spe 84287 decline curve analysis using type curves — … · 2018-11-18 · production data...

Copyright 2003, Society of Petroleum Engineers Inc.

This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in Denver, CO., 5-8 October 2003.

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Abstract This work provides the development, validation, and appli-cation of new decline type curves for a well with a finite conductivity vertical fracture centered in a bounded, circular reservoir. This work fills a significant void in the modern inventory of decline type curves. In particular, this work is directly applicable to production data analysis for cases taken from low permeability gas reservoirs.

Using an appropriate analytical solution for this case, we pre-pared "decline" type curves for FcD values from 0.1 to 1000 — individual type curves are generated for each FcD value using a range of reD values from 2 to 1000. The following "type curves" are provided:

l "Fetkovich" format rate-time decline type curves (con-stant pressure case): qDd versus tDd

l "Fetkovich-McCray" format rate-time decline type curves (equivalent constant rate case): qDd versus Ddt

l "Fetkovich-McCray" format rate-cumulative decline type curves: qDd versus NpDd

We provide an example demonstration of the methodology for decline type curve analysis using a field case of continuously measured production rate and surface pressure data obtained from a low permeability gas reservoir.

These solutions/type curves provide an analysis/interpretation mechanism that has not previously been available in the petroleum literature. Compared to field data, we find that the traditional type curve solutions for an infinite conductivity vertical fracture are typically inadequate — and, the new solu-tions for a well with a finite conductivity vertical fracture clearly show much more representative behavior. This vali-dation suggests that the proposed type curves will have broad

utility in the petroleum literature — particularly for appli-cations in low permeability gas reservoirs.

Objectives The following objectives are proposed for this work:

l To develop and validate a series of decline type curves for a well with a finite conductivity vertical fracture centered in a bounded, circular reservoir. l To provide a methodology for using decline type curves

to analyze and interpret production or injection well performance for a well with a finite conductivity verti-cal fracture. l To demonstrate these new type curves using continu-

ously measured production data (rates and pressures).

In considering these objectives we note that we are strongly motivated to provide these tools in light of the current high level of activity in the analysis and interpretation of reservoir performance data acquired from low permeability gas reser-voirs. We recognize that current methods based on the case of a vertical well with an infinite conductivity vertical fracture are overly-ideal for low permeability reservoirs — and we must reconcile the need for a new decline type curve for a finite conductivity vertical fracture. This rationale is the moti-vation for this work.

Prior Work In this section we address the prior work performed on topics relevant to the analysis/interpretation of rate and pressure be-havior for a well with a finite conductivity vertical fracture. The appropriate references are cited as follows:

Topic Refs. Historical Methods — Production Data Analysis 1-2

Production Data Analysis for Fractured Wells 3-5 Decline Type Curve Analysis — Fetkovich-McCray Format 6-10 Well Solutions — Finite Conductivity Vertical Fracture 11-15 Miscellaneous 16-18

Historical Methods — Production Data Analysis: We cite these references to orient the reader towards historical methods for the analysis of production data. The Arps work (ref. 1) is a compilation of empirical results and field practices up to about 1940. The Fetkovich work (ref. 2) addresses most production data analysis technologies from 1940-1980, and should be helpful as a general reference.

SPE 84287

Decline Curve Analysis Using Type Curves — Fractured Wells H. Pratikno, ConocoPhillips (Indonesia), J. A. Rushing, Anadarko Petroleum Corp., and T.A. Blasingame, Texas A&M U.

2 H. Pratikno, J. A. Rushing, and T.A. Blasingame SPE 84287 _________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Production Data Analysis for Fractured Wells: The first reference in this section (Gringarten3) addresses the analytical (and very tedious) solution for a well with an infinite conduc-tivity or uniform flux vertical fracture in a bounded rectan-gular reservoir. This was presented (and is still used) as an analysis mechanism for fractured wells — this solution is for-mulated for a well produced at a constant rate. The work presented by Carter4 creates a "Fetkovich"-style production decline type curve for gas well. It is relevant to this discussion because Carter addresses the question of fractured well ana-lysis (albeit in the form of a negative skin factor).

The final reference (Fraim, et al.5) presents a comprehensive suite of decline type curves for the case of a well with a finite conductivity vertical fracture. These type curves were derived using numerical simulation and the decline variables were established using regression to estimate a relation between the dimensionless decline variables and the reD and FcD variables.

Our present work uses analytical solutions to develop the required type curves, and we also establish a relation for the decline variables — although we derive our relations from the analytical solution. While there are other modern techniques developed in the present work (e.g., the rate integral and rate integral-derivative functions, as well as the rate-cumulative type curve), it is important to note that these are essentially refinements — the Fraim, et al. work is robust and functional, and should be regarded as a major contribution in this area.

Decline Type Curve Analysis — Fetkovich-McCray Format: The use of the "Fetkovich-McCray" type curve format (i.e., the rate integral (qDdi) and rate integral-derivative (qDdid) functions, and later, the material balance time function )( Ddt evolved as an effort to present data functions with better reso-lution (qDdi and qDdid), as well as to account for variable-rate/ variable pressure drop conditions ).( Ddt Refs. 6-10 provide the current technologies for the analysis of production data using decline type curve analysis.

Well Solutions — Finite Conductivity Vertical Fracture: This section recounts the solutions we have used or adapted in order to model the case of a well with a finite conductivity vertical fracture produced at a constant rate in a bounded reservoir. Cinco, et al.11 presented a semi-analytical, real domain solution for a well with a finite conductivity vertical fracture in an infinite-acting reservoir. This semi-analytical solution is tedious, but accurate.

Houze, et al.11 presented a Laplace domain solution for a well with an infinite conductivity vertical fracture in an in-finite-acting reservoir, while Cinco and Meng13 provide a Laplace domain solution for a well with a finite conductivity vertical fracture in an infinite-acting reservoir. The use of a Laplace domain solution offers some convenience, and while such solutions are not necessary (e.g., the work of Fraim, et al.5), we will use a Laplace domain solution in this work. Using the techniques presented by Ozkan and Raghavan,14-15 for the case of a well with an infinite conductivity vertical fracture, we develop a "desuperposition" solution for the case of a well with a finite conductivity vertical fracture in a finite (circular)

reservoir — which is based on the transient flow solution given by Cinco and Meng.13

Miscellaneous: Refs. 16 and 17 provide details of the "Gaver-Stehfest" algorithm, the most common tool used to numeri-cally invert Laplace transform solutions. Ref. 18 is the re-search thesis developed by the principal author on the subject of this work.

Desuperposition Solution We have developed and validated a new "desuperposition" solution for the case of a well produced at a constant rate with a finite conductivity vertical fracture in a finite-acting (cir-cular) reservoir. Schematically, this "desuperposition" solu-tion is expressed as:

(pD)bounded finite conductivity case = (pD)transient finite conductivity case + (pD)boundary effect (circular reservoir) ............................ (1)

We readily acknowledge that other solutions may exist for a well with a finite conductivity vertical fracture in a bounded system — our goal in using desuperposition was to develop a relatively fast and accurate routine for this particular case. The proposed solution is validated using numerical simulation and should be considered appropriate for the analysis of well performance data (flowrates and pressures) taken from wells which have under-gone hydraulic fracture stimulation treat-ments.

The desuperposition solution for the case of a well producing at a constant rate with a finite conductivity vertical fracture in a finite-acting (circular) reservoir18 is given below:

[ ]

++

−+

++−

+

=≤=≤

∫∫∫∫

∫ +−

dzzIxu

dzzIxu

ruI

ruK

us

s,xqsxsF

'dx"dxs,"xq'xx

F

'dxu|'xxKu|'xxK s,'xq

s,rr,y,|x|p

DD

eD

eD

DfDfDcD

fDD

cD

DDfD

eDDDDds,D

)()1(

0 )(

)1(

0)(

)(121

)( )(00

)| ()| ()(1

021

)01 (

001

1

00

ππ

..................................................................................... (2)

Where u=sf(s) — f(s) is the interporosity flow function for the dual porosity reservoir case (f(s)=1 for this work).

Validation of the Desuperposition Solution: In order to validate Eq. 2, we use results generated by numeri-cal simulation for FcD=0.2π, π, 2π, 10π. The comparison of solutions is shown in Figs. 1-4 — we note excellent agreement between our desuperposition solution and the numerical simu-lation results. We consider this validation sufficient to pro-ceed with the development of new decline type curves using the desuperposition solution.

Pseudosteady-State Performance Correlation: bDpss Recalling the general (dimensionless) identity for pseudo-steady-state performance flow, we have:

DADpssDApss,D tbtp π2)( += ..................................... (3)

SPE 84287 Decline Curve Analysis Using Type Curves — Fractured Wells 3 _________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Where we note that for the case of a well with a finite con-ductivity vertical fracture, the bDpss parameter is independent of time, but this parameter is a function of reD and FcD. Solv-ing Eqs. 2 and 3 for the dimensionless pseudosteady-state parameter, bDpss, we obtain:

DAds,DDpss tpb π2-= ................................................. (4)

We note that during pseudosteady-state Eq. 4 will yield a constant value of bDpss for a given case of reD and FcD. We have constructed plots of (pD,ds)Eq. 2 and (bDpss)Eq. 4 versus tDA for a given value of FcD and a range of values for reD. These various cases are presented in Figs. 5-9 for FcD=0.1, 1, 10, 100, and 1000, respectively. The bDpss values derived from this exercise are tabulated in Table 1 (for all cases) and are presented graphically in Fig. 10 (along with a correlating function for bDpss(reD,FcD)).

Given a particular pair of reD and FcD values, the bDpss(reD,FcD) values for that pair can be estimated using Fig. 10, or the cor-relation given below:

44

33

221

45

34

2321

1

4346400492980)(ln 2

ubububub

uauauauaa

r..rb eDeDDpss

++++

+++++

+−=−

............ (5)

Where,

)( ln cDFu =

a1 = 0.93626800 b1 = -0.38553900 a2 = -1.00489000 b2 = -0.06988650 a3 = 0.31973300 b3 = -0.04846530 a4 = -0.04235320 b4 = -0.00813558 a5 = 0.00221799

The correlation given by Eq. 5 yields an excellent approxima-tion for the input data (see Fig. 10) — this result should be more than sufficient for all applications where an estimate of bDpss(reD,FcD) is required.

Development of Decline Type Curves for a Well with a Finite Conductivity Vertical Fracture. The work to develop and correlate the bDpss(reD,FcD) variable in the previous section is essential for the development of decline type curves for this case. In particular, we define the dimen-sionless "decline" time and rate functions using the bDpss para-meter as the correlating variable (the traditional approach originally proposed by Fetkovich2).

General Decline Type Curve Construction: (qDd versus tDd) The general definitions of the base decline type curve vari-ables (tDd and qDd) as given by Fetkovich (ref. 2) are:

DpssDDd bqq = ........................................................... (6)

DADpss

Dd tb

tπ2

= ......................................................... (7)

The remainder of this effort is relatively straightforward — we simply create a "type curve" (using Eq. 2) for a particular case of dimensionless fracture conductivity, FcD, with a sampling of

reD "stems." The sequence of variables that we used in this work is:

FcD = 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000 reD = 1.2, 1.3, 2, 3, 4, 5, 10, 20, 30, 40, 50, 100, 200, 300,

400, 500, 1000

As a technical comment we note that we utilize the "material balance time" format for this work (see ref. 8) — as such, we use Eq. 2 as a constant rate solution, then use the 1/pD result as qD (obviously q is constant, so we actually use the q/∆p function). This procedure is addressed in detail in ref. 8, we mention this process for the sake of completeness. The "Fet-kovich-McCray" format is q/∆p versus ,t as well as the asso-ciated auxiliary functions for q/∆p.

Auxiliary Decline Type Curve Variables: Auxiliary "decline" variables have been proposed (ref. 6) in order to improve the resolution and function of matching production data against model responses (type curves). From a practical standpoint this is the most relevant issue for the practitioner who wants to be able to interpret and analyze production data. The auxiliary variables typically used for decline type curve analyses are:

l Rate integral function: qDdi

ττ dqt

tt

Nq Dd

Dd

DdDd

pDdDdi )(

0

1 ∫== ..................... (8)

l Rate integral-derivative function: qDdid

DdDdiDd

DdiDd

Dd

DdiDdid qq

dtdq

ttd

dqq −=−=−=

)ln(..... (9)

l Cumulative production function: NpDd

ττ dqt

N DdDd

pDd )(0∫= ........................................ (10)

Type Curve Inventory: To this point we have developed the reservoir model, cor-related the pseudosteady-state parameter (bDpss(reD,FcD)), and established the appropriate time and rate function variables. Our inventory of new type curves is as follows:

FcD Fig. qDd, qDdi, qDdid versus Ddt 1x10-1 11

1x100 12 1x101 13 1x102 14 1x103 15

qDd and qDdi versus NpDd 1x10-1 16

1x100 17 1x101 18 1x102 19 1x103 20


While the cumulative production case (i.e., the qDd and qDdi versus NpDd type curves) are unique, these curves do not provide a high resolution perspective of the data. Compared to the standard rate-time format time curves, the rate-cumulative production type curves are significantly "com-pressed" (by using Np) and typically fit on 3x3 log-log cycles, while the same rate-time data set generally requires twice the log scale (at least on the x-axis). In short, there is nothing wrong with the Np format plot — however this method should be used as a confirmation tool, as opposed to being a primary analysis mechanism.

The inventory of type curves presented in this work should be sufficient for all situations encountered in practice. For the practitioner who requires additional cases, the procedures and governing relations presented above are completely general and should be readily reproducible.

Application of the Decline Type Curve Method for a Well with a Finite Conductivity Vertical Fracture — An Illustrative Example

Orientation:

We present the procedures and application of the Fetkovich-McCray-style type curves for rate-time and rate-cumulative production (or injection) analysis. This discussion is specifi-cally tailored to the case of a well with a finite conductivity vertical fracture in a bounded homogeneous reservoir.

For the case of a well with a finite conductivity vertical frac-ture, we can estimate the following reservoir properties:

l Reservoir properties: — Effective permeability, ko (or kg) — Skin factor for near well damage or stimulation, s — Fracture half-length, xf — Fracture conductivity, FcD (dimensionless)

l Reservoir volumes: — In-place fluid volume, N (or G) — Oil or gas reserves, Np,res (Gp,res) — Reservoir drainage (or injection) area, A

Procedures for applying the decline type curve analysis for this particular case are given below. The working analysis relations are summarized in a later section.

Type Curve Matching Procedure:

1. Assemble the production (oil — STB/D or gas — MSCF/ D) and bottomhole pressure (psia) data versus time (in days). The "material balance time" function given by:

op qNt /= ............................................................... (10a)

For gas, we must use the "material balance pseudotime" function, given as:

τµ

µ τ dpcp

qt

q

ct

gg

g

tg

gigia

)()(0

)(

)( ∫= .......................... (10b)

2. Compute the pressure drop normalized rate and rate inte-gral functions — the pressure drop normalized rate func-tion is given by:

)(// wfioo ppqpq −=∆ ........................................... (11a)

For gas, the pseudopressure drop normalized rate function is given by:

)(// pwfpigpg ppqpq −=∆ .................................. (11b)

The pressure drop normalized rate integral function is given as:

τdp

qt

tpq o

io 0

1)/(

∆=∆ ∫ .......................................... (12a)

For gas, the pseudopressure drop normalized rate integral function is given as:

τdp

qt

tpq

p

ga

aipg

0

1)/(

∆=∆ ∫ ................................ (12b)

The pressure drop normalized rate integral-derivative function is given by:

[ ] [ ]ioioido pqtd

dt

pqtd

dpq )/(

1)/(

)ln()/( ∆−=∆=∆

................................................................................. (13a) For gas, the pseudopressure drop normalized rate integral-derivative function is given by:

[ ] [ ]ipgaa

ipga

idpg pqtdd

tpq

tdd

pq )/( 1

)/( )ln(

)/( ∆−=∆=∆

................................................................................ (13b)

3. The following data functions are plotted on a scaled log-log grid for type curve matching using the appropriate Fetkovich-McCray format type curve: (in our case a frac-tured well) a. )/( pqo ∆ versus t (or )/( pg pq ∆ versus at )

b. io pq )/( ∆ versus t (or ipg pq )/( ∆ versus at )

c. ido pq )/( ∆ versus t (or idpg pq )/( ∆ versus at )

4. We now "force" match the depletion data trends onto the Arps b=1 (harmonic) stems for each of the Fetkovich-McCray style type curves being used: qDd, qDdi, and qDdid.

Once a "match" is obtained, we record the "time" and "rate" axis match points as well as the reD transient flow stem. Recall that for this case, reD = re/xf.

a. Rate-axis "Match Point:" Any MPMP )()( Ddqpq −∆ pair

b. Time-axis "Match Point:" Any MPMP )()( Ddtt − pair

c. Transient flow stem: (reD) Select the )( pq ∆ , ipq )( ∆ , and idpq )( ∆ functions that best match the transient data stems.

d. Calculate/estimate the bDpss value — using Eq. 5.or the correlation plot (Fig. 10).

Estimation of Reservoir Properties: Using the results of the "match point," we can estimate the following reservoir properties:

l Original volume-in-place:


MP

MP

MP

MP)()/(

)()(1

Dd

o

Ddt qpq

tt

cN

∆= (oil) ................... (14a)

MP

MP

MP

MP)(

)/(

)()(1

Dd

pg

Dd

a

gi q

pq

tt

cG

∆= (gas) .................. (14b)

l Reservoir drainage area:

)1(61485

wirr

oSh

NB.A

−=

φ (oil) ................... (15a)

)1(61485

wirr

gi

Sh

GB.A

−=

φ (gas) .................. (15b)

l "Effective" drainage radius:

π/Are = ................................................................. (16)

l Formation permeability: (effective permeability)

∆=

MP

MP)()/(

2141Dd

oDpss

ooo q

pqb

hB

.kµ

(oil) ..... (17a)

∆=

MP

MP

)(

)/(2141

Dd

pgDpss

gigig q

pqb

h

B.k

µ (gas) .... (17b)

l Fracture half-length:

eD

ef r

rx = .................................................................. (18)

l Radial flow skin factor: ( )( ln cDFu = )*

33221

211

102769988 105137619 1013660421

105065321 1000271136485461

ln

u.u.u.

u.u..

xr

sf

w

−−−

−−

+++

+−+

≈

xxx

xx

................................................................................... (19)

* Unpublished correlation (P. Valko, Texas A&M U.).

Data Requirements and Analysis Overview: We provide the overall procedures that are used to analyze and interpret production well performance data. These procedures are:

1. Verification of pertinent rock, fluid, and completion data using available field records and fluid property cor-relations. The data required for the analysis include:

l Viscosity at pi, µo or µgi l Fluid compressibility at pi, co or cgi (or cti, if used) l Formation volume factor at pi, Bg or Bgi l Wellbore radius, rw l Porosity, φ l Net pay or production interval, h l Irreducible water saturation, Swirr

2. Initial screening of field production data using Cartesian, semilog, and log-log plots:

l Identify errors or anomalies in the production data, l Identify changes in the operational practices, and l Use data smoothing (rare).

3. Perform type curve analysis using the Fetkovich-McCray decline type curve for a particular FcD to determine the time and rate match points — as well as the transient stem match (i.e., reD). These match points are then used to estimate the following:

l Total system volume for production, G or N, l Transient stem match, reD = re/xf, and l Pseudosteady-state flow constant, bDpss.

These results are then used to estimate the reservoir drain-age area, formation permeability, and the near-well skin factor.

4. To estimate the gas reserves, Gp,res, at current producing conditions, we use the following approach:

l Plot (qg/∆pp), versus cumulative gas production, Gp,

and extrapolate to (qg/∆pp) = 0.

While the procedures given above may seem lengthy and tedious, we will demonstrate the utility and straightforward-ness of this approach as we apply these procedures to a field example. Implied, but not stated, we assume that the data are representative — commingled or allocated data must be sorted and assigned to individual wells. In addition, the pressure data should be as accurate as practical and must be taken on a comparable frequency as the production rate data.

Example 1: Low Permeability Gas Reservoir (Texas)

Fluid Properties and Production Data

Reservoir Properties: Wellbore radius, rw = 0.333 ft Estimated net pay thickness, h = 170 ft Average porosity, φ = 0.088 fraction Average irred. water saturation, Swirr = 0.131 fraction

Fluid Properties: (γg=0.7 (air=1), T=300 deg F) Gas form. volume factor at pi, Bgi = 0.5498 RB/MSCF Gas viscosity at pi, µgi = 0.0361cp Gas compressibility at pi, cgi = 5.1032x10-5 psi-1

Production Parameters: Initial reservoir pressure, pi ˜ 9,330 psia Initial reservoir pseudopressure, ppi ˜ 7,536 psia

Material Balance Analysis: (Fig. 21 and 22)

Fig. 21 shows a plot of semilog rate and Cartesian production pressure history versus time. The data appear generally well behaved and correlated, the only major "event" in this data sequence is the extended shut-in test that was conducted at about 75-100 days during production.

By plotting the productivity index, (qg/∆pp), versus the cumu-lative gas production, Gp, (Fig. 22), we can estimate the gas reserves for Example 1 to be ˜ 0.983 BSCF using a straight-line extrapolation of the date trend on Fig. 22. The cumulative production for the well to date is 0.691 BSCF.

Production Volume Summary:

Gp = 0.691 BSCF Gp,res = 0.983 BSCF


Data Function Analysis: (Fig. 23 and 24)

Fig. 23 shows the normalized rate function data, (qg/∆pp), versus the material balance pseudotime functions which have been calculated using Eqs. 11b and 10b, respectively. In this figure we note that the production data have been refined to eliminate transient data "spikes" which result from major transients in the rate history (e.g., recovery after shut-in). Fig. 24 shows the rate, (qg/∆pp), rate integral, (qg/∆pp)i, and rate integral-derivative, (qg/∆pp)id functions versus the material ba-lance pseudotime function. The calculation of these functions is addressed specifically in the previous section. From this figure we clearly note a linear trend predicted by material ba-lance.

Decline Type Curve Analysis Results: (Fig. 25)

We now match the rate function (qg/∆pp), the rate integral function (qg/∆pp)i, and the rate integral-derivative function (qg/∆pp)id — which are plotted versus the material balance pseudotime function, ,t on the Fetkovich-McCray type curve for a well centered in a bounded circular reservoir with a finite conductivity vertical fracture (FcD=5). The three rate func-tions are "forced matched" on the Arps b=1 (harmonic) de-cline stem (at late times) as this behavior is required by pseudosteady-state theory. After we complete the matching process, we then obtain the match points from the data/type curve overlay.

The production rate and pressure history provide strong evi-dence of boundary-dominated flow behavior. As such, we obtained a good match of the data on the depletion stem as well as a unique match on the transient stem for an reD value of 2 (and an FcD value of 5 — since we used the FcD=5 type curve). Using reD and FcD, and the time and rate match points, we then calculate estimates of gas-in-place, reservoir drainage area, effective permeability to gas, fracture half-length, and the skin factor.

Type Curve Match: FcD=5.0 Fetkovich-McCray rate versus material balance time de-cline type curve for a well with a finite conductivity verti-cal fracture.

Transient Matching Parameter: reD = 2.0

Match Points: (tDd)MP = 1.0 )( at MP = 58 days (qDd)MP = 1.0 (qg/∆pp)MP = 0.89 MSCF/D/psi

Calculations

We first calculate the pseudosteady-state flow constant, bDpss, which has been correlated to the transient match parameters, reD and FcD (i.e., Eq. 5):

Pseudosteady-State Flow Constant

44

33

221

45

34

2321

)5( )5( )5( )5( 1

)5( )5( )5( )5(

4346400492980)(ln 2

bbbb

aaaaa

r..rb eDeDDpss

++++

+++++

+−=−

. (5)

Where,

)( ln cDFu =

a1 = 0.93626800 b1 = -0.38553900 a2 = -1.00489000 b2 = -0.06988650 a3 = 0.31973300 b3 = -0.04846530 a4 = -0.04235320 b4 = -0.00813558 a5 = 0.00221799

or:

44

33

221

45

34

2321

2

)5( )5( )5( )5( 1

)5( )5( )5( )5(

)2( 4346400492980)2(ln

bbbb

aaaaa

..bDpss

++++

+++++

+−= −

1.00222=Dpssb

Gas-in-Place

We estimate the gas-in-place, G, using Eq. 14b (note that we require an estimate of cgi from our fluid properties table):

MP

MP

MP

MP)(

)/(

)()(1

Dd

pg

Dd

a

gi q

pq

tt

cG

∆= ............................. (14b)

Or:

BSCF 0121 MSCF 100121

1)MSCF/D/psi 890(

1days) 58(

)psi 1010325(

1

6

1-5

.

.

.

.G

==

=−

x

x

Reservoir Drainage Area and Equivalent Drainage Radius

The drainage area is estimated using Eq. 15b:

)1(61485

wirr

gi

Sh

GB.A

−=

φ........................................ (15b)

Therefore:

acres 515 ft 240,195

)13101( ft) 170( )0880(RB/MSCF) 5498(0 MSCF) 100121(

61485

2

6

.

....

.A

==

−=

x

We then calculate the "equivalent" drainage radius, re, using Eq. 16, which gives us:

π/Are = ................................................................ (16)

This yields:

ft 5762)/ft (240,195 2 .re == π


Using the bDpss parameter calculated from the transient reD value and the FcD value for a particular type curve, we can then solve for the effective permeability to gas, kg. Using reD we can also solve for the fracture half-length, xf, which, combined with the FcD value, can then be used to estimate the pseudoradial flow skin factor, s.

Effective Gas Permeability

Using the estimate of the net pay interval (˜ 170 ft), we calculate the effective permeability to gas using Eq. 17b:

∆=

MP

MP

)(

)/(2141

Dd

pgDpss

gigig q

pqb

h

B.k

µ............. (17b)

Therefore:

)psi

MSCF/D 0.89(1.00222)(

ft) 170(cp) .0361RB/MSCF)(0 54980(

2141

x

..k g =

Or:

md 0150.kg =

Fracture Half-Length

The fracture half-length is estimated from Eq. 18 using the effective drainage radius and the transient reD value as fol-lows:

De

ef r

rx = .................................................................. (18)

or:

ft 138)2(

ft) 5276(==

.xf

Radial Flow Skin Factor: ( )(ln cDFu = )

The pseudoradial flow skin factor is estimated using Eq. 19, as follows:

33221

211

102769988 105137619 1013660421

105065321 1000271136485461

ln

u.u.u.

u.u..

xr

sf

w

−−−

−−

+++

+−+

≈

xxx

xx

................................................................................... (19)

Therefore:( )5(ln =u )

075)5(ln 102769988)5(ln 105137619)5(ln 1013660421

)5(ln 105065321)5(ln 1000271136485461

ft) 138(ft) 3330(

ln

33221

211

....

...

.s

−=+++

+−+

≈

−−−

−−

xxx

xx

Summary of Results:

Gcgi = 51.62 MSCF/psi kgh = 2.5 md-ft G = 1.012 BSCF kg = 0.015 md A = 5.51 acres xf = 138 ft re = 276.5 ft s = -5.07

We believe that the interpretation and analysis of these data is both accurate and unique — and we readily acknowledge that most of the credit for the success of this case is due to the vigilance of the operator in acquiring production rates and (surface) pressures on a daily basis.

Summary and Conclusions We have presented and validated a new "desuperposition" solution to represent the case of a well with a finite conduc-tivity vertical fracture in a bounded homogeneous reservoir where the well is produced at a constant rate. We have used this solution to develop a new decline type curve for the analysis of production data. We specifically targeted the application of low permeability gas reservoirs, but the metho-dology is valid to any scenario where the base assumptions are applicable.

Conclusions:

The following conclusions are derived from this study:

1. We have successfully constructed, validated, and applied a new set of unified decline type curve solu-tions for the behavior of a well with a finite conduc-tivity vertical fracture producing from a closed homo-geneous reservoir.

2. This set of decline type curves can be applied to analyze and interpret production data from a fractured well to estimate the following: l Formation permeability, l Fracture half-length, l Fracture conductivity, l Volume of in-place fluids, and l Reservoir drainage area.

3. The proposed method is a product of the "Fetkovich-McCray" approach (i.e., using pseudosteady-state flow as a basis for analysis) — as such, the method is error tolerant and generally robust. This approach should be the preferred method of analysis/interpretation for pro-duction data acquired from low permeability gas reser-voirs.

Recommendation for Future Research:

We suggest the following tasks for future work in the development of decline type curves for a well with a finite conductivity vertical fracture in a bounded homogeneous reservoir.

1. Consider the effects of fracture cleanup and fracture-face skin on the production performance. Such an approach may be difficult to implement due to the erratic well performance behavior that occurs during cleanup.

2. Compare the solutions proposed in this work to responses for naturally-fractured (or dual porosity reservoirs) as a mechanism to assess the concept of near-well fracture networks.


Nomenclature

Dimensionless Variables: Real Domain

bDpss = Dimensionless pseudosteady-state constant FcD = Dimensionless fracture conductivity NpDd = Dimensionless decline cumulative production

pD = ?2141

1p

qBkh

. µ, Dimensionless pressure func-

tion for constant flowrate case

qD = qpphk

B

wfi

)(1

141.2−

µ, Dimensionless pres-

sure function for constant flowrate case qDd = qDbDpss, Dimensionless decline rate qDdi = Dimensionless decline rate integral qDdid = Dimensionless decline rate integral-derivative rD = re/xf, Dimensionless radius reD = Dimensionless reservoir drainage radius

tD =

0.006332

wrtc

kt

φµDimensionless time (rw)

tDA =

0.00633Atc

ktφµ

Dimensionless time (A)

tDd = ADpss

Dtb

π2, Dimensionless decline time

tDxf =

0.006332

fxtc

kt

φµDimensionless time (xf)

xD = x/xf, Dimensionless x-distance (fracture case) yD = y/xf, Dimensionless y-distance (fracture case)

Dimensionless Variables: Laplace Domain

ds,Dp = Laplace transform of the dimensionless pres-sure function (desuperposition solution)

s = Laplace domain variable, dimensionless

Field Variables:

c = Compressibility, 1/psi G = Gas-in-place, MSCF or BSCF Gp = Gas production, MSCF/D Gp,res = Gas reserves, MSCF or BSCF h = Net thickness, ft k = Effective permeability, md µ = Fluid viscosity, cp N = Oil-in-place, STB Np = Cumulative oil production, STB Np,res = Oil reserves, STB φ = Porosity, fraction pi = Initial reservoir pressure, psia ppi = Initial reservoir pseudopressure, psia

pp = ,dpz

pp

pp

z

gbasei

igi µ

µ

∫ Normalized pseudo-

pressure function, psia

q = Production rate STB/D or MSCF/D re = Drainage radius, ft rw = Wellbore radius, ft s = Skin factor t = Time, days

at = ,dpcp

qt

q

c

gg

g

tg

gigi τµ

µ τ )()(0

)(

)( ∫ Normalized

pseudotime function, days xf = Fracture half length, ft

Constants:

π = Circumference to diameter ratio, 3.1415926… γ = Euler’s constant, 0.577216…

Special Functions:

I0(x) = Modified Bessel function (1st kind, zero order) I1(x) = Modified Bessel function (1st kind, 1st order) K0(x) = Modified Bessel function (2nd kind, zero order) K1(x) = Modified Bessel function (2nd kind, 1st order)

Special Subscripts:

Dd = Dimensionless decline variable D = Dimensionless variable g = Gas i = Integral function (or initial value) id = Integral-derivative function o = Oil pss = Pseudosteady-state

References

Historical Methods — Production Data Analysis:

1. Arps, J.J: "Analysis of Decline Curves," Trans., AIME (1945) 160, 228-247.

2. Fetkovich, M.J.: "Decline Curve Analysis Using Type Curves," JPT (June 1980) 1065-1077.

Production Data Analysis for Fractured Wells:

3. Gringarten, A.C.: "Reservoir Limits Testing for Fractured Wells," paper SPE 7452 presented at the 1978 SPE Annual Technical Conference and Exhibition, Hou-ston, TX., 1-3 October.

4. Carter, R.D.: "Type Curves for Finite Radial and linear Gas Flow Systems: Constant Terminal Pressure Case," SPEJ (October 1985) 719-728.

5. Fraim, M.L., Lee, W.J., and Gatens, J.M., III: "Ad-vanced Decline Curve Analysis Using Normalized-Time and Type Curves for Vertically Fractured Wells," paper SPE 15524 presented at the 1986 SPE Annual Technical Conference and Exhibition, New Orleans, LA, 5-8 October.

Decline Type Curve Analysis — Fetkovich-McCray Format:

6. McCray, T.L.: Reservoir Analysis Using Production Decline Data and Adjusted Time, M.S. Thesis, Texas A&M U., College Station, TX (1990).

7. Palacio, J.C. and Blasingame, T.A.: "Decline Curve Analysis Using Type Curves — Analysis of Gas Well Production Data," paper SPE 25909 presented at the


1993 Joint Rocky Mountain Regional/Low Permeability Reservoirs Symposium, Denver, CO, 26-28 April.

8. Doublet, L.E., Pande, P.K., McCollum, T.J., and Blasingame, T.A.: "Decline Curve Analysis Using Type Curves — Analysis of Oil Well Production Data Using Material Balance Time: Application to Field Cases," paper SPE 28688 presented at the 1994 Petroleum Con-ference and Exhibition of Mexico held in Veracruz, Mexico, 10-13 October.

9. Doublet, L.E. and Blasingame, T.A.: "Evaluation of Injection Well Performance Using Decline Type Curves," paper SPE 35205 presented at the 1996 SPE Permian Basin Oil and Gas Recovery Conference, Midland, TX, 27-29 March.

10. Agarwal, R.G., Gardner, D.C., Kleinsteiber, S.W., and Fussell, D.D.: "Analyzing Well Production Data Using Combined Type Curve and Decline Curve Analysis Concepts," paper SPE 49222 prepared for presentation at the 1998 SPE ATCE, New Orleans, LA, 27-30 September.

Well Solutions — Finite Conductivity Vertical Fracture:

11. Cinco-Ley, H., Samaniego-V., F. and Dominguez, N.: "Transient Pressure Behavior for a Well with a Finite-Conductivity Vertical Fracture," SPEJ.(August 1978) 253-264.

12. Houze, O.P., Horne, R.N., and Ramey, H.J. Jr.: "Infinite Conductivity Vertical Fracture in a Reservoir with Dou-ble Porosity Behavior", paper SPE 12778 presented at the 1984 SPE California Regional Meeting, Long Beach, April 11-13.

13. Cinco-Ley, H. and Meng, H.Z.: "Pressure Transient Analysis of Wells with Finite Conductivity Vertical Fractures in Double Porosity Reservoirs," paper SPE 18172 presented at the 1988 SPE Annual Technical Conference and Exhibition, Houston, TX., 5-8 October.

14. Ozkan, O. and Raghavan, R.: "New Solutions for Well-Test-Analysis Problems: Part 1 — Analytical Considera-tions," SPEFE (September 1991) 359-368.

15. Ozkan, O. and Raghavan, R.: "New Solutions for Well-Test-Analysis Problems: Part 2 — Computational Consi-derations, and Applications," SPEFE (September 1991) 369-378.

Miscellaneous:

16. Gaver, D.P., Jr.: "Observing Stochastic Processes and Approximate Transform Inversion," Operations Re-search, vol. 14, No. 3 (1966), 444-459.

17. Stehfest, H.: "Numerical Inversion of Laplace Transforms," Communications of the ACM (January 1970), 13, No. 1, 47-49. (Algorithm 368 with correction (October 1970), 13, No. 10).

18. Pratikno, H.: Decline Curve Analysis Using Type Curves — Fractured Wells, M.S. Thesis, Texas A&M U., Col-lege Station, TX (2002).


Figure 1 – Constant rate solution for a well with a finite con-ductivity vertical fracture (FcD = 0.2π).

Figure 2 – Constant rate solution for a well with a finite con-ductivity vertical fracture (FcD = π).

Figure 3 – Constant rate solution for a well with a finite con-ductivity vertical fracture (FcD = 2π).

Figure 4 – Constant rate solution for a well with a finite con-ductivity vertical fracture (FcD = 10π).

Figure 5 – Cartesian plot (pD versus tDA) for a well with a finite conductivity vertical fracture (FcD = 0.1).

Figure 6 – Cartesian plot (pD versus tDA) for a well with a finite conductivity vertical fracture (FcD = 1).





Figure 10 – Correlation of bDpss = f(reD,FcD) values for a well with a finite conductivity vertical fracture produced at a constant rate.

Figure 11 – Fetkovich-McCray decline type curve — rate versus material balance time format for a well with a finite conductivity vertical fracture (FcD = 0.1).

Figure 12 – Fetkovich-McCray decline type curve — rate versus material balance time format for a well with a finite conductivity vertical fracture (FcD = 1).





Figure 16 – Fetkovich-McCray decline type curve — rate versus cumulative format for a well with a finite conduc-tivity vertical fracture (FcD = 0.1).

Figure 17 – Fetkovich-McCray decline type curve — rate versus cumulative format for a well with a finite conduc-tivity vertical fracture (FcD = 1).





Figure 21 – Semilog rate and Cartesian production pressure versus time — Example 1.

Figure 22 – Pseudopressure drop normalized rate function ver-sus cumulative production, extrapolation yields gas reserves at current condition — Example 1.

Figure 23 – Pseudopressure drop normalized rate function (with transient "spikes" edited — "data edit plot") versus material balance pseudotime function — Example 1.

Figure 24 – Pseudopressure drop normalized rate, integral, and integral-derivative functions versus material ba-lance pseudotime function — Example 1.


Figure 25 – Match of production data for Example 1 on the Fetkovich-McCray decline type curve (pseudopres-sure drop normalized rate versus material balance time format) for a well with a finite conductivity vertical fracture (FcD = 5).

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