Enhanced-Oil-Recovery Potential for Lean-Gas Reinjection ...download.xuebalib.com/15jaLRImVs1z.pdf · lDx a 6 mDy b 6 nDz h cos p 2 lx m a 6 my b 6 nz m h lDx a 6 mDy b 6 nDz h; ð8Þ

Enhanced-Oil-Recovery Potentialfor Lean-Gas Reinjection in Zipper

Fractures in Liquid-Rich BasinsOluwanifemi Akinluyi and Randy Hazlett, University of Tulsa

Summary

Production from liquid-rich shale has become an important contributor to US production, but recovery factors are low. Enhanced-oil-recovery (EOR) methods require injectivity and interwell communication on reasonable time scales. Herein, we investigate the develop-ment of fracture interference for the application of recycled-lean-gas injection to displace reservoir fluids between zipper fractures inliquid-rich shales. In condensate systems, the liquids produced from miscible displacement could be extracted at the surface and the gasreinjected. In unconventional oil systems, immiscible displacement would occur with arrest in the oil-rate decline upon the onset ofpressure support until immiscible front breakthrough, although this may never occur in a reasonable time. In either case, the time forinterference is critical in assessment of process feasibility.

Using superposition plus existing analytical solutions to the diffusivity equation for arbitrarily oriented line sources/sinks for pres-sure and new extensions for the pressure logarithmic temporal derivative, we analyze the time for interfracture-communicationdevelopment (i.e., interference) and productivity index (PI) for both classical biwing fractures in a zipper configuration and complex-fracture networks. As a novel contribution, we demonstrate the ability to map both pressure and pressure temporal derivative as a func-tion of time and space for production and/or injection from parallel motherbores under the infinite-conductivity wellbore and fractureassumption. The infinite-conductivity assumption could be relaxed later for more-general cases.

We present the results in terms of geometrical-spacing requirement for both horizontal wells and stimulation treatments to achievereasonable time frames for interfracture communication and sweep for parameters typical of various shale plays. Results can be used todetermine whether spacing currently considered for primary production is sufficient for direct implementation of EOR or if currentpractice should be modified with EOR in the field-development plan.

Introduction

The use of zipper fractures has recently become very prominent. This method has been known to increase production and save timewhen drilling multiple wells. According to Jacobs (2014), zipper fractures are increasing initial production and estimated ultimate-recovery rates. As documented by Sharma (Jacobs 2014), operators in south Texas have reported improved initial production rates rang-ing from 20 to 40% by use of the zipper-fracture method. Previous work was performed to compare and evaluate the performance ofzipper fractures and the modified-zipper-fracture arrangement. The zipper fractures do not overlap, but the modified-zipper fractures dooverlap. Sierra and Mayerhofer (2014) noted that overlapping zipper fractures have an incremental recovery factor in the range of 15 to20%.

In this paper, we study the performance of various zipper-fracture configurations with respect to PI and development of fracture in-terference. Unlike the purely production scenario, interwell communication is essential for concurrent injection and productionschemes, as depicted in Fig. 1 with a highlighted, idealized, biwing-fracture-repeating unit in context with alternating motherbores forinjection and production. In modeling, we assume cased-hole completions, so there is no production from the motherbore exceptthrough fractures. In the uniform-pressure (infinite-conductivity) assumption, fluid reaching the fracture is effectively removed with nopressure drop in the fractures or motherbore. The EOR methodology under consideration is the injection of lean gas in liquid-richunconventional reservoirs with recovery of liquids and gas recycling. No phase-behavior considerations are included in this current pro-cess-feasibility study. Knowing the time required for communication between these fractures will help evaluate the performance andalso the minimum-space requirements, with particular interest if current spacing for production is adequate for future EOR. The radiusof investigation helps in the determination of the drainage area, which can be related to the time. According to Datta-Gupta et al.(2011), “The concept of radius of investigation is fundamental to well test analysis and is routinely used to design well tests. … The ra-dius of investigation can also be useful in identifying new well locations, planning, designing and optimizing hydraulic fractures inunconventional wells.” Van Poolen (1964) presented the radius-of-investigation formula as

ri ¼ 0:029

ffiffiffiffiffiffiffiffiffiffikt

/lct

s: ð1Þ

Barree et al. (2014) noted that taking the affected drainage area at 5 years is a good first guess at the effective drainage area for a sin-gle fracture that can be used to forecast production from a composite horizontal well with multiple transverse fractures, but our solutionin this paper assumes no flow regime or location of first interference. The radius-of-investigation method gives only a proxy for theeffective drainage area in radial flow, and as such will not be used in our calculations and results.

Approach

Uniform-Flux Solutions. In Hazlett and Babu (2014), the time-dependent solution to the diffusivity equation for a uniform-flux linesink at arbitrary orientation in a rectangular box of dimension (a, b, h) through spatial integration of a point source along a parameter-ized vector is given as

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Copyright VC 2017 Society of Petroleum Engineers

This paper (SPE 179577) was accepted for presentation at the SPE Improved Oil Recovery Conference, Tulsa, 11–13 April 2016, and revised for publication. Original manuscript received forreview 14 November 2016. Revised manuscript received for review 11 July 2017. Paper peer approved 13 July 2017.

J179577 DOI: 10.2118/179577-PA Date: 9-October-17 Stage: Page: 1 Total Pages: 15

ID: jaganm Time: 15:43 I Path: S:/J###/Vol00000/170084/Comp/APPFile/SA-J###170084

2017 SPE Journal 1

PDðx; y; z; x1; y1; z1; a; b; c; L; tÞ � t

/lCt

� �þ 1

L�ðL0

X1l;m;n 6¼0

Clmn

p2

� cosplx

a

� �cos

pmy

b

� �cos

pnz

h

� �cos

plxo

a

� �cos

pmyo

b

� �cos

pnzo

h

� �D2

lmn

ds

� 1

L�ðL0

X1l;m;n 6¼0

Clmn

p2

E � cosplx

a

� �cos

pmy

b

� �cos

pnz

h

� �cos

plxo

a

� �cos

pmyo

b

� �cos

pnzo

h

� �D2

lmn

ds

E � exp�p2D

2

lmnt

/lCt

!; D2

lmn �kxl2

a2þ kym2

b2þ kzn

2

h2

� �; � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ð2Þ

with PD defined as the dimensionless integrated point-source solution:

PD �1

L�ðL0

Pi � P½x; y; z; xoðsÞ; yoðsÞ; zoðsÞ; t�qBl=ðabhÞ ds: ð3Þ

The parameterization in s follows the well trajectory by use of direction cosines a, b, and c.

½xoðsÞ; yoðsÞ; zoðsÞ� � ðx1 þ as; y1 þ bs; z1 þ csÞ: ð4Þ

The factor Clmn takes on values 2, 4, and 8 depending on the dimensionality of the infinite series. The first term represents materialbalance, the second gives the pseudosteady-state pressure at observation point (x, y, z), and the third term contains the transient decaycontribution. The point source at (xo, yo, zo) is integrated with respect to the parametric variable, s, starting from (x1, y1, z1) along thevector with direction cosines (a, b, c) and magnitude L to the endpoint (x2, y2, z2). Anisotropy is allowed through the directionallydependent permeability (kx, ky, kz). Furthermore, we introduced dimensionless time as the product of two dimensionless groups.

tD � tD �kx

a2

� �¼ t

/lCt

� �� kx

a2

� �: ð5Þ

As stressed by Hazlett and Babu (2014), the mathematical complexity comes in the pseudosteady-state-term evaluation because sin-gularities are in space and not time. Considerable effort must be taken to recast the slow convergent-series summation represented inEq. 2 into a computationally tractable form. Analogously, McCann et al. (2001) gave readily computable functional forms for a pointsource in a two-dimensional (2D) rectangle with Dirichlet or Neumann external-boundary conditions. The transient term in Eq. 2 hasthe benefit of exponential damping and can be more readily evaluated. Hazlett and Babu (2014) further demonstrated that the solutionto strictly 2D problems, such as in the case of fully penetrating vertical fractures, is contained within the more-general result.

Performing the spatial integration and taking the time derivative of Eq. 2, we obtain

dPD

dtD¼ 1� 1

2p

X1l;m;n 6¼0

Clmnexpð�p2D2

lmntDÞ � cosplx

a

� �cos

pmy

b

� �cos

pnz

h

� �� kðx1; y1; z1; x2; y2; z2; l;m; n; a; b; hÞ; ð6Þ

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

Low pressure

High pressure

Repeatingunit

Fig. 1—Communication between two long horizontal wells with repeating zipper fractures that overlap.



2 2017 SPE Journal

and the popularized logarithmic time derivative, as an indicator of flow regime, would then be

dPD

dlntD¼ tD �

dPD

dtD¼ tD � 1� 1

2p

X1l;m;n6¼0

k � Clmnexp �p2D2

lmntD

� �� cos

plx

a

� �cos

pmy

b

� �cos

pnz

h

� �" #; ð7Þ

where k represents the sum of four different terms over 6 signs [(þm,þ n), (þm, –n), (–m,þ n), (–m, –n)] in compact form as

k �X4

6

sinp2

lDx

a6

mDy

b6

nDz

h

� �cos

p2

lxm

a6

mym

b6

nzm

h

� �lDx

a6

mDy

b6

nDz

h

� � ; ð8Þ

with Dx ¼ x2 � x1; Dy ¼ y2 � y1; Dz ¼ z2 � z1; xm ¼ ðx2 þ x1Þ=2; ym ¼ ðy2 þ y1Þ=2; zm ¼ ðz2 þ z1Þ=2. This form is obtained by apply-ing standard trigonometric identities to simplify the expanded form containing endpoints. Note that with Eqs. 2 through 8 good for arbi-trary observation points (x, y, z), we are in the position to map pressure and pressure temporal derivative values across the solutiondomain, giving new insight into localized flow-regime behavior, not just that which is observable at a well.

Uniform-Pressure Solutions. Because the line-source Neumann function is produced through spatial integration of a unit-strengthpoint source, the final result is that with a uniform-flux internal-boundary condition. The pressure will then vary down the length of thefracture. A uniform-pressure internal-boundary condition can be recovered by segmenting the trajectory into multiple uniform-flux seg-ments with adjustable strength to yield a common pressure at control points, as done by Gringarten et al. (1974) and others. Hazlett andBabu (2014) demonstrated the ability to compute so-called infinite-conductivity, constant-rate solutions for 2D complex-fracture pat-terns interpreted from microseismic. Acuna (2016) gave an analytical solution for complex-fracture networks that contains differentflow regimes for interpretation of pressure and flow-rate behavior. Kuchuk et al. (2015) also used spatial finite- and infinite-conductivityfractures with arbitrary length and orientation to show different flow regimes exhibited by a horizontal well with multiple fractures.

Here, we make use of the solutions of Hazlett and Babu (2014) reduced to two dimensions for the time-dependent behavior or frac-ture systems, primarily with the uniform-pressure inner-boundary condition. Because the analytic solution is available for any observa-tion point, we make dense selection of such points to produce pressure and pressure-derivative maps to monitor the time evolution offracture interference in the interwell region. Kuchuk et al. (2016) used flow rate and pressure to transform production data to pressureand pressure derivatives. That paper also noted the importance to unconventional gas and oil reservoirs because the boundary-domi-nated pseudosteady-state flow regime takes years to develop. We furthermore generalize our result to include interacting injection/production-fracture sets to examine EOR potential. The long-time pressure behavior would reach a true steady state in these balanced-injection/production systems. Thus, the characteristic slope of the temporal pressure derivative would decay to zero rather than achievea value of unity.

For the pseudosteady-state term of Eq. 2, we use the dimensionally reduced form given in Hazlett and Babu (2014) as follows:

PD ¼ab2

p2Lða2ky þ b2kxÞ

2a � Hðx� x1Þ � Hðx2 � xÞ �X1m¼1

1

m2cos

pmy

b

� �cosp

m

by1 þ

baðx� x1Þ

� �

þX4

j¼1

X1m¼1

1

m2

� �Ej x; x2; mð Þcos

pmy

b

� � b

ffiffiffiffiffikx

ky

s� sinp

my2

b

� �

þ a � kj x; x2ð Þ � cospmy2

b

� �26664

37775

�X4

j¼1

X1m¼1

1

m2

� �Ej x; x1; mð Þcos

pmy

b

� � b

ffiffiffiffiffikx

ky

s� sinp

my1

b

� �

þ a � kj x; x1ð Þ � cospmy1

b

� �26664

37775

0BBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCA

þ a3

p3aLkx

� �F3

x2 þ x

a

� �þ F3

x2 � x

a

� �� F3

x1 þ x

a

� �� F3

x1 � x

a

� �h i; � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ð9Þ

with

E1ðx; xi; mÞ �e�pðxi þ xÞ

ffiffiffiffiffiffiffiffiffiffikym2

kxb2

sffiffiffiffiffiffiffiffiffiffikym2

kxb2

s ; ð10Þ

E2ðx; xi; mÞ �e�p½2a�ðxi þ xÞ�


kxb2


kxb2

s ; ð11Þ

E3ðx; xi; mÞ �e�pjxi � xj


kxb2


kxb2

s ; ð12Þ

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



2017 SPE Journal 3

E4ðx; xi; m; nÞ �e�pð2a�jxi � xjÞ


kxb2


kxb2

s ; ð13Þ

and

k1;2;3;4 ¼ ½�1; 1;�signðx� xiÞ; signðxi � xÞ�; i ¼ 1; 2; ð14Þ

where

F3ðzÞ �X1n¼1

sinðpnzÞn3

¼ p3 � z� 1

6� jzj

41� jzj

3

� �� ; 0 � jzj � 2: ð15Þ

For steady-state balanced injection and production, the leading term in Eq. 2 is dropped.For uniform pressure, we segment the fractures in a geometric progression from the midpoint outward to capture the contribution

from the tip of each fracture without the need to introduce so many segments. A separate systematic study indicated that more than 200segments were required before the solution no longer depended on the number of segments for equal segment lengths. With geometricprogression, we can achieve equivalent accuracy with only 5 to 10 segments on either side of the midpoint. By use of superposition, thesegment contributions to the pressure at any observation point are additive. For a system with N segments,

PDðx; y; z; tDÞ ¼XN

j

ljPDjðx; y; z; x1; y1jz1j; aj; bj; cj; tDÞ; ð16Þ

where lj¼ qj/qref are the source strengths related through a common reference flow rate and are subject to overall material balance.Note that contributions computed in Eqs. 2, 6, and 7 are for unit-source strength, and mulitpliers to such terms allow unequal contribu-tions in multiple source/sink formulations. In the uniform-flux inner-boundary condition, source strength is proportional to fracturelength. For the uniform-pressure inner-boundary condition, the initial guess is uniform flux, and flux strength is subsequently adjustedto yield the same pressure at selected control points. Here, the control points are chosen to be fracture-segment midpoints. Note that thePDj values need to be computed only once, enabling rapid updating of flux distribution for the uniform-pressure inner-boundary condi-tion. Because we adapt the development of Hazlett and Babu (2014) for injection and production, readers are referred to that work forvalidation exercises.

Results and Discussion

Production From Zipper Fractures. For illustration purposes and later comparison with pressure-maintenance cases, a single zip-per-fracture configuration is analyzed with respect to the previously described pressure and pressure derivative. The particular geome-try is a 2:1 length ratio in the zipper-representative repeating element. Thus, the spacing between fractures belonging to a singlemotherbore is half the horizontal-well spacing. In this case, the fracture length is 75% of the interwell spacing. This geometry isdepicted in the inset diagram of Fig. 2, which also gives plots of the pressure, pressure derivative, and the slope of the logarithmic pres-sure derivative in log-log space. One can then simply examine the plot of slope history to identify changes in the flow pattern betweenlinear flow and pseudosteady-state behavior. No wellbore-storage terms are presented in this model. We assume the horizontal wellsare cased and perforated for stimulation stages; thus, there is no contribution to inflow by the wellbore itself. Horizontal wells are sim-ply modeled as infinite-conductivity gathering systems for infinite-conductivity fractures. As such, all fractures connected to a wellmust have the same pressure. The uniform-pressure approximation is accomplished throughout a common fracture system throughadjustment of individual strengths of uniform-fracture-segment fluxes to yield the same pressure value at selected control points, cho-sen here as fracture-segment midpoints. In the repeating element for multiple-fractured horizontal wells, the border fractures are givenone-half the strength of the internal fracture because boundary fractures will gather one-half of their fluid allotment from the neighbor-ing cell.

With the industry movement toward more stages, we include an aspect ratio more in line with current practice (Fig. 3). Although thelarge-aspect-ratio problem indicates an even-more-encouraging early communication in overlapping zipper fractures, it also shows newand different behavior regarding flow regimes. We see a steep rise and then a fall in the slope of the temporal logarithmic derivativebefore settling into a steady state. The overlapping region quickly becomes a zone reaching a local steady state, while the interplaybetween radial flow at the injector toe and linear flow at the producer heel gives a complex evolving flow-pattern and pressure-transientbehavior. A third case without the overlapping region displays a similar behavior regarding this time domain, with shifted transitionaway from linear flow because of the alternative geometry (Fig. 4). Our conclusions are dependent on the time to departure from linearflow as an indicator of overlapping-fracture interference. The severe aspect-ratio problems are not likely to produce the ideal-biwing-fracture configuration, resulting in potential direct communication between injector and producer. This situation would result in very-fast communication between injector and producer that is not necessarily catastrophic. It would, however, result in increased cycling oflean gas. Fig. 5 shows centerline pressure-derivative traces normal to the fractures with the intent of examining the interfacture-spacingresponse and the time to fracture interference. Note we are not using an ad hoc square-root-of-time estimator, but an actual solution tothe diffusivity equation; the mathematical solution makes no assumption on flow regime. In fact, the enforcement of uniform pressureyields a result in which the fracture tip and internal fracture segments both contribute toward the observable pressure. Thus, we canobserve changes in flow regime as they naturally evolve from the governing equation for this 2D problem. Fig. 5 indicates the onset offracture interference in the neighborhood of tD¼ 0.001.

Rather than monitoring only a select number of points, as in Fig. 5, access to a rapidly computed, analytic solution valid for all ob-servation points affords the opportunity to construct dense maps of pressure and pressure temporal derivative. These images for a set ofdifferent tD values are presented in Fig. 6. Here, the development of fracture interference is visually apparent and coincides with the

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



4 2017 SPE Journal

initial deviation in slope in Fig. 2. Flow-field information is available in pressure spatial information even when flow-regime signatureis lost in transient response.

Balanced-Injection/Production Cases. For balanced-injection/production EOR-feasibility cases, material balance forces a no-accumulation relationship on the source strengths.

XN

j

lj ¼ 0: ð17Þ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.000010.000001

0.00001

0.0001

0.001

0.01

0.1

0.0001 0.001 0.01 0.1 1 100

0.25

0.5

0.75

1

1.25

Dimensionless Time (tD)

Dim

ensi

onle

ss P

ress

ure

or P

ress

ure

Der

ivat

ive

Slo

pe o

f the

Log

arith

mic

Pre

ssur

e D

eriv

ativ

e

1

0.75

0.075

PD

m

dPD /dlntD

Fig. 3—Pressure and pressure-derivative response for production from a repeating unit in a zipper-fracture configuration asshown with 75% penetration and well-spacing/fracture-spacing ratio of 1:0.075. Parallel motherbores would be on the left andright. Fractures are assumed to be infinite conductivity and fully vertically penetrating. The slope of the derivative curve is pre-sented separately, indicating early linear flow, the onset of interference, a complex-flow regime with linear flow in the heel interact-ing with radial flow from the toe, and the transition to pseudosteady-state flow.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0001

0.001

0.01

0.1

1

10

0.00001 0.0001 0.001 0.01 0.1 1 10

Slo

pe o

f the

Log

arith

mic

Pre

ssur

e D

eriv

ativ

e

Dim

ensi

onle

ss P

ress

ure

or P

ress

ure

Der

ivat

ive

1

1/20.75

m

dPD /dlntD


PD

Fig. 2—Pressure and pressure-derivative response for production from a repeating unit in a zipper-fracture configuration asshown with 75% penetration and well-spacing/fracture-spacing ratio of 2. Upper and lower fractures are one-half the strength ofthe intermediate fracture. Parallel motherbores would be on the left and right. Fractures are assumed to be infinite conductivityand fully vertically penetrating. The slope of the derivative curve is presented separately, indicating early linear flow, the onset ofinterference, and the transition to pseudosteady-state flow.



2017 SPE Journal 5

To accommodate uniform pressure in injection and production, the fracture sets must be subdivided into classes with flux adjustmenton each class to yield independent common values of pressure at segment control points. A series of case studies was conducted onrepeating units of ideal zipper configurations, as illustrated in Fig. 1, to gauge the effect of geometry and degree of partial penetration.Each boundary fracture provided one-half of the produced amount. Because these repeating-unit boundary fractures are modeled with auniform-pressure inner-boundary condition and share the same motherbore, the fractures, although spatially separate, must be at thesame pressure. We assign both injectors as belonging to the same fracture set. To facilitate uniform-pressure calculations, the three indi-vidual half-wing fractures were segmented into 10 contributing elements by use of a geometric progression of 2 in length, therebyappropriately capturing the tip contribution with a minimal number of elements. In Table 1, the average dimensionless pressures andproductivity indices are provided for the two fracture classes: injectors and producers. Because we compute the dimensionless pressuredrop per unit of fluid withdrawn, the dimensionless PI is simply the negative inverse of PD. We choose to monitor PD and changes inslope of the pressure response to unit production instead of tracking PI caused by the connection to classical interpretation methods,realizing that similar trends could also be found in PI and its various derivatives. This matrix of geometries shows that resulting averagepressures in both classes reach equal magnitude but opposite signs. It should be noted that if the dimensionless pressure per unit volumeof fluid withdrawn per unit thickness is divided by the drainage area, the fully penetrating cases (L/a¼ 1) all reduce to the same one-

0.000010.00001

0.0001

0.001

0.01

0.1

1

0.0001 0.001 0.01 0.1 1 100

0.25

0.5

0.75

1

1.25


Dim

ensi

onle

ss P

ress

ure

or P

ress

ure

Der

ivat

ive

Slo

pe o

f the

Log

arith

mic

Pre

ssur

e D

eriv

ativ

e

1

0.25

0.075

dPD /dlntD

Fig. 4—Pressure and pressure-derivative response for production from a repeating unit in a zipper-fracture configuration asshown with 25% penetration and well-spacing/fracture-spacing ratio of 1:0.075. Parallel motherbores would be on the left andright. Fractures are assumed to be infinite conductivity and fully vertically penetrating. The slope of the derivative curve is pre-sented separately, indicating an extended early-linear-flow period, partial radial growth from a sector, and the transition to pseu-dosteady-state flow after interference between injector and producer tips.

0.00

–5

–10

–15

–20

–25

–30

0.1 0.2

0.00001

0.0001

L = 0.75 a = 1

b = 1/2

0.001

0.01

0.1

0.3 0.4 0.5 0.6

Distance y/b

Pes

sure

Der

ivat

ive

dpD

/t D

0.7 0.8 0.9 1.0–0.8

–0.9

–1

–1.1

–1.2

–1.3

–1.4

Fig. 5—Pressure-derivative profiles for the centerline trace shown in the inset diagram for production from a repeating unit in a zip-per-fracture configuration as shown with 75% penetration. Note the profiles for the lowest three tD values use the left axis, whereasprofiles for tD values of 0.01 and 0.1 use the secondary y-axis. These figures show interference in the interfracture space startingnear tD 5 0.001, and derivative values at the centerline rise above the pseudosteady-state magnitude before settling to a steadyvalue of –1 in the neighborhood of tD 5 0.1.



6 2017 SPE Journal

dimensional (1D) flow problem. Fig. 7 is the equivalent production with balanced-injection version of Fig. 2. It is clearly seen and mod-eled that the long-time slope of the pressure temporal logarithmic derivative approaches zero for the balanced-injection/productioncase, as opposed to unity for pseudosteady state. The plot of slope indicates development of interference, again in the neighborhood oftD¼ 0.001. A representation of the evolving pressure and pressure derivative as a function of dimensionless time is given in Fig. 8.Again, we see that the pressure and derivative maps are complementary in that as information fades in one with time, it becomes moreapparent in the other. The same time progression for the 50%-penetration case is shown in Fig. 9. Here, we observe an interesting rever-sal of linear-flow direction between early- and late-time flow regimes and interaction in the tip region although there is no overlap.Fig. 10 shows steady-state pressure maps for b/a¼ 1 and various fracture lengths for the representative unit cell. Comparable mappingsfor b/a¼ 0.5 and 0.25 are given in Figs. 11 and 12, respectively. Note that in each geometry, a shift is observed in pseudosteady-state-flow regime from linear flow horizontally for short fractures to linear flow vertically for long fractures. Intermediate fracture lengthswill have much-more-complicated flow patterns in long-time flow behavior. Fig. 13 shows a direct comparison of steady-state pressuremaps under the uniform-flux and uniform-pressure inner-boundary conditions. Although the physics and flow fields are different, mate-rial-balance considerations require the average pressure along the fracture in uniform flux to be the same as that determined throughsegmentation and flux adjustment for the uniform-pressure inner-boundary condition.

P P ′tD = 10–5

10–1

10–2

10–3

10–4

Fig. 6—Dense pressure and pressure-derivative maps for production from a repeating unit in a zipper-fracture configuration withb 5 a/2 and 75% penetration. Note that the pressure derivative is insightful for flow-field structure at early times, whereas late-timeinformation is lost. The pressure field retains flow-regime information through spatial information even when no longer evident intransient signatures.



2017 SPE Journal 7

To cascade these results to application, we must look at typical values of variables used in the dimensionless normalization processfor unconventional reservoirs. Fig. 14 gives the previous dimensioned result for the zipper-fraction configuration where b/a¼ 0.5 with75% penetration by fractures with typical parameters for oil and gas systems cited by Barree et al. (2014) and given in Table 2. In Fig.14, we see departure from linear flow in 5 years or fewer for both oil and gas systems if medium effective permeability is 10�4 md orgreater. Fig. 15 also gives a similar result for the zipper-fraction configuration where b/a¼ 0.075. With this tighter spacing, fracture in-terference is developed in fewer than 5 years for the lowest permeability modeled (10�5 md). We find general agreement with conclu-sions of Barree et al. (2014) on the onset of interference and optimal spacing. Assuming that a reasonable time frame for developingfracture/fracture communication should be fewer than 5 years, we conclude that the process might be feasible for microdarcy material,but not with nanodarcy effective matrix permeability. Similar plots could be constructed for other b/a ratios indicative of relative frac-ture/well spacing, noting that the distance between interacting fractures in the zipper configuration is actually b/2.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.0001

0.001

0.01

0.1

0.00001 0.0001 0.001 0.01 0.1 1 10

Slo

pe o

f the

Log

arith

mic

Pre

ssur

e D

eriv

ativ

e

Dim

ensi

onle

ss P

ress

ure

or P

ress

ure

Der

ivat

ive

Dimensionless Time tD

1

1/20.75

m

PD

dPD /dlntD

Fig. 7—Pressure and pressure-derivative response for injection and production from a repeating unit in a zipper-fracture configu-ration as shown with 75% penetration. Upper and lower fractures are one-half the strength and of opposite sign with respect to theintermediate fracture. Parallel motherbores would be on the left and right. Fractures are assumed to be infinite conductivity andfully vertically penetrating. The slope of the derivative curve is presented separately indicating early linear flow, the onset of inter-ference, and the transition to steady-state flow.

Drainage-AreaAspect Ratio

(b/a)Fracture Length

(L/a)

DimensionlessPressure at

Injector (PD1)

DimensionlessPressure at

Producer (PD2)

DimensionlessPI (Injectivity)

(PI1)Dimensionless

PI (PI2) PD2 (ab/a2)

1 0.25 –0.5487 0.5487 –1.822 1.822 0.5491 0.5 –0.2709 0.2709 –3.691 3.691 0.2711 0.75 –0.1589 0.1528 –6.293 6.546 0.1531 1 –0.1250 0.1250 –8.002 8.002 0.125

0.5 0.25 –0.7353 0.7354 –1.360 1.360 1.4710.5 0.5 –0.2517 0.2517 –3.973 3.973 0.5030.5 0.75 –0.0889 0.0889 –11.25 11.24 0.1780.5 1 –0.0625 0.0623 –16.00 16.00 0.125

0.25 0.25 –1.2223 1.2225 –0.818 0.818 4.8900.25 0.5 –0.2516 0.2517 –3.974 3.973 1.0070.25 0.75 –0.0554 0.0529 –18.00 18.90 0.2120.25 1 –0.0312 0.0312 –32.01 32.01 0.125

Table 1—Dimensionless pressure (PD) and PI values for various ideal zipper-fracture representative-element geometries at steady state with

tD 5 1. Index 1 represents the injector, and 2 represents the producer. Drainage area is given by the product, ab, whereas penetration ratio is

the normalized fracture length, L/a. The characteristic well spacing is given by a, and the fracture spacing in either well is b. Note that PD

values are dimensionless pressure differential per unit fluid volume produced per unit thickness at steady state. The last column shows the

fully penetrating cases (L/a 5 1) to be the identical 1D solution in the x-direction when placed on a dimensionless pressure-differential per

unit-fluid-volume-produced per unit-drainage-volume basis.



8 2017 SPE Journal

The results could be recast to determine the spacing needed to produce interference within a target time frame or to ascertain if pri-mary development spacing is appropriate for EOR. Again, our conclusions in the examination of Figs. 14 and 15 are dependent on thetime for departure from linear flow, indicating the onset of fracture interference between injector and producer. It should be noted thatattempts to produce high fracture density would no doubt increase chances of bashing and departure from biwing structures. The pre-sented framework is not dependent on any simplified fracture pattern. It is a discrete-fracture model requiring only that the pattern bespecified. A probabilistic approach can also be entertained.

Finally, we demonstrate that although the modeling has focused on ideal zipper fractures, the method can be extended to complex-fracture cases. In Fig. 16, we show the developing pressure profile with two interacting complex fractures, one serving as a complex in-jector and the other as the producer. For this exercise, we took the complex-fracture pattern interpreted from microseismic by Fisheret al. (2004) for the Barnett Shale and duplicated it with a spatial shift. In working with this complex-fracture set in infinite-conductivity-production mode, Hazlett and Babu (2014) found that long-term behavior was characterized by an inactive core and only active perimeterfractures. Here, we find that only a fraction of the perimeter remains active in long-time behavior because we observe long-time linearflow from one fracture mass toward another. In repeated units, we would expect to see two active perimeter subsets remaining active oneither side. Interleaving such fractures without overlap, giving a direct conduit of communication between injector and producer, would

P P ′tD = 10–5

10–1

10–2

10–3

10–4

Fig. 8—Pressure and pressure-derivative maps for production from a repeating unit in a zipper-fracture configuration with b 5 a/2and 75% penetration. Note that the pressure derivative is insightful for flow-field structure at early times, whereas late-time infor-mation is lost. The pressure field retains flow-regime information through spatial information even when no longer evident in tran-sient signatures.



2017 SPE Journal 9

be difficult, but most probably only a subset of the fractures depicted act in infinite-conductivity mode. Undesirable gas cycling may bean eventuality, but it does not preclude a viable EOR process.

Conclusions

1. A lean-gas-injection process is envisioned for EOR in unconventional reservoirs between induced fractures in zipper and modified-zipper configurations. In condensate systems, recycled lean gas would miscibly displace reservoir fluids between fracture sets withproduced-liquid extraction. In unconventional oil systems, immiscible displacement would occur with arrest in the oil-rate declineupon the onset of pressure support until immiscible-front breakthrough, although this may never occur in reasonable time. In eithercase, the time for interference is critical in process feasibility.

2. Analytic solutions to the diffusivity equation in two dimensions for discrete-fracture systems allow unique opportunities to accessfracture interference quantitatively in the absence of ad hoc assumptions on the flow pattern.

3. Analysis of the forward-model solutions allow rapid and accurate quantification of pressure, PI, the temporal derivative of pressure,and the slope of the derivative at any dimensionless time without the need for time marching.

P P ′tD = 10–5

10–1

10–2

10–3

10–4

Fig. 9—Pressure and pressure-derivative maps for production from a repeating unit in a zipper-fracture configuration with b 5 a/2and 50% penetration. Note that the pressure derivative is insightful for flow-field structure at early times, whereas late-time infor-mation is lost. The pressure field retains flow-regime information through spatial information even when no longer evident in tran-sient signatures.



10 2017 SPE Journal

4. Flow regime and flow-regime changes can be readily diagnosed.5. The magnitude of the dimensionless pressure decreases and the dimensionless PI per unit area increases with increasing overlap in

fractures in the zipper configuration for all dimensionless drainage areas, represented by ab/a2. Thus, we have the intuitive resultthat longer generated fractures give better resulting interfracture communication between injectors and producers, so long as frac-tures can be adequately propped and bashing is not induced. Because results also scale with respect to interwell distance, we havethe anticipated result that the well spacing corresponding to the longest generated propped fracture possible would give the bestresults. This work, however, quantifies that intuitive result.

6. The magnitude of the dimensionless PI increases with tighter fracture spacing for fracture lengths greater than L/a¼ 0.5, anddecreases with tighter fracture spacing for fracture lengths less than L/a¼ 0.5. We also note that although pseudosteady state maynever be attained in many unconventional reservoirs, pseudosteady-state productivity is still an important parameter because tran-sient behavior still strides toward this limit. However, we see that the time to interwell communication through the fracture systemis the key in EOR potential and will dictate best spacing.

7. The onset of interfracture communication (i.e., the development of interference) can be seen and quantified from analytic solutions tothe diffusivity equation with particular attention to the interfracture region in both pressure and pressure temporal derivative. This onsetof interference can also be captured by the slope of the temporal derivative available from the analytic discrete-fracture forward model.

8. For the ideal-zipper-configuration case of b/a¼ 0.5 and L/a¼ 0.75, we find communication initiates at approximately tD¼ 10�3.This dimensionless time can be associated with a family of real times for different effective permeabilities for the stimulated reser-voir volume in the zipper configuration.

9. Using typical parameters for gas and oil, we find we can establish interwell communications between injectors and producers withina 1- to 5-year window, provided the effective permeability of the stimulated volume is sufficiently high, although a steady state isgenerally not attainable for k< 0.001 md.

10. As supported by the time of departure from strictly linear flow (one-half slope) indicating the onset of interference, the process maybe feasible for microdarcy material and reservoir matrix but not for nanodarcy effective matrix permeability.

11. The described EOR-feasibility methodology can be extended to complex-fracture scenarios, as demonstrated with the complex-fracture set interpreted from microseismic by Fisher et al. (2004).

(a) (b) (c) (d)

Fig. 10—Dimensionless pressure maps for b/a 5 1 at tD 5 1 for different zipper-fracture degrees of partial penetration. The colormap for each image is self-normalized with respect to maximum and minimum values for optimal viewing. Note that the referencecase of full penetration does not represent direct-well communication, because horizontal wells are expected to be cased-holecompletion. (a) L/a 5 0.25; (b) L/a 5 0.5; (c) L/a 5 0.75; (d) L/a 5 1.

(a) (b)

(c) (d)

Fig. 11—Dimensionless pressure maps for b/a 5 0.5 at tD 5 1 for different zipper-fracture degrees of partial penetration. The colormap for each image is self-normalized with respect to maximum and minimum values for optimal viewing. (a) L/a 5 0.25; (b) L/a 50.5; (c) L/a 5 0.75; (d) L/a 5 1.



2017 SPE Journal 11

(a) (b)

(c) (d)

Fig. 12—Dimensionless pressure maps for b/a 5 0.25 at tD 5 1 for different zipper-fracture degrees of partial penetration. The colormap for each image is self-normalized with respect to maximum and minimum values for optimal viewing. (a) L/a 5 0.25; (b) L/a 50.5; (c) L/a 5 0.75; (d) L/a 5 1.

(a) (b)

Fig. 13—Comparison of dimensionless pressure maps for b/a 5 1 at tD 5 1: (a) Uniform flux and (b) uniform pressure. A line graphof pressure along the y-value of the central fracture is superimposed.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.01 0.1 1 10 100 1,000

Slo

pe o

f Log

arith

mic

Pre

ssur

e D

eriv

ativ

e

Time (years)

k = 10–2 md

k = 10–3 md

k = 10–4 md

k = 10–5 md

Oil

Gas

Fig. 14—Development of interference as evidenced by deviation of the slope from a value of 0.5 for the logarithmic temporal deriva-tive of pressure as a function of time for various matrix permeabilities for the zipper-fracture configuration b/a 5 0.5 and typicalgas and oil parameters as given by Barree et al. (2014).



12 2017 SPE Journal

Parameter Gas Oil

Viscosity (cp) 0.02 0.5ct (psi–1) 1.90×10–4 2.25×10–5

Porosity φ 0.08 0.08

a (ft) 1320 1320b/2 (ft) 330 330

Table 2—Parameters used in the reduction of dimensionless

predictions for the b/a zipper-fracture configuration to actual time as

provided for oil and gas by Barree et al. (2014).

0.010.0010

0.2

0.4

0.6

0.8

1

1.2

1.4

0.1 1 10 100 1,000 10,000

Time (years)

Slo

pe o

f Log

arith

mic

Pre

ssur

e D

eriv

ativ

e

k = 10–2 md

k = 10–3 md

k = 10–4 mddPD /dlntD

k = 10–5 md

Oil

Gas

Fig. 15—Development of interference as evidenced by deviation from a value of 0.5 for the slope of the logarithmic temporal deriva-tive of pressure as a function of time for various matrix permeabilities for the zipper-fracture configuration b/a 5 0.075 and typicalgas and oil parameters as given by Barree et al. (2014).

(a)

(c) (d) (e)

(b)

Fig. 16—Dimensionless pressure maps for interacting complex fractures of Fisher et al. (2004). The color map for each image isself-normalized with respect to maximum and minimum values for optimal viewing. (a) tD 5 0.0001; (b) tD 5 0.001; (c) tD 5 0.01; (d)tD 5 0.1; (e) tD 5 1.



2017 SPE Journal 13

Recommendations

The considered ideal-zipper-fracture studies do not take into account the presence of natural fractures, a potential conjugate set of or-thogonal fractures. These would generate variation in communication patterns, and in the worst case, create channeling of injected flu-ids. Future studies should take into account a secondary set of fractures, their probability of occurrence, and the effect of heterogeneouseffective permeability in the stimulated volume.

The results supplied here fall into the domain of pressure-transient analysis with constant rate. Future results should recast these sol-utions as rate transients with bottomhole-pressure constraint as the more-likely scenario for unconventional reservoirs. Direct compari-son of pressure and rate transients for identical cases should be performed.

Nomenclature

a ¼ cell length, Lb ¼ cell width, LB ¼ formation volume factor, L3/L3

ct ¼ compressibility, P�1

Clmn ¼ constantDlmn ¼ dimensionless distance parameter

E ¼ dimensionless exponential factorEj ¼ dimensionless exponential factorF3 ¼ defined functionh ¼ cell height, L

k,km ¼ average matrix permeability, L2

kx,y,z ¼ directional permeability, L2

l,m,n ¼ dimensionless indicesL ¼ fracture length, LP ¼ pressure, PPi ¼ dimensionless initial pressure

PD ¼ dimensionless pressure differencePI ¼ dimensionless PIq ¼ well-flow rate, L3/tri ¼ radius of investigation, Ls ¼ distance parameter, Lt ¼ time

tD ¼ dimensionless time with length-scaling grouptD ¼ dimensionless time

x, y, z ¼ coordinates, Lxe ¼ reference length, L

xo, yo, zo ¼ source coordinates, Lx1, y1, z1 ¼ line-source endpoint coordinates, Lx2, y2, z2 ¼ line-source endpoint coordinates, L

a ¼ angle with x-axisb ¼ angle with y-axisc ¼ angle with z-axisd ¼ Dirac delta, L�1

k ¼ dimensionless sum of four trigonometric functionskj ¼ dimensionless vector of unit magnitude carrying sign informationl ¼ viscosity, P�tlj ¼ dimensionless source strength, (L3/t)/(L3/t)/ ¼ porosity, L3/L3

References

Acuna, J. A. 2016. Analytical Pressure and Rate Transient Models for Analysis of Complex Fracture Networks in Tight Reservoirs. Presented at the

Unconventional Resources Technology Conference, San Antonio, Texas, USA, 1–3 August. URTEC-2429710-MS. https://doi.org/10.15530/

URTEC-2016-2429710.

Barree, R. D., Cox, S. A., Miskimins, J. L. et al. 2014. Economic Optimization of Horizontal Well Completions in Unconventional Reservoirs. Presented

at the SPE Hydraulic Fracturing Technology Conference, The Woodlands, Texas, 4–6 February. SPE-168612-MS. https://doi.org/10.2118/168612-

MS.

Datta-Gupta, A., Xie, J., Gupta, N. et al. 2011. Radius of Investigation and its Generalization to Unconventional Reservoirs. J Pet Technol 63 (7): 52–55.

SPE-0711-0052-JPT. https://doi.org/10.2118/0711-0052-JPT.

Fisher, M. K., Heinze, J. R., Harris, C. D. et al. 2004. Optimizing Horizontal Completion Techniques in the Barnett Shale Using Microseismic Fracture

Mapping. Presented at the SPE Annual Technical Conference and Exhibition, Houston, 26–29 September. SPE-90051-MS. https://doi.org/10.2118/

90051-MS.

Gringarten, A. C., Ramey, H. J. Jr., and Raghavan, R. 1974. Unsteady-State Pressure Distributions Created by a Well With a Single Infinite-Conductivity

Vertical Fracture. SPE J. 14 (4): 347–360. SPE-4051-PA. https://doi.org/10.2118/4051-PA.

Hazlett, R. D. and Babu, D. K. 2014. Discrete Wellbore and Fracture Productivity Modeling for Unconventional Wells and Unconventional Reservoirs.

SPE J. 19 (1): 19–33. SPE-159379-PA. https://doi.org/10.2118/159379-PA.

Jacobs, T. 2014. The Shale Evolution: Zipper Fracture Takes Hold. J Pet Technol 66 (10): 60–67. SPE-1014-0060-JPT. https://doi.org/10.2118/1014-

0060-JPT.

Kuchuk, F., Biryukov, D., Fitzpatrick, T. et al. 2015. Pressure Transient Behavior of Horizontal Wells Intersecting Multiple Hydraulic and Natural Frac-

tures in Conventional and Unconventional Unfractured and Naturally Fractured Reservoirs. Presented at the SPE Annual Technical Conference and

Exhibition, Houston, 28–30 September. SPE-175037-MS. https://doi.org/10.2118/175037-MS.



14 2017 SPE Journal

https://doi.org/10.15530/URTEC-2016-2429710

https://doi.org/10.15530/URTEC-2016-2429710

https://doi.org/10.2118/168612-MS

https://doi.org/10.2118/168612-MS

https://doi.org/10.2118/0711-0052-JPT

https://doi.org/10.2118/90051-MS

https://doi.org/10.2118/90051-MS

https://doi.org/10.2118/4051-PA

https://doi.org/10.2118/159379-PA

https://doi.org/10.2118/1014-0060-JPT

https://doi.org/10.2118/1014-0060-JPT

https://doi.org/10.2118/175037-MS

Kuchuk, F., Morton, K., and Biryukov, D. 2016. Rate-Transient Analysis for Multistage Fractured Horizontal Wells in Conventional and Un-Conven-

tional Homogeneous and Naturally Fractured Reservoirs. Presented at the SPE Annual Technical Conference and Exhibition, Dubai, 26–28 Septem-

ber. SPE-181488-MS. https://doi.org/10.2118/181488-MS.

McCann, R., Hazlett, R. D., and Babu, D. K. 2001. Highly Accurate Approximations of Green’s and Neumann Functions on Rectangular Domains. Proc.Royal Soc. A 457: 767–772. https://doi.org/10.1098/rspa.2000.0690.

Sierra, L. and Mayerhofer, M. 2014. Evaluating the Benefits of Zipper Fracs in Unconventional Reservoirs. Presented at the SPE Unconventional Resour-

ces Conference, The Woodlands, Texas, 1–3 April. SPE-168977-MS. https://doi.org/10.2118/168977-MS.

Van Poolen, H. 1964. Radius-of-Drainage and Stabilization-Time Equations. Oil Gas J. September: 138–146.

Oluwanifemi Akinluyi worked as an EOR implementation engineer in 2016. Her research interests include EOR, reservoir simula-tion, well integrity, and reservoir-data analytics. Akinluyi holds bachelor’s and master’s degrees, both in petroleum engineering,and an MBA degree, all from the University of Tulsa. She is a member of SPE and currently serves as a technical reviewer for SPEJournal and a Petrobowl question writer.

Randy Hazlett is an associate professor at the University of Tulsa’s McDougall School of Petroleum Engineering. He worked in theresearch and development center of Mobil for 15 years and spent another decade conducting research as president of Poten-tial Research Solutions. Hazlett’s areas of expertise are in well testing and inflow performance of complex wells, core analysis,EOR, geomechanical properties, emulsions, and the fundamentals of multiphase flow in porous media. He has authored orcoauthored more than 30 publications and holds 12 patents. Hazlett holds a PhD degree in chemical engineering from the Uni-versity of Texas at Austin. He serves on the SPE Scholarship Committee.



2017 SPE Journal 15

https://doi.org/10.2118/181488-MS

https://doi.org/10.1098/rspa.2000.0690

https://doi.org/10.2118/168977-MS

本文献由“学霸图书馆-文献云下载”收集自网络，仅供学习交流使用。

学霸图书馆（www.xuebalib.com）是一个“整合众多图书馆数据库资源，

提供一站式文献检索和下载服务”的24 小时在线不限IP

图书馆。

图书馆致力于便利、促进学习与科研，提供最强文献下载服务。

图书馆导航：

图书馆首页文献云下载图书馆入口外文数据库大全疑难文献辅助工具

http://www.xuebalib.com/cloud/

http://www.xuebalib.com/

http://www.xuebalib.com/cloud/


http://www.xuebalib.com/vip.html

http://www.xuebalib.com/db.php

http://www.xuebalib.com/zixun/2014-08-15/44.html


Documents

Enhanced-Oil-Recovery Potential for Lean-Gas Reinjection ...download.xuebalib.com/15jaLRImVs1z.pdf · lDx a 6 mDy b 6 nDz h cos p 2 lx m a 6 my b 6 nz m h lDx a 6 mDy b 6 nDz h; ð8Þ