International Journal for Numerical and Analytical Methods in Geomechanics Volume 17 issue 10 1993...

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, VOL 17, 677-698 (1 993)

A FINITE ELEMENT SOLUTION OF A FULLY COUPLED IMPLICIT FORMULATION FOR RESERVOIR SIMULATION

Y. SUKIRMAN* AND R.W. LEWIS'

Civil Engineering Department, University College of Swansea, Singleton Park, Swansea, SA2 8PP. U K

SUMMARY

An iterative method is presented for solving a fully coupled and implicit formulation of fluid flow in a porous medium. The mathematical model describes a set of fully coupled three-phase flow of compressible and immiscible fluids in a saturated oil reservoir. The finite element method is applied to obtain the simultaneous solution ( S S ) for the resulting highly non-linear partial differential equations where fluid pressures are the primary unknowns. The final discretized equations are solved iteratively by using a fully implicit numerical scheme. Several examples, illustrating the use of the present model, are described. The increased stability achieved with this scheme has permitted the use of larger time steps with smaller material balance errors.

1. INTRODUCTION

It is now generally recognized that the solution of multiphase, multidimensional flow in porous media can be accomplished by approximating the non-linear partial differential equations using a numerical method. In the early development stage of reservoir simulation techniques, numerical methods were only applied for the solution of linear differential systems. However, the equations of multiphase fluid flow in a porous media are normally highly non-linear as the mobility and density terms are strong functions of saturation and pressure, respectively. The non-linear terms account for the effects of reservoir heterogeneity, relative permeability contrasts, rock and fluid compressibility factors, gravity, capillary pressure and viscosity variations. Therefore, the solution of such problems involves the solution of sets of discretized equations in time and space where the coefficients are both spatially and temporally dependent.

Normally, at least two steps are necessary in order to obtain the final solution; firstly, a technique to discretize the problem spatially and secondly, a choice of an appropriate time differencing scheme. The finite difference method has historically been widely used for solving many reservoir problems. However, in many cases, e.g. flow regimes which give rise to shock-front developement, or involve convective dispersion, there are inherent difficulties if the finite difference method is used. Recently, the finite element method has become increasingly popular due to its natural flexibility in being able to handle the boundary conditions for simulating complex geometric problems.

The second step in the solution process involves the temporal discretization of the already spatially discretized form of the equations. A suitable method is required in order to define the changes of the non-linear terms with time. Although explicit time discretizing techniques are computationally inexpensive, the maximum stable time step is normally very small. On the other

* Ph.D. Student. + Professor.

0363-9061/93/100677-22$16.00 0 1993 by John Wiley & Sons, Ltd.

Received 4 January 1993 Revised 6 April 1993

678 A FINITE ELEMENT SOLUTION FOR RESERVOIR SIMULATION

hand, implicit methods allow the use of larger time steps with smaller time truncation errors and a commensurate increase in the stability of the solutions. Recent research has concentrated on improving the implementation of implicit solutions in order to reduce the computing costs.' - 4 Thomas et aL2 and Forsyth et a1.j have consequently presented adaptive implicit procedures for their finite difference models. These approaches attempt to minimize the computation time by using implicit methods for a small number of cells in the flow model, while using an explicit method for the remaining cells.

The primary objective of this paper is to investigate the applicability of an iterative method for solving a fully coupled fully implicit formulation of three-phase flow using a finite element solution. The flow formulation, which is strongly coupled with the reservoir compressibility behaviour, is presented in detail. The non-linear coeffecients are calculated to each Gauss point of an element by using the calculated values of the unknowns at the nodes. This type of model is commonly suited for natural depletion recovery where rock compressibility can act as an effective energy drive to the reservoir. Finally, the convergence and stability characteristics of the present scheme were investigated and compared with those obtained by Nolen et aL4 and Chappelear et a1.'

2. FULLY COUPLED THREE-PHASE FLOW FORMULATIONS

The finite element model is utilized to obtain the simultaneous solution of the partial differential equations for the flow of oil, water and gas and hence the calculation of the pressure within each phase. The flowing fluids were considered to be compressible and immiscible which account for the effects of relative permeability contrasts, complex fluid properties and the effects of capillary pressure. The flow equations are strongly coupled with the reservoir rock compressibility and therefore can be used for simulating the production performance of a reservoir which is overlain by an unconsolidated formation.

The continuity equation for each flowing phase can be obtained in usual manner as found in the l i t e ra t~re ,~ . * i.e. by combining Darcy's flow law with the appropriate continuity equations. In general, the rate at which fluid accumulates in a unit volume, minus the divergence of phase velocities, must be equal to the net flow Qnet from any external source. The following expression can be applied for each of the flowing phase at standard conditions:

-V(MIV(Pl + p l g h ) ) + (rate of fluid accumulation) = Qn,,

1 = 0, g, w

where P I is the pressure, pl is the fluid density, Bl is the formation volume factor, k the absolute permeability matrix, p l is the dynamic fluid viscosity and h is the height above an arbitrary datum. The subscripts 0, g and w refer to the oil, gas and water phases, respectively.

Lewis and Schreflerg presented several factors which contribute to the rate of fluid accumulation of each flowing phase. Only those related to the reservoir compressibility and the saturation changes are considered in the present paper. In this case, the total reservoir compressibility Cr, is defined as the sum of the fluid compressibilities Crf and the effective rock compressibility Crm, as

Y. SUKIRMAN AND R.W. LEWIS 679

where

and K , is the bulk modulus of rock, 4 the average porosity, fraction, DT the tangential elastic stiffness matrix and mT = ( 1 , 1, 1, 0, 0, 0)'.

In practice, the effective rock compressibility is normally a constant value which is equal to the value of the matrix compressibility c, or bulk compressibility cb of the reservoir rock. In this paper, C,, is assumed to vary with different rock properties such as, Young modulus E and Poisson's ratio v, which determine the value of the matrix DT. Therefore, the present model is suitable for solving any multiphase flow problem in porous media where a significant influence of the reservoir compressibility is expected.

It is also assumed that the pore volume is completely filled by a combination of the fluids present, i.e.

The oil saturation S,, can be replaced with (1 - So - S,) . On differentiating equation (3) we obtain

so + s, + s, = 1.0 (3)

dS, - dS, dS, dt dt dt (4)

Some other fundamental concepts must now be introduced. First, the fluid pressures at any point in the reservoir are related by their capillary relationships. Therefore, for water wet oil reservoirs, the following expressions are used:

For oil-water system

P , , = Po - P , For oil-gas system

P,, = P, - Po (5b)

where P,, is the oil-water capillary pressure, P,, the gas-oil capillary pressure and Po, , , , the oil, gas and water pressure, respectively.

In the present paper, the effect of the gravity term is neglected and therefore the fluid pressure is equal to the flow potential of each flowing phase. In simultaneous solution procedures, the saturation changes must be expressed in terms of the capillary pressure and finally in terms of the fluid pressures. On differentiating equations (5) w.r.t time, then multiplying and dividing the LHS by dS, and dS,, respectively, we obtain

as, as, ap, ap, at - - K(X- - F)


as, as, ap, ap, at =&i- - z)

where S ; and S;! are the slopes obtained from plots of S, vs. P, , and S, vs. Pcg, respectively.

of the flowing phases can be written as follows: On incorporating equations (2)-(6) into equation (l), the final mass balance equation for each

For the oil phase

For the water phase

- V T - - V P , + kk,, - -+ [B", - S L + - C B, S W rm S" ,la: ~ PwBw

S W ap B W at

+ - C r m S C J + Q w = O

For the gas phase

B, S O go rm ,)la: +C, .CrmS; ]%+[Q,R S -(-)-'S;!+ a 1 - Q,S;!+Q,-RR: ,+C C S" ~ so ' P o B, B, BO

where s; = s, + 6PcwS& s; = so - 6 P c w s ; - dP,,s;

s; = s, + GP,,S;!

(9)

The parameters Q,, Q , and Q , are the source/sink terms of the oil, water and gas phases, respectively. In this context a positive flux implies a production well and vice versa. Note that the effective rock compressibility term C,, in equations (7)-(9) is strongly dependent on the average

Y. SUKIRMAN A N D R.W. LEWIS 68 1

pore pressure which was calculated from

P = SOP0 + S,P, + S,P, ( 1 la) In fact, the terms S;, Sb' and S: were derived from equation (1 la) which indicate the fraction of each fluid that 'wets' the rock surface. and therefore

If it is assumed that the sand grains are water 'wet' then, equations (1 la) and (1 1 b) become

The mobility terms in equations (7)-(9) are strongly dependent on the unknowns, e.g. the relative permeabilities depend on the fluid saturations. As initial conditions, it was assumed that the reservoir was in capillary pressure equilibrium with both the initial water and gas saturations above their critical values, i.e. all phases were mobile. In the present paper, the three-phase oil relative permeability calculations were based on a statistical probability model by Stone," viz;

where kro, is the oil relative permeability for oil-water system, krog the oil relative permeability for oil-gas system, k,, the water relative permeability for oil-water system and k,, the gas relative permeability for oil-gas system. The above method uses both imbibition and drainage oil relative permeabilities and is valid only if the value of k,,, at residual water saturation S,,, is unity.

Equations (7)-(9) are subject to the following boundary conditions. (a) Closed boundary: There is no flow of any phase across the impervious section of the

boundary system. In this case, the continuity of flow across the boundary is given by

71 = 0, g, w

where n is the unit normal vector and qn is the flow of n-phase per unit area of the boundary surface. In the case of impervious rock, zero permeability was assumed and therefore no flow across the boundary. (b) Open boundary: There is flow across the boundary which can either be in or out of the system. The flow qn of the 71-phase must satisfy the following condition:

The flow boundary can be defined in two ways; firstly be prescribing the flow rate, i.e.

682 A FlNITE ELEMENT SOLUTION FOR RESERVOIR SIMULATION

or, secondly, by specifying the value of pressure at the boundary surface as

where the values of qb and Pb are the prescribed flow rate and pressure at the flow boundary, respectively.

3. THE FINITE ELEMENT DISCRETIZATION METHOD

The finite element discretization of the flow equations may now be expressed in terms of the nodal fluid pressures, i.e. Po, P, and Pg by using the Galerkin Weighted Residual method. The unknowns are related to their nodal values by the following expressions:

where N is the shape function.

of the flowing phases as follows: Upon substitution of equations (13) into equations (7)-(9), we obtain the final discretized form

For water

- dP, dP, d P WPPW + w w - dt + wo- dt + W,J dt + qw = 0

For oil

- dP, d P d P o,p, + 0,- dt + 0 , L dt + O , B dt + qo = 0

For the gas flow equation

aP, dP, d P dt dt dt

G,P,+G,- + G o - + G , A + q g = O

A detailed explanation of the above coefficients is given in the appendix. Equations (15)-(17) represents a set of ordinary differential equations in time. For convenience, the equations are written in the following form:

-wp 0 -0, [ 0 - G , - G , -Go - G ,

The time discretization method used here is based on a Kantorovich ype approach" which has been described in detail by Lewis et aL9 When applied to equation (18), this method requires the solution of the equations

f, + Afn g F d t = 0 I"

683

where At, is the length for nth time step, F is a function representing equation (18) and 3 is an arbitrary function of time.

Y. SUKIRMAN AND R.W. LEWIS

By assuming a linear variation of P within each time step, as given by

where

N1 = ( I - a ) and N - t - - - ( ; in)

the integration of equation (18) gives the following result:

-w, -w, P, - WPW -wo -w, -Ow -HP, - O , ] k ; ] =[ -0, -Hpo -0,

-G, -Go -Gbg [Zw -Go -Gbg t m + At where

Wpw = (W, + WpaAt) Wbw = (Ww + Wp(l - ~ t ) A t )

4 w

q0 ]At .%

Hpo = (0, + 0 , a A t ) Hbo = (0, + 0,(1 - a)At) Gps = (G, + G,uAt) GbS = (G, + G,(1 - C I ) A ~ )

where CI depends on the form of the time-stepping method used, e.g for a Galerkin scheme, a is equal to 0.6667 and for a fully implicit solution scheme, CI is equal to unity.

4. NUMERICAL PROCEDURES

Equations (21) represent a fully coupled and highly non-linear system which require simultaneous solution. Since all the coefficients are dependent on the unknowns, iterative procedures are performed within each time step to obtain the final solutions. For this purpose, a fully implicit formulation was applied in programming the simulation code.

4. I. Fully implicit formulation

Before detailing the method for solving the fluid pressures simultaneously, some of the derivative terms in equations (7)-(9) require definition. In fully implicit schemes where fluid pressures are the unknowns, all the derivatives must be expressed in terms of pressures. The derivative of the fluid saturations in equations (6) can therefore be defined as

684

and

A FINITE ELEMENT SOLUTION FOR RESERVOIR SIMULATION

Similarly, we can define the derivative of ( l / B l ) and R,, by

and

(224 - - dR,, - Rso(P", f"k+l 1 - RSO(P3

p g + l , k + l - pg a p ,

where i = w, 0, g. The terms in equations (10) can be evaluated within each iteration level as

s; = s; + s;dP:,+' s; = s: - s;dP;;,+1 - s',dP:,+'

s; = s; + S',dP:,i' (224

where

The superscripts n + 1 and k + 1 are the new time and iteration levels, respectively. Conse- quently, these quantities in equations (22) represent implicit coefficients.

Equations (21) can be expressed, for a fully implicit scheme, in a convenient matrix form as

+ A (P - P " ) + Q = O (234 ~ ~ + l , k + l p n + l , k + l n + l . k + l n + l , k + l

-w, P W T = -0, At,

[ -Gp] '=[ill -w, -wo -wg -0, -0, -0.1, Q = E ] A t (234 -G, -Go -G,

where T is the flow coefficient matrix which is composed of the mobility terms of the flowing phase, A is the accumulation matrix composed of terms in equations (2) and (22), P is the vector of fluid pressures and Q is the source/sink vector which also accounts for the flow across the


boundaries. Equations (23) represent a non-linear, fully implicit formulation, for a fully coupled three-phase fluid flow system in porous media. The numerical solution of these equations is the main concern of this paper.

4.2. ClasslJication of non-linear terms

The non-linear coefficients in equations (23), which appear in matrices T and A, can be further specified as follows:

where T, is the constant part of the flow coefficient, fl = I / p B and f 2 = krl. These functions represent two type of non-linear terms which determine the mobility of an I-phase at a Gauss point 'g'. It is interesting to note the simplicity of the finite element method in approximating the average non-linear coefficients between elements, i.e. by calculating the coefficients at each Gauss point as illustrated in Figure 1. In this study, the pressure and saturation of the fluids are the unknowns at 'i', which implicitly determine the non-linear parameters. On the other hand, the finite difference method approximates the non-linear terms based on 'weighted' values obtained at different time levels and by alternative means between grid points.* In this respect, finite element approximations can result in higher-stability solutions even if applied on a complex geometric reservoir model. A similar treatment may be applied in approximating the matrix A.

Aziz and Settari' defined the non-linearities in equation (23) into two categories, namely, weak and strong non-linearities. The authors have classified parameters which are functions of pressure of a single phase as weak non-linearities, whilst all saturation-dependent terms are considered as strong non-linearities. The weak non-linear terms include By+', (l/BJ', C;" and R: which depend on the degree of pressure changes. The effects of these parameters are normally neglected when pressure remains constant, e.g. in steady-state problems. The strong non-linearities depend on the saturation or capillary changes, which approximates the values of krl and S ; . In practice, these approximations for any iteration level ( k + 1) are normally obtained from the curves, or tables, of PVT and rock properties data versus pressure and saturation, respectively.

For Pi > Pi+l Downstream mobility:

Upstream mobility: Ta = T+l

Ta E Ti

Ta = (Ti + T7+1)iZ Ta Average mobility:

a) Finite difference approximation

1 2

element 1 ekmnl2

j = 1.2. ..NP 4'+

4 5 6

b) Finite element method

Figure 1 . Approximation of nonlinear terms in (a) finite difference and (b) finite element methods


input data

Pi,Si.Kr,Pc. ..etc

calculate coefficient terms in equation (23)

I solve unknowns Po, PW and Pg

using direct solver

4.3. Simultaneous solution procedures

L

N

In the present paper, equations (23) were solved simultaneously using a direct solver. Many authors agree that simultaneous solution schemes involve a complex programming effort, higher computer storage and processing time^.^,^, l 2 However, this method is suitable for many difficult reservoir problems where stability and the use of larger time steps are the main concern.

Figure 2 shows a schematic process of the simultaneous solution procedures for solving a fully implicit formulation of equations (23). Note that all weak non-linear parameters were updated within each iteration level by using the most recent calculations of fluid pressures. The new values of fluid saturation were obtained from the saturation-capillary curves using the most recent calculations of capillary pressures.

4.4. Stability analysis

Equations (23) must provide a stable solution to be of any practical use. In the present model, the stability of the final solution was monitored by applying a convergence criterion and checking the material balance of the flowing phases.

A convergence criterion is set which is based on the maximum fluid pressure change since the last iteration, i.e.

IIP:+l - P")I < E, i = 1, 2 , . . . , NP (25)

where Pi is the fluid pressure at node i, NP is the total number of nodes in the mesh used, (k + 1) and k are the new and old iteration levels, respectively, and E , is the convergence limit. Peaceman13 presented a suficient condition for the convergence of such solution in equation (23) by using a maximum eigenvalue which has to be much less than unity. However, many author^".^ agree that this condition should not be used for the final solution but can be useful for monitoring the stability.

time step I

Figure 2. A schematic process of simultaneous solution procedures


A stable solution implies that the effect of an error made at one stage of the computation will not be propagated at later stages. The final solution of equation(23a) involves a non-linear approximation of the vector P" into another vector P"" which is applied at each node of the mesh. If this approximation is expressed in terms of a transformation vector T,, then equation (23a) can be written as

P"+ l = T,P" (26)

In this case the vector T, includes all the non-linear terms as already defined in equation (24). By introducing &"+l and E" as the error vectors at new and old time levels, respectively, we can write equation (26) as

(P"+1 + & " + I ) = TJP" + E " ) (27) Then a suficient condition for the stability of T, is

This condition guarantees that an error, E" introduced at time level t", will not be propagated in the later stages of the computation. Equation (28a) can be generally written for the case of an iterative method as follows:

In this study, the error vector is analysed at every iteration by introducing a residual term in equation (23a). Therefore, a residual Rk,, corresponding to a vector Pk, may be defined as

Rk, = T"Pk + A"(P"+' - P") + Q (29) where the subscript m indicates that the elements of the matrices T and A are evaluated as functions of a vector P". For the case of implicit schemes, the vector P" are the values obtained from the recent calculations which may be generally different from Pk. Using the same definition for R;+ 1, we can write equations (23) in the final form as

Therefore, the final solution satisfies the conditions

In practice, equation (31) is the same as the material balance calculation for each of the flowing phases which occur within two different time levels. This balance check requires that the rate of fluid accumulation minus the divergence of velocity must be equal to the net flow of the reservoir system, i.e. total flow from any external source minus the total outflow from the reservoir. A general material balance equation MBE, for each of the flowing phases can be given as

- kkrl (rate of I-phase accumulation) - VT ( ~ plBl V ( P I + p&)


The final solution also requires that

( 1 RHS - LHS ( 1 < E, (33)

where E, is material balance limit. This check is performed after ech iteration within a time step after the unknowns have been calculated. In this case, the most recent values of fluid saturation are used.

The conditions given in equation (28) were used for investigating the stability of the solution as obtained from equation (31). For a numerical stability analysis, an example involving free response was considered, i.e. when the source/sink terms are all ~ e r o . ~ , ' * ~ ~ For simplicity, it was assumed that the non-linear coefficients were 'locally' constant and therefore equation (30) becomes

(34) Ty+l ,k+lpn+l ,k+l + A n + l , k + l ( p n + l . k + l - p n ) = 0

The stability analysis of this form has been investigated by many a ~ t h o r s ' - ~ , ' ~ where in general the Fourier error analysis was used. Applying the same conditions to equations (34) will result in

= o (35) En+lTn+l,k+l + E n + l A n + l , k + l - E n ~ + l , k + l

On dividing by (PA"+ l * k + '), equations (35) become

which then gives

E n + 1 A n + l , k + l - --

En ~ + l , k + l + Tn+l,k+l

Note that the RHS term in equation (36b) is always less than unity for any value of the matrices and therefore the equation may be abbreviated to T n + l , k + l and A n + l , k + l

which again shows that the fully implicit formulation is unconditionally stable.

5. NUMERICAL EXAMPLES

The fully coupled finite element model described in the previous sections was employed to simulate two- and three-phase flow in a saturated oil reservoir. For this purpose, three numerical examples were solved in the present paper. The first example is the water injection problem presented by Nolen and Berry5 which is similar to the Buckley-Leverett linear analysis for a two-phase incompressible fluid flow problem. This example illustrates the stability of the fully implicit formulation described in the present paper where the finite element method was used to obtain the final solution.


The other two examples indicate the capability of the developed code for solving a complex reservoir simulation problem as discussed in Reference 6. For the second example, two simulations were performed for validating the three-phase flow model which involve the phenomenon of water and gas coning in a saturated oil reservoir. This example also investigates the problem of gas-phase appearance and disappearance as reported by Chappelear et ~ 1 . ~ and Forsyth et

The last application employs the same reservoir data as in the second example but was used to indicate the utility of the developed code in simulating the effects of reservoir compressibility on the ultimate recovery.

5.1. Example 1: water injection problem

This example is a Buckley-Leverett-type displacement of oil by water injection as discussed by Nolen and Berry. The truncation error characteristics of the developed fully implicit finite element code were evaluated and the results were compared with those obtained using an explicit method by Nolen and Berry. Thus, the model is one-dimensional with incompressible fluid flow as shown in Figure 3. Table I and Figure 3 give a complete description of fhe model employed. Figure 4 shows the time truncation error computed by subtracting the saturation profiles for At = 5 and At = 10 from those for At = I .

5.2. Example 2: a three-phase coning study

This is the second SPE comparative solution project as reported by Chappelear et ul.I3 The actual reservoir model has been modified in order to provide a challenging test problem. It is a single-well model which involves gas and water coning as shown in Figure 5. In the present paper, the results obtained using the fully implicit, fully coupled finite element model are

Table I. One-dimensional water injection problem

Length of the reservoir, m Grid increments Cross-sectional area, mz Porosity, fraction Absolute permeability, md Viscosity of oil, cp Viscosity of water, cp Initial Sw, fraction

S W K W 0.15 O~OOOO 0.25 0.0040 0.30 0.0102 0.45 0.0305 0.55 0.0497 0.65 0.0797 0.75 0.1244 0.90 0.2215 Well data

Oil production rate, Bbl/d Water injection rate, Bbl/d

610 20 93

500 0.20

0.5 0.5 0.15

K ro 1~0000 0.5880 0.4460 0.1770 0.0724 0.0163 oQ020 0-0

70 70

pc (Psi) 0.00 1 5 0.0013 0.0012 0.0009 0.0007 0.0005 0.0003 00


+ Prod.

I-D Buckley Leverett model

61 Om t P

f i t e element mest used

Figure 3. Oil displacement by water injection in one-dimensional model

Explii Scheme [ref.%!]

L I 600 1200

Distance from injection pt.(ft)

lrrplidt FEM

Figure 4. Saturation error for different schemes at t = 300 d

compared with those obtained from two companies namely; D & S Research and Development Ltd. and J. S . Nolen and Assocs. Tabe I1 summarizes the basic reservoir descriptions which include the initial reservoir conditions and the properties of the reservoir rock data.

A three-dimensional geometrical representation of the reservoir was used in all examples as shown in Figure 5. Two fully sealing faults were placed at the top and at the base of the reservoir model in order to simulate no-flow boundary conditions. It was assumed that no water was produced at the well and the produced gas oil ratio at the surface was calculated by using

Y. SUKIRMAN A N D R.W. LEWIS 69 1

Y

52 m

X

Figure 5. Finite element mesh used in Examples 2 and 3

Table 11. Data used in three-phase coning problem

Reservoir length, m 767.0 Reservoir width, m 400-0 Reservoir thickness, m 53.0 Average porosity, fraction 0.147 Residual oil saturation, fraction 020 Critical gas saturation, fraction 004 Bubble point pressure, psi 5600.0 Initial pressure, psi 3600.0 For rock data

Examples 2 and 3(i) Examples 3(ii) Exanple 3(iii)

E = 0.96E + 09N/m2 E = 0-96E + 06 N/mz E = 096E + 05 N/m2

C , = 0.20E - 09(N/mZ)-' C, = 0.20E - 06 (N/m2)-' C, = 0.20E - 05 (N/mz)-'

For all cases, the value of Poisson's ratio u used is 0.27. Well data

Times (ds) Production (STB/D) Injection (STB/D) 1-10 lo00 0

10-50 100 150 50-275 lo00 150

275-720 decline 150 720-900 100 150 Other data:

Layer Permeability (md) Swi S g i 1 100 25.3 0.001 2 300 27.8 OW05 3 450 28.7 0.0o05 4 215 32.0 O~ooO5


Similarly, the water cut ratio can be calculated from

In practice, the GOR and WOR produced at the surface were used as a criterion to close down a producing well, i.e. at the time that production was no longer economic. Several simulations were performed under two different reservoir conditions: (i) a closed system without an injection well (case 1) and (ii) dead oil (case 2) (i.e. no dissolved gas) was injected into well Y at the rate given in Table 11. For case 1, the results obtained are shown in Table111 and Figures 6 and 7. Table 111 shows the computational performance of the present model as compared to the results published by Chappelear et aL6 The answers depicted in Figures 6 and 7, show that the results obtained using the developed finite element code agree well with those obtained by D & S Research and Development Ltd. and J. S. Nolen and Assocs.

0 100 200 300 productiasl Time (Days)

Figure 6. Gas oil ratio vs. production time

Table 111. Comparison of calculated results

Model Computational parameters Matrix solution

(Company) Time step Time-step cut Non-linear update

D & S 158 7 290 iter-D4 FEM 82 0 85 iter-Gauss Nolen 33 3 95 D4-Gauss


For case 2, the results obtained are given in Table IV and Figures 8 and 9 which show the robustness tests on the developed finite element code. The 'dead oil' injection caused the occurrence of free-gas disappearance and appearance, especially during the period 10- 100 d (see Table IV). Figures 8 and 9 indicate the gas saturation profile (values at a mid-point between

0 100 200 300 productian Time Days)

Figure 7. Water cut vs. production time

0 50 100 mllctial Time (days)

Figure 8. Gas saturation vs. time for various schemes

150


Table IV. Distribution of free gas at different times

Layer Point Y Mid-W-Y Point W Point X

t = O d 1 2 3 4

t = l O d 1 2 3 4

t = 5 0 d 1 2 3 4

t=113d 1 2 3 4

0.06 10 0.0608 0.0605 06050

0.OOOo 0.OOOo 0.OOOo O~OOOO

0.OOOo o.OOO0 O~OOOO OOOOO

0.0650 0.0640 0.0640 0.0640

0.0020 0.0015 OG015 0001 5

0.0740 0.0720 0.0710 0.07 10

0.0990 0.0880 0.0860 0.0790

O.oo00 0.OOOo OOOOO 0.OOOo

0.06 13 0.06 12 00610 006 10

1650.0

+--- - - -- R

8 1500.0 1111 8 cr:

1350.0 0 50 100 150

Productiool Time (days)

Figure 9. GOR vs time for various schemes

Y. SUKIRMAN A N D R.W. LEWIS 695

4000

-& E - 8691E+10 4- E = 8631E+O6

m 2 2000 p: 0 0

0 0 100 200 300 400

Production Time (Days)

Figure 10. Gas oil ratio vs. time for different reservoir rocks

0.0

n

i -0.8

0 200 400 600 Producdon Time (days)

Figure 11. Effects of reservoir compressibility


Y and W) and the produced gas oil ratio at the surface, which correspond to this repressurization phenomena. The finite element model has handled the problem sucessfully without implementing any computational constraints as have been reported by Forsyth and Sammon.14 The finite difference ~ imula t ion '~ was terminated at 100 d, i.e. when all the free gas had dissolved into the oil phase, whilst this is not the case for the present solution.

5.3. Problem 3: efsects of reservoir compressibility

This type of problem was used to analyse the effect of reservoir compressibility on ultimate recovery. Production from a heterogeneous and unconsolidated saturated oil reservoir, as described in the second example, was simulated for two types of reservoir rocks as shown in Table 11. Two simulations were performed using a closed reservoir model to investigate the influence of different degrees of reservoir compressibility on the calculated water cut and gas oil ratio at the surface. Three values of Young's modulus, i.e. 0.9631E + 09, 0.9631E + 06 and 0.9631E + 05 were used in each simulation. The bulk compressibility of each type of rock was approximately calculated as being equal to 0 1 x E. The production data were indentical as for the second example.

The results obtained from all runs are shown in Figures 10 and 11. Figure 10 indicates that reservoir compressibility reduces GOR at the surface which therefore results in a higher oil recovery. Figure 1 1 shows the average pressure drops versus production time for different values of Young's modulus, E. The results obtained are in accordance with expected reservoir engineering performance.

6. CONCLUSIONS

A fully coupled fully implicit finite element model has been developed for simulating three immiscible and compressible fluids flowing in a saturated oil reservoir. The derivation of the continuity equations considered the effect of rock compressibility in which some rock parameters, e.g. Young modulus, Poisson's ratio and rock compressibility factor, were included. The mass balance equations have also taken into account the effect of capillarity and the variation of relative permeability.

The application of a finite element spatial discreitization resulted in a semi-discrete form of the equations in time. The developed fully implicit simulation code was unconditionally stable and therefore large time steps could be used in the present study. From the preliminary results presented, the first two examples show the stability and accuracy of the present model as compared to those published by Chapplear.' The robustness of the developed model was tested by simulating a difficult three-phase coning problem and predicting the performance of an unconsolidated reservoir. The code is under continuous development and will be used in future for solving single-well simulations, e.g. hydraulic fracturing simulation around both vertical and horizontal wells.

APPENDIX

+ Sw4 dt 5 ( l ) ] N d C l Bw

Y . SUKIRMAN AND R.W. LEWIS 697

where s, S B, B O

CG,, = - t R,,?

REFERENCES

1. A. Behie and P. K. W. Vinsome, Block iterative methods for fully implicit reservoir simulation, SPE 9303, 1980. 2. G. W. Thomas and D. H. Thurnau, Reservoir simulation using an adaptive implicit method, SOC. Pet. Eny. J . ,

3. P.A. Forsyth and P.H. Sammen, Practical considerations for adaptive implicit methods in reservoir simulation, J . 759-768 (1983).

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4. D. K. Ponting, B. A. Joster, P. F. Naccache, M. 0. Nicholas, R. K. Pollard, J. Rae, D. Banks, S. K. Walsh, An efficient

5. J. S. Nolen and D. W. Berry, Tests of the stability and time step sensitivity of semi-implicit reservoir simulation

6. J. E. Chappelear and J. S. Nolen, Second comparative solution project: a three-phase coning study, J. Pet. Tech.,

7. H. B. Crichlow, Modern Reservoir Engineering-A Simulation Approach, Prentice-Hall Englewood Cliffs, NJ 1977. 8. K. Aziz and T. Settari, Petroleum Reservoir Simulation, Applied Science Publishers, London, 1979. 9. R. W. Lewis and B. A. Schrefler, The Finite Element Method in the Deformation and Consolidation of Porous Media,

10. H. L. Stone, Probability model for estimating three-phase relative permeability, J. Pet. Tech., 20, 214-218 (1970). 11. 0. C. Zienkiewicz and R. L. Taylor, The Finite Element Method, Vol. 1/2,4th edn, McGraw-Hill, London, 1991. 12. P. P. Bansal et al., A strongly coupled, fully implicit, three dimensional three-phase reservoir simulator, SPE 8329,

13. D. W. Peaceman, A nonlinear stability analysis for difference equations using semi-implicit mobility, SOC. Pet. Eng.

14. P. Forsyth Jr. and P. H. Sammon, Gas phase appearance and disappearance in fully implicit black oil simulation,

15. T. J. R. Hughes, *Unconditionally stable algorithms for nonlinear heat conduction, Compur. Methods Appl. Mech.

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International Journal for Numerical and Analytical Methods in Geomechanics Volume 17 issue 10 1993...

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Transcript of International Journal for Numerical and Analytical Methods in Geomechanics Volume 17 issue 10 1993...