D. W. Peaceman. Fundamentals of numerical reservoir simulation. Elsevier Scientfic Publishing Company, Amsterdam-Oxford-New York, first edition, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....law and a degenerate parabolic equation expressing the conservation of the injected fluid (concentration equation) Let Omega be a bounded domain in the plane R with smooth boundary and T 0 a fixed number. Our differential system, under appropriate physical assumptions, is given by [1] div u = f in Omega Theta (0; T ) 1:1) u = Gamma rp with the boundary condition u Delta = 0 on and c div(cu) Gamma div(Drc) cf with the boundary and initial conditions Drc Delta = 0 on c(x; 0) c 0 (x) 1:6) where p and u are the pressure and Darcy s velocity ....

....and permeability of the medium, respectively, 1 ; 2 ) denotes the exterior normal to f the source and sink terms and c = c(x; t) is the injected concentration at injection wells and the resident concentration at production wells. The diffusion dispersion tensor D will be considered as in [1], i.e. D = D(u) ff m I jujfff l E(u) ff t E (u)g; 1:7) E(u) juj u Omega u; E (u) I Gamma E(u) 1:8) where ff m , ff l and ff t are, respectively, molecular diffusion, longitudinal and transverse dispersion coefficients. Normally dispersion is physically more important ....

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D.W. Peaceman, Fundamental of Numerical Reservoir Simulation, Elsevier, Amsterdan, 1977.

....ELLAM , that address the conservation problem locally; see [1, 2, 3, 4, 14] The cost of these methods appears to be signi cantly greater than for the MMOCAA procedure, particularly in three space variables; thus, it can be valuable to o er alternative approaches. The standard Muskat equations [5, 15] describing two phase incompressible, immiscible displacement in a gravity free environment are the two conservation equations s o r (kk ro (s ) o rp o ) q o ; 1:1a) s r (kk r (s ) rp ) q ; 1:1b) Department of Mathematics, Purdue University, West Lafayette, ....

D. W. Peaceman, \Fundamentals of Numerical Reservoir Simulation", Elsevier, New York, 1977.

.... from ours can be found in [2, 4, 8, 9, 37] A brief history of forerunners of the technique given in [23] and here can be found in [23] The model problem we have taken to test our numerical method corresponds physically to the waterflooding of a petroleum reservoir (two phase immiscible flow) [11, 40, 22]. We consider the system of partial differential equations governing the flow written in terms of the global pressure [1, 11] which leads to a system consisting of an elliptic parabolic coupled pair of equations. Since our main concern is to introduce and justify the new numerical method, we will ....

....media. The algorithm is extended in x9 to include the effects of gravity; the experimental calculations including gravity will appear in a paper [24] on fractured reservoirs, where the gravitational effects are critical. 2. The Two Phase Flow System The standard Muskat equations [11, 40] describing two phase incompressible, immiscible displacement in a gravity free environment are the two conservation equations OE s o Gamma r Delta (kk ro (s ) o rp o ) q o ; 1) s Gamma r Delta (kk r (s ) rp ) q ; 2) and the equation of state p c (s ) p o ....

D. W. Peaceman, "Fundamentals of Numerical Reservoir Simulation", Elsevier, New York, 1977.

....it respects the minimal regularity on the solution of the differential system. 1. Introduction This is the second paper of a series in which we analyze mathematical properties and develop numerical methods for the flow of two incompressible, immiscible fluids in porous media Omega ae , d 3 [6, 27]: 1:1) OE t s Gamma r Delta (w (s) rpw fl w ) q w (s) GammaOE t s Gamma r Delta ( o (s) rp o fl o ) q o (s) p c (s) p o Gamma pw ; where w indicates a wetting phase (e.g. water) o denotes a nonwetting phase (e.g. oil) OE and are the porosity and absolute permeability ....

D. W. Peaceman, Fundamentals of Numerical Reservoir Simulation, Elsevier, New York, 1977.

..... 2) The single phase model which describes the (saturated) flow of a (slightly compressible) fluid (water aqua) may be considered as a special case of the two phase system (1) 2) with S o 0. On the other hand, the system (1) 2) can be extended to the black oil model [32,31] See also [38,30,45,16,48,21,34,11]. In the black oil model in addition to water and oil there is a gas component with N G 0 which is dissolved in the oil phase. In addition at lower pressures there may exist a gas phase with S g 0. Here, the oil phase density depends on the pressure and on the amount of gas and we use the ....

....to minimum with a small #. A major disadvantage of the fully implicit method is the complexity of the Newtonian iteration, which may be costly. Other formulations known as IMPES (implicit pressures, explicit saturations) are attractive alternatives [7] Related are methods of total pressure [38] or streamlines streamtubes [8] Most rely on the splitting of the model equations into the time lagged (formally) elliptic part and the parabolic hyperbolic part. Some of difficulties which arise in these approaches involve treatment of compressibilities and of varying well patterns, and the ....

D.W. Peaceman, Fundamentals of Numerical Reservoir Simulation,1sted.(ElsevierScientfic,Ams- terdam, 1977).

....are naturally associated with the cell sides, the mixed nite element method combines naturally with the ux evaluations in the MUSCL scheme for the tracer conservation equation. The mixed nite element method has much in common with block centered nite di erences used in the petroleum industry [8, 75, 90], so there is signi cant experimental evidence that it can be used reliably in practice. Furthermore, the mixed method naturally involves harmonic averaging of series ow, so it produces zero velocities at the edges of ow barriers. In previous work, we have used the lowest order mixed method in ....

....of the hybrid mixed method is that the linear system involves an average of d unknowns per cell, where d is the number of spatial dimensions. Thus this system is larger than the usual system for cell centered pressure in standard petroleum simulation, such as block centered nite di erences [75]. Because the pressure unknowns are associated with cell sides, the stencil is non standard; in particular, we cannot use readily available incomplete factorizations for the linear system. Further, because the piecewise polynomial spaces for the Lagrange multipliers are not nested between levels ....

D.W. Peaceman. Fundamentals of Numerical Reservoir Simulation. Elsevier Scienti c Publishing Co., 1977.

....the superconvergence phenomenon. This has two advantages, first, a sparse, positive definite linear system results, and, second, the method can be relatively easily incorporated into existing standard cell centered finite difference reservoir or groundwater simulators that handle diagonal K [27]. We also address another computational diculty, namely, that in practice the conductivity K can be zero in a subdomain of . The standard mixed variational formulation requires inverting K, an impossibility in this degenerate case. Although our theory does not extend to the degenerate case, our ....

D. W. PEACEMAN, Fundamentals of numerical reservoir simulation, Elsevier, Amsterdam, 1977.

....can flow or be immobile. We distinguish between phases and components within these fluids which are denoted by subscripts c and C, respectively. Assume isothermal and equilibrium conditions and no chemical reactions and no adsorption. Use general conservation of mass equation for each component [27] as follows: 2.1) c9(qSNc ) V . Vc = qt. Ot where concentration No of component C is defined as No = c pcScncc, with phase density Pc and saturation c for phase c. Since a component can exist in more than one phase, we define nee as the mass fraction of component C in phase c. Source term ....

....i) which phases and components are present and flowing, ii) what restrictions are placed on nee, and iii) which application specific constitutive relationships are used. Some of such specific models are very well known, understood, and widely used, in both petroleum and environmental engineering [27, 18]. These include in particular 1) a popular single phase incompressible flow modeled by an elliptic equation, 2) a two phase flow model which is a parabolic hyperbolic system of 2 equations, and 3) a black oil model: a coupled system of 3 equations [8, 32, 19, 7, 6] and 4) a compositional model [8, ....

D.W. Peaceman. Fundamentals of numerical reservoir simulation. Elsevier Scientfic Publishing Company, Amsterdam-Oxford-New York, first edition, 1977.

....in the application of adaptive mesh refinement, we assume that the flow is incompressible, and that adsorption, hysteresis, capillary pressure, and physical dispersion are all negligible. The basic equations of two phase immiscible displacement are well known and treated in several sources [7, 8, 52, 55, 60]. The polymer flooding model is commonly presented as an extension of the classical two phase Buckley Leverett model [4, 61] and has been analyzed extensively from both mathematical and computational points of view [32, 36, 40, 43, 44, 45, 46, 77] We require that the mass of each of the fluid ....

....Note that the well boundary conditions are time dependent in that the mobility and density terms along the boundary depend on the fluid in the wells and the fluid within the reservoir. For further details on wells and boundary conditions used in reservoir simulation, see Lake [52] and Peaceman [60]. 2 Numerical Treatment of Flow in Porous Media Multi phase flow in oil recovery and aquifer remediation processes exhibits behavior typical of the solutions to both elliptic parabolic and hyperbolic partial differential equations. For example, pressure changes are felt quickly throughout the ....

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D.W. Peaceman. Fundamentals of Numerical Reservoir Simulation. Elsevier Scientific Pub- lishing Co., 1977.

....reservoirs and contaminant transport in aquifers have motivated research on numerical methods for the simulation of multicomponent miscible displacement problems in porous media. The numerical approximation of incompressible miscible displacement has been carefully investigated by numerous authors [2, 3, 4, 5, 6, 8, 9, 18, 19, 20, 21, 27, 30] over the past two decades. However, relatively little research has focused on compressible ow problems; comparisons between existing research and the models of this paper will be given in x2. The main objectives of this paper are as follows: The derivation of a hierarchy of models for ....

....injection rates used in forced recovery, the specie balance equations are partial di erential equations of almost degenerate parabolic type, having a dominant rst order hyperbolic part which is not well approximated by pure Eulerian methods. The momentum balance equation is given by Darcy s Law [27]. An equation for the pressure can be derived by combining Darcy s Law with a mass balance equation for the uid, as we shall see. Consider a uid at temperature T o and pressure P , consisting of n s chemical species with composition speci ed by the species mass fractions = 1 ; ns ....

D. W. Peaceman, \Fundamentals of Numerical Reservoir Simulation", Elsevier, New York, 1977.

....that gravity does not enter the concentration equations; also, 0 = r 0 = c 0 = 0. The boundary conditions u n j = D rc ) n j = 0 (3.3) require the periphery of the reservoir to be impermeable. Compatibility to incompressibility is expressed by Z q dx = 0: Following [12], we assume that D = D ( u) 1; N c , is thedim( dim( matrix D = F (x) d ;mol I juj(d ;long E(u) d ;trans E (u) 3.4) Nuclear Contamination in Fractured Porous Media 6 representing di usion and dispersion. In (3.4) d ;mol is the molecular di usion coe ....

D. W. Peaceman, \Fundamentals of Numerical Reservoir Simulation", Elsevier, New York, 1977.

....in the porous blocks, the ow of uids in such a fractured reservoir is seriously a ected by the existence of the fractures, since the resistance to ow in the fractures is much smaller than that in the blocks. Flow in the blocks will be described by means of the usual Darcy and conservation laws [17]. Flow in the fractures will also be described using Darcy s law; this implies that the fractures will be treated as if they have been redistributed over all the reservoir with a very small porosity. Implicit in this is the assumption that in a numerical approximation any spatial discretization ....

D. W. Peaceman, Fundamentals of Numerical Reservoir Simulation, Elsevier, New York, 1977.

.... ) where A is an appropriate in nite lattice. We shall let the time interval of interest be denoted by J = 0; T ] In each type of ow to be considered, our starting point is a microscopic model which consists of the proper equations describing Darcy ow in a porous medium (see, e.g. 10] [20], 23] posed over all of . The porosity, permeability, etc. will be discontinuous across . The equations on m will be scaled by appropriate powers of to conserve ow in some sense; consequently, the porous medium equations do not degenerate as tends to zero and the form of the ....

D. W. Peaceman, Fundamentals of Numerical Reservoir Simulation, Elsevier, New York, 1977.

....and v is the seepage velocity (the product of the saturation by the particle velocity of the phase ) Since the uid occupies the whole pore space, the saturations satisfy X s = 1: 2. 2) The theory of multiphase ow in porous media is based on the following form of Darcy s law of force [1, 8]: v = K x p ; for = w; g; o: 2.3) where K denotes the absolute permeability of the porous medium, is the mobility of phase , and p is the pressure of phase . The mobility is usually expressed as = k = the ratio of the relative permeability k and the viscosity ....

d. peaceman, Fundamentals of Numerical Reservoir Simulation, Elsevier, Amsterdam, 1977.

....o ; kw ; P c which are rock and uid speci c and are functions of the water saturation Sw and may also be x dependent. The ow is described by the classical equations of conservation of mass and momentum (Darcy s law) complemented by a set of algebraic constraints and constitutive equations (see [11]) The densities o ; w are known functions of pressure of each phase with known compressibility constants c w and c o . The rock is assumed to be incompressible, but this constraint is easily removed. N o t r ( o K k o o (rP o o GrD) 0; 1) N w t r ( w K kw w ....

....of the reservoir. The ow is driven by injection production wells which are implemented in the numerical model using the Peaceman model (see [12] and appear in the equations 1 and 2 as the right hand side terms q o (x) q w (x) respectively. Several numerical algorithms were proposed (see [11]) and numerous codes in petroleum industry, environment management, as well as in research labs exist which employ now standard discretization techniques and are validated against eld data and results. While these codes could be understood as the single block approach, the domain decomposition ....

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D. W. Peaceman (1977). Fundamentals of numerical reservoir simulation. Elsevier, Amsterdam.

....for interaction between di erent variables, are small compared to the respective entries of the diagonal block. Pressure governs saturations According to Assumption 1, our formulation is presented in terms of pressure and saturations. At least for the black oil isothermal models, the studies [2, 15, 5] show that: the pressure equation is essentially parabolic or elliptic and the saturation equations are hyperbolic or transport dominated parabolic. These features are expected to be inherited by compositional models as well [27] A well known consequence is that the pressure equation must be ....

Peaceman D., Fundamentals of Numerical Reservoir Simulation. Elsevier Scientic Publishing Company, 1977.

....model The di erential equations for each model typically form a subset of the following general set of equations. Consider isothermal ow in the absence of reactions, dispersion and adsorption. The conservation of mass is speci ed for each component denoted with subscript M as follows [4, 10, 14]: WM ) t r FM = q M : 1) Here WM is the total component concentration which is equal to the sum of mSmnmM over all phases m, with m ; Sm denoting density and saturation of phase m, respectively, and nmM denoting mass fraction of component M in phase m. Note that, for every given ....

....as in the two phase model. After discretization in space and time, the above system can be solved implicitely or explicitely. 3.3. The black oil model. The black oil model is a three component three phase (water, oil and gas) system subject to the following commonly used restrictions [14, 17]: 1) the water component exists only in the water phase and it is the only component in that phase, 2) the gas phase contains only the gas component and may be absent if pressure is high enough, and 3) the oil phase may contain both oil and gas components. These restrictions imply nwW = 1; nwO = ....

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D. W. Peaceman. Fundamentals of numerical reservoir simulation. Elsevier Scientc Publishing Company, Amsterdam-Oxford-New York, rst edition, 1977.

....is a production well located at the top of the formation and that, in order to maintain pressure, several water injection wells are placed in the bottom part of the region. Simulation of hydrocarbon production in this reservoir requires, with the traditional approach, the use of a black oil code [15, 20]. This is due to the need to account for phase behavior of light and heavy hydrocarbons. However, in the main part of the reservoir the uids operate under typical two phase or single phase conditions. Meanwhile, the black oil code is 6 10 times slower than the two phase code, while both, the ....

....oil and gas) model describing the ow in petroleum reservoir. It is assumed that the aqueous phase contains only water component and that the water component does not exist in other phases. Furthermore, the gas phase contains only the light hydrocarbon component. These are standard assumptions [15, 12], also [18, 6, 20, 13, 5] In the fully implicit black oil model considered here, the primary variables are water pressure Pw , oil component concentration NO , and gas component concentration NG . The discretized mass conservation equations are ( NW ) n 1 ( NW ) n 4t n 1 r U n 1 w = q ....

D. W. Peaceman. Fundamentals of numerical reservoir simulation. Elsevier Scientc Publishing Company, Amsterdam-Oxford-New York, rst edition, 1977. 16

....is pressure. Also, s and f represent the saturation and the fractional flux of water, respectively. q i is the volumetric source term for phase i, i = oil or water) and q = q oil q water is the total volumetric source term. Gravity, molecular diffusion and capillary pressure are neglected. See [9] for a discussion of these and more general equations. We consider a 2D cross section with wells at fixed pressure on the left and right sides of the computational domain. No flow boundary conditions are imposed on the top and bottom of the cross section. The equations above are thus solved in a ....

D. Peaceman. Fundamentals of Numerical Reservoir Simulation. Elsevier, Amsterdam-- New York, 1977.

....One major consideration of this paper is the treatment of the discontinuity in the coecient tensor K when discretizing the linear system (1.4) The motivation comes from the physical mass conservation interpretation of the second equation in (1. 4) the continuity of the pressure (cf. 10, 4] and [2, 29]) and the fact that mixed nite element approximation maintains mass balance in each element by imposing continuity of the velocity at the element boundaries in the normal direction [30] Therefore, given (1.4) the mixed nite element, cell centered nite di erence, or nite volume ....

D.W. Peaceman, Fundamentals of Numerical Reservoir Simulation, Elsevier Scientic Publishing Co., Amsterdam, 1977.

....main result shown in this section. The proof of the main result is presented in x2.3, and two of the lemmas needed for the main result are proven in x2.4. 2.1. The differential system. The usual equations describing two component, incompressible, miscible displacement are given by (see, e.g. [7, 19]) 2:1) Gammar Delta fk(x) rp Gamma aeg) c)g = q I Gamma q P ; OE(x) t c Gamma r Delta Gamma D(u)rc) u Delta rc q I c = cq I ; for (x; t) 2 Omega T j Omega Theta J with J = 0; T ] T 0) where OE and k are the porosity and absolute permeability of the porous ....

D. W. Peaceman, Fundamentals of Numerical Reservoir Simulation, Elsevier, New York, 1977.

.... of mass conservation for 324 An Accurate Approximation to Compressible Flow 325 the uid mixture and Darcy s law lead to the following coupled system of partial di erential equations (PDEs) that describes uid ow processes in a porous medium reservoir with injection and production wells [2, 5, 9] t ( r ( u) q; x 2 ; t 2 (0; T ] u = K (c) rp grd) x 2 ; t 2 (0; T ] 1:1) In many cases, the thickness of the medium is signi cantly smaller than its length and width. Hence, it is reasonable to average the medium properties vertically and to assume IR 2 with a ....

....r . Eq. 1.3) and its simpli ed versions have been widely used in modeling subsurface contaminant transport and remediation in the hydro science community. It can also be applied to compressible uid ow processes in reservoir simulation, unless the uids contain large quantities of dissolved gas [2, 9]. Due to the e ect of large pressure changes involved in porous medium uid ow processes and the type of the medium of the reservoir, the porous medium can deform. Let c (x) be the compressibility of the medium. The following constitutive relation is often used to model the porosity [1] ....

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Peaceman, D. W., Fundamentals of Numerical Reservoir Simulation, Elsevier, Amsterdam, 1977.

....1. 1 Underlying Mathematical Model The mathematical model for describing the miscible displacement of one fluid by another in reservoir simulations, the movement of solute in groundwater contaminant transport, and various other applications can be represented by a system of differential equations [2, 5, 28, 58] (OEp) t r Delta v = f ; x 2 Omega ; t 2 [0; T ] v = Gamma K (c) rp Gamma aegrz) x 2 Omega ; t 2 [0; T ] 1:1) This research was supported in part by DOE Grant No. DE FG05 95ER25266, by ONR Grant No. N00014 94 1 1163, and by the funding from the Mobil Technology Company. ....

....model saturated unsaturated flow and transport in cross sectional or vertical geometries as well as heterogeneous and anisotropic material properties. Allen and Murphy [1] developed a finite element collocation technique. Extensive studies have also been conducted in petroleum industry. Peaceman [58], Aziz and Settari [2] Ewing [28] and Russell and Wheeler [68] presented detailed descriptions on the finite difference methods and finite element methods that have been used in the petroleum industry. Because the advection in the transport equation (1.2) is governed by the Darcy velocity, one ....

D.W. Peaceman, Fundamentals of Numerical Reservoir Simulation, Elsevier, Amsterdam, 1977.

.... of high permeability.Thus, a fractured porous medium possesses an additional length scale compared to an unfractured medium.Such fractured reservoirs could be modeled by permitting the porosity and permeabilitytovary rapidly and discontinuously over the reservoir (the single porosity model)# see [6, 39, 41] for a standard description of flow in porous media. Both of these quantities are significantly larger in the fractures than in the matrix blocks, and therefore the computational and data requirements for the model would be impractical. Such difficulties could be overcome by replacing these ....

D. Peaceman, Fundamentals of Numerical Reservoir Simulation, Elsevier, New York, 1977.

....that the total uid velocity is independent of position; we take it to be constant in time as well as space. After nondimensionalizing the time and space variables, uid viscosities, and capillary pressures in the standard way, one obtains the following system of partial di erential equations [31, 6]: sw t x f w (s w ; s g ) Dw ; 2.1) s g t x f g (s w ; s g ) D g : 2.2) Here s w and s g denote the water and gas saturations, and the oil saturation is s o = 1 s w s g . The state of the uid is de ned by the saturation pair, denoted S = s w ; s g ) The ....

Peaceman, D., Fundamentals of Numerical Reservoir Simulation, Elsevier, Amsterdam, 1977.

....and c(x; t) the volumetric concentration of the invading fluid. Applying Darcy s law and conservation of mass for the fluid mixture and for the injected fluid, and assuming no flow at the boundary, we obtain the mathematical model as a coupled nonlinear system of partial differential equations [3, 4] u = Gamma k(x) c) 5p Gamma fl 5 d) x; t) 2 Omega Theta J; 1) div u = q(x; t) x; t) 2 Omega Theta J; 2) OE c t Gamma div(D(x; u) 5 c) u Delta 5c = c Gamma c)q(c) x; t) 2 Omega Theta J; 3) D. Q. Yang Miscible Displacement in Porous Media 3 with the boundary ....

.... source term q is a combination of point sources and sinks representing injection and production wells, respectively, and is the exterior normal to the boundary Gamma = The viscosity is determined by (c) 0) c 1=4 Gamma 1)M 1) Gamma4 ; 7) and the dispersion tensor D is given by [4] D = D(x; u) OE(x) d m I juj(d l E(u) d t E (u) 2 4 OEd m d l u 2 1 juj d t u 2 2 juj (d l Gamma d t ) u1u2 juj (d l Gamma d t ) u1u2 juj OEd m d l u 2 2 juj d t u 2 1 juj 3 5 ; 8) where M is the mobility ratio, 0) is the viscosity of oil, E(u) u ....

[Article contains additional citation context not shown here]

D. W. Peaceman, Fundamentals of Numerical Reservoir Simulation (Elsevier North-Holland, New York, 1977).

....ELLAM , that address the conservation problem locally; see [1, 2, 3, 4, 17] The cost of these methods appears to be signi cantly greater than the MMOCAA procedure, particularly in three space variables; thus, it can be valuable to o er alternative approaches. The standard Muskat equations [5, 18] describing two phase incompressible, immiscible displacement in a gravity free environment are the two conservation equations s o t r (kk ro (s ) 1 o rp o ) q o ; 1:1a) s t r (kk r (s ) 1 rp ) q ; 1:1b) Department of Applied Mathematics, National Sun ....

D. W. Peaceman, \Fundamentals of Numerical Reservoir Simulation", Elsevier, New York, 1977.

....air water system, numerical experiments AMS subject classifications. Primary 65N30, 76S05 1. Introduction. In this paper we develop and analyze a fully discrete finite element procedure for solving the flow equations for an air water system in groundwater hydrology, ff = a; w [3] 14] [32]: OEae ff s ff ) t r Delta (ae ff u ff ) f ff ; x 2 Omega ; t 0; 1.1) u ff = Gamma kk rff ff (rp ff Gamma ae ff g) x 2 Omega ; t 0; 1.2) where Omega ae d , d 3 is a porous medium, OE and k are the porosity and absolute permeability of the porous system, ae ff , s ....

.... 1: Finally, an example of the air density function ae a is given by the relation [3] 2.22) ae a (p a ) ae 0a 1 p a p 0a ; where ae 0a is the density of the air phase at the reference pressure p 0a . While the phase mobilities can be zero, the total mobility is always positive [32]. The assumptions (2.18) and (2.19) are physically reasonable. Also, the present analysis obviously applies to the incompressible case where c(s; p) 0. In this case, the analysis is simpler since we have an elliptic pressure equation instead of the parabolic equation (2.9) Thus we assume ....

D. Peaceman, Fundamentals of Numerical Reservoir Simulation, Elsevier, New York, 1977.

....a concluding remark is given in x9. 2. Single Phase Flow To motivate the later sections, in this section we describe the flow of a fluid in a porous medium Omega ae 3 with density ae and pressure p. The usual equations describing single phase flow in Omega are well understood [4] [31]: OEae) t r Delta (aeu) q; x 2 Omega ; t 0; 2.1a) u = Gamma k (rp Gamma ae g) x 2 Omega ; t 0; 2.1b) where OE and k are the porosity and absolute permeability of the porous medium, u is the volumetric velocity, is the viscosity of the fluid, g is the gravitational, ....

D. W. Peaceman, Fundamentals of Numerical Reservoir Simulation, Elsevier, New York, 1977.

....R Gamma gds = 0 holds in the case of a = 0. In general, we assume that a 2 L 1 ( Omega Gamma is a non negative known function. The coefficient K 2 L 1 is assumed to be a d by d positive definite matrix possibly with discontinuities in its entries. In the context of reservoir simulation [4, 22] K represents the permeability (actually, mobility) tensor. Heterogeneity of the porous medium is characterized by strong variations, by orders of magnitude, in the entries of K from one part of the region Omega to another. Anisotropy occurs if the ratio of any two eigenvalues of K is large. The ....

D. W. Peaceman. Fundamentals of Numerical Reservoir Simulation. Elsevier North-Holland, New York, 1977.

....homogeneous porous medium, without including the effects of molecular diffusion and dispersion. We neglect the vertical variations so that a two space dimensional problem will be considered. The equations of such a flow reduce to the system (see Douglas and Roberts [DR] Douglas [Do] and Peaceman [Pe]) OE t u i div(q u i ) OE z i u i t p q Gamma u i = s i q ; 1 i n 1; q = Gamma k grad p: x E mail : amirat ucfma.univ bpclermont.fr y E mail : ziani math.univ nantes.fr 1 2 Y. Amirat and A. Ziani Here u i denotes the concentration of the ith component of the fluid ....

D.W. Peaceman, Fundamentals of Numerical Reservoir Simulation, Elsevier, N.Y. (1977). Y. Amirat and A. Ziani

....the superconvergence phenomenon. This has two advantages: first, a sparse, positive definite linear system results, and, second, the method can be relatively easily incorporated into existing standard cell centered finite di#erence reservoir or groundwater simulators that handle diagonal K [27]. We also address another computational di#culty, namely, that in practice the conductivity K can be zero in a subdomain of# . The standard mixed variational formulation requires inverting K, an impossibility in this degenerate case. Although our theory does not extend to the degenerate case, our ....

D. W. PEACEMAN, Fundamentals of Numerical Reservoir Simulation, Elsevier, Amsterdam, 1977.

....MP , the decomposition of the Majda Pego unstable region = E [ s [ f , and the coincidence, BogdanovTakens, and in ection loci E , BT , and I for a quadratic model. 4.1. The basic equations. We consider one dimensional, horizontal ow of three immiscible uid phases in a porous medium [32]. For concreteness, we consider a uid composed of gas, oil and water, mixed at macroscopic level. The di erences among these phases lie in some ow properties. We assume that the whole pore space is occupied by the uid and that there are no sources or sinks. Compressibility, thermal and ....

....occupies the whole pore space, the saturations satisfy X j s j = 1: 4.2) As a consequence, any pair of saturations in the saturation triangle may be chosen to describe the state of the uid. The theory of multiphase ow in porous media is based on the following form of Darcy s law of force [32, 35, 4]: v i = K i x p i ; 4.3) where K denotes absolute permeability of the porous medium, i 0 is the mobility of phase i, and p i is the pressure of phase i. The mobility is usually expressed as i = k i = i , the ratio of the relative permeability k i and the viscosity i of phase i. ....

D. Peaceman, Fundamentals of numerical reservoir simulation, Elsevier, Amsterdan, 1977.

....high permeability. Thus, a fractured porous medium possesses an additional length scale compared to an unfractured medium. Such fractured reservoirs could be modeled by permitting the porosity and permeability to vary rapidly and discontinuously over the reservoir (the single porosity model) see [6, 39, 41] for a standard description of flow in porous media. Both of these quantities are significantly larger in the fractures than in the matrix blocks, and therefore the computational and data requirements for the model would be impractical. Such difficulties could be overcome by replacing these ....

D. Peaceman, Fundamentals of Numerical Reservoir Simulation, Elsevier, New York, 1977.

....equation and the second is correspondingly referred to as the pressure equation. The saturation equation is of hyperbolic nature whereas the pressure equation is elliptic; thus we consider a coupled hyperbolic elliptic system. The basic properties of the system are derived by e.g. Peaceman [17] and Ewing [7] Variants of the system have been analyzed by a series of authors; Alt and DiBenedetto [2] and Kruzkov and Sukorjanskii [15] proved existence and uniqueness results for smooth solutions of this system in the presence of capillary forces. For miscible ow, Feng [8] obtained similar ....

....nonlinear saturation equation holds. Finally, we derive stability estimates for the initial value problem. 2 The mathematical model In this section we will give the precise assumptions on the mathematical model under consideration. A derivation of the model can be found in the book by Peaceman [17]. We consider incompressible and immiscible two phase ow in a two or three dimensional reservoir R d , d = 2; 3. 2 Conservation of mass for the two phases reads ( w s w ) t r ( w v w ) q w ; o s o ) t r ( o v o ) q o ; where is the porosity and i ; s i ; v i ....

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Peaceman, D. W. 1977 Fundamentals of Numerical Reservoir Simulation. Amsterdam, Elsevier. 29

....ELLAM , that address the conservation problem locally; see [1, 2, 3, 4, 14] The cost of these methods appears to be significantly greater than for the MMOCAA procedure, particularly in three space variables; thus, it can be valuable to offer alternative approaches. The standard Muskat equations [5, 15] describing two phase incompressible, immiscible displacement in a gravity free environment are the two conservation equations OE s o t Gamma r Delta (kk ro (s ) Gamma1 o rp o ) q o ; 1:1a) OE s t Gamma r Delta (kk r (s ) Gamma1 rp ) q ; 1:1b) Department of ....

D. W. Peaceman, "Fundamentals of Numerical Reservoir Simulation", Elsevier, New York, 1977.

....ELLAM , that address the conservation problem locally; see [1, 2, 3, 4, 15] The cost of these methods appears to be significantly greater than the MMOCAA procedure, particularly in three space variables; thus, it can be valuable to offer alternative approaches. The standard Muskat equations [5, 16] describing two phase incompressible, immiscible displacement in a gravity free environment are the two conservation equations OE s o t Gamma r Delta (kk ro (s ) Gamma1 o rp o ) q o ; 1:1a) OE s t Gamma r Delta (kk r (s ) Gamma1 rp ) q ; 1:1b) and the equation of ....

D. W. Peaceman, "Fundamentals of Numerical Reservoir Simulation", Elsevier, New York, 1977.

....= 0 (Buckley Leverett Eqn) r Delta v = 0 (Incompressibility) where v is the seepage velocity, is a relative total mobility, and p is pressure. s and f represent the saturation and the fractional flux of water, respectively. Gravity, molecular diffusion and capillary pressure are neglected. See [8] for a discussion of these and more general equations. These equations are solved in a rectangle for two dimensional flow. We nondimensionalize the units by specifying the flow domain to be the unit square. The reservoir fluids only occupy the pore space between the rocks. In effect, all flow ....

D. Peaceman. Fundamentals of Numerical Reservoir Simulation. Elsevier, Amsterdam--New York, 1977. 12 Glimm et al.

....fluid, increasing the reservoir s effective permeability significantly over the permeability corresponding only to the rock matrix. The reservoir mechanism of fractured systems is significantly different from that of so called single porosity system as for instance in [11] 13] 14] 18] [21]. To describe the flow of the fluid in such a fractured reservoir several authors in the engineering literature as for instance [8] 12] 15] 26] showed that if there are many well connected fractures the fractures network behaves as an equivalent porous medium, described by the so called ....

- D.W. PEACEMAN, Fundamentals of numerical reservoir simulation, Elsevier, New-York, 1977.

....the superconvergence phenomenon. This has two advantages, first, a sparse, positive definite linear system results, and, second, the method can be relatively easily incorporated into existing standard cell centered finite difference reservoir or groundwater simulators that handle diagonal K [27]. We also address another computational difficulty, namely, that in practice the conductivity K can be zero in a subdomain of Omega Gamma The standard mixed variational formulation requires inverting K, an impossibility in this degenerate case. Although our theory does not extend to the ....

D. W. Peaceman, Fundamentals of numerical reservoir simulation, Elsevier, Amsterdam, 1977.

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D. W. Peaceman. Fundamentals of numerical reservoir simulation. Elsevier Scientfic Publishing Company, Amsterdam-Oxford-New York, first edition, 1977.

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D.W. Peaceman, Fundamentals of Numerical Reservoir Simulation. Elsevier, Amsterdam, 1977.

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D.W. Peaceman (1977) Fundamentals of Numerical Reservoir Simulation. Elsevier, Amsterdam.

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D. Peaceman, Fundamentals of Numerical Reservoir Simulation, Elsevier, New York, 1977.

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D.W. Peaceman. Fundamentals of Numerical Reservoir Simulation. Elsevier Scientific Publishing Co, 1977.

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D. W. Peaceman. Fundamentals of numerical reservoir simulation. Elsevier, Amsterdam, 1977.

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D.W. Peaceman, Fundamental of Numerical Reservoir Simulation (Elsevier, Amsterdan, 1977).

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D. W. Peaceman, Fundamentals of Numerical Reservoir Simulation, Elsevier Scienti c Publishing Co., Amsterdam, 1977.

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D. W. Peaceman, Fundamentals of Numerical Reservoir Simulation, Elsevier Scientific Publishing Co., Amsterdam, 1977.

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D. W. Peaceman. Fundamentals of Numerical Reservoir Simulation. Elsevier Science Publ., 1977.

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