It is possible to obtain more accurate finite difference solutions at a given grid resolution with the approaches Mallison describes, but the overall impacts of numerical dispersion are co
Trang 10.2
0.4
0.6
0.8
1.0
Pressure (psia) (a) Oil 1, Gas 1, Wang
Initial Tie Line Crossover Tie Line 1 Crossover Tie Line 2 Crossover Tie Line 3 Crossover Tie Line 4 Crossover Tie Line 5 Crossover Tie Line 6 Crossover Tie Line 8 Crossover Tie Line 9 Injection Tie line
Initial Injection
XO 1
XO 2
XO 3
XO 4
XO 5
XO 6
XO 7
XO 9
0.0 0.2 0.4 0.6 0.8 1.0
Pressure (psia) (b) Oil 2, Gas 2, Wang
Initial Tie Line Crossover Tie Line 1 Crossover Tie Line 2 Crossover Tie Line 3 Crossover Tie Line 4 Crossover Tie Line 5 Crossover Tie Line 6 Crossover Tie Line 8 Crossover Tie Line 9 Injection Tie line
Initial Inj.
XO 1
XO 2
XO 3
XO 4
XO 5
XO 6
XO 7
XO 8
XO 9
0.0
0.2
0.4
0.6
0.8
1.0
Pressure (psia) (c) Oil 1, Gas 1, Jessen
Initial Tie Line Crossover Tie Line 1 Crossover Tie Line 2 Crossover Tie Line 3 Crossover Tie Line 4 Crossover Tie Line 5 Crossover Tie Line 7 Crossover Tie Line 8 Crossover Tie Line 9 Injection Tie line
Initial Injection
XO 1
XO 2
XO 3
XO 4
XO 5
XO 6
XO 7
XO 8
XO 9
0.0 0.2 0.4 0.6 0.8 1.0
Pressure (psia) (d) Oil 2, Gas 2, Jessen
Initial Tie Line Crossover Tie Line 1 Crossover Tie Line 2 Crossover Tie Line 3 Crossover Tie Line 4 Crossover Tie Line 5 Crossover Tie Line 7 Crossover Tie Line 8 Crossover Tie Line 9 Injection Tie line
Initial Inj.
XO 1
XO 2
XO 3
XO 4
XO 5
XO 6
XO 7
XO 8
XO 9
Figure 7.11: Tie line lengths for condensing/vaporizing gas drive systems described by Zick [140]
(a)Crossover tie line 4 has zero length at 2169 psia (147.6 atm) (b) Crossover tie line 3 has zero length at 3013 psia (205.0 atm) (c) Crossover tie line 3 has zero length at 2256 psia (153.5 atm) (d) Crossover tie line 3 has zero length at 3070 psia (208.9 atm) (a) and (b) replotted from Wang
[128], used with permission (c) and (d) recalculated with a slightly different fluid description from Jessen [42], used with permission
Trang 2• Whether or not tie line rarefactions appear in a displacement can be determined easily from
the placement (on the vapor side or the liquid side of the two-phase region) and concavity of envelope curves
• Once the key tie lines are determined for a given pair of injection gas and oil compositions,
the full solution for a gas displacement can be determined relatively easily
• Effects of volume change as components transfer between phases are similar in
multicompo-nent systems to those observed for systems with two, three, and four compomulticompo-nents Use of scaled shock and rarefaction velocities allows straightforward construction of the complete solutions
• A displacement is multicontact miscible if any of the key tie lines is a critical tie line (a tie
line of zero length)
• In displacements in which the injection gas is a mixture or the injection gas does not have
the highest equilibrium K-value, it is likely that a crossover tie line will control miscibility Condensing/vaporizing gas drives, including most CO2 floods, are examples
7.7 Additional Reading
The first solutions for systems with more than four components were obtained by Dindoruk [19] and Johns [54] Dindoruk reported solutions for a constant K-value system with five components and outlined the extension of his approach to systems with variable K-values Johns obtained solutions with variable K-values for a five-component system and for a six-component system, both for injection of a pure component A systematic procedure for calculating solutions for multicomponent
systems with any number of components in the injection gas was reported by Jessen et al [48].
Jessen [42] and Ermakov worked simultaneously to develop the systematic approach for finding multicomponent solutions when effects of volume change as components change phase are absent
or present Numerous example solutions for systems with and without volume change are reported
by Jessen [42], and additional solutions that show the effects of volume change are described by Ermakov [23]
The analytical calculation of MMPs for multicomponent systems was considered first by Johns and Orr [55] Wang [128] and Wang and Orr [129] showed how to find the key tie lines for injection
of a single-component gas and then used the key tie lines to find the MMP They then considered
how to find the key tie lines and the MMP for a multicomponent injection gas [128] Jessen et
al developed a much more efficient algorithm for finding the key tie lines [45, 42] and showed
that predictions of minimum miscibility pressures based on the key tie line approach agreed very well with experimental data and with MMPs calculated by other methods Wang and Peck [131] reported results of additional tests of the accuracy of the key tie line approach They showed that predicted MMPs and MMEs for systems with widely varying injection gas compositions agreed very well with experimental observations
Trang 3Compositional Simulation
by F M Orr, Jr and K Jessen
The gas displacement problems considered in previous chapters can also be solved by numerical approaches, typically finite difference methods In fact, before the analytical solutions considered here were developed, the only way to solve such problems was to do so numerically In this chapter,
we compare analytical and numerical solutions and show that the numerical solutions converge
to the analytical ones In addition, we consider the limitations of each approach to solving the flow equations Interpretation of numerical solutions requires careful attention to the effects of numerical dispersion on the solutions, the subject of the next section
We also consider here numerical approaches to calculation of minimum miscibility pressure (MMP) or minimum injection gas enrichment for miscibility (MME) Several numerical schemes have been described, ranging from calculations based on mixing cells to extrapolation of numerical results from compositional simulations The analysis of composition paths for multicomponent displacements discussed in Chapter 7 can be used to determine whether these numerical approaches yield accurate estimates of MMP or MME That analysis shows that calculations based on a single mixing cell yield incorrect estimates of MMP and MME when a crossover tie line controls development of miscibility Numerical approaches based on multiple mixing cells or compositional simulation can yield accurate estimates of MMP and MME if appropriate attention is paid to effects
of numerical dispersion
In Section 8.1, we consider how numerical dispersion arises in finite difference computations Section 8.2 shows how calculated composition paths are modified by the presence of numerical dispersion The impact of numerical dispersion on a particular displacement depends on the phase behavior of the gas/oil system Section 8.3 describes how the magnitude of the effect of numerical dispersion depends on phase behavior and suggests a way to determine which systems will be sensitive to the effects of dispersion Finally, numerical schemes for calculation of MMP and MME are discussed in Section 8.4
8.1 Numerical Dispersion
Numerical solutions to material balance equations like Eqs 2.4.4 or 4.1.1 are usually obtained by finite difference methods Those solutions are always affected to some extent by truncation error
In this section, we examine the sources of that truncation error
Eq 4.1.1 can be solved easily with a fully explicit (forward time), backward space difference of the form
213
Trang 4C n+1
i,k = C i,k n − ∆τ
∆ξ (F
n i,k − F n
where k indicates the grid block and n the time level While there are several other differencing
options available, Eq 8.1.1 illustrates well the issues that arise in finite difference solutions (see
Mallison et al [73] for an evaluation of higher order computational schemes that improve accuracy
for one dimensional solutions over the simple differencing scheme discussed here) It is possible
to obtain more accurate finite difference solutions at a given grid resolution with the approaches Mallison describes, but the overall impacts of numerical dispersion are common to all the methods,
so we use here the simplest of methods to illustrate the ideas
In a displacement calculation, the compositions in each grid block are calculated at the new time step with Eq 8.1.1 A flash calculation is then performed for the composition in each grid block to determine the phase compositions and saturations, with which the fractional flows of the components can be calculated in preparation for the next time step
Lantz [64] showed that even when the conservation equation being solved is dispersion-free, the truncation error in the finite difference version of the differential equation mimics a second order term in the original differential equation He showed that for finite difference form of Eq 8.1.1, the numerical eqivalent of the Peclet number for single-phase flow is [64, 127]
P e −1 num= ∆ξ
2
1− ∆τ
∆ξ
and for two-phase flow is
P e −1 num= ∆ξ
2
df1
dS1
1− ∆τ
∆ξ
df1
dS1
The exact form of the numerical Peclet number depends on the finite difference formulation for the time and space derivatives Peaceman [96, pp 74-81] reports expressions appropriate to the various implicit and explicit finite difference forms in common use For the remainder of this discussion,
we will use the simple form of Eq 8.1.1, which permits calculation of compositions at the new time level without a matrix inversion and which illustrates in a straightforward way the impact of numerical dispersion on calculated composition paths
Eqs 8.1.2 and 8.1.3 indicate that the inverse numerical Peclet number will be greater than zero
as long as ∆τ < ∆ξ The P e −1
num must be positive if the numerical calculation is to be stable [96] If it is negative, the numerical solution will show nonphysical oscillations in compositions
and saturations Hence, ∆τ must be less than ∆ξ for single-phase flow, and much less than ∆ξ
if dS df1
1 > 1, as it will be for the S-shaped fractional flow curve appropriate to two-phase flow in
a porous medium, particularly when the injection gas has a viscosity that is lower than that of
the oil displaced As a result, effects of numerical dispersion can be reduced by reducing ∆ξ (and therefore ∆τ ), but they cannot be eliminated entirely for Eq 8.1.1.
The limitation on time step size is a version of the Courant-Friedrichs-Levy (CFL) condition [15, 69], which states that the finite difference scheme of Eq 8.1.1 is unstable if
++
++λ p ∆τ
∆ξ
++
for each of the p eigenvalues Because the nontie-line eigenvalues generally have values close to one,
it is the tie-line eigenvalue (df /dS) that determines the maximum stable value of the time step size.
Trang 5Table 8.1: Peclet Numbers for Finite Difference Simulations
Grid Blocks P e −1 P e
20 0.0462 21.6
100 0.00926 108
1000 0.000926 1080
10000 0.0000926 10800
Numerical dispersion arises from the way fluids are treated in the finite difference representation Consider what happens in a first-contact miscible displacement, for example An increment of solvent is injected into the first grid block in the first time step At the end of that time step the fluids are mixed so that the block has uniform composition In the next time step, a fraction of the solvent injected in the first time step flows out of grid block 1 to grid block 2, and that happens no matter how small the amount of solvent injected into grid block 1 Thus some solvent propagates the entire length of grid block 1 in the first time step, even though physical flow over the same distance would take longer than one time step In the second time step, some solvent leaves grid block 2 by the same sequence of events Thus, there is a smearing of the composition profile that comes from the treatment of the blocks as mixing cells with finite size In the limit as the grid blocks become infinitely small, that smearing disappears
8.2 Comparison of Numerical and Analytical Solutions
If the analytical solutions obtained in Chapters 4-7 are correct, then numerical solutions obtained with a finite difference calculation will converge to the analytical solutions as the number of grid blocks is increased To illustrate convergence and how numerical dispersion affects compositional
simulation results, we consider first a simple ternary system with constant K-values (K1= 2.5, K2
= 1.5, and K3= 0.05), which might represent a CH4/CO2/C10system at a relatively low pressure.
In this example, the initial oil mixture contains no CH4, and the injection gas is pure CH4 The
mobility ratio was fixed at M = 5, and the simulations were performed by solving Eq 8.1.1 with fixed ratio of time step size to grid block size, ∆τ /∆ξ = 0.1.
Figure 8.1 compares the composition path obtained by compositional simulation for finite dif-ference grids of 20, 100, 1000, and 10,000 grid blocks with the composition path of the analytical solution The corresponding saturation and composition profiles are also shown in Fig 8.1, and recovery curves for CO2 and C10 are shown in Fig 8.2 The analytical solution includes a leading
semishock from the initial composition to a point on the initial tie line, a very short rarefaction along the initial tie line, a long nontie-line rarefaction that connects the initial tie line to the injection tie line, and a trailing shock to the injection composition
Comparison of the finite difference (FD) and analytical (MOC) solutions shown in Figs 8.1 and 8.2 reveals several important points First, the FD solutions do converge to the analytical solution, confirming that the two approaches are consistent However, the rate of convergence is not high, and very fine computational grids are required for this problem if the details of the solution are to
be reflected accurately The FD solutions with 20 and 100 grid blocks show significant deviations
of the calculated composition paths and composition profiles In this displacement, the CO2 that
Trang 6a
a a
Oil
Dilution Line
Injection Gas
Injection Gas Tie Line
Initial Tie Line 20
100 1000
0
1
Sg
a a
a a
a
MOC
20 Blocks
100 Blocks
1000 Blocks 10,000 Blocks
0
1
a
a a
a
0
1
a a
a a
20 100 1000
0
1
C10
ξ/τ
a a a a
a
Figure 8.1: Effects of numerical dispersion on a vaporizing gas drive for a ternary system with
constant K-values, K1 = 2.5, K2 = 1.5, and K3 = 0.05 The initial composition is C1 = 0, C2 =
0.3760, C3 = 0.6240, and the injection gas is pure C1 For all simulations, ∆τ /∆ξ = 0.1, and M
= 5
Trang 70.0 0.2 0.4 0.6 0.8 1.0
τ
Analytical Solution
20 Grid Blocks
100 Grid Blocks
1000 Grid Blocks 10,000 Grid Blocks
C10
CO2
Figure 8.2: Component recovery for the displacements shown in Fig 8.1
is present initially in the oil is swept up into a bank (see the CO2 profile in Fig 8.1) That bank
is poorly resolved with 20 and 100 grid blocks but is much better resolved with 1000 grid blocks With 10,000 grid blocks, however, the FD solution is nearly indistinguishable from the the MOC solution on the scale of the plots in Fig 8.2 Thus, the FD solution does, in fact, converge to the analytical solution, but it is clear that this problem is relatively sensitive to the effects of numerical dispersion (see Section 8.3)
Second, the FD solutions can resolve the key tie lines, shocks and rarefactions that are impor-tant parts of the solution, but they do so only if the FD grid includes enough grid blocks Regions
of the solution where compositions are changing rapidly, the shocks and the nontie-line rarefaction, are the most difficult to capture in the FD solutions The leading and trailing shocks are resolved somewhat better at a given grid resolution because they are self-sharpening The nontie-line rar-efaction is smeared much more The composition gradient is significant along the nontie-line path, and numerical dispersion acts to reduce that gradient When that rarefaction is smeared, the FD composition path follows closely a path that resembles the nontie-line paths obtained in the ana-lytical solution As the grid is refined, the nontie-line portion of the FD solution approaches more closely the MOC nontie-line path
Calculated component recovery (Fig 8.2) is also quite sensitive to grid resolution Because re-covery at some late time in a displacement (often 1.1 or 1.2 pore volumes injected) is often used as
an indicator of multicontact miscibility, it is important that effects of numerical dispersion on recov-ery be assessed when compositional simulations are used to estimate minimum miscibility pressure (see Section 8.4) Here it is enough to note that when numerical dispersion alters composition path significantly, it can also have a quite significant effect on calculated recovery, particularly in multicomponent systems at pressures or enrichments near the MMP or MME
Trang 8The FD solutions reflect the interplay of dispersion (numerical in this case) and convection Fig 8.1 shows the dilution line that connects the initial oil composition to the injection gas composition The effect of dispersion is to move the solution composition path toward the dilution line If there were no flow at all, then mixtures of the initial and injection fluids would lie on the dilution line The effect of convection, and the accompanying chromatographic separations of components that take place as components partition between the flowing phases, is to push the composition path closer to the MOC solution, in which effects of dispersion are absent The Peclet number reflects the relative importance of the contributions of dispersion and convection (see Section 2.7) In this
example, the Peclet number can be estimated with Eq 8.1.2 The value of df1/dS1 is not constant throughout the solution but for the purposes of estimating the effects of dispersion, it is convenient
to use the maximum value of df1/dS1in Eq 8.1.2, because the resulting value of the Peclet number
will also determine whether the numerical computation is stable (it is if P e −1 is positive) That
value is about 2.45 for M = 5 and the relative permeability functions of Eqs 4.1.14-4.1.19 with S or
= S gc = 0, the values used for all the constant K-value solutions discussed in this chapter
Table 8.1 reports approximate Peclet numbers for the four FD simulations The solutions with
P e > 1000 are close to the analytical solution The estimate given in Section 2.7 of the Peclet number appropriate to slim tube displacements (P e = 2500) suggests that about 2500 grid blocks
would be required for the level of numerical dispersion in the FD calculation to approximate the physical value With that grid resolution, the FD solution would approximate closely the dispersion-free MOC solution, another indication that the use of the analytical solutions for 1D displacements like those performed in slim tubes is a reasonable approach
Systems with K-values that depend on composition display similar behavior To examine the impact of numerical dispersion in simulations with variable K-values, Eq 8.1.1 was solved using the Peng-Robinson EOS with the pressure at which the phase behavior was evaluated held con-stant Fig 8.3 compares the analytical solution for a six-component displacement with no volume change(see Fig 7.5) with FD solutions calculated with 50, 500, and 5000 grid blocks In this sys-tem, the FD solutions converge reasonably rapidly to the analytical solutions as the grid is refined With 500 grid blocks, much more limited smearing of the shocks is observed, and with 5000 grid blocks, the FD solution is almost indistinguishable from the MOC solution The agreement is simi-lar when the effects of volume change are included, as Fig 8.4 shows These examples demonstrate that FD compositional simulation can produce solutions that converge to the analytical solution
if sufficiently fine grids are used The computational cost is much higher for the FD solutions, of course The FD solutions are needed, however, to deal with situations in which the pressure at which phase behavior is evaluated is not constant or when the injection composition is not constant, because the analytical solutions derived here are for Riemann problems only in which the initial and injection compositions are constant Finite differences are also used for two-dimensional and three-dimensional compositional simulations, for which analytical solutions are not available While computational cost for these one-dimensional calculations is not a problem, corresponding two- and three-dimensional simulations are often too slow to allow use of large numbers of grid blocks in each dimension It is rare, for example, to see use of as many as twenty grid blocks between wells in field-scale calculations Hence, it is likely that effects of numerical dispersion on calculated composition paths will be significant in multidimensional FD compositional simulations One approach to dealing with the difficulties that arise from numerical dispersion in FD calculations
is to decouple the representation of the effects of reservoir heterogeneity, which control where low viscosity injected gas flows preferentially, from the kinds of chromatographic dtermination of
Trang 91
Sg
MOC
50 Grid Blocks
500 Grid Blocks
5000 Grid Blocks
0
1
0
1
0.0
0.2
C4
ξ/τ
0.0
0.2
C10
0.0
0.2
C16
0.0
0.2
C20
ξ/τ Figure 8.3: Saturation and composition (mole fraction) profiles for displacement of a six-component oil (Oil A in Table 7.2 by pure CO2 at 2940 psia (200 atm) and 160 F (71C) Effects of volume
change as components change phase are not included in this example The solid line is the analytical solution, and the dotted lines are the FD solution See the discussion of Fig 7.6 for a description
of the MOC solution for this fully self-sharpening example
Trang 101
Sg
0
1
0
1
0.0
0.2
C4
ξ/τ
0.0
0.2
C10
0.0
0.2
C16
0.0
0.2
C20
ξ/τ
0
1
500 Grid Blocks
5000 Grid Blocks
Figure 8.4: Saturation and composition (mole fraction) profiles for the displacement of a six-component oil (Oil A in Table 7.2 by pure CO2 at 2940 psia (200 atm) and 160 F (71 C) This
example includes the effects of volume change as components change phase The solid line is the analytical solution, and the dotted lines are the FD solutions
... computational cost for these one-dimensional calculations is not a problem, corresponding two- and three-dimensional simulations are often too slow to allow use of large numbers of grid blocks in each... class="page_container" data-page="6">a
a a
Oil
Dilution Line
Injection Gas< /small>
Injection Gas Tie... resembles the nontie-line paths obtained in the ana-lytical solution As the grid is refined, the nontie-line portion of the FD solution approaches more closely the MOC nontie-line path
Calculated