˝umerical simulatio˝ a˝d i˝terpretatio˝ of … · considered in this paper estimates formation...

SPWLA 52nd

Annual Logging Symposium, May 14-18, 2011

1

UMERICAL SIMULATIO AD ITERPRETATIO OF

PRODUCTIO LOGGIG MEASUREMETS USIG A EW COUPLED

WELLBORE-RESERVOIR MODEL

Amir Frooqnia, Rohollah A-Pour, Carlos Torres-Verdín, and Kamy Sepehrnoori,

The University of Texas at Austin

Copyright 2011, held jointly by the Society of Petrophysicists and Well Log Analysts (SPWLA) and the submitting authors.

This paper was prepared for presentation at the SPWLA 52nd Annual Logging Symposium held in Colorado Springs, CO, USA, 14-18 May, 2011.

ABSTRACT

Measurements of wellbore fluid dynamics such as fluid velocity and pressure are widely used to monitor downhole

production. There is a wealth of information in these measurements about static and dynamic petrophysical

properties of producing rock formations. We introduce a new method to interpret dynamic petrophysical properties

of rock formations from measurements acquired with production logging tools (PLT). The specific application

considered in this paper estimates formation permeability from production logging (PL) measurements. To that end,

we develop a coupled wellbore-reservoir model to simultaneously simulate the physics of fluid flow both in the

borehole and in permeable formations which are in hydraulic communication with the borehole.

Our interpretation method uses the concept of computational fluid dynamics to simulate fluid flow in the wellbore.

Even though the developed wellbore model is capable of simulating two-phase flow systems, in this paper, we

assume single-phase, Newtonian, and incompressible fluid flow through the borehole and solve one- and two-

dimensional versions of the Navier-Stokes equations in cylindrical coordinates. Subsequently, we interface the

borehole fluid flow model with a reservoir flow model and use the resulting coupled model to simulate PL

measurements. Permeability estimation is performed by minimizing the difference between measurements of fluid

pressure and velocity and their corresponding numerical simulations.

Synthetic cases are used to appraise the accuracy and reliability of the permeability estimation method. We find that

the accuracy of the estimation decreases in the presence of thin layering. Additionally, it is shown that unaccounted

two-phase fluid flow in the borehole yields estimates of permeability closed to those of relative permeability to the

dominant fluid phase. Finally, testing of the estimation method on field data acquired in the deepwater Gulf of

Mexico, yields formation permeabilities that are in agreement with well-log derived permeabilities. Differences

between well-log derived permeabilities and those estimated from PL measurements arise because of the differences

in the volume of investigation. It is also shown that our estimation method correctly predicts cross-flow taking place

during a shut-in well test.

ITRODUCTIO

Estimation of permeability is one of the major technical challenges in formation evaluation. Permeability can be

estimated from various methods such as core analysis, formation testing, and well-log interpretation. These methods

are reliable for calculations/estimations before the completion of the wellbore. However, they are not reliable after

casing or perforating the borehole. Well-testing analysis is the only practical method available to estimate formation

permeability after completing a wellbore. Even though well testing provides valuable information, it has limited

degrees of freedom to reliably estimate complex spatial distributions of permeability in the vicinity of the wellbore.

Additionally, the relatively large volume of investigation of well-testing measurements considerably limits their

vertical resolution, hence their application in the construction of detailed reservoir models. A pertinent example of

this limitation arises in the estimation of post-fracture permeability following hydro-fracturing operations, where

well-testing measurements do not have the spatial resolution needed to accurately appraise enhancements in

formation permeability.

SPWLA 52nd


2

Measurements acquired with PLT provide a unique opportunity to estimate petrophysical properties of rock

formations even after years of production. Production logging instruments measure wellbore fluid properties such as

fluid velocity, pressure, temperature, and in case of multi-phase flow, fluid hold-up. The main application of these

measurements is to assess borehole integrity and to diagnose adverse wellbore conditions during production. Eissa et

al. (2010) described contemporary applications of PL measurements and showed that PL could be used to quantify

cross-flow between layers in multi-layer formations. In Eissa et al.’s (2010) work, wellbore pressure data were used

to estimate downhole fluid density and to detect depth segments where water flowed into the wellbore. Sullivan et

al. (2007) described a method to estimate apparent permeability of multi-layer formations. The formation consisted

of three main production layers behaving as three different pressure compartments. They implemented Darcy’s

equation to separately estimate the apparent permeability of each layer. Estimated permeabilities were used to

improve the reservoir geological model, resulting in accurately simulated pressure responses during inter-well pulse

testing (Sullivan 2006). Rey et al. (2009) advanced a method to estimate permeability distributions based on PL

measurements which assumed steady-state flow and neglected borehole friction losses. They related the vertical

derivative of fluid velocity to fluid inflow profile.

The majority of the abovementioned contributions attempted to qualitatively interpret PL measurements without the

need of numerical simulations for quantitative verification of results. Interpretation of PL measurements requires

simulating the physics of fluid flow in the wellbore. A plethora of documented technical contributions exist

concerning the simulation and interpretation of single- and multi-phase fluid flow in pipes. Among those

contributions, Bendiksen et al. (1991) developed a so-called dynamic two-fluid model to simulate a wide range of

two-phase oil and gas flow conditions in pipelines. The two-fluid flow model accurately predicted steady-state fluid

pressure, temperature, hold-up, and flow-regime transitions for different flow patterns. Bonizzi et al. (2009)

introduced a model to simulate two-phase flow in horizontal and near-horizontal wellbores. Their model

successfully predicted the occurrence of different flow regimes without using transition maps or changing closure

relationships. Lahey and Drew (2001) formulated a complete set of volume-averaged conservation equations to

simulate complex multi-phase flow systems. Lahey (2005), Lahey (2007), Lahey (2009), Yeoh and Tu (2010),

Kolev (2007), Prosperetti and Tryggvason (2007), Ishii and Hibiki (2006), and Kleinstreuer et al. (2003) introduced

various expressions for the conservation equations used to describe multi-phase flow systems.

In the context of the petroleum industry, a great deal of research has been conducted to study fluid flow in wellbores.

Ouyang et al. (1999) formulated a homogeneous model to simulate bubbly flow in pipes. They showed that the

inflow or outflow of fluid through perforations imposed an extra pressure drop in the borehole. Shirdel and

Sepehrnoori (2009) developed a semi-steady-state, thermal, coupled, and compositional wellbore-reservoir model to

simulate fluid velocity and pressure drop along horizontal wells. They extended their work to advance a transient

two-fluid model for simulating fluid velocity, pressure drop, and hold-up during the transient-time cycle of

production wells (Shirdel and Sepehrnoori 2011).

Accurate and reliable numerical simulation of PL measurements provides untapped opportunities to estimate

relevant static and dynamic petrophysical properties of rock formations. In this paper, we introduce a model to

simulate measurements of PLT the honor the dynamic fluid-flow interactions between the wellbore and permeable,

producing rock formations. The wellbore fluid flow model is coupled to a near-wellbore reservoir model developed

by the Formation Evaluation Group of the University of Texas at Austin. Based on the model introduced to

numerically simulate PL measurements, we developed a new quantitative interpretation method to estimate multi-

layer formation permeability from cased-hole production logs. This is accomplished with nonlinear inversion, by

minimizing the difference between measured and numerically simulated production logs of fluid velocity and

pressure. Validation of the estimation method is performed with synthetic and field measurements. In what follows,

we introduce the modeling method of single-phase fluid flow in the wellbore. Subsequently, we describe the

wellbore-reservoir coupling strategy, and the inversion procedure to estimate permeability from production logs.

Thereafter, two synthetic cases and one field example with data acquired in the deepwater Gulf of Mexico are used

to appraise the performance of the new permeability estimation method. Finally we summarize the relevance of our

work and discuss its advantages and limitations for application to field measurements.

SPWLA 52nd


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0 0.5 1

2

4

6

8

10

12

14

Radius [in]

Dep

th [

ft]

-12

-10

-8

-6

-4

-2

0x 10

(a) Radial velocity [ft/sec] (b) Vertical velocity [ft/sec]

0x 10

-3

MD

[ft

]

MD

[ft

]

0 0.5 1

2

4

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Radius [in]

0.5

1

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Radius [in]

Dep

th [

ft]

-12

-10

-8

-6

-4

-2

0x 10

(a) Radial velocity [ft/sec] (b) Vertical velocity [ft/sec]

0x 10

-3

MD

[ft

]

MD

[ft

]

0 0.5 1

2

4

6

8

10

12

14

Radius [in]

0.5

1

1.5

2

2.5

Figure 1: Synthetic Case No. 1: 2D simulation of fluid velocity in the wellbore, distributions of (a) radial fluid velocity and (b) vertical fluid velocity.

Water saturation(a) (b)

200 400 600 800

0

20

40

60

80

100

120

140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

MD

[ft

]

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.25

0.5

0.75

1

Sw

[fraction]

Kr [

fra

cti

on

]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

5

10

15

20

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30

35

40

Pc [

psi]

Krw

Kro

Pc

Water saturation(a) (b)

200 400 600 800

0

20

40

60

80

100

120

140

0.1

0.2

0.3

0.4

0.5

0.6

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MD

[ft

]

0.3

0.4

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0.9

1.00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.25

0.5

0.75

1

Sw

[fraction]

Kr [

fra

cti

on

]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

5

10

15

20

25

30

35

40

Pc [

psi]

Krw

Kro

Pc

Figure 2: Synthetic Case No. 2: (a) vertical distribution of formation water saturation. (b) Relative permeability and capillary pressure curves vs. water saturation. Green lines identify the average value of water saturation in different layers.

METHOD

Production logs are acquired in the wellbore where the physics of fluid flow is different than that of porous and

permeable media. Inside the wellbore, the available flowing area decreases whereby fluid velocity drastically

increases, when compared to fluid flow in porous and permeable rock formations. To simulate fluid flow in

wellbores, we develop 1D and 2D models directly from Navier-Stokes equations. The model assumes single-phase,

incompressible, and isothermal fluid flow through a rough pipe that has a porous wall. The fundamental formulation

for this class of fluid flow is given by

0 ,V∇⋅ =r

(1)

( ) ( ) ,ext

V VV P g Ftρ ρ τ ρ

∂+∇⋅ = −∇ −∇⋅ + +

∂

r r r rr

(2)

( ) ,TV Vτ µ= − ∇ +∇r r

(3)

where Vr

is fluid velocity, P is fluid pressure, gr

is gravitational force, ρ is fluid density, µ is fluid viscosity, τ is shear

stress tensor,

extFr

is any external force per unit volume of the fluid (e.g. wall frictions), and I is the identity matrix.

The above equations do not have analytical solutions, and therefore, their solution needs to be approached with

specialized numerical methods. To that end, we develop a numerical solution based on the Finite Volume Method

(FVM). As the first step, 2D axial-symmetric versions of the above equations are derived in cylindrical coordinates.

Subsequently, volume integrals of the equations are taken over a finite control volume to transform them from their

original differential form to an integral form. With the application of the divergence theorem, the volume integral of

convective terms are changed to surface integrals, and the resulting equations are discretized on staggered grids. The

staggered gridding method defines all vector properties (e.g. velocity) at the faces of a control volume while all

scalar properties (e.g. pressure) are defined at the center of the control volume. This approach improves the stability

of the solution by enforcing the coupling between pressure and velocity fields. Finally, we solve the discretized

equations with a numerical method referred to as “SIMPLE-C”1 algorithm. This method uses the continuity equation

(Eq. 1) to treat the pressure equation, and iteratively solves the pressure equation together with the momentum

equation (Eq. 2). Versteeg and Malalasekara (1995) give additional details about the FVM-based SIMPLE-C

algorithm.

1 SIMPLE-C: Semi-Implicit Method for Pressure Linked Equations-Consistent

SPWLA 52nd


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0

2.5

5

7.5

10

12.5

15

MD

[ft

]

8900 8920 8940

Sand-shale model0 0.5 1 1.5

0

2.5

5

7.5

10

12.5

15

V [ft/sec]

Measured velocity

Simulated velocity

0 200 400 600 800

Permeability [mD]

Kact

KPL

0 200 400 600 800

0

2.5

5

7.5

10

12.5

15

Permeability [mD]

8804 8806 8808 8810

0

2.5

5

7.5

10

12.5

15

P [psi]

Measured pressure

Simulated pressure

(a) (b) (c) (d) (e)

Fluid velocity [ft/sec] Fluid pressure [psi]

0

2.5

5

7.5

10

12.5

15

MD

[ft

]

8900 8920 8940

Sand-shale model0 0.5 1 1.5

0

2.5

5

7.5

10

12.5

15

V [ft/sec]

Measured velocity

Simulated velocity

0 200 400 600 800

Permeability [mD]

Kact

KPL

0 200 400 600 800

0

2.5

5

7.5

10

12.5

15

Permeability [mD]

8804 8806 8808 8810

0

2.5

5

7.5

10

12.5

15

P [psi]

Measured pressure

Simulated pressure

(a) (b) (c) (d) (e)

Fluid velocity [ft/sec] Fluid pressure [psi]

Figure 3 : Synthetic Case No. 1: (a) Sand-shale model. (b) Numerically simulated fluid velocity (blue curve) and synthetic measurements of fluid velocity (red curve). (c) Numerically simulated fluid pressure (blue curve) and synthetic measurements of fluid pressure (red curve). (d) Estimated permeabilities from PL measurements (red line) along with actual permeabilities (blue line). (e) Estimated permeabilities with error bars.

Accurate and efficient simulation of wellbore fluid pressure and velocity is key to interpret PLT measurements.

However, relating these measurements to formation petrophysical properties requires coupling the physics of fluid

flow in the wellbore to that of porous and permeable media. The next section briefly discusses the method adopted

in this paper for efficiently coupling the wellbore and reservoir models.

Coupling Strategy In the developed algorithm, coupling between the wellbore and the formation takes place at the interface of the two

domains (i.e., perforations). At each time step, boundary conditions at the wellbore-side of perforations are updated

according to the solution of reservoir simulation obtained at the previous time step. Using the updated boundary

conditions, the algorithm finds the solution in the wellbore domain (i.e. wellbore fluid velocity and pressure) and

subsequently updates boundary conditions at the reservoir-side of perforations. By successive updating the boundary

conditions, the algorithm partially decouples the two domains and solves the problem separately. Furthermore, The

character of fluid flow is different in the two domains, and therefore, the required time steps are different and should

be determined appropriately. Normally, fluid velocity is much higher in the wellbore domain and consequently, time

steps should be shorter in the wellbore than in the reservoir domain.

An additional capability of the coupled simulation method is that allows the simulation of cross-flow between

reservoir compartments that exhibit different values of average pressure. Depending on both wellbore fluid pressure

and compartment pressure, fluid may preferentially flow from one compartment to another.

Minimization Procedure

The objective is to estimate the permeability vector that minimizes the difference between production measurements

of velocity and pressure and their numerical simulations. To that end, we minimize the quadratic cost function C

given by

( ) ( )( ) ( )( ) ( ) ( )2 2 2 2

2 2 1 22, , ... ,nC x d l G x d l e x d G x d e e e= − = = − = + + +

(4)

where G designates the simulation of PL measurements which combines Darcy’s law in porous media and Navier-

Stokes equations in the wellbore, x is formation permeability vector, d is the vector of production measurements, e is

the data residual vector defined as the difference between actual and simulated measurements. We minimize the cost

function iteratively using the Gauss-Newton gradient-based approach. At each iteration, the algorithm solves the

linear system of equations

SPWLA 52nd


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MD

[ft

]

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140

Pressure [psi]

MD

[ft

]

Measured pressure

Simulated pressure

0 0.2 0.4 0.6 0.8

20

40

60

80

100

120

Oil hold-up0 0.5 1 1.5 2

V [ft/sec]

Measured velocity

Simulated velocity

0.2 0.4 0.6

Oil hold-up [fraction]

(a) (b) (c) (d) (e)

0 100 200 300 400

Permeability [mD]0 200 400 600 800 1000

0

20

40

60

80

100

120

140

Estimated permeabilities

Permeability [mD]

Kact

Kr

KPL

Mixture velocity [ft/sec] Mixture pressure [psi]

0

0

20

40

60

80

100

120

140

MD

[ft

]

8880 8900 8920 8940

0

20

40

60

80

100

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140

Pressure [psi]

MD

[ft

]

Measured pressure

Simulated pressure

0 0.2 0.4 0.6 0.8

20

40

60

80

100

120

Oil hold-up0 0.5 1 1.5 2

V [ft/sec]

Measured velocity

Simulated velocity

0.2 0.4 0.6

Oil hold-up [fraction]

(a) (b) (c) (d) (e)

0 100 200 300 400

Permeability [mD]0 200 400 600 800 1000

0

20

40

60

80

100

120

140

Estimated permeabilities

Permeability [mD]

Kact

Kr

KPL

Mixture velocity [ft/sec] Mixture pressure [psi]

Figure 4 : Synthetic Case No. 2: (a) Numerically simulated oil hold-up. (b) Numerically simulated mixture velocity (blue curve) and synthetic measurements of mixture velocity (red curve). (c) Numerically simulated mixture pressure (blue curve) and synthetic measurements of mixture pressure (red curve). (d) Estimated permeabilities from PL measurements (red line) along with actual permeabilities (blue line) and relative permeabilities of the dominant fluid phase (green line). (e) Estimated permeabilities and corresponding error bars.

( ) ( )( ) ( ) ( )12

, , , , ,kT T

k k k kJ x d J x d I x J x d e x dα

+

+ ∆ = −

(5)

where xk is the solution at k

th iteration,α is a regularization (stabilization) parameter which is determined via

Hansen’s L-curve method (Aster et al. 2005), I is the identity matrix, and J is the Jacobian matrix containing the

first-order derivatives of the cost function, given by

( )( ),

,j ik

ij

j

C x dJ x d

x

∂= ⋅

∂

(6)

SYTHETIC CASES

The interpretation method is applied to two synthetic cases. The first case is a multi-layer formation which consists

of layers from 1.5 in to 4 ft, producing single-phase fluid at isothermal conditions. Figure 3a describes the Synthetic

Case No. 1. All layers share the same petrophysical properties except that the thickness of layers involved is

different. Blue-colored layers are assumed to be shale. The objective is to investigate the performance of the method

to estimate layer permeabilities in the presence of noise. To construct synthetic PLT measurements, we perform 2D

simulations of wellbore fluid velocity and pressure, and subsequently calculate the average fluid velocity along the

wellbore. Calculated vertical distributions of fluid velocity and pressure are contaminated with 5% zero-mean

additive Gaussian noise to yield the measurements input to the estimation. Figure 1 shows the 2D fluid velocity

distribution in the vertical and radial directions. Figures 3b and 3c compare input measurements with their

corresponding numerical simulations performed after completing the inversion of permeabilities. The plots indicate

that the inversion method reproduces the input measurements with an error lower than 1%. Figure 3d describes the

permeability values estimated from production logs and superimposed with actual values of permeability. The

algorithm successfully estimates the permeability of thick layers. However, significant errors arise in the estimation

of thin-layer permeability. Figure 3e shows that error bars of estimated permeabilities widen with a decrease of

layer thickness. The second synthetic case considers an isothermal two-phase fluid flow system. A multi-layer

formation simultaneously produces oil and water into the wellbore; therefore, two immiscible fluids coexist in the

wellbore during the PL test. The synthetic formation consists of 7 fluid-producing layers interbedded with shale

layers. Figure 2a shows the vertical distribution of water saturation in the formation. The water-oil-contact is

located 100 ft below the formation’s lower depth bound (i.e. MD=140 ft). Oil and water densities are assumed equal

to 0.8 and 1 g/cm3 respectively. At this condition, water saturation ranges from 74% to 23% from the bottom to the

SPWLA 52nd


6

MD

ft

x00

x10

x20

x30

x40

x50

x60

x70

x80

x90

[API/API]

Normalized

Gamma Ray

0---------------------1

[Ω.m/Ω.m]

Normalized

Resistivity

0.01--------------------1

[Ω.m/Ω.m]0.01--------------------1

Normalized

Density

0.8---------------------1

g/cm3

g/cm3[ ] [pu/pu]

Normalized

Neutron

1---------------------0.4[pu/pu]

Normalized

Porosity

0-----------------------1[su/su]

Normalized

Water

Saturation

0-----------------------1[mD/mD]

Normalized

Permeability

0.0001----------------1

MD

ft

x00

x10

x20

x30

x40

x50

x60

x70

x80

x90

[API/API]

Normalized

Gamma Ray

0---------------------1

[Ω.m/Ω.m]

Normalized

Resistivity

0.01--------------------1

[Ω.m/Ω.m]0.01--------------------1

Normalized

Density

0.8---------------------1

g/cm3

g/cm3[ ] [pu/pu]

Normalized

Neutron

1---------------------0.4[pu/pu]

Normalized

Porosity

0-----------------------1[su/su]

Normalized

Water

Saturation

0-----------------------1[mD/mD]

Normalized

Permeability

0.0001----------------1

Figure 5: Field Example: Normalized well logs for (from left to right) Gamma-Ray, shallow (red curve) and deep (blue curve) resistivity, bulk density, and neutron porosity. Tracks 6, 7 and 8 show porosity, saturation, and permeability calculated from well-log interpretation, respectively.

top of the formation. All layers belong to the same rock type, hence exhibit the same capillary pressure and relative

permeability curves shown in Figure 2b. All other petrophysical properties are uniform across the layers. The

objective is to examine the effect of unaccounted presence of a second fluid phase on the interpretation of

production logs. We use the 1D coupled two-phase flow model to synthesize PLT measurements. Distributions of

wellbore oil and water velocity, hold-up, and mixture pressure should be simulated to describe the two-phase

system. To estimate permeability, we use the 1D coupled single-phase flow model. Results of two-phase simulation

are averaged to calculate a mixture velocity across the wellbore. Mixture velocity and pressure are used as synthetic

measurements to test the inversion method. The following formula is used to calculate wellbore mixture velocity:

,o o o w w wmix

o o w w

AV AVV

A A

ρ α ρ α

ρ α ρ α

+=

+

(7)

where Vmix is mixture velocity, A is wellbore radius, αo and αw are oil and water hold-ups, respectively, ρo and’ ρw are

oil and water densities, respectively, and Vo and Vw are oil and water velocities, respectively. Figures 4b and 4c

describe the mixture velocity and pressure after the addition of 5%, zero-mean Gaussian random noise (red curves).

Figure 4a also describes simulation results for oil hold-up across the wellbore. As expected, oil hold-up increases

upward because water saturation decreases upward, whereby oil influx into the wellbore increases. Despite the

presence of noise in the synthetic measurements, there is a good match between velocity log and the numerically

simulated velocity. However, comparison of simulated and measured fluid pressure shows that the slope of

simulated pressure is different from the original value. This discrepancy is explained by the fact that the simulation

was performed under the assumption of single-phase flow (i.e. oil with density of 0.8 g/cm3), but as emphasized

SPWLA 52nd


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x00

x10

x20

x30

x40

x50

x60

x70

x80

x90

Normalized

Spinner Velocity

-0.2--- ----1

MD

ft

Normalized

Fluid Density

0----- -----1

Fluid

Temperature

64----[deg F]----72

Fluid Pressure

3100---[psi]---3400

Normalized

Wellbore Radius

-1-------[in/in]------1

Normalized

Fluid Hold-up

0----[fraction]----1cm/s

cm/s[ ]

g/cm3

g/cm3[ ]

x00

x10

x20

x30

x40

x50

x60

x70

x80

x90

Normalized

Spinner Velocity

-0.2--- ----1

MD

ft

Normalized

Fluid Density

0----- -----1

Fluid

Temperature

64----[deg F]----72

Fluid Pressure

3100---[psi]---3400

Normalized

Wellbore Radius

-1-------[in/in]------1

Normalized

Fluid Hold-up

0----[fraction]----1cm/s

cm/s[ ]

g/cm3

g/cm3[ ]

Figure 6: Field Example: Production logging measurements of (from left to right) normalized spinner average velocity, normalized wellbore fluid density, wellbore fluid temperature, wellbore fluid pressure, normalized wellbore radius, and normalized wellbore fluid hold-up. Each log was acquired with different tool speeds.

earlier, there were two immiscible fluids involved during the acquisition of PL measurements. Figure 4d describes

the estimated layer permeabilities (red curves). Clearly, estimated permeability values differ from original layer

absolute permeabilities. Close inspection of the inversion results indicates that estimated layer permeabilities are

instead close to corresponding values of relative permeability to the dominant fluid phase (green curves). It should

be noted that only oil was included in the simulation and interpretation of PLT measurements; consequently,

interactions of oil and water in both the wellbore and the formation were unheeded by the inversion method. Figure

4e describes the estimated permeabilities with corresponding error (uncertainty) bars. Results indicate that presence

of noise in the measurements did not appreciably affect the estimated permeabilities.

FIELD EXAMPLE

Data considered in this example were acquired in a Gulf of Mexico deepwater turbidite system. The objective of this

exercise is to use production and well logs to estimate layer-by-layer permeability along the wellbore and compare

the results to those obtained from interpretation of well logs. It should be noted that well logging was performed

SPWLA 52nd


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γγγγ-Ray

7000 8000 9000 10000 11000 12000

Pres [26 zones]

Pres [6 zones]

Pwell

Velocity gradient

Velocity profile

(a) (b) (c) (d)

Normalized

G-Ray [API/API]

0 1

Sand-shale modelNormalized velocity

and its gradient [fraction]

-1 -0.5 0 0.5 1

MD

[ft

]

x10

x20

x30

x40

x50

x60

x70

x80

x90

Wellbore and formation fluid pressure [psi]

2000 3000 4000 5000 6000 7000 γγγγ-Ray

7000 8000 9000 10000 11000 12000

Pres [26 zones]

Pres [6 zones]

Pwell

Velocity gradient

Velocity profile

(a) (b) (c) (d)

Normalized

G-Ray [API/API]

0 1

Sand-shale modelNormalized velocity

and its gradient [fraction]

-1 -0.5 0 0.5 1

MD

[ft

]

x10

x20

x30

x40

x50

x60

x70

x80

x90

Wellbore and formation fluid pressure [psi]

2000 3000 4000 5000 6000 7000

Figure 7: Field Example: (a) Gamma-Ray log. (b) Normalized distribution of fluid velocity (red curve) and its gradient (blue curve). (c) Multi-layer reservoir model constructed from the vertical distribution of fluid velocity. Blue and red depth zones identify shale and sand layers, respectively. (d) Estimated initial reservoir pressure based on PLT measurements for four different production rates; wellbore pressure (green curve) and formation pressure with 6 (red curve) and 26 (blue curve) pressure zones.

before completion of the wellbore while PL measurements were acquired after one year of production.

Well Logs

Figure 5 describes some of well logs acquired in the field. The normalized Gamma-Ray log (Track 1) indicates sand

layers interbedded with shale layers. Normalized deep and shallow resistivity logs (Track 2) show relatively high

resistivities across sand layers, thereby indicating presence of hydrocarbon. Interpreted (normalized) porosity,

saturation, and permeability logs are shown in Tracks 6 through 8. Well logs are used to initialize the reservoir

model for the subsequent simulation of PLT measurements.

Production Logs

Figure 6 shows PLT measurements. Normalized spinner velocity (Track 1) is used to calculate fluid velocity.

Remaining tracks describe measurements of wellbore fluid density, pressure, temperature, and hold-up as well as

wellbore radius. Note that fluid density, hold-up, and wellbore radius have been normalized, and that wellbore fluid

pressure and temperature have been shifted for their display in Figure 6. The purpose is to honor as much

information as possible from PLT measurements to construct a consistent wellbore-reservoir model. Details about

the initialization of reservoir pressure, detection of bed boundaries, and the estimation of layer-by-layer permeability

are given below.

Simulation of PLT Measurements

The first observation in Figure 6 is that water hold-up is almost zero. Furthermore, the temperature log suggests an

almost constant temperature across the production interval. Therefore, it is pertinent to assume a single-phase

isothermal flow condition throughout the model. Estimating permeability hence requires simulation of wellbore

fluid velocity and pressure. To construct a consistent coupled model, remaining properties such as formation

porosity, initial pressure, drainage radius, and fluid density and viscosity should be entered as known parameters.

SPWLA 52nd


9

0 1 2 3 4 5 6

x816-x742

x742-x718

x718-x669

x669-x598

x484-x455

x455-x400

Well

bo

re p

ressu

re [

psi]

1000

6000

5000

4000

3000

2000

7000

8000 x93.2-x78.4

x78.4-x73.6

x73.6-x63.8

x63.8-x49.6

x26.8-x21.0

x21.0-x10.0

0 0.08 0.16 0.24 0.32 0.4 0.48

Normalized incremental mass influx [fraction]

0 1 2 3 4 5 6

x816-x742

x742-x718

x718-x669

x669-x598

x484-x455

x455-x400

Well

bo

re p

ressu

re [

psi]

1000

6000

5000

4000

3000

2000

7000

8000 x93.2-x78.4

x78.4-x73.6

x73.6-x63.8

x63.8-x49.6

x26.8-x21.0

x21.0-x10.0

0 0.08 0.16 0.24 0.32 0.4 0.48

Normalized incremental mass influx [fraction]

Figure 8: Estimation of initial reservoir pressure based on PLT measurements for four different production rates. Wellbore pressure is plotted vs. incremental mass influx for different production intervals.

Basic assumptions: Simulating PLT measurements requires reliable assumptions to simplify the numerical

procedure. First, we tie the model to porosity yielded by well-log analysis. PLT measurements also provide

information about fluid density, which are input to both reservoir and wellbore fluid models. Fluid viscosity is

chosen based on existing PVT analysis. Initial reservoir pressure and bed boundaries are obtained from fluid

velocity and pressure logs. In addition, PL is a transient test; therefore, it is necessary to select appropriate initial

conditions and test duration in order to correctly simulate pressure and velocity distributions. The model assumes

axial-symmetry about the borehole axis; therefore, rotational flow is neglected in both the wellbore and the

formation. Furthermore, we assume single-phase, incompressible, and isothermal fluid flow conditions. In cases of

turbulent flow, an extra pressure drop is imposed to account for friction losses. At each depth interval, friction losses

are adjusted in such a way that the numerically simulated pressure matches exactly with measured pressure.

Moreover, all variations in the velocity log are assumed to originate from physical phenomena included in the simul-

ation model. This is tantamount to assuming that the velocity log is noise-free in the estimation of permeability.

Populating bed boundaries: The formation consists of sheet sands interbedded with shale layers; therefore, we

assume that a multi-layer formation with alternation of sand and shale layers is a pertinent model to simulate fluid

flow behavior during PL tests. Figure 7b shows the fluid velocity log and its depth gradient. Wellbore fluid velocity

changes across permeable media. Whenever fluid flows toward the wellbore the velocity increases, whereas the

velocity decreases when fluid flows toward porous media; therefore, the depth derivate of velocity is useful to

identify the boundaries of fluid-producing layers. When the absolute value of velocity gradient decreases below a

certain threshold, the interpretation method assumes that the corresponding layer is shale. Figure 7c shows the

multi-layer model constructed following the procedure described above. The model is consistent with the Gamma-

Ray log. It should be noted that the construction procedure is entirely based on dynamic production logs, whereupon

the multi-layer model contains information about after-completion features such as induced fractures or production

damages. Another important concern is that the velocity gradient is noise sensitive; special procedures are needed to

denoise the measurements prior to performing the estimation of permeabilities.

Estimating initial formation pressure: The formation consists of different compartments with different average

initial pressures. In addition, PLT measurements of shut-in test revealed cross-flow between formation layers.

Clearly, the assumption of a single compartment will lead to erroneous interpretations. In this example, velocity and

pressure logs obtained from four different production rates are used to estimate the initial reservoir pressure. Two

properties are calculated for each interval, (a) average wellbore fluid pressure, and (b) mass flux increment.

Subsequently, linear interpolation is used to calculate the average pressure necessary to produce a zero mass flux

increment. The resulting pressure is assumed to be the average initial pressure of the corresponding interval. Figure

8 plots the average initial pressure versus incremental mass influx when the formation is divided into 6 pressure

compartments. The intercepts represent average pressure for the corresponding compartment. Figure 7d shows the

constructed vertical distribution of initial reservoir pressure for 6 and 26 compartments. The same plot shows

wellbore fluid pressure. We observe that in some intervals, formation pressure is higher than wellbore pressure,

SPWLA 52nd


10

x400

x450

x500

x550

x600

x650

x700

x750

x800

γγγγ-Ray0 1 2 3 4

22400

22450

22500

22550

22600

22650

22700

22750

22800

KWell-log

KPL

(a) (b) (c) (d) (e)

0 0.5 1

2.245

2.25

2.255

2.26

2.265

2.27

2.275

2.28

x 10

8200 8250 8300 8350 8400

22400

22450

22500

22550

22600

22650

22700

22750

22800

Simulated pressure

Measured pressure

2 4 6 8 10 12

Measured velocity

Simulated velocity

Normalized

G-Ray [API/API]

0 1

MD

[ft

]

x10

x20

x30

x40

x50

x60

x70

x80

x90

Normalized

fluid velocity Fluid pressure [psi]

3200 3250 3300 3350 3400

ρo

0 0.15 0.3

Normalized

permeability [mD/mD]

10-4 10-3 10-2 10-1 100

ft/sec

ft/sec[ ]

g/cm3

g/cm3[ ]Normalized

0 0.3 0.6 0.9

x400

x450

x500

x550

x600

x650

x700

x750

x800

γγγγ-Ray0 1 2 3 4

22400

22450

22500

22550

22600

22650

22700

22750

22800

KWell-log

KPL

(a) (b) (c) (d) (e)

0 0.5 1

2.245

2.25

2.255

2.26

2.265

2.27

2.275

2.28

x 10

8200 8250 8300 8350 8400

22400

22450

22500

22550

22600

22650

22700

22750

22800

Simulated pressure

Measured pressure

2 4 6 8 10 12

Measured velocity

Simulated velocity

Normalized

G-Ray [API/API]

0 1

MD

[ft

]

x10

x20

x30

x40

x50

x60

x70

x80

x90

Normalized


3200 3250 3300 3350 3400

ρo

0 0.15 0.3

Normalized

permeability [mD/mD]

10-4 10-3 10-2 10-1 100

ft/sec

ft/sec[ ]

g/cm3

g/cm3[ ]Normalized

0 0.3 0.6 0.9

Figure 9: Field Example: (a) Gamma-Ray log. (b) Numerically simulated (blue curve) and measured (red curve) velocity distributions. (c) Numerically simulated (blue curve) and measured (red curve) pressure distributions. (d) Wellbore fluid density calculated from PLT measurements. (e) Permeability estimated from production logs (red curve) and well logs (green curve).

indicating fluid flows from the compartment into the wellbore. By contrast, wellbore pressure in other layers is

higher than compartment pressure, indicating that fluid is being injected into the formation.

Estimation of ear-Wellbore Permeability

We apply the minimization procedure described earlier to estimate permeability. The reservoir model consists of

200 numerical layers and 140 petrophysical layers. Petrophysical layers are chosen based on the absolute value of

velocity gradient. When the gradient changes remarkably, the algorithm detects additional petrophysical layers to

accurately reproduce the measured velocity distribution. Figure 9 shows simulated PLT measurements and

estimated layer permeabilities. Figure 9b and 9c show numerically simulated velocity and pressure logs together

with field logs. The error between simulations and measurements is lower than 1% in both cases. Figure 9d shows

wellbore fluid density calculated from PLT measurements. Figure 9e compares permeability distributions estimated

from production logs and well logs. The first observation is that the trends of the two distributions are the same;

however, absolute values are different. At several depths, PL interpretation indicates lower permeability compared

to that estimated from well logs; near-wellbore formation damages and formation compaction are possible reasons

for this behavior. In some depth intervals, production logs suggest higher permeability than the one derived from

well-log interpretation. This enhancement in permeability could be due to formation fracturing after well

completion. Figure 10 provides information about the performance of the estimation algorithm. Figure 10a and 10b

show the statistical distribution of velocity and pressure errors in the final simulation. We observe that both pressure

and velocity errors are within acceptable ranges. Figure 10c gives an overview of the minimization procedure.

After 4 iterations, both velocity and pressure errors reach an acceptable range (lower than 1% error). Figure 10d

shows estimates of permeability with error bars. At some points, error bars widen, thereby suggesting sensitivity to

noisy measurements.

Simulation of Additional PL Tests

It is of great interest to apply the estimated permeability model to simulate production logs acquired from other tests

in the same field. Four tests performed at different rates were considered for this exercise. To estimate permeability

we used the measurements acquired at the highest rate (i.e. PL test 1). We use the estimated model to simulate

SPWLA 52nd


11

1 2 3 40

10

20

30

Re

lati

ve

err

or

[%]

Iteration

Velocity

Pressure

0 1 2 3 40

5

10

15

20

Pressure error [psi]

Fre

qu

en

cy

(a)

(b)

(c) (d)

-0.2 -0.1 0 0.1 0.20

20

40

60

80

Velocity error [ft/sec]

Fre

qu

en

cy

MD

[ft

]

x10

x20

x30

x40

x50

x60

x70

x80

x90

10-4 10-3 10-2 10-1 100

Normalized permeability

[mD/mD]

-0.02 -0.01 0 0.01 0.02Velocity error ft/sec

ft/sec[ ]

1 2 3 40

10

20

30

Re

lati

ve

err

or

[%]

Iteration

Velocity

Pressure

0 1 2 3 40

5

10

15

20

Pressure error [psi]

Fre

qu

en

cy

(a)

(b)

(c) (d)

-0.2 -0.1 0 0.1 0.20

20

40

60

80

Velocity error [ft/sec]

Fre

qu

en

cy

MD

[ft

]

x10

x20

x30

x40

x50

x60

x70

x80

x90

10-4 10-3 10-2 10-1 100

Normalized permeability

[mD/mD]

-0.02 -0.01 0 0.01 0.02Velocity error ft/sec

ft/sec[ ]

Figure 10: Field Example: Histogram of (a) velocity error and (b) pressure error. (c) Relative norm of pressure (red curve) and velocity (blue curve) errors. (d) Estimated permeability and corresponding error bars.

-2 0 2 4 6 8 10 12 14

x10

x20

x30

x40

x50

x60

x70

x80

x90

MD

[ft

]

1000 1500 2000 2500

x10

x20

x30

x40

x50

x60

x70

x80

x90

MD

[ft

]

Measurements

PL run 1

PL run 2

PL run 3

PL run 4

3000 4000 5000 6000-0.15 0.15 0.45 0.7 1.1Normalized


(a) (b)

ft/secft/sec

[ ]

test

test

test

test

-2 0 2 4 6 8 10 12 14

x10

x20

x30

x40

x50

x60

x70

x80

x90

MD

[ft

]

1000 1500 2000 2500

x10

x20

x30

x40

x50

x60

x70

x80

x90

MD

[ft

]

Measurements

PL run 1

PL run 2

PL run 3

PL run 4

Measurements

PL run 1

PL run 2

PL run 3

PL run 4

Measurements

PL run 1

PL run 2

PL run 3

PL run 4

3000 4000 5000 6000-0.15 0.15 0.45 0.7 1.1Normalized


(a) (b)

ft/secft/sec

[ ]

test

test

test

test

Figure 11: Field Example: Numerically simulated distribu-tions of (a) fluid velocity and (b) wellbore fluid pressure acquired from different PL tests. Cross-flow takes place during well shut-in testing (PL test 4) from upper to lower layers.

production measurements of velocity and pressure for the remaining tests. Figure 11 shows the simulation of all

measurements in the same plot. Figure 11a indicates that the estimated permeability model predicts velocity logs

within acceptable errors. In addition, the model is reliable to predict cross-flow between different compartments

during shut-in well tests. Figure 11b shows the prediction of fluid pressure logs.

COCLUSIOS

We introduced and successfully tested a new method to estimate permeability based on the nonlinear inversion of

production logs. The numerical simulation model used in the estimation couples the physics of wellbore and

formation fluid flow and consequently, explicitly relates dynamic fluid conditions in the wellbore with petrophysical

properties of fluid-producing formations. One of the advantages of this interpretation method is that it provides

estimates of permeability after wellbore completion with an intrinsic spatial resolution comparable to that of well

logs. Furthermore, because the estimation method uses cased-hole production measurements it provides a unique

opportunity to estimate local enhancements or reductions of permeability even after years of production.

We considered two synthetic cases to study the effects of (a) layer thickness, (b) presence of a second fluid phase,

and (c) noisy measurements on permeabilities estimated from production logs. Inversion results indicated that the

method was accurate in thick layers. It was found that measurement noise caused erroneous permeability estimations

in thin layers. For layers thinner than 1 ft, presence of 5% random, zero-mean additive Gaussian noise in the

measurements biased the calculated permeabilities by more than 20% thereby rendering the estimation unreliable.

Additionally, inaccurate determination of bed boundaries can cause significant variations of estimated permeability,

especially in the presence of thin beds. Estimations of permeability were carried out from numerically simulated

two-phase PL measurements. Results from this exercise indicated that estimated permeabilities were close those of

relative permeability of the dominant fluid phase. Notwithstanding, there was a significant mismatch between

simulation and measurements of wellbore pressure because of the difference between density of oil and water.

Application of the interpretation method to field data acquired in the deepwater Gulf of Mexico yielded layer

permeabilities that compared well to permeabilities calculated from well logs. Permeabilities estimated from PL

measurements followed the trend of well-log permeability. Discrepancies between the two interpretations could be

SPWLA 52nd


12

associated with formation compaction, near-wellbore damage or stimulation that take place after completion of the

wellbore. For instance, near-wellbore formation permeability could drastically increase after hydro-fracturing

operations, whereby the estimated permeability could be significantly higher than the one calculated from well logs

in the vicinity of induced fractures. Another technical issue which is critical to the interpretation method advanced in

this paper concerns the presence of noise in the measurements. In the documented example with field data, selection

of bed boundaries was guided by the gradient of borehole fluid velocity. It is well known that gradients of fluid

velocity are highly sensitive to measurement noise, whereby spurious layers could be detected in the presence of

noise. It is recommended that PL measurements be critically examined and processed to rid them of biases and

deleterious noise prior to using them in the estimation of layer permeabilities.

LIST OF ACROYMS

FVM Finite Volume Method

MD Measured Depth

PL Production Logging

PLT Production Logging Tools

SIMPLE-C Semi-Implicit Pressure Linked Equations-Consistent

1D One-Dimensional

2D Two-Dimensional

LIST OF SYMBOLS

A

Wellbore radius, [in]

C Quadratic cost function

d Vector of measurements

e Vector data residuals

extFr

External force per unit volume of fluid, [psi/ft]

gr

Gravitational forces, [ft/sec2]

G Relationship between measurements and unknowns

J Jacobian matrix

k Iteration number

P

Pressure, [psi]

t

time

Vr

Fluid velocity, [ft/sec]

mixVr

Mixture velocity, [ft/sec]

oVr

Oil velocity, [ft/sec]

wVr

Water velocity, [ft/sec]

x

Vector of unknowns

α

Regularization parameter

oα

Oil hold-up, [fraction]

wα

Water hold-up, [fraction]

ρ

Fluid density, [g/cm3]

ρo

Oil density, [g/cm3]

ρw

Water density, [g/cm3]

µ

Fluid viscosity, [cp]

τ

Stress tensor, [psi/ft]

SPWLA 52nd


13

ACKOWLEDGEMETS

The work described in this paper was partially funded by The University of Texas at Austin’s Research Consortium

on Formation Evaluation, jointly sponsored by Anadarko, Apache, Aramco, Baker Hughes, BG, BHP Billiton, BP,

Chevron, ConocoPhillips, ENI, ExxonMobil, Halliburton, Hess, Maersk, Marathon, Mexican Institute for

Petroleum, Nexen, Pathfinder, Petrobras, Repsol, RWE, Schlumberger, Statoil, TOTAL, and Weatherford.

REFERECES

Aster, R. C., Borchers, B., and Thurber, C. H., 2005, Parameter Estimation and Inverse Problems: Burlington, MA:

Elsevier Academic Press.

Bendiksen, K., Maines, D., Moe, R., and Nuland, S., 1991, The Dynamic Two-Fluid Model OLGA: Theory and

Application: SPE 19451, SPE production Engineering, vol. 6, no. 2, pp. 171-180.

Bonizzi, M., Andreussi, P., and Banerjree, S., 2008, Flow Regime Independent, High Resolution Multi-Fluid

Modeling of Near-Horizontal Gas-Liquid Flows in Pipelines. International Journal of Multiphase Flow, vol. 35,

no.1, pp. 34-46.

Eissa, M., Joshi, S., Altajar, F., Sobeih, M., Alhussini, O., and Alsafadi, G., 2010, Production Logging Case Studies:

What Can We Say about the Well, in Addition to the Downhole Flow-Profile: SPE 133570, Proceedings of the

SPE Production and Operations Conference and Exhibition: Society of Petroleum Engineers, Tunis, Tunisia,

June 8-10.

Ishii, M. and Hibiki, T., 2006, Thermo-Fluid Dynamics of Two-Phase Flow: New York City, NY: Springer-Verlag.

Kleinstreuer, C., 2003, Two-Phase Flow: Theory and Application. London, Great Britain: Taylor and Francis Book

Inc.

Kolev, N.I., 2007, Multi-Phase Flow Dynamics I, third edition, New York City, NY: Springer-Verlag.

Lahey, R.T. and Drew, D.A., 2001, The Analysis of Two-Phase Flow and Heat Transfer Using a Multidimensional,

Four Field, Two-Fluid Model: /uclear Engineering and Design, vol. 204 no. 1-3, pp. 29-44.

Lahey, R.T., 2005, The Simulation of Multi-Dimensional Multi-Phase Flows. /uclear Engineering and Design, vol.

235 no. 10-12, pp. 1043-1060.

Lahey, R.T., 2007, On the Computation of Multi-Phase Flows: The 12th International Topical Meeting on /uclear

Reactor Thermal Hydraulics (/URETH-12), Pennsylvania, USA, September 30-October 4.

Lahey, R.T., 2009, On the Direct Numerical Simulation of Two-Phase Flows: /uclear Engineering and Design, vol.

239, no. 5, pp. 867-879.

Ouyang, L.B., Aziz, K., 1999, A Mechanistic Model for Gas-Liquid Flow in Pipes with Radial Influx or Outflux:

SPE 56525: Proceedings of the SPE Annual Technical Conference and Exhibition, Houston, Texas, October 3-6.

Prosperetti, A. and Tryggvason, G., 2007, Computational Methods for Multi-Phase Flow: New York City, NY:

Cambridge University Press.

Rey, A.C., Peres, A.M., Junior, A.B., Almeida, S.R., and Sombra, C.L., 2009, Direct Permeability Estimation Using

Production Logging: Paper WWW, Transactions of the SPWLA 47th

Annual Logging Symposium, Houston, USA,

June 21-24.

Shirdel, M. and Sepehrnoori, K., 2009, Development of a Coupled Compositional Wellbore/Reservoir Simulator for

Modeling Pressure and Temperature Distribution in Horizontal Wells: SPE 124806, Proceedings of the SPE

Annual Technical Conference and Exhibition, New Orleans, Louisiana, October 4-7.

Shirdel, M. and Sepehrnoori, K., 2011, Development of a Transient Mechanistic Two-Phase Flow Model for

Wellbores: SPE 142224, Proceedings of the SPE Reservoir Simulation Symposium, Woodlands, Texas, February

21-23.

Sullivan, M.J., 2007, Permeability from Production Logs - Method and Application: SPE 103486, Journal of

Petroleum Technology, vol. 59, no. 7, pp. 80-87.

Sullivan, M.J., Belanger, D., and Skalinski, M., 2006, A New Method for Deriving Flow-Calibrated Permeability

from Production Logs: Paper K, Transactions of the SPWLA 47th

Annual Logging Symposium, Veracruz, Mexico,

June 4-7.

Versteeg, H. K. and Malalasekara, W., 1995, An Introduction to Computational Fluid Dynamics, The Finite Volume

Method: New York, NY, Longman Scientific and Technical.

Yeoh, G.H. and Tu J., 2010, Computational Techniques for Multi-Phase Flows: Burlington, MA: Butterworth-

Heinemann.

˝umerical simulatio˝ a˝d i˝terpretatio˝ of … · considered in this paper estimates formation...

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