Simulation of Naturally Fractured Reservoirs with SimBestII

Transcription

1 Oil IT Journal Contributed Paper 0509_CP_0 Simulation o Naturally Fractured Reservoirs with SimBestII Mohammed S. Al-Jawad Assistant Proessor Petroleum Engineering Department College o Engineering/University o Baghdad mohaljawad@yahoo.com Abstract Simulation o naturally ractured reservoirs is one o the laborious techniques. This is because there are two dierent porous media in which luids exist and low. These porous media exhibit wide variations in there physical properties. The matrix represents the major storage capacity while the ractures system provides the main paths o the lowing luids. Several simulators have been constructed by some o the specialist companies. One o them is the SimBestII, which is the product o Scientiic Sotware-Intercomp. In order to simulate a reservoir perectly by such simulator, one should have an idea about the equations and the procedure encountered in it. In the current study, an attempt has been made to understand the general structure o SimBestII. The low equations that are employed in SimBestII are investigated and some o the calculation procedures are clariied. The exchange term in the low equations is discussed comprehensively and the actors that could be justiied to reduce or magniy the imbibition rate are detected. Introduction Warren and Root () classiied the porosity o reservoirs as - Primary porosity, which represents the pores constructed during precipitation o solid grains. Granular rocs, such as sandstone, exhibit this type o porosity. 2- Secondary porosity ormed by ractures, solution channels, or vugular voids in porous media. It is attributed to tectonic movements and chemical process. The porosity o carbonate rocs is oten secondary porosity. The volume o luids, which are reserved in secondary porosity, is oten less than that in the primary porosity but the ormer exhibit less resistance to low o luids. Naturally ractured reservoirs can be described as ractured homogenous reservoirs. They oten comprise high permeable, low storage volume ractures, and tight, high storage volume matrix blocs. The porosity o the matrix is the primary porosity, whereas secondary porosity is represented by ractures channels. Thereore, naturally ractured reservoirs are termed as dual porosity systems. Simulation o dual porosity systems is more laborious than conventional reservoirs simulation. In single porosity systems, one equation or each phase can describe the low o that phase in the system. However, in dual porosity systems, single equation is no longer suicient unless simpliying assumptions are invoed. In 963, Warren and Root () introduced the concept o dual porosity model. They presented a comprehensive solution or the problem o single phase low in naturally ractured reservoir. They idealized naturally ractured reservoirs as a system composed o continuum orthogonal ractures networ superimposed by a non continuum, identical, regular parallelepipeds matrix blocs The ractures networ provides the main path or luids to low rom the reservoir. The luids in the matrix are produced ater they are drained to the ractures by luid imbibition into the matrix in a counter low process. Accordingly, the low o water and oil in ractures with cylindrical coordinate system is governed by(2): Oil IT Journal

2 r Qw r ( Tw ( To r r Vb = t r Pw i, r Po t ) + z ( Tw ( ϕ Sw bw ) + z ( To z ) z Φw i, z Φo z Vbi, Qo = t ( ϕ So bo ) i, t while the matrix equations are Vb Tw t Vbi, Toα i ( Φo Φom ) = t ( ϕmso t ) Tw ) To αi, αi, ( Φw ( Φo i, α i, ( Φw Φwm ) = t ( ϕmswmbwm ) i,, mbom ) i, Φw Φo where the subscripts and m reer to racture and matrix respectively. m m ) ) () (2) (3) (4) Water and oil transmissibilities which controls luid transer between racture and matrix blocs are deined as (5) Tw To α i, α i, = Vb, K, i mi = Vb, K, i mi Krw bw σ µ w Kro bo σ µ o where σ is a shape actor and it is a property o the system. The shape actor (σ) plays an important role in the racture/matrix exchange term. During the last three decades, many attempts had been made to predict exactly the amount o luid transer between the racture and matrix blocs. Thereore, several techniques or handling the shape actor have been presented in the literature. Non o them can be considered as a precise procedure when considering the ambiguity accompanying such process. In the current study, the approach that has been adopted by SimBestII to simulate naturally ractured reservoir is investigated. One o the practical ways to perorm this job is by considering a hypothetical example and solving it by SimBestII and other simulator whose structure is quit recognized. The comparison between the output o the simulators will give a hint about the procedure adopted by SimBestII. Consequently, the simulator presented by Al-Jawad (2) is adopted and the results o the provided example are taen as a base or comparison. (6) The Flow Terms Equations, 2, 3, and 4 represent the equations that control the low o water and oil in racture and matrix systems written in inite dierence orm. These equations illustrate the concept o dual porosity model, which presumes that no low taes place between two adjacent matrix blocs. Moreover, the wells are opened to low only in the ractures blocs. The low o luids between the ractures and the matrix is summarized by the exchange term. Five dierent terms can be recognized in the racture equations (eq. and 2). Two o them are similar in nature, which are the low terms in radial and vertical directions. The sin/source term stands or the production or injection o luids applied at the bloc. The exchange term is dierent in that it represents the transer o luids between ractures and

3 matrix bloc. The right hand side term is the accumulation term that describes the rate o luid volume change with time. The last two terms are common in racture and matrix equations. The low terms in the racture equations have the major eect on the luid movement within the reservoir. Obviously, the racture permeability is the main actor that controls these terms. In this aspect, two types o racture permeability are deined in the literature (3). The irst one is a measure o the racture channel conductivity. In this case, only the racture void area represents the low cross section. This type is nown as the intrinsic racture permeability and or the single racture shown in igure it is deined as: 2 b K = (7) 2 The other type o racture permeability is the conventional racture permeability, which is based on the classic Darcy s deinition (3). Here, the racture and the associated roc bul orm a hydrodynamic unit. It is deined or a single racture as: 3 b K 2 h = (8) Either one could be used in a low equation with the suitable cross sectional area. The area corresponds to the intrinsic permeability is: A = a b (9) While that should be used with the conventional permeability is: A B = a h (0) All symbols are depicted in igure. 2

4 h b Figure - A single racture in a bul bloc The relationship between the two types o permeability could be deined through racture porosity. However, racture porosity is deined as the void volume o the racture per bul volume o the bloc. Accordingly, ϕ a b a h = = b h () and K = ϕ K (2) As shown in equation 2, the racture permeability is less than the intrinsic racture permeability. Naturally ractured reservoirs are usually characterized by small racture porosity. Keeping this act in mind, the intrinsic racture permeability may be several hundreds times greater than the conventional one. Thereore, the utilization o the suitable type o racture permeability is quite vital. One o the objects o this study is to realize rom the type o permeability that is adopted by SimBestII. However, it is anticipated that SimBestII presumes that the racture permeability data is the intrinsic racture permeability. On the 3

5 other hand, the conventional permeability is adopted in the low equations since the cross sectional area in these equations is the bloc area. The approval o the above hypothesis is accomplished through the ollowing administrations: - Run SimBestII in dual porosity mode with conventional racture permeability. 2- Run SimBestII in dual porosity mode with intrinsic racture permeability. 3- Run SimBestII in single porosity mode with conventional racture permeability value assigned to the porous media. 4- Run SimBestII in single porosity mode with permeability equal to the conventional permeability times the racture porosity. Other data or these runs are given in table. The plot o bloc pressure or these runs is depicted in igure 2. It is clear rom this igure that the results using the intrinsic permeability in ractured medium and conventional permeability in single porosity model are close to each other. In the single porosity model, only one type o permeability is deined and used in the low equations. This leads to the conclusion that in the dual porosity mode, SimBestII uses the intrinsic permeability in the low equations ater multiplying it by the racture porosity. In other words, the intrinsic permeability value should be put in the data ile and SimBestII will then multiplies it by the racture porosity, which is also given in the data ile, to get the conventional racture permeability value that would be employed in the low equations. The Exchange Term The exchange o luids between ractures and matrix has a great eect on the rise o water in the wells. This is attributed to the act that the water usually rising up in the ractures system due to its high permeability and connectivity. The result is a depleted ractures system and highly oil saturated matrix. The exchange o luids between ractures and matrix will manage the situation and i there is a higher rate o luid transer, the produced water cut will be less. This process is simulated by the exchange term in the low equations. 4

6 Table - Example data (ater re. 2) Basic data Total thicness o the reservoir, t Radial extent o the reservoir, t 2050 Well bore radius, t 0.25 Thicness o the perorated interval, t 28.2 Fractures and matrix properties Fractures Matrix Permeability, md Porosity Compressibility, psi Vertical / Horizotal permeability ratio 0.5. Shape actor, t Fluids properties Water Oil Compressibility, psi - 5.5x00-5 4x0-6 Stoc tan speciic gravity Viscosity, cp Formation volume actor, RB/STB Grids Speciications Number o radial grids 0 Number o vertical grids 7 Radius to the bounndary o the bloc, t 0.25, 2.0, 4.32, 9.33, 20.7, 43.56, 94., , , , Thicness o the blocs, t 28.2, 25.0, , , 32.4, 30.0, 35.0 Saturation unctions Fractures system Matrix system S w K rw K ro P cow S wm K rwm K rom P cowm

7 Bloc pressure, psi Dual porosity, K=25000 md Single porosity, K=000 md Dual porosity, K=000 md Single porosity, K=8 md Time, days Figure 2- Bloc pressure history or dierent runs Mathematically, the exchange term is simply a source or sin term composed o transmissibility and dierence in pressure (or potential) between the matrix and the surrounding ractures. This term appears in the racture and matrix equations to govern the low o the considered phases. This means that water will move to the matrix i the water pressure in the racture is higher than that in the matrix. Similarly, oil will move out o the matrix i its pressure in the matrix is higher. All o the actors in this term are amiliar except the shape actor, σ. However, this actor controls the rate o luid exchange between racture and matrix. Prediction o the shape actor value is not a simple matter. Kazemi, Merril, and 6

8 Zeman (4), have presented a method or estimating the value o σ. SimBestII adopts this method when it calculates the rate o luid exchange. It can be summarized as ollows: σ = σ = σ = 4 2 L x 4 L x L z 4 L + L + L 2 x 2 z 2 y For the low in one direction For the low in two directions For the low in three directions Where L s (s = x, y, or z) are deined by Kazemi et al. as the dimensions o the matrix bloc. However, SimBestII manuals ( 5,6 ) deined L s as the distance between the ractures in each o the coordinate directions and i they are identical in magnitude to the grid bloc dimensions then the model becomes that proposed by Warren and Root (5). For a highly ractured reservoir, the values o L s should be substantially less than the matrix grid bloc dimension (5). It is obvious that decreasing the values o Ls will increase σ and thus increases the racture ability to transmit luid to the matrix or vice versa. In the current study, the cylindrical coordinates system is adopted which is a special case o the low in two directions. Thus equation 4 has been used to calculate the value o σ. The example presented by Al-Jawad (2) is employed in SimBestII and the results are compared in order to understand the computation techniques o SimBestII. The data are given in table. The initial water saturation in ractures and matrix blocs adjoining the well is compared in igure 3. The movable water saturation or the same grids ater 25 days o production is depicted in igure 4. The value o σ in the example is t -2 and according to equation 4, this value corresponds to L x =L z = t. Figure 5 demonstrates a comparison between the results o SimBestII and that o reerence 2. The compatibility between the curves o igure 5 indicates that similar procedures are used to produce them. Thus, equations, 2, 3, and 4 are also the basic equations that implied in SimBestII. This act is also indicated in reerence 5 and 6. Exact values o L s are hard to be determined and usually, they assumed equal to the bloc dimensions and then increased or decreased according to the results o history matching. However, it is believed that value o σ governs the amount o luid transer. That is, various values o L x and L z could lead to the same value o σ and equal amounts o luid will transmit rom the racture to the matrix or vice versa. In igure 6 two sets o L x and L z are employed in addition to that proposed by the example. For the irst one, L x =30t, and L z =6.254t. Applying equations 4, the value o σ is 0.068, which is the same as in the example o reerence 2. This value is also duplicated i L x =0000t and L z =6.99t. Thereore, the water cut history curves coincide. However, SimBestII should be provided by L s rather than σ. The values o L s may be varied regionally or locally. Such variations produce various imbibition rates i and only i the value o σ varies. Consequently, one has to be very cautious when assigning values o L s, since some dierent combinations o L s may not lead to dierent exchange rate. (3) (4) (5) 7

9 Fractures system (SimBesst II) Matrix system (SimBest II) Fractures system (SimBest II) Matrix system (SimBest II) Ater re. 2 Figure 3- Initial water saturation in ractures and matrix blocs 8

10 Fractures system (SimBest II) Matrix system (SimBest II) Matrix system (Sim Best II) Fractures system (Sim Best II) Ater re. 2 Figure 4- Water saturation in ractures and matrix blocs ater 25 days 9

11 SimBestII with σ = t 2 Ater re. 2 Figure 5 Comparison o water cut history The inal article that will be discussed here is the eect o σ on the imbibition rate. When the value o σ vanishes, the low o luids taes place through the ractures system only and no luid could be produced rom the matrix. In this case, the results or ractures system obtained by SimBestII or dual porosity mode will be similar to that o the single porosity mode. On the other hand, as the value o σ increases, to a certain limit, the rate o imbibition also ampliies. Table 2 introduces the values o L s and the corresponding values o σ as calculated by equation 4. In igure 7, the water cut history is established or the values o σ presented in table 2. As can be seen rom the igure, the water cut decreases as the imbibition rate increases. This is true since when the reservoir exhibit higher rate o imbibition the water will be transmitted to the matrix bloc rather than produced. Table 2- Values o the shape actor or various values o L x and L z L x, L z, t σ, t Figu u re - W ater cu t h istory

12 Water cut, raction Lx=30, Lz=6.253 Lx=0000, Lz=6.99 Lx=Lz= Time, days Figure 6- Water cut history or three sete o Lx and Lz Conclusions The ollowing conclusions can be drawn rom this study: - In SimBestII, one should assign the intrinsic racture permeability value to the racture permeability in the data ile. 2- SimBestII multiplies the given permeability (intrinsic permeability) by the racture porosity to get the conventional racture permeability, which would be used in the low equations.

13 Water Cut, Fraction Figure ( The value o Lx and Lz Time. days ) - Water cut history or various values o Lx and Lz Figure 7- Water cut history or various values o L S 3- The method presented by Kazemi et al. (4) or estimating the shape actor is adopted by SimBestII. 4- The comparison between the results o SimBestII and that o Al-Jawad (2) assures the anticipated procedures o SimBestII. 5- The values o L s have no eect on the imbibition rate unless they change the value o σ. 6- The imbibition rate is directly proportional to the value o the shape actor that is, i σ is doubled, the volume transer between ractures and matrix will be two times, eeping other actors unchanged. While the variation o L 2 s in some instances may disagree with the imbibition rate. Thereore, the adjustment o σ value is more sensible than the alteration o L s. Nomenclature b= Shrinage actor, STB/RB K= Absolute permeability, md K r = Relative permeability L S = Dimension o the matrix bloc in s direction, t P= Pressure, psi Pc= Capillary pressure, psi Q= Flow rate, t 3 /day 2

14 Oil IT Journal Contributed Paper 0507_CP_0 S= Saturation, raction T= Transmissibility Vb= Bul volume, t 3 Symbols = Finite dierence operator Φ= Potential, psi µ= Viscosity, cp σ= Shape actor, t -2 φ= Porosity, raction Superscripts n= Time level Subscripts i= Grid index in radial direction = Grid index in vertical direction = Fracture m= Matrix o= Oil r= Radial direction t= Time index w= Water z= Vertical direction Acnowledgment The authors wish to express their appreciation to the Ministry o Oil/Reservoirs and Fields Development Directorate or their permission to use SimBestII. Reerences - Warren, J.E. and Root, P.J.: The Behavior o Naturally Fractured Reservoirs. SPEJ, Sept. 963, pp Al-Jawad, M.S.: Mathematical Modeling o Conning in Stratiied and Fractured Reservoirs. Ph.D Dissertation, University o Baghdad, Oct Van Gol-Racht, T.D.: Fundamentals o Fractured Reservoir Engineering. Elsevier Sc. Pub. Co., Amsterdam, Kazemi, H., Merril, JR., L.S., and Zeman, P.R.: Numerical Simulation o Water-Oil Flow in Naturally Fractured Reservoirs. SPEJ, Dec. 976, pp Reservoir Simulation in Practice Scientiic Sotware-Intercomp, Inc., Denver, Colorado, SimBestII TM Dual-Porosity, Blac Oil Reservoir Simulator; User s Manual Scientiic Sotware-Intercomp, Inc., Aug Oil IT Journal 3