Characterization of three-phase ow in porous media …scientiairanica.sharif.edu/article_4112_860e32230ef8e8e...implicit estimation of relative permeabilities and/or capillary pressures

Scientia Iranica C (2017) 24(3), 1281{1301

Sharif University of TechnologyScientia Iranica

Transactions C: Chemistry and Chemical Engineeringwww.scientiairanica.com

Research Note

Characterization of three-phase ow in porous mediausing the ensemble Kalman �lter

S. Jahanbakhshi, M.R. Pishvaie� and R. Bozorgmehry Boozarjomehry

Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, P.O. Box 11365-11155, Iran.

Received 29 May 2016; received in revised form 14 September 2016; accepted 21 November 2016

KEYWORDSThree-phase ow;Relative permeability;Capillary pressure;Ensemble Kalman�lter;Dimensionlesssensitivity.

Abstract. In this study, the ensemble Kalman �lter is used to characterize three-phase ow in porous media through simultaneous estimation of three-phase relativepermeabilities and capillary pressures from production data. Power-law models of relativepermeability and capillary pressure curves are used and the associated unknown parametersare estimated by assimilating the measured historical data. The estimation procedure isdemonstrated on a twin numerical setup with two di�erent scenarios, in which a synthetic2D reservoir under three-phase ow is considered. In the �rst scenario, all the endpointsare assumed to be known and only the shape factors are estimated during the assimilationprocess. In the second, all the endpoints and shape factors are estimated by assimilatingobserved data. Accurate estimation of the unknown model parameters is achieved byassimilating oil, water, and gas production rates of the producers and bottom-hole pressureof the injector. Moreover, sensitivity analysis of the observations with respect to theparameters de�ning the relative permeabilities and capillary pressures shows that for themost sensitive parameters, better estimation and lower uncertainty are obtained at the endof the assimilation process.© 2017 Sharif University of Technology. All rights reserved.

1. Introduction

Several practical applications, such as enhanced oilrecovery and solute transport in aquifers, involve si-multaneous ow of two or three phases in the reservoir.Hence, it is a fundamental requirement to accuratelymodel the multiphase ow and transport in the porousmedia [1]. Available models of multiphase ow aresimply the extension of Darcy's law of single phase ow. Apparently, the only way to put the real physicsof the ow into these conventional models is throughcapillary pressures and relative permeabilities. Thus, it

*. Corresponding author. Tel.: +98 21 6616 6429;Fax: +98 21 6602 2853E-mail addresses: [email protected] (S.Jahanbakhshi); [email protected] (M.R. Pishvaie);[email protected] (R. Bozorgmehry Boozarjomehry)

is of capital importance to correctly determine relativepermeability and capillary pressure curves.

Reservoir simulation is a tool for decision-makingprocesses involved in the development and managementof petroleum reservoirs. A reservoir simulation modeltakes into account the impact of rock and uid prop-erties on the uid ow through porous media. Themodel is then used to predict the future performanceof the reservoir and assess sensitivity of these forecaststo uncertain reservoir parameters [2]. To improve thereliability of the model, it is bene�cial to incorporatehistorical measurements in the simulation model, whichis known as history matching in the petroleum litera-ture [3].

Conventionally, two-phase relative permeabilitiesand capillary pressures are obtained from displacementexperiments of core samples [4]. However, it is alsopossible to utilize production data to estimate relative

1282 S. Jahanbakhshi et al./Scientia Iranica, Transactions C: Chemistry and ... 24 (2017) 1281{1301

permeabilities and capillary pressures through historymatching [5,6]. Archer and Wong [7] were amongthe �rst investigators who determined the relativepermeability curve by matching production data from acore displacement experiment. Many other researchersapplied nonlinear optimization techniques to obtain animplicit estimation of relative permeabilities and/orcapillary pressures from production data [8]. Forexample, Sigmund and McCa�ery [9] applied non-linear least-square procedure to determine relative-permeability curves for heterogeneous core samplestaken from Alberta carbonate reservoirs. In theirstudy, only shape factors of the relative permeabilitycurves together with their standard errors were esti-mated by matching the observed recovery and pressuredata. Kerig and Watson [10] extended the previousparameter estimation procedure by considering theuse of several alternative functional forms, includ-ing the power-law and cubic spline models. Theyeventually concluded that cubic splines were more exible than power-law models. These optimizationalgorithms require calculation of the Jacobian and/orHessian matrix of the objective function (that is de�nedas the di�erence between the simulated results andthe measurements). Henceforth, high computationalcost and convergence to the local minima are twocommon shortcomings of these gradient-based meth-ods [11]. Some investigators have also employedevolutionary algorithms, such as Genetic Algorithm(GA) and Simulated Annealing (SA) [12,13]. How-ever, in these cases, there is no guarantee to �ndan optimal solution in a �nite number of gener-ations, and the convergence rate is relatively low.Recently, the widely used method in assisted historymatching processes is the Ensemble Kalman Filter(EnKF), which is an e�cient, recursive, and unbiased�lter [14].

The EnKF is known as an attractive alterna-tive for history matching problems because of itsease of implementation and computational e�ciencyfor models with large number of variables. Also,EnKF is a sequential data assimilation scheme andis well suited for continuous reservoir model updatingand closed-loop reservoir management problems [15].Moreover, uncertainty assessment in reservoir char-acterization and performance predictions is becomingmore important and, lately, more researchers have gotinvolved in this area [13,16]. Since the EnKF canprovide a measure of uncertainty along with historymatching, it is also getting more attractive for theresearchers working on uncertainty assessment. TheEnKF is a sequential Monte Carlo method in whichmodel states and covariance matrix are stored andpropagated through an ensemble of model realiza-tions. Compared to the other available methods, theEnKF requires no information about the gradient,

which makes it more exible to adapt to di�erenttypes of model parameters and several commercialreservoir simulators [17]. The EnKF advantages aswell as its limitations are thoroughly discussed inEvensen [18].

The EnKF has been applied to estimate variousreservoir parameters, such as porosity and permeability�elds [19], uid contacts [20], and grid block transmissi-bility. Li and Yang [21] applied the EnKF to estimatemultiple petro-physical parameters for the PUNQ-S3model, which included estimation of vertical and hori-zontal permeability, porosity, and three-phase relativepermeability curves. These curves were generatedusing power-law models. Unknown parameters of thesemodels are of two types: shape factors and endpoints(which are discussed in Section 2.1). However, tosimplify the model, only shape factors of these modelswere considered in the estimation process. They con-cluded that relative permeability could be estimatedwith good accuracy when the absolute permeability�eld was known. When relative permeability wastuned simultaneously along with porosity and absolutepermeability, the accuracy of the estimated relativepermeability was poor.

Li et al. [22] used a B-spline model for the two-phase oil and water relative permeabilities. To ensuremonotonicity of the relative permeability curves, atransformation was implemented to map the originalparameters to the pseudo-parameters. These pseudo-parameters were sequentially updated and adjustedtoward their reference values, while the variance ofthe estimation decreased throughout the assimilationprocess using the EnKF. However, relative root-mean-square error of the parameters decreased more sig-ni�cantly before water breakthrough, in which theinjected water reached the producing wells. Therefore,they concluded that the observation data collectedprior to water breakthrough contributed more to theestimation of the relative permeability curve thanthose collected after water breakthrough. Seiler etal. [23] found that a signi�cant improvement in theresults of history matching was obtained by updatingthe relative permeabilities in addition to porosity andpermeability �elds and initial uid contacts. Oncemore, power-law models of relative permeability curveswere used and both the endpoints and the shape factorswere estimated through history matching with theEnKF.

Zhang et al. [24] used the EnKF to simultaneouslyestimate relative permeabilities and capillary pressuresthrough matching the historical data, including wa-ter saturation pro�le, cumulative oil production, andpressure drop. Again, power-law model was used forboth relative permeabilities and capillary pressures.They concluded, albeit qualitatively, that historicaldata were more sensitive to the relative permeabilities,

S. Jahanbakhshi et al./Scientia Iranica, Transactions C: Chemistry and ... 24 (2017) 1281{1301 1283

especially water relative permeability, than capillarypressures. Their study, however, was limited to water ooding experiments in two-phase oil-water systems.Later, Zhang and Yang [25] used the EnKF to esti-mate three-phase relative permeabilities and capillarypressures for tight formations by assimilating historicaldata including oil, water, and gas production ratesand cumulative oil production. Power-law modelsof relative permeabilities and capillary pressures wereused and these prior models were updated during theassimilation process. Nevertheless, they considered theentry capillary pressures (i.e., endpoints of the power-law model) to be �xed to prevent poor estimation,and proposed to predetermine these values accordingto the prior knowledge. Moreover, the estimated shapefactors of the capillary pressure curves were less accu-rate than the estimation of the relative permeabilityparameters, which was again qualitatively attributedto the less sensitivity of historical data to capillarypressure parameters.

This study is motivated by the work of Li etal. [26]. They considered a 2D, three-phase owsystem and estimated endpoints and shape factors ofthe relative permeabilities by assisted history matchingusing the EnKF. They also found that distribution ofthe updated parameters tends to be Gaussian even ifinitial parameters are uniformly distributed. This isdue to the fact that all the distributions are bound to beGaussian in the EnKF. Furthermore, they proposed tointegrate production data from multiple samples in theassimilation process to obtain more accurate estimationof multiphase ow parameters. The main contributionof this study is to advance the topic beyond the e�ortsof Li et al. [26] by including endpoints and shape factorsof three-phase capillary pressure curves in the unknownparameter set. In other words, this study demonstratesjoint estimation of three-phase relative permeabilityand capillary pressure curves by assimilating observa-tion data. Furthermore, sensitivity of the observationdata to the unknown model parameters is analyzedquantitatively, which clearly justi�es the estimationresults and the associated uncertainties of the unknownmodel parameters.

Here, an ensemble-based assimilation techniqueis developed and successfully applied in the simulta-neous estimation of relative permeabilities and capil-lary pressures under three-phase ow condition in theporous media. Power-law models are used for bothrelative permeability and capillary pressure curves.Unknown model parameters of these models are ad-justed progressively toward their reference values byassimilating observed historical data, including oil,water, and gas production rates of the producers andbottom-hole pressure of the injector. The estima-tion technique is validated in a synthetic �ve-spot2D three-phase reservoir with two di�erent scenar-

ios. In the �rst, all the endpoints are assumed tobe known and only the shape factors are estimatedduring the assimilation process. In the second, allthe endpoints (including endpoints of the capillarypressure curves) and shape factors are estimated byassimilating observed data. Furthermore, the impactsof availability of the endpoints and ensemble sizeare evaluated, and quantitative sensitivity analysis ofthe observation data to the unknown parameters isperformed.

2. Methodology

2.1. Relative permeability and capillarypressure curves

Direct measurement of three-phase relative permeabil-ities is di�cult and time consuming; hence, empiricalmodels are widely used [27]. A comprehensive descrip-tion of the available correlations for the three-phaserelative permeabilities along with their shortcomingsis provided by Oliveira and Demond [28]. Based onthe channel ow theory, the relative permeability ofwater and water-oil capillary pressure in a three-phasesystem are similar to those of the two-phase water-oil case [29]; and the relative permeability of gas andgas-oil capillary pressure are the same as those for thetwo-phase gas-oil case. The relative permeability ofoil is then determined by interpolating two-phase data.Therefore, appropriate models of two-phase relativepermeability and capillary pressure curves are initiallyneeded.

Representative models of these curves are clas-si�ed under two categories: parametric models andnon-parametric models [26]. Non-parametric models,such as B-splines, have larger degrees of freedom,which make them more adaptable. However, if thesemodels are used in the history matching process, avariable transformation is required to ensure that es-timated relative permeabilities and capillary pressuresare monotonic functions of saturation. Moreover, val-ues estimated for a B-spline model frequently deviatefrom the reference values and, thus, the estimationsu�ers from non-uniqueness [11]. Parametric models,such as power-laws, are the most widely used modelsin multiphase ow studies.

Furthermore, the parameters de�ning a power-law relation carry more engineering information, andone may discuss their uncertainty in regard to timeor sensitivity to the observation data. In this study,the empirical Corey-type (power-law) models are usedto represent the two-phase relative permeability andcapillary pressure curves. These models are simpleand they also capture the trends of many experimentalmeasurements.

For the oil-water system, relative permeabilityand capillary pressure functions are:


8>>>>>><>>>>>>:kro = ao (1� SwD)bo

krw = aw (SwD)bw

Pcow = P �cow (1� SwD)ncow

SwD = Sw�Swc1�Swc�Sorw

(1)

where, kro and krw are relative permeabilities of oil andwater, respectively; Sw is the water saturation; Swc isthe critical (or irreducible) water saturation; Sorw isthe residual oil saturation; and ao and aw are relativepermeabilities of oil at Sw = Swc and water at Sw =1� Sorw, respectively. Exponents bo and bw are shapefactors of relative permeability curves; P �cow is the oil-water capillary pressure at Sw = Swc; ncow is the shapefactor of the capillary pressure curve; and SwD is thedimensionless water saturation.

For the gas-oil system, relative permeability andcapillary pressure functions are:8>>>>>><>>>>>>:

krog = aog (1� SgD)bog

krg = ag (SgD)bg

Pcgo = P �cgo (SgD)ncgo

SgD = Sg�Sgc1�Swc�Sgc�Sorg

(2)

where, krog and krg are relative permeabilities of oiland gas, respectively; Sg is the gas saturation; Sgcis the critical gas saturation; Sorg is the residual oilsaturation; and aog and ag are relative permeabilitiesof oil at Sg = Sgc and gas at Sg = 1 � Swc � Sorg,respectively. On the other hand, exponents bog and bgare shape factors of relative permeability curves; P �cgois the gas-oil capillary pressure at Sg = Sgc; exponentncgo is the shape factor of the capillary pressurecurve; and SgD is the dimensionless gas saturation.Consequently, parameters of the relative permeabilitiesand capillary pressures are divided into two types:shape factors (bo, bw, ncow, bog, bg, and ncgo) andendpoints (ao, aw, P �cow, aog, ag, and P �cgo). Essentially,ao = aog = 1 [11,26], because relative permeabilityis de�ned as the ratio of the e�ective permeability tothe absolute oil permeability at the irreducible watersaturation.

The oil relative permeability for the simultaneous ow of oil, water, and gas in the reservoir is calculatedby Stone's model II method [30]. Stone proposed twoempirical models of three-phase relative permeabilitythat are the most popular in the petroleum indus-try [31]. These models are considered as the industrystandard and have become a benchmark against whichother models are compared. Stone's model II is amodi�ed version of Stone's model I and better predictsoil relative permeability in three-phase ow systems,especially for water-wet reservoir rocks [32]. Moreover,

it is not required to estimate the minimum residualoil saturation (Som) in this model. Stone [30] furtherveri�ed the validity of model II by comparing the resid-ual oil saturation data predicted from this model withthe experimental measurements given by Holmgren andMorse [33]. In addition, hysteresis e�ects can be easilytaken into consideration by using the appropriate two-phase data.

The form of Stone's model II used in this studyis the modi�cation proposed by Aziz and Settari [34],which is as follows [27]:

kro=ao��

kroao

+krw��

krogaog

+krg��(krw+krg)

�: (3)

Stone's model II gives relative permeability of oil asa function of Sw and Sg. Negative values may beobtained from this model, albeit they have no physicalmeaning. Thus, kro is set to zero whenever the modelpredicts a negative value [28].

2.2. State-space representation of systemThe discrete nonlinear state-space formulation of thesystem is represented as [35]:

xk = Mk (xk�1;uk) + wk; (4)

dk = Hkxk + Dk; (5)

where Mk is the reservoir simulation model; xk =[s;m; ~d]Tk is the (augmented) state vector wherein sis ns � 1 system states (which consist of pressures,water and gas saturations, and solution gas/oil ratiosfor each grid block); m is nm�1 model parameters thatencounter constant dynamics (i.e., mk = mk�1); and ~dis nd�1 predicted observations. uk is the control inputsor boundary conditions and wk is the dynamic systemnoise (or model error), which is a white Gaussian noisewith covariance matrix Qk of size nd�n (n is the size ofthe state vector, n = ns+nm+nd), i.e. wk � N(0;Qk).dk is the vector of real observations of size nd � 1 andmatrix Hk of size nd � n is the operator mapping thestate space to the observation space [18]:

Hk � �O I�k ; (6)

where O is the null matrix of size nd � (ns + nm)and I is the identity matrix of size nd � nd. Also,Dk = ["1; "2; � � � ; "nd ]k is the vector of measurementperturbations of size nd � 1, which are sampled froma white Gaussian distribution with covariance matrixRk of size nd � nd, i.e. Dk � N(0;Rk).

As the state vector also includes the predictedobservations, the relation between dk and xk canalways be expressed by Eq. (5), regardless of the typeof the relationship that exists between dk and sk. Inaddition, because there is little theoretical knowledge of


model error statistics, a rigorous procedure to properlyaccount for modeling errors has not been establishedfor multiphase ow in porous media [36]. Moreover,a common assumption in the parameter estimationapplications is that the dominating errors in the modelare due to the uncertainty in the parameters and,thus, dynamic model error is neglected (Q = On�n).Reservoir simulation model together with the initialand boundary conditions is described in Section 3.1.

2.3. The ensemble Kalman �lterThe EnKF is a sequential Monte Carlo method in whicha �nite number of ensemble members are used to fore-cast error statistics and to approximate Kalman gainmatrix for updating model variables [37]. The EnKF isthe most suitable for high-dimensional discrete systemswherein storage and manipulation of the error covari-ance matrix are impossible or impractical [38]. In theEnKF, model states and system error covariance arestored and manipulated implicitly through an ensembleof model realizations. The EnKF is an e�cient andeasy-to-implement data assimilation method and hasbeen successfully applied in history matching problemsto estimate various parameters of the reservoir. Detailsof the EnKF theory and algorithms are described byEvensen [18]. The EnKF contains two consecutivesteps; the �rst is the forecast step in which the reservoirsimulator uses the updated model realizations from theprevious time step to generate forecasted state vectorfor the current time where new observation data areavailable; in the second step, known as the assimilationstep, the forecasted state vector is assimilated with newobservations under a Bayesian framework.

For simplicity, time index of the variables isremoved. The forecast ensemble perturbations oranomalies are obtained based on calculation of theirdi�erence from the mean of the ensemble. Then, theith vector of anomalies becomes Af

i = xfi � xf :

Af = [xf1 � xf ;xf2 � xf ; � � � ;xfN � xf ]; (7)

in which xfi is the ith forecast vector, N is thenumber of ensemble members, and xf is the forecastof ensemble mean, as follows:

xf =1N

NXi=1

xfi : (8)

The forecast of error covariance matrix is then calcu-lated as:

Pf =1

N � 1

NXi=1

�xfi � xf

��xfi � xf

�T=

1N � 1

Af �Af�T ; (9)

wherein superscript T denotes matrix transpose. In

the assimilation step, the assimilated ensemble mean,ensemble perturbations, and ensemble members areobtained by employing Kalman �lter equation:

xa = xf + K(d�Hxf ); (10)

Aa = Af + K�D�HAf� ; (11)

xai = Aai + xa: (12)

Here, K is the Kalman gain matrix of size n � nd.Kalman gain is calculated based on the minimizationof the analysis error covariance matrix [35]:

Pa =1

N � 1

NXi=1

(xai � xa) (xai � xa)T

=1

N � 1Aa(Aa)T ; (13)

which results in [38]:

K = PfHT �HPfHT + R��1

; (14)

Pa = (I�KH) Pf : (15)

At each assimilation stage, the system states, modelparameters, and predicted observations are updatedsimultaneously.

Even though there is a great interest to usethe EnKF for reservoir history matching, the methodsu�ers from a couple of shortcomings. A basic assump-tion of the EnKF is that the updated state variables(primary reservoir-simulation variables) and updatedstatic model parameters are statistically consistent.That is, at each assimilation step, updated ensembleof state variables is same as the ensemble of statevariables obtained by running the reservoir simulatorwith updated ensemble of parameters from time zero.This consistency has been proved only for systems withGaussian statistics in which there is a linear relationbetween model parameters and state variables [39-41].However, the reservoir simulation equations are highlynonlinear and, consequently, this assumption can beinvalid. A procedure to avoid inconsistency is to re-run the simulator with the latest updated ensembleof model parameters from time zero to the currenttime after each assimilation step. This procedure isreferred to as half-iteration EnKF [20,40]. Obviously,the simulation re-run increases the computational cost.Thus, Wang et al. [20] proposed to do a re-run onlyif the relative change in the ensemble mean of themodel parameters is larger than a threshold value [17].However, in this study, we have considered a smallsynthetic reservoir model, which is described in Sec-tion 3.1. Subsequently, computational burden is nota critical challenge here and a simulation re-run isperformed after each assimilation cycle.


2.4. Half-iteration ensemble-basedassimilation procedure

The implementation road map for estimation of relativepermeability and capillary pressure curves is summa-rized below:

- Step 1: Representative models of relative perme-abilities and capillary pressures are selected andinitialized. A number of N realizations from theprior knowledge of the unknown parameters aregenerated;

- Step 2: All the ensemble parameter sets are propa-gated with the reservoir simulator to generate thepredicted observations for the next measurementtime. Therefore, the state vector components foreach ensemble member become:

s = [Po; Sw; Sg; Rs]T for each grid block; (16)

m=[bo; aw; bw; P �cow; ncow; bog; ag; bg; P �cgo; ncgo]T ;(17)

~d = [WOPRi;WWPRi;WGPRi;WBHPj ]T ;8<:i 2 fProduction wellsg

j 2 fInjection wellg (18)

- Step 3: When new observations become available,the EnKF algorithm is used to assimilate the realobservations with the predicted ones, meanwhile rel-ative permeability and capillary pressure parametersare updated;

- Step 4: The reservoir simulator is re-run with theupdated ensemble of model parameters from timezero to the current time. Consequently, updatedensemble of system states, s, and predicted obser-vations, ~d, which is consistent with the updatedensemble of model parameters, is obtained;

- Step 5: The assimilation procedure is repeated untilno more observations are available.

To evaluate the �lter performance, Relative Root-Mean-Square Error (RRMSE) is calculated, whichillustrates the dimensionless di�erence between theaverage estimated values and reference (or true) values,as:

RRMSE =

vuut 1Np

NpXi=1

�xavgi � xtrue

ixtruei

�2

; (19)

wherein xavgi and xtrue

i are the average and true valuesof the ith model parameter, respectively, and Np isthe total number of to-be-estimated parameters. Asdiscussed previously, a twin numerical setup has been

implemented for assessment purposes. Consequently,true values of to-be-estimated parameters are known(which will be discussed in Section 3.2) in order to ex-amine the estimation performance. If the assimilationconverges, RRMSE will decrease along the simulationtime interval. However, RRMSE is just a traditionaltool that has been employed in many other studies anddoes not guarantee the success of assimilation [5,42].

3. Case study

3.1. Reservoir model descriptionA 2D synthetic reservoir is considered to simulta-neously estimate relative permeabilities and capillarypressures under three-phase ow condition. Thesynthetic model is built in the commercial �nite-di�erence-based simulator, ECLIPSE. ECLIPSE is awell-established and industry-reference simulator thathas been widely applied as the forward model in the as-sisted history matching processes [12,15]. Accordingly,in this study, ECLIPSE software with a fully-implicitscheme is considered as the forward simulation model.

The heterogeneous reservoir model is divided intoa uniform grid system of size 21�21�1 with dimensionsof 100 � 100 � 50 ft. No- ow boundary conditionis assumed at all sides of the reservoir model, andthe phases present in the reservoir are oil, water, andgas. Absolute permeability map is generated usingsgsim module of SGeMS software [43] with isotropicGaussian covariance in which major and minor rangesare set to 5 grid blocks. Figure 1 depicts the absolutelog-permeability map, whose mean and variance areequal to 6 and 1 mD, respectively, and the wellpatterns. Porosity of the formation is 10%. Absolutepermeability and porosity are known and �xed duringthe assimilation process.

System states consist of (oil) pressure, water andgas saturations, and solution gas/oil ratio for each grid

Figure 1. Absolute log-permeability �eld (mD) and welllocations of the synthetic 2D reservoir.


Table 1. Well patterns and speci�cations.

I1 P4 P3 P2 P1 WellWell type Producer Producer Producer Producer InjectorX location 2 20 20 2 11Y location 2 2 20 20 11Constraint 1500 psi 1500 psi 1500 psi 1500 psi 5000 STB/D

block. The initial reservoir pressure is 3600 psia, whichis equal to the bubble point pressure of the reservoir uid. Thus, there is no free-gas initially present insidethe reservoir and initial gas saturation equals zero.Also, initial solution gas/oil ratio is 1.390 MSCF/STBand initial water saturation is 0.3.

The synthetic reservoir is a �ve-spot in whichthere are four production wells (P1, P2, P3, and P4)at the corners of the domain and a water injectionwell (I1) at the center of the reservoir. Water isinjected through the injection well at a constant rate of5000 STB/D and all producers have constant bottom-hole pressure of 1500 psia. Well (block) locations andspeci�cations are listed in Table 1. Oil, water, andgas production rates of the producers and bottom-holepressure of the injection well are used as the productiondata; these provide 13 data points for each assimilationcycle. Because real historical data from the �eld side isnot available, we have to generate synthetic historicaldata from the reservoir simulator using the true modelparameters. Total production history is 720 days andmeasurements are available every 60 days; hence, wehave 12 assimilation cycles. To mimic actual noisymeasurements [44], a white Gaussian noise is addedto the synthetic historical data. Standard deviation ofthis noise is 10 psi for bottom-hole pressure, 5 STB/Dfor oil and water production rates, and 8 MSCF/D forgas production rate.

3.2. Reference relative permeability andcapillary pressure curves

Throughout this study, endpoint saturations are as-sumed to be known. We consider Sorw = Sorg = 0:11,Swc = 0:152, and Sgc = 0. Conventionally, criticalgas saturation (Sgc) is assumed to be zero wheneverthe reservoir is initially undersaturated [25]. Similarvalues have been incorporated for endpoint saturationsby other researchers [45].

Mean values and standard deviations of the un-known model parameters are considered as the priorknowledge (prior probability) in Table 2. Also, sometypical arbitrary values are selected for the true valuesof these parameters. They, however, should be inthe ranges covered by Gaussian distributions whosemean and standard deviations are speci�ed in Table 2.Therefore, reference relative permeability and capillarypressure curves are generated with these true valuesand shown in Figures 2 and 3, respectively.

Table 2. Reference values of the relative permeabilityand capillary pressure parameters.

Parameter Averagevalue

Standarddeviation

Referencevalue

b0 3.00 0.80 3.70aw 0.55 0.15 0.40bw 5.00 1.00 4.00P �cow 100.00 10.00 90.00ncow 3.50 0.80 4.30bog 4.00 0.70 4.70ag 0.70 0.20 0.85bg 2.20 0.50 1.70P �cgo 50.00 5.00 55.00ncgo 4.00 0.70 4.70

Reference model is the simulation model in whichall the unknown parameters are set to their true values,and will be used to evaluate the assimilation process.True observations are also obtained from the reservoirsimulator using this reference model. Subsequently,estimated model parameters and updated observationsare compared with their true values.

4. Results and discussions

4.1. Scenario 1: Estimation of shape factorsIn this scenario, we assume all the endpoints to beknown. Hence, there exist only 6 parameters to beestimated by assimilation of the observation data withthe dynamic model:

m = [bo; bw; ncow; bog; bg; ncgo]T : (20)

A typical number of 100 ensemble members are con-sidered for this scenario. Each ensemble member isgenerated from a set of random numbers with Gaussiandistribution whose mean and standard deviation arelisted in Table 3. That is, a number of 100 randomnumbers are generated for bo from a normal distri-bution with mean and standard deviation equal to3 and 0.8, respectively, [bo � N(3; 0:8)]. The sameprocedure is repeated for bw, ncow, bog, bg, and ncgousing their own mean and standard deviation values.This provides a number of 100 di�erent realizations forthe model parameters, which are then used in the initialensemble.


Table 3. Initial, estimated, and reference values of the relative permeability and capillary pressure parameters inScenario 1.


STD Referencevalue

Averageinitial

Estimatedvalue

FinalSTD

Relativeerror (%)

bo 3.00 8:00� 10�1 3.70 3.06 3.70 1:36� 10�2 0.00bw 5.00 1:00� 100 4.00 5.16 4.00 1:42� 10�2 0.00ncow 3.50 8:00� 10�1 4.30 3.67 4.31 5:00� 10�2 0.23bog 4.00 7:00� 10�1 4.70 3.97 4.67 2:57� 10�1 0.64bg 2.20 5:00� 10�1 1.70 2.19 1.70 1:57� 10�2 0.00ncgo 4.00 7:00� 10�1 4.70 3.94 4.64 4:32� 10�1 1.28

Figure 2. Reference relative permeabilities and envelopesof initial ensembles for (a) the oil-water system and (b)the gas-oil system.

The boundary envelopes of the initial ensemblemembers are depicted in Figures 2 and 3. Theseenvelopes are generated with the maximum and mini-mum values of the ensemble members and all the otherinitial curves will be within the range covered by these

Figure 3. Reference capillary pressure and envelopes ofinitial ensembles for (a) the oil-water system and (b) thegas-oil system.

envelopes. As all of the endpoints (i.e., aw, P �cow, ag,P �cgo) are assumed to be known and constant in thisscenario, there is no spread at the endpoints of thecurves.

The averages of the estimated relative permeabil-


Figure 4. Estimated and reference relative permeabilitycurves for (a) the oil-water system and (b) the gas-oilsystem in Scenario 1.

ity and capillary pressure curves are compared withtheir corresponding references in Figures 4 and 5.Clearly, an accurate evaluation of the relative perme-abilities and capillary pressures is performed with themaximum relative error of 1.28%. The initial, esti-mated, and reference parameters are listed in Table 3.

Figure 6 displays evolution of the relative per-meability and capillary pressure parameters versusassimilation time. The dots represent mean valuescalculated by averaging ensemble members and errorbars show standard deviations. Before the assimilation,there is a large error and uncertainty for all the param-eters. As assimilation proceeds, parameters get closerto their reference values while standard deviations(uncertainties) of the updated parameters are usuallyreduced. This is in agreement with other investigationsin relation to the estimation of the relative permeabil-ities and capillary pressures [11,21,24].

Although the uncertainties in the parameters bo,bw, ncow, and bg have been reduced signi�cantly inFigure 6, the shape factors of the capillary pressurecurve and oil relative permeability in the gas-oil system

Figure 5. Estimated and reference capillary pressurecurves for (a) the oil-water system and (b) the gas-oilsystem in Scenario 1.

(ncgo and bog) still have large uncertainty. Actually, forthe most sensitive parameters, better estimation andlower uncertainty are obtained during the assimilationprocess. Thus, it can be inferred that parameters bo,bw, ncow and bg have more profound e�ects on the uid ow through porous media and the production data.Furthermore, larger uncertainty and larger relativeerror for the estimation of ncgo and bog may have rootin the fact that production data contain informationre ecting low free gas saturation inside the reservoir.Figure 7 accentuates that maximum free-gas saturationin the reservoir for all ensemble members does notexceed 0.18.

Moreover, in Section 4.3, the sensitivity of theobservation data to the unknown model parameterswill be discussed thoroughly and it will be illustratedthat estimation of the most sensitive parameters iscloser to their reference values and, also, uncertainty ofthese parameters is decreased signi�cantly at the endof assimilation.

Average initial values of the relative permeability


Figure 6. Estimation of the relative permeability and capillary pressure parameters versus assimilation time: (a) bo, (b)bw, (c) bog, (d) bg, (e) ncow, and (f) ncgo, in Scenario 1.

and capillary pressure parameters are listed in Table 3,which are relatively close to the mean values of theirdistributions. For the �rst assimilation cycle in Fig-ure 6, parameters bo, bw, and bg are overestimated whilethe other parameters are underestimated relative totheir corresponding reference values. This observationcan be better justi�ed using the Kalman �lter equation(Eq. (10)), in which the assimilated ensemble mean isequal to the forecasted ensemble mean plus the updateterm, K(d�Hxf ). This term makes the model param-eters adjust in some way that closer agreement betweenthe predicted observations and real measurements isobtained when assimilated parameters are utilized inthe reservoir simulator. Thus, any increase or decreasein the assimilated model parameters is interpretedthrough the update term in the Kalman �lter equation.

Figure 8 shows the trend of RRMSE variation, as

calculated by Eq. (19), versus assimilation time. Itis a decreasing trend, which means that the overallestimation error is reduced as the assimilation pro-ceeds. Results of the history matching process arepresented in Figure 9. At each assimilation cycle,updated ensemble of model parameters is used togenerate its corresponding relative permeability andcapillary pressure curves. Then, reservoir simulator isre-run with the updated curves to calculate oil, gas,and water production rates for all the producers andthe bottom-hole pressure of the injector. Also, beforethe assimilation, initial ensemble members were used inthe simulation model to forecast reservoir performanceand assess the uncertainty. The gray and the blackcurves denote the results from the initial and updatedensembles, respectively, and the red line representsthe true observations from the reference model. The


Figure 7. Maximum free gas saturation in the reservoirversus assimilation time.

Figure 8. Relative root-mean-square error versusassimilation time for Scenario 1.

updated ensemble members perform fairly well andclosely match the true observations for di�erent typesof production data.

Before the breakthrough of water, no water isproduced from the production wells and, thus, thereis no data of water production. However, water isinjected via the injection well throughout the produc-tion history. Bottom-hole pressure of the injectionwell is internally related to the relative permeabilityand capillary pressure curves through the simulationequations. As the bottom-hole pressure of the injectionwell is also included in the historical data used for theassimilation process, it will provide enough informa-tion for estimating parameters of water ow functionsduring the pre-breakthrough period.

4.2. Scenario 2: Estimation of shape factorsand endpoints

When the endpoints are also unknown, there exist10 parameters to be estimated by assimilation of theobservation data with the dynamic model, which are

shown as vector m in Eq. (17). This makes the historymatching problem more complicated and challengingdue to its higher degree of freedom. Apparently, thisrequires larger number of ensemble members to makethe �lter converge to appropriate estimates. If theensemble size is not large enough to approximate theprior covariance properly, the ensemble members maycollapse to a single one or the updated ensemble mem-bers may get improperly tuned with the observationdata; either case results in a poor data match and, thus,�lter divergence occurs. As Figure 10 depicts, using anumber of 100 ensemble members (N = 100), similar towhat used previously, led to the divergence of the �lter.This is despite the fact that RRSME was decreasing atthe early assimilation cycles; however, after a while,RRMSE started to increase, which ultimately resultedin �lter divergence. Subsequently, a twice largerensemble size was examined (N = 200), which madethe �lter converge and RRMSE decrease along theassimilation time. The same approach is implementedby some other researchers, such as Li et al. [26],in order to determine the appropriate large-enoughensemble size. Additionally, to our knowledge, noresearchers have provided an analytical error analysisto deterministically indicate bounds on the numberof realizations required for convergence, and this iscertainly application-dependent.

Each ensemble is generated from a Gaussiandistribution with the mean and standard deviationvalues listed in Table 4. As with the �rst scenario,the averages of the estimated relative permeabilityand capillary pressure curves have been comparedwith their corresponding reference values in Figures 11and 12, respectively. Obviously, more accurate estima-tion of the relative permeability and capillary pressurecurves is obtained for the oil-water system. This will bejusti�ed by calculation of dimensionless sensitivities inSection 4.3. For the endpoint of water and gas relativepermeabilities, cut-o� values of aw = 1:0, and ag = 1:0have been used and all the greater values are set to 1.0during simulation.

Except for P �cgo, the estimations of the otherunknown parameters at the end of assimilation aremuch closer to their corresponding reference valuesthan to the initial ones. The initial, estimated, andreference parameters are also listed in Table 4. Theestimation error after the assimilation is small enoughand its maximum relative error is about 8.89%.

Figures 13 and 14 represent evolution of therelative permeability and capillary pressure parametersversus assimilation time for the oil-water and gas-oilsystems, respectively. Before the assimilation, there isa large error and large uncertainty for all the param-eters. During the assimilation, model parameters getcloser to their reference values except for P �cgo in whichthe initial and estimated values are almost the same


Figure 9. History matching results for oil production rate, water production rate, gas production rate, and bottom-holepressure in Scenario 1.

(see Table 4). Furthermore, estimation of ncgo has beenonly improved at the �rst assimilation cycle, while noimprovement is achieved during the next cycles. Ingeneral, estimation of the most sensitive parametersis closer to their reference values and uncertainty ofthese parameters is decreased signi�cantly at the end

of assimilation. Therefore, it can be inferred thatparameters bo, aw, bw, ncow, and bg have more profounde�ects on the production data. Later, the sensitivity ofthe observation data to the unknown parameters willbe discussed. Also, the larger relative error for theestimation of the relative permeabilities and capillary


Table 4. Initial, estimated, and reference values of the relative permeability and capillary pressure parameters inScenario 2.


STD Referencevalue

Averageinitial

Estimatedvalue

FinalSTD

STDratio�

Relativeerror (%)

bo 3.00 8:00�10�1 3.70 3.01 3.71 9:10�10�3 1:14�10�2 0.27aw 0.55 1:50�10�1 0.40 0.55 0.40 4:40�10�3 2:93�10�2 0.00bw 5.00 1:00�100 4.00 5.07 3.99 3:73�10�2 3:73�10�2 0.25P �cow 100.00 1:00�101 90.00 98.79 89.56 3:97�100 3:97�10�1 0.49ncow 3.50 8:00�10�1 4.30 3.50 4.29 6:00�10�2 7:50�10�2 0.23bog 4.00 7:00�10�1 4.70 3.96 4.95 3:18�10�1 4:54�10�1 5.31ag 0.70 2:00�10�1 0.85 0.71 0.88 5:74�10�2 2:87�10�1 3.53bg 2.20 5:00�10�1 1.70 2.22 1.73 3:72�10�2 7:44�10�2 1.76P �cgo 50.00 5:00�100 55.00 50.02 50.11 4:83�100 9:66�10�1 8.89ncgo 4.00 7:00�10�1 4.70 4.03 4.91 4:98�10�1 7:11�10�1 4.47

�: Ratio of the �nal STD to the initial std.

Figure 10. Relative root-mean-square error versusassimilation time for Scenario 2.

pressure of the gas-oil system may be due to the factthat production data contains information re ectinglow free-gas saturation (see Figure 7).

Similar to Figure 9, at each assimilation cycle,reservoir simulator is re-run with the updated relativepermeability and capillary pressure curves to calculateoil, gas, and water production rates for all the pro-ducers and the bottom-hole pressure of the injector.In Figure 15, the gray and the black curves denotethe results from the initial and updated ensembles,respectively; and the red line represents the trueobservations from the reference model. It is clearthat updated ensemble members closely match the trueobservations for di�erent types of production data andthis further veri�es that relative permeabilities andcapillary pressures are well adjusted during assimila-tion using the EnKF.

4.3. Sensitivity analysisIn this section, the sensitivity of the observation datato the relative permeability and capillary pressure

Figure 11. Estimated and reference relative permeabilitycurves for (a) the oil-water system and (b) the gas-oilsystem in Scenario 2.

parameters obtained in the second scenario is analyzed.Dimensionless sensitivity coe�cients provide a relativemeasure of how di�erent data a�ect estimates of themodel parameters. Dimensionless sensitivity of obser-vation di to model parameter mj is given by [45,46]:


Figure 12. Estimated and reference capillary pressure curves for (a) the oil-water system and (b) the gas-oil system inScenario 2.

Figure 13. Estimation of the relative permeability and capillary pressure parameters versus assimilation time for theoil-water system: (a) bo, (b) bw, (c) aw, (d) ncow, and (e) P �cow, in Scenario 2.


Figure 14. Estimation of the relative permeability and capillary pressure parameters versus assimilation time for thegas-oil system: (a) bog, (b) bg, (c) ag, (d) ncgo, and (e) P �cgo, in Scenario 2.

Si;j =@di@mj

�mj�di

; (21)

wherein �2mj is the prior variance of model parameter

mj and �2di is the variance of measurement error for the

ith observation. The derivatives involved in Eq. (21)are calculated using a �ve-point stencil as [47]:

@di@mj

=�fi;1 + 8fi;2 � 8fi;3 + fi;4

12�mj; (22)

in which:fi;1 = di at

m=[m1; � � � ;mj�1;mj+2�mj ;mj+1; � � � ;mNp ]T ;

fi;2 = di at

m=[m1; � � � ;mj�1;mj+�mj ;mj+1; � � � ;mNp ]T ;

fi;3 = di at

m=[m1; � � � ;mj�1;mj��mj ;mj+1; � � � ;mNp ]T ;

fi;4 = di at

m=[m1; � � � ;mj�1;mj�2�mj ;mj+1; � � � ;mNp ]T ;(23)

and �mj = 0:01mj :fi;1 through fi;4 are evaluatedusing the reservoir simulator.

Actually, higher values of dimensionless sensitiv-ity result in better estimation of the model parametersand lower uncertainty (lower variance) at the end of


Figure 15. History matching results for oil production rate, water production rate, gas production rate, and bottom-holepressure in Scenario 2.

assimilation. For example, if Si;j > Sk;j , then use of difor history matching gives a better estimate of mj thanthe one obtained by history matching based on dk [45].

In this section, we choose production rates ofwell P1 and bottom-hole pressure of well I1 so asto perform the sensitivity analysis. Figure 16 showsthe dimensionless sensitivity of the oil production rateat well P1 to the model parameters as a function of

time. It shows that oil production rate at well P1 isvery sensitive to the parameters de�ning the oil-waterrelative permeabilities (bo, bw, and aw). Furthermore,water and gas production rates at well P1 (Figures 17and 18) and bottom-hole pressure at well I1 (Figure 19)are quite sensitive to the oil-water relative permeabilityparameters. Therefore, one expects a close agreementbetween the estimation of the oil-water relative perme-


Figure 16. Dimensionless sensitivity of well P1 oilproduction rate to relative permeability and capillarypressure parameters.

Figure 17. Dimensionless sensitivity of well P1 waterproduction rate to relative permeability and capillarypressure parameters.

ability parameters and the reference values with lowuncertainty at the end of assimilation. This is veri�edby the results presented in Figure 13 and Table 4. Oilproduction rate at early times is also sensitive to theshape factor of the gas relative permeability, bg.

Figure 17 represents the dimensionless sensitivitycoe�cients of water production rate at well P1. As it isshown, water production is sensitive to the parametersbo, bw, aw, and ncow. Figure 18 shows that at theearly times of assimilation, gas production rate of wellP1 is quite sensitive to bg, as expected. However, asassimilation cycle proceeds, when the free-gas satura-tion decreases and the gas production rate is reducedsigni�cantly, only bo, bw, and aw dominate the gas owrate.

Figure 19 represents the dimensionless sensitivitycoe�cients of owing bottom-hole pressure at well I1.Bottom-hole owing pressure is strongly in uenced bytotal mobility [45], which is the sum of the mobilitiesof each phase and is de�ned as:

Figure 18. Dimensionless sensitivity of well P1 gasproduction rate to relative permeability and capillarypressure parameters.

Figure 19. Dimensionless sensitivity of well I1bottom-hole pressure to relative permeability andcapillary pressure parameters.

�t =kro�o

+krg�g

+krw�w

: (24)

Grid blocks surrounding the injection well are almostsaturated with water and oil. Therefore, the bottom-hole pressure is expected to have high sensitivity tothe oil-water relative permeability parameters, as itis depicted in Figure 19. Around the injection well,there is a high ow rate in the reservoir, and so thebottom-hole pressure of the injection well is insensitiveto P �cow and ncow. As a result, a small relative error isobtained for estimation of parameters bo, bw, aw, ncow,and bg with low variances at the end of assimilation, asdepicted in Figures 13 and 14 and Table 4.

The dimensionless sensitivity of the productionrates of well P1 and bottom-hole pressure of well I1 tothe other parameters, including P �cow, bog, ag, P �cgo, andncgo, are all small. However, the information contentsof these sets of data are di�erent. Therefore, historymatching of all the production rates and bottom-hole


pressure yields an improved estimation of these param-eters whose maximum relative error is 8.89%, as listedin Table 4. However, as discussed above, parameterswith high values of dimensionless sensitivity usuallyresult in better estimation of the model parameters andlower uncertainties. Table 4 shows that uncertainty ofthe parameters P �cow, bog, ag, P �cgo, and ncgo at the endof assimilation is much greater than parameters withhigh dimensionless sensitivity.

5. Conclusions

In this study, three-phase relative permeabilities andcapillary pressures were simultaneously estimated fromhistorical data using the EnKF. The proposed tech-nique was validated by a 2D three-phase heterogeneousreservoir with two di�erent scenarios. Power-lawmodels were used for both relative permeability andcapillary pressure curves and the associated unknownparameters were adjusted sequentially toward theirreference values during the assimilation process. Allthe unknown model parameters were accurately esti-mated except for the gas-oil entry capillary pressure.This is explained by calculation of the dimensionlesssensitivities, which represent the sensitivity of theobservation data to the unknown model parameters.It was veri�ed that higher values of dimensionlesssensitivity resulted in better estimation of the modelparameters and lower uncertainty (lower variance) atthe end of assimilation. Nonetheless, this consequencecould be further approved by considering more complexand larger reservoir models in the future. Moreover, theimpacts of availability of the endpoints and ensemblesize on the �lter performance were evaluated.

The second scenario in which the endpoints wereunknown seemed to be a better representative ofan actual history matching problem performed tocharacterize a real three-phase ow reservoir throughsimultaneous estimation of relative permeabilities andcapillary pressures from production data. However,this scenario was more complicated and challenging dueto its higher degree of freedom. Hence, to avoid �lterdivergence, a larger number of ensemble members wererequired, which increased the computational burden.

Nomenclature

A Ensemble perturbations or anomaliesa Endpoint of relative permeability curveb Shape factor of relative permeability

curveD Measurement perturbations vectord Real observations~d Predicted observationsH Mapping operator

I Identity matrixK Kalman gain matrixM Reservoir simulation modelkr Relative permeabilitym Model parametersN Gaussian distributionN Number of ensemble membersnc Shape factor of capillary pressure curvend Number of observations at each

assimilation cyclenm Number of model parametersNp Number of to-be-estimated parametersns Number of model statesO Null matrixP Error covariance matrixPc Capillary pressure (psia)P �c Endpoint of capillary pressure curve

(psia)Q Model error covariance matrixR Measurement noise covariance matrixRs Solution gas/oil ratio (MSCF/STB)RRMSE Relative Root-Mean-Square ErrorS Saturations Model statesSd;m Dimensionless sensitivity coe�cient of

observation d to model parameter mSgc Critical gas saturationSgD Dimensionless gas saturationSorg Residual oil saturation in the gas-oil

systemSorw Residual oil saturation in the oil-water

systemSwc Critical water saturationSwD Dimensionless water saturationstd Standard deviationu Control inputsw Model errorWOPR Well Oil Production Rate (STB/D)WWPR Well Water Production Rate (STB/D)WGPR Well Gas Production Rate (MSCF/D)WBHP Well Bottom-Hole Pressure (psia)x State vector (augmented)

Greek letters

" Measurement perturbation�t Total mobility (cp�1)� Viscosity (cp)


�d Standard deviation of measurementerror

�m Standard deviation of model parameter

Subscripts

g Gas phasego Gas-oil systemo Oil phaseow Oil-water systemw Water phase

Superscript

a Analysis valueavg Average valuef Forecast valueT Matrix transposetrue True value

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Biographies

Saman Jahanbakhshi received his BS degree inPetroleum Engineering and his MS degree in ReservoirEngineering in 2010 and 2012, respectively, from SharifUniversity of Technology, where he is currently a PhDcandidate in Petroleum Engineering. His researchinterests include real-time reservoir characterization,uncertainty quanti�cation, and ow modeling in frac-tured reservoirs.

Mahmoud Reza Pishvaie received MS and PhD

degrees in Chemical Engineering from Sharif Universityof Technology in 1991 and 1999, respectively, where heis now working as Faculty Member. He has been amember of the Research Council at SPGC since 2009.His research interests include optimization, control,and simulation of processing plants.

Ramin Bozorgmehry Boozarjomehry received MSand PhD degrees in Chemical Engineering from SharifUniversity of Technology and the University of Cal-gary in 1990 and 1997, respectively. Since then, hehas been with Chemical & Petroleum EngineeringDepartment at Sharif University of Technology. Hisresearch interests include process simulation, control,and optimization.

Documents

Characterization of three-phase ow in porous media …scientiairanica.sharif.edu/article_4112_860e32230ef8e8e...implicit estimation of relative permeabilities and/or capillary pressures