81634

Copyright 2003, IADC/SPE Underbalanced Technology Conference and Exhibition This paper was prepared for presentation at the IADC/SPE Underbalanced Technology Con-ference and Exhibition held in Houston, Texas, U.S.A., 2526 March 2003. This paper was selected for presentation by an IADC/SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the International Association of Drilling Con-tractors or the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the IADC, SPE, their officers, or members. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the International Association of Drilling Contractors or the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836 U.S.A., fax 01-972-952-9435.

Abstract In this paper methodologies for reservoir characterization dur-ing underbalanced drilling is presented. In these methodolo-gies we are using a transient wellflow model coupled to a tran-sient reservoir model, and use estimation techniques to esti-mate reservoir properties. Our focus is to estimate the perme-ability and reservoir pressure along the well, using measured data usually available while drilling. The measured data are outlet rates, pump pressure and downhole pressure. The liquid injection and gas injection rates are used as input to the model. The methodologies are applied to synthetic cases.

Introduction

Underbalanced drilling is becoming increasingly popular. During an underbalanced drilling operation the well pressure should be kept below the formation pressure at all times. Since the well is producing while drilling, the well may be tested real time. Estimation of near wellbore characteristics of the formation gives important information when using smart com-pletions since highly productive zones can be located.

Several recent papers1-4 have addressed well-testing during underbalanced drilling. Methodologies have been presented in these papers where the permeability profile in the near well-bore region is estimated based on the assumption that the total flow rate from the reservoir is known and that the reservoir pressure is assumed to be constant and known. An alternative technique1 has been presented where the assumption of known and constant reservoir pressure is not needed, but known total flow rate from the reservoir is still needed. The total flow rate from the reservoir is however not usually measured during an underbalanced drilling operation. As it is pointed out in Ref. 1 it is not straightforward to determine the total flow rate from the reservoir on the basis of the surface flow measurements. Compressibility of the fluids in the system may lead to loading and unloading in the wellbore, and mass flow rate at the sur-

face is therefore affected by production, injection, and the change in mass stored in the wellbore as a function of time. It is therefore a need to develop methodology for reservoir char-acterization during underbalanced drilling that applies only data measured during the operations.

Some recent papers5,6 have addressed the challenge of calibrating a well flow model real time according to measured data. The advantage of having such a calibrated well flow model is that reliable predictions for the well conditions (like bottomhole pressure) can be given at all times. A limitation of these papers is that no reservoir models have been included.

In a previous paper7 a least square methodology to estimate near-well reservoir properties was presented. The present pa-per is an extension of this work. In the present paper the en-semble Kalman filter is introduced as an alternative method to estimate reservoir properties during underbalanced drilling. The importance of active tests during the drilling operation is also investigated.

The outline of the paper is as follows. The well flow model and the reservoir model are first described. Then two method-ology for interpreting the reservoir properties from the meas-ured data are described. The described methodologies are then applied to synthetic cases.

The description of the well flow model, reservoir model as well as the estimation methodologies, closely follows the presentations in Refs. 5-7, but are included here for the con-venience of the reader.

Dynamic Well Flow Model A dynamic model for describing the transient behavior of the two-phase flow conditions in LHD and UBD operations can be expressed with basis in the drift-flux formulation of the two-phase flow conservation laws8. Due to the complexity of the model, a numerical solution strategy is required.

The numerical scheme solves a set of three conservation equations, one for the mass of each phase and one for the mix-ture momentum. The mixture energy equation is not taken into account. Instead a fixed temperature profile in the well is used, which can be calculated in advance or provided by the data acquisition system. Conservation Equations. The drift-flux formulation of the conservation equations is given by

( ) ( ) mvzt GGGGG

=

+ ,............ (1)

IADC/SPE 81634

Reservoir Characterization during UBD: Methodology and Active Tests Erlend H. Vefring, RF-Rogaland Research, Gerhard Nygaard, Telemark University College, Rolf Johan Lorentzen, Geir Nvdal and Kjell Kre Fjelde, RF- Rogaland Research

2 E. H. VEFRING, G. NYGAARD, R.J. LORENTZEN , G. NVDAL AND K. K. FJELDE IADC/SPE 81634

( ) ( ) mvzt LLLLL

=

+ , ....... (2)

( ) ( )pvvz

vvt

2GGG

2LLLGGGLLL ++

++

( ) sin g Fdz

dpGGLL +

= . ........... (3)

The transient drift-flux model is a system of non-linear partial differential equations, which is hyperbolic in an ample region of physical multiphase parameters9. The model de-scribes the fully transient behavior of both pressure pulse propagation and mass transport. Closure relations for flow in drillstring. In the standard drift-flux approach, the closure of the system is achieved by specifying density models for each phase and a slip relation between the phases. Generally the slip relation can be pre-sented in the following form:

( ) 2LLGG1G CvvCv ++= , ............. (4) where C1 = 1 and C2 = 0 is adopted in the present work. In addition, it is necessary to provide an appropriate model for the frictional pressure loss term in the momentum equation. A frequently used expression for this term is:

MMM12

vv)d(d

2fFdz

dp

=

,.....(5)

The Equations (1)-(3) together with (4) and (5), constitute a model for the downward two-phase flow in the drillstring. Closure relations for flow in annulus. Mechanistic models have become quite popular for describing steady-state two-phase flow in producing wells. The mechanistic models pro-vide information about flow patterns, pressure drops, gas volumetric fractions and phase velocities based on knowledge of the superficial velocities, densities, viscosities, interfacial tension and well geometry. This relationship can be expressed as

( ) calculateGsLsGLGL21 v,v,,,,,,d,dM

( )drop pressure,v,v, LGG . ................ (6)

Mechanistic models have also been developed for two-phase flow in annuli10. Furthermore, it is possible to integrate mechanistic steady-state procedures into fully dynamic two-phase flow models to provide the necessary information re-garding phase velocities and pressure loss terms 11,12. Recently these models have been extended to include counter-current flow13,14, which is essential as backflow can occur in UBD and LHD operations when gravitational forces exceed pressure drop in the annulus.

Dynamic reservoir model The dynamic reservoir model is based on the transient con-stant terminal rate reservoir model15 and is similar to the res-ervoir model used in Ref. 1. When drilling, the reservoir in-flow is calculated in small segments of 1 meter length. We model the influx from the reservoir to each segment number j in zone i through the equation

))cre

t4Klog((2S

)ps(pK4)(tq

2w

ji,i

wires,iji,ji,

+

= . ...(7)

Several consecutively segments where the permeability and reservoir pressure are equal are referred to as a zone. The res-ervoir consists of several zones, and we assume that the per-meability and reservoir pressure varies in different zones. The other quantities such as porosity, etc., are kept constant over the whole reservoir. The reservoir zones are connected to the well as shown in Fig. 1. A least square methodology to estimate near-well reservoir properties The unknown reservoir properties may be estimated using the well flow simulator and measurements of pump pressure, downhole pressure and outlet rates. Using the simulator we compute values of the observables for a given set of reservoir properties. The estimation of the inflow profile is done by searching for reservoir properties that reconcile the measured data.

To estimate the unknown reservoir properties (permeabil-ity and pressure), we make the simplifying assumption that these properties are constant in n specified zones of the well. This means that the inflow profile can be expressed by a vec-tor c containing n2 parameters, or by assuming knowledge of the reservoir pressure for each zone n parameters. Hence, the simulated values are functions of the same parameters.

The estimation of the reservoir properties is done by find-ing the vector estcc = such that

2

1

i, )c(f=

m

i i

obsiy

..(8)

is minimized. In the expression above, obsiy , is the ith meas-urement, )(f ci is the simulated value for the corresponding measurement with inflow given by the parameter vector c and i is the standard deviation of the measurement i .

To avoid unphysical solutions, certain limitations must be put on the parameters. These limitations are implemented us-ing linear inequality constraints on the parameter vector, c .

The Levenberg-Marquardt method is applied to compute the unknown parameters represented by c , which minimizes the least squares expression (8) given the constraints. The im-plementation of the Levenberg-Marquardt method is based upon the description given in Ref. 16 and the handling of con-straints upon methods described in Ref. 17. Error analysis. The measurement errors will give rise to an uncertainty in the estimated inflow profile. This uncertainty can be estimated by linearization18,19 of the vector valued func-tion [ ]mii cfcF 1)()( == around the estimated point estcc = (linearized covariance analysis). In the linearized covariance analysis we assume that the errors are Gaussian and uses a linearization of )(cF around the estimated point estc in the computations.

IADC/SPE 81634 RESERVOIR CHARACTERIZATION DURING UBD: METHODOLOGY AND ACTIVE TESTS 3

The uncertainty in the estimated parameters is expressed by the covariance matrix of the estimated parameters

estc which is ( ) 11JJP = T . Here

nm

jij

i

c

,

1,1

fJ==

= ...................................(9)

is the sensitivity matrix of the simulated values with respect to a change in the reservoir properties, and is the covariance matrix of the measurement errors. We are assuming uncorre-lated measurement errors, therefore is a diagonal matrix where the ith diagonal entry is equal to 2i . The covariance matrix, P , depends both on the accuracy of the measurements and the sensitivity of the simulated values to a change in the reservoir properties. The ensemble Kalman filter to estimate near-well reservoir properties

As an alternative to the least-square methodology for esti-mating near-well reservoir properties, we evaluate the use of ensemble Kalman filter with an augmented state vector. The Kalman filter is more suitable for online estimations, as the parameter estimates are updated each time new measurements become available. In the least square approach presented pre-viously, the parameters have to be estimated after all the data are collected, i.e. after the drilling is completed.

The ensemble Kalman filter technique has been used for estimating model parameters in a well-flow simulator for un-derbalanced drilling6 and to estimate the permeability in a near-well reservoir model20,21. For the convenience of the reader we will recall basic facts about the ensemble Kalman filter, and discuss some details on the actual implementation for this study. This presentation follows closely the presenta-tion given in (Ref. 6).

The ensemble Kalman filter was first introduced in geo-physical sciences22 as an alternative to the extended Kalman filter for large non-linear models. Using the Kalman filter it is possible to combine the information obtained from the meas-urements with the model to get an improved estimate of the state vector of the system. As the state vector we use the un-known model parameters, which are the reservoir pressure and permeability for each of the reservoir zones the well is pene-trating, and discretized values for pressure, water mass flow rate and oil mass flow rate in the well. The model parameters are defined as

( ),....K,p,K,p res,2res,2res,1res,1= ,..(10) a vector of length two times the number of penetrated reser-voir zones. This gives a state vector on the form

( ) iL,iG,i Q,Q,ps = , (11) where i runs through a spatial grid defined by the numerical method.

Denote the state vector for the jth member of the ensem-ble after inclusion of the measurement by ajs . Each state vector is used as initial value to the simulator for a forward simula-tion that is run to the time when the next measurements are

taken into account. The jth state vector prior to the inclusion of the next measurements is

jaj

fj )sf(s += ,(12)

where )sf( aj denotes the updating of the state vector done by running forward the simulator to the time were new measure-ments becomes available, and j is a stochastic contribution representing the model error. The model error we use is nor-mally distributed with zero mean and covariance matrix . More details on the specification of will be given below.

To take into account the measurements we use the covari-ance matrix of the ensemble around the ensemble mean. The mean value of the ensemble is given by

==n

1jfjsn

1s ,.(13)

and the ensemble covariance matrix is T)s s()ss(

1-n1R fi

n

1jfj

n

1i= == ,..(14)

where n is the number of members in the ensemble. To combine the information from the measurements with

the model in a proper way, we need both to know the uncer-tainty in the current estimate of the state and the uncertainty in the measurements. We assume that the errors in the measure-ments are statistically independent, and with known variances. This gives a covariance matrix for the measurement errors.

For proper use of the filter an ensemble of observations is needed23. This is defined by

j+= dd j ,..(15)

where d is the actual observation and j is drawn from a nor-mal distribution with zero mean and covariance matrix . The observation vector d is related to the state vector s through the equation sHd = , for an appropriate matrix H. The state vectors in the ensemble are updated using the gain matrix 1TT )(HRHRHG += , ...(16) through the equation

)sHdG(ss fjjfj

aj += (17)

A major issue with the ensemble Kalman filter is the size of the ensemble. The optimal size of the ensemble for our ap-plication is a subject for further research. Experience in the oceanographic science24 has indicated that the filter may func-tion using a size of the ensemble in the range 100 500. We have chosen to use 100 members in the ensemble. This means that 100 forward simulations are needed. The size of this model is small compared to the models used in the oceano-graphic science, therefore it should be investigated if a reason-able performance of the filter can be achieved with smaller ensemble size than 100.

It is our experience that proper specification of the covari-ance matrix for the modeling error is crucial to get good per-formance of the filter, and we have investigated different forms of model noise. The covariance matrix for the model noise, , is diagonal, i.e. the noise added to one state is inde-pendent of the noise added to any other state.

The model noise added for the time dependent variables is very small ( 1610 ) in all the cases. Concerning the model


noise added to the model parameters we have tried different scenarios, the one described below is found as the most prom-ising, but this is a topic for further research.

In selecting the model noise of the reservoir parameters (pressure and permeability) of the zones, we take into account that there is no information in the measurements about the reservoir zones that is not penetrated. Therefore we use the covariance matrix for the modeling error, , to activate the estimation of the parameters in the actual zone. Since most information about the permeability and the reservoir pressure can be extracted from the measurements during the first time period after opening (30-90 minutes), the estimation of the parameters is nearly turned off after the zone is passed.

In the examples we present, all measurements are gener-ated synthetically by running the model with a given perme-ability and pressure, and adding noise to the obtained values to generate measurements. As covariance matrix, , for the measurement error we have used the same covariance matrix as used when generating the measurements. In a field imple-mentation, the covariance matrix for the measurement errors should take into account the uncertainty in the measurement devices, but also include uncertainty in the positioning of the measurement gauges, and inaccuracies due to the applied nu-merical method25. Examples We consider a total of four examples. The first two examples are presented to show the performance of two estimation tech-niques, the Levenberg-Marquardt methodology and the En-semble Kalman filter, both applied to a single-phase case with a reservoir consisting of 10 different zones.

In the first two examples the inflow of liquid occurs in 10 zones, each of length 30 m. Within each zone the permeability and the reservoir pressure are constant. Each zone is again divided into 1m long segments and the flow from each seg-ment is given by (7).

Table 1: Parameters used in Example 1 and Example 2 Compressibility of liquid 2.1810-9 Pa-1 Viscosity of liquid 0.05 Pa.s Wellbore radius 0.067 m Penetration rate in the formation

0.005 m/s

Uncertainty in pump pressure 0.15 % (standard deviation) Uncertainty in bottom hole pressure

0.5 % (standard deviation)

Uncertainty in flow rate 1 % (standard deviation)

In the last two examples, water and gas is injected into the drillstring and oil is produced from a reservoir which consists of 3 different zones, each of length 100 m.

For all the examples, the synthetic measurements repre-senting pump pressure, downhole pressure, liquid return rate and gas return rates are generated by adding normal distrib-uted noise to the simulation results. In all the figures presenting liquid mass flow rate, gas mass flow rate, pump pressure or bottom-hole pressure the dots

Table 2: Parameters used in Example 3 and Example 4 Compressibility of injection liquid

1.010-9 Pa-1

Compressibility of production liquid

1.1710-9 Pa-1

Viscosity of injection liquid 0.001 Pa.s Viscosity of production liquid

0.04 Pa.s

Wellbore radius 0.01 m Penetration rate in the formation

0.0083 m/s

Uncertainty in pressures 0.15 % (standard deviation) Uncertainty in flow rates 1 % (standard deviation) represents synthetic measurements and the solid line repre-sents the results from simulations with the estimated parame-ters.

In Ex. 1, Ex. 2 and Ex. 4 the downhole pressure is fluctu-ated, in an effort to oscillate the inflow from the reservoir into the well. With reference to the equation describing the tran-sient inflow from the reservoir (7), it should be possible to identify the reservoir permeability if the pressure difference is varied. This corresponds with the estimation theory within the field of adaptive control26, where the input signal to a plant is referred to as sufficiently rich of order n, if it consists of at least n/2 distinct frequencies. This means that if the system is oscillated with one frequency sinus oscillations with certain amplitude, then two parameters can be identified simultane-ously. The frequency applied to the downhole pressure is 0.5 mHz and the amplitude is 8 bar peak-to-peak. Example 1: Permeability and pressure estimation, large reservoir pressure variations. In this simple example, liquid is circulated in the well, and the same liquid is produced from the reservoir. There is no gas injection into the drillstring. A true state is generated syntheti-cally by using true reservoir pressure pres,i = [230, 260, 220, 220, 240, 280, 230, 230, 250, 220] bar . The true permeabil-ity values are Ki =[300, 600, 400, 100, 50, 800, 200, 300, 100, 400] mD. The choke opening is varied, giving sinus variations to the choke pressure. These variations to the choke pressure gives similar fluctuations in the bottom hole pressure. From Fig. 2 we can observe the fluctuations in bottom hole pressure. The fluctuations peak-to-peak is about 8 bar, with a mean of 204 bar. Since the flow into the well depends on the bottom hole pressure, the influx from the reservoir also shows these fluctuations, as presented in Fig. 3.

The actual and estimated permeabilities are presented in Fig. 4, and actual and estimated pressures are presented in Fig. 5. From Figs. 2 and 3 we observe that we are able to match the bottom hole pressure and liquid flow rate at outlet. However, we observe that the permeability and reservoir pressure not are well estimated for the different zones. Since we are able to reproduce the measured data with some discrepancy between estimated parameters and true parameters, additional informa-tion is needed to obtain reliable estimates. We observe that if the permeability of a zone is over estimated, the reservoir pressure is under estimated and vice versa.


Example 2: Permeability and pressure estimation, small reservoir pressure variations. This example is equivalent to Ex. 1, but the variations in res-ervoir pressure are smaller. As in Ex. 1, liquid is circulated in the well, and liquid is produced from the reservoir. There is no gas injection into the drillstring. A true state is generated syn-thetically by using true reservoir pressure pres,i = [215, 220, 225, 219, 220, 221, 222, 222, 218, 230] bar . The true per-meability values have not been changed, and Ki =[300, 600, 400, 100, 50, 800, 200, 300, 100, 400] mD. The fluctuations of the bottom hole pressure are generated in the same way as in Ex. 1. This causes the fluctuations of the liquid mass rate at the outlet.

The actual and estimated permeabilities are presented in Fig. 6, and actual and estimated pressures are presented in Fig. 7. Again, we observed that we were able to match the bottom hole pressure and the liquid flow rate at outlet. In this example we also observe that the estimates of reservoir permeability and reservoir pressure compares fairly well with the true val-ues. Example 3: Permeability and pressure estimation, no oscil-lations in downhole pressure. In this example, water and gas is injected into the drillstring. Oil is produced from the reservoir. A true state is generated synthetically by using true reservoir pressure pres,i = [212, 220, 216] bar . The true reservoir permeability values are Ki =[400, 100, 700] mD. The choke pressure is kept constant at 6 bar. The initial estimates of the reservoir pressures for all three zones are 215 bar, and the initial estimates of the reservoir permeabilities are 500 mD.

The ensemble Kalman filter produces on-line estimates during the drilling process. The reservoir permeability esti-mates for zone 1 while drilling is presented in Fig. 8. The res-ervoir pressure estimates for zone 1 is presented in Fig. 9. From Fig. 8 we observe that the permeability is approaching the true permeability value, but the estimate drifts away af-ter zone 2 is passed. From Fig. 9 we see that that the estimate does not approach the true value at all. Example 4: Permeability and pressure estimation, with oscillations in downhole pressure. This example is equivalent to Ex. 3, but here the choke pres-sure is oscillated. A true state is generated synthetically by using true reservoir pressure pres,i = [212, 220, 216] bar . The true reservoir permeability values are Ki =[400, 100, 700] mD. The choke pressure is varied, causing fluctuations in the downhole pressure. The initial estimates of the reservoir pres-sures for all three zones are 215 bar, and the initial estimates of the reservoir permeabilities are 250 mD.

The ensemble Kalman filter produces on-line estimates during the drilling process, and the reservoir permeability es-timates during drilling for zone 1 is presented in Fig. 10. The reservoir pressure estimates for zone 1 is presented in Fig. 11. From Fig. 11 we see that that the estimate converges to the true value.

The final estimates after all the zones have been drilled are shown in Figs. 12 and 13. We observe that the estimates in Ex. 4 fit better to the true values, than for the results in Ex. 3 with no oscillations.

Conclusions Novel methodology for reservoir characterization during un-derbalanced drilling has been presented. The methodologies are based on a transient well flow model coupled to a transient reservoir model and estimation techniques are applied to esti-mate permeability and reservoir pressure. An important aspect of the methodology presented here is that only the data usually measured during an underbalanced drilling is used as available data. The examples presented in this paper illustrate that active tests may improve the reservoir characterization during under-balanced drilling. The results from Example 1 and 2 did not show any significant differences between the results obtained with least squares estimation and with ensemble Kalman filter. The advantage of the ensemble Kalman filter is however that the results can be obtained during drilling. Nomenclature C1= Gas factor. C2= Gas holdup. c = Compressibility. c = Parameter vector. d = Measurement vector. d1 = Outer diameter of the annulus. d2 = Inner diameter of the annulus. f = Fanning friction factor. )c(fi = Simulated value at the ith sensor, as a function of

the inflow profile. )sf( aj = Forecasted value by simulator after next time step

using initial state ajs .

( ) [ ] == =mi 1i )c(fcF Simulated values as function of the parameters.

G = Kalman gain matrix. g = Gravity acceleration. H = Measurement matrix. J = Sensitivity matrix of simulated values with respect

to changes in the parameters. K = Permeability. M = Mechanistic model. m = Mass transfer between phases. n = Number of members in ensemble. p = Pressure. P = Covariance matrix of error in estimated

parameters. q i,j = Flow from segment j in zone i R = Ensemble error covariance matrix. S = Skin factor s = Member of ensemble. t = Time. ti,j= Time since opening segment j in zone i v = Velocity. obsiy , = Measured value at the ith sensor. z = Spatial coordinate. = Volumetric fraction. = 0.5772. = Measurement noise vector.


= Measurement noise covariance matrix.

i = Standard deviation of the measurement error of the ith sensor.

= Angle of inclination. = Viscosity. = Density. = Interfacial tension.

= Porosity. = Model noise vector. = Model error covariance matrix. = Covariance matrix of the measurement error.

= Model parameters.

s = Inflow segment length.

Subscripts est = Estimated.

F = Friction. G = Gas.

Gs = Gas superficial. i = Counting variable (zone/measurement). j = Counting variable (segment/ensemble member).

L = Liquid. Ls = Liquid superficial. M = Mixture. O = Oil.

res = Reservoir. w = Well.

W = Water. Superscripts a = Analyzed. f = Forecast.

Acknowledgements This work has been supported financially by the Norwegian Research Council.

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K1 K2 Ki...

Figure 1: Reservoir inflow.

0 200 400 600 800 1000 1200190

195

200

205

210

215

220Bottom hole pressure

Minutes

bar

MeasuredEstimated

Figure 2: Bottom hole pressure, Example 1: Permeability and pressure estimation, large reservoir pressure variations.

0 200 400 600 800 1000 12000

100

200

300

400

500

600

700Liquid mass rate at outlet

Minutes

l/min

MeasuredEstimated

Figure 3: Liquid mass flow rate at outlet, Example 1: Permeability and pressure estimation, large reservoir pressure variations.

1 2 3 4 5 6 7 8 9 100

100

200

300

400

500

600

700

800

900

Zone

Permeability

mD

TrueLM EstimateEnKF Estimate

Figure 4: True and estimated reservoir permeabilities, Example 1: Permeability and pressure estimation, large reservoir pressure variations.

1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

350

Zone

bar

Pressure


Figure 5: True and estimated reservoir pressures, Example 1: Permeability and pressure estimation, large pressure variations.


1 2 3 4 5 6 7 8 9 100

100

200

300

400

500

600

700

800

900

Zone

Permeability

mD


Figure 6: True and estimated reservoir permeabilities, Example 2: Permeability and pressure estimation, large pressure variations.

1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

350

Zone

bar

Pressure


Figure 7: True and estimated reservoir pressures, Example 2: Permeability and pressure estimation, large pressure variations.

0 100 200 300 400 500 600 7000

50

100

150

200

250

300

350

400

450

500

550

Minutes

mD

Reservoir Permeability, Zone 1

TrueEstimate

Figure 8: Reservoir permeability estimates in zone 1, Example 3: No oscillations in downhole pressure.

0 100 200 300 400 500 600 700200

205

210

215

220

225

230

Minutes

bar

Reservoir Pressure, Zone 1

TrueEstimate

Figure 9: Reservoir pressure estimates in zone 1, Example 3: No oscillations in downhole pressure.

0 100 200 300 400 500 600 7000

50

100

150

200

250

300

350

400

450

500

550

Minutes

mD

Reservoir Permeability, Zone 1

TrueEstimate

Figure 10: Reservoir permeability estimates in zone 1, Example 4: With oscillations in downhole pressure.

0 100 200 300 400 500 600 700200

205

210

215

220

225

230

Minutes

bar

Reservoir Pressure, Zone 1

TrueEstimate

Figure 11: Reservoir pressure estimates in zone 1, Example 4: With oscillations in downhole pressure.


1 2 30

100

200

300

400

500

600

700

800

900

Zone

Permeability

mD

TrueEx. 3, no osc.Ex. 4, with osc.

Figure 12: True and estimated reservoir permeabilities, Compari-sons between Ex. 3, no choke oscillations and Ex. 4, with choke oscillations.

1 2 3200

205

210

215

220

225

230

Zone

bar

Pressure

TrueEx. 3, no osc.Ex. 4, with osc.

Figure 13: True and estimated reservoir pressures, Comparisons between Ex. 3, no choke oscillations and Ex. 4, with choke oscilla-tions.

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