GRIDDING AND WELL MODELLING IN RESERVOIR SIMULATION
2. what is the relation between P e and the average grid block pressure, P ?
(assume ∆x = ∆y)
Pe
Pwf Pwf
rw r re
P Well at BHFP
= Pwf
re <-> ∆x, ∆y ??
re =>
Pe =P
∆x ∆y
h
P Grid block pressure
How do we choose re such that Pe =P ??
Relation re
∆x
∆y re is where P(re) =P and re = re (∆x, ∆y) but what is the formula ? re
rw
The issue is defined quite clearly in Figure 28. From this figure, we would like to choose re such that Pe coincides with the average grid block pressure,P. The latter quantity (P ) is calculated in the simulation itself. In fact, we need to know how to calculate re from the quantities Δx and Δy as indicated in Figure 29 where we show the re as function of Δx and Δy, i.e. re(Δx, Δy).
If we know the formula for re(Δx, Δy), then we can calculate the PI (equation 21) and use this in the simulation to couple the quantities Qo (oil flow rate) and P (average grid block pressure); i.e.
P
Q PI P P
r r
k h P r P Q
r x
r x y
Q PI P P Q PI P P
PI kh k S
r r
o wf
w
wf
e
e
o o wf w w wf
o
ro o
o e
w
= ( − )
= ( ) ( − )
≈
≈ +
= ( − ) = ( − )
= ( )
.
ln .
. ( )
.
.
. ; .
. .ln 2
0 2
0 14
2
2 2
π µ
π µ
∆
∆ ∆
(22)
Figure 29
The relation between re and the block dimensions, Δx and Δy.
Figure 28
Schematic indicating how the near well pressures relate to the corresponding quantities in a Cartesian grid block
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Gridding And Well Modelling
Here, we can set either Qo or Pwf and then calculate the other one from P (and the known PI ). This was achieved in a very simple but ingenious way in a classic paper by Donald Peaceman (1978), another pioneer of numerical reservoir simulation.
Peaceman did this by carrying out a 2D numerical solution of the pressure equation on a Cartesian (x, y) grid for a quarter five-spot configuration as shown in Figure 30.
But, from equation 19, we know that:
Hence, if we plot the pressure at grid blocks away from the well block vs. the well block spacing on a logarithmic scale as shown in Figure 31, then we can extrapolate back to find the equivalent radius where P = Pe in terms of the well block dimension (Δx). It turns out that the simple relation is (for Δx = Δy): Therefore, we have a very simple way of calculating the PI or “well connection factor” as it is sometimes called of a well in a simulation grid block.
The simple Peaceman formula applies to a well in a radial environment (the five-spot configuration is as close as we can get to radial using a common 2D Cartesian grid) and for Δx = Δy. In fact, some modification to the simple formula is required for wells in “corner” locations or set close to a boundary and these are shown in Figure 32. Also, if Δx ≠ Δy, but the well is isolated (radial flow), then: compare pressures with the expected radial profiles (from Peaceman, 1978)
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Gridding And Well Modelling
Figure 32
Well factors for wells in various positions relative to the boundaries; after Kuniansky and Hillstad
• On Edge: i = 0 or j = 0
• On Diagonal: i = j
• i ≠ j ≠ 0 0.6
0.5
0.4
0.3
0.2
0.1
0
0.1 0.2 0.4 0.6 1 2 3 4 5 6 (1.0)
(1.1) (2.1) (2.0)
(2.2)
From areal average pressure pij - po qµ / kh
ro / ∆x r Ao / ∆x
Factor in terms of (re / ∆x) (re / ∆x) = 0.2
(Peaceman's equation)
(re / ∆x) = 0.196
implies no flow boundary
(re / ∆x) = 0.433
(re / ∆x) = 0.193
(re / ∆x) = 0.72
For anisotropic permeabilities kx ≠ ky
ky kx
kx ky
ky kx
re = 0.28
+
.∆x2 + .∆y2
1/2 1/2 1/2
1/4 1/4
Figure 31
Log plot of the pressures from the 2D areal grid used to find re (from Peaceman, 1978)
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Gridding And Well Modelling
We may find that, in a given simulation of a field case, that we input all the known or estimated data but the well in our simulation does not perform like the real case.
It may produce more (higher PI) or it may produce less (lower PI) than expected.
The former case may be due to the well being stimulated and the second case may be due to well damage. Within limits, we may adjust the calculated well PI in the simulation model in order to reproduce the observed field behaviour. However, we should think carefully before making such changes since the simulated well productivity may be wrong because some (or several) other aspects of the reservoir simulator input data are wrong.
It is now relatively straightforward to extend our discussion on PI and simple well models in a homogeneous single layer system to the flow of two phases - say, oil and water - as shown in Figure 33. Since two phases are being produced, then the saturations of both oil and water (So and Sw) must be at values where their relative permeabilities are > 0 (i.e. = > So > Sor ; Sw > Swc).
We now apply the same ideas as were developed above for the single phase case.
From the radial two phase Darcy Law, the volumetric production rates of oil and water are given by:
where the PI0 and PIw are the oil and water productivity indices, respectively, and
Near well two-phase flows in a Cartesian grid block.
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Gridding And Well Modelling
where, again, re is calculated using Peaceman’s formula. In the above equation, we have not incorporated the separate phase pressure, Po and Pw, in the well block but these may be used in a given calculation.
4.3 Well Modelling in a Multi-Layer System
The most common case which is modelled is where we have multi-phase (e.g. two phase oil/water) flow in a layered system where the layers are of different permeability.
This situation is shown for a simple four layer system in Figure 34.
Clearly, there are additional issues in this system since all four layers may be producing both oil and water and the proportions of each phase may be changing as the saturations (and hence relative permeabilities) change. In addition, there is also a gravitational potential in each layer which we may have to take into account.
Using the notation in Figure 34, we note that the oil flows in layer k (k = 1, 2, 3, 4, in this example) are:
the well are:
where we have taken the mean block pressure and also the well flowing pressure in layer k as being the same for both oil and water phases. Again, in the layered case, we can specify the total flow or the flowing bottom hole pressure and then we can calculate the other one using the above relations. (In fact, we can also specify the flow of either phase - oil or water - and then find the flow of the other and the bottom hole flowing pressure). For example, suppose we specify the total flow, QT. We need to decide how this total flow is made up - i.e. what are the separate Qo and Qw (QT = Qo + Qw) and how this flow is allocated from each of the layers in the system.
For simplicity, we make the assumption that there is a single well flowing pressure, Pwf (i.e. Pwf1 = Pwf2 = Pwf3 = Pwf4). Hence, for each layer, k : Everything is known in the above equation which allows us to determine the allocation of all fluids from each layer and we can calculate the corresponding bottom hole flowing pressure, Pwf.
In a very similar manner, we could specify the well flowing pressure, Pwf, and then calculate the individual flows, Qok and Qwk etc. in each layer.
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Gridding And Well Modelling
Layer 1 2 3 4 Well
completion
h
QT = Qo + Qw Qo1
Qw1 Qo2 Qw2 Qo3 Qw3 Qo4 Qw4
k1, h1, P1, Sw1
k2, h2, P2, Sw2
k3, h3, P3, Sw3
k4, h4, P4, Sw4
(a) (b)
4.4 Modelling Horizontal Wells
Figure 35 shows the trajectory of a “horizontal” well in a reservoir simulation model. This is not well represented by the purely radial r/z model grid discussed in Section 2.1 above in the context of a vertical well. Hence, it is less likely that the well connection factors calculated as shown in previous sections will apply for a horizontal well. This is broadly true although the basic principle is very similar.
That is, each well sector intersects a grid block (i,j,k) even although the well may be going through this block in say the x(i) direction and the flows between the well sector and the grid are given by an expression of the form:
Q PI P P khk S
r r
P P
Q Q Q PI PI P P
Q Q Q PI PI P P
Q PI
ok ok k wfk
ro o k
o ek
w
k wfk
T ok wk
k ok wk
k k wfk
Tk ok wk ok wk k wf
o i j k
= ( − ) = ( ( ) )
( − )
= ( + ) = ( + ) ( − )
= ( + ) = ( + ) ( − )
=
= =
∑ ∑
2
1 3
1 3
π µ .ln
, , oo i j k i j k i j k
w inj o o
w
w
w inj o o
w
w
P P
Q B Q
B Q
Q B Q
B Q
wf , ,
. (
, ,−
, ,)
= +
> +
(32)
where the actual form of the productivity index expression, PIoi,j,k , may be rather more complex since (a) the well may intersect the block in a more complex way and (b) the aspect ratio of the block is rather different when a horizontal well intersects it in that the x-direction well is very close to the z-boundaries since Δz is often smaller than Δx or Δy.
Figure 34
Well modelling of two-phase flow in a multi-layered system
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Gridding And Well Modelling
Figure 35
Cartesian grid cut from a 3D reservoir model showing two horizontal wells going through the system; two vertical wells also shown.
4.5 Hierarchies of Wells and Well Controls
Simple well control can be understood in terms of the well models discussed above.
For a single well, we can essentially set the well flowing pressure and then calculate the flows or vice versa but not both and with certain constraints (see above). Alternatively, we may set the wellhead pressure and then calculate the Pwf from the - calculated or input - well formation to surface pressure drops etc. We now consider controls on pairs, then groups and then clusters of groups of wells in a field - indeed, we can even couple together the wells from several reservoirs and set more global controls and this will be described briefly.
For a simple injector/producer well pair, for example in a 2D x/z cross-section, we can apply a range of well controls. Suppose this is a simple waterflood in an oil/water system. One of the most common controls is to fix the water injection rate at the injector but with (upper) limits on the well flowing pressure. The corresponding producer is then controlled by setting the bottom hole flowing pressure and then allowing the calculation of the oil and water phase flows (Qo and Qw). The volumetric production will be approximately balanced with the total production volume being of the order of the injected water volume - but not quite the same. Do you know why this is?
Clearly the formation volume factors (Bo and Bw) will affect the exact production volumes; when we are injecting water and producing 100% oil, the reservoir volume of injected water per day is Qw.Bw and this will displace (virtually) the same reservoir volume of oil. The volume of oil produced per day is Qo stb which is actually Qo.Bo reservoir bbls, equating these reservoir volumes shows that if we inject water at a rate of Qw (stb/day), we produce oil at a rate of Qw.Bw/Bo stb/day. Since Bo > Bw, then the volumetric production rate of oil (in stb/day) is lower than the injected water injection rate (in stb/day). This must be taken into account in considering well control by voidage replacement as discussed below.
If we wish to set injected Qw to precisely voidage replace whatever is produced, then we can do so to a good approximation by noting that if the production rate of oil
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Gridding And Well Modelling
and water in our simulation is currently, Qo and Qw. What volume of injected water must we inject to exactly replace the reservoir volume of these two phases? This is now quite straightforward since and, from the above discussion, it is evident that the quantity of water that must be injected, Qw inj (in stb/day) is given by:
Hence, we would gradually adjust the volume of water injected in the simulation model based on what we have just produced (at the last time step say) to the above quantity in order to voidage replace. At the producer, the most common option would be to constrain by bottom hole flowing pressure as described above.
Note that it is possible but less common to constrain all wells by volumetric injection/
production rate. We can see why if we consider an incompressible fluid where it is clearly impossible for the injection and production rates to be different since the pressure would go to + ∞ or - ∞, depending on whether we over- or under-injected, respectively. Although it is possible to specify different volumetric flow rates at injector and producer for a compressible fluid, this can only be done within very tight limits and the pressures tend to go to unrealistic limits e.g. if we over-inject (i.e.
> +
), the pressure tends to rise to unphysically high levels well above fracture pressure of the reservoir rock.4.5 CLOSING REMARKS - GRIDDING AND WELL MODELLING
In this section, the student has been presented with a largely non-mathematical description of gridding and well modelling in reservoir simulation. A more mathematical treatment of these issues will be given as we develop the flow equations and consider their numerical solution in Chapter 5 and 6, respectively.CONTENTS
1 INTRODUCTION
2 THE SINGLE PHASE PRESSURE EQUATION