The Importance of Capillary Pressure Inclusion and Accuracy

Also see:

•  Capillary Pressure Adjustment

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 S₁ >> S₂, 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_oλ_g/(λ_o+λ_g) (P_cgo(S_g2)-P_cgo(S_g1))          (1)

for the gas-oil displacement, and

     q_o = τ λ_wλ_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, respectively¹. 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_g², 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 (L₂=L₁=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 = Swn²

  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).