Application of inflow control devices to heterogeneous reservoirs

Journal of Petroleum Science and Engineering 78 (2011) 534–541

Contents lists available at ScienceDirect

Journal of Petroleum Science and Engineering j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p e t r o l

Application of inflow control devices to heterogeneous reservoirs V.M. Birchenko a,⁎, 1, A.Iu. Bejan b, A.V. Usnich c, D.R. Davies a,⁎ a b c

Institute of Petroleum Engineering, Heriot–Watt University, Edinburgh, EH14 4AS, UK University of Cambridge, Department of Pure Mathematics and Mathematical Statistics, Statistical Laboratory, Wilberforce Road, Cambridge CB3 0WB, UK Institut für Mathematik, Universität Zürich, 190 Winterthurerstr., 8057 Zurich, Switzerland

a r t i c l e

i n f o

Article history: Received 1 November 2009 Accepted 1 June 2011 Available online 15 June 2011 Keywords: horizontal wells Inflow Control Devices ICD heterogeneity

a b s t r a c t The rate of inflow to a long well can vary along its completion length due to reasons such as frictional pressure losses or reservoir permeability heterogeneity. These variations often negatively affect the oil sweep efficiency and the ultimate recovery. Inflow Control Devices (ICDs) are a mature, well completion technology that can make the inflow profile more uniform by restricting the high specific inflow rate segments of the completion. The paper presents a mathematical model for effective reduction of the inflow imbalance caused by the second of the above mentioned reasons, that is reservoir heterogeneity. The model addresses one of the key questions of the ICD technology application — the trade-off between well productivity and inflow equalisation. The practical utility of the model is illustrated through a case study. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Increasing well–reservoir contact has a number of potential advantages in terms of well productivity, drainage area, sweep efficiency and delayed water or gas breakthrough. However, such long, possibly multilateral wells bring not only advantages but also present new challenges in terms of drilling, completion and production. One of these challenges is that the variation in rock properties (and hence fluid specific inflow rate) tends to increase with increasing well length. Completion intervals with a length of several thousands of feet have become very popular in the last few years. Such completions will often have significantly uneven specific inflow distribution along their length if special measures are not taken. These inflow variations often cause premature water or gas breakthrough and, hence, should be minimised. Advanced well completions have proved to be a practical solution to this challenge. Inflow Control Devices (ICDs) and Interval Control Valves (ICVs) are two established types of advanced completions. ICVs permit an active control of inflow (or outflow) of multiple completion intervals or laterals (Gai, 2002) while ICDs provide a passive form of inflow control (Henriksen et al., 2006). ICDs have been installed in hundreds of wells during the last decade and are now considered to be a mature well completion technology (Figs. 1 and 2). The steady-state performance of ICDs can be analysed in detail with the help of well modelling software (Johansen and Khoriakov, 2007; Ouyang and Huang, 2005). Most reservoir simulators

⁎ Corresponding authors at: Institute of Petroleum Engineering, Heriot–Watt University, Edinburgh, EH14 4AS, UK. E-mail addresses: [email protected] (V.M. Birchenko), [email protected] (A.Iu. Bejan), [email protected] (A.V. Usnich), [email protected] (D.R. Davies). 1 Now with BP. 0920-4105/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.petrol.2011.06.022

include basic functionality for ICD modelling while some of them (Neylon et al., 2009; Wan et al., 2008) also offer a practical means to capture the effect of annular flow. Current numerical simulation software provide engineers with the tools to perform the design and economic justification of an ICD completion. However, relatively simple analytical models still have a role to play in: (1) Quick feasibility studies (screening ICD installation candidates). (2) Verification of numerical simulation results. (3) Communicating best practices in a non-product specific manner. This paper proposes an analytical model for heterogeneous reservoirs that quantifies the reduction of inflow variation along a horizontal well with ICDs installed. This model allows one to estimate: (1) The ICD design parameters that substantially reduce the inflow variation caused by reservoir heterogeneity. (2) The impact of a specific ICD completion on Inflow Performance Relationship (IPR) of a long well completed in a heterogeneous reservoir. 2. Assumptions Our model invokes the following assumptions with respect to the inflow from the reservoir: (1) Flow through the reservoir can be described by Darcy's law. (2) Steady or pseudo-steady state flow into the well. (3) The distance between the well and the areal reservoir boundary is much longer than the well length (or the boundary is parallel to the well).

V.M. Birchenko et al. / Journal of Petroleum Science and Engineering 78 (2011) 534–541

535

Fig. 1. Channel ICD schematics (courtesy Baker Oil Tools).

(4) The perpendicular-to-the-well components of the reservoir pressure gradients are much greater than the along-hole ones. It is well known that near wellbore region of several feet accounts for most of reservoir pressure drop. This pressure drop is typically much greater than that along the entire completion interval. Only in thousand feet long horizontal wells completed in high permeability reservoirs are these two pressure drops comparable (Birchenko et al., 2010b), however, since the distances over which these two pressure drops occur differ by several orders of magnitude, the assumption remains valid even in such cases. The above simplifications are often introduced in analytical descriptions of coupling of reservoir and wellbore flow. They are required to introduce the notion of the “specific productivity index” (PI per unit length). This empirical parameter is determined largely by distribution of formation permeability and fluid saturation in reservoir. It indicates that the fluid inflow from reservoir to wellbore is proportional to the pressure difference between the external reservoir boundary and the annulus. Consider an arbitrary point along the wellbore at distance l from the toe. The fluid inflow from reservoir to wellbore at this point is given by: dq = jðlÞðPe ðlÞ−Pa ðlÞÞ: dl

ð1Þ

For details on notation used see the complete nomenclature at the end of the paper. The chosen assumptions for the description of the wellbore flow are that: (1) Friction and acceleration pressure losses between the toe and the heel are small compared to the drawdown. The validity of this assumption has been discussed by many authors (e.g. Dikken (1990) or Birchenko et al. (2010b)). (2) The fluid is incompressible. Note that we do not assume the completion interval to be perfectly horizontal. The true vertical depth (TVD) can vary along the completion

since reservoir pressure at the external boundary, Pe, is measured at the same TVD as the corresponding point l of the tubing. The above assumptions imply that the difference between the reservoir pressure Pe and the tubing pressure P is constant throughout the completion length: Pe ðlÞ−P ðlÞ = ΔPw = const:

ð2Þ

Our assumptions about the ICDs are as follows: (1) There is no flow in the annulus parallel to the base pipe, i.e. the fluid flows from reservoir directly through ICD screens into the base pipe. This assumption is reasonable when: (a) ICDs are combined with a number of intermediate packers or a gravel pack (Augustine et al., 2008), or (b) The wellbore has collapsed around the screen so that annular flow is no longer possible. (2) ICDs of the same “strength” are installed throughout the completion length. This is the most common type of ICD application due to the relative simplicity of its design and installation operation (Henriksen et al., 2006). Further, the operational risk is reduced for the case that the rig is unable to install the completion at the correct depth. However, variable “strength” ICD completions have also been reported (Helmy et al., 2006; McIntyre et al., 2006). These are more complex operationally (to design and install) and require a detailed, reliable description of variation of j along the wellbore length at the time of completion installation. Such ICD designs are required to mitigate extreme variations in the fluid inflow (e.g. fractured or “super k” zones in carbonate reservoirs). (3) The profile q(l) describing the flow distribution along the wellbore's internal flow conduit is a smooth function, i.e. it has a continuous derivative. Strictly speaking, this empirical distribution is a step-like function, since the fluid enters the wellbore through a number of point-like sources — the ICD

Fig. 2. Orifice ICD schematics (courtesy Weatherford).

536

V.M. Birchenko et al. / Journal of Petroleum Science and Engineering 78 (2011) 534–541

nozzles or channels. However, the discontinuity jumps will be relatively small if the ICD well completion has a large number of joints of a standard length (≈ 12 m). Thus, the flow distribution profile along the wellbore's internal flow conduit q(l) can be approximated by a smooth function. We make this assumption for convenience in order to simplify mathematical modelling.

inflow along the completion length due to the ICD. It is the objective of this work to develop and explore a framework linking the ratio of the two coefficient of variations with well parameters (such as ICD “strength”, drawdown, etc.) using the model described above (Eq. (6)). 4. Solution

3. Problem formulation Let us analyse the impact of an ICD completion on the well inflow profile when the well is completed in a heterogeneous reservoir. The specific inflow rate is the derivative of the flow rate with respect to the measured depth. We designate a separate notation U to this quantity since it is central to this paper: U ðlÞ≡

dqðlÞ : dl

ð3Þ

The Specific Productivity Index, j, and hence the inflow, U, change stochastically along the completion interval. We will use the coefficient of variation to quantify the degree of these changes. Recall that the coefficient of variation of a random variable is defined as the ratio of its standard deviation and its mean. The annulus pressure Pa is equal to the base pipe pressure P for a conventional completion (no ICD). Hence U ðlÞ = jðlÞΔPw

ð4Þ

where ΔPw is a constant independent of l. Eq. (4) shows that, in case of conventional completion (no ICD), the coefficient of variation of specific inflow is equal to that of the specific PI: ð5Þ

CoV U = CoV j:

In the case of an ICD completion the annulus pressure is not equal to the base pipe (tubing) pressure. The difference between these two is due to pressure drop across the ICDs. The relationship between the specific inflow rate and specific productivity index is no longer linear because flow through ICD is turbulent. Birchenko et al. (2010a) derived the following quadratic equation describing the specific inflow rate through an ICD completion as a function of l: 2

aU ðlÞ + U ðlÞ = jðlÞ−ΔP ðlÞ = 0

ð6Þ

where ΔP(l) is the pressure difference between the external reservoir boundary and the tubing (base pipe) at the depth l from the toe and a is a parameter specific to the ICDs used. Bear in mind that in this work we assume ΔP(l) = const = ΔPw (Section 2). The first term in Eq. (6) represents pressure drop across ICD while the second term accounts for the pressure drop in reservoir. The sum of these two is the “drawdown” (Eq. (2)). The annulus pressure can now be expressed as: 2

Pa ðlÞ = Pe ðlÞ−U ðlÞ = jðlÞ = P ðlÞ + aU ðlÞ:

ð7Þ

Our first claim is that under the model described by Eq. (6) the ICD application reduces the variation of inflow so that the following inequality holds: CoV U b CoV j:

ð8Þ

Inequality (8) may seem intuitively obvious to engineers familiar with the ICD technology, however its rigorous mathematical proof known to the authors is not straightforward (see Appendix A). Let us consider the ratio of the two coefficients of variation, CoV U/CoV j. This ratio equals unity for a conventional completion and decreases monotonically with increasing ICD strength. The magnitude of this decrease is a quantitative measure of the equalisation of the

Taking into account equality (2) one can readily write down the solution of Eq. (6):

U ðlÞ =

qw = ∫

−1 +

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 + 4aΔPw j2 ðlÞ

ð9Þ

2ajðlÞ L

−1 +

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 + 4aΔP ðlÞj2 ðlÞ 2ajðlÞ

0

ð10Þ

dl

where

a=

8  ρcal μ 1 = 4 ρ 2 2 > > > l B aICD > < ρ μcal ρcal ICD

for channel ICD

> Cu ρl2ICD B2 > > > : an = Cd2 d4

for nozzle or orifice ICD

ð11Þ

and L is the completion length. Formula (10) is of limited use for a “quick-look” analysis if the local specific productivity index varies substantially along the completion interval since: (1) The exact shape of the productivity profile j(l) is often unknown: (a) Detailed measurements (logging) are not always feasible. (b) The productivity index changes with time (e.g. due to fluid saturation changes). (2) In general one needs to evaluate the integral (10) numerically even if j(l) is known (or can be estimated). The engineering team that develops each well drilling proposal will normally define an expected range of values for the specific productivity index, j, as part of the proposal. These could be based on: (1) Well log data. (2) Reservoir models. (3) Production performance of similar wells in the same field. The proposed range of values may take a number of forms. For instance, in its simplest form it could comprise of only three values: pessimistic (P90), most probable (P50) and optimistic (P10). Ideally, a complete specification of j would be available in the form of a probability density function (p.d.f.). In the case when j depends on some other parameters and information about their distribution is available this density can be estimated, for example, via Monte-Carlo simulation. It is often easier to make a judgement about the statistical distribution of the specific productivity index, η( j), rather than its spatial distribution j(l). This allows us to transform formula (10) into

qw = L∫

j2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 + 4aΔPw j2

−1 +

2aj

j1

ηð jÞdj:

ð12Þ

Calculation of the coefficient of variation requires mean and mean square values of the specific inflow rate. The mean specific inflow rate is the ratio of the well flow rate to its length:

hU i = qw =L = ∫

j2

j1

−1 +

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 + 4aΔPw j2 2aj

ηð jÞdj:

ð13Þ

V.M. Birchenko et al. / Journal of Petroleum Science and Engineering 78 (2011) 534–541

The p.d.f. of a triangular distribution is as follows:

Similarly, its mean square value is calculated as follows: D

U

2

E

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12 0 −1 + 1 + 4aΔPw j2 A ηð jÞdj: =∫ @ j1 2aj j2

537

ð14Þ

The choice of the method for solving the integrals (13) and (14) depends on the functional form of η( j), the p.d.f. of the specific PI. Notably, these integrals can be solved analytically for a piecewise linear p.d.f. (e.g. a uniform or triangular distribution). The corresponding solutions are presented below in Sections 4.1 and 4.2. When the density function has a more complex form (e.g. a normal or log-normal distribution) the integrals have to be evaluated numerically.

8 < 2ð j−j1 Þ = ð j2 −j1 Þ = ð jm −j1 Þ ηð jÞ = 2ð j2 −jÞ = ð j2 −j1 Þ = ð j2 −jm Þ : 0

for j1 ≤ j ≤ jm for jm ≤ j ≤ j2 otherwise:

ð21Þ

Then qw = hU iL =

ðI ð j Þ−I ð j ÞÞ ð I ð j Þ−I ð j Þ−j + j −j j ðI ð j Þ−I ð j ÞÞ−I ð j Þ + I ð j Þ + Þ j −j 2L j2 −j1 2

Uj

m

Uj

1

1

m

U

2

U

m

Uj

2

U

m

U

1

1

2

Uj

m

ð22Þ

m

4.1. Uniform distribution of specific productivity index and Generally speaking, a uniform distribution of the specific productivity index is unlikely to be encountered in practice. In fact, petrophysical quantities are usually modelled by a normal or log-normal distribution. However, the data required to determine the distribution parameters with sufficient precision is often unavailable. A uniform distribution may be a sensible starting assumption when data is scarce. Assuming that j is uniformly distributed between two values j1 and j2, j1 ≤ j2, its density function is as follows:  ηð jÞ =

1 = ð j2 −j1 Þ 0

for j1 ≤ j ≤ j2 otherwise:

ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

D E  u 2 2 u − U 2 U h i CoV U j + jm + j1 t = 2 CoV j hU i j21 + j22 + j2m −j1 j2 −j1 jm −j2 jm

ð23Þ

where D

E U2 =

ðS ð j Þ−S ð j ÞÞ ð S ð j Þ−S ð j Þ−j + j −j j ðS ð j Þ−S ð j ÞÞ−S ð j Þ + S ð j Þ Þ + j −j 2 j2 −j1 2

ð15Þ

Uj

m

Uj

1

1

m

U

2

U

m

2

Uj

U

m

U

1

1

2

Uj

m

ð24Þ

m

with

In this case: I ð j Þ−IU ð j1 Þ qw = hU iL = U 2 L j2 −j1

ð16Þ

IUj

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0  pffiffiffiffiffiffiffiffiffiffiffiffi 1 j 1 + 4aj2 ΔPw arcsinh 2j aΔPw A 1 @ p ffiffiffiffiffiffiffiffiffiffiffiffi −j + + = 2a 2 4 aΔPw

ð25Þ

and CoV U = CoV j

2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi ð j2 + j1 Þ 3 U 2 −hU i2

SUj ð jÞ = ð17Þ

hU ið j2 −j1 Þ

U

2

E

=

5. Case study SU ð j2 Þ−SU ð j1 Þ j2 −j1

ð18Þ

with

IU ð j Þ =

ð26Þ

and functions IU and SU defined by Eqs. (19) and (20) respectively.

where D

ΔPj = 2−IU ð jÞ a

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 + 4aΔPw j2 −ln 1 + 1 + 4aΔPw j2 2a

ð

1 2 −1 + 2aΔPw j + SU ð j Þ = 2a2 j

ð19Þ

Þ

1. Prolific reservoir 2. Medium productivity reservoir. This is done to quantitatively illustrate the dependence between the specific PI and ICD “strength” required to reduce inflow variations.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 + 4aΔPw j2 −

pffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffi −2j aΔPw arcsinh 2j aΔPw :

This case study shows how our model for the uniform distribution of the specific productivity index can be used in practice. We consider two following cases:

5.1. Prolific reservoir ð20Þ

4.2. Triangular distribution of specific productivity index The most probable, or modal value is often known within reasonable error margins in addition to the knowledge about the minimum and maximum values of the specific PI. The specific productivity index j may then be modelled by the (more complex) triangular distribution. This is a legitimate approach if a triangular distribution can be fitted to the field data almost as well as more common normal or log-normal distributions.

Let us consider a 1 km long well completed in a prolific heterogeneous reservoir (Table 2). The anticipated PI of the well is 2000 Sm 3/day/bar. A drawdown, ΔPr, of 0.5 bar is required for a conventional completion to achieve the target well rate of 1000 Sm 3/day.

Table 1 Channel ICD strength. Industrial “bar” rating aICD, bar/(Rm3/day)2 aICD, psi/(Rbbl/day)2

0.2 0.00028 0.00076

0.4 0.00055 0.0015

0.8 0.00095 0.0026

1.6 0.0016 0.0044

3.2 0.0032 0.0087

538

V.M. Birchenko et al. / Journal of Petroleum Science and Engineering 78 (2011) 534–541

200.5

Table 2 Prolific reservoir case study data. L J j1 j2 qw ρ μ B lICD

1000 m 2000 Sm3/day/bar 0.5 Sm3/day/bar/m 3.5 Sm3/day/bar/m 1000 Sm3/day 800 kg/m3 1.7 cp 1.2 Rm3/Sm3 12.2 m

200.0

Pressure, bar

Well length Well Productivity Index (PI) Minimum value of specific PI Maximum value of specific PI Target well flow rate In-situ fluid density In-situ fluid viscosity Formation volume factor Length of the ICD joint

199.5

199.0

198.5

0

200

400

600

800

1000

Measured depth from the heel, m Reservoir

Pressure drop introduced by a conventional completion is usually negligible compared to the drawdown:

Tubing

Fig. 4. Pressure profiles in the prolific reservoir case with 1.6 bar ICDS.

ð27Þ

5.2. Medium productivity reservoir The specific productivity index is the key parameter in the ICD completion design. The majority of ICD installations to date are in reservoirs with an average permeability of one Darcy or greater

1.2 1

2000

0.8 1500 0.6 1000 0.4 500

0.2 0 0

0.001

0.002

0.003

0.004

0.005

0.006

CoV ratio

Well flow rate

Fig. 5. Dependence of inflow equalisation and well productivity on ICD strength for channel ICDs in a prolific reservoir.

3.5

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5 0

0.5 0

200

400

600

800

1000

Measured depth (from the heel), m Specific PI

Specific Inflow with '1.6 bar' ICDs and no annular flow

Fig. 3. Prolific reservoir: inflow equalisation with ICDs.

0

2500

1

CoV ratio

4

Specific inflow rate, Sm3/day / m

Specific PI, Sm3/day / bar / m

4

0 0.007

Channel ICD strength, bar/(Rm3/day)2

1.2

3.5

Well flow rate, Sm3/day

2500

2000

0.8 1500 0.6 1000 0.4 500

0.2 0 0

2

4

6

8

10

12

14

Well flow rate, Sm3/day

The inflow distribution along the completion is expected to be highly uneven and uncertain due to complex reservoir geology. The local specific productivity index is anticipated to be within the range of 0.5–3.5 Sm 3/day/bar/m. Subject to the assumptions stated in Section 2, the inflow to the conventional completion will be proportional to the local specific productivity index. This implies a 7-fold variation of specific inflow rate for the above case. A completion combining ICDs and annular flow isolation will improve oil recovery by smoothing out the specific inflow rate variations and increasing oil sweep efficiency along the above horizontal well (Birchenko et al., 2008). The uniform distribution model (formula (16)) predicts that “1.6 bar” ICD completion with ΔPw of 1 bar will produce 1070 Sm3/day. That is, the “1.6 bar” ICD completion reduced well productivity by approximately 50% (for the target rate), but also delivered an improved degree of inflow equalisation (Fig. 3). The grey line in Fig. 3 was obtained using formula (9). The specific inflow rate variation is considerably smaller than for a conventional completion. Namely, the CoV U/CoV j ratio of 0.52 for the “1.6 bar” ICD case can be interpreted as almost a 50% reduction of the difference between regions of high and low specific inflow rate. As stated in Section 2, modelled reservoir and tubing pressures will only vary if the well is not strictly horizontal, although the difference between these two pressures is constant even for inclined wells. Annulus pressure may be calculated using formula (7). Fig. 4 illustrates the pressure distribution for the above scenario. An increase in the ICD strength gives an even more uniform inflow at the cost of further reduction of well inflow performance. This is

illustrated in Figs. 5 and 6 which were derived using formulae (16) and (17) for ΔPw = 1 bar.

CoV ratio

ΔPw ≈ΔPr = 0:5 bar:

Annulus

0 16

Effective nozzle diameter per 40 ft joint, mm CoV ratio

Well flow rate

Fig. 6. Dependence of inflow equalisation and well productivity on ICD strength for nozzle/orifice ICDs in a prolific reservoir.

V.M. Birchenko et al. / Journal of Petroleum Science and Engineering 78 (2011) 534–541

1

CoV ratio

200 Sm3/day/bar 0.05 Sm3/day/bar/m 0.35 Sm3/day/bar/m 10 bar

J j1 j2 ΔPw

2000

0.8 1500 0.6 1000 0.4 500

0.2

(Birchenko et al., 2008). In order to illustrate the importance of this parameter let us now consider the case with 10 times lower PI (200 Sm 3/day/bar) and 10 times higher total pressure drop (10 bar). Such modifications (Table 3) would not change the inflow performance of conventional completion as it is the product of the PI and the pressure drop that determines the inflow rate. However, the performance of an ICD completion will be different since the inflow is no longer proportional to the above mentioned product in this case. According to formulae (16) and (17), the flow rate of a “1.6 bar” ICD well is 1720 Sm 3/day and the CoV U/CoV j ratio is 0.85 for the medium productivity reservoir. This implies that the well flow rate and specific inflow rate variation were reduced by 15% only (in contrast to 50% in the Prolific Reservoir case). Figs. 7 and 8 illustrate that the ICD's efficiency of inflow equalisation generally decreases with decrease in reservoir permeability. Medium productivity reservoirs require installation of “stronger” ICDs than prolific reservoirs.

6. Discussion The question of precision of this model ultimately depends on the validity of assumptions made in Section 2. For instance, the formulae for the CoV U/CoV j ratio should be regarded as its lower (optimistic) estimate since they were obtained by neglecting annular flow. Annular flow can technically be completely eliminated by using a large number of isolation packers or a gravel-pack. However, in practice annular flow occurs to a greater or lesser extent in almost all wells. The implications of annular flow are very case specific; requiring help from a numerical simulator if they need to be studied (e.g. Neylon et al., 2009). This remark especially applies to fractured reservoirs where the characteristic length of reservoir heterogeneity (or width of the fracture) is considerably smaller than the length of the ICD joint. Adequate modelling of such cases is an extremely challenging task for both analytical and numerical methods.

1.2

CoV ratio

1

2000

0.8 1500 0.6 1000 0.4 500

0.2 0 0

0.001

0.002

0.003

0.004

0.005

0.006

Well flow rate, Sm3/day

2500

0 0.007

Channel ICD strength, bar/(Rm3/day)2 CoV ratio

Well flow rate

Fig. 7. Dependence of inflow equalisation and well productivity on ICD strength for channel ICDs in a medium productivity reservoir.

0 0

2

4

6

8

10

12

14

Well flow rate, Sm3/day

2500

1.2

Table 3 Medium productivity reservoir case study. Well Productivity Index (PI) Minimum value of specific PI Maximum value of specific PI Total pressure drop at the heel

539

0 16

Effective nozzle diameter per 40 ft joint, mm CoV ratio

Well flow rate

Fig. 8. Dependence of inflow equalisation and well productivity on ICD strength for nozzle/orifice ICDs in a medium productivity reservoir.

The neglect of frictional pressure losses is a valid assumption in most practical cases. The model presented in this paper is not applicable when both reservoir heterogeneity and friction have substantial impact on the inflow distribution. In such cases numerical simulation should be used for proper completion design. With numerical simulators at hand, some engineers may question the practical utility of the present work. However, it is recognised to be a good practice to employ a number of models of different complexity rather than one complex model when solving a difficult engineering problem (Williams et al., 2004). It should also be borne in mind that the experience with and the availability of numerical simulation varies from company to company. Relying solely on numerical simulation can be an obstacle in transferring best practices. Our analytical model is coarser than numerical models possible today. However, the analytical solution is simpler, more transparent and provides one with an insight into underlying physics in a more easily understood form. Analytical and numerical approaches complement one another. 7. Conclusions An explicit analytical model for ICD application to heterogeneous reservoirs has been proposed. Formulae (16) and (22) allow one to estimate the IPR of an ICD completion in a heterogeneous reservoir while formulae (17) and (23) quantify the ICD's equalisation effect. The presented model addresses the question of the trade-off between well productivity and inflow equalisation for ICD applications in heterogeneous reservoirs and allows one to estimate the ICD completion parameters suitable for such applications. The proposed model was used in a case study which quantitatively illustrated why medium permeability reservoirs require “stronger” ICDs to achieve a given degree of inflow equalisation than prolific reservoirs do. Acknowledgements V. M. Birchenko would like to thank sponsors of the “Added Value from Intelligent Field & Well system Technology” JIP at Heriot–Watt University, Edinburgh for the financial support. A. Iu. Bejan wishes to acknowledge the support of the James Watt Scholarship (Heriot–Watt University) and Overseas Research Students Awards Scheme (ORSAS). A. V. Usnich is supported by the Swiss National Science Foundation. We also wish to thank AGR Group for providing access to their software.

540

V.M. Birchenko et al. / Journal of Petroleum Science and Engineering 78 (2011) 534–541

Now the theorem will follow if we manage to prove that

Appendix A. Proof of the coefficient of variation ratio inequality Expression (9) represents the greatest root of the following quadratic equation: 2

aU + U = j−ΔP = 0:

ðA:1Þ

Since all three coefficients (a, 1/j and ΔP) of the above equation are strictly positive and finite, its greatest root is a positive real-valued random variable. pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi We use the transformation Y = U = ΔP , X = j ΔP to rewrite Eq. (A.1) as follows: 2

ðA:2Þ

aY + Y=X−1 = 0:

Since the applied transformation is linear, it preserves the coefficients of variations unchanged. Thus, CoV Y = CoV U and CoV X = CoV j. It follows that the inequality (8) is identical to the following one:



 ∑ I 2m + 1 I 2ðn−mÞ + 1

0≤m≤n

I2 ≤ðn + 1ÞI 2n + 2 I 21

where equality holds if and only if h represents a one-point mass distribution. The latter inequality, however, automatically follows from the following inequalities: I 2m + 1 I 2ðn−mÞ + 1 I 2 ≤ I 2n

2 + 2I 1; m

= 0; …; n:

In order to show Eq. (A.5) we use the Cauchy inequality

2 2 2 ∫f ð yÞg ð yÞdhð yÞ ≤∫f ð yÞ dhð yÞ∫g ð yÞ dhð yÞ where equality holds if and only if f is proportional to g almost everywhere (with respect to h and its support). In particular, the following inequality holds: 2

CoV Y b CoV X

ðA:3Þ

where X and Y are positive random variables linked by Eq. (A.2). We will, therefore, prove the inequality (A.3) in order to prove (8). We define a map y ↦ x as a continuous bijection x=

y 1−ay2

ðA:4Þ

from the interval function: 

 1 0; a−2 to (0, ∞). We also define the following 2

∞ y ∫ dhð yÞ 0 1−ay2 F ðaÞ =  2 ∞ y dh ð y Þ ∫ 0 1−ay2

ðA:5Þ

I k ≤ I k + 1 I k−1 :

ðA:6Þ

Since Y is a positive random variable and h is a proper probability density function (it integrates to unity) the case I 2 = 0 is not possible. Hence I 2 N 0, and Eq. (A.6) implies that I k N 0 for all k. In this case the Cauchy inequality (A.6) takes the form of equality only when y k + 1 is proportional to y k − 1 on the support of h. This implies that Y can take only one non-zero value almost surely, that is to say h is concentrated on at most one point outside zero. We can rewrite Eq. (A.6) as follows: I2 Ik I ≤…≤ ≤ k + 1: I1 I k−1 Ik

ðA:7Þ

In particular, I2 I ≤ 2n I1 I 2n

where h(y) is the probability density function of Y. It follows that:

+ 2

ðA:8Þ

+ 1

and

∞

∫ y2 dhð yÞ 2 F ð0Þ = 0∞ 2 = 1 + CoVðY Þ ∫ y dhð yÞ

I 2m + 1 I I I 2n + 1 ≤ 2m + 2 ≤ 2m + 3 ≤ … ≤ I1 I2 I3 I 2ðn−mÞ + 1

0

∞

∫ xð y; aÞ2 dhð yÞ 2 F ðaÞ = 0∞ 2 = 1 + CoVð X Þ ∫ xð y; aÞdhð yÞ

ðA:9Þ

so that

0

where x(y ; a) is a parametric function defined by Eq. (A.4). The following theorem proves that F(0) ≤ F(a) for any nonnegative real value of a, and hence that CoV(Y) ≤ CoV(X). We also prove that equality is possible if and only if Y (and hence X) has a degenerate distribution. Theorem 1. The function F satisfies the inequality F(a) ≥ F(0) where, except for the case when a = 0, equality holds if and only if h is a point mass distribution.

I 2m + 1 I 2n + 1 ≤ ; m = 0; …; n: I1 I 2ðn−mÞ + 1

ðA:10Þ

By multiplying the inequalities (A.8) and (A.10) we obtain the key inequality 2

I 2m + 1 I 2ðn−mÞ + 1 I 2 ≤ I 2n + 2 I 1 which holds for any m = 0, …, n. Thus the inequalities (A.5) are proven and the proof is complete.

∞

Proof. We use the following notation I k = ∫ yk dhð yÞ and develop 0 the nominator and denominator of F(a) as series in a exchanging sums and integrals since all integrals are absolutely convergent: y n = ∑ a I 2n + 1 1−ay2 n≥0  2 y n ∫ = ∑ ðn + 1Þa I 2ðn + 1Þ 1−ay2 n≥0  2 y n ∫ = ∑ a ∑ I 2m + 1 I 2ðn−mÞ + 1 : 2 1−ay n≥0 0≤m≤n ∫

Nomenclature Fluid volumes are quoted at standard conditions (S) while fluid viscosity and density are in downhole (R) conditions. B Cd Cu IU( j) IUj( j)

formation volume factor discharge coefficient for nozzle or orifice unit conversion factor: 8/π 2 in SI units, 1.0858 ⋅ 10 − 15 in metric units, 7.3668 ⋅ 10 − 13 in field units an auxiliary function, see Eq. (19) an auxiliary function, see Eq. (25)

V.M. Birchenko et al. / Journal of Petroleum Science and Engineering 78 (2011) 534–541

J P Pa Pe(l) SU( j) SUj( j) U ΔP ΔPr ΔPw arcsinh η( j) 〈〉 μ μ cal ρ ρcal aICD d j1 j2 jm l lICD q qw ICD ICV IPR p.d.f. PI TVD

well's productivity index tubing (base pipe) pressure annulus pressure reservoir pressure at the external boundary at the same TVD as point l of the wellbore an auxiliary function, see Eq. (20) an auxiliary function, see Eq. (26) inflow per unit length of completion pressure difference between the external reservoir boundary and the tubing (base pipe). pressure difference between the external reservoir boundary and the annulus drawdown i.e. pressure difference between external reservoir boundary and base pipe (tubing) inverse hyperbolic sine probability density function of the specific productivity index angled brackets are used to denote average values of variables viscosity of produced or injected fluid viscosity of calibration fluid (water) density of produced or injected fluid density of calibration fluid (water) channel ICD strength (Table 1) effective diameter of nozzles or orifices in the ICD joint of length lICD minimum value of specific productivity index maximum value of specific productivity index the mode (peak) of the triangular p.d.f. distance between particular wellbore point and the toe length of the ICD joint (typically 12 m or 40 ft) flow rate in the tubing well flow rate q(L) Inflow Control Device Interval Control Valve Inflow Performance Relationship Probability density function well Productivity Index True Vertical Depth

541

References Augustine, J., Mathis, S., Nguyen, H., Gann, C., Gill, J., 2008. World's first gravel-packed inflow-control completion. SPE Drill. Complet. 23 (1), 61–67 URL http://dx.doi.org/ 10.2118/103195-PA. Birchenko, V., Al-Khelaiwi, F., Konopczynski, M., Davies, D., 2008. Advanced wells: how to make a choice between passive and active inflow-control completions. SPE Annual Technical Conference and Exhibition. URL http://dx.doi.org/10.2118/ 115742-MS. Birchenko, V., Muradov, K., Davies, D., 2010a. Reduction of the horizontal well's heel-toe effect with inflow control devices. J. Pet. Sci. Eng. 75 (1–2), 244–250 URL http://dx. doi.org/10.1016/j.petrol.2010.11.013. Birchenko, V., Usnich, A., Davies, D., 2010b. Impact of frictional pressure losses along the completion on well performance. J. Pet. Sci. Eng. 73 (3–4), 204–213 URL http://dx. doi.org/10.1016/j.petrol.2010.05.019. Dikken, B.J., 1990. Pressure drop in horizontal wells and its effect on production performance. J. Pet. Technol. 42 (11), 1426–1433 November. URL http://dx.doi.org/ 10.2118/19824-PA. Gai, H., 2002. A method to assess the value of intelligent wells. SPE Asia Pacific Oil and Gas Conference and Exhibition. URL http://dx.doi.org/10.2118/77941-MS. Helmy, M., Veselka, A., Benish, T., Yeh, C., Asmann, M., Yeager, D., Martin, B., Barry, M., 2006. Application of new technology in the completion of ERD wells, Sakhalin-1 development. SPE Russian Oil and Gas Technical Conference and Exhibition. URL http://dx.doi.org/10.2118/103587-MS. Henriksen, K., Gule, E., Augustine, J., 2006. Case study: the application of inflow control devices in the troll oil field. SPE Europec/EAGE Annual Conference and Exhibition. URL http://dx.doi.org/10.2118/100308-MS. Johansen, T.E., Khoriakov, V., 2007. Iterative techniques in modeling of multi-phase flow in advanced wells and the near well region. J. Pet. Sci. Eng. 58 (1–2), 49–67 URL http://dx.doi.org/10.1016/j.petrol.2006.11.013. McIntyre, A., Adam, R., Augustine, J., Laidlaw, D., 2006. Increasing oil recovery by preventing early water and gas breakthrough in a West Brae horizontal well: A case history. SPE/DOE Symposium on Improved Oil Recovery. URL http://dx.doi.org/10. 2118/99718-MS. Neylon, K., Reiso, E., Holmes, J., Nesse, O., 2009. Modeling well inflow control with flow in both annulus and tubing. SPE Reservoir Simulation Symposium. URL http://dx. doi.org/10.2118/118909-MS. Ouyang, L.-B., Huang, B., 2005. An evaluation of well completion impacts on the performance of horizontal and multilateral wells. SPE Annual Technical Conference and Exhibition. URL http://dx.doi.org/10.2118/96530-MS. Wan, J., Dale, B.A., Ellison, T.K., Benish, T.G., Grubert, M.A., 2008. Coupled well and reservoir simulation models to optimize completion design and operations for subsurface control. Europec/EAGE Conference and Exhibition. URL http://dx.doi. org/10.2118/113635-MS. Williams, G., Mansfield, M., MacDonald, D., Bush, M., 2004. Top-down reservoir modelling. SPE Annual Technical Conference and Exhibition. URL http://dx.doi.org/ 10.2118/89974-MS.