Copyright 2004, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in Houston, Texas, U.S.A., 26–29 September 2004. 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 The primary objective of hydraulic fracturing is to create a propped fracture with sufficient conductivity and length to optimize well performance. In permeable reservoirs, the design objective is to achieve a Dimensionless Fracture Capacity, CfD, of at least 2. In lower permeability applications, additional conductivity is required (CfD > 10) to allow effective fracture fluid cleanup and optimized well perfor-mance. In some tight formation gas applications conventional cross-linked gel fracture stimulations are not creating the de-sired fracture dimensions. The potential reasons for the shorter than desired effective fracture lengths are numerous with the most likely being reservoir heterogeneity, excessive fracture height growth, and poor fracture fluid cleanup.

In recent years, there has been much discussion regarding the causes for, or reasons that the dimensions of the hydraulic fracture are shorter than desired. These include: relative permeability effects, fracture fluid cleanup, multi-phase flow, and non-Darcy flow. The former causes and reasons have been investigated in some detail; however, little data has been pub-lished regarding the effects of non-Darcy flow on fracture con-ductivity and effective fracture length. Some in the industry have suggested that tight gas well performance is hindered significantly by non-Darcy flow effects. This view, though potentially correct, is supported by little actual data in the lit-erature. Further, to mitigate this effect tip screen-out fracturing techniques and larger fracture stimulation designs often utilizing much more expensive ceramic proppants have been recommended and executed even in very low permeability applications. These methods may not be effective in tight gas applications but they surely are more expensive, potentially eroding the economic benefits of fracturing these low deliv-erability applications. In addition, little actual well perfor-

mance data has been presented to justify the importance of non-Darcy flow in fractures with much of the justification coming from the use of semi-analytical calculations and spreadsheets.

This paper will document an investigation of non-Darcy flow to hydraulically fractured oil and gas well performance. The investigation will utilize both a three dimensional single-phase numeric finite difference simulator and actual well per-formance to investigate the importance of non-Darcy flow to hydraulically fractured oil and gas wells. This paper will demonstrate the following: 1) The importance or lack of importance of non-Darcy flow

on hydraulically fractured oil and gas well performance, 2) Compare and contrast actual well performance of off-

setting wells where sand and ceramics were utilized in East Texas, Trinidad, and North Sea applications,

3) Develop treatment guidelines and fracture design objectives to limit/mitigate the effects of non-Darcy flow across a broad spectrum of fracturing applications.

Introduction The industry has been aware of the potential for non-Darcy flow in propped fracture for many years – since the pioneering work by Cooke.1 Since that work, much additional technology has been added, and that history has been well covered and will not be reviewed here (except as appropriate below). The primary problem was that the importance of this behavior was, at best, difficult to prove or quantify.

The “problem” was that fracturing was traditionally (at least in the 70’s and 80’s when this idea was broached) ap-plied to low permeability formations. The traditional, “defini-tive” test for non-Darcy effects (multi-rate drawdown) was difficult to apply operationally to such wells, and, again, at best, difficult to interpret as normal fractured well transient flow tends to mask non-Darcy effects. More recently, several papers have dealt with new analysis approaches that may make analysis for these effects more definitive in the future, but that is outside the realm of this work.

Because of this “problem”, the bulk of the literature has dealt with theoretical (analytical and numerical and semi-num-erical) studies and extensive laboratory testing. However, very few papers have examined well test data over a range of con-ditions to compare the magnitude of the non-Darcy effects with these predictions.

SPE 90864

An Investigation of Non-Darcy Flow Effects on Hydraulic Fractured Oil and Gas Well Performance Smith, M. B., SPE, NSI Technologies, Bale, A., SPE, StatOil, Britt, L. K., SPE, Cunningham, L. E., SPE, NSI Technologies, Jones, J. R., SPE, BP, Klein, H. H., SPE, H K Technologies, Wiley, R. P., SPE, BP

2 Smith, M. B., Bale, A., Britt, L. K., Cunningham, L. E., Jones, J. R., Klein, H. H., and Wiley, R. P. SPE 90864

Thus, the goal of this paper was to collect well test data from a range of environments to examine the magnitude of non-Darcy flow (in the propped fracture) effects. The goal was not entirely met, with some planned test data not collected for various reasons. Still, several cases are examined ranging from micro-darcy gas wells, to 40,000 bopd oil wells, to 120 MMSCFD frac-pack wells. Theoretical Discussion Apparent Conductivity. Much current practice in dealing with non-Darcy flow (i.e., predicting performance, selecting proppant, etc.) is based on the early theoretical work by Guppy 2, 3 . This showed that effects of non-Darcy flow can be expressed as an “Effective” or “Apparent” conductivity (always less than the “True” or laminar, Darcy flow conductivity). They further showed that under certain conditions, non-Darcy effects can be calculated based on two dimensionless parameters:

f

fffD xk

bkC = , ……………….…..…… (1)

the dimensionless fracture conductivity, and

µβρ

hbqk

Qf

fD = , ………….………… (2)

basically the ratio of inertial (i.e., non-Darcy or “turbulent”) drag forces on flow in the fracture to the “Darcy” drag forces. NOTE that in oil field units, Eq. 2 is expressed as

µ

ρβhb

qkxQ

f

fD

15100273.1 −

=

where “kf” is fracture permeability (md), “β” is the non-Darcy factor (1/ft), “ρ” is the fluid density (lbm/ft3), “q” is the total well flow rate (stb/d), “bf” is the fracture width (ft), “h” is fracture height (ft), and “µ” is fluid viscosity (cp). For gas wells this becomes:

if

fD hb

qMkxQ

µβ161064.4 −

=

where “M” is the gas molecular weight (typically about 20 for a gas with a specific gravity of 0.7) and “q” is the total well flow rate (MSCFD).

Several relations were proposed for calculating the “Ap-parent Conductivity”, but the relation seeming to be most commonly used is given by:

( )D

fDAppfD Q

CC

55.01+= . ………….. (3)

While actually proposed as a correction for the fracture con-ductivity determined from pressure build-up tests, this simple relation has (and is) received wide spread use as a predictive tool for the effects of non-Darcy flow. HOWEVER, what is not apparently seeing such wide spread use is the fact that his empirical relation (derived from multiple numerical model simulations) is subject to severe limitations. That is, the rela-tion is only valid for:

CfD < 10, QD < 2, or

CfD > 10, QD = 1 to 10. Since a value of QD < 2 is a case for “minimal” non-Darcy effect (i.e., “only” a 50% reduction in kfbf), and since achiev-ing CfD > 10 is generally only possible for low permeability formations, this relation is effectively limited to low permea-bility zones. What, then, is the impact of violating these limi-tations?

Impact of Violating Limits. A partial answer comes from Settari and Bale. 4 An extensive finite difference model study found that

“…. All simulation data was below the straight line of Eq. 3, but with scatter, indicating that there is some other dependency. However, when the data was filtered to in-clude only k = 2 md, and the range of (CfD)True < 10, QD < 2, linear correlation was obtained but with slightly lower coefficient of 0.48 instead of 0.55. Stark 5 observed similar small discrepancy in an independent comparison.”

That is, for ALL the cases studied, the non-Darcy effects were less than predicted by Eq. 3, with the magnitude of the error varying from case-to-case. They eventually concluded that it required five variables to correlate the non-Darcy effects for the range of cases considered. Summed up, they found that “It’s not that simple.” when a realistic range of conditions are considered.

Example Case History. Another example is a brief synop-sis of a case where the authors first encountered a significant over statement of non-Darcy effects. This was for a low (0.12 md) permeability dry gas well in Canada. The zone was quite thick (120 feet of net pay in a 150 foot gross interval, TVD), and several treatments had been done using oil base fluid and Ottawa sand at a maximum concentration of 4 PPG. A review of these treatments had suggested in situ fracture coverage of about 0.5 lb/ft2 (estimated conductivity of about 550 md-ft, propped width of 0.05 inches, and β of 36,400 1/ft or 0.0012 atm-sec2/gram). Based on this review, recommen-dations had been made to the operator that non-Darcy (and multi-phase since the wells made small amounts of water, i.e., < 10 bwpd) flow was reducing effective conductivity by 92% (to about 50 md-ft) and reducing 1-year cumulative by 35% (1525 to 1024 MMCF).

Starting from this point in the data review, the above num-bers were used to simulate the expected productivity using a 3D finite difference based reservoir model – with quite differ-ent results. This simulation said non-Darcy effects “only” re-duced the apparent conductivity by 75%. Moreover, the im-pact on production (while significant) was “only” 16% (1525 to 1278).

Based on the gross formation thickness (150 ft) and an average first year rate of about 3.5 MMSCFD, QD is calculated to be about 20. Using Eq. 3 then predicts

ftmdbkQ

bkbfk Trueff

D

TrueffAppf −==

+= 46)(043.0

55.01)(

)( ,

- a 92% conductivity reduction. These results are summarized in Figure 1.

The prediction of a 35% reduction in 1-year cum was made using spreadsheet based calculations, so the calculations were examined. Indeed, using a conductivity of 50 md-ft in the numerical model showed a similar reduction in well per-

SPE 90864 An Investigation of Non-Darcy Flow Effects on Hydraulic Fractured Oil and Gas Well Performance 3

formance. That is, the “problem” with the simple prediction was in using Eq. 3 for a QD value far beyond valid limits for that relation. This led to overstating the non-Darcy effects on

well performance by a factor of 2.2X even in this simple geologic case (i.e., minimal multi-phase flow effect, no con-verging flow due to limited perforated interval, etc.). (Note that there were other differences between the spreadsheet solution and the numerical model, with the main item being ignoring the 30 feet of non-pay in the simple calculation. This thickness is there, and it does possess conductivity, and thus, also tends to reduce non-Darcy effects. However, the major cause of the wrong answer in this case was the use of Eq. 3.)

This might be summarized by – It’s not that simple! That is, except for very limited cases, a simple correction based on CfD and QD is not sufficient to allow accurate use of spread-sheet calculations and/or type curve production models for predicting non-Darcy flow effects. Did non-Darcy flow impact production from these wells? “Yes”, almost certainly. These were not “tight gas” wells, but wells with initial flow rates on the order of 10 MMscfd – non-Darcy effects would certainly be expected. Was this impact significant? “Yes”, a 16% (250 MMscf) reduction in 1-year cum is important. However, there is no need to use solutions that overstate the impact of non-Darcy flow by a factor of 220%.

Fracture Width Effects. Another factor often overlooked in making decisions regarding non-Darcy flow effects is the

dominant importance of fracture width. This was noted in the early work of Holditch

6 where results such as seen in Figure 2 were found – i.e., 40/60 mesh sand shows much less non-Darcy effect than 20/40. How is that possible?

If one assumes for a moment that Eq. 3 is valid for large

values of QD, then that relation can be rewritten as

( )

qhb

qkhbfbk

Qbk

bk

f

f

Trueff

D

TrueffAppff

βρµ

βρµ

55.055.0)(

55.0)(

2

=

==

That is, flowing or apparent conductivity is proportional to width SQUARED, but only related linearly to β (and not related to kf at all since this assumes flow dominated by non-Darcy effects).

Thus, for the figure, the basic assumption was that the two cases had EQUAL conductivity – kfbf. Since the smaller sand had lower kf, it had to have more width, bf. The greater width was more important than the reduced kf (and increased β), and thus, was “better” (J/JD = 1 in this plot indicates “0” non-Dar-cy effect). Of course, things would have been even better still to have increased the width and continued to use larger sand – i.e., both increase width and kf and reduce β!

Of course, Eq. 3 is NOT valid for large QD. Still, this em-phasizes the importance of propped fracture width to non-Dar-cy flow. The non-Darcy effect is related to velocity-squared. Thus, an increase in width reduces flow velocity along the fracture, causing a significant decrease in non-Darcy effects.

For example, in the 0.12 md case discussed above, increas-ing sand volume by 50% (probably by simply going to higher concentrations such as 6 to 8 PPG) easily increases proppant coverage to 0.75 lb/ft2. This 50% increase in width (easily/ cheaply accomplished) reduces non-Darcy effects by a factor of 2.25. This increases the expected 1-year cum to about 1,410 MMCF and has reduced the difference to an 8% reduc-tion in 1-year, and < 5% difference in 3-year cum.

Folds-of-Increase. As another example of the magnitude of non-Darcy effects, simple “Folds-of-Increase” simulations

Figure 1 – Effect of non-Darcy Calculations on Conductivity and Cumulative, 0.12 md Gas Well Example

200

400

600

800

1,000

1,200

1,400

KfW

(md-

ft)1-

Year

Cum

(MM

CF)

550 md-ftββββ = 0

KfW 1-Year Cum

550 md-ftββββ = 0.0012

or130 md-ftββββ = 0.0

50 md-ftββββ = 0

6 12 18 24 30

500

1,000

1,500

2,000

2,500

3,000

3,500

Time (months)

Cum

(MM

CF)

Cond = 550, ββββ = 0.0 Cond = 550, ββββ = 0.0012 Cond = 130, ββββ = 0.0 Cond = 50, ββββ = 0.0 Cond = 860, ββββ = 0.0012

Figure 2 – non-Darcy Flow Effects

4 Smith, M. B., Bale, A., Britt, L. K., Cunningham, L. E., Jones, J. R., Klein, H. H., and Wiley, R. P. SPE 90864

were made with, and without “β”. It is well known that for Darcy, pseudo-radial flow, the maximum “Folds-of-Increase (for a given volume of proppant) is achieved for a (CfD)True value of about 2. That is, given a fixed volume of proppant (and a simple fracture geometry) one can make the fracture longer/narrower (thus, a lower CfD), or one can make the fracture shorter/fatter (thus, a higher CfD). The maximum FOI is achieved for the fracture length/conductivity that gives di-mensionless fracture conductivity, CfD, of 2.

A simple case was simulated. This assumed a constant proppant volume of 7,000 lb, with a fixed fracture height of 50 feet and fixed formation permeability of 0.5 md. Thus, for a ½ length, xf, of 70 feet, this yields 1 lb/ft2 proppant coverage. Published long-term proppant data was then used, with no reduction for embedment or fluid damage, and fracture con-ductivity (and propped width and β) was determined for this 1 lb/ft2 case. The values used are included in Table 1, and were selected just as a “typical” range of proppant types and values under a nominal proppant stress of 4,000 psi. .

Table 1 – Fracture Properties for FOI Study 1 lb/ft2 kfbf (md-ft) β (atm-sec2/gram)

20/40 Ottawa 1540 0.00070 12/20 Ottawa 3250 0.00095 20/40 Ceramic 4150 0.00040 30/50 Ceramic 1750 0.00063

Propped length was then increased in steps, decreasing the propped width and conductivity proportionally (i.e., doubling the xf to 140 feet reduces propped width and conductivity by 50%). The numerical simulator was then run with, and without non-Darcy effects. The simulation assumed a constant, low rate of 500 Mscfd, with the goal being to have low to mod-erate values of QD for checking Eq. 3. The model was run to steady-state, and the required drawdown compared to the drawdown (at steady-state) for 500 Mscfd in radial flow (0 skin). This was used to calculate a “Folds-of-Increase as plotted in Figure 3 for the 20/40 Ottawa sand case.

As seen in this figure, for the Darcy flow, the maximum FOI is found at CfD of about 2 as expected. However, if non-Darcy effects are considered, more propped width is required, and the maximum FOI is found for a (CfD)True of about 4.3. For

a range of values of xf/kfbf, the apparent conductivity was calculated using Eq. 3, and the simulation was then run using this apparent conductivity (with β=0). For this case, using the (Cfd)Apparent calculated by Eq. 3, the maximum FOI was found at a (CfD)True of 6.2. That is, a 50% error (6.2 versus 4.3) even for this case with a moderate value of QD (3.74). Results for the remaining three cases are included in Table 2.

Table 2 – (CfD)True for Maximum FOI

Equation 3 Darcy Flow

Numerical non-Darcy QD (CfD)True

20/40 Ottawa 2 4.3 3.7 6.2 12/20 Ottawa 2 11.4 11.7 14.9 20/40 Ceramic 2 6.6 8.0 10.8 30/50 Ceramic 2 4.3 4.2 6.6

As expected, the non-Darcy effects increased the “opti-mum” conductivity for every case, with an increase of about 2X to about 5X. That is, significant effects. However, in every case use of a simple CfD/QD relation overstated the non-Darcy effects, even for these moderate values of QD.

Numerical Model. The example above and the case histories below used a single well, single phase finite difference model to examine the non-Darcy effect. The numerical procedures are similar to other models of this type (reference 4 for example) so the numerical details are not repeated here.

The reservoir is modeled by a numerical, finite difference simulator. The reservoir fluid is a considered as a single phase, compressible liquid or non-ideal gas. The numerical solution also includes the compressibility of the rock and the saturated water.

The reservoir can contain multiple vertical layers, with varying properties for horizontal/vertical permeability, poro-sity, pressure, water saturation, non-Darcy “β”. The non-Darcy β is a function of formation porosity and permeability.

The model simulates a single well in a rectangular or cir-cular drainage area. For rectangular cases, the aspect ratio of the formation can differ from unity, and the well need not be in the center of rectangle.

The model also has the capability including fractured and un-fractured vertical and horizontal wells. Multiple horizontal and vertical fractures are allowed, and for horizontal wells the fractures can be orthogonal to or parallel to the wellbore. The fracture conductivity (kfbf), fracture permeability (kf), propped width (bf), and non-Darcy β can be constant or vary when coupled to a fracture simulator.

For fractured cases the perforated height can differ from the pay height.

The external boundary conditions can be either constant pressure or zero flow. The wellbore boundary conditions can be constant bottom hole pressure, constant rate (where zero rate for shut-in is allowed), or constant surface pressure for gas wells. For the constant surface pressure cases, the well pressure solution is coupled to the reservoir pressure solution.

In the reservoir the temperature is assumed constant but the viscosity for gas calculations varies with changing pressure.

Figure 3 – CfD for Maximum FOI w & w/o non-Darcy Flow

1 2 3 5 10 20 30 50 100

6

7

8

9

(C )

BH

P (p

si)

Darcy Flow

fD True

non-Darcy Flowββββ = 0.0007

SPE 90864 An Investigation of Non-Darcy Flow Effects on Hydraulic Fractured Oil and Gas Well Performance 5

Case Histories Several case histories are discussed below, and some notes are needed in preparation. For each case, published, long-term lab conductivity data was used for kfbf and β. This was then, in some cases “corrected” for multi-phase flow. Several pub-lications were used to determine these values and corrections including the work of Penny 7, 8 and Milton-Taylor 9.

In addition, several of the cases below involved con-verging flow where the perforated interval did not cover the entire zone thickness. For such cases, a simple value for QD is not possible. Schulte 10 discussed a method of calculating the effect of such converging flow (including non-Darcy effects). However, for the converging flow cases considered here, frac-ture geometry was near radial (i.e., tip-to-tip length < twice the height), and Schulte’s methods are not applicable in such cases – so the use of this “correction” was not investigated.

Micro-Darcy Gas Well: Many in the industry have been advocating the use of intermediate and high strength proppant in tight formation gas reservoir applications. The argument is made that in such reservoirs well performance can be im-proved by increasing fracture conductivity and mitigating non-Darcy flow effects and/or improving fracture fluid clean-up. Though both of these effects can be influenced by fracture conductivity, studies have shown that effective fracture clean-up can be realized provided that dimensionless fracture con-ductivity in excess of 10 is achieved. Dimensionless fracture conductivities well in excess of this value can and are being achieved in tight formation gas reservoirs even when using sand as the proppant.

In 1992, Amoco (now BP) was approached by a major service company with the idea of using high strength proppant in the Cotton Valley Formation. This idea was based on the theory that non-Darcy flow was both reducing gas flow rates, and retarding clean-up and thus, possibly reducing effective fracture length. The idea was that the service company would substitute ISP (charged at the price for sand) with a return on their investment coming back based on specified performance enhancements. Since a fair sum of money was involved (roughly $1,200,000 in 2004 $), a great deal of time and care was used to select the two candidate wells, with both com-panies having to agree on this selection.

Eventually, acceptable candidate wells were found, and two direct offsets were fracture stimulated, one with ceramic and the other with Ottawa sand. The treatments in both wells consisted of two stages (i.e. one Taylor stage and one Upper Cotton Valley Stage). The Cotton Valley Formation is a well documented tight formation gas reservoir and with fracture closure stress on the order of 6,000 to 6,500 psi Ottawa sand is the predominate proppant used. Table 3 summarizes the reservoir properties determined through log interpretation for the wells of interest. Also included in the Table are the fracture half-length and conductivities used to evaluate the post fracture well performance with and without non-Darcy flow.

Table 3: Cotton Valley Reservoir Properties Parameter Taylor Sand Upper CV

Pressure, psi 5,200 4,500 Net Pay, ft 20 40 Avg Porosity 0.067 0.07 Avg Sw 0.52 0.48 Permeability, md 0.010 0.010 Drainage, acres 160 80 Temperature, oF 270 270 Gas Gravity 0.65 0.65 xf, feet 500 500 kfw, mdft 18.8 to 188 18.8 to 188

As previously discussed, two offset wells were fracture

stimulated in two stages. One of the wells utilized 20/40 Ottawa sand while the other well used 20/40 Ceramic. The completion and stimulation in each well included perforating the Taylor sand interval (Stage 1) and fracture stimulating. Following the Taylor stimulation, a sand plug was set and the Upper Cotton Valley (Stage 2) perforated, broken down, and fracture stimulated. Following the fracture stimulations, fluid loads were recovered and the wells were produced to sales.

Table 4 summarizes the fracture stimulations conducted in these wells. As shown, the well fracture stimulated with ceramic proppant utilized twelve percent less fluid, only three percent less proppant, and slightly larger pad fraction. Thus, by all accounts, the well that utilized ceramic proppant should have outperformed the well utilizing Ottawa sand if effective fracture clean-up and non-Darcy flow effects were important to the well performance in this tight gas application. However, through the end of 2003, Well 2 (i.e. the well utilizing Ottawa sand) had produced nearly twenty two percent more gas than the well with Ceramic proppant.

Table 4 – ETCV Fracture Stimulation Comparison

Fluid

M-Gal

Pad

M-Gal

20/40 Sand M-lb

20/40 ISP

M-lb

100 mesh

M-lb

Well 1: Stage 1 394 125 1914 21.7 Stage 2 214 69 275 359 10.0 Total 608 194 275 2273* 31.7 Well 2: Stage 1 398 110 1647 17.5 Stage 2 296 95 984 15.0 Total 694 205 2631** 32.5 * - 80 M-lb Resin Coated ** - 300 M-lb Resin Coated

Figure 4 shows a cumulative production plot comparing

the two wells performance as a function of time. As shown, the Ottawa sand well is significantly outperforming the well with ceramic proppant. It should be noted that at least for the first few months of production, the ceramic well produced slightly more gas than the Ottawa sand well but not nearly enough to offset the additional cost of the proppant.

6 Smith, M. B., Bale, A., Britt, L. K., Cunningham, L. E., Jones, J. R., Klein, H. H., and Wiley, R. P. SPE 90864

Why would the additional fracture conductivity not result in improved well performance? The first possibility of course is that even though these two wells are offsets, pay quality could still be varied. Lateral pay continuity, particularly in the Upper Cotton Valley, might account for the poor long-term

performance of the ISP well. However, before the ISP well be-gins to drop off, for 3 to 4 years (i.e., the time period where additional conductivity should be most important) perfor-mance of the two wells is essentially identical.

Secondly, and the favorite response, is that non-Darcy flow effects are unimportant in tight formation gas reservoirs where sufficient fracture conductivity for effective fracture clean-up is achievable with sand as the proppant.

Figures 5A and 5B compare the production declines from the two wells utilizing Carter Constant Terminal Pressure Type Curves. Note that the well that utilized ceramic proppant (Figure 5A) had a steeper early time transient decline than the Ottawa sand well (Figure 5B). This would indicate that the ceramic proppant resulted in a higher conductivity fracture as anticipated, but apparently, the Ottawa sand provided more cost effective conductivity in this application. Also note that the production decline analysis indicated that the well fracture stimulated with Ottawa sand is about “30% better” than the ceramic well based on Gas-In-Place and flow capacity considerations.

However, the fact that little or no benefit was derived from the use of ceramic proppant in this tight gas application should not come as any great surprise if non-Darcy flow is of little importance to this application. To highlight the non-Darcy effect or lack of effect, a series of simulations with and without non-Darcy flow were conducted. The simulations utilized the reservoir and fracture properties outlined in Table 1. Two fracture conductivities were considered (18.8 and 188 mdft). Results of these simulations are shown as Figure 6, a plot of cumulative gas recovery versus time.

This figure has several interesting components. First, note that the cumulative gas recovery over the twelve year (4,380 days) production period is 870 MMCF (18.8 mdft case) to 950 MMCF (188 mdft case) which is similar to the actual well recoveries of the ceramic (800 MMCF) and Ottawa sand (1,000 MMCF) wells over the same time period as shown in Figure 4. Secondly, note that non-Darcy flow had ZERO impact on these simulations (β of 0.0007 atm-sec2/gram used for 20/40 Ottawa sand). This result was realized regardless of the fracture conductivity and FCD.

Further review of this figure shows the effect of increasing the fracture conductivity by an order of magnitude (18.8 to

Figure 4: Gas Recovery Comparison-Sand versus Ceramic

0

200,000

400,000

600,000

800,000

1,000,000

1,200,000

0 20 40 60 80 100 120 140 160

Time, Months

Cum

ulat

ive

Gas

Rec

over

y, M

MC

F

Well 1: Ceramic Proppant

Well 2: Ottawa Sand

Figure 5A: Decline Analysis for Well With Ceramic We Na e

QdD

0.00

200.

0050

0.02

00.

050

0.10

0.20

0.50

1.0

2.0

5.0

TdD0.0020 0 .0100 0 .050 0 .20 0 .50 1 .0 2 .0 5 .0 10 20 50 100 200 500

Figure 5B: Decline Analysis for Well With Sand W e ll Nam e

QdD

0.00

200.

0050

0.02

00.

050

0.10

0.20

0.50

1.0

2.0

5.0

TdD0 .0020 0 .0100 0 .050 0.20 0 .50 1 .0 2 .0 5.0 10 20 50 100 200 500

Figure 6: Gas Recovery Comparison of Non-Darcy Flow

0

100

200

300

400

500

600

700

800

900

1000

0 500 1000 1500 2000 2500 3000 3500 4000 4500Time, days

Cum

ulat

ive

Rec

over

y, M

MC

F

Cotton Valley Simulation w/ and w/o Non-Darcy Flow (Ottawa Sand)

xf = 500 feet, kfw = 188 mdft, F CD = 37.6

Cotton Valley Simulation w/ and w/o Non-Darcy Flow (Ottawa Sand)

xf = 500 feet, kfw = 18.8 mdft, F CD = 3.76

SPE 90864 An Investigation of Non-Darcy Flow Effects on Hydraulic Fractured Oil and Gas Well Performance 7

188 mdft) only increased the cumulative gas recovery by 80 MMCF (only 8.5% of cumulative recovery or about 18 mscfpd) over the twelve-year production period. Such a small benefit from fracture conductivity in this application would be more than offset by the variability in pay quality from well to well.

In any event, non-Darcy flow has NO impact on well performance for this tight gas application even though the simple spreadsheet relations suggest that it does.

820 Sm3/d (5,200 bopd) Oil Well. This case is from a North Sea Cook Formation well – a “moderate” permeability (18.8 md) oil well produced at a maximum rate (during a multi-rate drawdown test) of about 5,200 bopd (820 Sm3/d). A log sec-tion of the well is included in Figure 7. The 55m (gross TVD thickness) zone was perforated over an 18m interval (as seen in the figure), and completed with a propped fracture treatment per Table 5.

Table 5: – Propped Fracture Treatment, 5,200 bopd Oil Well

Fluid: 40# HPG, Borate Cross-Link Prop: 16/20 Resin Coated Ceramic Pad Volume: 12 M-Gal Pad %: 10% Prop Laden Fluid Volume: 88 M-Gal Prop Concentration: 1 to 10 PPG Prop Volume: 487 M-lb Pump Rate: 40 bpm

Subsequent to the propped fracture treatment and an ex-tensive post-frac production period, the well was shut-in for a pressure build-up (with the well flowing at 734 Sm3/d prior to the shut-in/build-up). After the shut-in, a multi-rate drawdown

test was conducted. The pressure build-up provided a per-meability of 18.8 md, a net height of 44m, skin = -1.2, and a reservoir pressure of about 247 bar. The multi-rate drawdown results are plotted in Figure 8 showing minimal, if any, non-Darcy effects. This seems surprising, even for an oil well (where non-Darcy effects are generally expected to be smaller), given fairly high rates (820 Sm3/d) and converging flow (and a small, 1%, water cut). Note too, that converging flow effects are magnified since better quality formation is located above/below the perforations, with minimal pay ac-tually directly across the perforated interval. Finally, it is noted that the results of the PBU give a calculated PI of 8.68 Sm3/d/bar. However, that this would be a pseudo-steady-state PI. This is in reasonable agreement with the multi-rate drawdown PI of 9.3 Sm3/d/bar – particularly since with only 6 hour steps (too short to achieve stabilized, pseudo-steady-state flow), PI from the multi-rate test is expected to be somewhat high. This agreement (along with the straight line BHP-Rate behavior) suggests that non-Darcy effects are minimal.

Due to the converging flow effect, a simple value of QD is not possible. For a flow rate of 820 Sm3/d, based on the perf-orated height, QD is about 6. This is based on an estimated, average fracture conductivity of about 12,000 md-ft (based on special lab tests for this RC proppant with a complete gel break, and an estimated effective in-situ proppant coverage of about 2.5 lb/ft2) and an estimated β factor of 0.0008 (published data for non-RC proppant shows β of about 0.0004). Detailed post-frac evaluation and reservoir simulation

was not available for this case, but while not “huge”, a QD of 6 suggests that inertial drag forces (i.e., non-Darcy flow effects) are 6X greater than the viscous, Darcy flow drag forces. Thus, one might expect some evidence of non-Darcy behavior as compared to the straight-line drawdown seen in the actual test.

55 MMSCFD Gas Well. This is a frac-pack case from off-shore Trinidad where a thick, but relatively poor quality reser-voir was completed with two, stacked frac-packs as seen in the log section in Figure 9.

Figure 7: Log Section, 5,200 bopd Oil Well

Figure 8 – Multi-Rate Test Results, 5,200 bopd Oil Well

100 200 300 400 500 600 700 800 900

100

120

140

160

180

200

220

240

Rate (sm3/d)

BH

P (b

ar)

PI = 9.3 Sm /d/bar3

8 Smith, M. B., Bale, A., Britt, L. K., Cunningham, L. E., Jones, J. R., Klein, H. H., and Wiley, R. P. SPE 90864

The lower set of perforations was stimulated with a propped fracture treatment as detailed in Table 6. The some-what low proppant volume and low maximum PPG were due to operation problems. Valve failures on the stimulation vessel when switching to flush resulted in an early shut-in, leaving the 8 through 10 PPG stages in the wellbore. Despite this op-erational problem, a tip screenout (TSO) was achieved. Post-frac analysis suggested a fracture penetration or ½ length of about 48 feet, with total in situ proppant coverage of about 3.5 lb/ft2.

Table 6: – Lower Fracture Treatment, 55 MMSCFD Gas Well Fluid: 30# HPG, Borate Cross-Link Prop: 20/40 Mesh Ceramic Pad Volume: 2 M-Gal Pad %: 10% Prop Laden Fluid Volume: 15.2 M-Gal Prop Concentration: 0.5 to 7.4 PPG Prop Volume: 50.2 M-lb Pump Rate: 24 bpm

A few days later, the upper set of perforations was fractured as detailed in Figure 10. This treatment was pumped pretty much as planned, and a post-frac review of bottomhole pressure in the figure shows bottomhole pressure behaving as expected. This analysis suggests a fracture penetration of 72 feet, with total in situ proppant coverage of 4.8 lb/ft2.

Since only one PBU test was performed with flow co-

mmingled from both intervals, these results were averaged and treated as a single fracture with a penetration of 60 feet and total proppant coverage of 4.1 lb/ft2. Based on published long

term lab tests at the actual in situ proppant stress, and assum-ing a 50% regained permeability (i.e., damage factor of 0.5 from gel residue) and 1.0 lb/ft2 “lost”, this then predicted a laminar flow conductivity of about 3,900 md-ft, an effective propped width of about 0.31 inches, and β of about 36,420 1/ft (0.0012 atm-sec2/gram) (this β ignores the multi-phase flow effects in this condensate rich gas well).

After being placed on production, the well was opened for a short clean-up flow, and then shut-in. It was then opened up for an extended flow, and flowed for about 11 days at about 55 MMSCFD. The well was then shut-in for a surface pressure build-up, and the analysis of this post-frac build-up is seen in the Figure 11 log-log plot. The behavior here clearly indicates fractured well performance, and the analysis is included in Table 7. Since the PBU so clearly showed fractured well be-havior (and the associated negative skin), no multi-rate test was performed since it was felt that there would be no signi-ficant non-Darcy effects. However, while this may have been true, the post-frac well test results are significantly different for fracture conductivity from the simple “predicted” conduc-tivity based on the post-frac analysis. This difference was addressed by simulating the post-frac test flow and comparing the bottomhole drawdown for various simulations versus the actual drawdown just before the well was shut-in. The simula-tion was done with a finite difference reservoir model (briefly described elsewhere in this paper) connected with “nodal” type calculations for the near wellbore pressure drop due to flow through the packed perforation tunnels, etc.

Figure 9 – Log Section, 55 MMSCFD Gas Well

MD

SSTVD

13200

13300

12100

12200

13300

13400

13500

12300

Figure 10 – Post-Frac Analysis Using Design Parameters Fluid: 30# HPG, Borate Cross-Link Prop: 20/40 Mesh Ceramic Pad Volume: 2.7 M-Gal Pad %: 12% Prop Laden Fluid Volume: 24.1 M-Gal Prop Concentration: 0.5 to 10.0 PPG Prop Volume: 121.9 b Pump Rate: 25bpm A 1, 24 Sd, Upper

Net

Pre

ssur

e (p

si)

100

200

300

400

500

600

700

Time (min)7.0 14.0 21.0 28.0

Simulated Data

Measured Data

SPE 90864 An Investigation of Non-Darcy Flow Effects on Hydraulic Fractured Oil and Gas Well Performance 9

Table 7 – Reservoir/Fluid Properties, 55 MMscfd Gas Well

Reservoir/Fluid Properties PRes 7,100 psi BHFP at Shut-in 4,700 psi Gas Gravity 0.8 Condensate 40 bbl/MMSCF Formation Temperature 196° F Porosity 0.17 SW 40% Net Pay h 235 feet PBU Analysis Results Permeability 2,2 md Fracture ½ Length 75 feet kfw 710 md-ft CfD 4.3

The first simulation assumed a fracture penetration of 70 feet, a laminar flow conductivity of 3,900 md-ft, a propped fracture width of 0.31 inches, and a Beta of 0.0018 atm-sec2/g. As seen in Figure 12, the first bar shows this giving near perfect agreement with the final bottomhole flowing pressure of about 4,700 psi (2,400 psi drawdown). A second simulation used a conductivity of 660 md-ft with “0” β (i.e., the well test result), and this also agrees with the actual data. That is, non-Darcy flow had reduced apparent conductivity from 3900 to 710 md-ft – a pretty significant non-Darcy flow effect. How-ever, in this case, the effect on performance was small The third case assumed conductivity of 3.900 md-ft with no non-Darcy effect, and this only reduced the drawdown for the 55 MMSCFD rate from 2,400 psi to about 2,200 psi.

Due to the converging flow, a simple value for the non-Darcy parameter, QD, is not possible to calculate. Based on a rate of 55 MMSCD and a fracture height of 235 feet (i.e., the entire gross interval), QND = 16.9. If a “height” of 100 feet is used (i.e., length of the two perforated intervals), QND = 39.6. Using 16.9 (the minimal non-Darcy effect) in the effective conductivity relation

ftmdbkQ

bkbfk Trueff

D

TrueffAppf −==

+= 170)(043.0

55.01)(

)(

gives a very low effective conductivity. However, if this is further corrected (i.e., an increase in Beta) for multi-phase flow in the gas condensate well (40 bbl/MMSCF), the effec-tive conductivity is predicted to reduce to about 50 md-ft. As seen in the figure, this would have caused a 50% increase in the drawdown beyond the measured behavior.

Thus, for this high rate gas well, with multiphase and converging flow non-Darcy flow is important – reducing the effective conductivity from 3900 to 700 md-ft. This has no significant effect on productivity however, increasing the required drawdown from about 2,200 psi to the observed 2,400 psi. However, the simple apparent conductivity relation of Eq. 3 overstates this effect by a factor of 5, and if multi-phase effects are included, overstatement of the non-Darcy effect is nearly 10X.

High Rate Oil Well. This case is for an Indirect Vertical Frac-ture Completion (IVFC) as described by Bale, et. al.11 In this case, a lower zone was perforated (as seen in Figure 13), with the goal of propagating a propped fracture up, into the high permeability (1000+ md) Rannoch-3 formation. Estimated

Figure 11 – Post-Frac PBU Analysis, 55 MMscfd Gas Well

Figure 12 – 11 Day Drawdown, 55 MMscfd Gas Well

500

1,000

1,500

2,000

2,500

3,000

3,500

Dra

wdo

wn

(psi

)

Calculated DrawdownAfter 11 days at 55 MMSCFD

KfW = 3900ββββ = 0.0018

KfW = 710ββββ = 0.0

KfW = 3900ββββ = 0.0

KfW = 53ββββ = 0.0

KfW (md-ft)

ββββ (atm sec /gram) P = 7,100 psi

2

Res

ActualDrawdown

Figure 13 – Log Section, High Rate Oil Well

10 Smith, M. B., Bale, A., Britt, L. K., Cunningham, L. E., Jones, J. R., Klein, H. H., and Wiley, R. P. SPE 90864

permeability for the three zones ranged from about 25 md for the Rannoch-1, to about 500 md for the Rannoch-2, to 1000+ md for the Rannoch-3.

The well was completed through 10m of perforations in the lower Rannoch-1 zone with a propped fracture treatment per Figure 14. The post-frac analysis of the (near) bottomhole

pressure data is seen in the figure, with the treatment behaving much as designed. (Note that at least part of the departure be-tween the simulated pressure behavior and the actual is due to correcting gauge data to the perforations. The gauge was lo-cated 241m MD above the perforations, and the point where “actual” bottomhole pressure tends to level out is just about when flush passes the gauge depth.) This figure also plots the final, in situ proppant coverage (lb/ft2) as determined using a fully 3D fracture geometry model. Note that due to the high fluid loss at the top of the fracture, there is significant upward proppant movement after the end of pumping, resulting in ra-ther low proppant coverage over the perforations.

This gives an average proppant coverage of about 3 lb/ft2 out to a fracture penetration of about 40m (131 ft). Again using published long-term lab conductivity data, assuming a damage factor of 0.5, and assuming 0.5 lb/ft2 “lost”, the ex-pected fracture properties are a conductivity of 12,000 md-ft, effective propped width of 0.28 inches, and β = 0.0005.

Some months after the completion (producing at about 2,0000 Sm3/day, or 12,500 bopd), a multi-rate drawdown was run (after a short shut-in) with results as seen in Figure 15. The increasing drawdown with rate, i.e., non-Darcy flow ef-fects, is clear from the data – despite the use of 16/20 mesh ceramic proppant.

The multi-rate test was simulated using the 3D finite diff-erence model, utilizing the complex proppant distribution pic-tured above. Conductivity was apportioned (md-ft per lb/ft2) in order to achieve an average of 12,000 md-ft, with an average propped with of 0.28 inches. β was constant everywhere at 0.0005. With small adjustments to increase permeability in the Rannoch-3 from 1,000 to 1,200 md, increase average conduc-tivity to 14,000 md-ft, & decrease β to 12,140 1/ft (0.0004 atm-sec2/gram) – the simulation was quite close to the actual multi-rate data as seen in the figure. (Note that these changes possibly reflect less embedment in the harder formation in the critical near perforation region as opposed to a general in-crease in the “average”. However, it was treated compu-tationally as an increased average.) β was then set = “0”, propped width was kept constant per the proppant distribution seen above, and kf was varied to match the data. The final rate/pressure point was matched with an average conductivity of 9,000 md-ft – a 36% reduction in conductivity due to non-Darcy flow effects. Finally, the effect of the non-Darcy flow on well performance is seen in comparing the actual draw-

Figure 14 – Post-Frac Analysis for High Rate Oil Well

Fracture Penetration (m)20 40 60 80

42.00 min

1780m

TVD

1800

1820

1840

1860

Stress (Bar)350

Shal

eR

anno

ch3

Shal

e

0.000

0.500

1.000

1.500

2.000

2.500

3.000

3.500

4.000

4.500

5.000

PSF

psf

0.512 m/se

Table 8 – Upper Fracture Treatment, High Rate Oil Well

Fluid: 35# HPG, Borate Cross-Link Prop: 16/20 Mesh RC Ceramic Pad Volume: 10 M-Gal Pad %: 20% Prop Laden Fluid Volume: 33.5 M-Gal Prop Concentration: 0.5 to 12.0 PPG Prop Volume: 138 M-lb Pump Rate: 40 bpm

Figure 15 – Multi-Rate Drawdown Test Results for High Rate Oil Well

500 1000 1500 2000 2500

250

260

270

280

Rate (Sm3/d)

BH

P (b

ar)

PI = 117 Sm /d/bar3

PI = 84 Sm /d/bar3

PI = 71 Sm /d/bar3

500 1000 1500 2000 2500

250

260

270

280

Rate (Sm3/d)

BH

P (b

ar)

Data SimulationKfW = 14,000 md-ft

ββββ = 0.0004

SimulationKfW = 9,000 md-ft

ββββ = 0.0

SimulationKfW = 14,000 md-ft

ββββ = 0.0

SPE 90864 An Investigation of Non-Darcy Flow Effects on Hydraulic Fractured Oil and Gas Well Performance 11

down behavior to the simulation based on an average conduc-tivity of 14,000 md-ft with β = “0”. At 2,500 Sm3/d (15,470 bopd), the non-Darcy effects were increasing required draw-down by about 50%, i.e., about a 1/3 reduction in PI.

Given the complexity of the problem (converging flow, non-uniform propped width distribution, etc.), it is very diffi-cult to calculate any value for QD. Based on the perforated height (10m) and the maximum flow rate of 2,500 Sm3/d (15,470 bopd), QD is calculated as about 16. Using Eq. 3, this predicts a 90% reduction in conductivity – average conduc-tivity of only 1,380 md-ft – 3X the reduction actually ob-served.

Again, this case like the others shows that non-Darcy ef-fects are significant (a 36% reduction in Apparent Con-ductivity in a case where production is totally controlled by conductivity). However, again, a simple CfD & QD relation sig-nificantly overstates these effects.

Note, one might also conclude that having a propped frac-ture stimulated oil well that exhibits non-Darcy flow effects is a nice problem to have!

120 MMSCFD Gas Well. This is another frac-pack case from offshore Trinidad. This is from the same field as the well dis-cussed above, but for a different well and a shallower (much better quality) sand. This well was completed with two, stacked frac-packs (as seen in the log section in Figure 16), but for the purposes of this analysis, any relatively small pro-duction coming from the lower zone is neglected. Reservoir properties for the zone are also included in the figure.

The upper zone was completed through 40 feet of perf-orations (MD) from 9,621-9,659 feet TVD-RKB. Details of the treatment along with a post-frac review of the bottomhole treating pressure are included in Figure 17. The treatment was pumped as designed, placing 51,000 lb of proppant (81% of design) in the formation using 13.6 M-Gal of gel (94% of design) before terminating with a total, wellbore screenout with the final, 10 PPG, stage on the formation. Though the fin-al pressure matched expectations, overall behavior of pressure during the treatment was quite different from expected, pos-sibly indicating some difference in fracture behavior for this very “soft” (modulus of 0.3x106 psi), high fluid loss, unconsol-idated formation. The review of this data predicted a fracture covering the main pay sand, with about 40 feet of fully propped fracture penetration, with average in situ total prop-pant coverage of about 7 lb/ft2.

Using published long term lab tests at the actual in situ proppant stress, and assuming a 50% regained permeability (i.e., damage factor of 0.5 from gel residue) and 2.0 lb/ft2 “lost”, this predicted a laminar flow conductivity of about 10,500 md-ft, an effective propped width of about 0.56 inches, and a β of about 33,385 1/ft (0.0011 atm-sec2/gram) (this Beta includes multi-phase flow effects in this gas condensate well).

Some time after being placed on production, a multi-rate drawdown (and subsequent pressure build-up) test was per-formed utilizing data from a permanent bottomhole gauge lo-cated about 200 feet above the perforations. Analysis of these tests shows a permeability of about 650 md, with a total skin (at 120 MMscfd) of +40. The non-Darcy skin, D, was about 0.0003 (skin units/Mscfd). Thus, this gives (at 120 MMscfd) a non-Darcy skin of 36, and a mechanical skin of +4.

The rate/bottomhole pressure data from the multi-rate test is plotted in Figure 18, and the non-Darcy flow effects are clear. This figure also plots data from another well in the same sand. This second well was completed with a cased hole gra-vel pack completion due to the presence of a gas-water con-tact. In the analysis below, the gravel pack well will be used to “calibrate” the near wellbore pressure drop through the com-pletion (packed perforation tunnels, gravel pack, etc.), and this is used to try an isolate the non-Darcy flow effects in the propped fracture for the frac-pack completion. Reservoir prop-erties for the gravel pack well are included in Table 9.

Figure 16 – Formation Properties, 120 MMscfd Gas Well4

Y

RA 0.2 200

RP 0.2 200

RHOBg/cc1.65 2.65

NPHIPU0.6 0

PHIEDEC0.5 0

BVWDEC0.5 0

CALPERM 1 10 0 00

100 CALPERM

G R 0 150

V SHD EC0 1

P E R F 0 4

0PE RF

Depth TVD 9,621’ TVD-RKB

(to top perforation) Net h (TVD) 129 feet

Porosity 0.244 Sw 0.167

Temperature 169 F k ~650 md

Total skin (at 120 MMscfd) +40 D (skin units per Mscfd) 0.0003

Mechanical Skin +4 Original PRes 4,634

PRes at Time of Test 4412 psi Gas Gravity 0.629

Rate @ Time of Shut-In

Gas – 120 MMscfD Cond – 1440 STB

Condensate Yield 12 BBL/MMCF

Figure 17 – Post-Frac Analysis Using Design Parameters Fluid: 40# HPG, Borate Cross-Link Prop: 20/40 mesh ceramic Pad Volume: 1.5 M-Gal Pad %: 10% Prop Laden Fluid Volume: 11.6 M-Gal Prop Concentration: 0.5 to 10 PPG Prop Volume: 51 M-lb Pump Rate: 25 bpm pp

Net

Pre

ssur

e (p

si)

100

200

300

400

500

600

700

800

Time (min)5.0 10.0 15.0 20.0

Simulated Data

Measured Data

Figure 18 – Multi-Rate Results, 120 MMscfd Gas Well

20 40 60 80 100 120

4,250

4,300

4,350

4,400

4,450

4,500

4,550

4,600

4,650

Rate (MMscfd)

BH

P (p

si)

Gravel Pack Well

Frac-Pack Well

12 Smith, M. B., Bale, A., Britt, L. K., Cunningham, L. E., Jones, J. R., Klein, H. H., and Wiley, R. P. SPE 90864

Table 9 – Formation Properties, Gravel Pack Well

Perforated Interval 85 feet (MD) Net h (TVD) 164

Porosity 0.28 Sw 0.125

Temperature 164 F K (from BHP PBU test) ≈ 800 md

Total skin (at 60 MMscfd) +115 D (skin units per Mscfd) 0.0016

Mechanical Skin +19 Original PRes 4,634 psi

PRes at Time of Test 4634 psi Gas Gravity 0.656

Rate @ Time of Shut-In

Gas – 60 MMscfD Cond – 720 STB

BHFP @ Shut-In 4524 psi Condensate Yield ~12 BBL/MMCF

To try and study the effects of non-Darcy flow in the fracture, it was first necessary to separate the pressure drop across the completion from the reservoir drawdown – since clearly at a rate 114 MMscfd, there is a significant ∆p across the wellbore completion. The following procedure was fol-lowed to make this separation: • The gravel pack well as simulated to determine the

reservoir drawdown for the multi-rate drawdown steps in the figure. This was a radial flow simulation INCLUD-ING effects of non-Darcy flow in the formation. For the non-Darcy formation effects, the correlation discussed in reference 4 was used. This gave a βFormation value of 0.011 (atm-sec2/gram). Looking at the range of literature data for formation β factors, this is a fairly high value. However, that seemed appropriate for this very fine grain formation.

• The difference between this simulated reservoir draw-down and the measured wellbore drawdown was then assumed due to a ∆p across the completion. A rate depen-dent skin value was then calculated that, when added to the reservoir drawdown, matched the total wellbore draw-down data. Note that no particular effort was made to associate this rate dependent skin with the number of perforations connected to the formation, etc.

• The magnitude of the rate dependent skin was then de-creased by a factor of 6X to theoretically account for the use of 20/40 “gravel” in the frac-pack as compared to 40’60 “gravel” for the gravel pack. This is basically as-suming a similar flow area through the perforations (on a foot-by-foot) basis for the two wells, with a reduced pres-sure drop (for a similar velocity) due to higher permea-bility and lower β. This was then used to calculate the pressure drop across the completion for the multi-rate drawdown steps for the frac-pack well.

• Using this “fixed” rate dependent completion skin, the formation/frac properties were varied to match total well-bore drawdown. Only minor adjustments were required, reducing the conductivity to 10,000 md-ft and reducing β to 0.0008 atm-sec2/gram (from an “expected” value of

0.0011). (Note that to insure apples-to-apples, non-Darcy flow in the formation was also used for the frac-pack well simulations. However, one check run showed this to be having negligible effect on the results.)

The results are included in Figure 19, showing good agreement with the total wellbore drawdown for both cases –

with 90% of the drawdown for the gravel pack being across the completion. For the frac-pack, this shows about ½ the total drawdown across the completion.

This calculated reservoir drawdown (based on a conduc-tivity of 10,000 md-ft, effective propped width of 0.56”, and β = 24,280 1/ft, 0.0008 atm-sec2/gram) was then compared to simulations with β = 0.

As seen in Figure 20, for a conductivity of 10,000 md-ft,

with no non-Darcy effects, expected drawdown is about 60% of the actual drawdown. That is, non-Darcy flow has reduced well productivity by about 60%. Conductivity was then varied (with β =0), and a fracture conductivity of 1,100 md-ft was required to match the actual reservoir drawdown behavior. Non-Darcy effects have reduced the apparent conductivity by 90% from the “true” conductivity. Not an unexpected result for a well flowing at 114 MMscfd.

Again, due to converging flow, it is not possible to cal-culate a simple value for QD. Based on the total height of the zone (129’) and the maximum flow rate of 114 MMscfd, QD is

Figure 19– Multi-Rate Results, 120 MMscfd Gas Well

20 40 60 80 100 120

4,250

4,300

4,350

4,400

4,450

4,500

4,550

4,600

4,650

Rate (MMscfd)

BH

P (p

si)

Frac-Pack Data

Gravel PackData

SimulationTotal Drawdown

SimulationReservoir Drawdown

Frac-PackTotal Drawdown

Frac-PackReservoir Drawdown

Figure 20:Drawdown w/ and w/o Non-darcy Flow Effects

20 40 60 80 100 120

4,350

4,400

4,450

Rate (MMscfd)

BH

P (p

si)

Simulated Frac-PackReservoir DrawdownKfW = 10,000 md-ft

ββββ = 0.0008KfW = 10,000 md

ββββ = 0.0

KfW = 1,100 md-ftββββ = 0.0

KfW = 450 md-ftββββ = 0.0

SPE 90864 An Investigation of Non-Darcy Flow Effects on Hydraulic Fractured Oil and Gas Well Performance 13

about 38 – and this value will certainly tend to drastically underestimate the total non-Darcy effect since there is no con-sideration for the converging flow (QD based on the 40’ perforated interval is about 122). Still, use of this lower value in Eq. 3 shows a final, apparent conductivity of about 450 md-ft – i.e., a 96% reduction. The predicted reservoir drawdown based on this value for conductivity is also included in the fig-ure. This shows an expected drawdown about 20% greater than the actual, i.e., an additional 20% reduction in well performance. At this point, the fracture is basically ineffective, and the simulation is back to a radial flow (with converging flow) behavior.

Conclusions The conclusions of this study of field cases is clear – in every case (including the oil wells) excepting the micro-darcy gas well, non-Darcy flow effects have a clear, predictable effect on “Apparent Conductivity” and post-frac performance. For the cases studied, this ranged from a 10% reduction in well performance to a 35% reduction in well performance (for a high rate oil well).

However, the current industry practice of predicting non-Darcy effects, and designing treatments including proppant selection, based on spreadsheet and/or type-curve production models is, in many cases overestimating the magnitude of non-Darcy effects. This over estimation is due to the use of simple correlations for “Apparent Conductivity” such as

( )D

fDAppfD Q

CC

55.01+= .

beyond the valid limits for such relations. For the cases studied here (which included a fair range of conditions from a 0.12 md gas well to a 1,200 md oil well), this overestimated the effects of non-Darcy flow from a factor of 2X to factors > 10X.

Thus, the conclusion is “It’s not that simple!” Non-Darcy flow is a real effect, and for the cases studied, the smallest effect was about a 1/3 reduction in Apparent Conductivity (as compared to the expected, “True” conductivity). However, to predict the effects in some “general” case, rigorous numerical modeling is the only reasonable approach.

Nomenclature bf = fracture width (ft) CfD = Dimensionless fracture conductivity (CfD)True = “True” or “Laminar Dimensionless fracture conductivity (CfD)Apparent = Dimensionless fracture conductivity corrected for non-Darcy/multi-phase flow flow effects h = formation/fracture height (ft) k = formation permeability (md) kf = fracture permeability (md) kfbf = fracture conductivity (md-ft) q = total well flow rate (bopd or Mscfd) QD = Dimensionless non-Darcy flow parameter M = Gas Molecular weight xf = fracture ½ length (ft) ρ = fluid density (lbm/ft3)

β = non-Darcy flow parameter (1/ft) µ = fluid viscosity (cp) References 1. Cooke Jr., C.E., “Conductivity of Fracture Proppants in Multiple

Layers,” SPE 4117, JPT, 1973. 2. Guppy, Kern H., Cinco-Ley, H., Ramey Jr., H.J., “Pressure

Buildup Analysis of Fractured Wells Producing at High Flow Rates,” SPE 10178, JPT, 1982.

3. Guppy, K.H., Cinco-Ley, H., Ramey Jr., H.J., “Non-Darcy Flow in Wells With Finite-Conductivity Vertical Fractures,” SPE 9291, SPEJ, 1982.

4. A. Settari, A. Bale, R.C. Bachman, and V. Floisand, “General Correlation for the Effect of Non-Darcy Flow on Productivity of Fractured Wells,” SPE 75715, SPE Gas Technology Symposium, Calgary, Alberta, Canada, 30 April-2 May 2002.

5. Stark, A., “Investigation of the Influence of Turbulence on Deliverability of Gas Wells,” M.Eng. Thesis, Dept. of Chem. And Petrol. Eng., U. of Calgary, Dec. 1998.

6. Holditch, S.A., and Morse, R.A., “The Effects of Non-Darcy Flow on the Behavior of Hydraulically Fractured Gas Wells,” SPE 6417, JPT, 1976.

7. Penny, Glenn and S.Jin, Liang, “The Development of Laboratory Correlations Showing the Impact of Multiphase Flow, Fluid, and Proppant Selection Upon Gas Well Productivity,” SPE 30494, SPE Annual Technical Conference & Exhibition, Dallas, TX, U.S.A., 22-25 October, 1995.

8. L. Jin and G.S. Penny, SPE, “A Study of Two-Phase, Non-Darcy Gas Flow Through Proppant Pacs,” SPE 66544, SPEPF, Nov. 2000.

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