COMPARISON OF AN IMPEC AND A SEMI- IMPLICIT FORMULATION FOR COMPOSITIONAL RESERVOIR SIMULATION

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1 Brazilian Journal of Chemial Engineering ISS rinted in Brazil Vol. 3, o. 04, , Otober - Deember, 204 dx.doi.org/0.590/ s COMARISO OF A IMEC AD A SEMI- IMLICIT FORMULATIO FOR COMOSITIOAL RESERVOIR SIMULATIO B. R. B. Fernandes, A. Varavei 2, F. Marondes 3* and K. Seehrnoori 4 Laboratory of Comutational Fluid Dynamis, Federal University of Ceará, Brazil. 2 Center for etroleum and Geosystems Engineering, The University of Texas at Austin, USA. 3 Deartment of Metallurgy and Materials Siene and Engineering, Federal University of Ceará, Brazil. marondes@uf.br 4 Center for etroleum and Geosystems Engineering, The University of Texas at Austin, USA. (Submitted: ovember, 203 ; Revised: January 6, 204 ; Aeted: February 2, 204) Abstrat - In omositional reservoir simulation, a set of non-linear artial differential equations must be solved. In this work, two numerial formulations are omared. The first formulation is based on an imliit ressure and exliit omosition (IMEC) roedure, and the seond formulation uses an imliit ressure and imliit saturation (IMSAT). The main goal of this work is to omare the formulations in terms of omutational times for solving 2D and 3D omositional reservoir simulation ase studies. In the omarison, both UDS (Uwind differene sheme) and third order TVD shemes were used. The omutational results for the aforementioned formulations and the two interolation funtions are resented for several ase studies involving homogeneous and heterogeneous reservoirs. Based on our omarison of IMEC and IMSAT formulations using several ase studies resented in this work, the IMSAT formulation was faster than the IMEC formulation. Keywords: Comositional reservoir simulation; Segregated formulation; IMEC; IMSAT; Finite-volume method. ITRODUCTIO Several formulations have been develoed for solving the governing artial differential equations arising from modeling fluid flow for omositional simulations in orous media. In general, formulations an be lassified as Imliit ressure Exliit Comosition (IMEC), Imliit ressure and Saturation (IMSAT), or the Fully Imliit Method (FIM). hase saturation is defined as a volume of hase er void sae available for fluid flow. The IMEC formulation has the lowest ost in terms of omutational time er time-ste. However, due to the higher degree of exliitness in the alulation of omosition, this formulation annot use large time-stes when omared to the FIM and IMSAT aroahes. The IMSAT method an handle larger time-stes, omared to the IMEC aroah. It is also less exensive in terms of omutational time, er timeste, than the FIM aroah. Additionally, the IMSAT aroah is more stable than the IMEC formulation due to the redution in the degree of exliitness. Also, aording to Cao (2002), it has good erformane omared to other FIM aroahes, sine saturations are muh more ouled than omositions. Therefore, although the saturation alulation involves solution of a linear system of equations for the IMSAT formulation, the larger time-stes used by the formulation omensate the overall ost and render a CU time redution omared to IMEC aroahes. In this work, an IMSAT formulation roosed by *To whom orresondene should be addressed

2 978 B. R. B. Fernandes, A. Varavei, F. Marondes and K. Seehrnoor Watts (986) was imlemented into the UTCOM simulator. UTCOM was develoed at the Center for etroleum and Geosystems Engineering at The University of Texas at Austin for the simulation of enhaned reovery roesses. The UTCOM simulator is a multihase/multi-omonent omositional equation-of-state simulator, whih an handle the simulation of several enhaned oil reovery roesses. The original numerial roedure of the UTCOM simulator is an IMEC formulation based on Às et al. (985). Several roedures for solving ressure and saturation imliitly (IMSAT) have been roosed in the literature (Brano and Rodrigues, 996; Kendall et al., 983; Sillette et al., 973). However, the IMSAT aroah roosed by Watts (986) and adoted in this work has similar features to the original IMEC formulation of the UTCOM simulator. For instane, the IMSAT formulation solves simle sets of linear systems for both ressure and saturations, while only one flash roedure is erformed er time-ste, thus allowing the alulations er time-ste to be muh faster than the other IMSAT and IMEC aroahes that need iterations, thus erforming flash and solving the onservation equations until onvergene. The aroahes that do not iterate in a time level are alled one-iteration formulations; therefore, this work is based on a omarison of two one-iteration aroahes: an IMEC and an IMSAT formulation. Although the IMEC and IMSAT formulations imlemented and used in this work are not new, to the best of our knowledge this is the first time that these formulations are omared for three hase hydroarbon flow simulations (oil, gas, and a seond liquid hydroarbon hase). The seond liquid hydroarbon hase is imortant for CO 2 injetion roesses, where the CO 2 and some light omonents tend to form a CO 2 -rih hase. Another imortant feature inluded in this work is the use of a high-resolution TVD sheme to aroximate the fluxes for the Watts formulation. Also, only few erformane results are shown in the literature for this formulation. Some of these results an be found in Haukås (2006). In this work, results are omared in terms of volumetri rodution rates, saturation fields, and CU time. For most investigated ases, the formulation was able to use larger timestes than the IMEC formulation, ahieving the same results. HYSICAL MODEL The Watts formulation is basially an adatation of the method of Sillette et al. (973) ombined with the formulation of Ás et al. (985), where the volume error onstraint is added to ressure and saturation equations in order to use just only one flash alulation er time-ste. In this setion, we show the molar balane equations, the ressure equation, and the new saturation equations that are inluded in the original formulation of Ás et al. (985). If advetion is the only transort mehanism involved, the molar balane equations aording to Chang (990) are given by k k rj qk = xkjξj K Φ j b μ j= j b, V t V, () k =,...,, + where k is the moles of omonent k, V b is the bulk volume, x kj is the mole fration of omonent k in hase j, ξ j is the molar density, resetively, q k is the molar rate of omonent k through the well, is the number of hydroarbon omonents, + denotes the water omonent, k rj and µ j are the relative ermeability and visosity of hase j, resetively, K is the absolute ermeability tensor, and Φ j is the hydrauli otential of hase j, whih is defined by Φ j = ρjgd jr, (2) where is the ressure of the oil hase, ρ j is the mass density of hase j, g is the gravity, D is the deth, whih is ositive in the downward diretion, and rj is the aillary ressure between hases j and r. Fluid hase equilibrium between the hydroarbon hases is onsidered (water is not onsidered in any flash alulation). This assumtion onsiders that the hemial otential of all hases are the same. This an be exressed in terms of the equality of the fugaities (f) of the hases, whih an be stated as follows: o i g i f f = 0, i =,...,. (3) Fugaity and VT roerties (molar density, omressibility fators, and volume derivatives) are evaluated in this work using the eng-robinson Equation of State (EOS) (eng and Robinson, 978). The flash roedure used onsiders a fixed and known ressure, temerature, and global omositions (isothermal flash) in order to evaluate the hase omositions and fluid roerties. Further details of this roedure an be found in ershke (988). A Brazilian Journal of Chemial Engineering

3 Comarison of an IMEC and a Semi-Imliit Formulation for Comositional Reservoir Simulation 979 volume-shift aroah based on the work of Jhaveri and Youngren (988) is also available for liquid density orretion. Two hase stability test algorithms are imlemented in the UTCOM simulator: the stationary oint loation method (Mihelsen, 982) and the Gibbs free energy minimization algorithm, whih is similar to the Trangenstein (987) method and was modified by ershke (988) to deal with three hydroarbon hase equilibrium. In general, as ommented by ershke (988), the stationary method is faster than the Gibbs free energy method; therefore, the stationary method was used in this work. The flash alulation used in UTCOM is a ombination of the Aelerated Suessive Substitution (ACSS) method (Mehra et al., 983) with the modified version of the Gibbs free energy minimization method (ershke, 988). At the beginning of the flash roedure, we use the ACSS method in order to rovide a reasonable initial estimation, and then we swith to the Gibbs free energy minimization method in order to aelerate the onvergene. The swithing riterion to hange from one method to another is given by Chang (990) as: j i max ln f ln f 2 i ε swi for i =,..., and j = 3,...,, (4) where, the suersrit 2 denotes the oil hase. The swithing riterion (ε swi ) equal to 0.0 as suggested by Chang (990) is used. The ressure equation used for both formulations is based on the volume onstraint roosed by Ás et al. (985). The ressure equation is obtained from the equality between the formation ore volume (V ) and the total fluid volume (V t ): V ( ) = V (,,...,, + ), (5) t where the ore volume is given by: ( ) V = Vbφ 0 f + f, (6) where φ o is the orosity at the referene ressure ( f ), and f is the rok omressibility. Taking the derivative of Eq. (5) with reset to time, alying the hain rule to the right-hand side, substituting Eqs. () and (6), and dividing all by the bulk volume, we obtain V t V t t φ 0f = b + ξ Φ + + Vt krj qk xkj j K j k k μ, s( s i) j j V = = b (7), where denotes derivative evaluated by holding the number of moles onstant. The saturation equation is solved imliitly only for the IMSAT formulation. The aroah used here is desribed by Watts (986). By definition, the saturation of hase l (S l ), as stated before, is the ratio of the hase volume to the volume of the ore: V S =, (8) V where V l is the volume of hase l, whih is a funtion of ressure and number of moles. Equation (8) an be written as: SV ( ) = V (,,...,, + ). (9) Taking the derivative of Eq. (9) with reset to time, alying the hain rule on the right-hand side and substituting Eq. () and dividing it by the bulk volume, we obtain: V ( SV ) = V t V t b b + V q k + ( xkjξ jvj ) +. k k V =, s ( s i) j= b (0) Comaring Eqs. (7) and (0), we an observe that saturations and ressure equations have similar formats. As suggested by Watts (986) and Sillette et al. (973), the hase veloity that is needed in Eq. (0) should be evaluated as a funtion of total veloity. Aording to these authors, if this aroah is taken, the segregated solution of ressure and saturations will be equivalent to a fully imliit roedure in terms of ressure and saturations. ext, we resent the exression for the total veloity using the idea given in Watts (986), but now onsidering a full ermeability tensor. Extending the stes of Kaasshieter (999) for a three-hase flow, or even four-hase flow, yields: Brazilian Journal of Chemial Engineering Vol. 3, o. 04, , Otober - Deember, 204

4 980 B. R. B. Fernandes, A. Varavei, F. Marondes and K. Seehrnoor g( ρj ρm) D vj = f j vt + λmk, () + m ( jo = mo ) where krm λ =, (2) μ m m v = v = λ K ( + gρ D), (3) t i j jo j i= j= and, f j λ j = λ m= m, (4) where, v t is the total fluid veloity, λ m is the m-th hase mobility whih is defined as a ratio of relative ermeability and visosity of the m-th hase, and f j is alled the frational flow, whih is defined as above. Substituting Eq. () into (0) yields: + V V qk ( SV ) = + + V t V t V b b k= k, s ( s i) b ( ( ) ( )). + V xkj j f j vt mk g j m D jo mo ξ + λ ρ ρ + k= k, s ( s i) j= m= (5) Equation (5) is the final saturation equation in terms of total veloity. The next setion is devoted to showing the numerial disretization alied to the above equation. AROXIMATE EQUATIOS In order to obtain an aroximate equation for the saturation of hase l, we will integrate Eq. (5) over the ontrol volume of Figure and time. Figure : Control volume. The final form of the saturation equation for the three-dimensional ontrol volume as shown in Figure, is given by: Brazilian Journal of Chemial Engineering

5 Comarison of an IMEC and a Semi-Imliit Formulation for Comositional Reservoir Simulation 98 S V S V S V n n n+ n+ n n V, n+ n,,,,, ( ) + n n n,, n ξ, + n n+ k, k, k= =Δt V q + + n n n n+ n n n n+ +Δ ξ Δ ξ t V, x u k kj j j t V k, x u kj j j e w k= j= k= j= + + n n n n+ n n n n+ +Δt V, x ξ v Δ ξ k, kj j j t V x v k kj j j n s k= j= k= j= + + n n n n+ n n n n+ +Δ ξ Δ ξ t V k, x w kj j j t V k, x w kj j j = = f b k j k= j=, (6) where V is the total volume derivative with reset to total number of moles of omonent k, and tk semi-imliit veloity at the east interfae, whih is given by: n u + j e is a n+ λ n+ j e n+ n+ n n j = t + x λm ρ j E ρm E e e e e e n+ m= λm e m= { e ( ) ( ) u u T g D D g D D where, T x e n+ n+ n+ n+ ( moe, mo, ) ( joe, jo, ) } + 2ΔΔ y z = Δ x Δ x + K K x x E., (7) (8) The semi-imliit veloity, at the other interfaes, is given by equations similar to Eq. (7). ewton s method is used to treat the non-linearities involved in Eqs. (6) and (7) on aount of the imliit evaluation of the relative ermeabilities and aillary ressures. erforming a similar roedure for the saturation equation, we obtain the aroximate mole balane equations, in terms of the semi-imliit veloity, as given below: n+ n k, k, n+ n n n+ n n n+ = q +, ( x ξ ) u - k kj j j ( x ξ kj j) uj t e e w w j= j= Δ n n n+ n n n+ n n n+ n n n+ x vj x vj x wj x wj n n s s f f b b j= j= j= j= ( ξ kj j ) ( ξ kj j ) ( ξ kj j ) ( ξ kj j ) (9) Brazilian Journal of Chemial Engineering Vol. 3, o. 04, , Otober - Deember, 204

6 982 B. R. B. Fernandes, A. Varavei, F. Marondes and K. Seehrnoor The final form of the ressure equation is obtained (onsidering only the x diretion for simliity) as: V V t V q + n n 0 Vt, +, n n +, φ ( ) + n n j n n b f, =Δ,, tk k n ξ j= j, k= +Δt V x ξ λ T + n n n n n n tk, kj j = e k j= + + ( j x) ( E ) +Δt V x ξ λ T (( ) ( )) + n n n n n n n tk, ( kj j j x ) jo,, + ρ E jo g j DE D e e k= j=. (20) Δt V x ξ λ T + n n n n n n tk, kj j = w k j= + + ( j x) ( W ) ( ) + n Δ n n n n n n t V ξ λ + ρ tk, x T g D D kj j w = w k j= ( j x ) ( jo, jo, W ) j ( W ) Further details regarding the aroximate equations for the ressure and moles an be obtained in Chang (990). Fig. 2 shows the flow hart of the roedure used in the UTCOM simulator for the imlementation of the Watts formulation. As we an see in this figure, ressure is evaluated first; then, an iterative roedure is used to evaluate the saturations. START Evaluate ressure at the new time-ste Evaluate total veloity j= j < - false The mobilities, the densities, and the mole fration of eah omonent in eah hase at eah gridblok interfae are evaluated using a one-oint uwind interolation funtion. A total variation diminishing (TVD) method using the Koren s flux limiter (Koren, 993; Liu et al., 995; Fernandes et al., 203) was also imlemented for solving the saturation equations in the relative ermeabilities and hase omositions. In this situation, as a non-linear term is inluded in the relative ermeability due to the flux-limiter, it was treated semi-imliitly using the idea resented by Rubin and Blunt (99), in whih the higher order terms are only onsidered in the indeendent term of the linear system. For the one-oint uwind imlementation, the mobilities are evaluated as: True Evaluate residues and derivatives for hase j Solve the saturation differenes and evaluate the new saturations j=j+ Evaluate the last saturation as one minus the summation of the other saturations λ Φ >Φ λ = λ Φ Φ n+ n n if n+ j j, x j, x x j x+ /2 n+ n n j if j, x j, x x+. (2) o ewton's Convergene reahed? Yes Evaluate the number of moles at the new time-ste erform flash alulation and all roerties at the new time-ste ED Figure 2: Flow hart for Watts formulation. Solving the roblem using the one-iteration IMEC aroah is straightforward sine only one set of linear equations (for ressure) needs to be solved and the total number of moles is omuted semi-imliitly to onserve the moles. A flash alulation is erformed to obtain hase omositions and hase mole frations. Finally, the water and the hydroarbon saturations, resetively, are evaluated as: Brazilian Journal of Chemial Engineering

7 Comarison of an IMEC and a Semi-Imliit Formulation for Comositional Reservoir Simulation 983 S w and = Vw w V = ξ V, (22) w L ( Sw ) V S ξ = =, (23) V L j ξ j= 2 j where L is the hase mole fration obtained in the flash in the absene of water. In this work, relative ermeability is modelled by the Corey s model (Corey, 986) and aillary ressure is modelled aording to Chang (990) as: φ E ow = Cσwo ( Sw ) ; K φ S w = C σ K + y So Sg og og y E, (24) where C and E are user-inut arameters, σ is the interfaial tension between the two hases and S denotes a normalized saturation. For both aroahes used in this work, one needs to solve the same number of variables. All variables are listed in Table. Also, Table shows some funtional relations for some of the variables used in this work. Table : List of funtional relations. Variable IMEC IMSAT o. of Equations er grid blok V Eq. (6) Eq. (6) S j Eqs. (22-23) Eq. (6) ξ j EOS EOS x ij EOS EOS ( -) k rj Corey model Corey model (986) (986) μ j Lorenz et al. Lorenz et al. (964) (964) Eq. (20) Eq. (20) j Chang (990) Chang (990) - ρ j EOS EOS q i Well model Well model + i Eq. (9) Eq. (9) + Total The hase aearane for the IMEC and IMSAT aroahes develoed in this work is treated only by the hase stability test, as desribed earlier. Also, the hase disaearane for the IMEC aroah is based on the hase stability test. However, for the IMSAT aroah, beause hase saturation an be omuted as zero or negative during the solution of Eq. (6), we have to oule the stability test with another aroah. In this ase, if the solution of Eq. (6) rodues a negative saturation, we set that saturation to zero. When a negative saturation is alulated, it means that the hase disaears for that time-ste. Setting saturation to zero is a ommon aroah used for some fully imliit aroahes; see Coats (980), for instane. RESULTS AD DISCUSSIO In this setion, we investigate the numerial solutions, as well as the erformane in terms of CU time of the Watts formulation imlemented in this work and the original IMEC aroah of the UTCOM simulator. The omarison studies were arried out by emirially setting the maximum allowable time-ste for various ase studies, whih did not rodue osillatory results for both investigated formulations. The first ase investigated is CO 2 injetion in an isotroi heterogeneous reservoir. All reservoir data used for this ase are shown in Table 2. The water is not onsidered in flash alulations and is not injeted in the reservoir for this ase. The water mole numbers are estimated by density and initial saturation. The omonents, the initial fluid omositions, and the injeted fluid omositions are shown in Table 3. Table 2: Reservoir data - Case. roerty Value Length, Width and Height 52.4 m, m and m orosity 0.25 Initial Water Saturation 0.25 Initial ressure 7.58 Ma ermeability in Z diretion 9.87x0-5 m 2 Formation Temerature 33.7 K Injetor BH 8.62 Ma roduer BH 7.58 Ma Grid 20x40x Table 3: Comonent data - Case. Comonent Initial Reservoir Injetion Fluid Comosition Comosition CO C C C C C C Brazilian Journal of Chemial Engineering Vol. 3, o. 04, , Otober - Deember, 204

8 984 B. R. B. Fernandes, A. Varavei, F. Marondes and K. Seehrnoor The absolute ermeability fields in the x and y diretions are shown in Fig. 3. In order to better visualize the variation of the ermeability fields, Figs. 3b and 3 show two different zooms of the whole sale resented in Fig. 3a. We omare the results of the Watts formulation with the original IMEC formulation of the UT- COM simulator. The results in terms of oil and gas rodution are resented in Figs. 4a and 4b, resetively. From Fig. 4, we an observe a good agreement between the original IMEC formulation and the Watts formulation imlemented. () Figure 3: ermeability in x and y diretions. a) Whole sale limits; b) and ) sale zoom. Figure 4: Volumetri rodution rates - Case. a) Oil and b) Gas. In this ase, water saturation (volumetri fration) is lower than the so-alled onnate saturation (0.25) and hene water annot flow through the rok ores. Initially, only oil and a mixture of CO 2 and light hydroarbons injeted (gas) are resent in the reservoir. As a result, CO 2 dislaes oil first and forms a third non-aqueous hase. This hase is referred to as a seond liquid hase (not onsidering the aqueous hase) (see Figure 5). It is imortant to mention that this behavior is only ossible by onsidering three-hase flash alulations. Also, the resent ase showed a slow onvergene when we tried to run using a two-hase flash (only oil and gas). This reinfores the imortane of onsidering the three-hydroarbon flash alulations. The results for the seond liquid front at 844 days, obtained with the IMEC and the Watts formulation, are omared in Figs. 5. As an be seen in this figure, a good agreement at the two fronts is verified. Brazilian Journal of Chemial Engineering

9 Comarison of an IMEC and a Semi-Imliit Formulation for Comositional Reservoir Simulation 985 Table 4: CU time omarison - Case. Grid CU time for IMEC (s) CU time for Seed-u ratio Watts (s) 20x x Figure 5: Seond liquid front at 844 days. a) IMEC; b) Watts. Case 2 is similar to Case, but we relaed the uwind interolation funtion by a third-order TVD interolation funtion. Figs. 7a and 7b show the volumetri rate of oil and gas, resetively, and Fig. 8 resents the seond liquid saturation front at 844 days. One again, the Watts formulation results are in good agreement with the IMEC formulation. Desite the large differene when omaring the rodution rates for TVD solution (Figure 7) with that of uwind solution (Figure 4), the hase behavior obtained was similar for the two ases (the third hase does aear). The time-stes used by the original UTCOM aroah (IMEC) and the Watts aroah are shown in Fig. 6. Figure 6: Time-ste - Case. As we an observe in Fig. 6, the time-stes used by the Watts formulation were larger than those of the IMEC formulation for the entire simulation. Due to the omutational time sent for the solution of the linear system equations for the saturations, the Watts formulation is more exensive er time-ste. However, the large time-stes used by the Watts formulation allowed this formulation to be less exensive in terms of CU time than the IMEC formulation. This fat an be seen in Table 4, whih shows the total CU time used by both formulations. From this table, we observe that Watts formulation is about two times faster than the IMEC formulation. We also hek the formulations erformanes by doubling the number of gridbloks in eah diretion. As we an see in Table 4, the seed-u ratio of the Watts formulation is imroved when the number of gridbloks is doubled in eah diretion. Figure 7: Volumetri rodution rates - Case 2. a) Oil and b) Gas. The time-stes used in this ase study for the IMEC and the Watts formulations are shown in Fig. 9. Although the time-ste attern resented in Fig. 9 is different from the ones shown in Fig. 6, using the uwind sheme, aroximately the same seed-u ratio of the Watts formulation omared to the IMEC formulation for Case was obtained as shown in Table 5. Brazilian Journal of Chemial Engineering Vol. 3, o. 04, , Otober - Deember, 204

10 986 B. R. B. Fernandes, A. Varavei, F. Marondes and K. Seehrnoor resetively, and Fig. 2 shows the oil saturation front at 254 days using a 00x00x5 Cartesian grid. Table 6: Reservoir data Case 3. roerty Value Length, width and thikness m, m and 4.48 m orosity 0.63 Initial Water Saturation 0.65 Initial ressure 9.65 Ma Formation Temerature K Injetor BH 0 Ma roduer BH 6.89 Ma Grid 00x00x5 Figure 8: Seond liquid front at 844 days. a) IMEC; b) Watts. Table 7: Comonent data Case 3. Comonent Initial Reservoir Comosition Injetion Fluid Comosition CO C C C C C C Figure 9: Time-ste omarison - Case 2. Table 5: CU time omarison - Case 2. Formulation CU time (s) Seed-u ratio IMEC Watts Case 3 refers to a WAG (water-alternating gas) roess in a heterogeneous reservoir. For this ase study, in eah yle, we first injet CO 2, and then water. Eah fluid is injeted at a fixed ressure over the ourse of ten days. The total simulation time is 280 days, whih orresonds to fourteen yles. Tables 6 and 7 resent the reservoir data set and the initial and the injeted fluid omositions, resetively, emloyed for this ase study. Two hydroarbon hases were onsidered in this ase and water still is not aounted for in the flash alulations. The absolute ermeability field in the x and y diretions is shown in Figure 0, while the ermeability in the z diretion is onstant and equal to 7.89x0-5 m 2 (8 md). For this ase study, the uwind sheme was used to obtain the solution for both formulations. Figures a and b resent the oil and gas volumetri rates, Figure 0: Absolute ermeability in x and y diretions Case 3. Brazilian Journal of Chemial Engineering

11 Comarison of an IMEC and a Semi-Imliit Formulation for Comositional Reservoir Simulation 987 formulation is aroximately three times larger than that used by the IMEC aroah. This average timeste results in a seed-u ratio of.7, as we an see in Table 8. In general, at the beginning of eah yle, the time-ste size is very lose to the minimal time-ste designed for the whole simulation. This aroah should be in favor of the IMEC formulation. Figure : Volumetri rodution rates - Case 3. a) Oil and b) Gas. Figure 3: Time-ste omarison - Case 3. Table 8: CU time omarison - Case 3. Formulation CU time (s) Seed-u ratio IMEC Watts The fourth ase study is another WAG roess, but now a homogeneous reservoir is tested. Caillary ressure is now inluded in order to hek the algorithm erformane when this hysial henomenon is relevant. Exet for absolute ermeabilities, all of the revious data resented in Tables 6 and 7 were used. The absolute ermeabilities in the x and y-diretions are set to.97x0-3 m 2, and the one in the z-diretion is equal to 9.87x0-4 m 2. We use an 80x80x5 grid, and the roess is simulated for 00 days. The oil and gas rates are resented in Figs. 4a and 4b, resetively. One again, we an observe a good math between the results of the two aroahes omared. Figure 2: Oil saturation front at 254 days - Case 3. a) IMEC; b) Watts. The time-stes used by both formulations are shown in Fig. 3 and the total CU time and the seedu ratio are resented in Table 8. From the figures, we an observe that the average time-ste used by Watts Brazilian Journal of Chemial Engineering Vol. 3, o. 04, , Otober - Deember, 204

12 988 B. R. B. Fernandes, A. Varavei, F. Marondes and K. Seehrnoor Figure 4: Volumetri rodution rates - Case 4. a) Oil and b) Gas. The time-ste omarison is shown in Fig. 5. From this figure, we an verify that the maximum time-ste emloyed by the Watts formulation is about two times larger than the one used by the IMEC aroah. It is imortant to mention that the time-ste ratio, aforementioned, is relevant for only the seond and third yles, as we an verify from Fig. 5. and an immobile aqueous hase are onsidered. The urose of this ase is to see how Watts aroah will erform with a large number of omonents by looking at the imat of exensive flash alulations over the simulation erformane. Tables 0 and show the reservoir data and the initial and the injeted fluid omositions, resetively. It is worthwhile to mention that all the omonents equal or higher than C 25+ have idential hysial roerties. The main goal here was to verify the erformane of the omared aroahes with a very large number of omonents. Table 0: Reservoir data Case 5. roerty Value Length, width and thikness m, m and 6. m Absolute ermeability in x, y, and z-diretions 9.87x0-4 m 2, 9.87x0-4 m 2 and 9.87x0-5 m 2 orosity 0.25 Initial Water Saturation 0.25 Initial ressure 9.65 Ma Formation Temerature 400 K Injetor BH 20 Ma roduer BH 6.55 Ma Grid 40x40x Table : Comonent data Case 5. Figure 5: Time-ste omarison - Case 4. Table 9 resents the CU time and the seed-u ratio obtained for ase 4. From this table, we an verify that the erformane of Watts formulation is worse omared to the revious ases, but it still erforms better than the IMEC aroah. Further investigation of the urrent Watts imlementation needs to be erformed when still more omliated hysial arameters are involved. Comonent Initial Reservoir Injetion Fluid Comosition Comosition CO C C C C C Other omonents (C 25+ ) - Figure 6 shows the oil and gas volumetri rates. From this figure, we an see a good math for oil and gas volumetri rates for both aroahes. Table 9: CU time omarison - Case 4. Formulation CU time (s) Seed-u ratio IMEC Watts The last ase study refers to a 2D gas flood with twenty-five omonents. Only two-hydroarbon hases Brazilian Journal of Chemial Engineering

13 Comarison of an IMEC and a Semi-Imliit Formulation for Comositional Reservoir Simulation 989 erformane dereased when the gridbloks are inreased, this formulation is still about 2 times faster than the IMEC aroah. COCLUSIOS Figure 6: Volumetri rodution rates - Case 5. a) Oil and b) Gas. The time-stes used by both aroahes are shown in Fig. 7. From this figure, we an verify that Watts formulation was able to handle this ase study with several omonents using muh large time-stes omared to the IMEC formulation. In this work, we imlemented the Watts formulation for omositional reservoir simulation using Cartesian grids. This formulation was inluded into the UTCOM simulator. Also, two interolation funtions for evaluating the hysial roerties at eah interfae of the ontrol volume were imlemented: UDS and a third-order TVD sheme. At least for the ase studies investigated, the seed-u ratio of the Watts formulation did not hange when the TVD and UDS shemes were used. For most of the ase studies tested, the Watts formulation imlemented in this work was around two times faster than the original IMEC formulation of the UTCOM simulator. We also verify, for some ase studies, that the erformane of Watts formulation ersists when the mesh is refined. It was also onfirmed that the new imlemented formulation is more effiient than the IMEC for ases with a large number of omonents. OMECLATURE Figure 7: Time-ste omarison - Case 5. Table 2 resents the total CU time and seed-u ratio for both aroahes. We also show the seed-u ratios when gridbloks in eah diretion are doubled. From this table, we an infer that Watts formulation is about 2.25 times faster than the IMEC aroah. Table 2: CU time omarison - Case 5. Grid CU time for IMEC (s) CU time for Seed-u ratio Watts (s) 40x x This suggests that, for a large number of omonents, the Watts formulation should be used instead of the IMEC aroah. However, when the number of gridbloks is inreased, the seed-u ratio dereased to.93. Although the Watts formulation f Rok omressibility a - g Gravity m d -2 f Frationary flow or fugaity for the equilibrium onstraint K Absolute ermeability m 2 k r tensor Relative ermeability L hase mole fration umber of moles, mol umber of omonents umber of hases ressure a q Well mole rate mol d - S Saturation t Time s V Bulk volume m 3 b V ore volume m 3 V t Total fluid volume m 3 V Total fluid artial molar m 3 mol - tk volume V hase volume m 3 Brazilian Journal of Chemial Engineering Vol. 3, o. 04, , Otober - Deember, 204

14 990 B. R. B. Fernandes, A. Varavei, F. Marondes and K. Seehrnoor V hase artial molar volume m 3 mol - k v Veloity vetor m d - x Comonent mole fration in eah hase Greek Letters ξ Mole density mol m -3 ρ Mass density kg m -3 σ Interfaial tension m - φ orosity λ hase mobility a - d - Φ Hydrauli otential a μ Visosity a d Δ t Time ste size d Δ x Satial ste size in the x m Δ y Δ z Suersrits diretion Satial ste size in the y diretion Satial ste size in the z diretion n revious time ste level n+ ew time ste level Subsrits b B e E f F g i j k l n o r s S t w W Bak interfae Bak ontrol volume East interfae East ontrol volume Front interfae Front ontrol volume Gas hase Control volume hase Comonent hase orth interfae orth ontrol volume Oil hase Control volume Referene hase South interfae South ontrol volume Total Water omonent/hase or west interfae West ontrol volume m m ACKOWLEDGMETS The authors would like to aknowledge the Abu Dhabi ational Oil Comany for the finanial suort for this work. We would like to thank Dr. Chowdhury K. Mamum for his omments on this manusrit. Also, the third author would like to thank the Cq (The ational Counil for Sientifi and Tehnologial Develoment of Brazil) for finanial suort through the grant o / Finally, we would like thank the ESSS (Engineering Sientifi Software Simulating) for roving the Kraken to re and ost-roessing the results. REFERECES Ás, G., Doleshall, S. and Farkas, E., General urose omositional model. SE Journal, 25(4), (985). Brano, C. M. and Rodriguez, F., A Semi-imliit formulation for omositional reservoir simulation. SE Advaned Tehnology Series, 4(), (996). Cao, H., Develoment of Tehniques for General urose Simulators. h.d. Thesis, Stanford University (2002). Chang, Y.-B., Develoment and Aliation of an Equation of State Comositional Simulator. h.d. Dissertation, The University of Texas at Austin, Austin Texas (990). Chang, Y.-B., oe, G. A. and Seehrnoori, K., A higher-order finite-differene omositional simulator. Journal of etroleum Siene and Engineering, 5(), (990). Coats, K. H., An equation of state omositional model. SE Journal, 20(5), (980). Corey, A. T., Mathematis of immisible fluids in orous media. Water Resoures ubliation, Littleton, CO (986). Fernandes, B. R. B., Marondes, F. and Seehrnoori, K., Investigation of several interolation funtions for unstrutured meshes in onjuntion with omositional reservoir simulation. umerial Heat Transfer art A, Aliations, 64(2), (203). Jhaveri, B. S. and Youngren, G. K., Three-arameter modifiation of the eng-robinson equation of state to imrove volumetri redition. SE Reservoir Engineering, 3(3), (988). Kaasshieter, E. F., Solving the Bukley-Leverett equation with gravity in a heterogeneous orous medium. Comutational Geosienes, 3(), (999). Brazilian Journal of Chemial Engineering

15 Comarison of an IMEC and a Semi-Imliit Formulation for Comositional Reservoir Simulation 99 Kendall, R.., Morrell, G. O., eaeman, D. W. and Watts, J. W., Develoment of a multile aliation reservoir simulator for use on a vetor omuter. SE Middle East Oil tehnial onferene. SE, Manama, Bahrain (983). Koren, B., A Robust Uwind Disretization Method for Advetion, Diffusion and Soure Terms. In: C. B., Vreugdenhil, and B., Koren, (Eds.), umerial Methods for Advetion-Diffusion roblems. otes on umerial Fluid Mehanis, vol. 45,. 7-38, Vieweg, Braunshweig (993). Liu, L., Delshad, M., oe, G. A. and Seehrnoori, K., Aliation of higher-order flux-limited methods in omositional simulation. Transort in orous Media, 6(),. -29 (994). Lohrenz, J., Bray, B. G. and Clark, C. R., Calulating visosities of reservoir fluids from their omositions. Journal of etroleum Tehnology, 6(0), (964). Mehra, A. R., Heidemann, R. A. and Aziz, K., An aelerated suessive substitution algorithm. Canadian Journal of Chemial Engineering, 6(4), (983). Mihelsen, J. L., The isothermal flash roblem. art I. Stability, Fluid hase Equilibria, 9(),. -9 (982). eng, D. Y. and Robinson, D. B., The haraterization of the hetanes and heavier frations for the GA eng-robinson rograms. Gas roessors Assoiation (978). ershke, D. R., Equation of State hase Behavior Modelling for Comositional Simulator. h.d. Thesis, The University of Texas at Austin, Austin, Texas (988). Runbin, B. and Blunt, M. J., Higher-order imliit flux limiting shemes for blak-oil simulation. SE Symosium on Reservoir Simulation, SE, Anaheim, USA, (99). Sillette, A. G., Hillestad, J. G. and Stone, H. L., A high-stability sequential solution aroah to reservoir simulation. Fall Meeting of the Soiety of etroleum Engineers of AIME. SE, Las Vegas, evada, USA (973). Trangenstein, J. A., Customized minimization tehniques for hase equilibrium omutations in reservoir simulation. Chemial Engineering Siene, 42(2), (988). Watts, J. W., A omositional formulation based on ressure and saturation equations. SE Reservoir Engineering Journal, (3), (986). Brazilian Journal of Chemial Engineering Vol. 3, o. 04, , Otober - Deember, 204