fast estimation of permeability from formation …spwla 54th annual logging symposium, june 22-26,...

SPWLA 54th Annual Logging Symposium, June 22-26, 2013

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FAST ESTIMATION OF PERMEABILITY FROM FORMATION-TESTER MEASUREMENTS ACQUIRED IN HIGH-ANGLE WELLS

Hamid Hadibeik, Rohollah A-Pour, Carlos Torres-Verdín,

Kamy Sepehrnoori, and Vahid Shabro, The University of Texas at Austin

Copyright 2013, 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 54th Annual Logging Symposium held in New Orleans, Louisiana, June 22-26, 2013.

ABSTRACT Conventional static methods to estimate permeability are based on correlations between well logs and core data. In contrast, formation testing provides a dynamic approach to estimate permeability (or mobility) from transient measurements of pressure and fractional flow. We develop a new method based on streamlines to estimate formation permeability from formation-tester measurements. A previously developed finite-difference reservoir model is coupled with a streamline method to evaluate near-wellbore dynamic petrophysical properties in the presence of invasion and arbitrary fluid distributions. The streamline method is specifically developed to overcome the technical challenges associated with deviated wells in complex reservoirs. In this study, synthetic reservoir models are constructed based on measurements acquired in an offshore well. The streamline-based method is then implemented to simulate packer and probe-type formation-tester measurements and to estimate permeability of multi-layer reservoirs. Inversion results indicate that the accuracy of estimated permeability is higher for formations with larger mobility because more streamlines track flow into probes from large-mobility layers. In the presence of 5% zero-mean Gaussian additive noise, the uncertainty of permeability varies from 6% to 38% for layers with high and low mobilities, respectively. The uncertainty increases to 8% and 41% for high and low mobility cases, respectively, when 5% skewed-Gaussian noise contaminates the measurements. The coupled finite-difference and streamline-based inversion method (FDSM) is then compared to a previously validated finite-difference reservoir model (FDM). It is observed that FDSM is 8 times faster than FDM on average when estimating permeability of heterogeneous reservoirs using inversion on formation-tester measurements acquired in highly-deviated wells. The computational advantage of FDSM is due to the application of one-dimensional solutions of fluid

saturations and concentrations along streamlines instead of three-dimensional numerical calculations in the FDM. However, forward modeling of synthetic measurements indicates up to 5% difference between FDSM and FDM in water saturation distribution results. In high-angle wells, mud-filtrate invasion causes a non-symmetric distribution of invading fluid around the perimeter of the wellbore. Formation permeability, permeability anisotropy, wellbore inclination with respect to layers, and cross flow between layers affect fluid distribution during invasion and fluid withdrawal. Therefore, the concentration and saturation calculations in each layer demand more numerical iterations. Moreover, presence of large permeability-porosity contrast among layers increases the complexity of numerical computations and uncertainty. The streamline-based inversion method is an excellent candidate to overcome these difficulties in an efficient manner. INTRODUCTION Dual-packer and multi-probe formation testers have been used to determine reservoir properties such as pore pressure and permeability (Angeles et al., 2010; Elshahawi et al., 1999; Hadibeik et al., 2012a; Proett et al., 2004; Zazovsky et al., 2008). Several analytical models have been developed to analyze formation-tester measurements using fundamental pressure transient equations (Agarwal, 1979; Bourdet et al., 1989; Goode and Thambynayagam, 1987; Hadibeik et al., 2012b). However, these models do not account for complexity of layered and heterogeneous reservoirs, multi-phase multi-component fluid flow, and fluid compressibility (Clarkson, 2009; Guo and Toro, 2000), among other important factors. In this study, a new numerical inversion technique is developed to take into account realistic reservoir complexities and to estimate permeability from formation-tester measurements accurately. The inversion method, based on modeling formation-tester measurements, accounts for the general nonlinear relationship between measurements and reservoir properties. However, previous attempts to develop a numerical inversion method for complex reservoirs have shown that inversion can be time consuming, especially in highly-deviated wells (Angeles et al.,


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2007a; Frooqnia et al., 2011; Malik et al., 2007). Application of finite-difference algorithms to simulate pressure and saturation measurements is one of the shortcomings of previous works. To improve computational efficiency of pressure and fractional flow calculations, a previously developed streamline-based model (Hadibeik et al., 2011) is coupled with a finite-difference method (A-Pour, 2011). Then, an inversion technique is used to estimate the vertical distribution of permeability in synthetic models which are constructed based on field data collected in the North Sea. Both formation pressure and fractional flow measurements are used for inversion of input data. However, the joint inversion of pressure and fractional flow measurements remains an ill-conditioned problem (Alpak et al., 2011; Angeles et al., 2007b; Malik et al., 2009). As the number of unknown parameters in the inversion process increases, the problem becomes more complex and the solution becomes non-unique. We invoke regularization and weighting factors to balance the effects of pressure and contamination measurements in the combined cost function. INVERSION METHOD The Gauss-Newton minimization approach (Aster et al., 2005) is applied to solve the nonlinear inverse problem. The measurement vector is defined as

1 2 i MP P P P P , (1)

where P is vector of pressure differences, M is number of measurement points, Pi for the drawdown test is initial pressure at the start of pumpout minus pressure at any time; Pi for the buildup test is final stabilized pressure minus pressure at any time. Angeles et al., (2007b) reported that the use of pressure differentials substantially increases the stability of the minimization algorithm. The fractional flow of a contamination vector is given by

1 2 i Mfw fw fw fw fw , (2)

where fw is vector of fractional flow measurement points and fwi is the contamination sampled at the ith time. The data mismatch vector, e(x), is a vector whose ith element is the residual error of ith-normalized measurements and is defined as

1 2 i Me(x) e e e e , (3)

where ei is logarithm of Pi at each pressure measurement; for contamination measurements, ei is

sim measi ii

meas i

fw fwe

fw

, (4)

where fwsim|i is simulated fractional flow at the ith time, and fwmeas|i is measured fractional flow at that time, x

vector represents model parameters in the inversion process and is given by

1 2 i Nx x x x x , (5)

where N is the number of unknowns. Both pressure and fractional flow measurements are employed to construct a balanced multi-physics vector of residuals in a mismatch data vector. The cost function between measurements and simulation model results is calculated as

2 22d x

1C(x) [ W e(x) W x ]

2 , (6)

where is Tikhonov’s regularization parameter

(0<<∞), dW is the data weighting matrix, and xW is

the model weighting matrix. Based on the cost function, the Jacobian matrix is formulated as

mmn

n

eJ(x) J

x

, (7)

m 1,2,3,..., M;n 1,2,3,..., N , where M is the number of measurements and N is the number of unknown model properties. MODELING OF NORTH SEA FIELD MEASUREMENTS Well-log measurements and core samples acquired from the southern region of North Sea indicate existence of a siliciclastic sequence (Hagemann, 1969). Figure 1 shows the mineral composition of the core samples in a ternary diagram; vertices indicate clay, quartz, and carbonate components. Sand flats are rich in quartz, causing facies to change to mud flats as clay and carbonate content increases. Figure 2 describes petrophysical measurements acquired in a well in this environment. The earth model shows a facies transition between the two regions of the ternary diagram. Formation testing is performed at the lower section of the well, where high resistivity values indicate presence of hydrocarbon. Figure 3 describes both the tool configuration and the earth model used to construct synthetic models using North Sea field data. Next, streamline-based inversion is applied to these models to estimate permeabilities from formation-tester measurements in vertical and high-angle wells. After mud-filtrate invasion, layers with low permeability-porosity values exhibit a higher radius of invasion and larger pressures near the borehole region. Figure 4 describes pore pressure and water saturation after 24 hours of water-base mud-filtrate invasion. A drawdown test is then performed with a dual packer for 21.6 hours followed by a 2.4-hour buildup test. Table 1 describes the permeability and porosity of model layers.


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It is assumed that saturation-dependent relative permeability and capillary pressure data follow the Brooks-Corey model. Table 2 summarizes the assumed Brooks-Corey parameters. Consequently, as shown in Figure 5, the layer with larger porosity-permeability values has higher relative permeabilities and lower capillary pressures. Hadibeik et al. (2011) indicated that streamlines transport reservoir fluids in a way to

minimize travel time. Therefore, they bend when entering a new medium with permeability different from that of the current medium. Figure 6 shows the produced streamlines after 24 hours of fluid pumpout in the vertical well model. Abrupt changes in the streamlines of Figure 6 are due to the above-mentioned fact; these changes occur when a streamline passes through a petrophysical bed boundary.

Figure 1: Ternary diagram with three inorganic components recovered from core samples (Brumsack, 1989). Sand flats are rich in quartz; facies change to mud flats as clay and carbonate content increases.

GR [API]

0 200

T [ ]

00.3Rshallow .m

0 50

Rdeep .m500

Water

Oil

Matrix

VCL

VCL

Sst

Csh [ ]

e [ ]

BVWE [ ] 01

10

01

01SwT [ ]

Csh [ ] 0.80

T [ ] 1 0

Figure 2: Measured and calculated petrophysical properties in a vertical well in the North Sea field. Hydrocarbon-bearing zone: lower section of the log, where resistivities indicate larger values and gamma ray decreases significantly. Track 1: measured gamma ray (GR); Track 2: total porosity calculated with neutron-density logs; Track 3: measured deep and shallow resistivities; Track 4: reservoir model, volumetric concentration of shale, effective porosity, and effective bulk volume of water; Track 5: total water saturation; Track 6: earth model.


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Table 1: Porosity and permeability of layers analyzed with dual packer measurements.

Rock Type k [mD] [ ] 1 200 0.20 2 526 0.24 3 50.5 0.16 4 5.34 0.08

Table 2: Brooks-Corey parameters used to describe saturation-dependent capillary pressure and relative permeability in the North Sea formation.

Property

Unit

Rock Type

1 2 3 4

krw0 [ ] 0.4 0.3 0.45 0.5

kro0 [ ] 0.8 0.9 0.7 0.6

[ ] 5 5 5 5

Swr [ ] 0.15 0.1 0.2 0.25

Sor [ ] 0.1 0.05 0.15 0.23

Pce [psi] 1.5 1.5 1.5 1.5

GR [API]

0 200

T [ ]

00.3 Rshallow .m0 50

Rdeep .m500

Csh [ ]

e [ ]

BVWE [ ] 01

10

01

Csh [ ] 0.80

T [ ] 1 0

Water

Oil

Matrix

VCL

VCL

Sst

2.7 ft

5.65 ft

1.65 ft

Figure 3: Configuration of the dual packer-type formation tester and two monitoring probes used for pressure testing and sampling the synthetic model created using North Sea field data. Probe 1: top-monitoring probe, Probe 2: located at the bottom.


5

4450

4435

4360

4420

4405

4390

4375

4465

0

10

20

30

40

50

6010‐1 100 101 102

P [psi]

Y-dir [ft]

TV

D [f

t]

4450

4435

4360

4420

4405

4390

4375

4465

0

10

20

30

40

50

6010‐1 100 101 102

P [psi]

Y-dir [ft]

TV

D [f

t]

(a) Pressure distribution

0

10

20

30

40

50

6010‐1 100 101 102

0.2

0.3

0.7

0.9

0.6

0.8

0.4

0.5

Sw [ ]

Y-dir [ft]

TV

D [f

t]

0

10

20

30

40

50

6010‐1 100 101 102

0.2

0.3

0.7

0.9

0.6

0.8

0.4

0.5

Sw [ ]

Y-dir [ft]

TV

D [f

t]

(b) Water saturation distribution

Figure 4: Radial distribution of (a) pressure and (b) water saturation after 24 hours of water-base mud filtrate invasion. Overbalance and reservoir pressures are 200 and 4350 psi, respectively.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Kr[

]

Sw [ ]

RK1RK2RK3RK4

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Kr[

]

Sw [ ]

RK1RK2RK3RK4

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

Sw [ ]

Pc[p

si]

RK1RK2RK3RK4

Figure 5: Saturation-dependent relative permeability and capillary pressure for four assumed rock types.

100

0

5

10

15

Y-dir [ft]

TV

D [f

t]

P [psi]

43394340434143424343434443454346434743484349

Figure 6: Flow streamlines entering the dual-packer, predominantly from spatial regions with good-quality rocks. PERMEABILITY ESTIMATION FROM NORTH SEA FIELD MEASUREMENTS ACQUIRED IN A VERTICAL WELL The following assumptions are embedded in the inversion process: (1) relative permeability and capillary pressure curves obtained from the Brooks-Corey model depend on permeability, (2) radius of mud-filtrate invasion is a function of permeability, (3) gravity effects are neglected in the streamline-based inversion model, (4) porosity is calculated from neutron-density logs, except for the analysis in which porosity is calculated from Wyllie’s permeability correlation (Wyllie et al., 1950), and (5) fluid viscosity is constant, obtained from laboratory measurements. In the first analysis, it is assumed that porosity depends on permeability, based on an empirical correlation proposed by Wyllie et al. (1950) as follows:

2 2wi

wi

100 [1 S ]k

S

, (8)

where k is permeability, is porosity, and Swi is irreducible water saturation. The dual packer is set at the largest porosity-permeability layers in Figure 3 (rock types 1 and 2), while the two monitoring pressure probes are deployed at the lower section. Subsequently, a drawdown-buildup test is applied to measure pressure transients and contamination. Figures 7 and 8 describe the simulation of pressure, fractional flow measurements, and their counterparts numerically simulated with the streamlined-based inversion method. Based on Figure 8, pressure transients at the end of buildup indicate that initial reservoir pressure is 4350 psi. Permeability uncertainty is assessed with both 5% zero-mean random Gaussian and 5% skewed-Gaussian noise added to the


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measurements. Figure 9 indicates that both fractional flow and pressure objective functions should be considered to minimize the overall cost function. This analysis also implicitly states that analytical inversion methods based solely on pressure measurements do not match both pressure and fractional flow in the reservoir; hence, the estimated permeabilities may be unreliable. Figure 10 describes results from the joint inversion of pressure and fractional flow measurements when porosity is calculated from Wyllie’s permeability correlation. The inverted distribution of permeability exhibits up to 45% uncertainty. Permeabilities with initial guess and estimated values are also shown in Figure 10. The second analysis in Figure 11 assumes that porosity is calculated from neutron-density logs. For this case, uncertainty in permeability estimations decreased to 39% for the lowest permeability layer. Comparison of these two analyses indicates that the uncertainty of inverted permeabilities decreases when porosity values are calculated from neutron-density logs.

100

10-3

10-2

10-1

100

Time [hrs]

Fw

[ ]

Meas.Inv.

Figure 7: Comparison of fractional flow contamination from measurements and numerically simulated from inversion results.

0 5 10 15 204338

4340

4342

4344

4346

4348

4350

4352

Time [hrs]

Pac

ker

Pre

ssur

e [p

si]

Meas.Inv.

0 5 10 15 204340

4345

4350

4355

4360

4365

4370

Time [hrs]

Pac

ker

Pre

ssur

e [p

si]

Meas.Inv.

0 5 10 15 204338

4340

4342

4344

4346

4348

4350

4352

Time [hrs]

Pac

ker

Pre

ssur

e [p

si]

Meas.Inv.

(a) Packer location (b) Monitoring probe 1 (c) Monitoring probe 2 Figure 8: Pressure transients for a (a) dual packer-type formation test with drawdown and buildup sequences in the North Sea reservoir field example, compared to reproduced measurements with streamline-based simulation with (b) monitoring probe 1 and (c) monitoring probe 2. Reservoir pressure stabilizes to initial formation pressure (4350 psi) after buildup.

0 2 4 6 8

100.1

100.4

100.7

Iteration number [ ]

Obj

ectiv

e fu

nctio

n [

]

PFw

0 2 4 6 8

100.2

100.4

100.6

100.8


Obj

ectiv

e fu

nctio

n [

]

PFw

0 2 4 6 810

0


Obj

ectiv

e fu

nctio

n [

]

PFw

(a) Pressure cost function (b) Fractional flow cost function (c) Joint cost function Figure 9: Comparison of objective functions to estimate permeability with (a) only pressure cost function, (b) only contamination cost function, and (c) joint inversion of fractional flow and pressure measurements.


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101

102

103

0

2

4

6

8

k [mD]

Rel

ativ

e D

epth

[ft]

True ModelInitial GuessInverted ModelLower BoundUpper Bound

101

102

103

0

2

4

6

8

k [mD]

Rel

ativ

e D

epth

[ft]


(a) Gaussian noise (b) Skewed-Gaussian noise Figure 10: Inverted permeability values for six petrophysical layers analyzed with a dual-packer formation tester in the North Sea vertical well. In this case, porosity depends on permeability based on Wyllie’s correlation. Measurements are contaminated with additive (a) zero-mean Gaussian noise and (b) skewed-Gaussian noise to appraise the uncertainty of inverted permeability.

101

102

103

0

2

4

6

8

k [mD]

Rel

ativ

e D

epth

[ft]


(a) Gaussian noise

101

102

103

0

2

4

6

8

k [mD]

Rel

ativ

e D

epth

[ft]


(b) Skewed-Gaussian noise Figure 11: Vertical distribution of permeability estimated with regularized joint inversion of pressure and contamination measurements acquired in the North Sea vertical well. Uncertainty bounds (error bars) decrease when compared to the first test case. Porosity is calculated from neutron-density logs; permeability uncertainty is assessed with 5% (a) zero-mean Gaussian noise and (b) skewed-Gaussian noise added to the measurements.

STREAMLINE-BASED INVERSION OF PERMEABILITY IN HIGH-ANGLE WELLS We constructed a synthetic case based on logs and formation-tester measurements acquired in a deviated well in the North Sea. Figure 12 defines petrophysical properties of each layer and indicates the dual-packer location and its monitoring probes. Streamline-based inversion is applied to formation-tester measurements in a 15o and a 75o deviated well. In this section, permeabilities of layers where both the dual-packer and its monitoring probes are active, are estimated with the streamline-based inversion method. Formations are subjected to water-base mud-filtrate invasion for 24 hours, followed by a 4-day fluid pumpout, and a 2.4-hour buildup with a dual packer thereafter. Figure 13 shows a vertical cross-section of a 3D streamline model for the packer-type formation tester in a 15o deviated well; streamlines flow into the dual packer symmetrically. Figure 14 shows simulated pressure and water saturation distributions at the end of drawdown. The maximum difference between the modeled water saturation distribution with finite-difference and streamline-based inverted model is 4%, as shown in Figure 15. Inverted permeability of good- quality layers (large permeability and porosity) in Figure 16 results in less uncertainty than do low- permeability layers for a 15o deviated well. When the deviation angle increases, streamlines flow smoothly toward the producing interval. Figure 17 shows the streamlines for a 75o deviated well after 24 hours of fluid pumpout; streamlines are less spread than those in the case of the 15o deviated well and more concentrated toward the layers that have higher permeabilities. Figure 18 describes the difference between the water saturation distribution from the finite-difference and


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streamline-based models for a 75o deviated well. Smaller discrepancy in this case compared to the case of the 15o deviated well is due to higher frequency of pressure updates in the 75o deviated well to account for anisotropy and cross-flow. Conversely, the CPU time of the streamline-based method is longer in the 75o deviated well. Figure 19 shows results from permeability estimation from the inversion method, describing the initial guess, model values, and

uncertainty bounds for transient measurements in the 75o deviated well. Table 3 compares CPU times for streamline-based inversion in the 15o and 75o deviated wells. Finite-difference inversion CPU times are compared to streamline-based inversion CPU times in a single simulation case. The streamline model is 8.75 and 5.74 times faster than the finite-difference method for the 15o and 75o deviated wells, respectively.

z

rw = 0.25 ft

re = 1971 ft

k = 200.0 mD, = 20%

k = 526.0 mD, = 24%

k = 10.0 mD, = 12%

k = 4.65 mD, = 9%

k = 50.5 mD, = 16%

k = 5.34 mD, = 8%

z = 0.0 ft

z = 9.8 ft

z = 2.0 ft

z = 3.6 ft

z = 5.0 ft

z = 6.5 ft

z = 8.0 ft

z

rw = 0.25 ft

re = 1971 ft

k = 200.0 mD, = 20%

k = 526.0 mD, = 24%

k = 10.0 mD, = 12%

k = 4.65 mD, = 9%

k = 50.5 mD, = 16%

k = 5.34 mD, = 8%

z = 0.0 ft

z = 9.8 ft

z = 2.0 ft

z = 3.6 ft

z = 5.0 ft

z = 6.5 ft

z = 8.0 ft

Dual-Packer Module

Dual-packer measurement location

Probe measurement location

2.7 ft

5.65 ft

1.65 ft

Figure 12: Schematic of the formation model based on the North Sea field data, indicating assumed dual-packer locations. Permeable horizontal layers are penetrated by a deviated well. Formations are subjected to water-base mud filtrate for 24 hours. Table 3: Comparison of CPU time required for measured streamline-based (ST) inversion versus CPU time required for the calculated finite-difference (FD) model.

Type Deviation [angle]

Measured ST [hrs]

Calculated FD [hrs]

Packer 15o 283.4 2482.8

Packer 75o 662.2 3801.1


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-5 0 5

16

18

20

22

24

26

Y-dir [ft]

TV

D [f

t]

P [psi]

4344.543454345.543464346.543474347.543484348.543494349.5

Figure 13: Flow streamlines entering the dual-packer from the reservoir at the end of pumpout in a 15o deviated well.

-5 0 5

16

18

20

22

24

26

Y-dir [ft]

TV

D [f

t]

P [psi]

4344.543454345.543464346.543474347.543484348.543494349.5

(a) Pressure distribution

-5 0 5

16

18

20

22

24

26

Y-dir [ft]

TV

D [f

t]

Sw [ ]

0.2

0.3

0.4

0.5

0.6

0.7

(b) Water saturation distribution Figure 14: Vertical cross-section of (a) pressure and (b) water saturation distributions calculated with streamline simulation after 24 hours of fluid pumpout.

-5 0 5

16

18

20

22

24

26

Y-dir [ft]

TV

D [f

t]

Sw

Rel. Diff[%]

-0.2

-0.1

0

0.1

0.2

0.3

Figure 15: Relative difference between water saturation distributions calculated from synthetic field and reproduced measurements in the 15o deviated well.

101

102

103

0

2

4

6

8

k [mD]

Rel

ativ

e D

epth

[ft]


Figure 16: Inverted permeability values and error bars calculated by adding 5% skewed-Gaussian noise to the measurements acquired in a 15o deviated well.

-14 -12 -10 -8 -6 -4

20

25

30

35

40

Y-dir [ft]

TV

D [f

t]

P [psi]

4330

4332

4334

4336

4338

4340

4342

4344

4346

4348

Figure 17: Streamlines reaching the dual-packer formation tester after drawdown performed in a 75o deviated well.


10

-14 -12 -10 -8 -6 -4

20

25

30

35

40

Y-dir [ft]

TV

D [f

t]

SwRel.

Diff

[%]

-0.01

-0.005

0

0.005

0.01

Figure 18: Relative difference between water saturation distributions calculated from the synthetic field measurement and the reproduced model in the 75o deviated well.

101

102

103

0

2

4

6

8

k [mD]

Rel

ativ

e D

epth

[ft]


Figure 19: Actual permeability, initial guess, and inverted vertical permeability distributions along with lower and upper uncertainty bounds calculated with 5% skewed-Gaussian noise added to the measurements. In the final analysis, an oval pad-type formation tester is simulated in a deviated well. The North Sea field model is used to construct a synthetic reservoir model penetrated by a 45o deviated well. Figure 20 illustrates the oval pad used to perform drawdown and buildup tests at three different depths as the tool moves upward in the well. Prior to the tests, the formation is subjected to 24 hours of water-base mud-filtrate invasion. The upper layers exhibit better rock quality and higher permeabilities; therefore, most streamlines enter the probe from the top layers. Figure 21 shows the streamlines entering the probe from each layer. Hadibeik et al., (2009) and Angeles et al., (2011) remarked that the preferential location for sampling in a high-angle well probe is at the top of the well. Figure 22 describes the vertical permeability distribution obtained from inversion of oval-pad measurements in a 45o deviated well. Table 4 summarizes the CPU time

required to invert oval-pad transient measurements in a 45o deviated well. In this analysis, the streamline-based inversion method is 6.87 times faster than a conventional finite-difference method. Table 4: CPU time comparison for measured ST and calculated FD inversion of oval-probe formation testing in a 45o deviated well.

Type Deviation [angle]

Measured ST [hrs]

Calculated FD [hrs]

Probe 45o 460.7 3165.7

CONCLUSIONS A robust and efficient streamline-based inversion method was developed to estimate vertical distributions of permeability from transient measurements of pressure and fractional flow acquired with formation testers. The following conclusions stem from the study: 1. In complex heterogeneous reservoirs penetrated by

high-angle wells, the inversion using a streamline fluid-flow simulator reduces CPU computation time by as much as 90% of the time required by equivalent inversion methods based on finite differences.

2. The streamline method provides a visual tool to appraise dominant fluid flow paths in heterogeneous reservoirs.

3. Sensitivity analysis of inversion results in the presence of noise indicates that errors in the estimation of permeability decrease in layers with high mobility. The flow contribution from these layers to the measurements becomes dominant, whereby the increased number of streamlines through them decreases uncertainty in permeability estimates.

4. Lower uncertainty in permeability estimates was achieved by posing the estimation in logarithmic space.

5. Uncertainty in estimated permeabilities decreased by 6% when layer porosities were calculated from neutron-density logs instead of porosity-permeability correlations.

6. Joint inversion of pressure and fractional flow is necessary for reliable estimation of permeability. Separate inversion of these two parameters causes instability of inversion results.

7. Streamline-based inversion expedites permeability estimation in complex reservoir models penetrated by high-angle wells. Use of streamline-based techniques for estimating formation properties in highly deviated wells is crucial because of the effect of several factors in the cost function,


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including eccentric invasion, permeability, anisotropy, cross flow, and wellbore deviation. Therefore, a large number of iterations are needed to converge to an acceptable solution. Each iteration in the finite-difference method is time consuming and CPU intensive due to the physical complexity of solving saturation equations in 3D systems.

8. To further decrease the uncertainty associated with estimated permeabilities, one could incorporate a multi-physics joint/combined inversion of formation-tester measurements and well logs. For example, resistivity logs can provide an estimate of mud-filtrate invasion depth as well as in-situ water saturation; hence, the number of unknowns will decrease in the inversion process.

z

rw = 0.25 ft

re = 1971 ft

k = 200.0 mD, = 20%

k = 526.0 mD, = 24%

k = 10.0 mD, = 12%

k = 4.65 mD, = 9%

k = 50.5 mD, = 16%

k = 5.34 mD, = 8%

z = 0.0 ft

z = 9.8 ft

z = 2.0 ft

z = 3.6 ft

z = 5.0 ft

z = 6.5 ft

z = 8.0 ft

Probe measurement location

Figure 20: Cross-section of the multi-layer North Sea field model penetrated by a deviated well. The six-layer formation is subjected to water-base mud-filtrate invasion for 24 hours. Subsequently, an oval-type probe performs drawdown and buildup tests at the three locations indicated by blue ellipses.


12

-10 -8 -6 -4 -2 0

18

20

22

24

26

28

30

Y-dir [ft]

TV

D [f

t]

P [psi]

3950

4000

4050

4100

4150

4200

4250

4300

(a) First station

-10 -8 -6 -4 -2 0

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D [f

t]

P [psi]

4000

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4300

(b) Second station

-10 -8 -6 -4 -2 0

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Y-dir [ft]

TV

D [f

t]

P [psi]

4341

4342

4343

4344

4345

4346

4347

4348

4349

(c) Third station Figure 21: Streamline model of an oval-probe type formation tester performing drawdown and buildup tests in a 45o deviated well. The probe measures pressure and contamination in the (a) first station, (b) second station, and (c) third station.

101

102

103

0

2

4

6

8

k [mD]

Rel

ativ

e D

epth

[ft]


Figure 22: Simultaneous inversion of permeability for four petrophysical layers that were tested with an oval probe in a 45o deviated well. NOMENCLATURE : Regularization parameter P : Pressure difference, [psi] Porosity T Total porosity e Effective porosity : Standard deviation

Wd : Data weighting matrix

Wx : Model weighting matrix, [mD-1]

BVWE : Effective bulk volume water C(x) : Objective function CPU : Central processing unit Csh : Shale concentration e(x) : Data mismatch vector fw : Fractional flow of contamination FD : Finite-difference FDM : Finite-difference method FDSM : Coupled finite-difference and streamline method GR : Gamma ray, [API] J(x) : Jacobian matrix k : Formation permeability, [mD] krw0 : End point relative permeability of water kro0 : End point relative permeability of oil : Water-oil saturation exponent Kr : Relative permeability M : Number of measurements N : Number of unknowns P : Pressure, [psi] Pc : Capillary pressure, [psi] Pce : End-point capillary pressure, [psi] Rdeep : Deep resistivity, [.m] Rshallow : Shallow resistivity, [.m] ST : Streamline Sst : Sandstone Sw : Water saturation


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Swi : Irreducible water saturation SwT : Total water saturation Swr : Residual water saturation Sor : Residual oil saturation SwRel.Diff : Relative difference of water saturation tFD : Finite-difference simulation time, [s] tST : Streamline simulation time, [s] TVD : True vertical depth, [ft] VCL : Normalized clay volume ACKNOWLEDGEMENTS The work reported in this paper was funded by The University of Texas at Austin’s Research Consortium on Formation Evaluation, jointly sponsored by Afren, Anadarko, Apache, Aramco, Baker-Hughes, BG, BHP Billiton, BP, Chevron, ConocoPhillips, COSL, ENI, ExxonMobil, Halliburton, Hess, Maersk, Marathon Oil Company, Mexican Institute for Petroleum, Nexen, ONGC, OXY, Petrobras, Repsol, RWE, Schlumberger, Shell, Statoil, TOTAL, Weatherford, Wintershall, and Woodside Petroleum Limited. REFERENCES Agarwal, R., 1979. Real Gas Pseudo-Time: A New

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fast estimation of permeability from formation …spwla 54th annual logging symposium, june 22-26,...

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