TWO-PHASE SIMULATION AND CORRELATION OF NMR MAGNETIC DECAY IN EQUILATERAL TRIANGULAR PORES

Transcription

1 SCA008-5 /6 TWO-PHASE SIMULATION AND CORRELATION OF NMR MAGNETIC DECAY IN EQUILATERAL TRIANGULAR PORES Unn H. á Lað, IRIS; Aksel Hirth, IRIS; Jan Finjrd, UiS and Svein M. Skjæveland, UiS. This paper as prepared fr presentatin at the Internatinal Sympsium f the Sciety f Cre Analysts held in Abu Dhabi, UAE 9 Octber- Nvember, 008 ABSTRACT A simple representatin f a prus rck is a bundle f straight tubes. If the tubes, r pres, have sharp crners, mre than ne phase can frm stable cnfiguratins inside the pre. This mdel has been valuable in studying drainage and imbibitin prcesses ith multiple phases invlved. In this paper e analyze the magnetic signal frm t phases in tubes (i.e. pres) ith triangular crss sectinal area. The NMR signal frm a fluid-filled prus rck is in general a cmplicated functin f pre gemetry, surface relaxatin and prperties f the fluids. The magnetic signal frm a single pre can alays be ritten as an infinite sum ver expnential functins. At certain cnditins, the magnetic signal simplifies greatly and flls a mnexpnential curve. The cmpsite magnetic signal frm all the pres in the rck can then be expressed as a sum f expnential functins here there is ne functin fr each pre size. This special case is called the Fast Diffusin Limit (FDL). The lgarithm f the magnetic signal is then inversely prprtinal t the pre radius making it pssible t find a pre-size distributin. We investigate (numerically) h the NMR signal fr immiscible phases depends n surface relaxivity, cntact angle, and pre size. The fluid cnfiguratin is given by the Mayer-Ste-Princen (MS-P) thery f immiscible displacement in angular gemetries. Frm the numerical experiments, a crrelatin is develped fr the magnetic decay f t immiscible phases inside a triangular tube. We test the crrelatin against the FDL apprximatin and find that the crrelatin gives a better representatin f the magnetic decay. This can be due t the effects f crners and angles. These effects are incrprated in the crrelatin. INTRODUCTION In the cntext f ettability characterizatin NMR measurements are becming very interesting because f the surface sensitive nature f NMR. Cmpared t cnventinal ettability measurements, NMR is faster and in additin it is nn-invasive and can be perfrmed in-situ. Previus rk has been made t characterize the ettability by including a triangular pre mdel in the interpretatin f the NMR measurements, Al- Mahrqi et al. (006). In pre mdelling rk, angular pres are used t mdel mixedet cnditins, Øren et al. (998), Helland and Skjæveland (004a) and Helland and Skjæveland (004b). When cmbining pre mdelling ith NMR simulatins, sme assumptins have been made that need further investigatin. The magnetic signal frm ne phase in a single pre is given as a sum f expnential functins; see Brnstein and Tarr (979) fr details,

2 SCA008-5 /6 M () t pre = M ( 0 pre ) exp ( t T ) exp B I ( t T ) i= 0 i i. () When the pre is ithin the FDL, the average time fr a mlecule t diffuse acrss the pre τ D = a /D is much shrter than the average time fr a mlecule t relax τ ρ = a/ρ. We get that τ D /τ ρ = ρa/d = γ <<. The effect acrss the pre is that the magnetizatin is unifrm and the magnetic decay is unifrm and mnexpnential and given by, M t = M 0exp t T, () ( ) ( ) ( ) pre pre here T = TB + T0. (3) In NMR interpretatin, the FDL assumptin is ften valid and the magnetic decay caused by the surface relaxatin /T S is mnexpnential and given by T S T 0 = V ( Sρ). (4) This assumptin is based n calculatins fr a single phase in simple gemetries such as circles, spheres, plates, squares, Brnstein and Tarr (979), and recently e have published a slutin fr an equilateral triangle, Finjrd et al. (006). These gemetries all have a certain radial regularity and symmetry. We anted t see hat happens hen this symmetry is n lnger present, fr instance hen the etting phase placed in the crners f the triangle has cntact ith the pre all n t sides f the pre and is cnstrained by the arched il-ater interface. We als anted t explre hat happens if the FDL parameter γ is larger than, i.e. utside the FDL interval. Des Eq. 3 still hld r has the decay becme multiexpnential fr a single pre? Frm the results e anted t find a crrelatin beteen surface relaxatin T 0 and surface relaxivity ρ, cntact angle θ and the fluid distributin. See Fig. fr image f triangular pre and Table fr relevant definitins. In the case f t r mre phases inside the pre, n analytical slutin has been published, s the magnetic relaxatin decay prblem needs t be slved numerically. We ill use Randm Walk fr btaining the numerical slutin, see Finjrd et al. (006) fr details n the Randm Walk algrithm in an equilateral, triangular pre. We use a triangular grid ith 30 x 30 pints fr the triangle and 5 5 alkers fr each phase. The gemetry f the fluid phases is calculated using the thery f Mayer-Ste-Princen fr immiscible fluid in triangular pres, Masn and Mrr (99). We simulated cases fr 6 different pre sizes. The size f each pre as determined by chsing the apprpriate rati n f the length f the etting all a and the length f the pre a. In additin the chice f the diffusin parameter γ determined the pre size. The resulting decay curves fr each phase ere nrmalized and fitted t a sum f t expnential functins, M ( t) = I0 exp( t T0 ) + ( I0 ) exp( t T ), (5) =, here I 0,, T 0, and T are crrelated ith relaxivity, cntact angle and fluid distributin. The result is cmpared ith the riginal decay curves, hich als as cmpared ith the riginal decay curve in the FDL regime.

3 SCA /6 RESULTS Fitting the decay curves frm the il and the ater t a biexpnential sum gives the curves fr I 0, T 0 and T,. The intensities I 0 appear t be nly eakly dependent n the cntact angles θ and the crrelatin fr the intensities becmes a functin f cntact length a alng the pre all, diffusin cefficient D and surface relaxivity ρ. See Fig. and Eqs. and 3. We bserve in Fig. 3 an example f h the relaxatin times T 0 and T depend n cntact angle θ, relaxivity ρ and fluid gemetry. We find that the relaxatin time fr the nnetting phase decreases and the crrespnding relaxatin time fr the etting phase increases as the etting angle increases. The crrelatins fr T 0 and T are given in Eqs. 8, 9,, and 3 and are a sum f the relaxatin times fr the fast and the sl diffusin regime fr bth ater and il seen in Table. Cmparing the magnetic decay frm the simulatins ith the magnetic decay calculated frm T S = V/Sρ, shed that this assumptin becmes inaccurate as the relaxivity increases and at l cntact angles, see Fig. 4 fr an example. We bserve frm Fig. 4 that the fit is verall in accrdance ith the simulated results. This favurs the use f the presented crrelatin. The resulting equatin can be used fr mdelling NMR decay fr t phases in an equilateral triangular pre by assuming a pre size distributin P(R n ) and generate magnetic decay curves based n the NMR parameters: ρ, ρ, D, D, T B,, T B,, and the etting parameters φ and a. Changing the etting parameters changes the decay curves crrespndingly. We get that m (6) M t = P R M t, () ( ) () n= n n here M n is M pre fr a given pre size R n. and M pre is given by M t = V I exp t T t T + I exp t T pre () [ ( ) ( ) ( t T )] 0 0 B 0 B. =, With the btained fitting parameters fr etting () and nn-etting () phases, e get, T0 = r ρ + r D, (8) T = 0.5 r ρ r D, (9) I 0 = [ ( ρ a D ) ( ρ a D )], () T0 = r ρ + 9R 4Dπ 0. 00r D, () T = 0.5 r ρ r D, () 6.3 (. + (( a D ) 5) ) I ρ, (3) 0 = + here r and r are given by Eq. 9. This crrelatin is nly valid hen t phases are present in the pre. See Helland and Skjæveland (004a) fr details n capillary entry pressure and Finjrd et al. (006) fr magnetic decay fr ne phase in triangular pre. (7)

4 SCA /6 DISCUSSION AND CONCLUSIONS When interpreting the NMR decay curves and the T distributin, it is cmmn t assume mnexpnential decay fr each pre. Recently, there has been an increased interest in angular pre shapes, here it is pssible t mdel cexisting t r three phases in the same pre. We find that the FDL-assumptin fr these angular pres is inaccurate, especially at l cntact angles r high surface relaxivity. This is prbably caused by the inaccessibility f the pre all clse t the fluid-fluid meniscus. The bserved l surface area culd interfere ith calculatins f the pre size distributins r surface relaxivity. In additin the magnetic decay becmes multiexpnential earlier fr these irregular fluid gemetries. This might lead t the signal frm the higher intensities being interpreted as smaller pres r bund fluid. Fr added accuracy in mdelling ne can use a biexpnential sum fr the surface relaxatin in these angular pres. We have fund a biexpnential crrelatin fr the magnetic decay fr t phases in a triangular pre. The crrelatin is valid fr primary drainage fr cntact angles in the range 0 55 degrees, relative pre sizes ranging frm 50, and relative surface relaxivity f hen t phases are present. The crrelatin functin can be used as a biexpnential sum r it is an ptin t use a mnexpnential decay curve ith the crrespnding relaxatin times T 0. ACKNOWLEDGEMENTS The authrs acknledge CncPhillips and the Ekfisk Cventurers, including TOTAL, ENI, Hydr, Statil and Petr, fr financing the rk and fr the permissin t publish this paper frm the research centre COREC. NOMENCLATURE a = Length f side f triangle a = Length f fluid-pre cntact D = Diffusin cefficient I = Intensity M = Magnetizatin P(R n ) = Pre size distributin r = Radius f curvature r = Calculated fluid radius = V /S R = Inscribed radius f pre S = Length f fluid-pre cntact t = Time T i = Decay time fr fluid T = Ttal decay time f pre = Bulk decay time f fluid T B T S = Surface decay time f fluid V = Fluid vlume γ = Rρ/D, diffusin regime parameter θ = Cntact angle ρ = Surface relaxivity τ D = Average diffusin time τ ρ = Average relaxatin time Subscripts i = Relaxatin mde, i = 0, = fr nn-etting phase, fr etting phase

5 SCA /6 REFERENCES Al-Mahrqi, S.H., Grattni, C.A., Muggeridge, A.H. and Jing, X.D. Pre-scale Mdelling f NMR Relaxatin fr the Characterizatin f Wettability, Jurnal f Petrleum Science and Engineering Vl. 5 (-4): p.7-86 (006) Brnstein, K. and Tarr, C.: Imprtance f classical diffusin in NMR studies f ater in bilgical cells, Phys. Rev. A 9, (979). ; Spin-lattice relaxatin in a system gverned by diffusin. J. Mag. Resn. 6, 7 4 (977). Finjrd, J., Hirth, A., a Lad, U.H., and Skjaeveland, S.M.: ``NMR fr Equilateral Triangular Gemetry Under Cnditins f Surface Relaxivity - Analytical and Randm Walk Slutin,'' Transprt in Prus Media, 69, pp (006). Helland, J.O. and Skjæveland, S.M.: Physically based capillary pressure crrelatin fr mixed et reservir frm a bundle f tubes mdel, paper SPE 8948 presented at the 004a SPE/DOE Sympsium n Imprved Oil Recvery, Tulsa, April 7. Helland, J.O. and Skjæveland, S.M.: Three-phase, mixed-et capillary pressure curves frm a bundle-f-triangular-tubes mdel, paper presented at the 004b Internatinal Sympsium n Reservir Wettability, Hustn, May 6 8. Masn, G. and Mrr, N.: Capillary behaviur f a perfectly etting liquid in irregular triangular tubes, J. Cll. Int. Sci. 4, 6 74 (99). Øren, P.E., Bakke, S., and Arntzen, O.J.: Extending predictive capabilities t netrk mdels, SPE Jurnal (Dec. 998) TABLES Table : Gverning equatins fr the fluid cnfiguratins. a) Wetting phase in crners f pre a = rcs θ + π 6 (4) ( ) ( ) 4 V 3 = a f θ (5) S = 6 (6) a b) Nn-etting phase in the centre f pre V = 3a 4 3a f θ 4 (7) ( ) ( ) a ( ) S = 3a 6 (8) Definitins fr etting and nn-etting phase. r V S (9) γ r ρ D (0)

6 SCA /6 τ D r D () Where: π π f ( θ) θ cs( θ) 3 cs( θ) sin ( θ) cs θ () = Table : Table frm Finjrd et al. (006) giving the analytical slutin fr magnetic decay in a triangular pre. Fast Diffusin Limit: γ << Sl Diffusin Limit: γ >> ρ 9R 4Dπ T 0 R ( ) ( ) T 9R ( 4Dπ i ) 9 R ( 4Dπ ( i + ) ) FIGURES F I 0 fr γ * = a ρ/d fr nn-etting phase a a I0 [dim.less] R θ a Figure : Fluid cnfiguratin fr the triangular pre γ * [dim.less] Figure : Intensity fr nn-etting phase. Simulated data (Markers), Crrelatin data (Line). T 0/τ D fr γ = 0.-, fr etting phase M(t/τ D)/M(0) fr γ = 0. 5, θ = 0 fr etting phase T0/τD [dim.less] θ = 0 θ = 5 θ = 30 θ = 45 θ = 55 M(t/τD)/M(0) [dim.less] ρ rel = ρ rel = ρ rel = ρ rel = 3 ρ rel = 5 ρ rel = γ [dim.less] Figure 3: Relaxatin time fr etting phase. Simulated data (Markers), Crrelatin data (Lines) t/τ D [dim.less] Figure 4: Magnetic decay fr etting phase. Simulated data (Thick line), Crrelatin data (Thin line), FDL data (Brken line).