Paper C Exact Volume Balance Versus Exact Mass Balance in Compositional Reservoir Simulation

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1 Paer C Exact Volume Balance Versus Exact Mass Balance in Comositional Reservoir Simulation Submitted to Comutational Geosciences, December 2005.

2 Exact Volume Balance Versus Exact Mass Balance in Comositional Reservoir Simulation Jarle Haukås Ivar Aavatsmark and Magne Esedal Centre for Integrated Petroleum Research (CIPR), University of Bergen, Norway Edel Reiso Norsk Hydro Oil & Energy Research Centre, Bergen, Norway Abstract. A new comositional formulation which incororates both an exact volume balance aroach and an exact mass balance aroach has been develoed and tested. The formulation is based on combining the conventional mass balance requirement with volume/isochoric balance requirements. The rimary variables are ressure, saturations and isochoric variables, and both a fully imlicit version and an IMPSAT version (imlicit in ressure and saturations, exlicit in the isochoric variables) are included. We note that the formulation reduces to a fully imlicit black-oil formulation when used with saturated black-oil fluid roerties. A comarison to similar aroaches is rovided, and the erformance of the exact volume balance aroach versus that of the exact mass balance aroach is investigated. Our conclusions are in favour of the exact volume balance aroach. Numerical results are shown. Keywords: comositional, mass balance, volume balance, isochoric variables, fully imlicit, IMPSAT, comarison 1. Introduction A comositional model usually involves of a set of mass balance equations, volume balance constraints and hase equilibrium equations. The thermodynamic relations are most often based on an equation of state. For an isothermal model, the number of rimary variables and equations equals the number of (seudo) comonents in the system. In a black-oil model, the hydrocarbon hases are assumed to consist of only two (seudo) comonents. This assumtion simlifies the hase equilibrium comutations, but in cases with large comositional gradients, the black-oil fluid descrition is not sufficiently recise. Traditionally, black-oil and comositional simulation has been erformed with different simulators. The reason for this is that the conventional comositional formulation uses ressure and comonent mole numbers (or ressure, overall comosition and overall density) as rimary variables, e.g., [1], while the conventional black-oil formulation uses ressure and saturations as rimary variables.

3 2 Haukås et al. Fluid flow is governed by Darcy s law, which is basically a relation in ressure and saturations. In addition, the requirement that the fluids must fill the ore volume is naturally given in terms of saturations. Pressure and saturations are therefore convenient rimary variables. A comositional formulation which includes ressure and saturations among the rimary variables will reduce to the conventional black-oil formulation when used with saturated black-oil fluid roerties. Having a unified black-oil and comositional formulation leads to reduced simulator develoment and maintenance costs. Several such formulations have been roosed in the literature, e.g., [2 7]. Furthermore, the ressure and saturations art can be decouled from the rest of the system, by combining the mass balance equations into ressure and saturation equations, and using an IMPSAT aroach (interblock flow terms imlicit in ressure and saturations only). Consequently, the black-oil art of the comositional system is identified. Different IMPSAT aroaches have also been roosed, e.g., [4, 6, 7]. The combination of mass balance equations has been discussed by several authors. Young and Stehenson, [1], Coats, [2], Coats et al., [5] and Cao, [6], use Gaussian elimination of the Jacobian of the linearized system. The volume balance method of Watts, [3], originally roosed by Ács et al., [8], bases the derivation on hysical rinciles. Wong et al., [9], showed that Gaussian elimination and the volume balance method yield the same system of finite-difference equations at the start of a timeste. Coats, [10], commented that the ressure equation is unique, indeendent of the manner of derivation. However, for the mentioned formulations, two different aroaches are used. Some formulations, e.g., [2, 6], involve a mass balance residual, but yield exact volume balance. Others, e.g., [1, 3, 5, 8], involve a volume balance residual, but yield exact mass balance. Coats et al., [5], reort that the use of a relaxed volume balance instead of a mass balance residual leads to better convergence roerties. Unfortunately, no numerical evidence has been given to suort that statement. This aer aims at clarifying the distinction between exact volume balance and exact mass balance. Motivated by the ideas of the volume balance method and the ideas of isochoric variables and equations introduced by Haukås et al., [7], we relace the conventional linearized mass balance scheme by a volume/isochoric balance scheme which still takes the mass balance into account. The scheme can be used for either exact volume balance or exact mass balance. The rimary variables are ressure, saturations and isochoric variables, and both a fully imlicit version and an IMPSAT version are included. The formulation is comared to similar aroaches, and the erformance of exact volume balance versus exact mass balance is investigated.

4 Exact Volume Balance Versus Exact Mass Balance 3 2. Background In the following, we consider the N c mass balance equations, and their combination into hysically interretable, weighted sums (e.g., volume balance equations, [3]). Here, N c is the number of comonents. The starting oint is the integral form V b t (φ n VT ) dv + S b ˆf n ds ˆq dv = 0, V b (1) where φ is the orosity, n is the N c vector of comonent amounts, V T is the total fluid volume, ˆf contains the Nc comonent fluxes, ˆq is the N c vector of source density rates, while V b is some (bulk) volume with surface S b. The unit normal of S b is denoted n. We note that each comonent flux in ˆf is a vector in sace. The dot roduct in (1) is taken for each comonent flux, so that ˆf n is a N c vector. We observe that (1) is given in terms of both extensive and intensive quantities. Extensive quantities, for instance volumes and mole numbers, deend on the amount of material resent, while intensive quantities, for instance orosity and densities, do not. To describe the intensive roerties of an isothermal system, N c indeendent intensive arameters are needed. If extensive roerties are to be determined, N c + 1 arameters are needed, of which at least one must be extensive. We first show how the differential form of the equations can be weighted and combined, and then consider the control-volume finitedifference forms. For the latter, the concets of mass balance, volume balance and isochoric balance are defined, and it is shown that the weighting rocedure is not straight-forward Differential form The differential form of (1) is based on an assumtion of continuity, and must be given in terms of intensive quantities only. This leads to ( φ z ) + t v ˆf ˆq = 0, (2) T where z = (1/n T ) n is the overall comosition, v T = V T /n T is the total secific volume and n T is the total mass (sum over all comonents). We note that the divergence is taken for each comonent flux in ˆf. Watts, [3], showed that a weighting of (2) by the total artial molar volumes ( V T /) yields a ressure equation of the form φ t φ v T ( ) ( ) vt z t + VT ( ˆf ) ˆq = 0, (3)

5 4 Haukås et al. while a weighting by the artial volumes of hase j, ( V j / ), yields a saturation equation of the form (φs j) φ ( ) ( ) v j V j ( t v T t + ˆf ) ˆq = 0. (4) z Here, is ressure, S j is the saturation of hase j, V j is the hase volume and v j = V j /n T is the hase secific volume. We note that derivatives of extensive variables with resect to extensive variables are intensive quantities. Consequently, (3) and (4) are given in terms of intensive quantities only, as required. Equation (3) can be interreted as the differential version of the requirement that the fluids must fill the ore sace, i.e., V T = V, where V denotes the ore volume. Similarly, equation (4) can be interreted as the differential version of V j = V S j. In other words, (3) and (4) are volume balance equations. Since the saturations are suosed to sum to unity, and the hase volumes sum to the total volume, a sum of (4) over all hases yields the ressure equation (3). Consequently, the volume balance equations constitute N indeendent equations, where N is the number of hases. If used in ractice, the volume balance equations relace N of the mass balance equations. Since it is not clear which mass balance equations should be relaced, N c N additional equations relacing the rest of the mass balance equations should also be introduced. If these equations are comlementary to the volume balance equations, the roerties of the original system are reserved. This corresonds to the idea of isochoric equations introduced by Haukås et al., [7]. However, Haukås et al. did not consider the differential form of the equations. Furthermore, the differential form is seldom used for real roblems, as the flow terms ˆf deend on a discontinuous ermeability field. The integral form of the equations is therefore required Integral (control-volume discretized) form For the solution of (1), a control-volume discretization is convenient. The bulk volume V b is assumed to be a known, time-indeendent arameter, and the roerties within V b are assumed to be constant over that volume. This allows for the form t (φ n VT V b ) + f q = 0, (5) where f is the N c vector of discretized comonent flow rates, while q is the N c vector of comonent source rates. In addition, we use the total

6 Exact Volume Balance Versus Exact Mass Balance 5 volume balance requirement, V T = V = φv b, to obtain where t + θ = 0, (6) θ = f q (7) is introduced for notational convenience. Before considering a weighting of (6), we resent some ossible conservation requirements Mass balance Conventionally, the time derivative in (6) is aroximated by a backward difference, leading to the form where n + θ t = 0, (8) t = t n t n 1, n = n n n n 1. (9) The form (8) alies to each control volume, and common flow terms are used for flow through a common surface of two control-volumes. This ensures local mass balance. Furthermore, the sum of (8) over all control-volumes exresses a global mass balance, i.e., that the net change of mass equals the net mass introduced into the system. Note that, for generality, we do not secify the timelevel at which θ is evaluated, i.e., θ can be either fully imlicit or artly exlicit in time Volume balance The requirement that the fluids must fill the ore sace is imortant. We let the N vector V reresent the fluid volumes, and let the N vector V reresent the corresonding arts of the ore volume. Examles of reresentations of V and V for a three-hase system are V o V = V g, V w and V T V = V g, V w S o V = φv b S g, S o + S g + S w = 1, (10) S w 1 V = φv b S g, S o + S g + S w = 1. (11) S w Here, suerscrits o, g and w denote the oil, gas and water hases, resectively. In the latter reresentation, S g and S w have been chosen to be rimary saturations. Other choices are of course ossible.

7 6 Haukås et al. Based on this notation, we introduce the general volume balance requirement V V = 0, S o + S g + S w = 1. (12) Isochoric balance As a sulement to the volume balance requirement (12), we roose an isochoric balance requirement, x W x n = 0, (13) where the N c N vector x is referred to as the isochoric variables, while the rows of the (N c N ) N c matrix W x san the nullsace of the N N c matrix W V = ( V /) of artial volumes. During a timeste, W x is ket fixed, usually the revious time level. The motivation for (13) is due to Haukås et al., [7]. As ( ) V V = n = W V n, (14) and W x sans the nullsace of W V, which is the orthogonal comlement of the row sace of W V, the isochoric variables are comlementary to volumes. Consequently, information rovided by the mole numbers n is reserved by the alternative set (V, x ). We now turn to the weighting of the mass balance equations (6). By differentiating through V T = φv b, assuming that V T = V T (, n), we find that φ V b t = V T t = ( ) ( ) VT n t + VT t. (15) Consequently, if we multily (6) by ( V T /), and use (15), we obtain ( ) ( ) φ V b t VT n t + VT θ = 0, (16) which is similar to (3). Furthermore, by differentiating through V j = φv b S j with the assumtion that V j = V j (, n), we find that V b (φs j) = V j ( ) ( ) V j V j = t t t + t. (17) If we multily (6) by ( V j / ), and use (17), we obtain ( ) ( ) ( V b φs j) V j V j t t + θ = 0, (18) n n

8 Exact Volume Balance Versus Exact Mass Balance 7 which is similar to (4). By introducing V and V, we may reresent (16) and (18) by the N indeendent equations V t ( ) ( ) V V n t + θ = 0. (19) In addition, motivated by Haukås et al., [7], we multily (6) by the time-indeendent (N c N ) N c matrix W x to obtain x t + W x θ = 0. (20) We aroximate the time derivatives in (19) and (20) by backward differences, leading to ( ) ( ) V V V + θ t = 0, (21) n x + W x θ t = 0. (22) However, (21) and (22) are not consistent with the conventional mass balance (8), but rather with + S + x + θ t = 0, (23) S x where S contains N 1 rimary saturations, see aendix A. If we solve (23) instead of (8), the conventional mass balance is obtained only when t 0. This inconsistency arises from exressing time derivatives of mass in terms of time derivatives of other variables, as for instance is done in Watts derivation of the differential form of the volume balance equations, [3]. Such an aroach only reserves the original mass balance in the continuous (differential) case. Watts does not address this roblem directly. However, for the urose of interretation, he uses V T = φv b to derive the form [ ( ) ] φ n 1 V b VT + n ( ) n 1 VT θ t = (V T V ) n 1, (24) which is similar to (16), excet for the term on the right hand side. Furthermore, Watts reorts that he uses a solution of the ressure and saturation equations to determine θ, followed by n n = n n 1 θ t. (25) Subsequent to (25), Watts uses a hase equilibrium calculation with (, n) to recalculate the fluid volumes, so that ( V j V S j) and thereby

9 8 Haukås et al. (V T V ) become non-zero. The latter is taken into account by (24), and Watts states that the hase volume discreancies can be resolved in much the same way. Consequently, Watts overcomes the mass balance inconsistency, and actually obtains an exact mass balance, (25). Watts aroach is a ossible solution to the mass balance inconsistency roblem, but it requires that the flow rates and source terms are determined as functions of ressure and saturations only. Otherwise, additional equations, e.g., isochoric equations, are needed. Furthermore, the use of an iterative scheme would increase control of the volume discreancies. The latter corresonds to relacing the timeste suerscrits of (24) by iteration ste suerscrits, and is also suggested by Wong et al., [9]. However, due to the right hand side of (24), Watts aroach does not corresond to a weighting of the mass balance equations. Actually, a weighting of the mass balance equations is of no interest if the residuals of those equations are zero. In order to obtain an alternative, weighted mass balance aroach, we should exclude the direct udate (25) and the volume discreancies ( V j V S j), but include a (weighted) mass balance discreancy. 3. New Princiles In the following, we roose a formulation which incororates an aroach similar to that of Watts, but including isochoric equations, and an aroach without volume discreancies. The former yields exact mass balance, while the latter yields exact volume balance, and is a weighted mass balance aroach. In order to clarify the distinction between exact mass balance and exact volume balance, we seek a common form of the iterative scheme. Rather than using a weighting of the mass balance equations as a starting oint, we consider a combination of the linearized mass balance requirement (8) and the linearized volume/isochoric balance requirements (12) and (13), where we make sure that the artial volumes aear. Consequently, in the case of no volume/isochoric discreancies, the scheme actually reduces to a weighted mass balance scheme. The linearization is with resect to the rimary variables. However, due to the thermodynamic relations, which cannot be written exlicitly in terms of the rimary variables, a set of secondary variables is also required. For each secondary variable we need a secondary equation, and the secondary equations will be fulfilled at every iteration level. As will become clear, the choice of secondary equations is a choice between exact volume balance and exact mass balance.

10 Exact Volume Balance Versus Exact Mass Balance Primary and secondary variables As mentioned in section 2, we need N c + 1 indeendent arameters, including an extensive one, to determine all intensive and extensive roerties of the system. We use the set (, S, x, V b ), where S contains N 1 rimary saturations, and x denotes the N c N isochoric variables, defined by (13). Omitting one saturation ensures that the saturations sum to unity, as required by (12). We treat V b as a known arameter, and let u = (, S, x ) be the set of N c rimary variables. The thermodynamic roerties of the hases are conventionally given in terms of ressure and the comonent mole numbers of the hases, denoted n j. We assume that water and hydrocarbons do not interact, i.e., that water does not exist in the hydrocarbon hases, and that hydrocarbons may not dissolve in the water hase. This means that there are N s = N h N hc + N wc = (N h 1) N hc + N c hase mole numbers. Here, N h is the number of hydrocarbon hases (zero, one or two), N hc is the number of hydrocarbon comonents, while N wc is the number of water comonents (zero or one). The hase mole numbers form a convenient set of N s secondary variables, u s = n j Primary and secondary equations In addition to the 2N c conservation requirements (8), (12) and (13), we imose chemical equilibrium between the hydrocarbon hases, f o f g = 0, (26) where f o and f g are N hc vectors containing the comonent fugacities of the oil and gas hases, resectively. The fugacity equalities (26) are very imortant. Actually, at any level where we want to calculate the volume derivatives ( V / ) n and ( V /), the hase equilibrium requirement (26) must be fulfilled, see aendix B.1. If there is only one hydrocarbon hase, (26) is redundant. Consequently, (26) constitutes (N h 1) N hc equations. Together with the conservation requirements, we have 2N c + (N h 1) N hc equations, which is the same as the number of rimary and secondary variables. The iterative scheme for determining the rimary variables must include N c equation residuals. The remaining N c +(N h 1) N hc equations are used to determine the secondary variables, and will be fulfilled at every iteration ste. To be able to calculate recise volume derivatives, the fugacity equalities (26) must be included among the secondary equations. The N c additional secondary equations will be either the mass balance requirement (8) or the volume/isochoric balance requirements (12) and (13).

11 10 Haukås et al Iterative scheme In order to form a common iterative scheme, we consider residuals of all of the conservation requirements, even though N c of them will be fulfilled at every iteration level. The derivation is based on a linearization of the conservation requirements with resect to u. We omit subscrits on derivatives with resect to these variables, assuming that they reresent total derivatives, see aendix B.2. A linearization of the mass balance equations (8) yields + ( ) ( ) (k (k+1) + S (k+1) + S ( θ t u (k+1) u ( x x (k+1) = ( n + θ t. (27) Furthermore, assuming that V = V (, n), we find that ( ) ( ) V V u = V u + u, (28) n for any u (, S, x ). We also assume that V = V (, S ). Consequently, a linearization of the volume balance requirement (12) yields [ ( V ( V ( V ( V ( ( x ) ] (k) n (k+1) + ( V (k+1) x (k+1) ( V S ( S (k+1) S S (k+1) = (V V. (29) With the use of (27), we arrive at the form + [ ( V ( V ( ) ] V (k) (k+1) + n ( θ t u (k+1) u = (V V ( V S S (k+1) ( ) V (k) ( n + θ t. (30)

12 Exact Volume Balance Versus Exact Mass Balance 11 In addition, a linearization of the isochoric balance (13), ( ) (k) x (k+1) W x (k+1) ( ( ) (k) W x S (k+1) + x (k+1) S x = (x W x n, (31) combined with (27), yields x (k+1) + W x ( θ u t u (k+1) = (x W x n W x ( n + θ t. (32) Equations (30) and (32) are N c equations written in the form of a Newton-Rahson iterative scheme, and can be solved with resect to the rimary variables u. We note that, if the volume/isochoric balance residuals vanish, the scheme corresonds to a weighted mass balance aroach, i.e., a weighting of (27) by ( V /) and W x IMPSAT scheme As an alternative to (30) and (32), we may use an IMPSAT scheme, in which the interblock flow terms f in θ = f q are treated exlicitly with resect to the isochoric variables x. This means that the derivatives of interblock flow terms with resect to x vanish. Consequently, (30) reduces to + [ ( V ( V ( V ( ) ] ( ) V (k) (k) (k+1) V + S (k+1) n S ( ) ( ) θ (k θ (k+1) + S (k+1) t S ( q t x (k+1) x = (V V ( ) V (k) ( n + θ t. (33) For all gridblocks that do not contain fully imlicit and variable source terms, q/ x vanishes. Consequently, the equations (33) are ressure and saturation equations, decouled from the rest of the system. Actually, (33) identifies the black-oil art of the comositional system.

13 12 Haukås et al. Furthermore, the equations (32) can be written x (k+1) W x ( q x t x (k+1) = (x W x n W x ( n + θ t ( ) ( ) W x θ (k θ (k+1) + S (k+1) t. (34) S Here, (k+1) and S (k+1) can be determined in advance by (33) in gridblocks that do not contain fully imlicit and variable source terms. Consequently, (34) can be solved one gridblock at a time (exlicitly). Elsewhere, (33) and (34) must be solved simultaneously Choice of secondary equations For the resented iterative scheme to roduce non-trivial changes of the rimary variables, all of the requirements (8), (12) and (13) cannot be fulfilled at every iteration level, i.e., we cannot have both exact mass balance and exact volume balance. However, as mentioned, N c of the conservation requirements can be included among the secondary equations, and thus be fulfilled at every iteration level Exact volume balance The volume/isochoric balance set of secondary equations is ( ) ( ) V u (k) V (k), u (k) s = 0, ( ) x (k) W x n u (k) s = 0, ( ) ( ) f o (k), u s (k) f g (k), u (k) s = 0, (35) where variables laced in arentheses indicate assumed deendencies. Equations (35) can be solved with resect to the secondary variables u s by a Newton-Rahson scheme, with the excellent initial estimate u (k,0) s = u (k 1) s + ( dus du ) (k 1) u (k). (36) Here, du s /du are total derivatives. The fulfilment of (35) at level k 1 ensures that total derivatives can be determined, see aendix B.2. We note that, due to (36), only a few iteration stes are required. By solving (35), we obtain volume balance and isochoric balance at every iteration level, so that the right hand sides of (30) and (32) only

14 Exact Volume Balance Versus Exact Mass Balance 13 contain mass balance discreancies. The iteration is used to reduce the mass balance discreancies below some tolerance. Actually, the volume balance can be made exact, by calculating using V j = n j ξ j (, u s ) (37) n j = ξ j (, u s ) V S j, (38) instead of n j = n j (u s ). Here, ξ j is the molar density of hase j, while n j is the total mole number of hase j Exact mass balance The mass balance set of secondary equations is ( ) ( ) n u (k) s n n 1 + θ u (k) t = 0, ( ) ( ) f o (k), u (k) s f g (k), u s (k) = 0. (39) These equations may also be solved with resect to u s. Here, the mass balance can be made exact, by first determining n (k) directly from the mass balance requirement. We then solve for the remaining arts u s of u s, for instance the gas hase mole numbers, using the initial estimate u (k,0) s = u (k 1) s + ( ) (k 1) us n (k). (40) However, the treatment of θ as an exlicit function of u, and not also as a function of u s, and the calculation of total derivatives of θ with resect to u in the iterative scheme, requires some exlanation. Due to the interblock flow terms, θ deends on variables of several gridblocks. Consequently, if a satial deendence of u s is considered in (39), the size of the corresonding linearized system will be unsuitable. In addition, the satial deendencies make it inconvenient to calculate total derivatives from the N s relations (39), see aendix B.2. To get around these roblems, we regard the rimary variables u to always be volume/isochoric balance consistent in themselves. This is lausible, since the volume/isochoric balance requirements (12) and (13) can be interreted as definitions of saturations and isochoric variables. Consequently, having determined the rimary variables at some iteration level, we roceed by solving (35) with resect to the secondary variables u s, and use this solution to udate θ and the (total) derivatives of θ. This defines θ as a function of u only. In the exact mass balance aroach we subsequently solve (39) with resect to u s, but θ and the derivatives of θ are not udated.

15 14 Haukås et al. By solving (39), we obtain mass balance at every iteration level, so that the right hand sides of (30) and (32) only contain volume balance and isochoric balance discreancies, resectively. The iteration is used to reduce the volume/isochoric discreancies below some tolerance. We note that the exact mass balance aroach is somewhat similar to the aroach of Watts, [3]. However, our aroach includes isochoric variables and equations, and is therefore more general. In addition, we use an iterative scheme, while Watts only uses a single iteration. The iteration imroves the quality of the solution. Due to the solution of both (35) and (39) at every iteration level, the resented exact mass balance aroach is more costly er iteration ste than the resented exact volume balance aroach. However, Coats et al., [5], reort that their exact mass balance aroach has better convergence roerties than the exact volume balance aroach of Coats, [2]. Increased costs er iteration ste could thus ossibly be comensated by a reduced number of iteration stes. We must use numerical tests to investigate the erformance of the two resented aroaches. However, in order to comare our results to those reorted by Coats et al., [5], we first comare our formulation to that of Coats, [2], and to that of Coats et al., [5]. 4. Comarisons to the Aroaches of Coats and Coats et al. In the following, the resented exact volume balance aroach is comared to the aroach of Coats, [2], while the resented exact mass balance aroach is comared to the aroach of Coats et al., [5] The aroach of Coats (1980) Coats, [2], includes hydrocarbon hase mole fractions among the rimary and secondary variables, instead of isochoric variables and hase mole numbers. The isochoric balance (13) is therefore not required. In addition, a different framework is used to set u the iterative scheme, see Aendix C. Coats scheme is a combination of the linearized mass balance requirement (8) and the linearized fugacity equalities (26). The linearization is in terms of both rimary and secondary variables, imlying that total derivatives are not calculated. Instead of solving the fugacity equalities at each iteration level, Coats iterative scheme is used to reduce both mass balance and hase equilibrium discreancies. This is in contrast with our aroach. We solve the secondary equations at every iteration level, so that we are

16 Exact Volume Balance Versus Exact Mass Balance 15 able to calculate recise volume derivatives and total derivatives. Admittedly, the hase equilibrium calculations increase the comutational costs on a er-iteration basis, but they also exclude thermodynamic difficulties from the Jacobian. The latter may be advantageous in challenging thermodynamic cases. In the fully imlicit case, Coats uses the linearized mass balance scheme without reformulation. In non-fully imlicit cases, Gaussian elimination of the Jacobian is used to form imlicit equations. In [10], Coats states that the ressure equation is unique, and shows that the Gaussian elimination is equivalent to a weighting by the total artial volumes. However, no equations of the form (32) are derived. In other words, one must choose which mass balance equation(s) to relace. We always reformulate all of the mass balance equations. For the exact volume/isochoric balance aroach, the reformulation is equivalent to a reconditioning of the linearized mass balance equations by artial volumes and the matrix W x. The use of comlementary equations and variables leads to a system that is better conditioned The aroach of Coats et al. (1998) Coats et al., [5], use the same variable set as Coats, [2], and the same framework, but include the requirement that the saturations should sum to unity as an extra secondary equation. Having determined the rimary variables, Coats et al. obtain exact mass balance by solving a system similar to (39) with resect to the hase mole numbers. The solution of these equations yields a set of saturations that do not sum to unity. However, being a secondary residual, the volume discreancy is taken care of by the iterative scheme, see aendix C. An exact mass balance aroach requires that the term θ is fixed when (39) is solved. Coats et al. udate θ as a function of the rimary and secondary variables rovided by the iterative scheme. Here, hase equilibrium residuals aear. Consequently, θ does not necessarily corresond to a hase equilibrium. However, by the subsequent solution of (39), all other terms corresond to a hase equilibrium. In our exact mass balance aroach, we solve the fugacity equalities twice, so that all terms are always in accordance with a chemical equilibrium. This is more costly on a er-iteration basis, but our aroach excludes hase equilibrium roblems from the iteration, thus imroving the robustness of the formulation. In addition, we note that the aroach of Coats et al., [5], like the aroach of Coats, [2], does not involve a general reformulation (reconditioning) of the system.

17 16 Haukås et al. 5. Numerical Comarison of the Exact Mass Balance Aroach To the Exact Volume Balance Aroach In the following, we comare the numerical roerties of the resented exact mass balance aroach, referred to as, to those of the resented exact volume balance aroach, referred to as. The main objective is to investigate whether exact mass balance leads to better convergence roerties than exact volume balance, as suggested by Coats et al., [5], and, if so, whether the comutational costs of may be less than those of. The latter is not obvious, since the comutational costs er iteration ste are larger for. The comarison of to is based on two numerical test examles, for which we study convergence, recision, comutational costs and the effect of relaxed convergence criteria Comared roerties Convergence and recision We monitor convergence by the normalized rimary solution changes, ds (k) = dx (k) = δs (k) δx (k) and by the normalized residuals r (k) c = r (k) V = r (k) x = r (k) f = r (k) c r (k) V r (k) x r (k) f d (k) < ɛ d, d (k) = N (k) max (k), (41) < ɛ ds, δs (k) = max S (k), (42) ( ) (k) < ɛ dx, δx (k) x = max, (43) < ɛ c, < ɛ V, < ɛ x, < ɛ f, r (k) c r (k) V r (k) x r (k) f x (k) ( ) n + θ t (k) = max n, (44) ( ) V V (k) = max φv b, (45) ( ) (k) = max x W x n x, (46) ( f o f g ) (k) = max. (47) Here, N denotes the number of gridblocks, and max are taken over all gridblocks, and fractions of vectors are to be interreted as fractions of the vector comonents. f g

18 Exact Volume Balance Versus Exact Mass Balance 17 The rimary solution changes aear in the iterative scheme used to determine the rimary variables. Provided that the Jacobian is calculated correctly, quadratic convergence is exected as the solution of the equations is aroached. Consequently, (41), (42) and (43) should decrease quadratically, regardless of the aroach used. The quantities (44), (45), (46) and (47) are normalizations of the rimary and secondary equation residuals. The normalization of (44) imlies that the mass balance of comonents aearing in small amounts is regarded to be relatively more imortant than the mass balance of comonents aearing in larger amounts. The same idea is reflected in (46) and (47). However, for the volume balance (45), we do not want to give a hase aearing in small amounts greater imortance. We therefore use the ore volume for normalization. We note that, with, (45) is exact, (46) and (47) are below their convergence limits at every iteration level, while (44) is decreased during the iteration. With, (44) is exact, (47) is below its convergence limit at every iteration level, while (45) and (46) are decreased during the iteration. The recision of and is associated with the limits below which the residuals can be reduced Comutational costs We let the average time sent on each nonlinear iteration ste for and be denoted α and α, resectively. The average is over each timeste, and we have argued that α < α. In addition, we let the number of nonlinear iteration stes er timeste for and be denoted β and β, resectively. According to Coats et al., [5], we could exect that β > β. Consequently, if which is equivalent to α β > α β, (48) κ = β β > α α = κ, (49) the total costs of for a timeste are less than the total costs of. However, if the inequality is turned the other way, is the less costly aroach. We comare the comutational costs of versus by monitoring κ and κ for each timeste.

19 18 Haukås et al Test cases We use two different test cases. The first case is similar to the Third SPE Comarative Solution Project, [12], while the second is similar to the Fifth SPE Comarative Solution Project, [13] Case 1: Third SPE Comarative Solution Project Case 1 includes 9 hydrocarbon comonents and water, and a grid of dimensions The descrition of comonents, rock and grid roerties is given in [12], where we use the fluid characterisation data rovided for the roject by Arco Oil and Gas Comany. However, we neglect caillary ressure and water and rock comressibility. In addition, we use analytical relative ermeability relations, k o r = k g r = k w r = ( S o ) Sco w, (50) ( S g ) Sco w, (51) ( S w Sco w ) Sco w, (52) where k j r is the relative ermeability of hase j. The connate water saturation is set to S w co = In [12], gas cycling, with rates secified for both injector and roducer, is used. We use another scenario. A constant rate of 0.25 std m 3 /s of methane is injected in gridblock (1,1,1) and a fixed bottom hole ressure roducer ( bars) in located in gridblock (9,9,4). The initial conditions rescribed in [12] corresond to a rich retrograde gas condensate reservoir. The ressure at datum deth is bars (3550 sia), the temerature is K (200 F) and only gas and connate water are resent initially. However, from the start of the simulation, oil aears in the reservoir. As a recise caturing of volumes is imortant when hases reaear/disaear, is more robust than in such cases. Consequently, a roer comarison cannot be done with the original initial conditions. We therefore let the state immediately after the oil hase reaearance eriod constitute the initial conditions for Case Case 2: Fifth SPE Comarative Solution Project Case 2 includes 6 hydrocarbon comonents and water, and a grid of dimensions The descrition of comonent, rock and grid roerties is given in [13], but we neglect caillary ressures and rock

20 Exact Volume Balance Versus Exact Mass Balance 19 and water comressibilities. In addition, we use analytical relative ermeability relations, ( S kr o o ) 2.75 = Sco w, (53) ( S kr g g ) 1.9 = Sco w, (54) k w r = ( S w Sco w ) (55) 1 S w co where the connate water saturation is set to S w co = 0.2. The initial conditions rescribed in [13], with a datum ressure of bars (4000 sia) and a temerature of K (160 F) corresond to a reservoir of connate water and undersaturated oil. The fifth SPE Comarative Solution Project describes a wateralternating-gas (WAG) injection cycle. We use another scenario. A constant gas rate of std m 3 /s (12000 Mscf/day) with comosition secified in [13] is injected into gridblock (1,1,1) and a fixed bottom hole ressure roducer ( bars) is located in gridblock (7,7,3). We choose Case 2 so that oil remains undersaturated throughout the simulation run. Consequently, there is no hase reaearance or disaearance, and the fugacity equalities are redundant Test scenario For both cases, we use a simulation run of 30 timestes, governed by { [ (1 + λ) u t n+1 = min t n n ] } min u u + λ u n, 30 days, (56) where t n+1 is the next timeste, t n is the revious timeste, u n is the change in the variable u over the revious timeste, u is the target variable change during the next timeste and λ is a tuning factor. The formula is due to Aziz and Settari, [14]. We use λ = 0.5, = 15.0 bars and S j, = 0.1. The initial timeste is 1 day, and the simulator is run in fully imlicit mode. Each case is run several times. The first run is with very strict convergence criteria, and is used to investigate the convergence behaviour, including the ultimate limits below which the residuals can be reduced. Subsequent runs are based on relaxed convergence criteria. We here study comutational costs and discreancies of ressure, saturations and normalized mole numbers from a reference solution obtained with strict convergence criteria.

21 20 Haukås et al. Pressure change Pressure change CVCRIT Iteration number Saturation change CVCRIT Iteration number a. Case 1, d. b. Case 1, ds. Isochoric variable change Iteration number CVCRIT Iteration number c. Case 1, dx CVCRIT Isochoric variable change CVCRIT Iteration number d. Case 2, d. e. Case 2, dx. Figure 1. Tyical behaviour of rimary solution changes during a nonlinear iteration for Case 1 and Case 2. The notions d, ds and dx refer to (41), (42) and (43), resectively. Convergence limits are also lotted. Note that there are no changes to the saturations (connate water and undersaturated oil) in Case Results Primary solution changes Figure 1 shows tyical behaviour of the rimary solution changes during a nonlinear iteration for Case 1 and Case 2. The notions d, ds and dx refer to (41), (42) and (43), resectively. Here, we have used the same convergence limit for all solution changes and for all residuals (10 10 for Case 1 and for Case 2). We note that the convergence limit can be set more strict for Case 2, as no saturation changes occur and no two-hase equilibrium has to be determined. We observe a quadratic convergence behaviour for both and, indicating that the Newton-Rahson scheme is imlemented roerly. In Case 1, the convergence rate of is somewhat better, thus suorting the statements of Coats et al., [5]. However, for relaxed convergence criteria, the differences are less significant.

22 Exact Volume Balance Versus Exact Mass Balance 21 Mass balance error Isochoric balance error Mass balance error CVCRIT Iteration number Volume balance error CVCRIT Iteration number a. Case 1, r c. b. Case 1, r V. CVCRIT Iteration number Chemical equilibrium error CVCRIT Iteration number c. Case 1, r x d. Case 1, r f. CVCRIT Iteration number Volume balance error CVCRIT Iteration number a. Case 2, r c. b. Case 2, r V. Isochoric balance error CVCRIT Iteration number c. Case 2, r x Figure 2. Tyical behaviour of residuals during a nonlinear iteration for Case 1 and Case 2. The notions r c, r V, r x and r f refer to (44), (45), (46) and (47), resectively, and the convergence limits are also lotted. Note that the fugacity equalities are redundant in Case Precision Figure 2 shows tyical behaviour of the equation residuals (44), (45), (46) and (47) during a nonlinear iteration for Case 1 and Case 2. We observe that the mass balance error r c of and the volume balance error r v of both flat out at about 10 12, which is reasonably close to exact. The recision level obtained for the isochoric balance and the hase equilibrium is also similar for and.

23 22 Haukås et al. κ κ κ κ Timeste number Timeste number a. Case 1, convergence limit b. Case 2, convergence limit κ 1.8 κ 2 κ κ Timeste number Timeste number c. Case 1, convergence limit d. Case 2, convergence limit κ κ 2 κ 1.5 κ Timeste number Timeste number e. Case 1, convergence limit f. Case 2, convergence limit Figure 3. Comarison of comutational costs for Case 1 and Case 2, measured by κ and κ as defined by (49). Three different convergence limits are used for all solution changes and residuals: 10 8, 10 4 and Comutational costs Figure 3 shows a comarison of the comutational costs of versus during the simulation runs, i.e., a comarison of κ and κ, as defined by (49). We recall that κ is a measure of the ossibly larger number of iteration stes required by, while κ is a measure of the larger comutational costs er iteration ste for. Three different convergence limits used for all solution changes and residuals are tested: 10 8, 10 4 and For Case 1, is more costly throughout the run, regardless of the convergence limit. For Case 2, the redundant fugacity equalities lead to a smaller κ, and κ is both below and above κ during the simulation run. For a convergence limit of 10 2, the results are in favour of, but no strong conclusion can be made. We note that relaxing the convergence limit only influences κ and not κ. This is what we should exect.

24 Exact Volume Balance Versus Exact Mass Balance Ref Ref (bars) 235 S o Time (days) n 1 /V (kmol/m 3 ) n 8 /V (kmol/m 3 ) Time (days) a. Case 1,. b. Case 1, S o. Ref Time (days) n 2 /V (kmol/m 3 ) Ref Time (days) c. Case 1, n 1/V. d. Case 1, n 2/V. Ref Time (days) n 9 /V (kmol/m 3 ) Ref Time (days) e. Case 1, n 8/V. f. Case 1, n 9/V. Figure 4. Plots of the discreancy between the solution obtained with a convergence limit of 10 2 and a reference solution (convergence limit 10 8 ) of the roduction block for Case 1. The lots show the variations in ressure, oil saturation and ore volume normalized mole numbers of comonents 1, 2, 8 and 9 with time Effect of relaxed convergence criteria When the convergence limit is relaxed, we exect discreancies from a more recise solution. We use the solution for which the residuals are less than 10 8 as a reference solution, and comare it to the and solutions for which the residuals are required to be less than Figure 4 shows the comarison for Case 1, in terms of ressure, oil saturation and the ore volume normalized mole numbers of comonents 1, 2, 8 and 9 of the roduction block with time. For Case 2, the discreancies are insignificant (do not show in lots). We observe that the solution differs more from the reference solution than the solution does. However, this results clearly deends on the convergence criteria, which have imlicitly been assumed

25 24 Haukås et al. to be equivalent (same limit used for all residuals). Using different convergence limits for different residuals is not straight-forward, and is not considered here. 6. Conclusions and Further Work A new comositional formulation which incororates both an exact mass balance aroach () and an exact volume balance aroach () has been develoed and tested. The form of the common iterative scheme clarifies the distinction between exact mass balance and exact volume balance, and motivates a comarison of the two aroaches. The comarison is seen in the light of the results reorted by Coats et al., [5], i.e., that exact mass balance leads to better convergence roerties than exact volume balance, and that exact mass balance therefore is referable. However, our conclusions are in favour of exact volume balance. We have found some indications that may have a better convergence rate than, but this cannot comensate the fact that requires more comutational effort er iteration ste. In addition, we have noted that is more robust than with resect to hase disaearance and reaearance. This is because a recise caturing of volumes is imortant in such cases. The different conclusions are clearly imlementation deendent. We note that Coats et al. comared an exact mass balance aroach where the hase equilibrium is considered at every iteration level, [5], to an exact volume balance aroach where hase equilibrium only is obtained as a result of the iteration, [2]. With our formulation, both aroaches aim at hase equilibrium at every iteration level. This may be one of the reasons why we obtain results different from those reorted by Coats et al., [5]. With resect to the ultimate recision that can be obtained, the two resented aroaches are similar. can be used to obtain almost exact mass balance, while can be used to obtain almost exact volume balance. However, under relaxed convergence criteria, the discreancies from a more recise solution seem to be larger for than for. This result is obtained by using the same convergence limit for all equation residuals. Using different limits for different residuals is not straight-forward, and is an interesting subject for further research.

26 Exact Volume Balance Versus Exact Mass Balance 25 Aendix A. Mass Balance Inconsistency To illustrate the inconsistency between the equations (21) and (22) and the conventional mass balance (8), we use (, S, x, V b ) as a comlete set of arameters. Here, S are N 1 rimary saturations and a constant V b is used to account for extensive quantities. In the following, subscrits on differentiation with resect to the variables (, S, x ) are omitted, while on differentiation with resect to (, n) they are not. For simlicity, we assume that orosity is constant, so that V is a linear function of the rimary saturations S. Consequently, using V = V and assuming that V = V (, n), we obtain 0 = V = V ( ) ( ) V V = + n, (57) V = V S = V ( ) V S = S, (58) S S S O = V x = V x = ( V ) x. (59) We also note that W x = x = 0, W x = x = O, W x = x = I. S S x x (60) Here, O and I are zero and identity matrices of suitable dimensions. Inserting these relations into (21) and (22), we obtain ( V ) [ + S + S x x + θ t [ W x + S + ] x + θ t S x Consequently, equations (21) and (22) lead to ] = 0, (61) = 0. (62) + S + x + θ t = 0, (63) S x which is inconsistent with the conventional mass balance (8).

27 26 Haukås et al. B. Calculation of Derivatives B.1. Calculation of volume derivatives The hase volumes V are thermodynamic roerties, conventionally given in terms of ressure and the comonent mole numbers n j of the hases, i.e., ( V = V, n j) (, V j = V j, n j). (64) Note that n j, containing the mole numbers of all hases, is different from n j, which only contains the mole numbers of hase j. The volume derivatives ( V / ) n and ( V /) are calculated with the additional assumtion that n j = n j (, n). (65) Consequently, ( ) ( ) ( V V V = + n n j j ( ) ( ) V V = j ) ( j ( ) j ) n, (66), (67) which require the calculation of derivatives of n j with resect to (, n). For the water hase and for the single hydrocarbon hase case, these derivatives are trivial, since n j = n. In the case of two hydrocarbon hases, we differentiate through [ f o (, n o ) f g (, n g ] ) n = 0, (68) j=o,g nj with resect to (, n), assuming the relation (65). This leads to a set of linear systems of equations that can be solved with resect to the derivatives of n j with resect to (, n). We note that the fulfilment of the hase equilibria relations is essential for the calculation of the volume derivatives (66) and (67). B.2. Calculation of total derivatives For a variable or relation that can be written exlicitly as a function of the rimary variables u, differentiation with resect to u is straightforward. However, for some variable or relation h = h [u, u s (u )] we must calculate the total derivative ( ) ( ) h h dh du = u u s + u s u du s du. (69)

28 Exact Volume Balance Versus Exact Mass Balance 27 Here, the calculation of du s /du requires that we have access to N s fulfilled secondary relations r s = r s (u, u s (u )) = 0, where N s is the number of secondary variables u s. Then, which leads to dr s du = ( ) rs u u s + ( ) rs u s u du s du = 0, (70) [ ( ) ] 1 ( ) du s rs rs =. (71) du u s u u u s We note that the fulfilment of N s secondary relations is essential for the calculation of total derivatives. C. The Framework of Coats In the following, we resent the framework roosed by Coats, [2], for deriving an iterative scheme to be solved for the rimary variables u. The N c mass balance equations are referred to as the rimary equations, and are written in the form r = r (u, u s ) = 0. (72) The rest of the equations (constraint equations) are referred to as secondary equations, denoted r s = r s (u, u s ) = 0. (73) Here, u are the rimary variables, and u s are the secondary variables. A linearization of the two sets of equations yields ( r u ( rs u u s u s u (k+1) + ( r u s From the latter, it is deduced that [ ( ) ] (k) 1 u (k+1) rs rs s = ( u s u u (k+1) s u = r (k), (74) ( u (k+1) rs + u (k+1) s = r s (k). (75) u s u u u s u (k+1) + r (k), (76) s