horizontal fractures in single in multilayer reservoir

Although the mathematical solution used to generate the data in Figure 1 is correct, given the inner-boundary condition of uniform flux, it is clear that after nearby formation boundarie

CSUG/SPE 147004

Horizontal Fractures in Single and Multilayer Reservoirs

Leif Larsen, SPE, Kappa Engineering, University in Stavanger

Copyright 2011, Society of Petroleum Engineers

This paper was prepared for presentation at the Canadian Unconventional Resources Conference held in Calgary, Alberta, Canada, 15–17 November 2011

This paper was selected for presentation by a CSUG/SPE program committee following review of information contained in an abstract submitted by the author(s) Contents of the paper have not

been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s) The material does not necessarily reflect any position of the Society of Petroleum Engineers,

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Abstract

Type curves for circular uniform-flux fractures in unbounded single-layer reservoirs were presented in the literature several

decades ago Unfortunately, these results have restricted validity since the simple uniform-flux inner-boundary condition will

cease to be physically meaningful when significant boundary effects from the top and bottom of the formation start to affect

the data This is especially important for large fractures in formations of limited thickness For horizontal fractures it is

therefore important to include finite conductivity as part of the fracture model Solutions addressing these issues are presented

in this paper along with examples highlighting effects of different modeling options The solutions include both compressible

and incompressible flow inside the fractures

The solutions have also been extended to multi-layer reservoirs through a decoupling approach, with both unbounded and

bounded circular models used in examples Solutions for layered reservoirs are important because layering is common with

obvious effects on productivity and because layering can create unusual pressure-transient characteristics during buildups if

there are shale streaks or other low-permeability flow barriers between layers acting as partially sealing boundaries

Horizontal fractures are becoming more important due to an increasing number of shallow-depth disposal wells, for

instance as part of carbon capture and storage developments Shallow gas developments with fracturing needed to obtain

sufficient productivity are also adding to the importance of these fracture models Both for producers and disposal wells,

layering will be an issue and has to be addressed

Introduction

Pressure-transient solutions for circular uniform-flux fractures in unbounded single-layer reservoirs were presented in the

literature by Gringarten and Ramey (1974) The results, which were based on Gringarten’s dissertation (1972), were also

reproduced as one of the type curves presented by Earlougher (1975) for cases with a centered fracture Figure 1 shows an

example of such pressure-transient data from a centered uniform-flux fracture with radius 10 times the thickness of a low

permeability isotropic formation An important observation here is that the solution exhibits two log-cycles of apparent simple

depletion (pseudosteady state) behavior between early linear flow and late radial flow This behavior was also discussed by

Gringarten and Ramey (1974)

Although the mathematical solution used to generate the data in Figure 1 is correct, given the inner-boundary condition of

uniform flux, it is clear that after nearby formation boundaries (top and bottom) start affecting the flow pattern, the uniform

flux condition cannot be maintained for large fractures Only infinite-conductivity and finite-conductivity fractures are

therefore realistic physical models in a strict sense Even so, since the uniform-flux and infinite-conductivity solutions will

have identical early data, and both will approach late radial flow, the uniform-flux solutions will be adequate for many

high-conductivity cases However, with radial flow within fractures, it follows that fracture high-conductivity will tend to be more

important for horizontal fractures than for vertical fractures Solutions for finite-conductivity fractures should therefore be

used unless high-conductivity is likely Effects of compressible flow within horizontal fractures should also be considered if

there is a possibility that this might be important For early data the latter case can be treated as a transient double-porosity

model of the “slab” type, but this approach is only valid when flow from the formation to the fracture can be treated as purely

one dimensional and perpendicular

Outer boundaries, such as circular no-flow or constant pressure boundaries, or single and other boundaries that can be

generated through simple image-well techniques based on line-source solutions, can be added in both the single and

multi-layer models if the fracture does not extend too close to the boundary or boundaries The restriction concerning nearby

boundaries is caused by a lack of symmetry in these models

Figure 1 Log-log diagnostic plot of data from a large horizontal uniform-flux fracture in a low-permeability formation

Mathematical Models for Horizontal Fractures with Uniform Flux

For circular horizontal fractures with uniform flux we only need the fracture radius and location along with the

permeability ratio Gringarten (1972) used a Greens function-based approach to develop real-space solutions directly

for this model, while the main solutions presented in this paper are based on integrals of Laplace-transformed point-source

solutions and superposition schemes to generate formation boundaries, with the Stehfest (1970) algorithm used to generate

real-space data This is quite straightforward for horizontal uniform-flux fractures with pressures computed on the axis of the

fracture, i.e., at for any z location Solutions for fractures with uniform flux can in turn be used to generate solutions for

horizontal fractures with infinite or finite conductivity by an approach similar to that used for vertical fractures The wellbore

pressure for horizontal fractures with infinite conductivity can also be approximated by using an equivalent pressure point on a

uniform-flux fracture, similar to the approach used for vertical fractures

f

/

z r

k k

0

=

r

Details of the development of Laplace transformed solutions for uniform-flux horizontal fractures (radial symmetry always

assumed) based on integrals of point-source solutions for unbounded 3D models are presented in Appendix A, with

superposition schemes used to generate upper and lower boundaries with no flow or constant pressure It is also shown how

integrals of point-source solutions in formations of finite thickness can be used to the same end

In dimensionless form the solutions for points on the axis of the fracture, i.e., at for any , can be expressed in

either of the two “closed forms”

0

D

2 ( 0, , ) D z D z fD jh D s z D z fD jh D s r fD z D z fD jh D s r fD z D z fD jh D s

D D D

j fD

h

s sr

=−∞

(1)

and

2

1

D D D D n fD n

s s

ω

∞

=

r

(2)

if both formation boundaries are sealing (top and bottom), where and The definition of

is standard, while and the choice of cosine or sine in Eq 2 depend on the type of formation boundaries Eq 1 can also be

modified to change the type of formation boundaries by specific choices of signs of the terms This, which is a standard

approach, is described in detail in Appendix A Note also that the ratio has been used explicitly in Eq 2, while it is only

used indirectly in Eq 1 through an initial coordinate scaling Also note that the definitions of dimensionless pressure and time

used for the solutions above are based on radial flow and formation properties with horizontal permeability ( ) as reference

2 2 2 /

n n h D

/

z r

/

n s n k z k

n

λ

r

k

For cases with , i.e., for observation points away from the axis of the fracture, which we need to generate solutions

for fractures with finite conductivity, the integral expressions cannot be evaluated in closed form We therefore have to resort

to numerical integration, which for the current study solely have been based on point-source solutions in unbounded 3D

models, with superposition used afterwards to include boundaries Details to this end are described in Appendix A

0

D

r >

Examples and Analyses of Data from Horizontal Fractures with Uniform Flux

From data sets that exhibit the three flow regimes early (vertical) formation linear, pseudo-steady state and late radial, it is possible to determine the three key unknowns: k r, k z and , provided the formation thickness and other basic formation and fluid parameters are known The point is that the pseudo-steady state (pss) data should correspond to the identity

f

r

t f t f

Δ = + = t+a (5)

in oil field units for some constant a, where represents the affected rock volume Either the derivate from the log-log plot at a chosen time or the slope from a Cartesian plot can therefore be used to determine the fracture radius

2

f

r h

π

f

r from the slope

2

0.074466

t f

qB m

c r h

φ

′ = (6)

Since early linear-flow data should correspond to the identity

z t z t

μ

π

×

(7)

in oil fields units if there is no fracture skin, the slope of a square-root of time plot can be used to determine k z if is known

Eq 7 is obtained by recognizing that the product

f

r

f

hx appearing in the equation for vertical fractures represents the area of one

side of half the fracture Analysis of late radial data, if present, is standard and will yield k r and the pseudo-radial skin value Although straight-line analyses will yield the key results, matching the data with a complete model should always be tried,

if available, to check the consistency of both the analyses and the data To check results from Fig 1, the data were generated with k z =k r =1 md, h=30 ft, r f =300 ft, q=100 STB/D, B=1 RB/STB, μ=1 cp, φ=0.1 and 5 1

3 10 psi

t

With reduced vertical permeability the ratio between the fracture radius and the effective thickness h′ =h k r /k z will be

reduced, and hence also the likelihood of observing the pss period This is illustrated in Fig 2 with cases based on data from

Fig 1 with different k z/k r ratios Since r f /h= 01 if k z /k r =1, note that k z /k r =0.01 corresponds to the effective ratio

This is essentially the lower end of the ratios that might indicate the presence of pss data for centered fractures For off-center fractures the ratio must be higher for the pss period to appear For shorter fractures it will not be present, and for very short fractures we might instead observe a period with spherical flow between early linear and late radial periods

/

f

Figure 2 Sensitivity of horizontal uniform-flux fractures to anisotropy

kz/kr = 0.0001

0.001 0.01

0.1

1

The importance of the ratio r f /h on the pressure response is also illustrated in Fig 3 by changing only r in the basic f

isotropic model with thickness 30 ft For the middle case with r f /h= we see again that the onset of the pss period is just 1

indicated, while the case with has a clear segment of spherical flow data between the early linear and late radial

flow periods Note that the shift in the data between the Figs 2 and 3 is caused by delayed onset of boundary effects (top and

bottom) when the

/

f

r h= 0.1

/

z r

k k ratio is reduced in Fig 2, while the onset of radial flow shows up earlier in Fig 3 for shorter fractures, but only after the onset of boundary effects (top and bottom)

Figure 3 – Sensitivity of horizontal uniform-flux fractures to fracture radius (isotropic data)

rf = 3 ft

rf = 30 ft

rf = 300 ft

Another situation of interest is how the pressure response is affected by off-center fractures This is illustrated in Fig 4

with the fracture moved to just 1 ft from the nearest formation boundary, corresponding to a radius of investigation at slightly

less than 0.004 hrs in any direction The data illustrate the key points – that very early data exhibit linear flow from both sides,

that data between 0.1 and 3 hrs are dominated by linear flow from just one side (other side trapped), with the onset of pss

behavior afterwards and the rest similar to the behavior of the centered reference fracture (markers)

Figure 4 – Effects of moving the fracture close to a formation boundry

Linear flow from one side

Linear flow from both sides

Markers: Centered fracture Curves: Fracture 1 ft from bottom

By using the solution for fractures with uniform flux we can also approximate the wellbore pressure for fractures with

infinite conductivity by computing the pressure at the equaivalent pressure point r D =0.628r fD in dimensionless coodinates

Choosing the fraction 0.628 for horizontal fractures differs from the fraction 0.74 normally used for vertical fractures Fig 5

illustrates the difference between wellbore pressures obtained from fractures with uniform flux (markers) and infinite

onductivity (curves) by this approach To justify the choice 0.628r for the equivalent pressure point to approximate the fD

wellbore behavior for horizontal fractures with infinite conductivity we need to compare with solutions for fractures with fracture conductivity included in the model

Figure 5 – Infinite conductivity handled through the use of an equivalent pressure point

Markers: Uniform flux Curves: Infinite conductivity

Horizontal Fractures with Finite Conductivity

There are two main differences between the treatment of horizontal and vertical fractures with finite conductivity, but both are based on subdivision and determination of the rate to each element with the constraint that the relevant flow equation is honored inside the fracture The differences involve radial vs linear flow inside the fracture and the use of uniform-flux circular flat rings vs uniform-flux vertical strips Even so, the computational set-ups are quite similar

Similar to vertical fractures the fracture conductivity is defined by the product F c =k w f , where k denotes the fracture f

permeability and w its width, or thickness With denoting horizontal permeability in the formation, we shall also define the

dimensionless fracture conductivity by the expression

r

k

f c

cD

r f r f

k w F

F

= = , (8)

which is similar to that used for vertical fractures If flow within the fracture is treated as incompressible, then the fracture radius, location and dimensionless conductivity fully define the model However, if flow within the frature is treated as compressible, then we also need to include the fracture diffusivity

f

f

f tf

k

c

η

φ μ

= (9)

as part of the model, either directly, or through the dimensionless fracture diffusivity

f f t f

fD

f tf r f tf r

η

= = = (10)

Note that with fracture conductivity F and c r given we cannot vary f ηfD arbitrarily in a physical model, but we can let ηfD

approach infinity in a mathematical model corresponding to incompressible flow within the fracture as a limiting solution

Fig 6 illustrates the effect of fracture conductivity on the pressure response for the case with incompressible flow assumed

in the fracture Two key observations can be made – that the effective length is reduced when the conductivity is reduced, and that early data exhibit a semi-log response The latter corresponds to half the slope we would observe in early data with

compressible flow inside the fracture The reason is explained in Appendix C Fig 7 illustrates how the data are affected by

changes in the diffusivity ratio with compressible flow inside the fracture Note that the response approaches that of

incompressible flow when ηfD is increased, and that we for low values of ηfD start observing a discrepancy during the pss

period This, which is barely indicated for the case ηfD =100 in Fig 7, is caused by increased fracture storativity in addition

to that of the formation, and hence an artifact of the approach for unrealistic diffusivity ratios

Figure 6 – Effect of conductivity on fractures with incompressible flow

Fc = 5 md·ft

50

500

5000

5E4

5E5

Figure 7 – Effect of the diffusivity ratio with compressible flow and finite conductivity

Diffusivity ratio 100 1000

10000

Fc = 500 md·ft

In Appendix C it is shown how a double-porosity slab model can be used to model early data frm fractures with

compressible flow These results serve to explain the bahavior of early data from fractures with incompressible flow, and in

addition can be used to verify the validity of the full model This is illustrated in Fig 8, with data based on Fig 7 with a

fracture of length 300 ft and ηfD values 100 and 1000

Figure 8 – Comparison of data from full solutions and from infinite-acting double-porosity type early data

Markers: FC with compressible flow Curves: “Double-porosity” model

Diffusivity ratio 100 1000

Horizontal Fractures in Layered Reservoirs

Only cases with a single fracture in a layered reservoir have been considered in this paper, with mathematical solution presented in Appendix E in terms of Laplace transforms The approach is based on decoupling as described by Larsen (1999), with standard dimensionless model description used in terms of κ’s (flow capacity fractions), ω’s (storativity fractions) and λ’s (cross-flow parameters) The approach can be used for any fracture type, and with outer boundaries of standard types if the boundaries do not come too close to the fracture (there is a symmetry problem for small distances)

Fig 9 has been used to verify that the mathematical approach is valid, with three identical, isotropic layers of thickness 30

ft (each) with permeability 5 md used in the layered model with a 300 ft fracture in the middle of the bottom layer What we see is that the layered model with default lambdas based on the layer properties reproduces almost exactly the single-layer solution with thickness 90 ft and the fracture 15 ft from the bottom With a very low lambda value (1E-8) the layered model follows that of just the bottom layer for almost 1000 hours before we start seeing the effect of crossflow With lambda 1E-6 the layered solution starts deviating from that of just the bottom layer after less than 10 hours

Figure 9 – Comparing single-layer and multi-layer solutions

Markers: Single layer, 30 and 90 ft Curves: Three layers, varied lambdas

Default isotropic Lambdas = 1E-6 Lambdas = 1E-8

Fig 10 illustrates a re-charge scenario during an extended buildup following 300 hours production The case is based on two fracture types, one with uniform flux and one with finite conductivity ( md.ft), both of length 300 ft in the middle of a 3-layer 90 ft model The crossflow parameter between the middle layer and the others is very low, at 1E-9 The outer radius is 500 ft and closed The key point to make is that both fractures exhibit pss behavior during late buldup data, similar to what can appear in buildup data from wells in segmented reservoirs with partially sealing boundaries

50000

c

Figure 10 – Re-charge effect in buildup data in a closed 3-layer model with fracture in middle layer and low lambdas

Uniform flux Finite conductivity

Circular 3-layer model

Conclusions

1 Analytical solutions can be derived and used to model and analyze data from horizontal fractures of the basic types such

as uniform flux, infinite conductivity, and finite conductivity with both incompressible and compressible flow

2 Outer boundaries can be generated by standard image-well techniques based on line-source wells or Bessel function

solutions, provided the boundaries do not come too close to the fracture

3 The solutions can be extended to layered models

4 Boundaries can be added in layered models by the same techniques used for single layers

Nomenclature

A = tri-diagonal matrix, Eq E-1

a ij = elemenst of A

B = formation volume factor, RB/STB

c t = total compressibility, psi-1

H = orthogonal matrix, Eq E-3

K = diagonal matrix flow capacity fractions, Eq E-1

k = permeability, md

p = pressure, psia

r = radius, ft

S = radius, ft

t = time, hours

w = fracture width (thickness), ft

z = vertical coordinate, ft

η = diffusivity, md·psi/cp

μ = viscosity, cp

φ = porosity

σ = s / η Df

ν = Special Laplace variable, Eq C-4

κj = flow capacity fraction, Layer j

ωj = storativity fraction, Layer j

λj = crossflow parameters

Subscripts

D = dimensionless

d = damage

r = radial

Acknowledgment

The author would like to thank Kappa Engineering for support to publish this paper

References

Earlougher, R.C., Jr 1977 Advances in Well Test Analysis SPE Monograph Series, Vol 5

Gringarten, A.C 1971 Unsteady-State Pressure Distributions Created by a Well with a Single Horizontal Fracture, Partial Penetration, or Restricted Flow Entry PhD dissertation , Stanford U

Gringarten, A.C and Ramey, H.J., Jr 1974 Unsteady-State Pressure Distributions Created by a Well with a Single Horizontal Fracture Soc Pet Eng J: 413–426; Trans., AIME, 257

Hartsock, J.H and Warren, J.E 1961 The Effect of Horizontal Hydraulic Fracturig on Well Performance J Pet Technology:

1050–1056; Trans., AIME, 222

Larsen, L 1999 Determination of Pressure-Transient and Productivity Data for Deviated Wells in Layered Reservoirs

SPEREE: 95–103

Ozkan, E and Raghavan, R 1991 New Solutions for Well-Test-Analysis Problems: Part 1–Analytical Considerations

SPEFE: 429–438

Stehfest, H 1970 Algorithm 368: Numerical Inversion of Laplace Transforms Communications of the ACM 13 (1): 47–49

Appendix A–Horizontal Fractures with Uniform Flux

Two somewhat different approaches will be presented, starting with the Laplace transformed solution for a point source in an unbounded 3D model, which in cylindrical coordinates can be expressed in the form

( , , )

D D fD

r z z s D

D D D

D D fD

h

− + −

=

+ − (A-1)

with the source at z=z f and h a chosen reference thickness For an unbounded 3D model we can determine the solution for a

uniform-flux disc of radius r f in the plane z=z f by integration In particular, for any point on the z axis we just get

0

( 0, , )

fD D D fD

D fD fD D fD

r r z z s

z z s r z z s

Duf D D D D

fD D D fD fD

− + −

For observation points with radial coordinate r o >0 it is more challenging For such cases we can use the solution

2

fD oD D D wD

fD oD D wD

D wD

fD oD

r r r z z s

r r z z s

z z s

fD r r

sr

π

− −

−

if r o≤r f, where

2

fD D oD fD

D oD fD D oD fD

oD D D oD D oD D

f

(A-4)

represents the angle of the arc in the upper half of the fracture area If r o = , then the first expression on the right-hand side r

will be 0 and it is also possible to simplify the expression for θ(r D) If , then we can then use Eq A-3 without the first expression on the right-hand side if we replace the lower bound of integration by

o

r >r f

oD fD

r −r , along with θ(r D) from Eq A-4 The integrals above must be computed numerically This can be done by using the subsitution

D D D wD

r r z z s u

u

D D D

e

− + −

−

=

∫ ∫ (A-5)

with u= r D2+(z D−z wD)2 s and f u( )= u2/s−(z D−z wD)2

The solutions above apply to single circular fractures in a 3D infinite model, with image “wells” used to generate

formation boundaries at z= 0 (bottom) and z= h

1

(top) This can be done with standard “image fractures” patterns at

z or all integer values of j The boundary type is controlled by the signature of neighboring fractures, with + for

producers and – for injectors, i.e., with multipliers

2jh z w

+ for producers and −1for injectors

for both images at z=2j h for each j

even, and 1 a− t +z w and +1 at −z w if j is odd for each j

It is also possible to follow the general approach of Ozkan and Raghavan (1991) and start with a Laplace-transformed

point-source solution for a formation of finite thickness in the form

0 0

1

D D D D n wD n

n

K r

ω

∞

=

= + ∑ (A-6)

for unbounded models, where ωn = +s λ ρn with ρ =k z/k r and the parameters λ along with the choice between the sine and n

cosine functions in Eq A-6 are used generate no-flow or constant pressure boundaries at the top and bottom of the formation

Note also that the first term in Eq A-6 is only included if both boundaries are sealing

To generate a solution for a fracture with uniform flux from Eq A-6 we can use the identity

[

0

1

b

] 1

a

∫ (A-7)

to derive the solution

1 0

2

fD

r

fD fD fD n fD n

wD D D wD D D wD n

fD

s

)

λ

ω

∞

=

for the transformed wellbore pressure, again with the first term of the final expression only included if both the formation

boundaries are sealing

Note that the first term on the “right-hand side” of Eq A-8 represents the solution for a fully penetrating full cylindrical

“well” with production at all points of the cylinder (on the surface and inside) This is not a meaningful physical model, but it

is a useful mathematical solution as just demonstrated It is also used below for layered reservoirs A similar real-space model

also played a role in the approach used by Gringarten (1972)

Appendix B–Horizontal Fractures with Finite Conductivity and Compressible Flow

Only radial-symmetric horizontal fractures will be considered with isotropic internal properties and hence flow governed by

the equation

2

2

f

r

∂

1 p f

.(B-1)

for the fracture medium with thickness w in the z direction If we for the time being assume that the fracture is placed in the

plane defined by z=0, then we get the boundary condition