Download - History Matching and Rate Forecasting in … Matching... · History Matching and Rate Forecasting in Unconventional Oil Reservoirs With an ... cal reservoir simulation. ... tially

History Matching and Rate Forecasting inUnconventional Oil Reservoirs With anApproximate Analytical Solution to the

Double-Porosity ModelB. A. Ogunyomi, T. W. Patzek, and L. W. Lake, University of Texas at Austin; and C. S. Kabir, Hess Corporation

Summary

Production data from most fractured horizontal wells in gas andliquid-rich unconventional reservoirs plot as straight lines with aone-half slope on a log-log plot of rate vs. time. This productionsignature (half-slope) is identical to that expected from a 1D lin-ear flow from reservoir matrix to the fracture face, when produc-tion occurs at constant bottomhole pressure. In addition,microseismic data obtained around these fractured wells suggestthat an area of enhanced permeability is developed around thehorizontal well, and outside this region is an undisturbed part ofthe reservoir with low permeability. On the basis of these observa-tions, geoscientists have, in general, adopted the conceptual dou-ble-porosity model in modeling production from fracturedhorizontal wells in unconventional reservoirs. The analytical solu-tion to this mathematical model exists in Laplace space, but itcannot be inverted back to real-time space without use of a nu-merical inversion algorithm. We present a new approximate ana-lytical solution to the double-porosity model in real-time spaceand its use in modeling and forecasting production from uncon-ventional oil reservoirs.

The first step in developing the approximate solution was toconvert the systems of partial-differential equations (PDEs) forthe double-porosity model into a system of ordinary-differentialequations (ODEs). After which, we developed a function thatgives the relationship between the average pressures in the high-and the low-permeability regions. With this relationship, the sys-tem of ODEs was solved and used to obtain a rate/time functionthat one can use to predict oil production from unconventionalreservoirs. The approximate solution was validated with numeri-cal reservoir simulation.

We then performed a sensitivity analysis on the model parame-ters to understand how the model behaves. After the model wasvalidated and tested, we applied it to field-production data by par-tially history matching and forecasting the expected ultimate recov-ery (EUR). The rate/time function fits production data and alsoyields realistic estimates of ultimate oil recovery. We also investi-gated the existence of any correlation between the model-derivedparameters and available reservoir and well-completion parameters.

Introduction

Many studies were published that focus on the solution of thedouble-porosity model for flow in hydraulically fractured horizon-tal wells. Barenblatt and Zheltov (1960) presented the first formu-lation of the double-porosity model. Warren and Root (1963)presented the first application of the double-porosity model toflow problems in the petroleum industry. Since then, manyauthors (de Swaan-O. 1976; Ozkan et al. 1987; Carlson and Mer-cer 1991; El-Banbi 1998; Mayerhofer et al. 2006) have presentedapplications of the model.

All the analytical solutions presented were in Laplace spaceand have had to be numerically transformed to real-time spacewith some form of inversion algorithm of which the Stehfest algo-rithm (Stehfest 1970) is the most popular. More recently, Belloand Wattenbarger (2008) presented the solution to the double-po-rosity model for linear flow in which they were able to obtainclosed-form analytical solutions for certain ranges of time. To dothis, they broke their Laplace-space solution into smaller bits withspecial properties of the solution that they could invert to real-time space. This piece-wise solution would have to be appliedsequentially. Samandarli et al. (2011) presented the application ofthis solution to history matching and forecasting the performanceof shale-gas wells. Song (2014) presented a finite-difference solu-tion to this problem and its application to oil production fromhydraulically fractured wells.

In this study, we present an approximate analytical solution tothe double-porosity model in real-time space that is valid acrossall time domains; that is, it is a continuous function that is validduring the transient and late-time flow from the fracture and ma-trix. We validate our solution against numerical simulation andalso show that our solution reproduces the production behaviorobtained from the inverted Laplace-space solution. We also pres-ent example applications of our solution to field data.

Model Development

Fig. 1a is a schematic of a hydraulically fractured horizontal wellin which the fractures are perpendicular to the wellbore, and Fig.1b is a 3D schematic of the same well. Between successive frac-tures is low-permeability reservoir matrix. We assumed that thefracture face is at a constant pressure that is equal to the bottom-hole well pressure of the well. The dashed red lines represent theno-flow boundaries created by the interference of flow from thematrix into the fracture face.

We made the following assumptions in the modeldevelopment:

• Flow is single phase and slightly compressible.• Flow occurs in the reservoir isothermally.• The reservoir is isotropic and homogeneous in each

compartment.• There is no direct communication between the matrix and

wellbore.• There is a large contrast in permeability between the fracture

and matrix compartments.• Secondary effects such as stress-dependent permeability

(porosity) and desorption are neglected.The system of equations that describes this conceptual model

is presented as follows—for the low-permeability reservoir ma-trix. The governing PDEs, initial condition, and boundary condi-tions are summarized as

@2pm

@z2þ @

2pm

@y2þ @

2pm

@x2¼ ð/lctÞm

km

@pm

@t; ð1Þ

pmðx; y; z; 0Þ ¼ pi; ð2Þ

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

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

Copyright VC 2015 Society of Petroleum Engineers

This paper (SPE 171031) was accepted for presentation at the SPE Eastern RegionalMeeting, Charleston, West Virginia, USA, 21–23 October 2014, and revised for publication.Original manuscript received for review 22 July 2014. Revised manuscript received forreview 20 September 2015. Paper peer approved 24 September 2015.

REE171031 DOI: 10.2118/171031-PA Date: 20-November-15 Stage: Page: 1 Total Pages: 13

ID: jaganm Time: 15:19 I Path: S:/REE#/Vol00000/150046/Comp/APPFile/SA-REE#150046

2015 SPE Reservoir Evaluation & Engineering 1

@pm

@x

��x¼xe

¼ 0; ð3Þ

kf

l@pf

@x

��xwf

¼ km

l@pm

@x

��xwf

; ð4Þ

@pm

@y

��y¼ye

¼ 0; ð5Þ

@pm

@y

��y¼ywf

¼ 0; ð6Þ

@pm

@z

��z¼z0

¼ 0; ð7Þ

and

@pm

@z

��z¼ze

¼ 0: ð8Þ

Eq. 1 is the diffusivity equation for the reservoir matrix, andEq. 2 is the constant-pressure initial condition. Eq. 3 means thatthere is a no-flow boundary at the external boundary of the reser-voir matrix. Eq. 4 states that the matrix flow into the fracture faceis equal to the outflow from fracture face (x ¼ xwf ). Eq. 5 statesthat there is a no-flow boundary at the external boundary of thereservoir matrix in the y-direction, and Eq. 6 states that there is nointeraction between the reservoir matrix and the wellbore; that is,there is no crossflow from the matrix into the wellbore. Eqs. 7 and8 are no-flow boundary conditions, and they model the fact thatthe reservoir is sealed at the top and bottom boundaries.

For the fracture, we have

@2pf

@z2þ @

2pf

@y2þ @

2pf

@x2¼ð/lctÞf

kf

@pf

@t; ð9Þ

pf ðx; y; z; 0Þ ¼ pi; ð10Þ

@pf

@y

��y¼ye

¼ 0; ð11Þ

pf ðx; ywf ; z; tÞ ¼ pwf ; ð12Þ

@pf

@x

��x¼0

¼ 0; ð13Þ

kf

l@pf

@x

��xwf

¼ km

l@pm

@x

��xwf

; ð14Þ

@pf

@z

��z¼z0

¼ 0; ð15Þ

and

@pf

@z

��z¼ze

¼ 0: ð16Þ

Eq. 9 is the diffusivity equation for the fracture for which Eq.10 is the initial condition. Eq. 11 is the no-flow boundary at thefracture tip. Eq. 12 states that at the wellbore, the fracture pressureis equal to the wellbore pressure and is constant. Eq. 13 states thatthere is no flow across the center of the fracture (symmetry), Eq.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Fracture

(a)

(b)

ye

ye

ywf

ywf

wf

Fracture

qf

xwf

xwf

L

L

xf

zo

ze

Matrix

Matrix

Horizontal Wellbore

Wellbore

qm

xe

xe

No-flow boundary Fracture

Fracture

xf

Fig. 1—(a) Schematic of a fractured horizontal well with planar fractures. (b) 3D schematic of a fractured horizontal well with planarfractures.



2 2015 SPE Reservoir Evaluation & Engineering

14 is identical to Eq. 4, and they have the same physical meaning.Eqs. 15 and 16 are no-flow boundary conditions at the top andbottom of the reservoir; they represent the fact that the reservoir issealed at the top and bottom boundaries.

Eqs. 1 and 9 form a coupled system of PDEs because of theboundary condition defined by Eqs. 4 and 14. We are interested indeveloping a rate–time relation for forecasting production ratefrom a system described by these sets of equations. To achievethis goal, we eliminate the spatial dependence in Eqs. 1 and 9 byintegrating over the spatial (x, y, and z) domains, respectively, andusing the boundary conditions (Walsh and Lake 2003) to obtain

ðvpmctÞm

dpm

dt¼ �qm ð17Þ

and

ðvpfctÞf

dpf

dt¼ �qf þ qm: ð18Þ

One can find the details of this derivation and its solution inOgunyomi (2015) and Appendix A. We thus transformed the sys-tem of PDEs into a system of ODEs. The problem is now a two-compartment problem.

One advantage of transforming the system of PDEs into a sys-tem of ODEs is that it is easier to solve for the producing rate.Another advantage is that it eliminates the need to know the spe-cific location and geometry/dimensions of the fracture(s).Nobakht et al. (2013) and Ambrose et al. (2011) presented amethod of forecasting production from a multifractured horizontalwell that considered planar hydraulic fractures of differentlengths. The new model from this study applies to fractures of anyarbitrary shape or geometry (planar or otherwise). Fig. 2 is a sim-plified representation of the new problem, which is a schematicrepresentation of the reservoir as a two-compartment system in se-rial flow. One can think of the first compartment as the aggregatedvolume of all the fractures in the reservoir. It is the only compart-ment connected directly to the wellbore. The average pressure inCompartment 1 is pf , and the flow rate from this compartmentinto the wellbore is qf . The second compartment is the aggregatedvolume of the reservoir matrix. The average pressure in the sec-ond compartment is pm. The matrix compartment does not com-municate directly with the wellbore; it only has a crossflow term,qm, into the fracture compartment.

The next step in the solution is to eliminate the average pres-sures in Eqs. 17 and 18 with a relationship between the averagereservoir pressure and flow rate. This step is achieved with an ana-lytical solution to the 1D linear-flow problem (Wattenbarger et al.1998; Bello 2009; Patzek et al. 2014) with constant pressure at the

fracture face—from which, the average reservoir pressure, asshown in Appendix B, is given by

p ¼ pwf þ8

p2ð pi � pwf Þ

X1n¼1

e�ð2n� 1Þ2k

4/lctL2p2t

ð2n� 1Þ2: ð19Þ

Eq. 19 is the complete solution that includes the transient andlate-time solutions. This is an important point because flow inunconventional formations exhibits long periods of transient-lin-ear flow, and a useful model must be able to predict productionfor early and late-time flow. Writing Eq. 19 for the fracture andmatrix compartments, respectively, we have

pf ¼ pwf þqfi

Jf

X1n¼1

qDfn

ð2n� 1Þ2ð20Þ

and

pm ¼ pf þqmi

Tx

X1n¼1

qDmn

ð2n� 1Þ2; ð21Þ

where

qDfn ¼ 2e�ð2n� 1Þ2kf

4ð/lÞf ctx2f

p2t

: dimensionless production rate for thenth normal mode for the fracture compartment;

qDmn¼ 2e

�ð2n� 1Þ2kf

4ð/lÞf ctx2f

p2t

: dimensionless production rate for thenth normal mode for the matrix compartment;

qfi ¼kf Af

lxfð pi � pwf Þ: initial production rate from the frac-

ture’s nth normal mode;

qmi¼ kmAm

lLð pi � pwf Þ: initial production rate from the

matrix’s nth normal mode;

Jf ¼p2

4

qfi

ð pi � pwf Þ: fracture productivity index (PI) and

Tx ¼p2

4

qm

ð pm � pf Þ: transmissibility between the fracture and

matrix compartments.

Eq. 19 was derived with the assumption that the pressure at thefracture face is constant and equal to the wellbore pressure pwf . Inwriting Eq. 21 for the matrix compartment, we have assumed thata constant pwf solution is valid even when it is changing. Thisassumption is a good approximation when there is a large contrastin permeability between the two adjacent compartments becausethe high permeability of the fracture compartment ensures a quickpressure equilibration with the wellbore pressure in the fracturecompartment, and, hence, the pressure at the interface betweenthe two compartments is approximately constant. This assumptionwas crucial in attaining the final solution.

By substituting Eqs. 20 and 21 into Eqs. 17 and 18 and aftermaking some mathematical manipulations (Ogunyomi 2015),we obtain

qfi

X1n¼1

dqfn

dt¼ � qfi

sf

X1n¼1

ð2n� 1Þ2qfn þqmi

sf

X1n¼1

ð2n� 1Þ2qmn

� � � � � � � � � � � � � � � � � � � ð22Þ

and

qmi

X1n¼1

dqmn

dt¼ Txqfi

sf Jf

X1n¼1

ð2n� 1Þ2qfn� 1

smþ Tx

sf Jf

� �qmi

�X1n¼1

ð2n� 1Þ2qmn; � � � � � � � � � � � � ð23Þ

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

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

. . . . . . . . .

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

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

qf

qm

Fracturecompartment

Horizontal wellbore

Matrix compartment

Fig. 2—A simplified representation of the double-porositymodel as a series model with two compartments (tanks) inwhich the first compartment represents the volume of the frac-ture, and the second compartment represents the volume ofthe reservoir matrix.




where

sf ¼ðvpf

ctÞfJf

: fracture time constant;

sm ¼ðvpm

ctÞmTx

: matrix time constant; and

Tx: crossflow transmissibility factor,where sf and sm are the fracture and matrix time constants,respectively. The parameters in the solution are now time con-stants and transmissibilities, not pore volumes (PVs) and perme-abilities, as in the original problem statement. Cao (2014)presented a detailed discussion of the physical meaning of timeconstants for immature and mature waterfloods; in this study,these time constants are for primary recovery and, physically,they indicate how fast the volumes in each compartment would bedrained. The index n in Eqs. 22 and 23 are the normal modes (in-dependent solutions). Consequently, we can solve the system ofODEs represented by Eqs. 22 and 23 for each mode. We then sumthese solutions to obtain the complete solution to the problem. Werewrite this system of ODEs in matrix-vector form for each nor-mal mode, as shown next, and then solve it with eigenvalue-diagonalization:

dqfn

dtdqmn

dt

0B@

1CA ¼ ð2n� 1Þ2

� 1

sf

1

sf

Tx

sf Jf� 1

sm� Tx

sf Jf

0BB@

1CCA qfn

qmn

� �:

� � � � � � � � � � � � � � � � � � � ð24Þ

The initial conditions to solve the system represented by Eq.24 are

qf ðt ¼ 0Þ ¼ qfi ð25Þ

and

qmðt ¼ 0Þ ¼ 0: ð26Þ

Eq. 25 simply states that, at time, t ¼ 0, the production ratefrom the fracture volume is equal to a finite value of qfi . AlthoughEq. 26 states that, at time, t ¼ 0, the production rate from the ma-trix volume is equal to zero, qmi

¼ 0. This is because, at time zero,the pressure everywhere in the formation is equal to the initial res-ervoir pressure, and, as a result, there is no pressure gradient forflow from the matrix into the fracture because the pressure of thefracture/matrix interface is still at the initial reservoir pressure.

We solved Eq. 24 by the eigenvalue-decomposition method, asshown in Appendix C, to obtain the following expression:

qf ¼ qfic

c� qek2t � qfi

qc� q

ek1t

þ qfi

4

cc� q

ffiffiffippffiffiffiffiffiffiffiffiffiffi�k2tp erfcð3

ffiffiffiffiffiffiffiffiffiffi�k2t

pÞ

� qfi

4

qc� q



pÞ: ð27Þ

Eq. 27 is the approximate analytical solution to the double-porosity model. The negative sign under the square root of theeigenvalues in Eq. 27 is necessary because the eigenvalues arealways negative. One can find details of the derivation in Ogu-nyomi (2015). The definition of the model parameters are sum-marized next:

k1 ¼1

2

�� 1

sf� 1

sm� Tx

sf Jf

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

sf� 1

sm

� �2

þ 2Tx

s2f Jfþ 2Tx

smsf Jfþ Tx

sf Jf

� �2s �

; � � � � ð28Þ

k2 ¼1

2

�� 1

sf� 1

sm� Tx

sf Jf

�


sf� 1

sm

� �2

þ 2Tx

s2f Jfþ 2Tx

smsf Jfþ Tx

sf Jf

� �2s �

; � � � � ð29Þ

q ¼ sf Jf

2Tx

�� 1

sfþ 1

smþ Tx

sf Jf

þ


sf� 1

sm

� �2

þ 2Tx

s2f Jfþ 2Tx

smsf Jfþ Tx

sf Jf

� �2s �

; � � � � ð30Þ

and

c ¼ sf Jf

2Tx

�� 1

sfþ 1

smþ Tx

sf Jf

�


sf� 1

sm

� �2

þ 2Tx

s2f Jfþ 2Tx

smsf Jfþ Tx

sf Jf

� �2s �

: � � � � ð31Þ

Eqs. 28 and 29 are the eigenvalues of the A matrix of the sys-tem of ODEs in Eq. 24. These equations show the mathematicalrelationship between the eigenvalues and the fracture, matrix timeconstants, the transmissibility coefficient between the fracture andthe matrix compartment, and the ratio of their permeabilities.Note that q and c are the first components of the eigenvector cor-responding to the eigenvalues in Eqs. 28 and 29, and the othercomponents are unity. Physically, k1 and k2 are the time constantsof an equivalent parallel flow model that will yield the sameresults as the original problem when appropriately weighted withthe eigenvectors. One can regard Eqs. 28 through 31 as expres-sions for scaling parameters that one can use to transform a two-compartment series flow model into a two-compartment parallel-flow model without crossflow. A generalization of this solution tothree compartments is available in Ogunyomi (2015).

Model Validation

We validate the approximate analytical solution to the double-porosity model, represented by Eq. 27, with a synthetic case. Wedeveloped it with a commercial black-oil, finite-difference simu-lator. The model used in the synthetic case was 2D and has twoadjoining reservoir compartments in which the compartment con-taining the producing well has a higher permeability than the sec-ond compartment (Fig. 3).

One can think of the compartment with the high permeabilityas the aggregated collection of the fracture volume whereas thesecond compartment represents the reservoir matrix with lowerpermeability. The simulation uses spatially resolved permeability

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

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

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

0 1,000 2,000

Permeability l (md) 2016-12-01

3,000 4,000 5,000

0 1,000 2,000 3,000 4,000

0.00 615.00 1230.00 ft

0.00 190.00 380.00

5,000

2,00

01,

000

00

–1,0

00

–2,000–1,000

1,0000

Test-1

91

100

82

73

64

55

46

37

28

19

10

K layer: 1

md

meters

Fig. 3—Reservoir grid for the synthetic case showing the per-meability field. The gridblocks in green are the high-permeabil-ity compartment, and the blue grids are the low-permeabilitycompartment.




cells, which is the main difference between it and the approximateanalytical solution. The volume of the fracture (high permeability)compartment is equal to 25% of the volume of the matrix (lowpermeability) compartment. All other properties are identical forthe two compartments. Table 1 summarizes the other inputs forthe synthetic case.

To validate the approximate analytical model, the productionrate obtained from running the synthetic case is matched to theproduction rate obtained from the approximate analytical solution.Fig. 4 presents a comparison of the production rate obtained fromthe synthetic case and the approximate analytical solution.

The production history in Fig. 4 exhibits two time scales; thefirst time scale initially starts as a straight line with a slope of one-half, which indicates transient-linear flow in one dimension. Thisflow regime is followed by an exponential decline that indicatesboundary-dominated flow (BDF) from the first compartment. Af-ter the dissipation of BDF from the first compartment, the secondcompartment starts with an expected straight line with half-slopesignature. This transient flow regime is then followed by an exter-nal BDF regime from the second compartment. The agreementshown in Fig. 4 is well within engineering accuracy. Table 2 sum-marizes the fitting parameters for the analytical solution.

Comparison of Approximate Analytical Solution With the

Laplace-Space Solution. In this section, we present a compari-son of the analytical solution derived in the previous section withthe Laplace-space solution to the double-porosity model. The so-lution to Eqs. 1 through 14 with Laplace transforms is given as

pfDðyD; sÞ ¼cosh½

ffiffiffiffiffiffiffiffiffiffisf ðsÞ

pyD�

scosh½ffiffiffiffiffiffiffiffiffiffisf ðsÞ

p�; ð32Þ

where

f ðsÞ¼xþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikð1�xÞ

3s

rtanh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð1�xÞs

k

r" #: interporosity trans-

fer function;

pD ¼p� pi

pwf � pi: dimensionless pressure;

tD ¼kf t

l½ð/ctÞf þ ð/ctÞm�x2f

: dimensionless time;

xD ¼x� xe

xwf � xe: dimensionless distance in the x-direction;

yD ¼y� ye

yf � ye: dimensionless distance in the y-direction;

k ¼ 12km

kf

xf

L

� 2

: interporosity transfer parameter; and

x ¼ð/ctÞf

ð/ctÞf þ ð/ctÞm: storativity ratio.

The solution given by Eq. 32 is the constant-pressure solution;that is, it assumes an instantaneous constant pressure at the frac-ture face. One can find the details of the derivation of this solutionin Bello (2009). A problem with using this solution is that it can-not be inverted back into the real-time space to obtain a closed-form analytical solution; hence, we use a numerical-inversionalgorithm to compute pressures and rate from this solution. Fromthis solution, we obtain the production rate at the fracture face bytaking the derivative of Eq. 32 and evaluating its value at the frac-

ture face; that is,dpfD

dyD

��yD¼1

qDðyD ¼ 1; sÞ ¼ffiffiffiffiffiffiffiffif ðsÞ

s

rtanh½

ffiffiffiffiffiffiffiffiffiffisf ðsÞ

p�: ð33Þ

Bello (2009) provided a detailed sensitivity analysis on Eq. 33to understand how the model parameters affect its productioncharacteristics. We summarize the result of this sensitivity analy-sis next. Fig. 5 is a plot of the dimensionless rate vs. dimension-less time in which the interporosity transfer parameter and thestorativity ratio were varied. One can explain the physical mean-ing of the general characteristic observed on this plot as follows:

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

Table 1—Summary of input parameters for the synthetic case as used in the commercial simulator.

1

1,000

1,000,000

1 100 10,000

Oil

Rat

e (s

tb/d

)

Time (days)

Fracturetransient flow

Fractureboundary

effect

Matrixtransient

flow

Matrix/systemboundary

effect

Simulation Data

Approximate Analytical

Fig. 4—Comparison of the production rate from the syntheticcase and the approximate analytical model. The graph in redrepresents the production rate predicted by the analytical solu-tion, whereas the graph in blue is the production rate from thesynthetic case. Table 2—Model parameters used in the validation case.




At the start of production, flow is predominantly linear with aslope of one-half, which represents transient flow from the frac-ture. Thereafter, an exponential decline period sets in when theeffect of the fracture boundary is felt. After this flow period,another linear-flow period starts (also characterized by a one-halfslope), representing transient flow from the reservoir matrix. Afterthis transient-flow period, another exponential decline period isobserved, and this is the effect of the external boundary of the ma-trix (Walsh and Lake 2003).

We now compare the production rate from the approximateanalytical solution, Eq. 27, to the production rate from theinverted Laplace-space analytical solution, Eq. 33. For the ap-proximate analytical solution to be useful, it should reasonablyreproduce the observed characteristics in Fig. 5. The results ofthis comparison for three cases are shown in Fig. 6.

In Fig. 6, Case 1 shows the history match for x¼ 10�3 andk¼ 10�5; Case 2 shows the history match for x¼ 10�1 andk¼ 10�5, and Case 3 shows the history match for x¼ 10�1

and k¼ 10�9. Clearly, Fig. 6 suggests that the production ratepredicted by the approximate analytical solution provides agood match to the production rate predicted by the Laplace-space solution.

Analysis of Model Parameters

Physical Meaning of the Model Parameters. The definition ofthe time constants in the approximate analytical solution is identi-cal in definition to that defined in the capacitance/resistancemodel (CRM); one can find details about the CRM in Sayapour(2008), Nguyen (2012), and Cao et al. (2014). As a result, we con-clude that it has a similar physical meaning. In the CRM, Nguyen(2012) and Cao et al. (2014), following the work of Seborg et al.(2003), defined the time constant to be the time it takes for 63.2%of a pressure pulse input to be observed as the output. The inputpulse for our model would be the pressure difference that is re-sponsible for flow. In the CRM, it is the injection rate.

Inferring Fracture and Matrix Volume From Model

Parameters. In this subsection, we investigate the possibility ofestimating the size (volume) of the fractures induced by the hy-draulic fracturing and the reservoir matrix from the model param-eters with the approximate analytical solution. To accomplish thistask, we took the following steps:

1. Built a numerical model with two compartments in whichone compartment has a high permeability and the secondcompartment has a low permeability with a commercial res-ervoir simulator.

2. Performed a history match of the production rate from thenumerical model to the approximate analytical solution toobtain the model parameters.

3. Changed the relative volume of each compartment in thenumerical-simulation model while keeping the total PVconstant and repeated Step 2.

4. After obtaining the model parameters for a few cases wemade a crossplot of each model parameter with the volumeof each compartment, as defined in the numerical simula-tion model.

The model parameters considered for this analysis are the frac-ture-compartment time constant (sf ), the matrix-compartmenttime constant (sm), and the initial-fracture production rate (qfi).The numerical model used for this analysis is identical to that pre-sented in Fig. 3, and the model used is Eq. 27. The result of thisnumerical experiment is summarized in Table 3; it presents asummary of the cases considered and the model parametersobtained from the history-matching exercise.

Fracture Time Constant. Fig. 7a presents the crossplot of thefracture time constant and the PV of the high-permeability com-partment, whereas Fig. 7b presents the same for the low-perme-ability compartment. From Fig. 7a, as the PV of the high-permeability compartment increases, the fracture time constantincreases, indicating a positive correlation between them. Thecoefficient of determination is large, R2¼ 0.98. Recall that thefracture time constant is defined as sf ¼ ðvpf

ctÞf =Jf , where vpfis

the fracture PV. This definition of the fracture time constant sug-gests that we can infer the size of the fracture volume from thevalue of the fracture time constant. In contrast, Fig. 7b suggests

10–1010–15

10–10

10–5

100

105

10–5

Dimensionless Time (tD)100 105 1010

ω = 1E–1 & λ = 1E–3ω = 1E–1 & λ = 1E–5ω = 1E–1 & λ = 1E–7ω = 1E–1 & λ = 1E–9ω = 1E–3 & λ = 1E–3ω = 1E–3 & λ = 1E–5ω = 1E–3 & λ = 1E–7ω = 1E–3 & λ = 1E–9

Dim

ensi

onle

ss P

rodu

ctio

n R

ate

(qD

)

Fig. 5—Effect of storativity ratio (x) and interporosity transferparameter (k) on the production rate from the double-porositymodel.

1.E–20

1.E–15

1.E–10

1.E–05

1.E+00

1.E+05

1.E–12 1.E–08 1.E–04 1.E+00 1.E+04 1.E+08

Oil

Rat

e (s

tb/d

)

Time (days)

Laplace-space solution (ω = 1E–3, λ = 1E–5)

Approximate analytical solution





Fig. 6—Comparison of production rate from the approximateanalytical solution to that from the Laplace-space solution.

τ τ λ λ

Table 3—Summary of model parameters obtained from the numerical experiments performed to investigate the possibility of inferring

fracture and matrix volumes by use of the approximate analytical solution.




that the fracture time constant decreases with increasing PV of thelow-permeability compartment. This figure also has a high coeffi-cient of determination, R2¼ 0.98. This observation is because ofthe fact that the fracture volume shares a boundary with the matrixvolume, and this shared boundary was held constant during thisexperiment whereas the other boundaries changed.

Matrix Time Constant. Fig. 8a is the crossplot of the matrixtime constant and the PV of the high-permeability compartment,whereas Fig. 8b represents the same for the low-permeabilitycompartment.

Fig. 8a suggests that, as the PV of the high-permeability com-partment increases, the matrix time constant decreases. This rela-tionship indicates a negative correlation between them. Thecoefficient of determination is high, R2¼ 1.0, suggesting thatthere is a relationship between the size of the fracture volume andthe matrix time constant. This transmissibility factor is a functionof the fracture dimension. From Fig. 8b, we observe that as thePV of the low-permeability compartment increases, the matrixtime constant increases. This observation is expected because, inthe definition of the matrix time constant, sm ¼ ðvpm

ctÞm=Tx, thematrix time constant is directly related to the matrix PV, vpm

. Thiscrossplot also has a high coefficient of determination.

Initial Production Rate. Fig. 9a presents a crossplot of theinitial production rate and the PV of the high-permeability com-partment. Fig. 9b is the crossplot of the initial production rate andthe PV of the low-permeability compartment. From Fig. 9a, as thePV of the high-permeability compartment increases, the initialproduction rate from the fracture decreases, indicating a negative

correlation between them. The coefficient of determination ishigh, R2¼ 0.84, suggesting that we can infer the size of thefracture volume from the initial production rate from the fracture.The definition of the initial production rate is given as qfi ¼ðpi � pwf Þkf Af =lLf . We note that Af ¼ hwf and vpf

¼ /Af Lf inthe numerical-simulation model. In the experiments conducted,when we increased the fracture PV, we increased Lf and as thisvariable is in the denominator of the definition of qfi . From thisdefinition, there is an inverse relationship between the initial pro-duction rate and the fracture volume, which explains the observa-tion in Fig. 9a.

From Fig. 9b, we see that, as the PV of the low-permeability(matrix) compartment increases, the initial production rate fromthe fracture increases. This observation is consistent with the factthat the total PV of the reservoir was kept constant, which impliesthat, by increasing the fracture PV, we decrease the matrix PV,vpm¼ /Af ðL� Lf Þ ¼ vpT

� vpf. Therefore, the initial production

rate should increase as the matrix PV is increased. With the goodcorrelation, we can estimate the matrix PV from the initial pro-duction rate.

Model Application to Field Data

We present example applications of the approximate solution tofield data and demonstrate how to use it to estimate reserves fromhydraulically fractured horizontal wells in liquid-rich unconven-tional formations. For convenience, we rewrite Eq. 27, as shownnext:

(b)(a)

R² = 0.98

0

100

200

300

Mod

el T

ime

Con

stan

t (t f(

t–1))

Mod

el T

ime

Con

stan

t (t f(

t–1))

High-Perm Pore Volume (ft3)

R² = 0.98

0

100

200

300

0.E+00 2.E+07 4.E+07 0.E+00 3.E+07 5.E+07

Low-Perm Pore Volume (ft3)

Fig. 7—(a) The crossplot of the fracture time constant and the PV of the high-permeability compartment in the numerical model. (b)The crossplot of fracture time constant and the PV of the low-permeability compartment in the numerical model.

R² = 1.00

0

2500

5000

7500

10000R² = 1.00

0

2500

5000

7500

10000

(b)(a)

Mod

el T

ime

Con

stan

t (t m

(t–1

))

Mod

el T

ime

Con

stan

t (t m

(t–1

))


0.E+00 2.E+07 4.E+07 0.E+00 3.E+07 5.E+07


Fig. 8—(a) The crossplot of matrix time constant and the PV of the high-permeability compartment in the numerical model. (b) Thecrossplot of matrix time constant and the PV of the low-permeability compartment in the numerical mode.




qf ¼ q1ie�k1t þ

q01iffiffitp erfc

3

2pffiffiffiffiffiffiffiffiffiffi�k1t

p� �þ q2i

e�k2t

þq02iffiffi

tp erfc

3

2pffiffiffiffiffiffiffiffiffiffi�k2t

p� �; � � � � � � � � � � � � � � ð34Þ

where q1i¼ qfi

cc� q

, q2i¼ qfi

cc� q

, q01i¼ qfi

4

cc� q

ffiffiffippffiffiffiffiffiffiffiffiffi�k2

p , and

q02i¼ qfi

4

cc� q

ffiffiffippffiffiffiffiffiffiffiffiffi�k2

p .

The model was fitted to production rate data from 88 hydrauli-cally fractured horizontal wells (Ogunyomi 2015). All the fitsobtained were within the limits of engineering accuracy. To applythe model to field data from a well, we fit the model to availablehistorical production-rate data from the well to obtain the modelparameters by minimizing the squared difference between themodel estimates and the field-production data—that is,

minðqData � qmodelÞ2—by changing sf ; sm, andkf Tx

kmJf. After obtain-

ing the model parameters, we proceed to forecast future productionrates and cumulative production until 100,000 days. We presenttwo example applications of the model to this data set. The proce-dure/work flow to apply the model is summarized next.

Application Work-Flow.

1. Make log-log diagnostic plots of rate vs. time and, when avail-able, flowing bottomhole pressure vs. time. One can use tubing-head pressure as a proxy for flowing bottomhole pressure.

2. Analyze the diagnostic plots from Step 1 to identify usefulinformation such as when the fracture boundary is felt orthe onset of transient linear flow from the matrix.

3. To apply the model to field data from a well, we fit themodel to available historical production-rate data from thewell to obtain the model parameters by minimizing the sumof squared difference between the model estimates and the

field-production data; that is, minðqData � qmodelÞ2 by

changing sf ; sm andkf Tx

kmJf.

a. First, manually change model parameters

�sf ; sm and

kf Tx

kmJf

�until there is a good match on the exponential

decline portion of the first time scale. This step providesa good starting point.

b. Then, call on solver to perform the minimization.4. Repeat previous steps until a satisfactory match is obtained.5. Make forecast of rate and EUR with the obtained model

parameters.

Example 1. For this example, we summarized the well details inTable 4. This well has been on production for 1,136 days. Fig. 10apresents the production rate on a log-log plot. The figure suggeststhat the production rate is relatively constant until approximately90 days after which the production rate from the well declinedexponentially until it started declining with a slope of one-half. Ifwe assume that production during the first 100 days is from thefracture volume and the production after 100 days is from the ma-trix, then we can match the model to these data, making sure thatwe match the exponential decline and the half-slope portions ofthe data. Fig. 10b presents the results of the rate history match andfuture performance. This figure contains three plots, the originalproduction data (red markers), the history match (green-coloredmarkers), and the forecast (black markers). We summarized themodel parameters obtained from the history-match exercise inTable 5. The mismatch at the start of the production history isbecause the well was produced at a variable bottomhole pressureduring this period whereas the model presented is based on theassumption that the wellbore pressure is constant. After obtainingthe model parameters from the history-match exercise, we use themodel to forecast future production rate and reserves until 10,000days. Fig. 10c presents the performance-forecasting results.

Example 2. We summarized the well details for this example inTable 4. This well has been on production for 531 days. The pro-duction rate from this well is shown in Fig. 11a on a log-log plot.This figure suggests that the production rate is relatively constantuntil approximately 10 days after which the well declined expo-nentially until it started declining with a slope of one-half. Again,if we assume that production during the first 100 days is from thefracture volume and the production after 100 days is from the ma-trix, then we can match the model to these data, making sure wematch the exponential decline and the half-slope portions of thedata. The result of the production-rate history match is shown inFig. 11b and that of future performance in Fig. 11c. This figurecontains three plots: the original production data (red markers),the history match (green-colored markers), and the forecast (blackmarkers). From this figure, we matched the exponential declineportion of the rate data, and we also matched the linear declineportion of the rate data. We summarized the model parametersobtained from the history-match exercise in Table 5. After

R² = 0.84

0

2500

5000

7500

10000

Initi

al P

rodu

ctio

n R

ate

(stb

/d)

Initi

al P

rodu

ctio

n R

ate

(stb

/d)

R² = 0.84

0

2500

5000

7500

10000

(b)(a)


0.E+00 2.E+07 4.E+07 0.E+00 3.E+07 5.E+07


Fig. 9—(a) The crossplot of initial production rate and PV of the high-permeability compartment in the numerical model. (b) Thecrossplot of initial production rate and the PV of the low-permeability compartment in the numerical model.

Table 4—Well data for Examples 1 and 2.




obtaining the model parameters from the history-match exercise,we used the model to forecast future production rate and cumula-tive production until 8,000 days. The result of the forecasting pro-cess is shown in Fig. 11c.

The model parameters obtained from these two examples andothers not shown here are all functions of the well and reservoirproperties. Consequently, the forecasted results have a high degreeof confidence, particularly when the fracture/reservoir interfacewas observed in the production-rate data, which provided an op-portunity for defining the geometry of the adjoining compartment.

Discussion

A generally accepted conceptual model for fractured horizontalwells is that a stimulated reservoir volume (SRV) develops aroundthe fractured well, and there is a region of undamaged reservoirbeyond the SRV (Miller et al. 2010). The SRV is expected to becomposed of a complex network of fractures of different geome-tries, ranging from (for example) planar, curved, and slanted andof different lengths. However, for ease of solution, existing“physics”-based models assume that the hydraulic fractures areplanar and perpendicular to the wellbore. The new solution pre-sented in this work overcomes this limitation of existing modelsbecause the assumption of planar fractures is not necessary.

Most empirical models do not account for the second timescale, and the end of transient linear flow must be determinedarbitrarily before switching to a BDF model. The new solutionpresented here also eliminates this limitation of empirical models.For cases in which there are no production data from the secondtime scale, one should use the single-porosity solution. This solu-tion is shown next:

qD ¼ 2e�

p2tD

4 þerfc

3

2pffiffiffiffiffitD

p� �ffiffiffiffiffiffiffiptDp : ð35Þ

In Eq. 35, the first term accounts for BDF and the second termis the transient solution. The dimensionless time tD is defined ast s= , where s is the time constant of Compartment 1 and t is time.The details of the derivation of Eq. 35 are in Ogunyomi (2015).

We showed that the model parameters derived from the use ofthe new solution are functions of the reservoir and well-comple-tion properties. Particularly, one can use the model parameters toestimate the drainage volume of a well; this characteristic of themodel could have potential application in infill-drilling and well-spacing optimization studies. Because the model has a closed ana-lytical form, it is especially suited for optimization studies thataccount for uncertainties in reservoir properties and the outcomeof well-stimulation (hydraulic-fracturing) treatments.

Conclusions

The main goal of this work was to develop a rate-vs.-time rela-tionship to predict realistic future performance from hydraulicallyfractured horizontal wells in unconventional formations. We sum-marize the major findings next:• A simple rate-vs.-time relationship is presented to predict well

performance in stimulated unconventional reservoirs with a dou-ble-porosity model. The model developed is valid for all flowregimes (transient and pseudosteady state, including the transitionperiod), and it is a function of reservoir and well properties.

• The solution presented, although approximate, was validatedwith numerical flow-simulation results and was shown to accu-rately reproduce the production-rate history and the EUR.

• The model identifies different flow regimes observed both inthe synthetic and field-production data, thereby largely over-coming limitations of other empirically derived models. One ofthe model’s attributes is that it always extrapolates to a finitecumulative-production volume.

• We also demonstrated the utility of the model for practicing engi-neers by presenting example applications to field-production data.

Nomenclature

f ðsÞ ¼ Laplace-space interporosity-transfer function,dimensionless

Jf ¼ fracture productivity index, B/D/psikf ¼ effective fracture permeability, mdkm ¼ effective matrix permeability, md

n ¼ index for normal modepD ¼ dimensionless pressurepf ¼ fracture pressure, psipf ¼ average pressure in fracture compartment, psipi ¼ initial reservoir pressure, psi

pm ¼ average pressure in matrix compartment, psipm ¼ matrix pressure, psi

pwf ¼ bottomhole flowing well pressure, psi

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

(a)

(b)

(c)

10

100

1,000

1 10 100 1,000 10,000

Oil

Rat

e (s

tb/d

)

Time (days)

0.01

0.10

1.00

10.00

100.00

1,000.00

10,000.00

Oil

Rat

e (s

tb/d

)

Time (days)

q-Data

q-Model (Forecast)

q-Model (History match)

0

100

200

300

400

500

1 10 100 1,000 10,000 100,000

0 50,000 100,000 150,000 200,000

Oil

Rat

e (s

tb/d

)

Cumulative Production (stb)

DataModel -forecastModel -history match

Effect ofvariable Pwf

half slope

Matrixtransient flow


Fracture boundary effect

Fig. 10—Summary of production profiles for Example 1: (a)presents raw data on the log-log diagnostic plot, (b and c)show history-matching and forecasting results on log-log andCartesian plots.

Table 5—Summary of model parameters for Examples 1 and 2.




qfi ¼ initial production rate from the fracture’s nth normalmode, dimensionless

qfn ¼ production rate for the nth normal mode for the fracturecompartment, dimensionless

qmn¼ production rate for the nth normal mode for the matrix

compartment, dimensionlessqmi¼ initial production rate from the nth normal mode of the

matrix, dimensionlesstD ¼ dimensionless timeTx ¼ transmissibility factor between fracture and matrix com-

partment, B/D/psixD ¼ dimensionless distance in the x-directionyD ¼ dimensionless distance in the y-directionc ¼ first element of the eigenvector corresponding to k2; the

other element is unity, dimensionlessk1; k2 ¼ eigenvalues of the A matrix for the system of ODEs,

days�1

k ¼ interporosity transfer parameter, dimensionlessx ¼ storativity ratioq ¼ first element of the eigenvector corresponding to k1; the

other element is unity, dimensionlesssf ¼ fracture time constant, dayssm ¼ matrix time constant, days

Acknowledgments

This work is supported by the sponsors of the Center for Petro-leum Asset Risk Management at the University of Texas at Aus-tin. We thank Hess Corporation for providing data for this workand the Computer Modelling Group (CMG) for use of their black-oil simulator. Larry W. Lake holds the Sharon and Shahid UllahChair at the Center for Petroleum and Geosystems Engineering at

the University of Texas at Austin. Tad Patzek holds the CockrellFamily Regents Chair in Engineering and the Lois K. and RichardD. Folger Leadership Chair at the Department of Petroleum andGeosystems Engineering.

References

Ambrose, R. J., Clarkson, C. R., Youngblood, J. E. et al. 2011. Life-Cycle

Decline Curve Estimation for Tight/Shale Gas Reservoirs. Presented at

the SPE Hydraulic Fracturing Technology Conference and Exhibition,

The Woodlands, Texas, USA, 24–26, January. SPE-140519-MS.

http://dx.doi.org/10.2118/140519-MS.

Barenblatt, G. I. and Zheltov, Y. P. 1960. Fundamental Equations of Fil-

tration of Homogeneous Liquids in Fissured Rocks. In Soviet PhysicsDoklady Vol. 5, page 522.

Bello, R. O., and Wattenbarger, R. A. 2008. Rate Transient Analysis in Natu-

rally Fractured Shale Gas Reservoirs. Presented at the CIPC/SPE Gas

Technology Symposium 2008 Joint Conference, Calgary, Alberta, Can-

ada, 16–19 June. SPE-114591-MS. http://dx.doi.org/10.2118/114591-MS.

Bello, R. O. 2009. Rate Transient Analysis in Shale Gas Reservoirs With

Transient Linear Behavior. PhD dissertation, Texas A&M University,

College Station, Texas, USA.

Cao, F. 2014. Development of a Coupled Two-Phase Flow CapacitanceResistance Model. Unpublished PhD dissertation, The University of

Texas at Austin, Austin, Texas, USA.

Cao, F., Luo, H., and Lake, L. W. 2014. Development of a Fully Coupled

Two-Phase Flow Based Capacitance-Resistance Model (CRM). Pre-

sented at the SPE Improved Oil Recovery Symposium, Tulsa, Okla-

homa, USA, 12–16 April. SPE-169485-MS. http://dx.doi.org/10.2118/

169485-MS.

Carlson, E. S. and Mercer, J. C. 1991. Devonian Shale Gas Production:

Mechanisms and Simple Models. J Pet Technol 43 (4): 476–482. SPE-

19311-PA. http://dx.doi.org/10.2118/19311-PA.

de Swaan-O., A. 1976. Analytic Solutions for Determining Naturally Frac-

tured Reservoir Properties by Well Testing. SPE J 16 (3): 117–122;

Trans., AIME 261. SPE-5346-PA. http://dx.doi.org/10.2118/5346-PA.

El-Banbi, A. H. 1998. Analysis of Tight Gas Wells. PhD dissertation,

Texas A&M University, College Station, Texas, USA.

Mayerhofer, M. J., Lolon, E. P., Youngblood, J. E. et al. 2006. Integration of

Microseismic Fracture Mapping Results With Numerical Fracture Net-

work Production Modeling in the Barnett Shale. Presented at the SPE An-

nual Technical Conference and Exhibition, San Antonio, Texas, 24–27

September. SPE-102103-MS. http://dx.doi.org/10.2118/102103-MS.

Miller, M. A., Jenkins, C., and Rai, R. 2010. Applying Innovative Production

Modeling Techniques to Quantify Fracture Characteristics, Reservoir

Properties and Well Performance in Shale Gas Reservoirs. Presented at

the Eastern Regional Meeting, Morgantown, West Virginia, USA, 12–14

October. SPE-139097-MS. http://dx.doi.org/10.2118/139097-MS.

Nguyen, A. P. 2012. Capacitance Resistance Modeling for Primary Re-covery, Waterflood, and Water/CO2 Flood. PhD Dissertation, The Uni-

versity of Texas at Austin, Austin, Texas, USA.

Nobakht, M., Ambrose, R., Clarkson, C.R. et al. 2013. Effect of Completion

Heterogeneity in a Horizontal Well With Multiple Fractures on the

Long-Term Forecast in Shale-Gas Reservoirs. J Can Pet Technol 52 (6):

417–425. SPE-149400-PA. http://dx.doi.org/10.2118/149400-PA.

Ogunyomi, B. A. 2015. Simple Mechanistic Modeling of Recovery From

Unconventional Reservoirs. Unpublished PhD dissertation, The Uni-

versity of Texas at Austin, Austin, Texas, USA.

Ozkan, E., Ohaeri, U., and Raghavan, R. 1987. Unsteady Flow to a Well Pro-

duced at a Constant Pressure in a Fractured Reservoir. SPE Form Eval 2

(2): 186–200. SPE-9902-PA. http://dx.doi.org/10.2118/9902-PA.

Patzek, T. W., Male, F., and Marder, M. 2014. Gas Production From Bar-

nett Shale Obeys a Simple Scaling Theory. In Proc., National Acad-

emy of Sciences of the United States of America. PNAS, December 3,

2013. 110 (49): 19731–19736. http://dx.doi.org/10.1073/

pnas.1313380110.

Samandarli, O., Al-Ahmadi, H., and Wattenbarger, R. A. 2011. A New

Method for History Matching and Forecasting Shale Gas Reservoir

Production Performance With a Dual-Porosity Model. Presented at the

SPE North American Gas Conference and Exhibition, The Woodlands,

Texas, USA, 12–16 June. SPE-144335-MS. http://dx.doi.org/10.2118/

144335-MS.

(a)

(b)

(c)

10

100

1,000O

il R

ate

(stb

/d)

Time (days)

1.E–05

1.E–03

1.E–01

1.E+01

1.E+03

1.E+05

Oil

Rat

e (s

tb/d

)

Time (days)

q-Dataq-Model (Forecast)q-Model (History match)

0

200

400

600

800

1 10 100 1,000 10,000

0 1 10 100 1,000 10,000

0 20,000 40,000 60,000 80,000

Oil

Rat

e (s

tb/d

)

Cumulative Production (stb)

DataModel -forecastModel -history match

Matrixtransient flow


Fracture boundary effect

Fig. 11—Summary of production-history profiles for Example 2:(a) presents raw data on the log-log diagnostic plot, (b and c)show history-matching and forecasting results on log-log andCartesian plots.




http://dx.doi.org/10.2118/140519-MS




http://dx.doi.org/10.2118/19311-PA






http://dx.doi.org/10.1073/pnas.1313380110

http://dx.doi.org/10.1073/pnas.1313380110



Sayapour, M. 2008. Development and Application of Capacitance-Resist-ance Models to Water/CO2 Floods. PhD dissertation, The University

of Texas at Austin, Austin, Texas.

Seborg, D. E., Edgar, T. F., and Mellichamp, D. A. 2003. ProcessDynamic and Control, second edition. New York: John Wiley & Sons,

Inc. ISBN-13: 9780471000778.

Song, D. H. 2014. Using Simple Models to Describe Production FromUnconventional Reservoirs. Master’s thesis, The University of Texas

at Austin, Austin, Texas, USA.

Stehfest, H. 1970. Numerical Inversion of Laplace Transforms. Communi-

cations of the ACM 13 (1) 47–49. http://dx.doi.org/10.1145/

361953.361969.

Walsh, M. P. and Lake, L. W. 2003. A Generalized Approach to Primary

Hydrocarbon Recovery. Elsevier. ISBN: 978-0-444-50683-2.

Warren, J. E. and Root, P. J. 1963. The Behavior of Naturally Fractured

Reservoirs. SPE J. 3 (3): 245–255. SPE-426-PA. http://dx.doi.org/

10.2118/426-PA.

Wattenbarger, R. A., El-Banbi, A. H., Villegas, M.E. et al. 1998. Produc-

tion Analysis of Linear Flow Into Fractured Tight Gas Wells. Pre-

sented at the 1998 SPE Rocky Mountain Regional/Low-Permeability

Reservoirs Symposium and Exhibition, Denver, USA, 5–8 April. SPE-

39931-MS. http://dx.doi.org/10.2118/39931-MS.

Appendix A: Conversion of the Coupled PDEs to aSystem of Coupled ODEs

Here, we present the details of how the coupled system of PDEswas converted into a system of coupled ODEs that is independentof the spatial variables. We eliminate the spatial dependence inEq. 1 by integrating with respect to x, y, and z with x varying fromxwf to xe, y from ywf to ye, and z from z0 to ze:

ðze

z0

ðye

ywf

ðxe

xwf

@2pm

@z2þ @

2pm

@y2þ @

2pm

@x2

� �dxdydz

¼ ð/lctÞmkm

ðze

z0

ðye

ywf

ðxe

xwf

@pm

@tdxdydz: ðA-1Þ

With the fact that t is independent of position, we can bringthe time derivative on the right side of Eq. A-1 outside the integralto obtain

ðze

z0

ðye

ywf

ðxe

xwf

@

@z

@pm

@z

� �dxdydzþ

ðze

z0

ðye

ywf

ðxe

xwf

@

@y

@pm

@y

� �dxdydz

þðze

z0

ðye

ywf

ðxe

xwf

@

@x

@pm

@x

� �dxdydz ¼ /lctð Þm

km

@

@t

ðze

z0

ðye

ywf

ðxe

xwf

pmdxdydz

0B@

1CA:

� � � � � � � � � � � � � � � � � � � ðA-2Þ

We define the average pressure in the matrix as

pm ¼

ðze

z0

ðye

ywf

ðxe

xwf

pmdxdydz

ðze

z0

ðye

ywf

ðxe

xwf

dxdydz

ðA-3Þ

and

)ðze

z0

ðye

ywf

ðxe

xwf

pmdxdydz¼ðze

z0

ðye

ywf

ðxe

xwf

dxdydz

0B@

1CApm ¼ vbm

pm;

� � � � � � � � � � � � � � � � � � � ðA-4Þ

where vbmis the bulk volume of the reservoir matrix. With the

substitution of Eq. A-4 into Eq. A-2 and by carrying out the inte-gral on the left, we achieve

ðxe

xwf

ðye

ywf

@pm

@z

��ze

� @pm

@z

��z0

!dydxþ

ðze

z0

ðxe

xwf

@pm

@y

��ye

� @pm

@y

��ywf

!dxdz

þðze

z0

ðye

ywf

@pm

@x

��xe

� @pm

@x

��xwf

!dydz ¼ /lctð Þmvbm

km

dpm

dt:

� � � � � � � � � � � � � � � � � � � ðA-5Þ

With the boundary conditions defined by Eqs. 3, 5, 6, 7, and 8,we can simplify Eq. A-5 to

�ðze

z0

ðye

ywf

@pm

@x

��xwf

dydz¼ /lctð Þmvbm

km

dpm

dt: ðA-6Þ

Multiplying both sides of Eq. A-6 bykm

l, we simplify Eq.

A-6 to

�ðze

z0

ðye

ywf

km

l@pm

@x

��xwf

dydz¼ /ctð Þmvbm

dpm

dt: ðA-7Þ

In Eq. A-7, we note that, from Darcy’s law, qm ¼ðze

z0

ðye

ywf

km

l@pm

@x

��xwf

!dydz and vpm

¼ /mvbm, where vpm

is the effec-

tive matrix PV. Therefore, we can rewrite Eq. A-7 as

�qm ¼ vpmct

�m

dpm

dt: ðA-8Þ

In Eq. A-8, pm is the average pressure in the reservoir matrixand qm is the net flow rate from the reservoir matrix.

Performing the same integration over the x, y and z domains ofthe fracture (Eq. 7), x from x ¼ 0 to xwf , y from ywf to ye, and zfrom z0 to ze, one can obtain

ðze

z0

ðye

ywf

ðxwf

0

@2pf

@z2þ @

2pf

@y2þ @

2pf

@x2

� �dxdydz

¼/lctð Þf

kf

ðze

z0

ðye

ywf

ðxwf

0

@pf

@tdxdydz: ðA-9Þ

Evaluating the integrals in Eq. A-9, we obtain:

ðye

ywf

ðxwf

0

@pf

@z

��ze

� @pf

@z

��z0

!dxdyþ

ðze

z0

ðxwf

0

@pf

@y

��ye

� @pf

@y

��ywf

!dxdz

þðze

z0

ðye

ywf

@pf

@x

��xwf

� @pf

@x

��0

!dydz ¼

/lctð Þf vbf

kf

dpf

dt:

� � � � � � � � � � � � � � � � � � � ðA-10Þ

In Eq. A-10, we made use of

ðze

z0

ðye

ywf

ðxe

xwf

pf dxdydz ¼ðze

z0

ðye

ywf

ðxe

xwf

dxdydz

0B@

1CApf ¼ vbf

pf .

Multiplying Eq. A-10 bykf

land applying the boundary condi-

tions from Eqs. 11 and 13, we obtain

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

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

. . . . . . . . . .

. . . . . . . . .

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

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




http://dx.doi.org/10.1145/361953.361969

http://dx.doi.org/10.1145/361953.361969




�ðze

z0

ðxwf

0

kf

l@pf

@y

��ywf

!dxdzþ

ðze

z0

ðye

ywf

kf

l@pf

@x

��xwf

!dydz

¼ /ctð Þf vbf

dpf

dt: ðA-11Þ

Noting that qf ¼ðze

z0

ðxwf

0

kf

l@pf

@y

��ywf

!dxdz and vpf

¼ /f vbf, one

can write Eq. A-11 as

�qf þðze

z0

ðye

ywf

kf

l@pf

@x

��xwf

!dydz ¼ vpf

ct

�f

dpf

dt: ðA-12Þ

From the boundary condition given by Eq. 14,kf

l@pf

@x

��xwf

¼km

l@pm

@x

��xwf

; substitute this into Eq. A-12, and it becomes

�qf þðze

z0

ðye

ywf

km

l@pm

@x

��xwf

!dydz ¼ vpf

ct

�f

@pf

@t: ðA-13Þ

By noting that qm ¼ðze

z0

ðye

ywf

km

l@pm

@x

��xwf

!dydz, one can rewrite

Eq. A-13 as

vpfct

�f

dpf

dt¼ �qf þ qm: ðA-14Þ

By eliminating the spatial dependence in Eqs. 1 and 9, we trans-formed a microscopic mass-balance equation to a macroscopicmass-balance equation. The microscopic mass-balance equationdescribes mass balance at a point, whereas the macroscopic equa-tion describes the mass balance for a finite system; see Walsh andLake (2003). Therefore, the model parameters in the macroscopicequation are the average properties in the finite volume.

Appendix B: Derivation of a MathematicalRelationship Between the Average ReservoirPressure and Flow Rate

The solution to the 1D linear-flow problem (1D diffusivity equa-tion) with constant pressure at the fracture face and a no-flowouter boundary is given by Wattenbarger et al. (1998) and Ogu-nyomi (2015):

pD ¼ 1þ 4

p

X1n¼1

ð�1Þn

2n� 1e�ð2n� 1Þp

2

� �2

tD

� cosð2n� 1Þp

2xD

� �;

� � � � � � � � � � � � � � � � � � � ðB-1Þ

where pD is the dimensionless reservoir pressure, tD is the dimen-sionless time, and xD is the dimensionless distance in the x-direc-tion. From Eq. B-1, we obtain the production rate at the fracture

face as qD ¼dPD

dxD

��xD¼1

; therefore,

qD ¼ �2X1n¼1

ð�1Þne�ð2n� 1Þp

2

� �2

tD

� sinð2n� 1Þp

2: ðB-2Þ

By inspection, all the terms of the series in Eq. B-2 are posi-

tive. The exponential term has two coefficients, sinð2n� 1Þp

2and

ð�1Þn. For odd values of n, sinð2n� 1Þp

2is þ1 (positive unity)

and ð�1Þn is –1 (negative unity). As a result, the product of thesetwo coefficients is always negative. When the product of thesetwo coefficients is multiplied by the negative sign outside thesummation sign, we realize that all the terms of this solution are

always positive. With the same reasoning, we conclude that, foreven values of n, all terms of the solution are positive. Therefore,we can write Eq. B-2 as

qD ¼ 2X1n¼1

e�ð2n� 1Þp

2

� �2

tD

� : ðB-3Þ

Again, with Eq. B-1, we obtain the volume-weighted average

reservoir pressure p ¼

ðv

pdv

ðv

dvas

pD ¼ 1þ 8

p2

X1n¼1

ð�1Þn

ð2n� 1Þ2e�ð2n� 1Þ2p2tD

4 sinð2n� 1Þp

2

� �:

� � � � � � � � � � � � � � � � � � � ðB-4Þ

In Eq. B-4, pD is the dimensionless average reservoir pressure.As with the rate equation, the terms of the infinite series arealways negative; therefore, we can rewrite Eq. B-4 as

pD ¼ 1� 8

p2

X1n¼1

e�ð2n� 1Þ2p2tD

4

ð2n� 1Þ2: ðB-5Þ

If we write Eq. B-5 in dimensional form, we obtain

p ¼ pwf þ4

p2ð pi � pwf Þ

X1n¼1

2e�ð2n� 1Þ2k

4/lctL2p2t


With the definition of productivity index ðpi � pwf Þ ¼p2qi

4J,

and by insertion of Eq. B-3 into Eq. B-6, we obtain an expressionthat relates the average reservoir pressure to the well rate as

p ¼ pwf þqi

J

X1n¼1

qDn


One can write Eq. B-7 for the fracture and matrix compart-ments, respectively, as

pf ¼ pwf þqfi

Jf

X1n¼1

qDfn

ð2n� 1Þ2ðB-8Þ

and

pm ¼ pf þqmi

Tx

X1n¼1

qDmn


By writing Eq. B-9 for the matrix compartment, we assumedthat the solution to the diffusivity equation with a constant-pres-sure inner-boundary condition is valid even when the pressure atthe inner boundary is varying. This assumption is reasonablewhen there is big contrast in transmissibility between two adjoin-ing reservoir compartments.

Appendix C: Solution of the System of ODEs

With insertion of Eqs. B-8 and B-9 into Eqs. A-8 and A-15, we,after some simplification, obtain the following system of ODEs:

X1n¼1

dqfn

dt¼ � 1

sf

X1n¼1

ð2n� 1Þ2qfn þ1

sf

X1n¼1

ð2n� 1Þ2qmn

� � � � � � � � � � � � � � � � � � � ðC-1Þ

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

. . . . .

. . . . .

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

. . . .

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

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

. . . . . . .

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

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

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




and

X1n¼1

dqmn

dt¼ Tx

sf Jf

X1n¼1

ð2n� 1Þ2qfn

� 1

smþ Tx

sf Jf

� �X1n¼1

ð2n� 1Þ2qmn: � � � � � ðC-2Þ

In Eqs. C-1 and C-2, we used qfn¼ qfiqfDn and q

mn¼ qmiqmDn.

The solution to this system of differential equations is obtainedwith eigenvalue diagonalization for each index n and then sum-ming these solutions. For index n, the system is given as

dqfn

dtdqmn

dt

0B@

1CA ¼ 2n� 1ð Þ2

� 1

sf

1

sf

Tx

sf Jf� 1

sm� Tx

sf Jf

0BB@

1CCA qfn

qmn

� �:

� � � � � � � � � � � � � � � � � � � ðC-3Þ

And the initial conditions are qf ðt ¼ 0Þ ¼ qfi and qmðt ¼0Þ ¼ 0. The eigenvalues of the A matrix of Eq. C-3 are given by

k1n¼ ð2n� 1Þ2

2

(� 1

sf� 1

sm� Tx

sf Jf

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

sf� 1

sm

� �2

þ 2Tx

s2f Jfþ 2Tx

smsf Jfþ Tx

sf Jf

� �2s )

and

k2n¼ ð2n� 1Þ2

2

(� 1

sf� 1

sm� Tx

sf Jf

�


sf� 1

sm

� �2

þ 2Tx

s2f Jfþ 2Tx

smsf Jfþ Tx

sf Jf

� �2s )

:

And the corresponding matrix of the eigenvectors is given by

q c

1 1

� �

¼sf Jf

2Tx� 1

sfþ 1

smþ Tx

sf Jfþ


sf� 1

sm

� �2

þ 2Tx

s2f Jfþ 2Tx

smsf Jfþ Tx

sf Jf

� �2s( )

1

0B@

� sf Jf

2Tx� 1

sfþ 1

smþ Tx

sf Jf�


sf� 1

sm

� �2

þ 2Tx

s2f Jfþ 2Tx

smsf Jfþ Tx

sf Jf

� �2s( )

1

1CA:

Therefore, the solution to this system in Eq. C-3 is given as

qfn ¼qfi

c� qcek2n t � qek1n t�

þ qmi

c� qcqek1n t � cqek2n t�

� � � � � � � � � � � � � � � � � � � ðC-4Þ

and

qmn¼ qfi

c� qek2n t � ek1n t�

þ qmi


:

� � � � � � � � � � � � � � � � � � � ðC-5Þ

With the initial conditions, the solution simplifies to

qfn ¼qfi


ðC-6Þ

and

qmn¼ qfi

c� qek2n t � ek1n t�

: ðC-7Þ

Therefore, the complete solution to Eqs. C-1 and C-2 is given by

qf ¼X1n¼1

qfn ¼X1n¼1

qfic

c� qek2n t �

X1n¼1

qfiq

c� qek1n t

� � � � � � � � � � � � � � � � � � � ðC-8Þ

and

qm ¼X1n¼1

qmn¼X1n¼1

qfiek2n t

c� q�X1n¼1

qfiek1n t

c� q: ðC-9Þ

To arrive at the final form of the solution, we eliminate the infinitesum in Eq. C-8 by converting the discrete sum to an integral to obtain

qf ¼ qfic

c� qeð2n�1Þ2k2t � qfi

qc� q

eð2n�1Þ2k1t

þ qfi

4

cc� q



pÞ

� qfi

4

qc� q



pÞ: ðC-10Þ

Babafemi A. Ogunyomi is currently a petroleum engineer withChevron Energy Technology Company in Houston. Previously,he was a petroleum engineer with Caesar Systems. Beforethat, Ogunyomi worked as a drilling engineer with KoreanNational Oil Corporation and as a production engineer withElf Petroleum Nigeria. He has authored or coauthored morethan 10 technical papers. Ogunyomi is interested in reservoir-engineering research, and holds a BS degree in petroleum en-gineering from the University of Ibadan, Nigeria, as well as MSand PhD degrees (also in petroleum engineering) from the Uni-versity of Texas at Austin. He is a technical editor for SPE Jour-nal, SPE Reservoir Evaluation and Engineering, and SPEEconomics and Management. Ogunyomi is a member of SPE.

C. Shah Kabir has just retired from Hess Corporation in Houston.His experience has spanned 39 years in the areas of transient-pressure testing, fluid and heat flow in wellbores, and reservoir en-gineering. Besides coauthoring more than 130 papers, Kabircoauthored the 2002 SPE textbook Fluid Flow and Heat Transfer inWellbores and contributed to the 2009 SPE monograph TransientWell Testing. He has served on SPE editorial committees for SPEProduction and Facilities, SPE Reservoir Evaluation and Engineer-ing, and SPE Journal. Currently, Kabir serves as an associate edi-tor for SPE Reservoir Evaluation and Engineering. He has chairedSPE Applied Technology Workshops and SPE Forum Series Meet-ings. Kabir was an SPE Distinguished Lecturer during 2006–2007and became an SPE Distinguished Member in 2007. He receivedthe 2010 SPE Reservoir Description and Dynamics Award.

Tadeusz Patzek is a professor of Chemical and Petroleum Engi-neering and Director of the Upstream Petroleum EngineeringResearch Center at the King Abdullah University of Science andTechnology in Saudi Arabia. Between 2008 and 2014, he waschair of the Petroleum and Geosystems Engineering Depart-ment at the University of Texas at Austin. Earlier, Patzek was pro-fessor of geoengineering at the University of California, Berkeley(1990 to 2008) and a senior research scientist for Shell Develop-ment at the Bellaire Research Center in Houston (1983 to 1990).His research interests involve multiscale modeling of multiphasefluid flow and fluid/rock interactions, and thermodynamics ofsurvivability of human kind. Patzek is an SPE Honorary Memberand a coauthor of more than 200 papers and one book.

Larry W. Lake holds the W. A. (Monty) Moncrief CentennialChair in the Department of Petroleum and Geosystems Engi-neering at the University of Texas at Austin. He has publishedmore than 100 peer-reviewed-journal articles and has taughtindustrial and professional short courses in enhanced oil recov-ery and reservoir characterization around the world. Lake isthe author or coauthor of four textbooks and the editor ofthree bound volumes. He has been teaching at the Universityof Texas at Austin for 30 years; before that, he worked for ShellDevelopment Company in Houston. Lake holds BS and PhDdegrees in chemical engineering from Arizona State Universityand Rice University, respectively. He is a member of the USNational Academy of Engineering.

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

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

. . . . . . .

. . . . . . . . .




Download - History Matching and Rate Forecasting in … Matching... · History Matching and Rate Forecasting in Unconventional Oil Reservoirs With an ... cal reservoir simulation. ... tially

Top Related