#### The Importance of Capillary
Pressure Inclusion and Accuracy
Also see:
Simulation models from the late 1950s
to the present have accounted for viscous pressure drop, gravity forces, and
capillary pressure. The relative importance of each is problem-dependent,
rendering general rules of questionable value. Nevertheless, such “rules”
have arisen, persist, and deserve some consideration. A general rule can
require such tedious exposition or qualification as to be questionably
“general”. Discussion and examples below illustrate the complex mental
exercise frequently required to explain observed simulation results in terms
of a reasonably brief rule satisfying intuition and basic mechanistic
understanding.
*Static vs dynamic P*_{c}
effects. If a reservoir has initial water-oil and/or gas-oil contacts
then initial water, oil, and gas in place can strongly depend upon the capillary pressure (Pc) curves. This is because the initial
saturation distributions are calculated assuming capillary-gravity
equilibrium. Thus it is often, if not generally, true that P_{c} has
a significant static effect – but only if initial water-oil or gas-oil
or gas-water contacts exist. If no initial contact(s) exist, P_{c} has no
(static) effect on initial fluids in place.
P_{c} will have some dynamic
effect on results (e.g. rates, cums, gor, wcut) regardless of whether the
above static effect exists. The significance of the dynamic effect alone can
be examined using research-type datasets or field study prototype datasets
(“representative” cross-sections or 3D sectors) where no initial contacts
exist. Such purely dynamic effects are examined here.
**The stratified reservoir.** A
stratified reservoir has a number of layers with significant or complete
lateral continuity. The layer permeabilities vary greatly and adjacent
layers communicate vertically. Simulations of oil displacement by water or
gas will result in displacing fluid advancing rapidly in highly permeable
layers and slowly in tight layers. This results in cell-pairs near a front
in a permeable layer where S_{1} >> S_{2}, where S is
displacing fluid saturation and subscipts 1 and 2 refer to the permeable
layer cell and to the cell in the tighter adjacent layer above or below,
respectively. This in turn leads to counter-current imbibition, normal to
the displacement direction, between the two cells. This is also
referred to as "capillary crossflow".
The rate of oil flow from the tight
cell 2 to the permeable cell 1 due to this P_{c}-driven
counter-current imbibition (capillary crossflow) is
q_{o} = -τ λ_{o}λ_{g}/(λ_{o}+λ_{g})
(P_{cgo}(S_{g2})-P_{cgo}(S_{g1}))
(1)
for the gas-oil displacement, and
q_{o} = τ λ_{w}λ_{o}/(λ_{w}+λ_{o})
(P_{cwo}(S_{w2})-P_{cwo}(S_{w1})) (2)
for the water-oil displacement. These
expressions neglect gravity and compressibility effects. The mobilities λ
are upstream values. The true definition of capillary pressure is Pnw - Pw,
where nw denotes the non-wetting phase and w denotes the wetting phase, but,
to avoid confusion here and in our model, the capillary pressures are
defined independently of wettability as: Pcgo = Pg - Po, and Pcwo = Po - Pw.
Capillary pressure tends to favor
imbibition of the wetting phase into a cell or the cell’s retention of the
wetting phase. Intuition might then indicate that P_{cgo} should
decrease recovery in the gas-oil displacement because P_{cgo} will
act to retain oil in the tight layer(s). Eq. 1 indicates that is
wrong: q_{o}
is positive because the cell-pair has S_{g1} >> S_{g2} and P_{cgo}(S_{g})
increases with increasing S_{g}. However, Example 1 below shows
that P_{cgo} can increase or decrease recovery in a gas-oil
displacement.
Intuition indicates that P_{cwo}
should increase oil recovery in the water-oil displacement because water
will imbibe into the the tight layer from its adjacent permeable layer,
causing oil to flow out of the tight layer by counter-current imbibition.
Eq. 2 above confirms that: q_{o} is positive because S_{w1}
>> S_{w2} and P_{cwo}(S_{w}) decreases with
increasing S_{w}. However, Example 2 below shows that P_{cwo}
can increase or decrease recovery in a waterflood.
*The highly heterogeneous reservoir*.
Arguably, the directional effect of Pc on oil recovery is best addressed in
the context of highly heterogeneous examples. The SPE10 Model 1 and Model 2
descriptions offer high heterogeneity in fine grids of 2000 and 1.12 million
blocks, respectively^{1}. Both Models have zero Pc with no initial
contacts so running them with Pc gives directional dynamic Pc effects for
publicly available problems with widely accepted 0-Pc recoveries.
Model 1 has a correlated,
geostatistically generated permeability description and Model 2 has a
geostatistical permeability description claimed to reflect actual North Sea
formations. Such highly heterogeneous descriptions are argued here to be
subject to the same Pc-effect arguments given above for stratified
reservoirs. Geostatistical descriptions frequently have permeable lenses or
channels of lateral extents which are significant but much less than
complete. For example, the Model 1 100x20 xz cross-section has a permeable
layer 13 of 500-1000 md extending 20 gridblocks laterally from the injector.
Adjacent layer 12 is tight with permeabilities roughly 100 times smaller.
*Effect of permeability on capillary
pressure*. Early laboratory studies showed that measured capillary
pressure curves were larger in tighter rocks. Leverett proposed a
proportionality between P_{c} and √(ф/k), the Leverett
J-function. This results in
P_{cwo} = L * entered P_{cwo}(S_{w})
P_{cgo} = L * entered P_{cgo}(S_{g})
where L = √{(ф/k)/(ф/k)_{ref})}.
The entered P_{c}(S) is the P_{c} curve for a rock of (ф/k)_{ref}.
The P_{c} value for a gridblock is then dependent upon saturation S
and the gridblock ф/k value.
**Example 1** **– effect of P**_{cgo}
on a gas-oil displacement. Example 1 is the SPE10 Model 1 gas-oil
displacement. The grid is a 100x20 2D xz cross-section. Each gridblock is 25
ft long by 2.5 ft thick. Permeability varies over the grid by six orders of
magnitude from 0.001 to 1000 md with an arithmetic average value of 162 md.
Porosity is constant at 0.2 and S_{wc} = 0. Three pore volumes of
gas are injected in about 21 years.
Three runs were made:
** **__Table 1. Effect of Pc
on Recovery, Gas-Oil displacement Example 1__
P_{cgo} increases recovery by
7.2 % (x9a vs. x9) if the k-dependence of P_{c} is neglected, in
accordance with Eq. 1 above. But if the k-dependence is included, P_{cgo} decreases
recovery by 12.1 % (x9b vs. x9). If entered P_{cgo} is 20S_{g}^{2},
then P_{c} increases recovery by 21 % (to 46.9 %) without
k-dependence and decreases recovery by 13.7 % (to 33.4 %) with k-dependence.
Why does k-dependent P_{cgo}
decrease oil recovery? Eq. 1 shows that the counter-current flow induced by
P_{cgo} acts toward an equilibrium state at which
P_{cgo2} = P_{cgo1}
**If P**_{cgo} is dependent only
upon S_{g} (L_{2}=L_{1}=1), then this gives S_{g2}
-> S_{g1} at large time (equilibrium). That is, the tight cell S_{g2}
approaches the permeable cell (large flood-induced) S_{g1} value, tight cell 2 oil
saturation is low and oil recovery is high. However, if P_{cgo} is
dependent upon L and S_{g} then equality of the cells’ P_{c}
values occurs with S_{g2} < or << S_{g1}, tight cell 2 oil
saturation is high and recovery is low, as shown in Figure 1.
*
***Example 2 – effect of P**_{cwo}
on a water-oil displacement. Example 2 is the same as Example 1 except
three pore volumes of water are injected over 20 years.
The drainage Pcwo curve used is 20Sw2
but it has no effect because the entire displacement reflects the imbibition
Pcwo curve (increasing Sw). Three different imbibition Pcwo curves are used
– water-wet, mixed-wettability, and oil-wet curves. All three curves are
similar at low Sw but reach 0 Pc at lower Sw values as water-wetness
decreases. The entered Pcwo curves are assumed to represent a rock having
the arithmetic average permeability of 162 md. The entered imbibition Pcwo curves for
the water-wet, mixed-wet, and oil-wet cases are shown in Figures 2, 3, and
4, tabulated in the outfiles
x10a.out, x10b.out, and x10c.out, and are given by:
water wet:
Pcwo = 20*(1-Swn)**2 -3.67*Swn**2
(3)
intermediate: Pcwo = 20*(1-Swn)**2 -
10*Swn**2
(4)
oil wet:
Pcwo = 20*(1-Swn)**2 - 100*Swn**2
(5)
where Swn = (Sw-Swc)/(1-Swc) = Sw for
this case (Swc=0).
Seven runs gave the following oil
recoveries (% of original oil in place):
** **__Table 2. Effect of Pc on
Recovery, Water-Oil Displacement Example 2__
Again, as in Example 1, the effect of
Pc on recovery can be queried by looking at the “equilibrium” Sw2 in a tight
cell 2 adjacent to a permeable cell 1 where Sw1 has reached a ‘large’ value
by displacement. The higher this equilibrium tight cell Sw2, the higher the
recovery. From Eq. 1 above, the Pc-driven counter-current imbibition alone
drives this tight cell Sw2 toward an equilibrium value dictated by
Pcwo2 = Pcwo1 = Pceq
Figures 2, 3, and 4 show Pc curves for the
water-wet, intermediate wetness, and oil-wet cases. The blue curves
represent k-independence and the pink and yellow curves reflect the two cells’
different Pc curves which exist due to k-dependence. Permeable block
Sw1 is a large value (due to displacement) approaching 1-Sorw.
With no k-dependence, tight block equilibrium saturation Sw2 equals Sw1 in
all 3 cases. **With k-dependence, Figures 2 and 4 show that (a) for
water-wet systems, Pceq is slightly positive (relative to Pcwo2),
tight block equilibrium
Sw2 > permeable block Sw1 and oil recovery is high, and (b) for oil-wet
systems, Pceq is slightly negative (relative to Pcwo2),
tight block equilibrium Sw2 << permeable block Sw1 and oil recovery is low.
** **However, Example 3 shows that oil-wet water/oil capillary
pressure can increase recovery compared to the case with no capillary
pressure. In general, the tight block equilibrium saturation Sw2 in
response to permeable block flood-induced high saturation Sw1 is very close
to the saturation at which the tight block 2 Pcwo is equal to zero:
equilibrium Sw2 is slightly lower than Sw at Pcwo2=0 for water-wet systems,
and slightly higher than Sw at Pcwo2=0 for oil-wet systems.**
**Example 3 – effect of P**_{cwo}
on a water-oil displacement**.** Example 3 is a water-oil displacement in a
140000-block upscaled grid of the SPE10 1.12 million block, Model 2
waterflood problem. The geostatistical permeability description is that of
an actual North Sea reservoir. Permeability varies from about .001 to 20000
md with an average kx of 276 md. Swc = Sorw = .2, average porosity is
0.1707, and
krw = Swn^{2}
kro = (1-Swn)^{2}
Swn= (Sw-Swc)/(1-Swc-Sorw),
and 0.917 hcpv of water are injected in
2000 days. Capillary pressure curves are given by Equations 3, 4, and 5, and
differ from those in Example 2 only because Swc=.2 here and was zero in
Example 2. Recovery results for five runs are shown in Table 3.
** **__Table 3. Effect of Pc on
Recovery, Water-Oil Displacement Example 3__
spe10_case2.inc (include file needed with above data files,
remove .dat extension after downloading)
With no k-dependence, both water-wet
and oil-wet Pc increase recovery by about 10 %. The effects of
Pc on water-oil displacement recovery for Example 2 (Table 2) and Example 3
(Table 3) are similar (Figures 2, 3, and 4, and the Example 2 discussion in
bold above apply here) except for the oil-wet Pc case with k-dependence. For
that case, recovery decreases for Example 2 and increases for Example 3,
compared to recovery for Pc=0. This results from the effect of water-oil
mobility ratio **M** on the magnitude of permeable cell Sw1 values and
associated value of Pcwo1 (Pceq). **If Sw1 is large, Pceq is negative and Fig.
4 indicates low recovery. If Sw1 is sufficiently low (less than Sw at
Pcwo2=0), Pceq is positive
and oil recovery is higher than the case without capillary pressure. Comparatively speaking, Sw1 is large in Example 2
and low in Example 3 because M is much lower in Example 2 than in Example 3.
The two examples have the same viscosity ratio but quite different relative permeability
curves.**
1.
Christie, M.A., and
Blunt, M.J., "Tenth SPE Comparative Solution Project: A Comparison of
Upscaling Techniques", SPE Reservoir Engineering and Evaluation, 4,
308-317, (2001). |