Journal of Petroleum Science and Engineering 165 (2018) 946–961

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Journal of Petroleum Science and Engineering

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

High-resolution visualization of flow velocities near frac-tips and flowinterference of multi-fracked Eagle Ford wells, Brazos County, Texas

Ruud Weijermars *, Ibere Nascentes Alves

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:Drained rock volumeMulti-staged fractured wellsEagle Ford Formation

* Corresponding author.E-mail address: r.weijermars@tamu.edu (R. Wei

https://doi.org/10.1016/j.petrol.2018.02.033Received 21 October 2017; Received in revised form 5 FeAvailable online 17 February 20180920-4105/© 2018 Elsevier B.V. All rights reserved.

A B S T R A C T

This study outlines a workflow that combines reservoir characterization and decline curve-based productionanalysis with a physics-based drainage model that quantifies where fluid is drained from, based on the fracturetreatment architecture. Decline curve analysis, applied to production data from four multi-fracked wells in theEagle Ford formation (Brazos County, East Texas), provides forecasts for the estimated ultimate recovery (EUR).About 50% of EUR is realized in the first 1000 days (~3 years) of the well-life. The drainage model shows wherein the reservoir the produced fluid is actually drained from, based on the estimated reservoir parameters. Thisstudy includes several fundamental assessments of factors that may impact any drainage model, such as (1) thepressure front propagation time responsible for the depth of investigation, (2) pressure effects due to up-scaling offracture patterns into a reduced number of fractures, and (3) interaction of fluid velocity patterns with pressuredepletion zones. The drainage model for the Eagle Ford wells in our case study suggests that the first generation ofhydraulic fractures recovers less than 1% of the original oil in place. With recovery factors so low, a repetitiveschedule of periodic refracking the wells - provided the first refracks prove successful - is highly recommended.

1. Introduction

Horizontal wells tapping into sub-horizontal reservoirs with very lowpermeability are stimulated with hydraulic fracture treatment to improvethe well productivity. Models aimed at studying flow patterns in suchreservoirs need to account for at least two processes that may criticallyaffect the applicability of any reservoir model. The first process is thepropagation distance into the matrix (depth of investigation, and speed)of the pressure front, imposed by the well and its fractures, uponcompletion of the wellbore and its fracture treatment. The second processthat must be accounted for in the reservoir model is to match the declinein the reservoir pressure commensurate with the observed production.The observed pressure drop must correspond to the productivity prop-erties in that part of the reservoir, where the produced fluid was drainedfrom.

The drainage model applied here allows detailed visualization of thefluid velocity and pressure field changes in hydraulically fractured res-ervoirs. Our models, based on gridless and meshless complex analysismethods (CAM), provide a high-resolution complement to reservoirsimulators based on gridded and meshed methods, which limits theirspatial resolution. Our reservoir drainage model shows the flow towards

jermars).

bruary 2018; Accepted 12 February

four sub-parallel, horizontal wells in a reservoir section of the Eagle Fordformation in Brazos County, Texas. The study (1) highlights theadvancement of the depth of investigation and nuances its meaning, (2)addresses reservoir pressure changes related to upscaling effects, and (3)compares fluid velocity patterns with pressure depletion zones, usingnewly adapted analytical methods. Decline curve analysis (DCA) is usedto history match 40 months production data of multi-fracked Eagle Fordwells in Brazos County, Texas. The DCA provides the estimated ultimaterecovery (EUR) for the wells, and the rate transient analysis (RTA) givesthe flow rates at each time increment over the life of the well. Subse-quently, the transient flow rates are used as inputs for the time-discretized CAM model to reconstruct both temporal and spatialchanges in the flow velocity and pressure field of the stimulated reser-voir, especially near each frac stage.

Until now the transient nature of the so-called stimulated rock volumehas been little emphasized, but is highlighted in our study, applying DCA,RTA and original-oil-in-place (OOIP) estimations in combination withCAM-based time-of-flight reconstructions. The drained region aroundeach the frac stages is visualized and quantified using time-of-flightcontours for fluid flow toward the well and its transverse fractures. Therecovery factor (RF), as it evolves over the life of the wells, is quantified

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R. Weijermars, I. Nascentes Alves Journal of Petroleum Science and Engineering 165 (2018) 946–961

combining estimations of OOIP with the EUR for the progressivelydrained region. The RF plot versus time is given for the visualizeddrained reservoir volume, also advancing over time. The proposedworkflow provides a practical method to establish the transient reservoirvolume drained over time.

2. Reservoir parameters

What is most relevant for the drainage model presented here are es-timations of the target payzone thickness and representative values of theflow space parameters, such as porosity, permeability, fluid saturation,fluid composition, viscosity and initial pressure of the reservoir. First, thereservoir characterization is briefly presented (Section 2.1). Completiondetails of the wells and fracture treatment program, also relevant for ourstudy, are discussed (Section 2.2), followed by a brief analysis of historicproduction data (Section 2.3).

2.1. Reservoir characterization

For model simulations, reproduction of global features takes prece-dence over local accuracy. Upscaling of reservoir properties for modelinputs as spatial inputs in our gridless and meshless drainage model, maybe a simple geometric average resulting in an effective permeability andporosity, or, if more data are available, a directional tensor quantity. Inwhat follows, no attempt is made to define geo-statistical procedures(e.g., Kelkar and Perez, 2002); only the range of relevant properties isbriefly outlined. At this stage, our drainage model is based on discrete

Fig. 1. Regional map showing Eagle Ford depth contours with superposed oil andstratigraphic correlations of well logs given in Fig. 2. Eagle Ford trend map basedpresentations (Halcon, 2014a,b).

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values, complemented with a sensitivity analysis for certain parameterranges. The main focus of our current study is to provide an overview ofthe methodology and work flow, illustrated with field data from fourEagle Ford wells.

2.1.1. Study areaThe Eagle Ford lease area studied is located in Brazos County, Texas

(Fig. 1). The stratigraphy of the Cenomanian-Turonian Eagle Ford for-mation has been described in detail for the southern region of itsoccurrence between the San Marcos Arch and the Maverick Basin(Donovan et al., 2012). The NE continuation of the Eagle Ford iscommonly referred to as the Eaglebine, a portmanteau for the undiffer-entiated Eagle Ford and Woodbine formations occurring between theBuda formation below and the Austin Chalk above (Martin et al., 2011;Bowman, 2014). Recent work suggests that Woodbine Group sedimentsthicken from 50 ft in Brazos County to over 500 ft in the NE adjacent LeonCounty (Vallabhaneni et al., 2016).

2.1.2. Eagle Ford sectionTwo cross-sections based on 12 well logs (Fig. 2a and b) were released

in the public domain by way of investor presentations by Halcon Re-sources, before the sales of its acreage to Hawkwood Energy (Q1 2017).The surface traces of the relevant cross-sections are outlined on the insetmap (Fig. 1), covering parts of Brazos, Burleson and Robertson Counties.The interpreted thickness of the Eagle Ford formation increases fromabout 300 ft in the West to about 500 ft in the Eastern part of thesectioned region. The lease area studied here contains the Reveille well

gas windows based on cumulative production data. Inset map shows regionalon Roth and Roth (2014) and inset map based on Halcon Resources investor

Fig. 2. a &b: Regional correlations of well logs showing Eagle Ford section thickening toward the SE and NE. Section lines are marked on inset map of Fig. 1. Uppersection (a) uses lower reservoir boundary as reference level; lower section (b) uses top of Eagle Ford as a reference datum. c: 3D model of well trajectories for leasearea studied. Two landing zones appear closely spaced: Six wells tap into the Austin Chalk (Riverside 1–6) and the four remaining wells (R,O, H, M) are in the EagleFord. Sections (a) and (b) are based on Halcon Resources Investor presentations (Halcon, 2014a,b). The well trajectories are constructed based on public wellcompletion data in DrillingInfo.

R. Weijermars, I. Nascentes Alves Journal of Petroleum Science and Engineering 165 (2018) 946–961

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R. Weijermars, I. Nascentes Alves Journal of Petroleum Science and Engineering 165 (2018) 946–961

(see inset map, Fig. 1), with three more parallel wells piercing the EagleFord landing zone, and 6 additional wells located nearby in the overlyingAustin Chalk (Fig. 2c). Noteworthy, Reveille initially produced 1000 stb/day in its first month and is still the best oil well on record in the EastTexas Eagle Ford, with a cumulative production of over 175,000 stb inthe first 6 months of well-life. In our study area, the Eagle Ford sectionmakes no distinction of any Woodbine being present. The base of theAustin Chalk occurs at 7800 ft and the base of the Eagle Ford is picked at8200 ft. The Eagle Ford section is about 400 ft thick, with a landing zonefor the Eagle Ford producers near the base of the Eagle Ford, about 70 ftabove the so-called false Buda (Fig. 2b). The local dip of the formation isabout 2.4� toward the SE, confirming the slope of regional sedimentaryprovenance toward the Gulf of Mexico basin.

2.1.3. Pay-zone thicknessEstimations of the target pay-zone thickness are made using several

data sources. A sub-horizontal shaly sequence of 400 ft is underlain bythe Buda and overlain by the Austin Chalk, as can be inferred fromproprietary logs for the wells in our study area. The vertically orientedhydraulic fractures are assumed to be arrested by upper and lower fracbarriers, the lower barrier occurring near the base of the Eagle Ford, andthe upper barrier is made up by the upper Eagle Ford (Fig. 2a). The lowerand upper frac barriers isolate a 150 ft thick column marked as theprincipal production zone. The four Eagle Ford wells studied here werecompleted in 2014, at which time sand damaged some of the sucker rodpumps of the nearby Austin Chalk wells, which suggests frac containmentwas limited. On the other hand, any post-stimulation production increaseremained absent in the Austin Chalk wells. Interestingly, jurisprudencefrom the Texas Supreme Court on subsurface trespassing by hydraulicfracturing (in a 2008 decision of Coastal Oil and Gas Corporation versusGarza Energy Trust), indicates that fracking beyond the boundaries ofone's tract currently does not permit an adversary to recover for a loss ofoil or gas that might have been produced (Rodgers, 2013).

2.1.4. PermeabilityCore and drill cuttings from the Eagle Ford in South Texas have

provided permeability estimations ranging between 1 nanoDarcy andabout 1 microDarcy (Walls and Sinclair, 2011). The variability inpermeability of Eagle Ford rocks covers at least 3 orders of magnitude(Rosen et al., 2014; Kosanke and Warren, 2016), with fine-grained marlnear the lower range (1nD), marl with coarser laminations (50–100 nD),laminated marl and limestone with microfractures (200nD), and massivediagenetically altered limestone (1000 nD or greater) with organicmatter lined by microfractures. In addition, permeability may exhibithysteresis, with permeability increasing during early production, butdecreasing in later well life, which is attributed to stress dependency(Rosen et al., 2014). In our study, we consider the observed stratificationin permeability, but neglect any hysteresis effects for simplicity. Forreference, the overlying Austin Chalk typically has a permeability rangeof 0.03–1.27 milliDarcy (Martin et al., 2011). In reservoir quality EagleFord section, organic matter may be averaging some 10% of total volume,and the degree of kerogen maturity determines its production potential.It has been suggested that kerogen may behave as a viscous liquid atreservoir conditions, which would result in much higher permeabilitiesthan assumed based on core tests (Walls and Sinclair, 2011). However,most contemporary interpretations (Dr. Yucel Akkutlu, personalcommunication, 28 Sep 2017) do not consider kerogen to behave as abitumen, but instead is considered an organic solid that needs pyrolysisfor mobilization.

2.1.5. PorosityPrevious work has highlighted the need to distinguish between

various types of pores, such as pores in organic matter and pores betweenmineral grains (Rine et al., 2013; Schieber, 2013; Schieber et al., 2016).Organic matter pores are in the range of 10–50 nm for so-called foampores and 100–1000 nm for bubble pores. Pores between the mineral

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framework also range between 10 and 1000 nm. In summary, the EagleFord lithology exhibits a highly heterogeneous porosity, being as high as50% in micro-fossil infill chambers (Schieber et al., 2016), but moregenerally ranging between about 2 and 10% (Walls and Sinclair, 2011),with clustering between 4 and 6% (Kosanke and Warren, 2016). Theporosity in shales being relatively high as compared to the ultra-lowpermeability is due to both isolated pores and high tortuosity betweenconnected pores (Rubinstein and Torquato, 1989). Proprietary data forEagle Ford samples from our study area indicate the permeability andporosity ranges are similar to those quoted from South Texas.

2.1.6. Fluid saturationThe presence of organic matter, established in petrophysical studies

of Eagle Ford samples, and the enhanced pore structure due to generation(and partly migration) of liquid hydrocarbons is supportive of liquidproduction potential (Fisher, 2016). Production data give an indicationof GOR (gas to oil ratio) and WOR (water saturation), but the actual fluidsaturation of pore space is generally poorly constrained. In any case,capillary pressure is assumed negligible.

2.1.7. Fluid composition and viscosityMapping of thermal maturity windows in the Eagle Ford has lead to a

classification of four zones (also termed populations or families; Clarkeet al., 2016): (1) light oil, (>1000 bbl/MMcf), (2) high-yield condensateand light oil (500–1000 bbl/MMcf), (3) rich condensate and wet gas(50–500 bbl/MMcf), and (4) lean condensate, wet gas and dry gas (<50bbls/MMcf). Our lease area is in the light oil window (family 1). Thecorresponding oil viscosity used as input for our drainage model is μ¼ 1cPoise (1mPa.s), assuming 40� API oil at a reservoir temperature ofabout 230 �F.

2.1.8. Initial reservoir pressureThe initial reservoir pressure, Po, for a hydrostatic reservoir is

determined by the hydrostatic pressure gradient ΔPHydrostatic=ΔL timesreservoir depth (d):

PHydrostatic ¼ ΔPHydrostatic

ΔL⋅d ½psi� (1)

The hydrostatic pressure gradient typically is ΔPHydrostatic=ΔL ¼ 0:45psi/ft. For an overpressured reservoir like the Eagle Ford, a pressuregradient of ΔPOverpressured=ΔL ¼ 0:6 psi/ft is common; the initial pressure isthen given by:

POverpressured ¼ ΔPOverpressured

ΔL⋅d ½psi� (2)

For the Eagle Ford reservoir section studied, we assume ΔPOverpressured=

ΔL ¼ 0:6psi=ft and an average reservoir depth, d, of 8000 ft, which givesPo.¼ 4800 psi. Actual well data suggest a bottom-hole pressure (BHP) of5087 psi, but our 4800 psi is closer to the average pressure when theregional slope of the formation is taken into account.

2.1.9. Original oil in placeWe used the standard formula for volumetric estimation of original

oil-in place (OOIP), using field units:

OOIP ¼ 7758Ahnð1� SwÞB

¼ 1:93x108 ½stb� (3)

Based on the following input parameters:

7758¼ reservoir barrels (bbls) of oil per (acre.ft)A¼ area in acres, taken as 10,000 ft by 10,000 ft¼ 108 ft2¼ 2296acres.h¼ height in feet (150 ft).n¼ porosity (0.08)Sw¼water saturation (0.05)

Table 1Effective lateral length, fracture spacing, fracture number and total proppantinjected for four Eagle Ford wells, Brazos County, Texas.

WellName

LateralLength

Numberof Stages

StageSpacing

TotalPerf

Fracture/Cluster

TotalProppant

R. Weijermars, I. Nascentes Alves Journal of Petroleum Science and Engineering 165 (2018) 946–961

B¼ volume factor, reservoir bbls/stocktank barrels (stb) ~1.05

The OOIP will be used to determine the progressive recovery factor ofthe developed acreage. The formation volume factor increases when thegas-to-oil ratio (GOR) increases as summarized in Fig. 3.

ft ft Clusters Spacingft

lbs

WellR

8630 35 250 139 63 12,282,550

WellO

2942 13 240 52 60 4,090,160

WellH

6550 22 300 131 50 10,664,970

WellM

5950 20 300 119 50 9,398,600

Table 2Fitting parameters for Duong model.

Well Name a(day�1)

m q1(stb/day)

q∞(stb/day)

Starting dates

Well R 1.5422 1.1985 1084.7 0 1 April 2014Well O 1.2020 1.1683 577.35 0 1 April 2014Well H 1.9158 1.2350 345.99 0 1 Nov 2014Well M 1.3879 1.1759 591.05 0 1 Nov 2014

2.2. Fracture treatment

The effective lateral length, fracture spacing, fracture number andtotal proppant injected for each well, based on company completion re-ports, are listed in Table 1. Each frac was initiated from a perf cluster overa length of 2 ft with 3 perfs/ft, and we assume each perf cluster createdonly one major hydraulic fracture. Fracture treatment data were not yetfully analyzed at the time of our study; a 3D fracture propagationmodel isunder development for use in a future expansion of our present analysis.In our current drainage model, we assume a constant transverse fracturehalf length of 500 ft (based on well interference reports). For example, awell with 20 stages will have nearly 120 fractures (Table 1), each havinga lateral length of 1000 ft and an assumed effective height of 150 ft(corresponding to the payzone thickness of Section 2.1). The total contactarea of all 120 fractures combined, with the reservoir matrix, wouldamount to 360 million ft2.

2.3. Production analysis

Historic production data (~24 months) of four Eagle Ford wells wereused to match a type curve using the Duong model (Duong, 2010, 2011).Several other curve fits were compared (Hu et al., 2018), but the Duongmodel gave the best fit. The general expression for oil production rate inthe Duong Model is:

qðtÞ ¼ q1tða;mÞ þ q∞; ½stb=d� (4a)

with tða;mÞ ¼ t�mea

ð1�mÞðt1�m�1Þ: ½d� (4b)

Cumulative oil production is given by:

Gp ¼ qatm ½stb� (5)

The Duong DCA parameters in Table 2 (a [/day], m [�], q1 [stb/day],and q∞ [stb/day]) were obtained from a least squares fit on historicproduction data. The corresponding production forecast over a maximumwell life of 40 year well life is shown in Fig. 4a and b.

Fig. 3. Correlations of formation volume factor and GOR, using compilationsindicated (Al-Marhoun, 1985, 1988; Al-Shammasi, 2001; Farshad et al., 1996;Glasø, 1980; Kartoatmodjo and Schmidt, 1994; and Standing, 1947; imageafter Petrowiki).

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3. Drainage model

Numerous models of flow interference of closely spaced fractures andwells have been published (Cipolla and Wallace, 2014; Lalehrokh andBouma, 2014; Yu et al., 2016, 2017) in order to investigate the effects ofvarying frac spacing and well spacing to achieve the most economicproductivity of shale oil and gas wells. Our model expands the range oftools available for such studies with a unique, high-resolution analysis offlow around hydraulic fractures, based on gridless and meshless complexanalysis methods (CAM). Before discussing our model, we first present abrief review of the pressure front propagation distance, which is criticalto know whether and when a reservoir model can be applied.

3.1. Pressure front propagation

The propagation distance into the matrix (depth of investigation) andspeed of the pressure front, imposed by the well and its fractures, uponcompletion of the wellbore and its fracture treatment, are investigated inour study. Several authors have reviewed the depth of investigation(Odeh and Nabor, 1966; Matthews and Russell, 1967; Gringarten andRamey, 1973; Earlougher, 1977; Kucuk and Brigham, 1979; Lee, 1982;Hagoort, 2006; Kuchuk, 2009). More specifically, the radius of investi-gation (ri) refers to the advance of the pressure drawdown front, whichexpands from the well outward. After a certain travel distance, thepressure front “senses” the proximity of the well's drainage boundary,and flow is assumed to change from transient/unsteady state topseudo-steady state, during which the average pressure in the reservoircontinues to decline at a constant rate with time until the reservoirpressure can no longer lift fluid to the surface. True steady-state flow mayonly occur when the pressure profile around the well becomes stationary,due to either natural influx of aquifer fluid or engineered water injectioncountering any drawdown in the reservoir's average pressure.

An original expression for the transient radius of investigation is derivedin Appendix A, starting from basic principles:

riðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kt1688:7nμct

s: ½ft� (6a)

Expression (6a) assumes use of imperial units (Appendix A). Thetransient radius of investigation, ri, at time t, varies with the reservoirproperties (porosity, n, and permeability, k, fluid viscosity, μ, and

Fig. 4. a &b: History matched production forecasts for the four Eagle Ford wells (R, O, H, M) and total daily production (a) and cumulative production (b). Time hereis the real time during reservoir operation, hence the jump in production/flow rate at 7 months (or 213 days). Historic production data courtesy Halcon Resources.

Table 3Values assumed for depth of investigation (Fig. 5).

Quantity Symbol Value Units

Total compressibility- c 1.5� 10�5 1/psiViscosity μ 1 cPPorosity n 0.02–0.08 –

Permeability k 10�7-10�4 Darcy

R. Weijermars, I. Nascentes Alves Journal of Petroleum Science and Engineering 165 (2018) 946–961

compressibility, ct). The compressibility must be negligibly small tojustify a simple Darcy flow description for incompressible fluids, but isneeded for unit integrity with ct, expressed as a fractional volume changeper applied pressure unit. Fig. 5 shows the plot of Eq. (6a) using the inputparameters given in Table 3. Appendix A gives a complete derivation ofour proposed solutions of the diffusivity equation for transient andpseudo-steady state flow.

Transient or unsteady state flow can persist in a reservoir without anylateral boundaries. The radial diffusivity equation is traditionally used todescribe such transient flow (Matthews and Russell, 1967). Transientflow solutions would apply (without any limitations) to continuous(unconventional) reservoirs, unless neighboring wells impose a finitedrainage region for each well equal to the applied well spacing, as isclearly the case in most practical field development programs. Pressuredrawdown of the “pressure transient” in any confined flow space will beaccelerated when the depth of investigation starts to sense the presenceof the lateral boundaries of the reservoir cell. For a point source model of

Fig. 5. Depth of investigation (ft) versus time (days) lapsed since the instan-taneous application of the pressure step function at the well location, for variousreservoir permeabilities. Note that the propagation speed of the pressure front isgiven by the gradient of the curves, being faster for the higher permeability ascompared to lower permeability reservoirs. Left scale shows depth of investi-gation for porosity n¼ 0.02; right scale assumes n¼ 0.08.

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a vertical well with constant flow rate, the critical radius of investigationoccurs after time t> tc, as follows from rearranging Eq. (6a):

tc ¼ 1688:7nμctr2ek

; ½hours� (6b)

where re is the finite radius of the cylindrical reservoir space. Fig. 5 can beused to estimate for any critical depth of investigation limited by acertain well-spacing, the time required to reach pseudo-steady-state flow.If the production space is not approximated by a cylinder shape,correction factors have been published for reservoir dimensions differentfrom cylindrical shapes (Earlougher, 1977). Various interpretations onthe precise occurrence of tc were reviewed in Odeh and Nabor (1966). Inany case, it follows from Eq. (6b) that when reservoir permeability is verylow (nD), such as in shale reservoirs, it takes much longer to reach thecritical depth of investigation and transient flow may persist for quitesome time.

The transient depth of investigation can be connected with the so-called diffusive time of flight used in fast-marching methods (Kinget al., 2016). More specifically, Eq. (6a) can be rewritten using the hy-draulic diffusivity, α ¼ k=ðnμctÞ:

riðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiαt

1688:7

r: ½ft� (7a)

The time required for the radius of investigation to expand to anylocation ri is given by:

t ¼ 1688:7r2iα

; ½hours� (7b)

The diffusive time of flight, τ, is then:

R. Weijermars, I. Nascentes Alves Journal of Petroleum Science and Engineering 165 (2018) 946–961

τ ¼ ffiffit

p ¼ ri

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1688:7

r; ½hours1=2� (7c)

α

Note that τ has the dimension of square root of time and connects tophysical time by t ¼ τ2, which is why we prefer to use Expression (7b) fortime involving estimations of the depth of investigation.

For practical applications the distinction between transient andpseudo-steady state flow remains an important one, because the produc-tivity index is affected. The difference is readily seen from a comparisonof the transient flow productivity index (adopting a formulation using thetransient radius of investigation, see Appendix A) with the formulationfor the pseudo-steady state flow index (for radial flow):

Transient flow: JðtÞ ¼ kh141:2μB

⋅1

ln�riðtÞrw

�� 3

4 þ s

�stb=daypsi

�(8)

Pseudo� steady state flow: J ¼ kh141:2μB

⋅1

ln�rerw

�� 3

4 þ s

�stb=daypsi

�(9)

The difference between Eqs. (8) and (9) is in the use of radius ofinvestigation (ri) in Eq. (8) instead of the finite reservoir radius (re) in Eq.(9). Note that Eq. (8) becomes identical to Eq. (9) when ri →re, whichoccurs exactly at tc, given by Eq. (7). Remember the productivity index, J,relates to the well rate, q, by J¼ q/(Pe-Pwf), with pressures Pe andPwf atrespectively, the boundary of the drainage region and wellbore in thereservoir, and negative skin factors, s, account for production enhance-ment by the fracture network.

3.2. Pressure-gradient domains in multistage fractured reservoir models

The pressure-gradient in a reservoir is established as the depth ofinvestigation advances outward from the well and its fractures. Thepressure front propagating radially outward from a well point source isslower than for a fracture source, but is used here, for simplicity, as alower limit for the depth of investigation from any pressure step functionimposed on the reservoir, either by a well or its connected fractures. Theimplication of Fig. 5 is that 100 nD and 2% porosity reservoirs, with 50 ftof hydraulic fracture spacing, the 25 ft depth of investigation required tocover the full spacing between each fracture is only reached after about130 days. For 1 μD reservoirs and the same frac spacing, the pressure-gradient between the fracs is already fully established after about 13days. For the upper porosity range of 8%, the time required to establishthe pressure gradient will double (Fig. 5, right-hand scale).

There can be no doubt that the drainage depth is lagging far behindthe depth of investigation, as can be quickly inferred from the followingexample. Assume a single well with a fracture contact area between thereservoir matrix and the fracture interface of 360 million ft2 (as detailedin Section 2.2) and an initial daily production on the order of 1000 bbls(5614 ft3). The corresponding matrix depth drained per day by eachfracture face with a porosity of 8% would be no more than 8.8� 10�6 ft/(2� 0.08) or 16.5 microns. Obviously, as the well rate drops, the incre-mental increase in the depth of matrix rock drained by the fractures willwane, as quantified later in our study.

The next question is whether any reservoir model based on thediffusivity equation can be applied to study the transient, early drainageepoch of shale reservoirs, when the local pressure gradients are stillpatchy and not interconnected. A prior study by our group has empha-sized that the streamlines are identical for wells during transient andpseudo-steady state flow, only the rate of flow is affected and wanesslower during pseudo-steady state flow than during transient flow(Weijermars et al., 2017a). For fracs producing with a constant pressureof an assumed finite reservoir without fluid replenishment, the drainagecontours will drain additional reservoir volumes that become progres-sively smaller as the flux into the fracs declines over time. Maintaining aconstant bottom-hole pressure (with pumps) and withdrawing fluid gives

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a concurrent decrease of the average reservoir pressure and the pressuredrawdown leads to a rapid decline in the production rate. In what fol-lows, our reservoir model assumes a pressure-gradient that is fullyestablished throughout the reservoir. Later in this study, we argue thatpressure gradients in the actual reservoir of our case studymay propagatefaster than estimated initially on the basis of reservoir parameters. Ourinterpretation of the faster pressure propagation can be inferred from acareful assessment of the pressure decline in the reservoir commensuratewith the known historic production rates in our field example.

3.3. Complex analysis method (CAM)

Our method for visualizing the progressive drainage (reservoirdepletion) of the so-called stimulated rock volume (SRV) near the fracstages is based on complex analysis. The analytical particle path solutionsobtained with our CAM based drainage model have been benchmarkedand validated in prior studies, using independent streamline solutionsbased on a numerical reservoir simulator (ECLIPSE; see Weijermars et al.,2016; Weijermars et al., 2017a,b).

3.3.1. AlgorithmsThe complex potential Ω(z) for an interval-source with time-

dependent strength m(t) [positive for an interval-source; negative for asink] along the real axis with the real interval [a,b] is:

Ωðz; tÞ ¼ mðtÞ2πðb� aÞ ½ðz� aÞlogðz� aÞ � ðz� bÞlogðz� bÞ�: ½m2:s�1�

(10a)

Differentiating this expression with respect to z, yields the corre-sponding velocity potential V(z):

Vðz; tÞ ¼ mðtÞ2πðb� aÞ ½logðz� aÞ � logðz� bÞ�: ½m2:s�1� (10b)

Alternatively, using the center (xc) and total length (L) of the interval-source results in:

Vðz; tÞ ¼ mðtÞ2πL

½logðz� xc þ 0:5LÞ � logðz� xc � 0:5LÞ�: ½m2:s�1� (10c)

For an interval-source (Fig. 6a) between end-points za (¼ zc – 0.5L ∙eiβ) and zb (¼ zc þ 0.5L ∙ eiβ), the complex potential Ω(z) is given by:

Ωðz; tÞ¼mðtÞ2πL

�ðz� zaÞlog�e�iβðz� zaÞ

�ðz� zbÞlog�e�iβðz� zbÞ

: ½m2:s�1�

(11a)

The velocity potential is again obtained by differentiating the aboveexpression with respect to z:

Vðz; tÞ ¼ mðtÞ2πL

�log

�e�iβðz� zaÞ

�� log�e�iβðz� zbÞ

�: ½m2:s�1� (11b)

Employing the center and total length of the interval-source (Fig. 6a),Eqs. (3) and (4) are readily rewritten into the following generalized ex-pressions for N interval-sources:

Ωðz; tÞ ¼XNk¼1

mkðtÞ2πLk

e�iβk��

e�iβk ðz� zckÞ þ 0:5Lk

�log

�e�iβk ðz� zckÞ þ 0:5Lk

� �e�iβk ðz� zckÞ � 0:5Lk

�log

�e�iβk ðz� zckÞ � 0:5Lk

: ½m2:s�1�

(12a)

Vðz; tÞ ¼XNk¼1

mkðtÞ2πLk

e�iβk�log

�e�iβk ðz� zckÞ þ 0:5Lk

� log�e�iβk ðz� zckÞ

� 0:5Lk

:

�m2:s�1

(12b)

Fig. 6. a: General fracture element at location zc, with end-points za and zb, totallength L and angle β. b: Example of fracture network (yellow) with stream lines(blue) and time of flight contours (red). After Van Harmelen and Weijermars(2018). (For interpretation of the references to colour in this figure legend, thereader is referred to the Web version of this article.)

R. Weijermars, I. Nascentes Alves Journal of Petroleum Science and Engineering 165 (2018) 946–961

Where zc is the center of an interval-source, L is its total length, and β is itsangle with the x-direction (Fig. 6a). Eq. (12b) can be applied to stringtogether multiple segments or into any arbitrary fracture network(Fig. 6b). The progressive drainage for different times can be visualizedby time-of flight contours (red in Fig. 6b) along streamlines that drain thereservoirs as fluid is carried off via the fractures (blue in Fig. 6b).

3.3.2. ScalingIn our previous studies (Weijermars et al., 2016, 2017a,b) we scaled

the 3D volumetric flux, q(t), of the source bym(t)¼ q(t)/(2πh). However,Eqs. (10)-(12) used in the present study assume the strength is simplyscaled by m(t)¼ q(t)/h, because we have written the factor 2π explicitlyin the denominator of the complex potentials for the velocity field. For aporous medium, the porosity n of the reservoir and the formation volumefactor, B, need to be accounted for in the scaling of the strength:

mðtÞ ¼ BqðtÞhn

:�m2:s�1

(13)

The strengths in Eqs. (10a,b)-(12a,b) are scaled using Expression (13).For dynamic similarity between any pair of models (or real field data),scaling rules that include rheological similarity are detailed inWeijermarsand Schmeling (1986). Additionally, inputs of q other than in m3/s (suchas using imperial field units of bbls/day or ft3/day), geometric scaling of

953

well distances and reservoir dimensions if not metric, m (but in ft as forreservoir thickness), all need to be converted in a consistent fashion.

3.3.3. Particle trackingThe velocities (vx, vy) of all fluid particles can be mapped using the

real and imaginary parts of the velocity potential:

VðzÞ ¼ vx � ivy: ½m2:s�1

(14)

The generic form of Eq. (14) will yield a specific velocity field solutionwhen a specific velocity field potential is used, such as Eq. (12b) in ourstudy.

Using the solution of Eq. (14) combined with a first order Eulerianscheme znþ1� zn þ v(zn) ∙ Δt, the streamline trajectories and time offlight contours can be calculated for any flow with well defined V(z).Tracing each streamline is accomplished by first choosing an initial po-sition, z0 (using complex coordinates), from which the tracing starts attime t0¼ 0, and calculating initial velocity v(z0(t0)). Next one selects atime step, Δt. The position of the tracer at time t1, i.e. after one time stepΔt, is denoted by z1(t1) and can be calculated as:

z1ðt1Þ ¼ z0ðt0Þ þ vðz0ðt0ÞÞ⋅Δt: ½m� (15a)

Generalizing this concept, the position of a tracer at time tj is given by:

zj�tj� ¼ zj�1

�tj�1

�þ v�zj�1

�tj�1

��Δt: ½m� (15b)

The principle of particle path tracking using time-series of continuoussolutions (gridless and meshless) to obtain high-resolution images basedon CAM simulations of transient flow processes is explained in Fig. 7.Consider a well for which the drainage region is modeled by flowreversal. The well acts as a source flow with strength, q, with a super-posed far-field flow, V∞, due to an aquifer (Fig. 7a). The complex po-tential and velocity potentials of the complex vector field for source andsink flows are (Weijermars et al., 2016; Weijermars and Van Harmelen,2017):

ΩðzÞ ¼XNk¼1

mk

2πlogðz� zkÞ:

�m2:s�1

(16a)

VðzÞ ¼XNk¼1

mk

2π1

ðz� zkÞ:�m2:s�1

(16b)

The width of the drainage region (Fig. 7a, blue streamlines) is smallerfor waning Rankine numbers, defined as Rk¼ q/(V∞hd0), with typicallength scale, d0, and reservoir thickness, h. The upper row in Fig. 7, showsa well producing with a certain constant rate superposed by a far-fieldflow. Fig. 7b shows how Rk declines over time due to a constant V∞,when q wanes due to pressure depletion. The transient pathlines forreservoir fluid can be modeled by our code, using time-stepping (Fig. 7c).

Streamlines are everywhere tangential to the velocity vectors, andstrictly speaking are only defined at a certain time, because the stream-line solution is an integral at a certain moment in time. Although inreservoir modeling we talk about streamline models, when time-dependent changes in pressure gradients occur - as is commonly thecase, velocity field solutions will vary for each time instant. What wereally construct is pathlines, obtained by Eulerian integration of the fluidparticle displacements for discrete time-steps (Fig. 7c). For the waningwell, a specific closed-form solution exists for each time increment as theRk number changes. Our method accommodates such time-dependencyby tracking particle flow paths using the Eulerian particle trackingscheme, essentially moving from one steady-state Rk-solution to the next,over very small time steps. The discretization time-steps become visibleas short travel path segments collocated from time-step to time-step(Fig. 7c).

Fig. 7. a: Well drainage pattern (blue streamlines) for well placed in an aquifer flow (red streamlines) scaled by the Rankine number (Rk). For a constant Rk,streamlines are actually streamlines that remain steady over time. b: Transient Rk numbers due a waning well. c: Pathlines for Rk changing from 10 to 5, with width ofdrained region (blue pathlines) narrowing over time to green outline. Inset image shows time-stepping segments of pathlines. Modified from Weijermars et al. (2014).(For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 8. Map view of well layout (axes in ft). See also details in Table 1.

R. Weijermars, I. Nascentes Alves Journal of Petroleum Science and Engineering 165 (2018) 946–961

4. Model results for field case

4.1. Well architecture

The four wells used in our field case study are from the Eagle Fordreservoir in East Texas, described in Section 2.1, with fracture treatmentdetails given in Section 2.2, and production performance analyzed inSection 2.3. The wells are sub-horizontal and parallel, with well spacingand reservoir model space as indicated in Fig. 8. Well O is a relativelyshort lateral due to reported circulation loss during drilling, beyondremediation even after two side-tracks. The possible cause may havebeen fracture communication with the overlying Austin Chalk, and lostcirculation lead to drill-bit damage and a stuck bottom-hole assemblythat required costly fishing operations.

The horizontal well separation of Wells H and M is 1234 ft; we used1250 ft in our model. The shortest distance between Wells R and O is754 ft near the heel, but interwell spacing increases rapidly toward to1250 ft, which is what we used in our model. The horizontal separationbetween the inner wells (O and M) of the two well pairs is about 4000 ft.The slope of the reservoir layer has an azimuth (normal to strikeline) of130 and average dip of 2.5�.

954

Table 4Unit conversion of reservoir attributes.

μ¼ 1 cPoise (1 mPa.s) [for 40 API oil at about 200F (reservoir temperature)]¼ 1.4503774389728� 10�7 psi.s¼ 1.4503774389728� 10�7� 0.0000115741 psi.days¼ 1.67868� 10�12 psi.days;Note: 1 s is 0.0000115741 Days

k¼ 200 nD¼ 200� 10�9� 10�12� 10.7639 ft2¼ 2.15278� 10�18 ft2

Note: 1 Darcy is equivalent to 9.869233� 10�13 m2 or about 10�12 m2 or 1 (μm)2

R. Weijermars, I. Nascentes Alves Journal of Petroleum Science and Engineering 165 (2018) 946–961

4.2. Flux allocation

A key step in the drainage model development is flux allocation toindividual fracture segments, which in our study is based on the historicproduction data and the related production forecasts (Fig. 4a and b). Thedrainage model applies the principle of flow reversal, putting producedfluid back in to the reservoir at the same rate as it was produced, toestablish where the fluid was drained from, quantifying the time-of-flightto the hydraulic fractures and its well. The flux allocation algorithm in-cludes a critical assumption about the payzone height, h, and the fracturelength, L:

qkðtÞ ¼ Cð1þWORÞqwellðtÞ

hkLkPkhkLk

�½ft3:day�1� (17)

with C¼ 5.61458333, a conversion factor accounting for stb/day inputunits of qwellðtÞ to output units of qkðtÞ in ft3/day. The water to oil ratio isgiven as WOR¼S_water/S_oil.

Eq. (17) essentially determines, based on fracture diagnostics or usingan assumed hydraulic fracture pattern, howmuch fluid was produced viaeach fracture segment. Due to the large number of perf clusters (Table 1),we upscaled the fractures in the flux allocation, reducing 5 fractures perstage to 1. This upscaling assumption requires justification. Using 1fracture instead of 5 potential fractures with their surfaces contacting thematrix rock, is dynamically similar to shortening the overall fracturelength by a factor 5. The effect is that reservoir pressure increasescorrespondingly, a scaling effect that will be explained and investigatedfurther based on our drainage simulations. In order to have a meaningfuldiscussion, the pressure scaling of our drainage model will be detailedfirst.

4.3. Scaling of the pressure field

The pressure change in any location z at a given time t may becalculated analytically using the following generic formula:

ΔPðz; tÞ ¼ �ϕðz; tÞμk

: ½psi� (18a)

where ϕðz; tÞ is the potential function, which is the real part of thecomplex potential given in Eq. (10a), μ the viscosity of the fluid

(expressed in psi.day, not Pa.s nor Poise), and k the permeability of the

reservoir (expressed in our model in ft2, not m2 nor Darcy). The genericform of Eq. (18a) will yield a specific pressure profile when a specificpotential function is defined (and μ and k are known). Note that ϕðz; tÞincludes the porosity as scaling factor for the well strength m(t)accounted for in expression (13). The actual pressure field at any giventime can be obtained using expression (18a) when the initial pressurestate P0 of the reservoir is known:

Pðz; tÞ ¼ P0 þ ΔPðz; tÞ ¼ P0 � ϕðz; tÞμk

: ½psi� (18b)

The pressure plots in this study were generated applying the so-calledflow reversal principle in our simulations: injecting the produced fluidback into the reservoir at the same rate as produced, and the effectivedrained region is visualized for a given time (highlighted by drainagecontours). For this reason all pressure plots show positive pressure, whichcan be translated to pressure depletion plots by taking the negative of thepressure scale. In our pressure computations, we express viscosity inpsi.day and permeability in ft, because the potential function used hasdimension of ft2/day; Table 4 gives the unit conversion.

4.4. Drainage model results

The algorithms of Section 3.3, with flux allocation of Section 4.2 andpressure field algorithms of Section 4.3 were used to simulate the

955

regional velocity field (Fig. 9a and b) and pressure field (Fig. 9c and d) inthe Eagle Ford reservoir. The regional velocity field (Fig. 9a) and corre-sponding pressure field (Fig. 9b) are shown for the 1st day of productionfor the first well pair (R and O) that started producing on April 1, 2014(Table 2). The second well pair (H and M) in our study was drilled,completed, fracture treated, and ready for first production 213 days afterthe first production of the earlier well pair. Fig. 9c and d shows thecorresponding velocity and pressure fields. The velocity fields at first dayof production for Well Pair 1 (day 1, April 1, 2014) and Well Pair 2 (day213, November 1, 2014) are given in Fig. 10a and b.

One may object that the pressure gradients shown in Fig. 9b,d are notrealistic, because the required depth of investigation implied by Fig. 5has not yet been established on the 1st day of production. However, WellR was fully drilled by 24 Dec 2013. The fracture treatment aimed for azipper frac with Well O and all stages of Well R were fractured by March6, 2014, after which flowback operations started. There would be at least24 days between completion of the last fracture stage and first produc-tion, which leaves enough time to establish the pressure gradient in thematrix space between the fractures.

A more nuanced analysis of the reservoir pressure depletion is war-ranted. For example, the pressure in the reservoir required to establishproduction in the model that matches the actual production, would be onthe order of 108 psi, while the actual reservoir pressure is know to beclose to 5� 103 psi. Assuming that the fluid viscosity is relatively wellconstrained (Table 1), there are three principal explanations for thediscrepancy between our model pressure and the actual reservoir pres-sure. One is capillary effects, which are not considered in our presentmodel. The other two factors that may raise the reservoir pressure arepermeability in Eq. (18) and fracture density, specified as the aggregatedfracture length in the denominator of Eq. (17).

The observed production rate of oil of viscosity 1 cP delivered by thereal well cannot be produced from a 200 nD matrix with an initialreservoir pressure of 5� 103 psi. The only way we could match both thewell productivity and the observed BHP (keeping oil viscosity at 1 cP), isto assume that the actual fracture density is larger and the equivalentpermeability much higher than assumed in our initial drainage model.For example, if the fracture density is 100 times denser than assumed inthe flux allocation of Eq. (17), the pressure would drop inversely by twoorders of magnitude. Additionally, if the matrix permeability used in Eq.(18) is enhanced, due to micro-fractures induced by the hydraulic frac-ture treatment, to an equivalent permeability of say 20–100 microDarcy,then our drainage model will generate realistic pressure ranges in theorder of 103 psi.

We emphasize that the observed trade-off between permeability,fracture density, and well productivity is not unique to our model, butwould occur in any reservoir model using the same parameters as we doin our current study. Perhaps an advantage of our CAM-based modelapproach is that every step in our analysis can be validated, which is whatlead us in the first place to critically evaluate the relationship betweenreservoir pressure, fracture density and well productivity. Assuming thatmicro-fractures occur and enhance reservoir permeability, the time toestablish the depth of investigation would also shorten dramatically(Fig. 5). The conclusion is that our drainage model can be confidentlyapplied because the pressure gradient is likely to be fully established inthe vicinity of the well pair when first production starts (day 1 of ourmodel) after about 1 month of flowback following the fracture treatmentcompletion.

Fig. 9. Left-hand images: Velocity field on day 1 (top) and day 213 (bottom). Color bar units in ft/day; length axes in ft. Right-hand images: Corresponding pressurefield at day 1 (top) and day 213 (bottom). Color bar units in psi, with P0¼ 4680 psi. (For interpretation of the references to colour in this figure legend, the reader isreferred to the Web version of this article.)

R. Weijermars, I. Nascentes Alves Journal of Petroleum Science and Engineering 165 (2018) 946–961

4.5. Drained reservoir sections and recovery factor

4.5.1. Velocity field versus pressure plotsThe velocity field is rarely visualized in commercial reservoir simu-

lators, which are commonly based on discrete element solution methods.Pressure contour plots are the more common way to show how pressurelows and highs vary across the reservoir (Fig. 9b,d). Such plots show theevolving pressure gradients as compared to the reference pressure beforeproduction started when any pressure gradients are negligible. At theonset of production, the steepest pressure gradients (Fig. 9b,d) occuralong the periphery of the fractured zone occupied by the respectivewellpairs. The steeper pressure gradient zones coincide with the higherflow velocities in the region peripheral to the frac tips (Fig. 9a,c). Higherresolution plots in Fig. 10a and b, show the flow in the central fracturezone is slowest due to relatively shallow pressure gradients. Contouredfor minor pressure changes, we see that pressure gradients still exist nearthe center of the fractured region (Fig. 9b,d), but the pressure differencesare much smaller in the center of the fractured region than in its moreperipheral zones, which explains why the flow in the central region be-tween the frac clusters is nearly stagnant (Fig. 10a and b). Arguably,velocity field plots show better than pressure plots where fluid is movingand, additionally, at what speed. Stagnation point positions depend onthe rate at which the individual fractures drain their adjacent matrixregion (Weijermars et al., 2017a,b).

4.5.2. Implications for production efficiencyWe used four Eagle Ford wells with historic production data to match

956

decline curves for each well using a Duong model (Fig. 4a and b), whichallows production forecasts over the 40 year well life. We have estimatedthe OOIP for the reservoir space, using Eq. (3) in Section 2.1, to be 193MMstb. For the lease area of 100 MMsqft or 2296 acres (Figs. 8 and 9c),the OOIP is 84 Mstb/acre. Our drainage model for the 2296 acre regionshows the recovery factor as it evolves over the well life (Fig. 11). Weconclude the recovery factor is less than 1% of the OOIP. Note that theregion occupied by the four wells in Fig. 9c leaves enough space for infillwells between the two existing (2014) well pairs, in which case the OOIPestimate used for the recovery factor in Fig. 11 would be reduced pro-portionally to the acreage drained by the infill wells. The infill drillingwould increase the recovery factors for the existing wells, purely due tothe downward adjustments of OOIP due to a lesser portion of totalacreage being allocated to the existing wells.

Integrating the total well rate decline over 40 years and allocatingproduction proportionally to each frac stage allowed us to visualize thewidth of the drained matrix region around the fracs (Fig. 12a and b).Evidently, the drained zones are quite narrow and leave large portions ofthe matrix between the fracs un-drained even after 40 years of produc-tion. The narrow region drained after such a long time corresponds with atotal cumulative production based on the type curve (Fig. 4b). InFig. 12ab, as follows from the 10 year TOF contour, fluid is drained up tojust 10 ft away from the fracture, but remember we used only 1 fractureper stage in our drainage model, as a proxy for the 5 fractures per stage inreality. Consequently, in reality just 2 ft would be drained by each frac-ture after 10 years of production. In the next 30 years, the matrix adjacentto such fracs is only drained an additional 1 ft deeper.

Fig. 10. a: Velocity field at day 1 near fracs in right part of earlier well pair (Rand O). b Velocity field at day 213 near fracs in right part of the later well pair(H and M). Color bar unit in ft/day, length axes in ft. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the Webversion of this article.)

Fig. 11. Cumulative produced by all four Eagle Ford wells after 10,000 days is1.1� 106 stb. The recovery factor increases over time, but stays below 1%of OOIP.

Fig. 12. a: Time-of-flight (TOF) contours toward the fracture spaced for 10years of drainage/production (red) for 3 outermost stages at the heel side ofReveille, after 40 years. b: Magnified portion of right most stage, showingbungling of TOF contours (axes in ft). (For interpretation of the references tocolour in this figure legend, the reader is referred to the Web version ofthis article.)

R. Weijermars, I. Nascentes Alves Journal of Petroleum Science and Engineering 165 (2018) 946–961

Our conclusion is that unless micro-fractures are present to draindeeper matrix regions between the fracs, large dead zones occur betweenthe principal frac zones. One implication would be that re-fracking of thewells between the original perf zones after about 2–3 years of productionmay contribute significantly to improve the recovery factors and thus willboost well economics accordingly.

5. Discussion and conclusions

We advocate the use of closed-form solution-methods to visualizeflow near hydraulic fractures at high resolution (Weijermars et al.,2017a,b; Van Harmelen andWeijermars, 2018). In the present article, we

957

translate our methodology into a practical workflow for the first time andapply our method using new field data to history match drainage patternsfor four adjacent horizontal wells in Brazos County, East Texas. Our highresolution visualization of flow in the reservoir reveals that most fluidmoves into the tips of the hydraulic fractures. The central region betweenthe fractures keeps hydrocarbons entrapped in stagnant flow zones (deadzones), which remain outside the reach of the hydraulic main fractures,even over the economic life of the typical well (30–40 years). Strandedoil may occur between the hydraulic fractures, but not at the tips. Armedwith such insight, operators may find recovery factors can still beimproved. We have shown that the Eagle Ford wells in our case studyregion recover, with the first generation of hydraulic fractures, less than1% of OOIP. Our models suggest that the long-term recovery factor of thewells may significantly improve by refracking existing wells with a sec-ond generation of hydraulic fractures, placed midway between the firstgeneration of fracs.

The conceptual paradigm shift proposed in this study boils down to akey question: “Does the flow system set up by the pressure step functionimposed by the creation of the well and associated hydraulic fractures evenknow there is anything beyond the SRV?” Our concept tries to avoidequating the SRV with the region where fluid is moving. Fluid movesbasically everywhere in the reservoir where the depth of investigationhas advanced, because the pressure gradient then is established in suchregions. The region penetrated by the depth of investigation is vast, andnot necessarily limited by an SRV rectangle defined by well spacing orother means.

Once a pressure gradient has been established, fluid in the reservoirtypically is moving everywhere but there is an important difference be-tween where fluid moves (drainage region) and what fluid actuallyreaches a well (drained region) after a given time. What our flow simu-lation method is capable of showing, in high resolution, is the subtlevelocity gradients which reveal fluid moves fastest peripheral to thefractures and a significant portion of the fluid moves in from the far-field.However, the flow is extremely slow and consequently the drained region(different from drainage region) remains extremely narrow, even overthe field life. The drained region grows over time and so does the re-covery factor. However, in unconventional plays, the rapid decline offlow velocities after the initial months, results in fluid, from farther awayfrom the well and its connected fractures, never reaching the well.

Analysis of production data for the Texas Eagle Ford since 2008(Fisher, 2016) suggests that the time-averaged (first three months) pro-duction per well has not significantly improved since 2010 and, in spiteof longer laterals and more fracture stages per well, hovers around 500

R. Weijermars, I. Nascentes Alves Journal of Petroleum Science and Engineering 165 (2018) 946–961

Boe/day. According to the latter study, only a small number of positiveoutliers, wells producing up to 3000 Boe/day, seems to have grown overtime. However, looking further than only the first three months of pro-duction, improvements in the total well length and closer fracturespacing will result in production sustained for longer. Our suggestedrefracking of the wells may further improve the recovery factor.

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Acknowledgements

Completion and production data for our field case were provided byHalcon Resources. Arnaud van Harmelen is thanked for running matlabto produce the plots of Figs. 9–12. This study was financially supportedby funds from the Texas A&M Engineering Station (TEES) and the Cris-man-Berg Hughes Consortium.

Appendix A. Transient radius of investigation

The objective of this appendix is to derive an expression for the transient flow of under-saturated oil in a radial reservoir, introducing the concept of thetransient radius of investigation.A1. Standard solution diffusivity equation

The general equation of the diffusivity, that describes the pressure profile in a cylindrical reservoir with an infinite-acting pressure gradient source inthe center filled with a slightly compressible fluid of constant viscosity (Carslaw and Jaeger, 1959), can be rewritten as:

1r∂∂r

r∂p∂r

�¼ ∂2p

∂r2 þ1r∂p∂r ¼

nμctk

∂p∂t ½Pa m�2� (A1)

A valid solution of the diffusivity equation, for any time, t, at any radial position, r, in the reservoir is:

pðr; tÞ ¼ pi þ qμ4πkh

Eið�xÞ ½Pa� (A2)

with the inverse of dimensionless time x ¼ nμct r24kt .

The exponential integral function Ei(x) is defined as:

EiðxÞ ¼ �Eið�xÞ ¼ ∫ ∞x

e�u

u

�du ½ � � (A3)

So the solution of the diffusivity equation (A1) can be rewritten as:

pðr; tÞ ¼ pi � qμ4πkh

EiðxÞ ⇒pðr; tÞ ¼ pi � qμ4πkh

∫ ∞x

e�u

u

�du ½Pa� (A4)

There is no analytical solution for the exponential integral Ei(x), so a numerical integration is required (or use of tables). For certain ranges of theargument x, approximations can be used such as for x> 10¼>Ei(x) is close to zero. Another important approximation was proposed by Abramowitz andStegun (1964), for longer dimensional times, limited by inverse dimensionless time x< 0.025, the exponential integral function of Eq. (A3) can beapproximated, with an error less than 1%, by:

Eið�xÞ � lnðγxÞ ⇒EiðxÞ � �lnðγxÞ ¼ ln 1γx

�½ � � (A5)

Note that γ ¼ e0:57722… ¼ 1:78108:::, where 0.57722 … is the Euler constant. The solution of the diffusivity Eq. (A2) for x< 0.025 can now besimplified to:

pðr; tÞ ¼ pi � qμ4πkh

EiðxÞ ⇒pðr; tÞ ¼ pi � qμ4πkh

ln 1γx

�½Pa� (A6)

Eq. (A6) can be applied to obtain the pressure for any lateral depth of investigation smaller than the bounding radius, re, at any time t after the onsetof production, provided that x< 0.025. The condition of x< 0.025 is fulfilled in conventional reservoirs just after a few minutes of production.However, in unconventional reservoirs, with well spacing in order of 1000 ft (re), and permeability in the order of 100 micro Darcy to 100 nanoDarcy,the critical radius of investigation may still takes months or years to establish.

One of the most practical locations to determine the pressure transient in well tests is at wellbore where r¼ rw and p(r,t)¼ pwf. For this specific radiusthe requirement of x< 0.025 will be fulfilled just after a few minutes of production and the expression for pwf can be simplified into:

pwf ðtÞ ¼ pi � qμ4πkh

ln

4ktγnμctr2w

�½Pa� (A7)

Reorganizing Expression (A7) in terms of the flow rate yields:

qðtÞ ¼ 4πkhμ

�pi � pwf

�ln�

4ktγnμct r2w

� ½m3:s�1� (A8)

The standard solutions for the pressure transient and corresponding productivity are given by Eqs. (A7) and (A8).

R. Weijermars, I. Nascentes Alves Journal of Petroleum Science and Engineering 165 (2018) 946–961

A2. Alternative formulation for transient solution diffusivity equation

Next instead of using the standard expression for the transient solution (Eq. (A8)), we apply a different approach introducing the transient radius ofinvestigation, defined as the radius up to the point where the pressure wave still did not disturb the reservoir. The region outside the transient radius, ri,has a reservoir pressure that is still equal to the original reservoir pressure. During transient flow, the transient radius of investigation is given by:

r2i;t ¼4ktγnμct

½m� (A9)

Inserting Eq. (A9) into Eq. (A8) gives:

qðtÞ ¼ 4πkhμ

�pi � pwf

�ln�r2i;tr2w

� ¼ 4πkhμ

�pi � pwf

�2 ln

�ri;trw

� ½m3:s�1� (A10)

Rewriting the equation in terms of field units: (q - bbl/d, K - mD, h – ft, p – psi, μ – cp and r-ft), and (t-h and ct – 1/psi), also introducing the concept ofskin, gives:

qðtÞ ¼ kh141:2μ

�pi � pwf

�ln�ri;trw

�þ s

½bbl=d� (A11)

r2i ¼ 2:637� 10�44γ

ktnμct

or r2i ¼kt

1688:7 nμct½ft2� (A12)

Eq. (A11) applies to transient flow, but appears very similar to the standard steady state solution, given by:

qðtÞ ¼ kh141:2μ

�pe;t � pwf

�ln�

rerw

�þ s

½bbl=d� (A13)

The physical difference between Eqs. (A11) and (A13) is that the initial reservoir pressure (transient) at ri in Eq. (A11) is replaced by the (now time-dependent) reservoir pressure at the boundary (steady state), re. We note that different conversion factors are used among different authors, as reviewedby Kuchuk (2009), which is partly attributed to the inclusion of empirical measurements. However, we prefer to use the theoretical relationship for thetransient solution give in Eqs. (A11) and (A12).

A3. Comparison with pseudo-steady state solution diffusivity equation

Our new, alternative solution for the transient solution of the diffusivity equation in Eq. (A10) can also be shown to be nearly similar to the standard

pseudo-steady state solution using the average reservoir pressure, given by:

qðtÞ ¼ 2πkhμ

�pðtÞ � pwf

�hln�rerw

�� 3

4 þ si ½m3:s�1� (A14)

Rearranging the radius term in the denominator of Eq. (A10), multiplying and dividing by e3/2, transforms to:

ln r2ir2w

e3=2

e3=2

�¼ 2

�ln rie3=4

rw

�� 34

�⇒ ri pss ¼ rie3=4 ½m� (A15)

so that the transient flow solution of Eq. (A10) can be rewritten as:

qðtÞ ¼ 4πkhμ

�pi � pwf

�2hln�ri;t pss

rw

�� 3

4

i ½m3:s�1� (A16)

Adjusting Eq. (A16) with a conversion factor to account for the use of field units: (q - bbl/d, K - mD, h – ft, p – psi, μ – cp, r-ft, t-h, ct – 1/psi), alsointroducing the concept of skin, s:

qðtÞ ¼ kh141:2μ

�pi � pwf

�hln�ri;t pss

rw

�� 3

4 þ si ½bbl=d� (A17)

r2i;t pss ¼ r2i e3=2 ¼ kt

1688:7 nμct⋅e3=2 ⇒ r2i;t pss ¼

kt376:8 nμct

½ft2� (A18)

Eq. (A17) applies to transient flow but is very similar to the pseudo-steady state expression given by:

qðtÞ ¼ kh141:2μ

�pðtÞ � pwf

�hln�

rerw

�� 3

4 þ si ½bbl=d� (A19)

959

R. Weijermars, I. Nascentes Alves Journal of Petroleum Science and Engineering 165 (2018) 946–961

The physical difference between Eqs. (A17) and (A19) is that the initial reservoir pressure (transient) of Eq. (A17) is substituted by the averagereservoir pressure (pseudo steady state) in Eq. (A19), using re instead of ri,t pss.Nomenclature

p Pressurepi Initial reservoir pressurepwf Wellbore flowing pressurer Reservoir radiusrw Wellbore radiusre Reservoir boundary radiusri Investigation radius, steady stateri pss Investigation radius, pseudo steady statet Timen Porosityμ Viscosityct Total compressibility of the system (rock and fluids)q Flow Ratek Permeabilityh Reservoir height (net paid)s Skin factorx Grouping variables parameteru Substitute variable for integrationγ Constant equal to e0.57722, where 0.57722 is the Euler constant

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