NEW COREFLOOD INTERPRETATION METHOD BASED ON DIRECT PROCESSING OF IN-SITU SATURATION DATA

Transcription

1 NEW COREFLOOD INTERPRETATION METHOD BASED ON DIRECT PROCESSING OF IN-SITU SATURATION DATA Matthe Gdfield, Stephen G Gdyear and Peter H Tnsley AEA Technlgy plc Winfrith, Drchester, Drset, DT2 8ZE, United Kingdm Abstract Generatin f gd quality relative permeability data depends n carefully designed prgrammes f cre flding, cmbined ith apprpriate interpretatin f the basic labratry data. Where capillary pressure is imprtant, the JBN technique can give misleading results. In this case, simulatin methds can be used t mdel the influence f capillary pressure. Hever, experience shs that the cnventinal fixed capillary pressure utlet bundary cnditin may fail t capture the develpment f the in-situ saturatin prfiles satisfactrily. This paper describes a ne apprach t crefld interpretatin, hich fully utilises in-situ saturatin data and reduces the sensitivity t assumptins abut the cre bundary cnditins. The ne methd extends previusly published techniques fr the interpretatin f gravity drainage flds by including viscus and capillary frces. Relative permeabilities are calculated directly frm in-situ saturatin and pressure drp data, ithut the need t use simulatin methds, by applying the Darcy fl equatins fr the mbile phases at each psitin and time in the cre. Darcy s equatins are recast t give separate pressure and fractinal fl equatins. The fractinal fl equatin is slved lcally, independently f any bundary cnditin assumptins, and the pressure equatin (ritten in a glbal integral frm) is slved fr the mbility. The algrithm is applicable t a ide range f ater-il displacements r gas-il displacements ith an immbile initial ater saturatin, including multi-rate and cnstant pressure drp unsteady-state flds, steady-state and gravity drainage flds. Synthetic saturatin and pressure drp data sets have been generated using a cnventinal simulatr fr a selectin f ater/il and gas/il cre flds. The algrithm is applied t the data sets t cnstruct the underlying relative permeabilities, including a significant fractin f the il relative permeability tail. The presence f capillary end effects alls relative permeabilities at sme saturatins bel the Bucley- Leverett shc frnt t be calculated in unsteady-state displacements. A range f smthing methds have been develped t all the basic algrithm t be successfully applied t datasets in hich nise has been added t the basic saturatin and pressure prfiles generated by simulatin. Intrductin Labratry special cre analysis (SCAL) ater r gas flding studies t determine relative permeabilities need t be perfrmed in such a ay that they are representative f reservir cnditins. The imprtance f preparatin methds t cnditin the cre t an apprpriate initial ater saturatin and ettability, tgether ith the chice f fluids and flding rates, are increasingly recgnised as ey elements f definitive SCAL prgrammes. The results frm cre flds need t be interpreted t determine relative permeabilities. Fr example, the cnventinal Jhnsn Bssler and Naumann (JBN) interpretatin methd [1] uses il prductin and pressure drp data t calculate relative permeabilities, assuming that the influence f capillary pressure n the displacement can be neglected. Hever, here capillary pressure is imprtant, (Figure 1), the JBN methd can give misleading results, verestimating the residual il saturatin and giving artificially suppressed relative permeabilities. The influence f capillary pressure can be reduced by increasing the displacement rate. Hever, high fl rates can change the character f the displacement, especially at intermediate-et cnditins, maing results unrepresentative f the reservir. In-situ saturatin mnitring is a perful tl fr identifying displacements influenced by capillary pressure. Where capillary pressure is shn t be imprtant mre sphisticated interpretatin methds are required, if meaningful relative permeabilities are t be calculated. Current best practice uses simulatin

2 methds t include the influence f capillary pressure. Separately measured static capillary pressure and/r limited in-situ saturatin data are used t cnstrain regressin methds t fit t prductin and pressure drp data [2-5]. This apprach is an indirect methd, using a series f simulatins cmbined ith infrmatin abut the gdness f fit t the chsen data t adjust the chice f relative permeabilities and capillary pressure. Where an independent measurement f capillary pressure is made n a secnd cre, it may be difficult t ensure that it is representative f the riginal reservir rate displacement test. The regressin prblem is made difficult by the large number f strngly cupled parameters that characterise the relative permeabilities and capillary pressure. Experience shs that the results ften fail t capture the develpment f in-situ saturatin data satisfactrily, especially at the cre utlet here a fixed saturatin (crrespnding t the cnventinal zer capillary pressure bundary cnditin) may nt be bserved. Gd quality in-situ saturatin data is ften measured during SCAL studies. A number f methds have been develped t mae use f this data, fr example in [6] relative permeabilities are calculated fr saturatins n the displacement frnt. An alternative apprach t interpreting in-situ data has been applied t gravity drainage cre flds, hich alls relative permeabilities t be deduced directly frm in-situ saturatin data. This uses the change in saturatin prfiles ith time t calculate the lcal phase Darcy velcity, hich can then be related directly t relative permeability in regins f the cre here capillary frces are small cmpared t gravity frces. The methd has been successfully applied in a number f studies f gravity drainage [7-9]. This r builds n these ideas by fully accunting fr the influence f viscus and capillary frces, in additin t gravity frces. Frmulatin f basic equatins It is assumed that the cre can be treated as a hmgeneus ne dimensinal system. The basic equatins are used t describe ater/il displacements r gas/il displacements ith a unifrm immbile initial ater saturatin. In this paper the equatins are derived in ntatin apprpriate fr a vertical ater/il displacement, but they can easily be translated t hrizntal cres (by setting g = 0) r gas/il displacements. Distance alng the cre is measured frm the inlet, hich is assumed t be at the base f the cre. The starting pint fr the ne interpretatin methds is Darcy s la, here the phase velcities are expressed accrding t: pα Uα = λ α + ρα g (1) here λ and λ are the ater and il phase mbilities respectively, defined by: λ = / (2) α r α µ α c ( S c( S ) p ( S ) p ( S and the ater/il capillary pressure p ) is defined in the standard cnventin by: p = ). (3) Creflding sequences are nrmally designed s that any change in the fluid vlumes due t the pressure drp alng the cre can be neglected s that the ttal Darcy velcity U ( is nn. This alls the Darcy equatin fr each phase t be re-cast t separate the pressure and fractinal fl terms. Saturatin equatin The pressure independent equatin frms an extensin f the familiar fractinal fl equatin: λ g( ρ ρ ) λλ dpc S U = U ( λ λ λ 1+ U t. (4) + ( ) λ + λ ds This cmprises a cnvective and dispersive term hich can be summarised as: S U = U ( f ( S, U ) + dcp ( S ), (5) here the gravity fractinal fl f is given by f ( S, U ) = f ( S ) + G ( S ), (6) U

3 f is the cnventinal viscus dminated fractinal fl curve: f S λ ( ) =, (7) λ + λ and G is the gravity cunter-current fl velcity functin defined by λλ G ( S ) = g( ρ ρ ), (8) λ + λ In the absence f gravity the gravity fractinal fl f reduces t the cnventinal viscus dminated fractinal fl curve f. The capillary dispersin rate functin d cp is given by d cp λλ dpc ( S ) = (9) λ + λ ds and crrespnds t the cunter-current fl term assciated ith capillary imbibitin. This cnstructin f the il Darcy velcity equatin in terms f f and d cp represents a natural parameterisatin f the equatin ith the separate terms having a clear physical meaning. Where gravity can be neglected cmpared t viscus and capillary frces, r here fl is at a cnstant ttal rate, the evlutin f saturatins ithin the cre is cntrlled explicitly just by these t independent functins. This cntrasts ith the cnventinal apprach here three separate functins are cnsidered (t relative permeabilities and a capillary pressure) even hen handling just the saturatin data. Pressure equatin Since pressure data is usually nn at a fe pints, ften nly at the ends f the cre, a mre glbal frm f the pressure equatin has t be frmulated t mae use f this infrmatin. One f the difficulties encuntered here is a pssible ambiguity in h pressures measured in the inlet and utlet lines are related t phase pressures in the cre at the inlet and utlet face (since the il and ater phase pressures are different by the capillary pressure). It is reasnable t assume that the inlet pressure crrespnds t the ater phase pressure at the inlet face f the cre (subject t gravity head crrectins) and that at sme pint after ater breathrugh the utlet pressure crrespnds t the ater phase pressure at the utlet face f the cre. In this case the pressure difference can be determined by re-arranging and integrating Equatin 1 alng the length f the cre: L U = + ( z, p( gρ L dz. (10) 0 λ ( S( z, ) Prir t ater breathrugh the utlet pressure ill crrespnd t the il phase pressure at the utlet face f the cre. The pressure drp can then be expressed in the flling frm: ( ) z p p( = gρ p c z ( S ( + gρ ) + ( L z z ( here ( is the psitin f a fixed saturatin ( ) U ( z, dz + λ ( S( z, ) p) 0 z ( S L U ( z, ( ( (, )) dz λ S z t at time t (nn frm the saturatin data). These equatins may als be expressed in terms f the ttal mbility and the gravity fractinal fl instead f the phase mbilities. Slutin f equatins The (dynamic relative permeability - capillary pressure technique SCAL) algrithms handle the saturatin and pressure data separately. Using infrmatin abut lcal phase Darcy velcities and saturatin gradients, the gravity fractinal fl curve and capillary dispersin rate functin are cnstructed t satisfy equatin 5. The pressure equatin is then slved fr the ater mbility. Finally, the relative permeabilities and capillary pressure are decnvluted frm the results f the analysis f the saturatin and pressure equatin. This sectin describes these stages in detail fr the algrithms applied t saturatin data (11)

4 defined cntinuusly in space and time. Fr prcessing actual data, discrete representatins f the algrithms are needed, ith the requirement fr apprpriate data smthing and interplatin techniques. Lcal phase Darcy velcity Analysis f equatin 5 requires that the lcal Darcy phase velcities are nn. These may be calculated frm the in-situ saturatins and the injectin histry using a material balance apprach: inj V ( z, U ( z, = U ( F (, (12) t here the phase vlumes per unit crss-sectinal area beteen the cre inlet and psitin z are given by and ( is the fractinal fl f il injected at time t. F inj z V ( z, = φ( ζ ) S ( ζ, dζ (13) 0 Direct slutin f saturatin equatin Direct slutin f the saturatin equatin 5 is made at a series f fixed saturatins. Fr a given saturatin S the crrespnding psitins fr this saturatin in the cre as a functin f time, are evaluated accrding t: S z (, = S (14) ( thereby determining the trajectry f the saturatin S in z-t space. Reriting equatin 5 in terms f z and t gives the flling frm: S U ( ) = U ( f ( S, U ) + d cp ( S ) (15) Except in the special case f zer ttal Darcy velcity (cnsidered bel), equatin 15 can be rearranged t give: U ( ) S = f ( S, U ) + d cp ( S ) (16) U ( U ( here U ( ) and S are calculated frm the bserved saturatin data. Prvided gravity frces are negligible, r fr perids f time hen the ttal flrate is cnstant, this equatin gives S functin f U ( pltting the bserved values f ( g ) ( U U, ) U( z t as a linear (, here the intercept is given by f ( S, U ) and the gradient by d cp ( S ). Thus U, ) U( z t S against, and perfrming a linear regressin alls ( U ( f S, ) and d cp ( S ) t be evaluated. Figure 2 shs this prcess schematically. In the case f n capillary pressure, a line f zer slpe ill be fund, crrespnding t the cnventinal Bucley-Leverett slutin. Where capillary pressure is imprtant, a line ith nn-zer slpe ill be fund. It is als rth nting the pssibility f nt being able t separate ut infrmatin abut f ( S, U) and d cp ( S ) in the special case f the saturatin crrespnding t ne n a stabilised frnt, here all the data uld grup arund a single pint. Hever, t hat extent this uld actually arise in practice is pen t questin, since data early in the fld may be t early fr the stabilised frnt t have been established, and in regins at the end f the cre influenced by capillary end effects the stabilised zne may be disrupted.

5 Repeating this prcedure fr different values f S alls the gravity fractinal fl and capillary dispersin functin t be cnstructed. A number f imprtant features f this algrithm are rth nting: the mst imprtant reservir engineering functin, f, determining the lcal seep efficiency f a ater fld is btained the calculatin f f and d cp is purely lcal, and n assumptins abut the inlet and utlet bundary cnditins are required f and d cp are cnstructed explicitly n a pintise basis, s that n particular functinal frms need t be assumed the methd can be applied t data irrespective f the type f flding prcedure used, i.e. multi-rate tests, unsteady-state tests, steady-state tests and gravity drainage (subject t the assumptin f a nn-zer ttal Darcy velcity and either negligible gravity frces r perids f cnstant ttal fl rate) here a cmpsite cre is used, the regressin can be applied t each plug separately, alling d cp t be evaluated fr each plug f and the methd can be applied t the dynamic data beteen established steady-state cnditins (the special case f analysis f the steady-state data is discussed bel) the cnstructinal nature f the algrithm demnstrates that an independent measurement f capillary pressure is nt required t interpret insitu saturatin data in the presence f capillary end effects. Pure cunter-current fl and gravity dminated slutin In the case f pure cunter-current fl the ttal Darcy velcity is zer and the frmulatin f the basic saturatin equatin is made in terms f G and d accrding t: cp U( ) S = G ( S) + dcp( S) (17) The gravity dminated ( U << λ ρ ρ ) ) versin f the saturatin equatin als taes this frm. U z t Pltting (, ) against g ( S and perfrming a linear regressin alls G ) and d ( ) t be ( S cp S evaluated. Where the slutin is gravity dminated and the additinal cnditin that the il mbility is l cmpared t the ater mbility hlds ( λ << λ ), G S ) reduces t: S ( G ( ) g( ρ ρ ) λ ( S ) (18) alling the il relative permeability t be calculated directly. Steady-state tests If nly the steady-state cnditins are t be analysed, there is n need t calculate lcal phase Darcy velcities, since these are nn frm the impsed fractinal fl. Equatin 16 simplifies t: inj S F ( = f ( S, U ) + dcp ( S ) (19) U ( Prvided capillary frces are sufficiently imprtant fr a given saturatin t be present in the cre n a number f equilibrium states this equatin may be analysed in the standard ay, subject t the nrmal assumptins that gravity frces are negligible r the ttal flrate is cnstant.

6 Multi-rate tests here gravity is nt negligible In the case here gravity frces are nt negligible, and f ( S, U) has been deduced frm data crrespnding t perids f cnstant fl at a number f ttal Darcy velcities U i, it is pssible t deduce f ( S ) and G S ) accrding t the flling analysis: ( f ( S, Ui) = f( S) + G ( S) (20) U Then by pltting f ( S, U i ) against / U i and perfrming a linear regressin alls f ( S ) and G ( S ) t be evaluated. Fr saturatins here this is pssible, this determines three independent functins and s alls the relative permeabilities and capillary pressure gradient t be determined ithut using pressure drp data (aviding the cmplicatins f bundary cnditin effects). This result is pssible because the different fl rates are cmbined ith a nn lcal gravitatinal pressure gradient. Gravity and generally varying fl rates In the case f a generally varying flrate it is pssible t rite the saturatin equatin in the flling frm: U ( z, is then a linear functin f cp S i U ( ) U ( S = G ( S ) + f ( S ) + dcp ( S ) (21) U ( S and S, in principle alling G ( ), f ( ) and S d ( ) t be determined frm a 2-D linear regressin. Hever, fr practical cases, insufficient data in U ( S and space may be available t mae this generally applicable. Direct slutin f pressure equatin Slutin f the pressure equatin (equatin 10) requires the ater mbility t be determined. In practice, pressure drp data is ften quite nisy, especially hen flds are cnducted at reservir cnditins. The slutin algrithm is based n a least squares regressin n the pressure equatin, by varying a parameterised versin f the ater mbility. Nte that hile the fractinal fl curve cnstructed by the methds f the previus sectin is ell defined, the data fr the pressure equatin and its interpretatin is mre uncertain. Hever, this ambiguity ill nly feed thrugh int the ttal mbility functin and the full capillary pressure curve, hich in many cases ill mae less impact n predictins f reservir perfrmance. Cnstructin f relative permeability and capillary pressure The il mbility, and hence relative permeabilities, are determined by slving equatin 6. The capillary pressure gradient can then be calculated using the nn frm f d and the mbility functins. The capillary pressure curve is then evaluated t ithin a cnstant by integratin. The abslute level f the capillary pressure uld be nn if it prves pssible t determine p c ( S ) frm the pressure equatin regressin if equatin 11 is used. Alternatively, the saturatin at the cre utlet after breathrugh uld crrespnd t zer capillary pressure if the cnventinal capillary pressure bundary cnditin is assumed. If phase selective pressure transducers ere utilised [10], that alled a direct dynamic measurement f capillary pressure t be made, it uld be pssible t cnstruct the relative permeabilities using this infrmatin tgether ith f and d cp deduced frm the saturatin data. In this scenari the ttal pressure drp data uld n lnger be required, cmpletely aviding the ambiguities assciated ith this measurement. cp

7 Smthing f saturatin data The algrithms described in this paper are based n cntinuum saturatin and pressure data. In practice saturatin and pressure data is nly measured at discrete spatial psitins and times, and the data is subject t statistical nise and lcal fluctuatins assciated ith small scale hetergeneity. Therefre, the algrithms have been recast in a discrete frm, and apprpriate smthing techniques develped. Nisy saturatin data is first smthed befre being interpreted using the discrete frm f the algrithms. A number f smthing methds have been develped, ranging frm parametric nn-linear least squares fitting t nn-parametric cnstrained spline methds. These methds have been applied t the saturatin-psitin, saturatin-time, psitin-time and saturatin-psitin-time spaces (S-z, S-t, z-t and S-z-. The smthing methds used in each case depend n the type f data being analysed and the nature f the displacement prcess. Validatin f methds A cnventinal blac-il reservir simulatr as used t generate synthetic saturatin and pressure drp data. Synthetic data has been generated fr a number f different test cases including: gas-il gravity drainage, unsteady-state ater-il, steady-state ater-il and cnstant pressure drp gas-il displacements. A gas-il gravity drainage displacement and an unsteady-state ater-il displacement are described here. In bth these cases a cnventinal zer capillary pressure bundary cnditin as applied at the utlet. A ttal f 40 equally spaced saturatin mnitring psitins ere assumed alng the cre and saturatin measurements ere taen at time intervals f 5 minutes r greater. Uncrrelated Gaussian randm nise as then added t the synthetic saturatin data t represent the effect f measurement errr and small-scale hetergeneity. Unsteady-state ater-il test In the unsteady-state test, ater as injected in a hrizntal 10 cm cre at a cnstant rate f 8 ml hr -1. The synthetic saturatin prfiles ith nise are illustrated in Figure 1. This test as interpreted using the general frm f the methd hich uses bth the saturatin and pressure drp data. As a basic validatin f the apprach, the synthetic saturatin data befre the additin f nise as prcessed. The interpreted fractinal fl curve is shn in Figures 3 and 4, here it can be seen that the entire fractinal fl curve is successfully recnstructed fr saturatins abve abut 0.5 (crrespnding t a ater fractinal fl f abut 0.3). The errr bars sh the 90% cnfidence interval btained frm the linear regressin. It shuld be nted that each data pint n the fractinal fl and dispersin rate functins is individually calculated, entirely independently f the ther pints and the assumptins made abut the cre inlet and utlet bundary cnditins. The presence f a capillary end effect alls fractinal fls significantly bel the Bucley-Leverett shc frnt t be calculated. In cntrast, the JBN interpretatin f the data shs unrepresentatively early ater breathrugh and t high a residual il saturatin. The capillary dispersin rate is als successfully recnstructed fr saturatins abve abut 0.5 (Figure 5). Fr the data set including nise, the interpreted il gravity fractinal fl and capillary dispersin rate functins are cmpared ith the riginals in Figures 6 t 8. Figure 6 shs a gd match t the shc frnt saturatin and that sme data are btained at ler saturatins. This figure als shs that the methd becmes less reliable n steep sectins f the saturatin prfiles (in this case at saturatins less than 0.55). The lg-plt f the fractinal fl (Figure 7) shs that l values f the fractinal fl can still be recvered. The relative permeability functins are shn in Figures 9 and 10. Since the ater relative permeability as mdelled by a parametric frm, a smth curve is btained. The il relative permeability is evaluated as a functin f the fractinal fl and s breas dn at the same pint as the fractinal fl curve. The ateril capillary pressure functin is shn in Figure 11. The saturatin at the utlet has been used t define the zer pint n the capillary pressure functin. The il recvery factr is pltted against pre vlumes f ater injected in Figure 12. There is a very gd match fr up t t PV injected and a gd match abve this. This shuld be cntrasted ith the basic cre fld recvery factr hich is ler due t the hld up f il in

8 the cre by capillary pressure. If a bump fld ere incrprated in the test sequence, additinal data at the high ater saturatins uld be available fr prcessing, hich uld be expected t further imprve the match t the underlying recvery curve (based n the viscus dminated fractinal fl curve). In cntrast, the JBN methd fails t match the recvery factr in this case. Gas-il gravity drainage test In the gravity drainage test, gas as injected at the tp f the vertically riented 60 cm cre at a cnstant rate f 2 ml hr -1. The synthetic saturatin prfiles ith nise are illustrated in Figure 13. This test as interpreted using the gravity dminated frm f the methd hich des nt use the pressure drp data. Applicatin f the gravity drainage versin f the algrithm alls the il relative permeability t be recvered, even frm data in regins f the cre strngly influenced by capillary pressure. As a basic validatin f the apprach, the synthetic saturatin data befre the additin f nise as prcessed. The interpreted il gravity cunter-current fl velcity is cmpared ith the riginal functin in Figure 14. This figure shs that the functin is successfully recnstructed fr saturatins abve abut 0.3. Fr the data set including nise, the interpreted capillary dispersin rate functin is cmpared ith the riginal functin in Figure 15. The crrespnding il relative permeability curves are shn in Figures 16 and 17. At l saturatins the gas mbility ill becme l enugh fr the displacement t cease t be gravity dminated and the gravity dminated slutin ill n lnger be valid. In this case, the accuracy f the results reduces (ith increasing saturatin gradien befre this pint is reached. A gd match is btained t the capillary pressure functin (Figure 18). This is because nly the rati f the cunter-current fl velcity t the dispersin rate is required in the evaluatin f the capillary pressure functin and it can be shn that this rati is generally less sensitive t the accuracy f the linear regressin fr gravity dminated displacements. Cnclusins 1. The equatins fr crefld simulatin have been refrmulated in terms f a natural parameterisatin f the multi-phase functins each ith a clear physical meaning. 2. The evlutin f the insitu saturatins is determined by nly t independent functins, the fractinal fl and capillary dispersin functin. 3. The in-situ saturatin data can be prcessed using an explicit cnstructinal methd () t determine the fractinal fl and capillary dispersin functin, independently f bundary cnditin assumptins. 4. The explicit cnstructin demnstrates that independent measurements f capillary pressure are nt required t interpret cre fld data. 5. The pressure equatin is slved using lcal Darcy velcity data ithut the need fr cnventinal simulatin. 6. The methdlgy is applicable t a ide range f cre flding methds including unsteady-state, steady-state and gravity drainage flds. 7. The methds have been successfully applied t synthetic data sets, hich include nise, by the develpment f a range f smthing techniques. Acnledgements AEA Technlgy is grateful fr the financial supprt fr this prject frm BP Amc, Shell Expr and the UK Department f Trade and Industry.

9 Nmenclature d capillary dispersin rate [s -1 ] cp f viscus dminated fractinal fl [dimensinless] f gravity fractinal fl [dimensinless] F inj ( injected fractinal fl f il [dimensinless] g gravity cnstant [ms -2 ] G gravity fl functin [m -1 s -1 ] permeability [m 2 ] rα relative permeability f phase α [dimensinless] L length f cre [m] p ater/il capillary pressure [Pa] p c α pressure f phase α [Pa] S α ( z, saturatin f phase α [dimensinless] S fixed ater saturatin used fr algrithm [dimensinless] S fixed ater saturatin used fr prebreathrugh pressure equatin [dimensinless] t time [s] U( ttal Darcy velcity [m s -1 ] U α Darcy velcity f phase α [m s -1 ] V (z, il vlume per unit area in cre beteen inlet and psitin z [m] z ( distance frm inlet f S at time t [m] ( ) z p ( z Gree: p φ(z) distance frm cre inlet f S at time t [m] distance frm cre inlet [m] cre pressure drp [Pa] prsity [dimensinless] λα mbility f phase α [Pa -1 s -1 ] µ viscsity f phase α [Pa s] α ρ α density f phase α [g m -3 ] ζ integratin variable [m] Subscripts: α phase label g gas phase il phase ater phase References 1. Jhnsn, E.F., Bssler D.P. and Naumann, V.O., Calculatin f Relative Permeability frm Displacement Experiments, Trans., AIME (1959), Richmnd, P.C. and Watsn, A.T., Estimatin f Multiphase Fl Functins frm Displacement Experiments, SPE Reservir Engineering, Feb 1990, Chardaire-Riviere, C., Jaffre, J. and Liu J., Multiscale Representatin fr Simultaneus Estimatin f Relative Permeabilities and Capillary Pressure, SPE 20501, presented at 65 th Annual Technical Cnference, Ne Orleans, LA, September, Chardaire-Riviere, C., Chavent, G., Jaffre, J. and Liu J. and Burbiaux, B.J., Simultaneus Estimatin f Relative Permeabilities and Capillary Pressure, SPE Frmatin Evaluatin, (Dec 1992), pp Mejia, G.M., Mhanty K.K., and Watsn, A.T., Use f In-Situ Saturatin Data in Estimatin f T- Phase Fl Functins in Prus Media, Jurnal f Petrleum Science and Engineering (1995) 12 pp Helset H.M., Nrdtvedt J.E., Sjaeveland S.M. and Virnvsy G.A., Relative Permeabilities frm Displacement Experiments ith Full Accunt fr Capillary Pressure, SPE Reservir Evaluatin & Engineering, April 1998, pp92-98 (SPE 36684). 7. Fulser, R.W.S., Naylr, P. and Seale, C., Relative Permeabilities fr Gravity Stabilised Gas Injectin, Trans IChemE, July 1990, 68, Part A. 8. Gdyear, S.G. and Jnes, P.I.R., Relative Permeabilities fr Gravity Stabilised Gas Injectin, 7 th Eurpean Sympsium n Imprved Oil Recvery, Msc, Russia, Octber Naylr, P., Sargent, N.C., Crssbie, A.J. and Gdyear, S.G., Gravity Drainage During Gas Injectin,8 th Eurpean Sympsium n Imprved Oil Recvery, Vienna, Austria, May van Wunni, J.N.M., Oedai, S, and Masalmeh, S., Capillary Pressure Prbe fr SCAL Applicatins, SCA-9908, 1999 Sciety f Cre Analysts Internatinal Sympsium, Glden, 1-4 August.

10 1.0 Water saturatin Oil held up by capillary pressure Bucley-Leverett U ( ) U ( f g 1 - Sr PV 0.5 PV 0.25 PV Distance frm inlet (m) Stabilised frnt S U ( Figure 1 Synthetic ater saturatins fr unsteady state test Figure 2 Schematic f direct slutin f saturatin equatins using the methd E E-01 Water fractinal fl Oil fractinal fl 1.0E E Figure 3 Capillary dispersin rate (s -1 ) JBN Water saturatin 2.5E E E E E E+00 Figure 5 Cmparisn f interpreted ater fractinal fl fr unsteady state test ithut nise Water saturatin Interpreted capillary dispersin rate fr unsteady state test ithut nise 1.0E E-05 Figure 4 Water fractinal fl Figure 6 JBN Water saturatin Cmparisn f interpreted il fractinal fl fr unsteady state test ithut nise Welge tangent Water saturatin Interpreted ater fractinal fl fr unsteady state test

11 1.0E E+05 Oil fractinal fl 1.0E E E E-04 Capillary dispersin rate (s -1 ) 2.0E E E E E-05 Water saturatin Figure 7 Interpreted il fractinal fl fr unsteady state test 0.0E+00 Figure 8 Water saturatin Interpreted capillary dispersin rate fr unsteady state test E E-01 Relative permeability Relative permeability 1.0E E Figure 9 Water saturatin Interpreted il and ater relative permeabilities fr unsteady state test 1.0E E-05 Figure 10 Water saturatin Interpreted il and ater relative permeabilities fr unsteady state test Capillary pressure (Pa) 2.0E E E E E+03 Recvery factr (%OIIP) N influence f cap. pres. -3.0E+03 Figure 11 Water saturatin Interpreted ater-il capillary pressure fr unsteady state test 0.0 Figure 12 Affected by cap. pres Prevlumes injected Cmparisn f interpreted il recvery factr

12 Gas saturatin PV 0.5 PV 6. PV Oil gravity cunter-current fl (m -1 s -1 ) 3.0E E E E E E Distance frm inlet (m) 0.0E+00 Gas saturatin Figure 13 Synthetic gas saturatins fr gravity drainage test Figure 14 Interpreted cunter-current fl velcity fr gravity drainage test ithut nise 1.0E E Capillary dispersin rate (s -1 ) 8.0E E E E E E E+05 Relative permeability E E+00 Figure E+00 Gas saturatin Interpreted capillary dispersin rate fr gravity drainage test 0.0 Figure 16 Gas saturatin 5.0E+03 Interpreted il relative permeability fr gravity drainage test Relative permeability 1.0E E E E E-05 Figure 17 Gas saturatin Interpreted il relative permeability fr gravity drainage test Capillary pressure (Pa) 4.0E E E E E+00 Figure 18 Gas saturatin Interpreted gas-il capillary pressure fr gravity drainage test