Post on 04-Jun-2018
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
Annual Logging Symposium, May 14-18, 2011
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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
Annual Logging Symposium, May 14-18, 2011
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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
6
8
10
12
14
Radius [in]
0.5
1
1.5
2
2.5
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
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
25
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
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
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
Annual Logging Symposium, May 14-18, 2011
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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
Annual Logging Symposium, May 14-18, 2011
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0
0
20
40
60
80
100
120
140
MD
[ft
]
8880 8900 8920 8940
0
20
40
60
80
100
120
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
120
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
Annual Logging Symposium, May 14-18, 2011
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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
Annual Logging Symposium, May 14-18, 2011
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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
Annual Logging Symposium, May 14-18, 2011
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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
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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
Annual Logging Symposium, May 14-18, 2011
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
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
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
Annual Logging Symposium, May 14-18, 2011
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
fluid velocity Fluid pressure [psi]
(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
fluid velocity Fluid pressure [psi]
(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
Annual Logging Symposium, May 14-18, 2011
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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
Annual Logging Symposium, May 14-18, 2011
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.
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