fluxo no darcyano
TRANSCRIPT
8/3/2019 Fluxo No Darcyano
http://slidepdf.com/reader/full/fluxo-no-darcyano 1/17
8/3/2019 Fluxo No Darcyano
http://slidepdf.com/reader/full/fluxo-no-darcyano 2/17
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
8/3/2019 Fluxo No Darcyano
http://slidepdf.com/reader/full/fluxo-no-darcyano 3/17
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
8/3/2019 Fluxo No Darcyano
http://slidepdf.com/reader/full/fluxo-no-darcyano 4/17
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).
115T. Friedel, H.-D. Voigt / Journal of Petroleum Science and Engineering 54 (2006) 112 – 128
8/3/2019 Fluxo No Darcyano
http://slidepdf.com/reader/full/fluxo-no-darcyano 5/17
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
116 T. Friedel, H.-D. Voigt / Journal of Petroleum Science and Engineering 54 (2006) 112 – 128
8/3/2019 Fluxo No Darcyano
http://slidepdf.com/reader/full/fluxo-no-darcyano 6/17
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.
117T. Friedel, H.-D. Voigt / Journal of Petroleum Science and Engineering 54 (2006) 112 – 128
8/3/2019 Fluxo No Darcyano
http://slidepdf.com/reader/full/fluxo-no-darcyano 7/17
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.
118 T. Friedel, H.-D. Voigt / Journal of Petroleum Science and Engineering 54 (2006) 112 – 128
8/3/2019 Fluxo No Darcyano
http://slidepdf.com/reader/full/fluxo-no-darcyano 8/17
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.
119T. Friedel, H.-D. Voigt / Journal of Petroleum Science and Engineering 54 (2006) 112 – 128
8/3/2019 Fluxo No Darcyano
http://slidepdf.com/reader/full/fluxo-no-darcyano 9/17
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).
120 T. Friedel, H.-D. Voigt / Journal of Petroleum Science and Engineering 54 (2006) 112 – 128
8/3/2019 Fluxo No Darcyano
http://slidepdf.com/reader/full/fluxo-no-darcyano 10/17
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-
121T. Friedel, H.-D. Voigt / Journal of Petroleum Science and Engineering 54 (2006) 112 – 128
8/3/2019 Fluxo No Darcyano
http://slidepdf.com/reader/full/fluxo-no-darcyano 11/17
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 .
122 T. Friedel, H.-D. Voigt / Journal of Petroleum Science and Engineering 54 (2006) 112 – 128
8/3/2019 Fluxo No Darcyano
http://slidepdf.com/reader/full/fluxo-no-darcyano 12/17
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).
123T. Friedel, H.-D. Voigt / Journal of Petroleum Science and Engineering 54 (2006) 112 – 128
8/3/2019 Fluxo No Darcyano
http://slidepdf.com/reader/full/fluxo-no-darcyano 13/17
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 .
124 T. Friedel, H.-D. Voigt / Journal of Petroleum Science and Engineering 54 (2006) 112 – 128
8/3/2019 Fluxo No Darcyano
http://slidepdf.com/reader/full/fluxo-no-darcyano 14/17
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
125T. Friedel, H.-D. Voigt / Journal of Petroleum Science and Engineering 54 (2006) 112 – 128
8/3/2019 Fluxo No Darcyano
http://slidepdf.com/reader/full/fluxo-no-darcyano 15/17
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).
126 T. Friedel, H.-D. Voigt / Journal of Petroleum Science and Engineering 54 (2006) 112 – 128
8/3/2019 Fluxo No Darcyano
http://slidepdf.com/reader/full/fluxo-no-darcyano 16/17
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
127T. Friedel, H.-D. Voigt / Journal of Petroleum Science and Engineering 54 (2006) 112 – 128
8/3/2019 Fluxo No Darcyano
http://slidepdf.com/reader/full/fluxo-no-darcyano 17/17
(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.
References
Alvarez, C.H., Holditch, S.A., McVay, D.A., 2002. Effect of non-
Darcy flow on pressure transient analysis of hydraulically fractured
gas wells. SPE-77468. Ann. Tech. Conf., San Antonio, Texas.
Sept.
Bennett, C.O., Reynolds, A.C., Raghavan, R., Elbel, J.L., 1986.
Performance of finite-conductivity, vertically fractured wells in
single-layer reservoirs. SPE-11029. SPEFE 399–412 (Aug.).
Cooke, J.C.E., 1973. Conductivity of fracture proppants in multiple
layers. SPE-4117. JPT 25, 1101–1107 (Sept.).
Davies, J.P., Holditch, S.A., 1998. Stress dependent permeability in
low permeability formation, East Texas gas reservoirs: TravisPeak. SPE-39917. Rocky Mountain Regional/Low-Permeability
Reservoirs Symp. and Exhib., Denver, Colorado. April.
Forchheimer, P., 1901. Wasserbewegung durch Boden. Z. Ver. Dtsch.
Ing. 45 (5), 1782–1788.
Fourar, M., Lenormand, R., 2000. Inertial effects in two-phase flow
through fractures. Oil & Gas Sci. and Techn.-Rev. IFP, vol. 55 (3),
pp. 259–268.
Friedel, T., Häfner, F., 2004. Development and application of a
hydraulic fracture reservoir simulation tool. European Conf. on the
Mathematics of Oil Recovery (ECMOR 9), Cannes, France. Aug.
Friedel, T., Behr, A., Voigt, H.-D., Mtchedlishvili, G., Häfner, F., 2003.
Simulation des Produktionsverhaltens gefracter Bohrungen in
geringpermeablen Gaslagerstätten. Erdöl Erdgas Kohle (7/8),
274–278.
Geertsma, J., 1974. Estimating the coefficient of inertial resistance in
fluid flow through porous media. SPE-4706. SPEJ 445–450 (Oct.).
Gil, J.A., Ozkan, E., Raghavan, R., 2001. Fractured-well-test design
and analysis in the presence of non-Darcy flow. SPE-71573. Ann.
Tech. Conf., New Orleans, Louisiana. Sept.
Guppy, K.H., Cinco-Ley, H., Ramey, J.H.J., 1981. Effect of non-Darcy
flow on the constant pressure production of fractured wells. SPE-9344. SPEJ 390–400 (June).
Guppy, K.H., Cinco-Ley, H., Ramey, J.H.J., Sameniego, V.F., 1982a.
Non-Darcy flow in wells with finite-conductivity vertical fractures.
SPE-8281. SPEJ 681–698 (Oct.).
Guppy, K.H., Cinco-Ley, H., Ramey, J.H.J., 1982b. Pressure buildup
analysis of fractured wells producing at high flow rates. SPE-
10187. JPT 2656–2666 (Nov.).
Holditch, S.A., Morse, R.A., 1976. The effects of non-Darcy flow on
the behavior of hydraulically fractured gas wells. SPE-5586. JPT
1169–1179 (Oct.).
Jin, L., Penny, G., 1998. A study on two phase, non-Darcy gas flow
through proppant packs. SPE-49248. Ann. Tech. Conf., New
Orleans, Louisiana. Sept.
Katz, D.L., Lee, R.L., 1990. Natural Gas Engineering: Production andStorage. McGraw-Hill Publishing Company.
Li, D., Engler, T.W., 2001. Literature review on correlations of the
non-Darcy coefficient. SPE-70015. Permian Basin Oil and Gas
Recovery Conf., Midland, Texas. May.
Li, D., Engler, T.W., 2002. Modeling and simulation of non-Darcy
flow in porous media. SPE-75216. IOR Symposium, Tulsa,
Oklahoma. April.
Martins, J.P., Milton-Taylor, D., Leung, H.K., 1990. The effect of non-
Darcy flow in propped hydraulic fractures. SPE-20709. Ann. Tech.
Conf., New Orleans, Louisiana. Sept.
Millheim, K.K., Cichowicz, L., 1968. Testing and analyzing low-
permeability fractured gas wells. SPE-1768. JPT 193–198 (Feb.).
Roberts, B.E., van Engen, H., van Kruysdijk, C.P.J.W., 1991.
Productivity of multiply fractured horizontal wells in tight gas
reservoirs. SPE-23113. Offshore Europe Conf., Aberdeen, U.K.
Sept.
Schlumberger GeoQuest, 2000. Weltest 200 User Manual.
Schlumberger GeoQuest, 2003. Eclipse 100 Technical Description
2003a.
STIMLAB-Consortium, 2003. URL http://www.corelab.com/stimlab/
default.asp.
Umnuayponwiwat, S., Ozkan, E., Pearson, C.M., Vincent, M., 2000.
Effect of non-Darcy flow on the interpretation of transient pressure
responses of hydraulically fractured wells. SPE-63176. Ann. Tech.
Conf., Dallas, Texas. Oct.
Vairogs, J., 1971. Effect of rock stress on gas production from low-
permeability reservoirs. SPE-3001. JPT 1161–1167 (Sep.).Vincent, M.C., Pearson, C.M., Kullman, J., 1999. Non-Darcy and
multiphase flow in propped fractures: case studies illustrate the
dramatic effect on well productivity. SPE-54630. Western
Regional Meeting, Anchorage, Alaska. May.
Wattenbarger, R.A., Ramey, J.H.J., 1969. Well test interpretation of
vertically fractured gas wells. SPE-2155. JPT 625–632 (May).
Wong, S.W., 1970. Effect of liquid saturation on turbulence factors for
gas–liquid systems. Can. Pet. Technol. 274 (Oct.–Dec.).
Zeng, Z., Grigg, R., Ganda, S., 2003. Experimental study of
overburden and stress influence on non-Darcy gas flow in Dakota
sandstone. SPE-84069. Ann. Tech. Conf., Denver, Colorado. Oct.
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
128 T. Friedel, H.-D. Voigt / Journal of Petroleum Science and Engineering 54 (2006) 112 – 128