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Chapter 17 Reservoir Simulation V-1398 Petroleum Engineering Handbook-Vol. V Joseph, C. and Pusch, W.H.: "A Field Comparison of Wet and Dry Combustion," JPT (Septem­ ber 1980) 1523. Koch, R.L.: "Practical use of combustion drive at West Newport field," Pet. Eng. (January 1965). Martin, W.L., Alexander, J.D., and Dew, J.N.: "Process Variables of In Situ Combustion," Trans., AIME (1958) 213, 28. Meldau, R.F., Shipley, R.G., and Coats, K.H.: "Cyclical Gas/Steam Stimulation of Heavy-Oil Wells," JPT (October 1981) 1990. Moss, J.T., White, P.O., and McNeil, J.S. Jr.: "In Situ Combustion Process-Results of a Five­ Well Field Experiment in Southern Oklahoma," Trans., AIME (1959) 216, 55. Olsen, D. and Sarathi, P.: "Field application of in- situ combustion," Report No. NIPER/BDM 0086, U.S. Dept. of Energy, Washington, DC (1994). Showalter, W.E. and Maclean, A.M.: "Fireflood at Brea Olinda Field, Orange County, Califor­ nia," paper SPE 4763 presented at the 1974 SPE Improved Oil Recovery Symposium, Tulsa, 22-24 April. Chapter 17 Showalter, W.E.: "Combustion-Drive Tests," SPEJ (March 1963) 53; Trans., AIME, 228. Terwilliger, P.L. et al.: "Fireflood of the P2-3 Sand Reservoir in the Miga Field of Eastern Reservoir Simulation Venezuela," JPT (January 1975) 9. Rod P. Batycky, Marco R. Thiele, StreamSim Technologies Inc.; K.H. Widmeyer, R.H. et al.: "The Charco Redondo Thermal Recovery Pilot," JPT (December 1977) Coats, Coats Engineering Inc.; Alan Grindheim, Dave Ponting, Roxar 1522. Software Solutions; John E. Killough, Landmark Graphics; Tony Settari, Williams, R.L., Jones, J.A., and Counihan, T.M.: "Expansion of a Successful In-Situ Combus­ tion Pilot in Midway Sunset Field," paper SPE I 6873 presented at the 1987 SPE Annual U. of Calgary and Taurus Reservoir Solutions Ltd.; L. Kent Thomas, Technical Conference and Exhibition, Dallas, 27-30 September. ConocoPhillips; John Wallis, Wallis Consulting Inc.; J.W. Watts, SI Metric Conversion Factors Consultant; and Curtis H. Whitson, Norwegian U. of Science and 0API 141.5/(131.5 + 0 API) =g/cm 3 Technology and Pera bar x 1.0* E+05 = Pa bbl X 1.589 873 E-01 = !TI3 Btu x 1.055 056 E +00 =kJ 17.1 lntroduction-K.H. Coats Cp X 1.0* E-03 =Pa·s The Merriam-Webster Dictiona,y defines simulate as assuming the appearance of without the ft X 3.048* E-01 =m reality. Simulation of petroleum reservoir performance refers to the construction and operation ft3 X 2.831 685 E-02 =m3 of a model whose behavior assumes the appearance of actual reservoir behavior. The model OF (°F -32)/1.8 = oc itself is either physical (for example, a laboratory sandpack) or mathematical. A mathematical OF (°F + 459.67)/1.8 =K model is a set of equations that, subject to certain assumptions, describes the physical process­ kW-hr x 3.6* E + 00 = J es active in the reservoir. Although the model itself obviously lacks the reality of the reservoir, lbm x 4.535 924 E-01 =kg the behavior of a valid model simulates-assumes the appearance of-the actual reservoir. psi X 6.894 757 E + 00 =kPa The purpose of simulation is estimation of field performance (e.g., oil recovery) under one *Conversion factor is exact. or more producing schemes. Whereas the field can be produced only once, at considerable ex­ pense, a model can be produced or run many times at low expense over a short period of time. Observation of model results that represent different producing conditions aids selection of an optimal set of producing conditions for the reservoir. The tools of reservoir simulation range from the intuition and judgment of the engineer to complex mathematical models requiring use of digital computers. The question is not whether to simulate, but rather which tool or method to use. This chapter concerns the numerical math­ ematical model requiring a digital computer. The Reservoir Simulation chapter in the 1987 edition of the Petroleum Engineering Handbook1 included a general description of reservoir simulation models, a discussion related to how and why they are used, choice of different types of models for different-reservoir problems, and reliability of simulation results in the face of model assumptions and uncertainty in reservoir-fluid and rock-description parameters. That material is largely omitted here. Instead, this chapter attempts to summarize current practices and trends related to development and application of reservoir simulation models. V-1400 Petroleum Engineering Handbook-Vol. V Chapter 17-Reservoir Simulation V-1401 TABLE 17.1-SPE COMPARATIVE SOLUTION PROJECT PROBLEMS 17.1.1 The Generalized Model. Any reservoir simulator consists of n + m equations for each of N active gridblocks comprising the reservoir. These equations represent conservation of 11 11+ 1 SPE1 Three-phase black oil mass of each of n components in each gridblock over a timestep t,.t from 1 to 1 • The first n (primary) equations simply express conservation of mass for each of n components such as oil, 1 Ox 10x3 300-block grid 3,650-day depletion with gas injection gas, methane, CO2, and water, denoted by subscript I= 1,2,... ,n. In the thennal case, one of the SPE2 Three-phase black oil "components" is energy and its equation expresses conservation of energy. An additional m 10x1x15 150-block r-z grid 900-day single-well coning depletion (secondary or constraint) equations express constraints such as equal fugacities of each compo­ SPE3 Nine-component retrograde gas nent in all phases where it is present, and the volume balance Sw + S0 + Sg + Ssolid= 1.0, where 9x9x4 324-block grid Ssolid represents any immobile phase such as precipitated solid salt or coke. 5,480-day cycling and blowdown There must be n + m variables (unknowns) corresponding to these 11 + m equations. For SPE4 Cyclic steam injection and steam displacement of heavy oils example, consider the isothennal, three-phase, compositional case with all components present in all three phases. There are m = 211 + l constraint equations consisting of the volume balance SPE5 Six-component volatile oil and the 2n equations expressing equal fugacities of each component in all three phases, for a 7 x7x 3 147-block grid total of n + m = 311 + I equations. There are 3n + I unknowns: p, S11,, S°' Sg, and the 3(n - I) 20-year WAG injection independent mo! fractions xu, where i = 1,2, ... ,n - l; j= 1,2,3 denotes the three phases oil, gas, SPE6 Three-phase black oil and water. For other cases, such as thennal, dual-porosity, and so on, the 111 constraint equa­ Single-block and cross-sectional dual porosity with drainage and gas and water injection tions, the n + 111 variables, and equal numbers of equations and unknowns can be defined for cases each gridblock. Because the m constraint equations for a block involve unknowns only in the given block, SPE7 Three-phase black oil they can be used to eliminate the m secondary variables from the block's n primary or conser­ 9 x9 x 6 486-block grid with horizontal wells vation equations. Thus, in each block, only 11 primary equations in n unknowns need be Eight 1,500-day injection-production cases considered in discussions of model fonnulation and the linear solver. The n unknowns are de­ SPE8 Two-phase gas-oil black oil noted by P;1, P;2, •••, P;,,, where P;,, is chosen as pressure P; with no loss of generality. These 10x1Qx4 400-block grid primary variables may be chosen as any 11 independent variables from the many available vari­ ables: phase and overall mo! fractions, mo! numbers, saturations, p, and so on. Different Comparison of 2,500-day 400-block grid results with 20-block unstructured and locally 12 15 authors choose different variables. - Any sensible choice of variables and ordering of the refined grid results primary equations gives for each gridblock a set of n equations in n unknowns which is suscep­ SPE9 Three-phase black oil tible to nonnal Gaussian elimination without pivoting. The (Newton-Raphson) convergence rate 24x25x15 9,000-block 25-well grid with geostatistical description for the model's timestep calculation is independent of the variable choice; the model speed 900-day depletion (CPU time) is essentially independent of variable choice. SPE10 Model 1: Two-phase gas-oil case with a 2,000-block 100x1 x20 grid and gas injection to The /th primary or conservation equation for block i is 2000 days (j = N + 1 ) Model 2: Two-phase water-oil case with a 1.12-million block 60x220x85 grid and water = I=1,2, ... ..................................... (17.1) M;� -M;'; t,.t L qij 1 -qil n, injection to 2,000 days ; = I Both models have geostatistical descriptions where M;1 is mass of component I in gridblock i, %, is the interblock flow rate of component I from neighbor block j to block i, and q;1 is a well term. With transposition, this equation is represented by f, = 0, the Ith equation of gridblock i. All n equations f,= 0 for the block can Models have been referred to by type, such as black-oil, compositional, thermal, general­ be expressed as the vector equation F; = 0 where f;1 is the Ith element of the vector F;. Finally, ized, or JMPES, Implicit, Sequential, Adaptive Implicit, or single-porosity, dual-porosity, and the vector equation more. These types provide a confusing basis for discussing models; some refer to the applica­ tion ( e.g., thennal), others to the model fonnulation ( e.g., implicit), and yet others to an F(P,, Pz, ..., PN)=o .......................................................
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