Post-Fracture Performance Diagnostics for Gas Wells With ... · test and use variable-rate pressure...

Copyright 2005, Society of Petroleum Engineers This paper was prepared for presentation at the 2005 SPE Eastern Regional Meeting held in Morgantown, W.V., 14–16 September 2005. This paper was selected for presentation by an SPE Program Committee following review of information contained in a proposal submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to a proposal of not more than 300 words; illustrations may not be copied. The proposal must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

Abstract This paper presents an integrated technique for evaluating the production performance of gas wells with finite-conductivity vertical fractures. Our methodology combines conventional pressure transient test analysis with new material balance decline type curves developed specifically for gas wells with finite-conductivity, vertical fractures. We utilize short-term pressure buildup test analysis to enhance the production data analysis, particularly for interpretation of early-time transient flow behavior. We illustrate—with several field cases—that both techniques can be integrated to provide not only a more consistent and systematic analysis methodology, but also a more accurate assessment of stimulation effectiveness.

Introduction Wells producing from tight gas sands require stimulation to achieve economic rates and to maximize ultimate recoveries. The most common stimulation technique is hydraulic fracturing. Depending on the type and size of the treatment, hydraulic fracturing may be expensive—often representing a significant percentage of the total completion costs. Since the economic viability of wells completed in tight gas sands depends on minimizing costs, then it is essential that we optimize fracture treatments, i.e., find the proper balance between stimulation costs and well productivity. A key component in achieving this balance is a post-fracture diagnostics program to determine stimulation effectiveness.

Many diagnostic techniques for evaluating hydraulically-fractured gas well performance have been documented in the petroleum industry, but theoretical model assumptions, model applicability and simplicity, data requirements, and/or data quality and quantity may limit the effectiveness of any single analysis technique. Therefore, we employ an integrated approach in which we capture the benefits and utilize the strengths of several types of hydraulically-fractured well

diagnostic techniques.1-8 Cipolla and Wright9,10 and Barree, et al.11 have identified and grouped fractured-well diagnostic techniques into three general categories—direct far-field, direct near-wellbore, and indirect. Our methodology focuses on two indirect diagnostic techniques—pressure transient testing and production data analysis. Specifically, we illustrate how short-term pressure buildup testing integrated with long-term production data analysis can be an effective method for evaluating stimulation effectiveness.

Indirect Fractured-Well Diagnostic Techniques Although pressure buildup testing is the most effective indirect technique for evaluating the stimulation effectiveness of hydraulically fractured gas wells, knowledge of reservoir permeability—either from the well test or from an independent source—is required to compute fracture properties. If a well is shut in for a sufficient duration to reach the pseudoradial flow period, then we can uniquely determine reservoir permeability from the test data (and very likely also estimate effective fracture half-length and fracture conductivity since these are dependent on the permeability estimate). Wells completed in tight gas sands require very long shut-in times to reach pseudoradial flow, but operators are reluctant to shut in a well for extended periods. If, however, we have an independent estimate of reservoir permeability, then shorter duration pressure buildup tests become practical.

Decline type curve analysis of production data has become a common alternative for estimating reservoir permeability without shutting in the well. Fetkovich12,13 was the first to incorporate transient flow models with decline curve analysis. He developed the standard “decline type curves” by combining an analytical model for transient, radial flow at constant bottomhole pressure with Arps’14 empirical exponential and hyperbolic rate decline models. We note for completeness that the exponential rate decline model is the analytical solution for a well produced at a constant bottomhole flowing pressure and boundary-dominated flow conditions.

The original Fetkovich decline type curves are useful for a range of reservoir pressure conditions, but we have observed cases where the boundary-dominated rate-time data changes evaluation curves over time (i.e., changes from one empirical model or stem to another). These changes have been attributed to changes in gas properties as a function of reservoir pressure. To address the impact of pressure-dependent gas properties on the evaluation of gas production

SPE 97972

Post-Fracture Performance Diagnostics for Gas Wells With Finite-Conductivity Vertical Fractures J.A. Rushing, SPE, and R.B. Sullivan, SPE, Anadarko Petroleum Corp., and T.A. Blasingame, SPE, Texas A&M U.

2 SPE 97972

data, Carter15 presented decline type curves for gas reservoirs. Although his method is theoretically more rigorous than simply using the empirical exponential and hyperbolic type curve matching, his approach is not universal. Additionally, both the Fetkovich and Carter models suffer from the constant bottomhole flowing pressure assumption. Note that this assumption primarily affects evaluation of the early-time transient flow data from which we assess both reservoir properties and stimulation effectiveness.

Crafton16 addressed the limiting constant bottomhole flowing pressure assumption with decline type curves developed using a rate normalization of the pressure responses, i.e., conversion of variable-rate responses to a constant-rate response. Because of the rate normalization process, Crafton’s Reciprocal Productivity Index (RPI) Method incorporates both transient and pseudo-steady state flow models developed for constant-rate production conditions. Araya and Ozkan,17 however, noted some limitations of the RPI Method, especially when used for wells exhibiting cyclic flow rates or variable-rate production during boundary-dominated flow.

Palacio and Blasingame18 developed the material balance decline type curve methodology that also overcomes the constant bottomhole pressure constraint by using rate (or pressure) normalization and a material balance time (i.e., cumulative production divided by instantaneous rate). They demonstrated that, when plotted in terms of the material balance time variable, the pressure-drop, rate-normalized data merge into a single harmonic decline curve during boundary-dominated flow (i.e., the reciprocal of the equivalent constant-rate pressure response). They also combined their new pseudosteady-state model with several transient models, including gas wells with infinite-conductivity vertical fractures. We note that gas-dependent properties are addressed using the appropriate pseudopressure and pseudotime definitions as given in Reference 18.

Pratikno, et al.19 combined the pseudosteady-state superposition concept (i.e., material balance time) with the transient flow model for a finite-conductivity vertical fracture. The value of this work is that it distinguishes the decline type curve behavior for the case of a vertical well with a finite-conductivity fracture. Although presented in a different format, Agarwal, et al.20 also developed decline type curves—for both the infinite- and finite-conductivity vertical fracture case—using a similar material balance time function.

Other production data analysis techniques21-24 consider the flow rate and pressure histories to be an extended drawdown test and use variable-rate pressure transient testing theory and superposition plotting functions to analyze the data. Unlike most conventional decline type curves, these other methods allow us to identify specific flow regimes characteristic of hydraulically-fractured wells—e.g., bilinear, formation linear, and pseudoradial flow.25,26 Similar to traditional pressure transient testing, these production data analysis methods also use specialized plotting techniques to determine the fractured properties from each flow regime. The ability to identify specific flow patterns (regimes) offers a significant advantage over conventional decline type curves. However, we still

cannot uniquely quantify fracture properties unless we have an independent estimate of reservoir permeability or the well has reached pseudoradial flow.

Regardless of the production data analysis technique employed, problems with production data quantity and/or quality—e.g., incomplete or infrequent data sampling; poor quality data; and/or erroneous data—affect the accuracy of the analyses. This impact is especially problematic for accurate assessments of both fracture and reservoir properties using transient data which are typically changing frequently and rapidly during early-time flow periods. Therefore, our method utilizes short-term pressure buildup test evaluation to improve the production data analysis—principally for interpretation of early-time transient flow behavior from which we can evaluate effective fracture conductivity and half-length.7,17,27,28

New Material Balance Decline Type Curves (MBDTC) The new material balance decline type curves (MBDTC)19 were developed using a “desuperposition” technique which mathematically combined transient and boundary-dominated flow models. The transient model is that for a well with a finite-conductivity, vertical fracture producing at a constant rate, while the boundary-dominated model is for a well centered in a finite, volumetric circular reservoir. The type curves are correlated using dimensionless fracture conductivity, FCD, and dimensionless reservoir radius, reD.

Because of the rate-normalization technique implemented, the type curve method is also applicable to variable flow rate, variable bottomhole flowing pressure, or combinations of these flowing conditions. Further, the type curves use three different pressure-drop, normalized-rate plotting functions which allows us to match several different model and data functions simultaneously, thereby providing more representative (and more accurate) matches of field production data. We can estimate effective permeability to gas and the near-wellbore flowing efficiency (either in terms of fracture half-length or a skin factor) from analysis of transient data. Furthermore, analysis of the pseudosteady-state or boundary-dominated data provides estimates of contacted gas-in-place and drainage area.

An example of the new type curves is shown in Fig. 1 which is a log-log plot of dimensionless pressure-drop normalized-rate (red curves, qDd), dimensionless pressure-drop normalized-rate integral (green curves, qDdi), and dimensionless pressure-drop normalized-rate integral-derivative (blue curves, qDdid) as a function of dimensionless material balance time. The particular type curves shown in Fig. 1 were generated for a dimensionless fracture conductivity (FCD) value of 5 (which is a fairly low conductivity value). The dimensionless reservoir radius, reD, is used as a correlating parameter for the transient portion of the type curves.

Integrated Analysis Methodology We have developed an iterative technique that integrates pressure transient testing with production data analysis. We begin with initial estimates of reservoir permeability (kg), effective fracture half-length (Lf), and dimensionless fracture conductivity (FCD)—all estimated using production data

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analysis based on the material balance decline type curves (MBDTC).19

Fig. 1—Example Material Balance Decline Type Curves (MBDTC)19 for Finite-Conductivity Vertical Fractures (FCD=5)

In the next step, we evaluate the pressure buildup test. We identify flow regimes characteristic of hydraulically fractured wells (i.e., bilinear, formation linear, pseudoradial flow)25,26 from a log-log plot of the pressure derivative. Depending on the type of flow regimes present, we may then use special plotting functions and analysis techniques to compute fracture properties. For example, if bilinear flow is present, then we can estimate effective fracture conductivity from the slope of the straight line on a fourth-root of time function plot. Or, if the formation linear flow period is present, then we can estimate effective fracture half-length from the slope of the straight line on a square-root of time function plot.

In the final step of our analysis procedure, we use an automatic history-matching process to analyze the complete well test data. Estimates of reservoir permeability from the MBDTC analysis and fracture properties from the special well test analyses are used to establish initial estimates and reasonable ranges for the history-matching parameters. We compare the history-match to results from the MBDTC and special well test analyses, and we iterate between the analyses until consistent results are obtained.

Production Data Analysis Procedure Using MBDTC. In this section, we outline a procedure for using the new material balance decline type curves (MBDTC), including the required input data.

1. Gather all of the production data—i.e., gas production and flowing pressure histories. Daily production data is preferred since it will allow the most accurate evaluation. Bottomhole flowing pressures are also preferred, but we can still utilize surface flowing pressures by computing bottomhole values using nodal analysis.

2. Gather details of the wellbore diagram and tabulate all reservoir rock and fluid data, including reservoir pressure (either initial pressure or average pressure at the onset of production), static bottomhole reservoir temperature, net sand thickness, average effective porosity and connate water saturation in the net thickness interval, and fluid properties.

3. Compute the field material balance pseudotime30 function:

τµ

τµd

pcp

qtq

ct

t

gg

g

g

gigia ∫=

0 )()(

)()(

................................(1)

4. Compute the field pseudopressure-drop normalized-rate functions:

a. Pseudopressure-drop normalized-rate function:

)( pwfpi

g

p

g

ppq

pq

−=

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

b. Pseudopressure-drop normalized-rate integral function:

τdp

qtp

q at

p

g

aip

g ∫ ∆=⎟⎟⎠

⎞⎜⎜⎝

⎛

∆0

1 .............................................(3)

c. Pseudopressure-drop normalized-rate derivative-integral function:

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

∆=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

∆=⎟

⎟⎠

⎞⎜⎜⎝

⎛

∆ip

g

aaip

g

aidp

g

pq

tdd

tpq

tdd

pq 1

)ln(

...................................................................................(4)

5. Overlay the three field pseudopressure-drop rate-normalized functions on MBDTC for a specific value of FCD. We recommend that you start with a low value of FCD as an initial estimate. Force the field data in the boundary-dominated flow regime to match the depletion stems for each of the respective dimensionless functions (i.e., match (qg/∆pp) against qDd, (qg/∆pp)i against qDdi, and (qg/∆pp)id against qDdid).

6. Evaluate the match of the transient field data on each of the three dimensionless transient functions. If a good match is not obtained, then choose another set of type curves by increasing the value of FCD.

7. Once a good match is obtained, record time and rate match points (indicated by subscript MP) for both field and type curve parameters. In addition, record values of the type curve correlating parameters, FCD and reD. Compute the well and reservoir parameters as follows:

a. Contacted gas-in-place (G):

MPDd

MPpg

MPDd

MPa

gi qpq

tt

cG

)()/(

)()(1 ∆

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

b. Reservoir drainage area (A) and effective drainage radius (re):

)1(615.5

wi

gi

ShGB

A−

=φ

.............................................(6a)

π/Are = ..............................................................(6b)

c. Effective gas permeability (kg):

MPDd

MPpgDpss

gigig q

pqb

hB

k)(

)/(2.141

∆=

µ.................(7)

where bDpss is the dimensionless pseudosteady-state constant (see Reference 19 for a definition).

q Dd,

q Ddi

, and

q Ddi

dq D

d, q D

di, a

ndq D

did

4 SPE 97972

d. Effective fracture half-length (Lf):

eD

ef r

rL = ................................................................. (8)

Note that we have implemented a “spreadsheet” approach such that much of the type curve matching can be accomplished interactively, while many calculations are computed automatically.

Pressure Buildup Test Analysis Procedure. The procedure outlined below assumes pseudoradial flow will not be present since we are using short-term pressure buildup testing. If, however, the well test data exhibits this flow period, then that data should be incorporated into the analysis. We should note also that the working equations are derived based on a pseudopressure and normalized pseudotime formulation utilized in the well testing software Pansystem,34 but the equation format shown below will change if different pressure and time functions are used.

1. Prepare a plot of pseudopressure29 change and derivative of pseudopressure change against normalized equivalent or superposition30,31 pseudotime function using the pressure buildup test data. Identify all flow regimes25,26 characteristic of hydraulically fractured wells from the shape of the derivative data.

3. If the bilinear flow period25,32,33 is present, prepare a Cartesian plot of pseudopressure against the fourth root of normalized pseudotime function. Compute the slope (mB) of the line drawn through the bilinear flow period identified from Step 2. Using kg from the MBDTC analysis and mB from the bilinear plot, compute effective fracture conductivity, wfkf:

gtgB

gff kchm

Tqkw

φµ175.444 2

⎟⎟⎠

⎞⎜⎜⎝

⎛= .......................... (9)

4. If the formation linear flow period25,32,33 is present, prepare a Cartesian plot of pseudopressure against the square root of normalized pseudotime function. Compute the slope (mL) of the line drawn through the formation linear flow period identified from Step 2. Using kg from the MBDTC analysis and mL from the formation linear plot, compute effective fracture half-length, Lf:

gtgL

gf kchm

TqL

φµ1925.40

⎟⎟⎠

⎞⎜⎜⎝

⎛= .............................. (10)

5. Compute the dimensionless fracture conductivity, FCD, using estimates of wfkf, Lf and kg:

gf

ffCD kL

kwF = ........................................................ (11)

Automatic History Matching Procedure. The next step in our integrated procedure is the use of an automatic history-matching process to analyze the well test. To help converge to the correct solution, we use estimates of reservoir permeability from the MBDTC analysis and fracture properties from the well test special analyses. These estimates are used to

establish initial estimates and reasonable ranges for the history-matching parameters. We then compare history-matched results to the MBDTC and special well test evaluations. If needed, we iterate between the analyses until we obtain consistent results.

Validation of Integrated Evaluation Technique To assess the validity of our integrated evaluation technique, we analyzed several pressure buildup tests numerically and then compared the results to our integrated evaluation approach. We illustrate the results of the numerical well test analysis with Field Example 1. Because this well has a very low-conductivity fracture, it represents an extreme test case for our integrated (analytical) evaluation technique.

Analytical Well Test Analysis—Field Example 1. The first field example is from a well completed in the low-permeability Lower Cotton Valley Sands in a field located in the North Louisiana Salt Basin in Jackson Parish, LA. The productive sands are deep (13,000 to 15,000 ft) and abnormally over pressured (pore pressure gradients > 0.90 psi/ft). Productive intervals have effective porosities ranging from 4% to 12%, while absolute permeability ranges from 0.005 md to 0.02 md. Because of the thick sand intervals, the wells are often stimulated with multiple stages. This particular well was stimulated in three stages—each stage composed of 8,500 to 10,000 bbl slick water and 170,000 to 230,000 lbs 40/70 proppant.

The well produced for about two years before being shut in for a two-week pressure buildup test (Fig. 2). Initial estimates of permeability and fracture half-length were obtained from the material balance decline type curve (MBDTC)19 analysis of the production data. The type curve match shown in Fig. 3 is a log-log plot of dimensionless pressure-drop normalized-rate (red curves, qDd), dimensionless pressure-drop normalized-rate integral (green curves, qDdi), and dimensionless pressure-drop normalized-rate integral-derivative (blue curves, qDdid) against dimensionless material balance time. The discrete data points in Fig. 3 represent the corresponding dimensional field data (i.e., red points qg/∆pp, green points (qg/∆pp), and blue points (qg/∆pp)id).

Fig. 2—Post-fracture gas production history, Field Example 1.

From our initial analysis, we estimate kg=0.0084 md and Lf =236 ft. The best match was obtained with the type curves for

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SPE 97972 5

FCD=2 and reD=2. We also estimate the contacted gas in place and drainage area for this well are 3.7 Bcf and 14.7 acres, respectively. We do note that the shapes of the dimensionless rate and integral functions during transient flow are very similar for a wide range of reD values. Although we do observe some differences, the dimensionless derivative function curve shapes are also somewhat similar, especially for reD values between 2 and 5. Incomplete or inaccurate pressure and rate data tend to exacerbate these problems, thus often making it very difficult to obtain unique estimates of kg and Lf from the early-time data analysis. We can, however, quantify a range of values represented by the range of reD. For this case, we would estimate kg and Lf ranges of 0.0084 md to 0.0137 md and 225.7 ft to 90.7 ft, respectively, for reD from 2 to 5.

Fig. 3—Material balance decline type curve analysis of post-fracture gas production, Field Example 1.

Because of the significant rate changes prior to shutting in the well, we analyzed the two-week pressure buildup data using both pseudopressure29 and pseudotime superposition30,31 plotting functions. A log-log plot of the pseudopressure change and pseudopressure derivative functions against the normalized pseudotime superposition function is shown in Fig. 4. For this particular case, the solid black line with a one-quarter slope drawn through the pseudopressure derivative function indicates a significant portion of the test was dominated by bilinear flow. Note that we do not observe either a formation linear or a pseudoradial flow period during the two-week shut-in period.

Fig. 4—Log-log plot of two-week pressure buildup test identifying fractured-well flow regimes, Field Example 1.

Next, we use special plotting functions to validate various flow patterns identified from the log-log derivative plot. The bilinear flow regime is validated by the straight line on a plot of pseudopressure against the fourth root of pseudotime superposition function shown in Fig. 5. If we have identified the correct straight line on the curve, then we can use the bilinear flow line slope to estimate effective fracture conductivity as defined by Eq. 9. On the basis of the line slope from Fig. 5 and kg from the MBDTC analysis, we estimate wfkf is 2.8 md-ft. Moreover, if we substitute Lf and kg from the MBDTC analysis and wfkf from Fig. 5 into Eq. 11, we compute FCD=1.4. Note that this value is consistent with FCD=2 from the initial MBDTC analysis.

Fig. 5—Fourth-root-of-time plot showing bilinear flow regime from two-week pressure buildup test, Field Example 1.

On the basis of the derivative plot in Fig. 4, we did not observe the pseudoradial flow regime. As a result, we cannot use conventional semilog analysis techniques to compute reservoir permeability from the well test. Instead, we use an automatic history-matching process during which we allowed kg, Lf, and FCD to vary. To assess the importance of non-Darcy flow in the well test analysis,35,36 we also included a rate-dependent skin factor defined by the non-Darcy flow coefficient, D. Estimates of kg from the decline type curve analysis and Lf and FCD from the special analyses were used to establish initial estimates and reasonable ranges for the parameters during the history-matching process. In fact, we used the range of kg and Lf corresponding to the range of reD values discussed previously. We then iterated several times between the automatic history matching, the MBDTC analysis, and the well test analysis until we obtained consistent results.

The final and best history match is shown in Fig. 6. The solid lines drawn through the discrete well test data represent the final history-matched solution. Note that results from the history-matched pressure buildup test analysis are very similar to that from the initial MBDTC analysis. We estimate kg and Lf are 0.0068 md and 221 ft, respectively, from the history-matched well test and kg=0.0084 md and Lf=236 ft from the MBDTC analysis. We also computed wfkf=2.96 md-ft corresponding to FCD=1.97 from the well test analysis as compared to wfkf=3.96 md-ft and FCD=2 from the initial MBDTC analysis. Both analyses suggest the stimulation

Bilinear FlowRegion

Fourth-Root of Pseudotime Superposition Function, (hr)1/4

Pse

udop

ress

ure

Func

tion,

psi

a2/c

p

Bilinear Flow Analysiskg = 0.0084 md (MBDTC)

Lf = 236 ft (MBDTC)wfkf = 2.8 md-ft (Eq. 9)

FCD = 1.4 (Eq. 11)

Bilinear FlowRegion


Pse

udop

ress

ure

Func

tion,

psi

a2/c

p Bilinear FlowRegion


Pse

udop

ress

ure

Func

tion,

psi

a2/c

p


Lf = 236 ft (MBDTC)wfkf = 2.8 md-ft (Eq. 9)

FCD = 1.4 (Eq. 11)

BilinearFlow

Pseudotime Superposition Function, hr

Pse

udop

ress

ure

Func

tions

, psi

a2 /cp/

MM

scfd

One-quarter slope line indicative of

bilinear flow

Pseudopressure functionPseudopressure derivative function

BilinearFlow


Pse

udop

ress

ure

Func

tions

, psi

a2 /cp/

MM

scfd


bilinear flow


Material Balance Pseudotime Function

Pseu

dopr

essu

re-D

rop

Nor

mal

ized

-Rat

e Fu

nctio

ns MBDTC Resultskg = 0.0084 md

Lf = 236 ftwfkf = 3.96 md-ftFCD = 2; reD = 2

G = 3.7 Bcf; A = 14.7 acres


Pseu

dopr

essu

re-D

rop

Nor

mal

ized

-Rat

e Fu

nctio


Lf = 236 ftwfkf = 3.96 md-ftFCD = 2; reD = 2

G = 3.7 Bcf; A = 14.7 acres

6 SPE 97972

treatment generated a fracture with a very low effective conductivity.

Fig. 6—Results from automatic history-match of two-week pressure buildup test, Field Example 1.

Numerical Well Test Analysis—Field Example 1. To demonstrate the validity of our analysis technique, we also present results of the numerical well test analysis for Field Example 1. We developed a two-phase (gas-water), three-dimensional, multi-layer finite-difference model. To reduce computational time, we simulated one quarter of the reservoir with a Cartesian grid system. Moreover, we used a Cartesian rather than a radial grid system so that we could more accurately model the linear flow patterns characteristic of hydraulically fractured wells.

We used data from a comprehensive core description and evaluation program to populate the grid blocks vertically. The core data suggested significant reservoir heterogeneity in the vertical direction, so we built the model using five hydraulic rock types and 120 hydraulic flow units distributed over an interval of about 300 ft. The flow units were generated using a methodology described in References 37 and 38. Water saturations were distributed vertically using core-derived, rock-type-specific capillary pressure curves. We also measured gas-water relative permeability data for each hydraulic rock type.

We input the gas production history shown in Fig. 2 and history-matched both the well flowing and shut-in pressures by varying fracture properties—including effective fracture half-length and fracture permeability. Our best match (Fig. 7) was obtained with an effective fracture permeability of 10 md, an effective fracture half-length of 270 ft, and an average effective reservoir permeability, kg=0.0071 md. On the basis of the modeled fracture grid width, we compute an effective fracture conductivity of 5 md-ft, which is very close to that estimated from the history-matched pressure buildup test analysis (i.e., wfkf=2.96 md-ft). Similarly, both the simulated effective fracture half-length of 270 ft and dimensionless fracture conductivity of 2.6 agree with estimates from the integrated well test and decline type curve analyses. Although not shown in this paper, we have successfully matched results from the analytical and numerical techniques on several other wells. Therefore, we feel that our technique is valid for evaluating hydraulically-fractured gas well performance.

Fig. 7—Numerical well test analysis of two-week pressure buildup test, Field Example 1.

Field Example 2 The second example illustrates a large conventional slick water-frac stimulation treatment for a well producing from the Bossier Sands in the East Texas Basin. Specifically, this well is completed in the Bald Prairie Field located in Robertson County, TX. The well was hydraulically fractured in early May 2001 with 9,710 bbl slick water and 135,000 lbs 40/70 proppant. As shown by the production history in Fig. 8, the initial gas rate was slightly greater than 1,500 Mscf/day. The well was shut in for a two-week pressure buildup test after about 18 months of production.


Initial estimates of permeability and fracture half-length were obtained from the MBDTC analysis of the production data. The best match (Fig. 9) was obtained with FCD=5 and reD=5. A match of the boundary-dominated data indicates the contacted gas-in-place is 0.52 Bcf corresponding to a drainage area of 23.7 acres. We also estimate kg=0.0195 md, Lf =114.5 ft, and wfkf=11.2 md-ft. Unlike Field Example 1, the early-time field derivative data trend appear to be following a single curve for reD=5, thus allowing more unique estimates of kg and Lf from the transient data analysis.

Similar to Field Example 1, we analyzed the pressure buildup data using both pseudopressure and pseudotime superposition functions since the well exhibited significant changes in flow rate prior to the two-week shut in period. A log-log plot of the

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Pse

udop

ress

ure

Func

tions

, psi

a2/c

p/M

Msc

fd


History-Matched Resultskg = 0.0068 md

Lf = 221.3 ftwfkf = 2.96 md-ft

FCD = 1.97D = 4.0x10-5 (Mscf/d)-1


Pse

udop

ress

ure

Func

tions

, psi

a2/c

p/M

Msc

fd




FCD = 1.97D = 4.0x10-5 (Mscf/d)-1

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mho

le S

hut I

n Pr

essu

re, p

si

Measured Bottomhole Pressure

Simulated Bottomhole Pressure

Simulated Resultskg = 0.0071 md

Lf = 270 ftwfkf = 5 md-ft

FCD = 2.6500

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0 50 100 150 200 250 300 350Elapsed Shut In Time, hr

Botto

mho

le S

hut I

n Pr

essu

re, p

si

Measured Bottomhole Pressure

Simulated Bottomhole Pressure

Simulated Resultskg = 0.0071 md

Lf = 270 ftwfkf = 5 md-ft

FCD = 2.6

SPE 97972 7

pseudopressure change and pseudopressure derivative functions against the pseudotime superposition function is shown in Fig. 10. Note that, similar to the first example, we did not observe either a formation linear flow or a pseudoradial flow period.



Next, we use special plotting functions to validate various flow patterns identified from the log-log derivative plot. The bilinear flow regime is validated by the straight line on a plot of pseudopressure against the fourth root of pseudotime superposition function shown in Fig. 11. Using the line slope from Fig. 11 and kg from the MBDTC analysis, we estimate wfkf is 7.8 md-ft. Moreover, if we substitute Lf and kg from the MBDTC analysis and wfkf from Fig. 11 into Eq. 11, we compute FCD=3.5 which is reasonably consistent with FCD=5 from the initial MBDTC analysis.

Like the first field example, we did not see evidence of pseudoradial flow on the log-log plot of the pressure derivative. Consequently, we could not use conventional semilog techniques to compute permeability, so we again used an automatic history-matching process to analyze the well test. The final and best history match for Field Example 2 is shown in Fig. 12. Note that results from the pressure buildup test analysis are very similar to those estimated from the initial material balance decline type curve analysis. We estimate kg=0.0185 md, while the fracture properties are Lf=101.1 ft, wfkf=16.4 md-ft, and FCD=8.8.



Field Example 3 The third example illustrates a small but very effective conventional slick water-frac stimulation treatment for a well producing from the Bossier Sands in the East Texas Basin. The particular well is completed in the Mimms Creek Field located in Freestone County, TX. The well was hydraulically fractured in early October 1999 with 8,220 bbl slick water and 37,000 lbs 20/40 sand proppant. As shown by the production history in Fig. 13, the initial gas rate was almost 12,000 Mscf/day. The well was shut in for a two-week pressure buildup test after more than two years of production.



Pseu

dopr

essu

re F

unct

ions

, psi

a2/c

p/M

Msc

fd




FCD = 8.77D = 1.1x10-14 (Mscf/d)-1


Pseu

dopr

essu

re F

unct

ions

, psi

a2/c

p/M

Msc

fd




FCD = 8.77D = 1.1x10-14 (Mscf/d)-1

Bilinear FlowRegion


Pse

udop

ress

ure

Func

tion,

psi

a2/c

p


Lf = 114.5 ft (MBDTC)wfkf = 7.8 md-ft (Eq. 9)

FCD = 3.5 (Eq. 11)

Bilinear FlowRegion


Pse

udop

ress

ure

Func

tion,

psi

a2/c

p

Bilinear FlowRegion

Bilinear FlowRegion


Pse

udop

ress

ure

Func

tion,

psi

a2/c

p



FCD = 3.5 (Eq. 11)


Pseu

dopr

essu

re-D

rop

Nor

mal

ized

-Rat

e Fu

nctio


Lf = 114.5 ftwfkf = 11.2 md-ftFCD = 5; reD = 5

G = 0.52 Bcf; A = 23.7 acres


Pseu

dopr

essu

re-D

rop

Nor

mal

ized

-Rat

e Fu

nctio



G = 0.52 Bcf; A = 23.7 acres


Pse

udop

ress

ure

Func

tions

, psi

a2/c

p/M

Msc

fd


bilinear flow



Pse

udop

ress

ure

Func

tions

, psi

a2/c

p/M

Msc

fd


bilinear flow


bilinear flow


0

2,000

4,000

6,000

8,000

10,000

12,000

10/8/99 10/8/00 10/8/01 10/8/02 10/8/03 10/8/04

Date

Gas

Pro

duct

ion

Rat

e, M

scf/d

ay


Buildup Test

0

2,000

4,000

6,000

8,000

10,000

12,000

10/8/99 10/8/00 10/8/01 10/8/02 10/8/03 10/8/04

Date

Gas

Pro

duct

ion

Rat

e, M

scf/d

ay


Buildup Test


Buildup Test

8 SPE 97972

Again, the first step in our procedure was to evaluate the production history with the MBDTC. The best match (Fig. 14) was obtained with FCD=10 and reD=2. A match of the boundary-dominated data indicates the contacted gas in place and drainage area are 2.1 Bcf and 15.2 acres, respectively. We also estimate kg=0.0191 md, Lf = 229.8 ft, and wfkf=43.9 md-ft. Note that the early-time field derivative data trend appears to be following a single curve for reD=2.


The next step in our procedure is the evaluation of the pressure buildup test. A log-log plot of the pseudopressure change and pseudopressure derivative functions against the pseudotime superposition function is shown in Fig. 15. Unlike the first two field examples, we observe both bilinear and formation linear flow periods, as indicated by the solid black lines with slopes of one-quarter and one-half, respectively. We do not, however, see evidence of pseudoradial flow.


The bilinear and formation linear flow regimes are also identified and validated by the plots of pseudopressure against fourth-root and square-root of pseudotime superposition functions shown in Figs. 16 and 17, respectively. Using mB from the line drawn through the bilinear flow period in Fig. 16 and kg estimated from the MBDTC analysis, we compute wfkf=77.4 md-ft. Further, if we use kg and Lf estimates from the MBDTC analysis of the production data, we estimate FCD=17.6 which is similar to that estimated from the MBDTC.

As indicated by the straight-line on the plot of pseudopressure against the square root of pseudotime superposition function

shown in Fig. 17, the formation linear flow period is very well defined. Using mL from the line drawn through the formation linear flow period and kg estimated from the MBDTC analysis, we compute Lf=239.8. Again, the fracture half-lengths estimated from the MBDTC and square-root-of-time analyses are generally in agreement. We also compute a dimensionless fracture conductivity of 16.9 corresponding to an effective fracture conductivity of 77.4 md-ft estimated from the bilinear flow analysis.


Fig. 17—Square-root-of-time plot showing formation linear flow regime from two-week pressure buildup test, Field Example 3.

Like the previous two examples, the absence of a pseudoradial flow period precludes use of conventional semilog analysis techniques to compute permeability from the well test. Alternatively, we used an automatic history-matching process to analyze the well test data. The best history match is shown in Fig. 18. We estimate kg and Lf are 0.0257 md and 261.1 ft, respectively. In addition, these results indicate the stimulation treatment generated a much more conductive fracture than Field Examples 1 and 2. We compute a fracture conductivity of 131.5 md-ft corresponding to a dimensionless fracture conductivity of 19.6.

Field Example 4 The fourth field example is a another well completed in the Bossier Sands in the Mimms Creek Field in Freestone County, TX. The well was hydraulically fractured in early April 2001 with 8,571 bbl slick water and 170,000 lbs 40/70 sand proppant. The well was shut in for a two-week pressure buildup test after about 18 months of production (Fig. 19).

Formation Linear Flow

Region

Square-Root of Pseudotime Superposition Function, (hr)1/2

Pse

udop

ress

ure

Func

tion,

psi

a2 /cp

Formation Linear Flow Analysiskg = 0.0191 md (MBDTC)

Lf = 239.8 ft (Eq. 10)wfkf = 77.4 md-ft (Bilinear Flow Analysis)

FCD = 16.9 (Eq. 11)


Region


Pse

udop

ress

ure

Func

tion,

psi

a2 /cp


Region


Region


Pse

udop

ress

ure

Func

tion,

psi

a2 /cp



FCD = 16.9 (Eq. 11)

Bilinear FlowRegion


Pseu

dopr

essu

re F

unct

ion,

psi

a2/c

p



FCD = 17.6 (Eq. 11)

Bilinear FlowRegion


Pseu

dopr

essu

re F

unct

ion,

psi

a2/c

p

Bilinear FlowRegion


Pseu

dopr

essu

re F

unct

ion,

psi

a2/c

p



FCD = 17.6 (Eq. 11)


Pseu

dopr

essu

re F

unct

ions

, psi

a2/c

p/M

Msc

fd

One-half slope line indicative of formation linear flow



bilinear flow


Pseu

dopr

essu

re F

unct

ions

, psi

a2/c

p/M

Msc

fd





bilinear flow


bilinear flow


Pseu

dopr

essu

re-D

rop

Nor

mal

ized

-Rat

e Fu

nctio



G = 2.1 Bcf; A = 15.2 acres


Pseu

dopr

essu

re-D

rop

Nor

mal

ized

-Rat

e Fu

nctio



G = 2.1 Bcf; A = 15.2 acres

SPE 97972 9

The MBDTC analysis of the production history is shown in Fig. 20. The best match was obtained with FCD=10 and reD=2. A match of the boundary-dominated data indicates the contacted gas-in-place and drainage area are 2.3 Bcf and 23.7 acres, respectively. We also estimate kg=0.0070 md, Lf =286.3 ft, and wfkf=20.4 md-ft. Although less certain than Field Example 3, the early-time field derivative data trend again appears to be following a single curve for reD=2.




A log-log plot of the pseudopressure change and pseudopressure derivative functions against the pseudotime superposition function is shown in Fig. 21. Similar to Field Example 3, we observe both bilinear and formation linear flow

periods, as indicated by the solid black lines with slopes of one-quarter and one-half, respectively. Again, we do not see any indications of pseudoradial flow developing during the two-week pressure buildup test.


The bilinear and formation linear flow regimes are also validated by the plots of pseudopressure against fourth-root and square-root of pseudotime superposition functions shown in Figs. 22 and 23, respectively. Using mB from the line drawn through the bilinear flow period in Fig. 22 and kg estimated from the MBDTC analysis, we compute wfkf=27.7 md-ft. Further, if we use kg and Lf estimates from the MBDTC analysis of the production data, we estimate FCD=13.8.


As indicated by the straight-line on the plot of pseudopressure against the square root of pseudotime superposition function shown in Fig. 23, the formation linear flow period is very well defined. Using mL from the line drawn through the formation linear flow period and kg estimated from the MBDTC analysis, we compute Lf=238.3. Again, the fracture half-lengths estimated from the MBDTC and square-root-of-time analyses are generally in agreement. We also compute dimensionless fracture conductivity of 16.6 corresponding to an effective fracture conductivity of 27.7 md-ft estimated from the bilinear flow analysis.

The best history match of the well test data for Field Example 4 is shown in Fig. 24. We estimate kg and Lf are 0.0082 md and 282.8 ft, respectively. Similar to Field Example 3, these

Bilinear FlowRegion


Pseu

dopr

essu

re F

unct

ion,

psi

a2/c

p



FCD = 13.8 (Eq. 11)

Bilinear FlowRegion

Bilinear FlowRegion


Pseu

dopr

essu

re F

unct

ion,

psi

a2/c

p



FCD = 13.8 (Eq. 11)


Pse

udop

ress

ure

Func

tions

, psi

a2/c

p/M

Msc

fd




bilinear flow


Pse

udop

ress

ure

Func

tions

, psi

a2/c

p/M

Msc

fd





bilinear flow


bilinear flow


Pseu

dopr

essu

re F

unct

ions

, psi

a2/c

p/M

Msc

fd


History-Matched Resultskg = 0.0257md

Lf = 261.1 ftwfkf = 131.5 md-ft

FCD = 19.6D = 8.6x10-20 (Mscf/d)-1


Pseu

dopr

essu

re F

unct

ions

, psi

a2/c

p/M

Msc

fd


History-Matched Resultskg = 0.0257md

Lf = 261.1 ftwfkf = 131.5 md-ft

FCD = 19.6D = 8.6x10-20 (Mscf/d)-1

0

500

1,000

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2,000

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4,000

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Rat

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ion

Rat

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scf/d


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Buildup Test


Pseu

dopr

essu

re-D

rop

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ized

-Rat

e Fu

nctio



G = 2.3 Bcf; A = 23.7 acres


Pseu

dopr

essu

re-D

rop

Nor

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-Rat

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G = 2.3 Bcf; A = 23.7 acres

10 SPE 97972

results indicate the stimulation treatment generated a much more conductive fracture than Field Examples 1 and 2. We compute a fracture conductivity of 44.3 md-ft corresponding to a dimensionless fracture conductivity of 19.1. Note again that these results are not too different from the results obtained from the MBDTC analysis of the production data.

Fig. 23—Square-root-of-time plot showing formation linear flow regime from two-week pressure buildup test, Field Example 4.


Summary and Conclusions We have developed an integrated approach for evaluating the post-fracture production performance of gas wells producing from tight gas sands. Although we focus on wells with finite-conductivity vertical fractures, the methodology is also valid for any vertically fractured well case (i.e., finite-conductivity, infinite-conductivity or uniform-flux fractures). We have validated our technique with a simulated case, and we have illustrated the applicability and utility of our integrated technique with several field examples. On the basis of the results, we offer the following conclusions:

1. The material balance decline curves developed for wells with finite-conductivity vertical fractures are generally very useful for evaluating the production performance in hydraulically-fractured gas wells. Agreement between the MBDTC and pressure buildup test analysis in our evaluations ranges from good to excellent.

2. The accuracy of these material balance type curves for evaluating stimulation effectiveness depends on the quality and quantity of production data—particularly the early-time transient data. Therefore, we strongly

recommend that operators strive to improve their data acquisition and gathering efforts. Moreover, we recommend the use of daily production for the production decline type curve analysis.

3. Although most production decline type curve methods are valuable tools for evaluating well production performance, no "history analysis" approach can completely replace conventional pressure transient testing. The value of pressure transient tests for establishing current flow capacity and flow efficiency can not be overstated.

4. Production data quality and/or quantity may preclude accurate evaluation of production data. The material balance decline type curve approach used in this work is both robust and error tolerant, and should be expected to perform well in practice. We believe that production data analysis can (and should) be able to "stand alone" in the absence of pressure transient tests.

5. The results of our study also demonstrate the value and function of short-term pressure buildup testing in tight gas sands. These pressure transient tests are quite useful when integrated with production data analysis with decline type curves—particularly for interpretation of early-time transient flow behavior. Therefore, we recommend that operators incorporate short-term pressure transient testing with production data analysis to evaluate the stimulation effectiveness of wells producing from tight gas sands.

Acknowledgements We would like to express our thanks to Anadarko Petroleum Corp. for permission to use the data and to publish the results of our study.

Nomenclature

Dimensionless Variables bDpss = dimensionless pseudosteady-state constant FCD = dimensionless fracture conductivity = wfkf/kgLf qDd = dimensionless pseudopressure-drop normalized rate

function qDdi = dimensionless pseudopressure-drop normalized rate

integral function qDdid = dimensionless pseudopressure-drop normalized rate

integral-derivative function reD = dimensionless reservoir radius = re/Lf

Dt = dimensionless material balance pseudotime function

Field Variables A = well drainage area, acres Bgi = gas formation volume factor evaluated at initial reservoir

pressure, RB/Mscf cg = gas compressibility, psia-1

cgi = gas compressibility at initial reservoir pressure, psia-1 ct = total system compressibility, psia-1

cti = total system compressibility at initial reservoir pressure, psia-1

G = contacted gas-in-place, Mscf h = net sand thickness, ft kf = fracture permeability, md kg = effective gas reservoir permeability, md Lf = effective or propped fracture half-length, ft


Region


Pse

udop

ress

ure

Func

tion,

psi

a2/c

p



FCD = 16.6 (Eq. 11)


Region


Pse

udop

ress

ure

Func

tion,

psi

a2/c

p


Region


Region


Pse

udop

ress

ure

Func

tion,

psi

a2/c

p



FCD = 16.6 (Eq. 11)


Pse

udop

ress

ure

Func

tions

, psi

a2/c

p/M

Msc

fd




FCD = 19.1D = 4.8x10-8 (Mscf/d)-1


Pse

udop

ress

ure

Func

tions

, psi

a2/c

p/M

Msc

fd




FCD = 19.1D = 4.8x10-8 (Mscf/d)-1

SPE 97972 11

mB = slope of line drawn through bilinear flow period on fourth-root of time plot

mL = slope of line drawn through formation linear flow period on square-root of time plot

ppi = normalized pseudopressure function evaluated at initial reservoir pressure, psia

ppwf = normalized pseudopressure function evaluated at bottomhole flowing pressure, psia

qg = gas flow rate, Mscf/day re = reservoir radius, ft rw = wellbore radius, ft Swi = initial connate water saturation, fraction T = bottomhole reservoir temperature, oR

at = normalized material balance pseudotime function, hr wf = fracture width, in wf kf = effective fracture conductivity, md-ft φ = effective porosity, fraction µg = gas viscosity, cp µgi = gas viscosity at initial reservoir pressure, cp

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12 SPE 97972

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