. Well O and Well R are denoted as Well Pair 1, while Well H andWell M are denoted as Well Pair 2. Each red line in Fig. 6 stands for acompletion stage, and each completion stage has several hydraulicfractures. Before production, each well was hydraulically fractured and alarge amount of proppant was injected (Table 5).

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Comparing well designs of Well Pair 1 (Well R and Well O), we notethat the effective lateral length of Well O is only about 1/3rd that of WellR, due to completion problems. Consequently, Well R has nearly 3 timesmore completion stages, fractures, and total proppant injected than WellO (Table 5). Because the fracture spacing of the two wells is about thesame, one would expect Well O to have a EUR of about 1/3rd that of Well

Fig. 5. EUR deviations and weighted residuals for history matched EUR (a) and production rates (b) relative to the numerical solution (CMG) for the Eagle Ford synthetic reservoir data.

Table 4Forecasted EUR using the MHD model over effective lateral length for the Eagle Ford synthetic CMG reservoir model (Cases 1–4).

Cases EUR of MHD Model EUR per Effective Lateral Length EUR per Fracture

Stb Stb/ft Stb

Case 1 1,740,932 300 58,031Case 2 1,745,757 301 58,192Case 3 2,746,554 474 91,552Case 4 3,447,609 594 114,920

Table 5Effective lateral length, fracture spacing, fracture number and total proppant injected for the Eagle Ford Brazos County field data.

Well Name Effective Lateral Length Number of Completion Stage Completion Stage Spacing Number of Fracture Fracture Spacing Total Proppant

– ft – ft – ft lbs

Well R 8630 35 250 139 63 12,282,550Well O 2942 13 240 52 60 4,090,160Well H 6550 22 300 131 50 10,664,970Well M 5950 20 300 119 50 9,398,600

Fig. 6. Map view of Eagle Ford field for (a) Well Pair 1 and (b) Well Pair 2. Any well interference by fracture hits is unconfirmed for Well Pair 2, but fracture hits were reported to theoperator during completion of the Well Pair 1.

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R, which was confirmed in our DCA analysis (see later).For Well Pair 2 (Well H and Well M), Well H has slightly longer

effective lateral length than Well M. Moreover, the completion stage,fracture number and total proppant injected is marginally greater forWell H than for Well M. The fracture spacing of the two wells is identical.

4.2.2. DCA fitting parameters and production forecastIn order to examine the inter-relationship of the fitting parameters

used on the Eagle Ford wells, the ranges of fitting parameters for the five

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DCA methods are summarized in Table 6. The exponential terms in thecorresponding models have the most significant impact the shape and fitof the history matched decline curves. Table 6 (a) to (e) show the fittingparameters for the MHD, Duong, LGM, and PLE (D∞≠0, and D∞ ¼ 0)models, respectively. Outliers were removed for the data set.

For the MHD model the b parameter lies between 1.15 and 1.25(Table 6a). For the Duong model, the a parameter lies between 1.20day�1 and 1.95 day�1 and the m parameter is within the range of1.15–1.25 (Table 6b). For LGM model, the n parameter is within the

Table 6Fitting parameters of four DCA models for Eagle Ford field data.

(a) Fitting parameters for MHD model

WellName

qi Di b Dmin

– Stb/day /year (Nominalrate)

– /year (Nominalrate)

Well R 1285 6.763 1.170 0.05Well O 512 11.903 1.159 0.05Well H 668 5.751 1.115 0.05Well M 587 6.916 1.237 0.05

(b) Fitting parameters for Duong model

WellName

a m q1 q∞

– (day�1) – (Stb/day)

(Stb/day)

Well R 1.5422 1.1985 1084.7 0Well O 1.2020 1.1683 577.35 0Well H 1.9158 1.2350 345.99 0Well M 1.3879 1.1759 591.05 0

(c) Fitting parameters for LGM model

WellName

K n ba– Stb – dayn

Well R 477,974 0.7860 211.49Well O 132,359 0.7400 145.43Well H 239,252 0.8470 260.86Well M 203,653 0.7984 200.76

(d) Fitting parameters for PLE model with D∞≠0

WellName

bqi bDi D1 bn D∞

– (Stb/day)

(day�1) (day�1) – (day�1)

Well R 2168.65 0.397 0.108 0.273 8.47E-04

Well O 770.82 0.334 0.114 0.340 4.70E-04

Well H 647.71 0.078 0.039 0.503 4.75E-04

Well M 706.08 0.138 0.062 0.449 6.45E-05

(e) Fitting parameters for PLE model with D∞ ¼ 0

WellName

bqi bDi D1 bn D∞

– (Stb/day)

(day�1) (day�1) – (day�1)

Well R 1913.30 0.248 0.092 0.370 0Well O 770.75 0.346 0.118 0.342 0Well H 669.84 0.080 0.042 0.522 0Well M 912.97 0.270 0.096 0.357 0

Y. Hu et al. Journal of Petroleum Science and Engineering 162 (2018) 617–632

range of 0.70–0.85 (Table 6c). For the PLE model with D∞≠0, the bnparameter lies between 0.25 and 0.45, the bDi parameter is between 0.05and 0.40, and the D∞ parameter is between 4.70E-04 and 8.50E-4(Table 6d). For the PLE model with D∞ ¼ 0, the bn parameter lies between

0.30 and 0.55, and the bDi parameter is between 0.05 and 0.35 (Table 6e).Similar as observed in the synthetic cases (Table 2d and e), forcing theD∞ term to be zero has a significant impact on the other fitting param-eters for PLE model (Table 6d and e), and the corresponding historymatches for each method therefore produces type curves with differentfits. The fitting parameters of Table 6 were used to compare the declinecurve fits when history matching the historic production data of the EagleFord wells.

To help develop a visual understanding of how each DCA methodhistory matches the actual production data from the wells, Fig. 7 plots theoil production rate and cumulative production for Well Pair 1 using thefive DCA methods. Fig. 8 plots the oil production rate and cumulative

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production for Well Pair 2. The oil production rate for all wells decreasesrapidly in the early life and much slower in later life. This trend is alsoreflected in the cumulative production. Comparing the production plotsfor Well Pair 1, we confirm that Well R has markedly greater cumulativeproduction than Well O. According to the production plots for Well Pair2, Well H and Well M have close production forecasts.

The actual daily production rate and the given cumulative productionof the wells are included in Figs. 7 and 8. The most optimistic EUR es-timate is provided by the MHD model for Well R and Well H and by theDuong model for Well O. For Well M, the EUR estimates from the MHDand Duong models are nearly identical (Table 7). The PLE model withD∞≠0 displays rapid decline and yields the most conservative EUR esti-mates. The LGM model and PLE model with D∞ ¼ 0 generate EURforecasts smaller than with the MHD and Duong models, but greater thanthe PLE model with D∞≠0.

4.2.3. History matching resultsIn order to examine the relationship between oil production and

characteristics of hydraulic fractures, we summarize the results of fore-casted EUR in Table 7. In Table 7, the highest EUR and the lowest EURestimates are highlighted in yellow and green, respectively. Based on theresults in Tables 5 and 7, we observe that the effective lateral length, thefracture number and the total proppant injected are directly proportionalto the EUR estimation. In other words, longer effective lateral length,more fractures, and more total proppant lead to higher EUR estimation.In this study, Well R has the longest effective lateral length, largestnumber of fractures, and largest volume of proppant injected, whichresulted in the highest EUR. Well O has the shortest effective laterallength with the least amount of fractures and proppant injected, whichresulted in the lowest EUR.

TheMHDmodel gives the highest EUR estimations for Well R, Well H,and Well M and the second highest EUR estimation for Well O. TheDuong model gives the highest EUR for Well O and the second highestEUR estimations for Well R, Well H, and Well M.

Fig. 9 summarizes the reliability of each DCA method by plotting theweighted residual of production rate for each history matched well. TheMHDmodel has the lowest weighted residual of production rates for WellR and Well H, and the second lowest ones for Well O and Well M. TheDuong model gives the lowest weighted residual of production rates forWell O andWell M and the second lowest ones for Well R andWell H. ThePLE model with (D∞≠0) provides the highest weighted residual of pro-duction rates for Well R, Well O and Well M with the second largestmatching error for well Well H. The LGM model yields the highestweighted residual of production rate for Well H. In general, the LGMmodel and the PLE model with (D∞ ¼ 0) give the medium values ofproduction forecasts and weighted residual of production rate. The EagleFord field data has a mismatch in the DCA weighted residuals of pro-duction rate (Fig. 9) larger than the mismatch for Eagle Ford syntheticdata (Fig. 5b). The explanation is that the field data include mechanicaladjustments (well workovers, choke and pump adjustments) leading tonoise that is lacking in the production data from the reservoir model. Ingeneral, the weighted residual of production rates for Well Pair 1 ishigher than the errors for Well Pair 2, which we ascribe to the highernumber of well shut-ins and bottomhole pressure adjustments aftercompletion, particularly for Well O (Fig. 9).

Table 8 summarizes the measurements of EUR using several hydraulicfracturing characteristics, including effective lateral length, completionstage, number of fracture, and total proppant injected. The EUR from theMHD model is used in this table since the MHD model providedreasonable history match results. For Well Pair 1, Well R has more EURestimated, longer effective lateral length, more completion stage, andmore fractures than Well O. The corresponding EUR per effective laterallength, EUR per completion stage and EUR per fracture for Well R isgreater than those for Well O. For Well Pair 2, Well H has slightly moreEUR estimated, longer effective lateral length, more completion stage,and more fractures than Well M. However, the corresponding EUR per

Fig. 7. Map view of Eagle Ford field Well Pair 1 (a) and related production forecasts (b)–(e).

Y. Hu et al. Journal of Petroleum Science and Engineering 162 (2018) 617–632

effective lateral length, EUR per completion stage and EUR per fracturefor Well R is slightly less than those for Well O.

It is expected that longer effective lateral length gives higher EUR perlength, but effective lateral length is not a decisive factor. Instead, it hasrelatively small influence on EUR compared to the other factors. More-over, more completion stages and higher fracture numbers providehigher EUR for Well Pair 1, which demonstrates the same behavior asthat of Eagle Ford synthetic Cases 3 and 4. Well R produces more becausethe short length of Well O limits the drainage region of Well R (by anywell interference to matrix and fracture hits) for only 1/3 of the welllength. Focusing on the synthetic well data (Fig. 5b), the MHD and Duongmodels matched with 1–5% accuracy. However, field data are matchedwith 11–15% accuracy for 3 wells (Well R, Well H, and Well M; Fig. 9).The residual for Well O is as high as 37% (Fig. 9) due to considerablespread in producing data even after removing extreme outliers for thehistoric data (Fig. 7c).

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4.3. Austin Chalk field data

4.3.1. Data descriptionThe oil production data of six wells in the Austin Chalk formation

located at the Texas A&M Brazos County campus were obtained fromDrillingInfo. In this study, the production data of six wells were inte-grated in a single set of data as only the aggregate data set is reliable,because several wells are reported as one production unit and individualdata are sketchy. We argue this is acceptable due to the vicinity of thewells, the similar geology features, and the analogous production con-ditions. The production data are from November 1st, 1992 to February1st, 2006, which covers 160 months. The total effective lateral length forthe 6 wells is 22,529 ft.

The Austin Chalk formation is an extremely heterogeneous carbonatereservoir with low permeability. This formation is composed of a dualporosity, naturally fractured production system. The permeability of the

Fig. 8. Map view of Eagle Ford field Well Pair 2 (a) and related production forecasts (b)–(e).

Table 7Forecasted EUR using five DCA methods for the Eagle Ford field data.

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matrix rock usually ranges from 0.001 to 0.01 md, while the fracturepermeability is about 10–100 time greater (Poston and Chen, 1991). Forthe studied wells, “water-fracs” were applied before production. Thisinexpensive process uses high volumes of water but no proppant, and hasobtained success in the Austin Chalk formation (Meehan, 1995). Due tolarge number of subsurface natural fractures, there is most likely only 1

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hydraulic fracture for each well, but fracture diagnostics arenot available.

4.3.2. DCA fitting parameters and production forecastThe fitting parameters for the five DCA methods are shown in Table 9

as an intermediate process to obtain the production rate and cumulative

Fig. 9. Weighted residuals for history matched production rates relative to the actualproduction rates for the Eagle Ford field data.

Table 9Fitting parameters of five DCA methods using Austin Chalk field data.

(a) Fitting parameters for MHD model

qi Di b Dmin

Stb/month /year (Nominalrate)

– /year (Nominalrate)

950 3.674 1.109 0.05

(b) Fitting parameters for Duong model

a m q1 q∞(month�1) – (Stb/

month)(Stb/month)

0.9777 1.2429 25308.57 0

(c) Fitting parameters for LGM model

K n baStb – monthn

837,904 0.5852 19.37

(d) Fitting parameters for PLE model with D∞≠0

bqi bDi D1 bn D∞

(Stb/month) (month�1) (month�1) – (month�1)

2,495,124.74 4.421 0.526 0.119 1.66E-05

(e) Fitting parameters for PLE model with D∞ ¼ 0

bqi bDi D1 bn D∞

(Stb/month) (month�1) (month�1) – (month�1)

2,555,555.88 4.444 0.527 0.119 0

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production plots. Table 9 (a) to (e) show the fitting parameters for theMHD, Duong, LGM, PLE (with D∞≠0), and PLE (with D∞ ¼ 0) models,respectively. The forecasts are based on stock tank barrel of oil permonth, while the previous two data sets are based on stock tank barrel ofoil per day. By forcing the D∞ term to be zero seems to have little impacton the other fitting parameters for PLE model for this data set. Thus, theexpected fitting curves behave the same.

Fig. 10 depicts the oil production rate and cumulative production toshow how each DCA method fits with the historic production data. Thisdata set has some distinct outliers which are included in Fig. 10 but wereexcluded for the DCA fit. Unlike the previous two data sets, the oil pro-duction rate decreases in a relatively consistent speed. Therefore, thecumulative oil production increases more smoothly in the early lifecompared to the Eagle Ford field and synthetic cases. The productionforecast curves of the various DCA methods using Austin Chalk data setappears to give close matches. A possible explanation is that the AustinChalk produce from a network with many natural fractures, which resulstin a slower and smoother productions decline. In terms of EUR, theDuong model gives the highest cumulative production at the end offorecast period while the other models yield similar EUR.

4.3.3. History matching resultsIn Table 10, the highest EUR and the lowest EUR estimates are

highlighted in yellow and green, respectively. As expected, the Duongmodel has the highest EUR. The lowest EUR is given by the PLE modelwith (D∞≠0). The MHD, LGM, and PLE with (D∞ ¼ 0) models give me-dium EUR values. Overall, three of the DCA methods predict nearlyidentical outcomes (LGM and PLE models), with MHD EUR within 1%difference, but the Duong EUR is almost 5% higher (Table 10).

In order to determine the reliability of each DCA method, the corre-sponding EUR forecasts are plotted in Fig. 11. The weighted residual ofproduction rate is within the range from 13% to 15%. The lowestweighted residual of production rate is given by the PLE model with(D∞≠0) and the highest weighted residual is from the Duong model. This

Table 8Forecasted EUR over various hydraulic fracturing characteristics for the Eagle Ford field data.

Well Name EUR of MHD Model EUR per Effective Later

– Stb Stb/ft

Well R 492,171 57Well O 126,427 43Well H 264,405 40Well M 245,385 41

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observation is opposite to the ones from the Eagle Ford cases.As the MHD model gives adequately reasonable EUR estimates, the

EUR from this method is used to compute the EUR per effective laterallength. The value obtained is 23 Stb/ft, about half of the EUR pereffective lateral length of the Eagle Ford field wells, which were muchcostlier to complete and stimulate due to the large number of hydraulicfractures (Table 5). Thus, the productivity of Austin Chalk wells per feetof effective lateral length is less than the productivity of Eagle Ford wells.

5. Discussion

This study focuses on determining the suitable DCA methods for hy-draulically fractured wells. For the Eagle Ford field cases and the EagleFord synthetic cases, the MHD model and the Duong model provide thetwo highest EUR estimations among the five DCAmethods, while the PLEmodel with (D∞≠0) yields the lowest EUR estimations for all cases. Inaddition, the MHD model and the Duong model give the two lowestmatching errors among the five DCA methods, while the PLE model with(D∞≠0) yields the highest matching errors for all cases. Therefore, theMHD model and the Duong model are the most preferable methods forEagle Ford oil production, while the PLE model with (D∞≠0) is the leastpreferable method.

Although the Duong model is developed for tight gas and shale gasreservoirs, its application for shale oil reservoirs gives fairly reasonableforecasts. The LGM model and the PLE model with (D∞ ¼ 0) generateneither optimistic nor conservative results with acceptable matchingerrors. Moreover, the Eagle Ford synthetic cases indicate that the number

al Length EUR Completion Stage EUR per Fracture

Stb Stb

14,062 35419725 243112,018 201812,269 2062

Fig. 10. Production forecast for the Austin Chalk field data.

Table 10Forecasted EUR of five DCA methods for the Austin Chalk data.

Fig. 11. Weighted residual of production rate for the Eagle Ford field data.

Y. Hu et al. Journal of Petroleum Science and Engineering 162 (2018) 617–632

of fracture hits does not affect the accuracy of the DCA method used.However, at the same time, our analysis suggests that any DCA methodcannot detect the existence of any fracture hots. The only distinguishingfactor is that more fracture hits result in higher EUR. The Eagle Ford fieldcases imply that with longer effective lateral length, more fractures, andmore total proppant injected, higher EUR estimation may be expected.

As discussed previously, Table 8 shows that Well R of Well Pair 1 hasmore EUR estimated, longer effective lateral length, more completionstage, and more fractures than Well O. The corresponding EUR pereffective lateral length, EUR per completion stage and EUR per fracturefor Well R are greater than those for Well O. The EUR estimated, effectivelateral length, completion stage, and fracture number of Well H of Well

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Pair 2 are all slightly higher than those of Well M, but the correspondingEUR per effective lateral length, EUR per completion stage and EUR perfracture for Well R are slightly less than those for Well O. The number ofcompletion stages and number of fractures have a greater impact on theEUR forecast than effective lateral length. Well Pair 1 displays the sameresult as Eagle Ford synthetic Cases 3 and 4, indicating that morecompletion stages and more fractures provide higher EUR. Well Pair 2and synthetic Cases 1 and 2 give similar behavior that more completionstages and more fractures give slightly less or almost the same EUR es-timate. We suggest that for horizontal wells with ample hydraulic frac-tures, longer effective lateral length, more completion stages and morefractures, higher EUR can be expected.

For the Austin Chalk oil production, the Duong model has the highestEUR. The lowest EUR is provided by the PLE model with (D∞≠0). Thelowest weighted residual of production rate is given by the PLE modelwith (D∞≠0) and the highest error is from the Duong model. The pref-erence of the five DCAmethods for this data set is less distinct because allmethods yield close forecasting curves and matching errors. The AustinChalk wells were completed with only one hydraulic fracture with manynatural fractures present, while the Eagle Ford wells are in a lithologywith much fewer natural fractures, completed with multiple hydrau-lic fractures.

6. Conclusion

In this study, five empirical DCA methods are reviewed (MHD,Duong, LGM and two PLE models). The data investigated are for fore-casting oil production in a shale reservoir (Eagle Ford) and a carbonatereservoir (Austin Chalk). Our study provides a reference for operators tohelp select the appropriate DCA method for reserve estimations. Keyconclusions drawn from this study are:

1. For Eagle Ford wells, the MHD model and the Duong model generallygive the best history matching results (for EUR and production rate),and therefore are the most preferable methods in this study. These

Y. Hu et al. Journal of Petroleum Science and Engineering 162 (2018) 617–632

two models also yield the highest EUR for all data sets studied. ThePLE model with (D∞≠0) gives the most conservative EUR estimationand the least reliable historymatching results. Therefore, it is the leastpreferable method. The LGMmodel and the PLEmodel with (D∞ ¼ 0)give production forecasts neither too optimistic nor too conservativewith moderate matching errors.

2. For Austin Chalk wells, the Duong model has the highest EUR and thehighest weighted residual of production rate. The PLE model with(D∞≠0) provides the lowest EUR and the lowest weighted residual ofproduction rate. However, all methods generate fairly similar EURpredictions and matching errors so that the preference of the DCAmethods to be used becomes indifferent.

3. If fracture hits are present where two parallel wells with horizontaldrilling and hydraulic fractures that include ample fracture hits, theDCA methods still work well but cannot recognize the occurrence offracture hits.

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4. Special attention needs to be paid when dealing with natural fracturesand hydraulic fractures, as they generate different production be-haviors. The different patterns of forecast for the Austin Chalk fielddata and the Eagle Ford field and synthetic data indicate that natu-rally fractured wells (Austin Chalk) give similar EUR predictionsusing any of the five DCA methods, while hydraulically fracturedwells (Eagle Ford) seem to be matched better with the MHD andDuong models.

Acknowledgement

This study was supported by start-up funds for the Texas A&M En-gineering Experiment Station to the senior author. Data were provided byHalcon Resources and Hawkwood Energy.

Nomenclature

qi initial production rate, volume/timeb decline exponent, dimensionlessDi initial nominal decline rate, 1/timea intercept constant defined by Eq. (7), 1/timem slope defined by Eq. (7), dimensionlesstða;mÞ time function defined by Eq. (6), dimensionlessq∞ production rate at infinite time, volume/timeGp cumulative production, volume (Duong model)Q cumulative production, volume (LGM model)K carrying capacity, volumeba constant of tn, timen

n hyperbolic exponent, dimensionlesst timeq production rate, volume/timebqi the rate “intercept”, volume/timebDi the initial decline constant, 1/timeD∞ the decline constant at infinite time, 1/timebn the time exponent, dimensionless

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