GRIDDING AND WELL MODELLING IN RESERVOIR SIMULATION
4 WELLS IN RESERVOIR SIMULATION .1 Basic Idea of a Well Model
Qp
Qp = - kA. krp µp.Bp
∆P .∆x
Harmonic average
Arithmetic averages Upstream value of krp
4 WELLS IN RESERVOIR SIMULATION 4.1 Basic Idea of a Well Model
The only way fluids can be produced from or injected into a reservoir is through the wells and we must therefore include them in our reservoir simulation model. As you may know, the area of Well Technology is vast and in addition to the long wellbore between the reservoir and the surface, there are many other technical features of wells that can have a major impact on the flows into and out of the reservoir. For example, there will be safety valves at the surface and many different types of completion in the well construction itself. Here, we will simplify things as much as possible in order to extract the central functions of the well that we will have to model in the simulator. A schematic of the total well is shown in Figure 25 where the details of the near well formation are shown inset. The near wellbore flows are thought to be radial in an ideal vertical well and this will have some relevance in modelling the near-well pressure behaviour, as discussed in Chapter 2 and elaborated upon below.
In addition to these near-well pressure drops, there are several other identifiable Figure 24
The correct inter-block averages for all terms in the two-phase block-to-block flows in a reservoir simulator.
4 4
Gridding And Well Modelling
pressure drops between the fluids in the reservoir and the surface oil storage facilities and we may have to model at least some of these. Indeed, it is this topside pressure behaviour that “links” or “couples” the surface with the pressure and flows that we are trying to model in the reservoir using reservoir simulation. The main decision is to determine how much of the formation to surface well assembly we will actually have to model. The main pressure drops are shown in Figure 26 (based on Figure 25 of whole well + ΔPs) and are associated with:
(i) Formation → wellbore flow, ΔPf→w: where fluids flow from a “drainage radius”, re, at pressure, Pe, to the wellbore. Figure 26 shows the near-well pressure profile, in the near-sandface region with bottom hole flowing well pressure (BHFP), Pwf. Thus the formation → wellbore pressure drop, ΔPf→w , is:
∆
∆
∆
∆
P P P
P P P
Q PI P P
Q PI P
P r Q
k h r r
P r P P Q
k h r r
Q k h
f w e wf
wf atm r s
o e wf
o e
w
e e wf e
w
→
→
= ( − )
= +
= ( − )
=
( ) = ( )
( ) = ( − ) = ( )
= ( )
.
.
. ln
. ln
. .
max .
µ π
µ π
π µ
2
2
2 ln ln
.
. .ln
r r
P P
PI k h r r
e w
e wf
e w
( − )
= ( )
2π µ
(14) (ii) The pressure drop, ΔPwell, that may occur along the completed region of the wellbore from the bottom of the well (or the “toe” of a horizontal well) to the wellbore just at the top of the completed interval. In very long wells, this pressure drop along the wellbore due to friction may be quite significant although there will be cases where it can be ignored;
Well tubulars Well
Reservoir showing two geological layers
Well completed in reservoir
Fluid flow from reservoir layers to wellbore Well head
To storage / export Surface facilities
(separator etc...)
Figure 25
Schematic of the fluid flows into a well in a grid
block model of the reservoir through to their storage or export from the field.
4 4
Gridding And Well Modelling
Well tubulars Well
Reservoir showing two geological layers
Well completed in reservoir
Fluid flow from reservoir layers to wellbore Well head
To storage / export Surface facilities
(separator etc...)
∆Pwh -> export
∆Pr -> s pressure drop from reservoir top to surface
Pwf
rw r re
Pe
∆Pf -> w
∆Pwell
Near wellbore formation to wellbore ∆Pf -> w
pressure drop along the well within reservoir section
(iii) Reservoir → surface pressure drop, ΔPr→s : the pressure drop from the well at the top of the completed formation just above the reservoir to the wellhead which is at pressure, Pwh . This ΔP is quite significant and, locally in any sector of the well, there will be a local pressure drop vs. flow rate/fluid composition relationship. This may be calculated from models (often correlations) of multi-phase flow in pipes.
As the fluids move up the wellbore, the pressure drops in oil/water production and free gas may also appear; thus, we can have three phase flow in the well tubular to the surface and we may have to incorporate this flow rate/pressure drop behaviour in our modelling.
(iv) There will frequently be further pressure drops as the fluids flow from the wellhead through the surface facilities such as the separators, various chokes, etc.
We will not consider this in detail here although it can be an important consideration in some field cases e.g. if the well is feeding into a network gathering system which other wells are also feeding into. This could be a complex surface gathering system network or a multiple-well manifold of a subsea production system.
In this section, we will mainly focus on the formation to wellbore pressure drops. Thus, our main task is to either calculate or set the well flowing pressure (Pwf) although we will return briefly to the issue of calculating the pressure drops between the reservoir and the surface in the discussion below.
To set the scene in modelling wells in a simulator, we will first consider a very simple model well producing only oil into the wellbore. How do we decide what Figure 26
Schematic of the fluid flows from the well through to storage or export showing the associated pressure drops that occur in the system.
28 Institute of Petroleum Engineering, Heriot-Watt University 29
4 4
Gridding And Well Modelling
the volumetric flow rate, Qo, of this well is? Indeed, do we decide or is it set for us by the reservoir and well properties? We will start with the simplest case where we basically “take what we can get” by drawing the wellhead pressure, Pwh, down as low as possible. Suppose that we simple open it up such that the oil pressure drops essentially to atmospheric. There is then the additional reservoir to surface pressure drop, ΔPr→s, to consider. Thus, the well flowing pressure, Pwf, would be given by:
we can now calculate how much. As we will see, this will depend on the physical properties of the system such as the permeability of the rock, the viscosity of the oil, the precise geometry of the well etc. However, in our simple conceptual well, we will take all of these quantities as “givens” for the moment. Suppose the well does flow at a volumetric flow rate, Qo, for reservoir pressure, Pe, and well flowing pressure, Pwf . We can then define a productivity index, PI, of the well as follows: where possible units of PI could be bbl/day/psi, for example. The above equation basically states how much oil is produced per psi of drawdown. This simple equation takes us back to our original question on “what/who decides on Qo?”. In our simple case, the answer is now clear; i.e. some things are “givens” - e.g. PI and Pe in a virgin oil producing system - and some we can set within limits - e.g. Pwf by setting the wellhead pressure. However, you may be able to set a flowrate, Qo, by installing a downhole pump (an ESP - electrical submersible pump). In that case, you would set Qo and then calculate Pwf where we are still considering the reservoir pressure (Pe) as a given. But clearly we cannot set Qo to any arbitrarily large value since the lowest possible value of Pwf = 0 (a vacuum), which would then set a maximum value of Qo given by: So, in summary, we can set either a pressure or a flowrate but (a) not both and (b) in either case, within limits.
But, can’t we affect the well PI or the reservoir pressure, Pe? We can actually affect the PI of a well by “stimulating” it possibly by locally hydraulically fracturing the well or by acidising it to increase the effective permeability of the near well region.
In addition, we can increase (or more commonly maintain) the reservoir pressure (which relates to the “reservoir energy”) to some extent by injecting a fluid - usually water or gas in another injector well. However, the basic well controls are either setting pressure or flow rate and this must be kept in mind when we model wells in reservoir simulation. We elaborate on these ideas in the following section where we introduce the central idea of a well model.
4 4
Gridding And Well Modelling
4.2 Well Models for Single and Two-Phase Flow
We now consider how a well model can be developed, firstly in our simple conceptual reservoir producing only oil. Figure 27 shows the local pressure profiles in a simple homogeneous well system in single phase flow (see Chapter 2, section 3.5). The pressure profile close to the wellbore, assuming radial flow, was derived in Chapter 2 and is given by (section 3.5):
Taking the pressure at radius re as being the reservoir pressure, Pe, then gives:
Well at BHFP
(assume ∆x = ∆y)
which can easily be arranged to obtain:
This now demonstrates exactly how the quantities k, h, μ, rw and re affect the well productivity. All of these factors behave as we might expect them to physically e.g.
as k↑, PI↑; as μ ↑, PI↓ etc.
Now consider how this relates to the pressures in the simulation block shown in Figures 27 and 28. In a grid block, the pressure is thought of as being constant throughout the block although we know that it should be varying continuously across the block;
we will refer to this as the average block pressure, P. The size of the grid block in Figure 27
Schematic showing (a) the near well pressure profiles that occur in a radial system and (b) corresponding quantities in a Cartesian grid block
30 Institute of Petroleum Engineering, Heriot-Watt University 31
4 4
Gridding And Well Modelling
our example is (Δx, Δy) and, for simplicity, we will assume that, Δx = Δy. Looking at the expression for PI in equation 21 and the quantities we have in the grid block, it is easy to make direct relations for some of them - obviously k, μ and h and also possibly rw and Pwf , although these latter two do not seem to appear in the grid model.
The drainage radius, re, and the reservoir pressure, Pe, which appear in the radial model do not appear in the grid model - instead, the block size (Δx, Δy) and average block pressure (P ) appear. This immediately suggests the following 2 questions: