SIMULATED ANNEALING – ADVANCES, APPLICATIONS AND HYBRIDIZATIONS pdf

55 4.5 Example system to illustrate the solution method for coupling the pro- duction facilities to the reservoir model... Solution of production facilities and reservoir as ties and res

Simultaneous

of Reservoir

and Surface Facilities

Simultaneous Simu1atio:n of

Reservoir and Surface Facilities

March 1994 Reservoir Simulation Industrial Affiliates Program

(SUPRI-B)

SURFACE FACILITIES

BY Denis Josi: Schiozer

March 1994

bY

Denis Josk Schiozer

11

The main objective of this work is to develop efficient techniques to couple solu- tions for performance of reservoir and surface facilities, which is a fundamental step for the development of management routines and for optiinization of total system periormance

Use of conventional techniques for full-field models can be very computation time intensive and, therefore, such models are either rarely used or they are simplified by handling surface facilities explicitly For this reason, a model with domain decorn- position where the degree of implicitness and the timestep size in each domain are related to its own characteristics was developed

The reservoir is represented by a Black-Oil simulator with three important fea- tures: local grid refinement, domain decomposition, and local timestep in each do- main The combination of these three features allows for irnproved predictions over conventional techniques, with no significant increase of the computation time

The model for surface facilities is treated as a network problem where nodes, which are defined as wells or groups, are connected by different units which can be wellbores, pipeline segments, or chokes Steady-state multip’hase flow correlations arc used to calculate pressure drop through these units

The interaction between reservoirs and surface facilities can have a great influence

on the performance of the whole system, and this interaction is greatly affected by the use of chokes in surface facilities Existing choke model:; are evaluated and their effect on the performance of the whole system is studied and discussed

iv

Different approaches have been investigated to couple production systems to reser- voirs Advantages and disadvantages of each method, as well as acceleration tech- niques, where domain decomposition is used to improve the performance of implicit methods, are presented The efficiency of each method is investigated under different conditions and some examples are presented

With these acceleration techniques, full-field simulation can be achieved more effectively and accurately than is possible with conventional techniques The efficiency

of this new approach increases for large systems and is naturally suited for parallel machines

I would like to express my deep gratitude and sincere appreciation t o Dr Khalid Aziz for his support, encouragement, and guidance throughout the course of this study The appreciation is extended to Dr Roland Horne and Dr Thomas A

Hewett for participating on the reading and examination committees, and to Dr Nathan Meehan for participating on the examination committee

I also wish to thank the faculty, staff, and colleagues of the Petroleum Engi- neering Department for their contributions to this work Special thanks to Les Dye, Hikari Fujii, Raphael Guzman, K.T Lim, Evandro Nacul, Cesar Palagi, Deniz Sumnu- Dindoruk, Santosh Verma, Chick Wattenbarger, Terry Wong, and Christina Del Vil- lar

I am grateful to UNICAMP and FAPESP for their financial support Reservoir Simulation research at Stanford University is supported by the SUPRI-B Industrial Affiliates Program

Finally, I would like to express my eternal gratitude to my family for their love, encouragement, and understanding I dedicate this work to my very precious wife

Ad riana *

vi

Contents

2.1 Reservoir Model 4

2.1.1 Domain Decomposition 4

2.1.2 Local Timestep 5

2.2 Surface Facilities Model 6

2.2.1 Multiphase Flow in Pipes 6

2.2.2 Chokes 6

2.3 Coupling of Reservoir and Surface Facilities 7

3 Reservoir Model 9 3.1 Introduction 9

3.2 Mathematical and Numerical Models 10

3.2.1 Flow Equation 10

3.2.2 Generalized Black-Oil Formulation 12

3.2.3 Additional Equations 14

3.2.4 Discretization of Flow Equation 15

3.2.5 Solution of Nonlinear Equations 21

3.3 Well Model 24

vii

3.3.1 Basic Well Model 25

3.3.2 Well Constraints 28

3.3.3 Horizontal Wells 30

3.4 Domain Decomposition 31

3.4.1 Description 31

3.4.2 Automatic Domain Decomposition 36

3.5 Local Timestep 39

3.5.1 Timestep and Coupling of Surface Facilities 40

3.6 Local Grid Refinement 42

3.6.1 Description 42

3.6.2 Restrictions 46

4 Surface Facilities 48 4.1 Introduction 48

4.2 Multiphase Flow in Pipes 49

4.2.1 Wellbore Model 50

4.2.2 Pipeline Model 51

4.3 Flow Through Restrictions 52

4.4 Solution Method - Balancing the System 52

4.4.1 Single.Wel1, Single-phase Production System 53

4.4.2 Production Network System 56

4.4.3 Group Node Constraints 63

4.4.4 Discussion 64

5 C h o k e M o d e l 67 5.1 Introduction 67

5.2 Sachdeva et al Model 69

5.3 Effect of Chokes on Coupling of Systems 73

5.4 Safety Valves 75

6 Coupling of Reservoir and Surface Facilities 77 6.1 Introduction 77

6.2.2 Coupling of Surface Facilities t o the Reservoir 81

6.2.3 Timestep Control and Local Timestep (LTS) 91

Method 2 Implicit Surface Facilities and Reservoir as Separate Domains 97 6.3.1 Acceleration Techniques 98

6.3.2 Results 106

6.3.3 Interface Boundary Conditions 112

6.3.4 Discussion 113

6.4 Method 3 - Fully Implicit 114

6.4.1 Coupling the Well Model to the Reservoir 114

6.4.2 Coupling Surface Facilities to the Reservoir 116

6.4.3 Results 120

6.5 Discussion 121

6.5.1 Handling Discontinuities in the Surface Model 123

6.5.2 Management Routine 124

6.3 7 Discussion and Recommendations 127 8 Conclusions 130 Nomenclature 133 Bibliography 138 A Program Description 147 A l Flow Charts 147

A.2 Subroutines and Functions Description 152

B Multiphase Flow Correlations 160 B l Principles of Multiphase Flow in Pipes 160

B.2 Vertical Flow 164

B.3 Horizontal and Inclined Flow 164

ix

2.1 Coefficients for two-phase critical flow correlations 7

3.1 3.2 Vector of connections for the coarse grid of the previous example

Vector of connections for the Cartesian refinement of the previous ex- ample 45

Vector of connections for the coarse grid of the previous example 46

Vector of connections for the cylindrical refinement of the previous example 47

44 3.3 3.4 B.l Multiphase correlations for vertical flow 164

C.l Reservoir data for Exa mple 1 168

C.2 Fluid properties for Example 1 168

C.3 Production facilities data for Example 1 169

C.4 Surface facilities data for Example 2 170

C.5 Reservoir data for Exa.mple 3 171

C.6 Surface facilities data for Example 3 172

xi

3.1 Connections (I) of gridblock ( b ) with its neighbors ( n ) 16

3.2 Cartesian grid system 17

3.3 Cylindrical grid system 18

3.4 Schematic cross-sectional representation of a vertical well 26

3.5 Schematic cross-sectional representation of a horizontal well 30

3.6 Decomposition of initial domain (a) in subdomains (0;) and interface boundary conditions (7.)

Order of Newtonian and domain iterations (after K.acu1, 1991)

Overlapping boundaries technique used to decompose initial domain (0) into subdomains (0.)

Schematic representation of domains to illustrate the automatic do- main decomposition technique

3.10 Example of timestep sizes for two different subdomains

3.7 3.8 3.9 3.11 Example of grid refinements

3.12 Example of Cartesian refinement

3.13 Example of cylindrical refinement

32 33 35 38 41 43 44 45 4.1 Single-well production system 49

4.2 Example of a more complex production system 50

4.3 Determination of bottom-hole pressure for a single-well production sys- tem 54

4.4 Determination of well flow rate for a single-well production system 55

4.5 Example system to illustrate the solution method for coupling the pro- duction facilities to the reservoir model 56

x11

Jacobian for Group Node 3 flow rate specified

68

pressure 72 Effect of choke size on oil flow rate

Sensitivity of the system to variations in flow rates

74

7 5

Method 1 (explicit) Coupling method with balanced surface facilities Well flow rate error for Method 1 (Example 1) Well flow rate for Example 1

Production facilities net work

Well flow rates and bottom-hole pressures as interface boundary con- ditions

Well flow error for explicit method with Dirichlet boundary conditions (Example 2, no choke)

Explicit method; Dirichlet boundary conditions; Example 2; no choke Bottom-hole pressure error for explicit method with Neumann bound- ary conditions (Example 2, no choke) Explicit method; Neumann boundary conditions; Example 2; no choke

89

20 90

X l l l

6.13 Explicit method; Neumann boundary conditions; Example 2; choke size20

6.14 Well flow rate error for explicit method with Dirichlet boundary con-

ditions (Example 2, with various choke sizes) 6.15 Bottom-hole pressure error for explicit method with Neumann bound-

ary conditions (Example 2, with various choke sizes) 6.16 Explicit method; Dirichlet boundary conditions; Example 2 6.17 Explicit method; Neumann boundary conditions; E)xample 2 6.18 Error control by limiting the timestep size (Example 2, no choke) 6.19 Error control by limiting the timestep size (Example 2, choke size 20) 6.20 Method 2 (implicit) Solution of production facilities and reservoir as

ties and reservoir as different domains using domain decomposition 101 6.23 Example of well subdomain 102 6.24 Automatic domain decomposition with a critical well-head choke on well 1 105 6.25 Comparison of cornputration times for Example 3 Effect of number of gridblocks 107 6.26 Comparison of computation times for Example 2 (no choke) Effect of tolerances on convergence among subdomains 6.27 Example 1 Effect of convergence tolerances on oil recovery 6.28 Use of overlapping boundaries with Method 1 to reduce the well flow

6.30 Jacobian matrix 118

x1v

6.31 Comparison of computation time for different methods (Examples 1

and 2) 120

121 122 6.32 Comparison of computation time for different methods (Example 3) 6.33 Oil recovery for Examples 1, 2 and 3

6.34 Determination of well flow rate with discontinuities in the multiphase flow correlation 124

6.35 Management routine 126

A.l Main program flow chart 148

A.2 Reservoir model 149

A.3 Subroutine presim 150

A.4 Subroutine simul 151

C l Example 2 169

C.2 Example 3 171

xv

Introduction

The normal approaches used to forecast petroleum production rates do not prop- erly take into account the interaction between fluid flow in the reservoir and in other components of the production system However, when complex gathering systems are present, models to handle surface facilities, wells, and reservoirs as integrated systems are necessary to improve the accuracy of predicting reservoir deliverability

An efficient method for coupling surface facilities to reservoir simulators is an important step for the development of management routines and optimization of system performance

In existing management routines and in some full-field cas,e studies, the production facilities model is normally treated explicitly and, frequently, operating conditions must be estimated before the simulation run At times, well flow rates are just dia- tributed according to pre-calculated well potentials and pressure drop in the surface facilities may even be ignored Chokes are also rarely considered in such models Op- timization routines simplify the reservoir model, which is often just a simple material balance combined with well inflow performance relationships

Due to the fact that the total system performance can be greatly affected by these simplifications, a model was developed where the accuracy of the reservoir model is improved by local grid refinement, where the production facilities are represented by correlations which can be used simultaneously with the reservoir model, and where

1

is used to improve the accuracy in the near-well region Domain decomposition, which was first introduced to reduce computation time for large and sparse matrix generated by LGR, is used here at the reservoir level It is the key idea behind the accelerations techniques used to improve the convergence between surface facilities and reservoirs With the local timestep technique, different timesteps can be used for each subdomain, so if timesteps of subdomains connected to surface facilities are reduced, timesteps of other subdomains can remain unchanged The combination of these three features leads to accurate solutions without a significant increase of the computation time over conventional techniques

Chapter 4 describes the production facilities which are treated as a network model where new units can be easily implemented Pipelines, wellbores and chokes are the units included in this work Multiphase flow correlations are used to calculate the pressure drop through these production units A brief overview of these correlations

is presented in Appendix B

Although the choke model is a part of the production facilities systems, it is addressed in a separate chapter (Chapter 5) because the performance of the entire system, as well as the techniques used to couple surface facilities to the reservoir, are greatly affected by surface chokes

Wells can be included as parts of either the reservoir or parts of the surface facilities model In this work, wells are normally treated as parts of the surface facilities system (which is then called production facilities) because such a procedure separates the reservoir from the parts that are modeled using multiphase flow correlations However, wells and surface facilities can have different degrees of implicitness as explained in Chapter 6

The most important part of this work is presented in Chapter 6 where different

methods to couple surface facilities with reservoirs are investigated Conventional techniques are compared with a new approach where domain decomposition is used to improve the performance of the coupling In the proposed approach, well subdomains are created inside the reservoir to minimize the computation time needed to balance' the inflow and outflow between subdomains The remaining part of the reservoir (reservoir subdomain) is simula,ted after the system is balanced, and therefore, taken out of the loop that performs iterations over domains A method where equations for surface facilities are incorporated in the system of reservoir equations is also presented

in Chapter 6 A discussion of the conditions under which e x h method is efficient is also presented

To avoid confusion between domain decomposition inside the reservoir, i e., phys- ical decomposition of the reservoir into smaller regions, and the treatment of surface facilities and reservoir as separate domains, the first will be referred to as reservoir domain decomposition (RDD), and the second as surface-reservoir domain decompo- sition (SRDD)

Finally, Chapters 7 and 8 summarize the contributions of this work and present some recommendations for future research and ideas for further improvement in full- field simulation

l A system is considered balanced when, at a given time, outflow from one subdomain boundary

is exactly balanced by the inflow t o another subdomain that shares the same boundary

4

Some other authors have proposed met hods which physically decompose the reser- voir domain into subdomains Pedrosa and Aziz [60] have reported slow convergence

in their approach where well regions were solved first by setting boundary condi tions to the remaining reservoir region Nacul et al [49] obtained better results using the same type of approach with some acceleration techniques Deimbacher and Heinemann [27] presented a dynamic grid where windows (physically defined refined subdomains) were used for a specific time period in regions of interest

Domain decomposition at the reservoir level instead of at the matrix level is used

in this work Even though the convergence among reservoir subdomains is not very fast, this approach has been found to be favorable when surface facilities are coupled with the reservoir

2.1.2 Local Timestep

Pressure and saturation changes vary in different parts of the reservoir and, there- fore, different timesteps are required to resolve these changes accurately Local (or multiple) timestepping techniques can be used to apply smaller timesteps in particular regions instead of limiting the timestep of the entire reservoir

Stability and convergence problems may be related to the application of local timestep Quandalle and Besset [62] applied this technique in conjunction with local grid refinement using extrapolation of boundary conditions Ewing e t al [35] have

reported restrictive conditions for stability of this type of approach

This technique has often been associated with domain decomposition where dif- ferent timesteps are used for each subdomain Deimbacher and Heinemann [27] have used multiple timesteps for different windows (subdomains) where a global timestep

is split into a number of local timesteps

A similar approach is used in this work to improve accuracy of explicit methods

to couple surface facilities to reservoir simulators This technique is discussed in Chapter 3

CHAPTER 2 LITERATURE REVIEW 6

2.2 Surface Facilities Model

This section gives a very brief review of multiphase flow correlations used in the surface facilities model of this work

2.2.1 Multiphase Flow in Pipes

Several authors have developed multiphase flow correlations to calculate pressure drop in wellbores and pipelines Due to the complexity of the flow mechanism, no available correlation is completely accept able under all conditions Therefore, these correlations must only be used under certain conditions where they have been tested

by comparison with field or laboratory data

A common problem related to currently used correlations is that, due to a variety

of reasons, but mainly due to the existence of different flaw regimes, they are not smooth over the entire range of operating conditions (Appendix B) When disconti- nuities exist in such correlations, convergence problems do olccur in implicit methods

2.2.2 Chokes

Tangeren et al [75] introduced the concept of critical flow for compressible fluids Gilbert [39], followed by several others, presented an empirical relationship between upstream pressure and liquid rate of the form:

where dG4 is the choke diameter (in 64th ’ s of a a inch) and the constants A, B, and

C are presented in Table 2.1

Ros [64] extended Tangeren’s work and showed that acclelerational pressure drop dominates the choke behavior and slippage effects are negligible Fortunati [37] was one of the first to develop a correlation for both critical and subcritical flow He also stated that for low values of gas-liquid ratio, the criticad ratio of downstream to

Table 2.1: Coefficients for two-phase critical flow correlations

Correlation Gilbert [39]

upstream pressure, which results in sonic flow, could be as low as 0.2 The same kind

of results but with different values were obtained by Ashford [a]

Some other models have also been developed for specific cases, like the Omaha e t

for subsurface safety valves, and the Surbey e t al [73] [74] model for multiple orifice

valves (MOV)

The Sachdeva e t al [65] model, which was used in this work, is described in Chapter 5 They presented a new set of equations to determine the critical ratio of downstream to upstream pressure and proposed a method for calculating the flow rate through a choke for various pressure conditions The most important advantage of this method is that the transition from subcritical to critical flow is smooth which is essential for the success of implicit methods for coupling surface facilities to reservoir simulators

The placement of chokes in the surface facilities has a very strong influence in reservoir performance and in the type of interface boundary conditions used in the coupling of systems This is discussed in Chaper 6 and in Schiozer and Aziz [66] However, despite the importance of chokes, a good comparative study of the few choke models existing in the literature is not available

2.3 Coupling of Reservoir and Surface Facilities

Among the first published results of simultaneous simulation of reservoir and production facilities are some studies of gas fields by Dempsey et al [as] Extension

CHAPTER 2 LITERATURE REVIEW 8

of these models to oil reservoirs was reported by Startzman c)t al [70], Emmanuel and Ranney et al [33], and Breaux and Monroe [17] Since then the application of large- scale simulation with the capability to integrate reservoir depletion processes with operating constraints at surface facilities has increased This type of approach has been used in full-field case studies (e.g., Wallace and Spronsen [78], Stanley e t al [69],

Stoisits [71], and Stoisits e t al [72]), in the development of management routines

Bohannon [15], Cain and Shehata [21], Carroll [22], Ravindran [63], and Fujii [38]) Unfortunately, many of the existing models are not fully described and others have reported that simplifications were necessary to reduce the amount of computation time required Such simplifications are also applied in commercial simulators w h i d have options for full-field simulation (e.g., Eclipse 200 [32]) Some of the most common simplifications are:

0 explicit treatment of surface facilities, which may lead to large errors if condi- tions change rapidly;

0 use of inadequate reservoir inflow performance relatioiiships (IPR’s) instead of

a reservoir simulator;

0 table look-up for production units, where interpolation and, at times, extrapola- tion have to be used, thus resulting in loss of accuracy of multiphase correlation; and

0 use of well potentials to allocate well production rates, which may provide good results when the total rate is specified but not when a group node pressure is specified

This work shows how domain decomposition can be used as an acceleration tech- nique for implicit methods so these simplifications are not necessary In such cases, the accuracy of the model can be improved without substantial increase of computation time

Reservoir Model

The model developed in this work is divided in four parts: reservoir, wells, surface facilities, and coupling of systems This chapter concentrates on the reservoir model which is a Black-Oil numerical simulator The other parts will be described in the next two chapters

In this work, the reservoir is represented by a Black-Oil reservoir simulator with three important features The first is local grid refinement which is used to improve accuracy in the near-well region The second is domain decomposition which was first introduced because the banded structure of the Jacobian is destroyed by extra termti created by local refinements The idea of dividing the region of interest in subdomains leads to the third feature which is local timestep, so each subdomain can be treated with a timestep related to its own characteristics These three techniques are very useful for handling the coupling of wellbores and surface facilities, and they lead to improved accuracy and efficiency of full-field models

The reservoir simulator used in this work is based on a previous program developed

by Nacul [48]; therefore, some aspects of the reservoir model which are described in his dissertation will not be repeated here However, basic concepts and all new developments will be described in the following sections

9

CHAPTER 3 RESERVOIR MODEL 10

Multiphase, multicomponent, isothermal flow in porous media is represented by nonlinear partial differential equations which are derived from physical laws of mass conservation and empirical relationships The numerical model is formulated by divid- ing the reservoir into gridblocks and applying the discrete form of these conservation laws to each fluid component in each block

The flow equation is obtained by applying mass conservation equation to an ar-

bitrary component c in a control volume V with an external area A , assuming that diffusion and dispersion can be neglected and that there are no chemical reactions

The resulting equation, considering the net mass flow rate of component c into volume

V , the net rate of mass injection of component c into volume V , and the net rate of mass increase of component c in volume V, is

where subscript p refers to phase, N p is the total number of phases, xEP is the fraction

of component c in phase p , p p is the density of phase p , v' is the velocity vector, n' is the unit vector normal to the area A, qp is the rate of phase p injected into V , q5 is the reservoir porosity, and Sp is the saturation of phase p

The first term of the left hand side of Eq 3.1 represents the net mass flow rate of

component c out of the control volume V through its boundaries The second term of the left hand side is the mass flow due to sources and sinks (positive for production) The right hand side represents the accumulation of component c in all phases of the system

For a stationary control volume, the derivative on the right hand side can be moved to inside the integral (Reynolds Transport Theorem) With this modification and with the application of the divergence theorem (Bird et al., 1960 pages 731-2) to the left hand side of Eq 3.1, the flow equation becomes

Since the control volume V is arbitrary, the integrands on both sides of Eq 3.2

must be equal, and the final form of the differential equation is

where the diagonal terms represent absolute permeabilities along the principal axes

of the system of coordinates applied to the problem

Substituting Eq 3.5 into Eq 3.3 yields

NP

p = l P P p = l p = l

CHAPTER 3 RESERVOIR MODEL 12

3.2.2 Generalized Black-Oil Formulation

For the Black-Oil formulation, phase equilibrium equations can be substituted by relationships to express phase properties as developed in Aziz [6] The equations of formation volume factor ( B ) , solubility ( R E P ) , mass fractions ( x E p ) , and phase densities

XE, = -

(3.9)

(3.10)

(3.11)

where V, is the volume of phase p at some specified pressure and temperature, Vzp

is the volume of component c at standard conditions dissolved in phase p at some given pressure and temperature, and p E is the density of the phase associated with component c at standard conditions

The phase viscosities, p p , are assumed to be known functions of pressure, water

and gas relative permeability are functions of water and gas saturations respectively, and oil relative permeability is function of both, water and gas saturations In this work, Stone’s models I (Stone, 1970) or I1 (Stone, 1973) as normalized by Aziz and Settari [8] (pages 35-6), are used to calculate the three phase relative permeabilities Substituting Eqs 3.8 to 3.11 into the final form of the differential equation, Eq 3.3, and dividing by the density of component c at standard conditions, p F , yields

where qp is the flow rate (at reservoir conditions) of phase p per unit reservoir volume,

reservoir volume, and QEp is the flow rate (at standard conditions) of component c in

phase p per unit reservoir volume

The generalized Black-Oil model can be obtained from the substitution of Darcy's law (Eq 3.4) in Eq 3.12, yielding

p = l p = l p = l

v ( R z p A p e @ p ) - (RcpQpp) = 5 (2 Os,) (3.16) The term A, is the phase mobility which is defined as

Standard Black-Oil equations can be obtained from Eq 3.16 if it is assumed that the solubility of gas and oil into the water phase and the solubility of oil in the gas phase are neglected, Le.,

R,, = R,, = R,, = 0 (3.18)

C H A P T E R 3 RESERVOIR MODEL 14

3.2.3 Additional Equations

At this point, the number of dependent variables exceeds the number of equations (flow equations) per gridblock ( N c ) , and therefore, some additional equations are nec- essary For the generalized isothermal Black-Oil model, the equations that complete the description are:

0 capillary pressure relationships,

0 phase equilibrium relationships,

0 phase constraints, and

0 saturation constraints

In this model, there are three phases: a liquid hydrocarbon phase (o), a gas

hydrocarbon phase (9) and water (w), and three components : oil (O), gas (g) and

water ( W ) Using this notation, the additional equations can be written as:

3.2.4 Discretization of Flow Equation

The discretized form of the flow equation can be obtained from finite-difference approximation of each of its terms The discretization will be performed in the origind form of Eq 3.1 followed by its transformation to the Black-Oil formulation

The right hand side of Eq 3.1 (accumulation term) can be discretized over the stationary reference block as

(3.23)

where the operator At is defined by

where superscript n represents the time level

The terms on the right and left hand sides of Eq 3.23 are not exactly equal because

of the truncation error that result from the discretization oE the accumulation term Truncation errors, which also result from the discretization of the other terms, are discussed in detail by Aziz and Settari [8]

Discretization of the Flux Term

The flux term represents the summation of flow rates across all faces of a given block b, i e , the summation of flow rates between block b and its neighbors (see

Fig 3.1) The connection of block b to its neighbors is represented by the index 1

Considering a generic connection 1, the flow rate of component c in phase p through

1 is given by

( 3 2 5 )

CHAPTER 3 RESERVOIR MODEL

n2

Figure 3.1: Connections ( I ) of gridblock ( b ) with its neighbors ( n )

The above equation can also be written as

Q ~ p , l Tcp,/ ( Q p p - Q ' p , b ) 7

16

(3.26)

where TFp,l is the transmissibility coefficient for the connection 1

TEp,l has a common term which is function of pressure and saturation, and a second term called geometrical factor (fG), which depends on the geometry of the problem,

z.e.,

(3.27)

Although Eq 3.26 can be applied for any grid system, the geometric factor intro- duced in the definition of the transmissibility coefficient makes the finite-difference approximation of the flux term dependent on the type of grid system Only Cartesian and cylindrical grid systems are available in the model used for this work

The derivation of Tzp,l for Cartesian and cylindrical systems can be found in Pe- drosa [59] For these two system, the connections l can be expressed in three positive

directions (i+,j+, IC+) and three negative directions ( i - , j - ; I C - ) A summary of the values for the positive directions is presented next Similar expressions can be writ- ten for the negative directions Fig 3.2 and 3.3 show the location of grid point and boundaries for these two systems of coordinates

Figure 3.2: Cartesian grid system

The geometric factor for Cartesian and cylindrical systems can be expressed as:

Cartesian Grid

0 x-direction

(3.28)