Fluxo No Darcyano

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pressure production, introducing new dimensionless

parameters for the reservoir non-Darcy flow.

In the first section, a brief historical overview is

presented concerning numerical investigations of non-

Darcy flow in fractured gas wells. Secondly, the simulator

implementation is illustrated as well as correlations usedin this work to derive the non-Darcy flow coefficients. In a

synthetically derived typical production scenario for a

tight-gas reservoir (0.01 mD), the impact of non-Darcy

flow in fractures and the reservoir is evaluated, assuming

constant and stress-dependent parameters. New type-

curves are developed for both the constant pressure as

well as constant rate production to examine the non-Darcy

flow characteristics. Neglecting non-Darcy flow in well-

test analysis can lead to an erroneous interpretation, as

demonstrated by means of a drawdown test.

2. Historical background

An early study of non-Darcy flow in fractured wells

was presented by Millheim and Cichowicz (1968).

Using a radial model, they investigated non-Darcy flow

in the reservoir. Later, Wattenbarger and Ramey (1969)

considered non-Darcy reservoir flow in an infinite-

conductivity fracture by means of a finite-difference

model. They concluded that non-Darcy flow in the

reservoir particularly affects short fractures. Holditch

and Morse (1976) restricted their investigations on non-

Darcy flow within the fracture and proved thesignificance for large flow velocities using non-Darcy

flow coefficients β t based on the experiments of Cooke

(1973).

Guppy et al. (1981, 1982a,b) introduced type-curves

for finite and infinite-conductivity fractures, taking into

account non-Darcy flow in the fracture. The authors

addressed constant rate production as well as the

constant pressure condition at the wellbore using a

semi-analytical model. In the case of constant rate

production the authors pointed out that non-Darcy flow

reduces the true fracture conductivity to a constant,apparent conductivity.

Roberts et al. (1991) analyzed the productivity of

multiple-fractured horizontal wells in tight-gas reser-

voirs with choked transverse fractures. The limited

communication between the fractures and the wellbore

created a choking effect near the fracture offset,

reducing the apparent fracture conductivity in light of

significant inertial pressure drops.

Jin and Penny (1998) presented an empirical model

that used the liquid to gas ratio to predict the effective

permeability or conductivity of a proppant pack under

two-phase non-Darcy flow conditions. Further papers

investigating inertial effects in two-phase flow through

fractures were published, e.g., by Vincent et al. (1999)

and Fourar and Lenormand (2000).

Umnuayponwiwat et al. (2000), Gil et al. (2001) and

also Alvarez et al. (2002) analyzed the impact of non-

Darcy flow on the evaluation of (fractured) well tests,concluding that disregarding non-Darcy flow will lead

to an overestimation of production. The authors

conducted their investigations at a reservoir permeabil-

ity of 0.1 mD for different production scenarios and

reported losses up to 25% after 10 years of production.

3. Simulator implementation of non-Darcy flow

Darcy's law, which describes velocity as a linear

function of the pressure gradient, has been traditionally

used in petroleum reservoir engineering. However, itsvalidity is restricted. Darcy's law takes only viscous

forces into account, neglecting inertial forces (its upper

limit). Those are captured with the Forchheimer equation

(Forchheimer, 1901), where large velocities cause

deviations from the linear Darcy flow. This is, as

Geertsma (1974) stated, primarily caused by the

continuous deceleration and acceleration of fluid

molecules traveling along a tortuous flow path through

the interconnected pores and also in the proppant pack.

In vector form, it can be written:

−gradU ¼ lk

þ bt qj uY

j

uY

; ð1Þ

where k is the permeability of the porous medium, μ is

the dynamic viscosity and β t denotes the non-Darcy flow

coefficient. A value β t = 0 marks the transition to Darcy's

law. The numerical investigations are conducted by

means of an in-house Black-Oil reservoir simulator

(Friedel and Häfner, 2004), which is based on a control-

volume method with finite differences using structured

and unstructured grids. The numerical solution is

obtained with a fully-implicit Newton linearizationmethod and a numerical Jacobian matrix calculation.

Although fully-implicit schemes that consider non-

Darcy flow have been used (Schlumberger GeoQuest,

2003), the current code utilizes an implicit realization by

iteration (Li and Engler, 2002). In order to calculate the

new solution at iteration level k , the velocity u pk is

derived using the control parameter from the previous

nonlinear iteration:

u

k

p ¼−

d

k −1

p

kk k r ; p

lk p gradU

k

p:

ð2Þ

113T. Friedel, H.-D. Voigt / Journal of Petroleum Science and Engineering 54 (2006) 112 – 128



Subsequently, a control parameter δ pk ★

is computed

for the next nonlinear iteration (k +1):

dk 1

p ¼1

1 þbk

t qk

pkk k r ; p

lk p

juk pj

: ð3Þ

To avoid oscillations, it was found that the control

parameter for the next iteration needs to be damped:

dk p ¼ dk −1

p þ dk 1

p −dk −1 p

⁎RP; ð4Þ

where RP denotes a relaxation parameter. This simple

relaxation technique significantly improves the conver-

gence behavior for solving the nonlinear system of

equations. Best results were achieved with values

between 0.4 and 0.6. The total number of Newton

iterations required for non-Darcy flow is just slightlyincreased compared to Darcy flow (δ p=1). Contrary to

the fully-implicit treatment, where non-Darcy effects are

not supposed to be large, the present approach facilitates

the consideration of extremely inertial flow with control

parameters 0bδ pbb1.

The fractured well and the reservoir are discretized

using structured grids. Grid construction is based on the

algorithms of Bennett et al. (1986). Because of the

common character of the investigations, a single layer

reservoir with homogeneous and isotropic properties is

considered. The hydraulic fracture spans the complete

thickness of the reservoir. Due to the symmetry of the flow

pattern (and to reduce the number of required grid blocks),

discretization is restricted to a quarter of the domain.

To ensure the reliability of the non-Darcy implemen-

tation and the discretization scheme, the simulator has

been validated against analytical solutions as well as

results from a commercial simulator, see Figs. 1 and 2.

4. Non-Darcy flow coefficients

Besides the flow velocity u, the magnitude of the

non-Darcy flow coefficient β t is the crucial factor for the

actual productivity restriction owing to inertial forces.

The coefficient is a characteristic of the morphology of

the porous medium, i.e., a measure of its tortuosity;

hence, its magnitude differs in matrix and fracture.

The non-Darcy flow coefficient of the reservoir

depends on the pore geometry and can be correlated tothe rock permeability, see Fig. 3. The correlation for the

experimental data from different sources and authors is:

br ¼ 4:1d 1011k −1:5res ; ð5Þ

where k is in mD and β r in 1/m. Nonetheless, the data

scatters within a magnitude of order. The relationship

β f = f (k f ) is also illustrated in Fig. 3 for a variety of

proppants and can be approximated:

bf ¼ 1d 1011k −1:11f ; ð6Þ

where, again, k f is in mD and β f in 1/m.

Fig. 1. Verification of non-Darcy and Darcy flow with analytical solution and stationary, turbulent skin factor for constant rate production in aunfractured vertical well.

114 T. Friedel, H.-D. Voigt / Journal of Petroleum Science and Engineering 54 (2006) 112 – 128



Fig. 2. Verification of non-Darcy flow in fractured vertical wells with Guppy's type-curves for constant pressure production.

Fig. 3. Non-Darcy flow coefficients in the reservoir (left) and coefficients for fracture proppant pack (right).




It is very common in reservoir simulation to use a

fictitious, enlarged fracture width bf

⁎ to discretize the

fracture plane:

b⁎f ¼ bf

k f

k ⁎f

; ð7Þ

where transmissibility of the fracture grid blocks is

maintained. Using bf ⁎ in Eq. (6) will tend to give

erroneous results, e.g., non-Darcy flow effects are likely

to be underestimated. Thus, the original β f of the true

fracture permeability has to be corrected to the artificial

width. For both cases, the quotient β f / bf 2 must be

constant.

Apart from those general correlations, several researchinstitutions and proppant manufacturers provide non-

Darcy flow coefficients under specific pressure–temper-

ature and damage conditions, such as the Stim-Lab

Proppant Consortium (STIMLAB-Consortium, 2003).

This is an ongoing industry project to characterize

commercially available proppants used in oil and gas

well fracture stimulation.Areas of interests are, e.g., (i) the

long-term conductivity of numerous proppant types asfunction of temperature and closure, (ii) baseline

conductivity of proppants vs. type, size, concentration,

embedment, closure and temperature, as well as (iii)

leakoff and conductivity of proppants with fracturing

fluids. Proppants under investigation include sands, resin-

coated sands, ceramics and resin coated ceramics.

Besides these experimental sources, several theoret-

ical models are available in the literature to determine

the coefficient (Li and Engler, 2001).

If there is a residual water saturation, or even a real

multi-phase flow, the non-Darcy flow coefficients

increase dramatically. There are additional collisions between the gas molecules and the residual fluid

molecules which slow the gas molecules down,

Table 1

Input parameters for the simulation example

Permeability (mD) 0.01

Porosity 0.1

Net thickness (m) 10

Rock compressibility (1/Pa) 7.5·10−10

Fracture half-length (m) 75

Fracture width (m) 0.005

Dimensionless fracture conductivity 50

Reservoir non-Darcy coeff. (1/m) 1014

Initial reservoir pressure (Pa) 600·105

Reservoir temperature (K) 423.15

Gas specific gravity 0.6

Standard conditions

Pressure (Pa) 1.013·105

Temperature (K) 288

Fig. 4. Influence of proppant type on the production rate with non-Darcy flow and constant non-Darcy coefficient.

Table 2

Non-Darcy flow coefficient of several proppant types (STIMLAB-

Consortium, 2003)

Type Carbo-Lite Carbo-HSP Carbo-HSP PRB

Mesh size 16/20 18/30 20/40 16/30

Bulk density (g/cm3) 1.6 1.9 2 1.6Median porosity 0.08 0.24 0.52 0.29

Permeability (D) 90 200 190 30

β t , undamaged (1/m) 3.3·105 9.1·104 7.4·104 3.9·106

β t , damaged (1/m) 3.2·106 8.9·105 7.2·104 3.8·107




requiring an increase of energy to accelerate the

molecules again. This coincides with raising inertial

pressure drops. Compared to single-phase predictions,

theoretical models suggest an increase of up to a factor

of 10 (Wong, 1970; Geertsma, 1974). Multi-phase non-

Darcy flow can be taken into account in the simulator

but is neglected in the results presented here.

5. Impact of non-Darcy flow on a typical gas

production scenario

The impact of non-Darcy flow on the productivity is

investigated by means of a typical production scenario

for a fractured tight-gas well. Initially, the well produces

at a constant rate until the lower well pressure limit is

reached. The entire production lasts 10 years.

5.1. Production with constant non-Darcy coefficients

Several simulation runs are conducted with two

different proppant types: (i) 16/20 C-Lite (ceramics

based proppant); concentration 2 lb/ft 2, and (ii) 18/30

Carbo-HSP (high strength sintered bauxite proppant);

concentration 2 lb/ft 2. Properties of the simulation

model are summarized in Table 1. Reservoir permeabil-

ity is assumed 0.01 mD with a dimensionless fracture

conductivity:

F CD ¼

k f bf

k r xf ð8Þ

of 50 and fracture width bf = 5 mm. The non-Darcy flow

coefficients of the proppants are based on experimental

data (Fig. 4). The reservoir non-Darcy coefficient is

derived from Fig. 3 with 1014 l/m, using the lower limit

of the measured data, and can therefore be considered

moderately low.

According to the experimental data, the permeabil-ities of the undamaged proppant pack under in-situ

conditions are 976 D for 16/20 C-Lite and 691 D

for 18/30 Carbo-HSP. Such permeabilities are rarely

achieved in tight-gas environments. Hence, the β t -factor

needs to be corrected. A frac fluid damage factor accounts

for the lower “true” permeability or damage by fracturing

fluid residuals. The factor specifies the ratio of ideal to real

permeability of the proppant pack. Assuming a dimen-

sionless fracture conductivity of 50, the real fracture

permeability is about 7.5 D — just 1% of the theoretically

predicted permeability.The non-Darcy flow coefficients are summarized in

Table 2. Resin-coated sand proppants exhibit distinctly

higher non-Darcy flow coefficients (see proppant type

16/30 PRB). The simulation results are illustrated in

Figs. 4 and 5. If non-Darcy flow effects are neglected,

the well produces for about 600 days at a gas rate of

670 m3/h, until a well flowing pressure of pwf = 100 bar

is reached. Subsequently, the rate drops to 380 m3/h after

10 years of production. If undamaged 18/30 Carbo-HSP

proppant is considered, productivity loss is low, even

though the plateau phase already ends after 300 days.

The final rate drops to 347 m3/h.

Fig. 5. Influence of proppant type on the cumulative production with non-Darcy flow.




If proppant damage is taken into account, the initial

rate can be sustained for just 50 days, with a final rate of

323 m3/h at the end of the production term. Using 16/20

Carbo-Lite proppant will cause an increase of inertial

effects. There is almost no plateau phase and the

terminal rate is about 280 m3/h. Using a β f -value fromthe correlation Eq. (6) provides similar results.

5.2. Production with permeability dependent non-

Darcy coefficients

Tight-gas reservoirs are stress-sensitive (Vairogs,

1971; Davies and Holditch, 1998; Friedel et al., 2003).

Productivity is further affected when including effects of the stress-sensitivity of the reservoir rock and proppant

Fig. 6. Stress-dependency of reservoir permeability and fracture conductivity.

Fig. 7. Influence of non-Darcy flow on the production rate with permeability (stress) dependent non-Darcy coefficients.




pack and, hence, the permeability dependency of the non-

Darcy coefficients, β t = f (k ). To investigate this effect, the

stress-dependency of the reservoir and fracture parameters

needs to be taken into account. In the simulations, this

phenomenon is modeled (in a simplified manner) by

means of pressure-dependent transmissibility multipliers

for fracture and reservoir permeability, see Fig. 6.

A decrease in reservoir permeability during produc-tion (as a consequence of increasing the effective stress)

is accompanied by a reduction of flow velocities. Hence,

the non-Darcy flow effects will be lowered (they depend

quadratically on the velocity). On the contrary, non-

Darcy flow coefficients increase with reduced perme-

abilities. Zeng et al. (2003) presented an experimental

study of overburden and stress influence on non-Darcy

effects on Dakota sandstone core. The authors found an

almost linear correlation between permeability, non-Darcy coefficients and effective stress.

Fig. 8. Influence of proppant type on the cumulative production with permeability (stress) dependent non-Darcy coefficients.

Fig. 9. Type-curves for fractured wells with constant pressure production and Darcy flow.




To capture the permeability dependency of the β t -

coefficients, stress-dependent non-Darcy flow coeffi-

cients were implemented into the simulation model using

the correlations, Eqs. (5) and (6). The results are

summarized in Figs. 7 and 8. As expected, there is no

linear superposition if stress-dependency, fracture clo-sure and non-Darcy flow effects are considered simul-

taneously. Instead, the overall reduction is distinctly

lower than the sum of the single effects due to their

mutual interaction. According to Fig. 8, a total reduction

of 40% is possible in a 10 year production period.

However, permeability dependency of non-Darcy flow

coefficients is mainly masked by the stress-dependency

of the reservoir permeability and the fracture closure.

6. Type-curve development

Traditionally, type-curve analysis has been utilized as a

powerful tool for well-test analysis. As a basic principle,

type-curve analysis utilizes dimensionless parameters. In

doing so, it is possible to apply graphical methods for

interpretation, as well as to provide a general solution for a

broad range of parameters. Type-curve analysis is therefore

useful to illustrate the impact of diverse parameters on the

behavior of the subject under consideration, e.g., to predict

the performance of a fractured vertical well under various

conditions. In addition, type-curves are also suitable for

determining the non-Darcy flow coefficients.

Initially, the existing type-curves from Guppy et al.(1981, 1982a,b) were used for verification of the

transient non-Darcy flow period in the developed

simulator. However, it was recognized that these type-

curves, restricted to inertial flow within the fracture, can

be generalized to include also the reservoir non-Darcy

effect, as presented in the following section.

6.1. Constant pressure production

The main production period in tight-gas reservoirs

attributes to the constant pressure flow regime. The

dimensionless rate qD for real gas is calculated as:

qD ¼CTQ

khjmð piÞ−mð pwf Þj: ð9Þ

The dimensionless time is defined:

t Dxf ¼

kt

/ct l x2f

; ð10Þ

where total compressibility ct and the dynamic viscosity

μ of the gas are considered at initial conditions. The real

gas pseudo-pressure m( p) accounts for the variation of

viscosity and density with pressure. In the case of a

slightly compressible fluid, the dimensionless rate is

calculated using:

qD ¼QBl

2kkhð pi− pwf Þ: ð11Þ

Given a pressure above 140 bar, gas essentially

behaves as a slightly compressible fluid, since p / (μ z ) isde facto constant. Provided that further conditions for its

Fig. 10. Type-curves for fractured wells with constant pressure production with non-Darcy flow ( F CD=50).




applicability are fulfilled, Eq. (11) can be used for the

calculation of the dimensionless gas rate in tight-gas

reservoirs under typical pressure conditions (Katz and

Lee, 1990). In the following, we refer to the applicability

of this condition; however, type-curves can be extended to

m( p)-real gas potential representation.To account for the non-Darcy flow, Guppy et al.

(1981) introduced a dimensionless parameter, ( pDND)f ,

as a kind of additional pressure drop:

ð pDNDÞf ¼2kqbf k f k resð pi− pwf Þ

bf l2

T 0 pm

Tz m p0

: ð12Þ

All fluid parameters are inserted at initial conditions.

The latter quotient on the right hand side, a modification

of Guppy's original parameter ( pDND)f , is used in order

to apply the methodology to real gas. The index ‘0’

represents the reference state (e.g., standard conditions).

According to this, Eq. (12) is multiplied with the initial

formation volume factor.

The ( pDND)f parameter is an equivalent Reynolds-

number (Geertsma, 1974):

ð pDNDÞf f

qu

lbf k f ð Þ; ð13Þ

where the product β f k f reflects the characteristic length

of the system (Martins et al., 1990). As obvious from

Eq. (6), the product can be considered almost constant

despite the dependency of β f k f on the proppant type,

pressure, temperature or any kind of damages. Begin-

ning from the limiting case ( pDND)f = 0, where solely

viscous Darcy flow prevails within the fracture, the

portion of the inertial pressure losses increases with

ascending Reynolds-Numbers.

To take account of the non-Darcy flow in the reservoir,

a second dimensionless parameter ( pDND)r is introduced,

similar to the dimensionless parameter for the fracture:

ð pDNDÞr ¼2kqbr k 2resð pi− pwf Þ

xf l2

T 0 pm

Tz m p0

: ð14Þ

Here, the maintenance of fracture conductivity (k f / bf )

is substituted with the “reservoir conductivity” (k res / xf ).

The characteristic length of the reservoir in Eq. (14) is

β r k res. In contradiction to the fracture system, this product

is nonlinear due to the relationship β r = f (k res), Eq. (5).

The ratio of fracture non-Darcy flow to reservoir non-

Darcy flow can be derived by comparison of Eqs. (12)

and (14):

ð pDNDÞf

ð p DNDÞr ¼

x2f

b2f

bf

br F CD:

ð15Þ

Evaluation of Eq. (15), using typical tight-gas para-

meters, proves the established fact that non-Darcy flow in

the fracture affects the productivity of the well to a

distinctly higher degree than inertial effects in the reservoir.

The ratio depends quadratically on the fracture half length

and linearly on the dimensionless fracture conductivity.At the same time, the relationship between Eq. (15)

and the reservoir permeability is comprised via the term

β f / β r . This term is a function of the corresponding

permeabilities in fracture and reservoir. The non-Darcy

flow coefficient of the reservoir increases by more than

the coefficient of the fracture. Hence, the influence of

the reservoir on the overall inertial pressure drop will

rise with increasing permeabilities.

Fig. 9 shows the relationship between dimensionless

time and rate for a wide range of fracture conductivities

F CD, neglecting non-Darcy flow effects. The bilinear flow period is of special interest when analyzing

fractured wells. This period is characterized by a quarter

slope for viscous Darcy flow if dimensionless fracture

conductivity is low. During this period, linear flow

occurs both within the fracture and, predominantly, from

the reservoir into the fracture. It typically lasts up to

several days in a tight-gas environment.

In Fig. 10, non-Darcy flow in fracture and reservoir is

included for F CD=50. Guppy et al. (1981) presented their

type-curves for values ( pDND)f = 0…1, which can be

considered sufficient for slightly compressible fluids.

However, for real gas, the range is too small and should beextended to ( pDND)f = 0…100 to reflect more typical

conditions. At first, only non-Darcy flow in the fracture is

considered, i.e., ( pDND)r = 0. Large values of ( pDND)f affect the fractured well dramatically; the bilinear flow

period is completely masked due to the increase of the

inertial effects. Hence, the real fracture conductivity is

lowered. The resulting apparent conductivity varies with

time (and flow rate). The reduction in productivity is most

distinct for small values of t D xf and is primarily caused by

the large pressure gradients and the corresponding flow

velocities within the fracture. Due to the higher pro-ductivity, non-Darcy flow effects are generally more

pronounced in highly conductive fractures.

The impact of reservoir non-Darcy flow is also

illustrated in the constant pressure type-curves of Fig. 11.

As previously mentioned, its influence is less severe

than inertial effects within the fracture: see the cases

( pDND)f = 0 with ( pDND)r N0. Nonetheless, the produc-

tivity can be further decreased as a consequence of the

reservoir inertial effects. Unlike the fracture flow, the

bilinear flow period is practically unaffected. After the

end of the bilinear flow period and the beginning of

linear reservoir flow, the influence of the reservoir non-




F i g .

1 1 .

N o n - D a r c y f l o w t

y p e - c u r v e s f o r c o n s t a n t p r e s s u r e p r o d u c t i o

n .




Darcy flow on the well becomes more pronounced.

Simultaneously, the impact of non-Darcy fracture flow

declines. This continues in the subsequent pseudoradial

flow period. The reason for this behavior is the increase

of reservoir domination due to the increasing drainage

area, while the fracture reduces to a point source.

As a result of the analysis it can be concluded that non-Darcy flow in the fracture is of secondary impor-

tance for the longterm behavior of the well. In contrast,

neglecting the non-Darcy reservoir flow may result in an

overestimation of fractured well potential.

6.2. Constant rate production

Constant rate production is typically restricted to ashort time period in tight-gas reservoirs. Again, results

Fig. 12. Type-curves for fractured wells with constant rate production and Darcy flow.

Fig. 13. Type-curves for fractured wells with constant rate production and non-Darcy flow ( F CD=50).




F i g .

1 4 .

N o n - D a r c y f l o w

t y p e - c u r v e s f o r c o n s t a n t r a t e p r o d u c t i o n .




are presented using the traditional dimensionless log

pD= f (log t D xf ) plots. The dimensionless pressure pD is

calculated as follows:

p D ¼2kkhð pi− pwf Þ

QBl: ð16Þ

To account for non-Darcy flow within the fracture,

Guppy et al. (1982a) introduced a dimensionless

parameter called the Dimensionless Flow Rate Con-

stant, (qDND)f , where:

ðqDNDÞf ¼k f qbf Q

bf hl: ð17Þ

In the original work, the density was replaced with

the molecular weight when considering real gas. The

constant (qDND)f characterizes the transition from

laminar Darcy flow (qDND)f →0 to the inertial non-

Darcy flow. Therefore, (qDND)f is equivalent to theconstant ( pDND)f .

To account for reservoir non-Darcy flow, a parameter

(qDND)r is introduced:

ðqDNDÞr ¼k resqbr Q

xf hl: ð18Þ

All fluid parameters in Eqs. (16), (17) and (18) are

again taken at initial conditions. Neglecting any non-

Darcy effects, type-curves are presented for a typical

range of F CD in Fig. 12.

Non-Darcy flow lowers the true fracture conductiv-

ity. Contrary to the constant pressure case, the apparent

fracture conductivity is not a function of time but a

function of (qDND)f and F CD (Guppy et al., 1982a):

ð F CDÞtrue

ð F CDÞapp

¼ 1 þ 0:31ðqDNDÞf : ð19Þ

Fig. 13 shows the reduction of real fracture

conductivity ( F CD

= 50) to an apparent conductivity

F CD= 12.2 for (qDND)f = 10 and (qDND)r = 0.

Fig. 14 presents type-curves including the reservoir

non-Darcy flow effect. The general trend is identical for

different values of (qDND)f,r , according to the shape of

the dimensionless pressure and its derivative in Fig. 12.

Consequently, the degree of non-Darcy flow does not

affect the principal slope of the curve during any relevant

flow periods (Guppy et al., 1982a). However, inertial

pressure losses increase with (qDND)f and (qDND)r .

7. Example

The impact of non-Darcy flow effects in fracture and

reservoir flow on the evaluation of a well-test is

Fig. 15. Drawdown example.

Table 3

Results of well-test analysis

Case 1 Case 2 Case 3

β f (1/m) 0 7.2·105 7.2·105

β r (1/m) 0 0 1·1014

k r (mD) 0.01 0.0098 0.014 0.012

F CD 50 49 1.6 1.02

xf (m) 75 76 54.3 71.8




demonstrated for a drawdown test. The well produces for

500 days with a constant rateof 104 m3/day(0.353 MMscf/

day). Formation and fracture parameters are according to

Table 1. The fracture non-Darcy flow coefficient equals

7.2·105 1/m (20/40 Carbo HSP); the reservoir non-Darcy

flow coefficient is 1014 1/m. The wellbore pressure

development is presented in Fig. 15 for the cases (i) solely

Darcy flow, (ii) non-Darcy flow in the fracture and Darcy

flow in the reservoir, and (iii) non-Darcy flow in bothdomains. Stress-dependency is ignored.

The test data is numerically generated. A commercial

well-test package (Schlumberger GeoQuest, 2000) is

then used to analyze the wellbore flowing pressure during

the production, facilitating both manual type-curve

matching and its automatic regression mode to identify

the best parameter set. To verify this procedure, a first

well-test analysis is conducted which assumes only Darcy

flow in the fracture and reservoir. The regression results

are in very close agreement to the input parameters, see

Case 1 in Table 3.

In Case 2, the pressure data only includes the non-Darcy flow in the fracture. As expected, both F CD and xf are too low if evaluated with Darcy flow type-curves. To

check the dimensionless conductivity, the well-test

Fig. 16. Well-test match Case 2 (non-Darcy flow in fracture and Darcy flow in reservoir).

Fig. 17. Well-test match Case 3 (non-Darcy flow in fracture and reservoir).




results are compared with Guppy's apparent conductiv-

ity. Using Eq. (17), (qDND)f =96 in the current case. The

corresponding apparent fracture conductivity, Eq. (19),

is 1.63, which fits very well to the well-test evaluation

result (Fig. 16).

In Case 3, non-Darcy flow is considered in bothdomains, with (qDND)r = 1.2. The regression of the

pressure data suggests a dimensionless fracture conduc-

tivity even lower than in Case 2 (as expected) but a

fracture half-length close to the actual value of 75 m

(Fig. 17). Additionally, the reservoir permeability match

is more accurate compared to the real value of 0.01 mD.

A second test with a higher non-Darcy flow coefficient

for the reservoir, supports this opposite trend.

8. Conclusions

Based on the results of the present study, the

following conclusions are offered:

(i) The implementation of non-Darcy flow in a fully-

implicit in-house Black-Oil simulator facilitates the

evaluation of flow with highly inertial character-

istics, i.e., control parameters 0bδ pbb1. Perme-

ability-dependent non-Darcy flow coefficients due

to stress-sensitivity of reservoir permeability in

tight-gas reservoirs and fracture closure can be

taken into account.

(ii) Non-Darcy flow is important when determining productivity of fractured wells in tight-gas reser-

voirs. Inertial flow affects the productivity despite

the low gas rates. Hence, non-Darcy flow effects in

tight-gas reservoirs should not be disregarded

during field development to avoid, e.g., an

overestimation of the predicted gas production or

the length of the plateau phase.

(iii) Regarding a realistic scenario, the total gas

production is reduced by 21% to 33%, depending

on the proppant type. Following the correlations for

the proppant non-Darcy flow coefficients (based onthe mean of a various proppants), a reduction of

even 40% may be expected due to non-Darcy flow

effects.

(iv) Although fracture non-Darcy flow dominates

the entire inertial pressure drops, it appears

essential to include both components of non-

Darcy flow into the simulation model to ensure

its accuracy.

(v) Non-Darcy flow in the fracture is of secondary

importance on the longterm behavior of the well,

in particular in the case of constant pressure

production. In contrast, neglecting the non-Darcy

reservoir flow contributes to an overestimation of

longterm fractured well potential.

(vi) New type-curves are presented for non-Darcy

flow in fracture and reservoir, facilitating the

prediction of future well performance. New

dimensionless parameters are introduced for thereservoir non-Darcy flow representing equivalent

Reynolds numbers.

Nomenclature

bf Fracture width m

B Formation volume factor res m3/

stock-tank m3

ct Total compressibility 1/Pa

C Constant

dmp difference in pseudo-pressures Pa2/Pa s

F CD Dimensionless fracture

conductivity

h Thickness m

k Permeability m2, mD

m( p) Real gas potential Pa2/Pa s

p Pressure Pa

( pDND) Dimensionless non-Darcy

flow parameter

qD Dimensionless flow rate

Q Wellbore flow rate m3/s

(qDND) Dimensionless flowrate parameter

RP Relaxation parameter

t Time s

t D Dimensionless time

T Temperature K

u Velocity m/s

xf Fracture half length m

z Compressibility factor

Indices

app Apparent D Dimensionless

DND Non-Darcy

f Fracture

i Initial

m Average

r Relative, reservoir

res Reservoir

p Phase

true True

wf Well flowing

0 Reference state⁎ Fictitious




(vii) Evaluation of well-tests using conventional Darcy

flow type-curves can result in erroneous parameters

in the well-test analysis such as fracture conductiv-

ity, fracture half-length and reservoir permeability.

The presence of reservoir non-Darcy flow further

decreases the apparent conductivity but also coun-

teracts the shortening of the fracture half-length.

Acknowledgements

The authors would like to thank the German Society

for Petroleum and Coal Science and Technology

DGMK (Hamburg), Exxon Mobil Production Germany

(Hannover), Gaz de France Produktion Exploration

Deutschland GmbH (Lingen), Wintershall AG (Kassel),

RWE DEA AG (Hamburg) and Erdöl-Erdgas GmbH

(Berlin) for funding parts of this work and the

permission to publish this paper.

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Greek

β t Non-Darcy flow coefficient 1/m

δ Control parameter

ρ Density kg/m3

μ Dynamic viscosity Pa s, cp

ϕ PorosityΦ Potential Pa


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