4. Flow towards horizontal wells
"Horizontal wells" are called those which continue along the reservoir in horizontal direction. Horizontal wells usually have longer completion (zone) interval in comparison with the vertical wells. This means more productive well. Moreover, horizontal wells are placed in the most possible distance from the oil-water and gas-oil contacts. This makes horizontal production wells less exposed to early water or gas breakthrough.
The advantages of horizontal wells are obvious and have been realized a long time ago. The ability of drilling horizontal wells completed along the reservoir was first reached in 1990s. This significantly contributed to the good oil supply, such that in the last half of the 1990s prices for oil were low.
Figure 4.1 illustrates a subsea field development. A relatively limited amount of wells are drilled and completed, often through different reservoir layers
Figur 4.1: Åsgard field, surface and og subsea systems (Statoil 2009)
Figure 4.2 illustrates well Q5 completed in the “Smørbukk” reservoir on the Asgard . The completed intervals are in the order of 1500 m, thus corresponding to several tens of vertical wells. Well branches pass through several layers and reservoir segments separated by faults
Figur 4.2 Well Smørbukk Q-5 Y (Statoil 2009)
4.1 Flow towards long horizontal wells
4.1.2 Mathematical solution
We will assume that the well is located at the centre of the reservoir and so long that we can ignore end effects. The reservoir is homogenous and has similar characteristics in all directions (isotropic) and inflow to the wells is even along the completed interval.
Equation (4-1) below provides the complex potential applicable to fluid flow in porous medium, electrical flow, heat transfer and other potential flows.
(4-1)
x: distance along the reservoir
y: distance perpendicular to the reservoir
z: the distance vector z = x + iy
p: pressure (hydrostatic potential)
Y: flow function (constant along the flow line)
h: height of layer
C: proportionality factor
Figure 4.3 shows calculated pressure curves (isobars) and the flow lines
Figure 4.3: Flow towards the horizontal well
To estimate the pressure drop towards the wells, it is sufficient to consider pressure along the x-axis, in other words setting y = 0 in 4-1. This provides the following relationship
(4-2)
Figure 4.4 illustrates the relationship above. We see that near the wellbore area the pressure changes by the radial influx (logarithmic). At the greater the distance from the well, the pressure change is linear.
Figure 4.4 Pressure distribution along the x-axis (for: h = 10m and: rw = 0.1m)
To determine the proportionality constant: C, we consider flow distance from the well. Figure 4.3 already indicates that when x> h, the influx pressure loss is approximately linear. When x> h, the negative exponent in (4-2) is so small that it can be neglected:
From derivation, we get: ; Comparing with the linear flow:
If we assume as much influx from both sides: Q = qoBo / 2, from the relationships above we get
The natural exponential may be expressed by series ezpansion
Providing the approximation
When xw <h, we are neglecting higher order part of the ranks above, and approaching (4-2)
(4-3)
The first link in (4-3) can be interpreted as pressure by linear to flow toward the well
(4-4)
The second link in (4-3) represents the additional pressure because of radial convergence. If we consider near wellbore area, we can set: xw = rw
(4-5)
We shall soon see that these results of (4-4), (4-5) also are important in other contexts.
4.1.2 Skin
Near the well the influx in horizontal wells is, as for a vertical well. Therefore the skin will correspond to the vertical wells. For vertical wells the skin pressure is associated with the layer height, which usually corresponds to the completion length. For horizontal wells, the skin pressure is associated with the completion length: Lw
(4-6)
The skin factor is related to the flow per meter, for both vertical and horizontal wells.
4.1.3 Steady-state productivity index for long, horizontal wells
The radial late convergence will only affect the pressure in a small area near the well and therefore will only have marginal importance for the average pressure. From figure 4.3, we see that the average reservoir pressure corresponds to the reservoir pressure which is between the well and the outer boundary: xe / 2. This pressure will be expressed as
This gives the steady-state productivity index
(4-7)
4.1.3 Pseudo-steady-state productivity index for long, horizontal wells
When the reservoir is finite, fluid flow at the outer boundary will be zero. When pressure falls constantly with time, the influx between the outer boundary and well increases. This can be expressed as
(4-8)
Integration of darcy’s equation, (4-4) has given the pressure distribution for pseudo-steady-state flow
We can now get an average linear pressure loss between the wells and drainage boundary
The average pressure will then be equal to: well pressure, plus skin pressure loss, plus the inflow pressure loss because of radial convergence, plus the average linear pressure loss
This provides the pseudo stationary productivity index equation
(4-9)
4.2 Shorter horizontal wells
4.2.1 Mathematical solution
Influx into a "long horizontal well" be perpendicular to the well. Thus, convergence toward the beginning and end of the completed interval is neglected. This is correct if the well is completed through the entire length of the reservoir, but an approach if the well is shorter.
The formula of "short horizontal wells" is based on the mathematical solution for a line source in two-dimensional space (4-10). It was originally used to calculate the outtake of river water through layers of sand, illustrated Figure 4.5, to obtain usable water from polluted rivers
Figure 4.5 Outtake of river water via sand layer (Muskat/1937/)
(4-10)
LF: length of the fracture (line source)
x: direction along the cracks
y: the direction perpendicular to the crack
K: proportionality constant
The solution also describes inflow to a hydraulically fractured well, for the limiting condition of negligible pressure loss along the fracture.
Pressure distribution over the reservoir area calculated by (4-10) is illustrated below
Figure 4.6: Pressure curves and flow lines
Solution gives elliptic pressure lines around the line source. The pressure along the x-axis follows from (4-10), by setting: y = 0. Hyperbolic inverse cosine (cosh-1) can be expressed as equivalent below
(4-11)
pF: pressure in the fractures
At sufficient distance from fractures: x> LF, (4-11) gives the following approach
(4-12)
Figure 4.7 compares the pressure profiles according to the fracture solution (4-11), and the radial approach (4-12). We see that approach is acceptable when the distance from the fracture is higher than the fracture length. In other words flow is almost radial.
Figure 4.7 Comparison of pressure profile along the fracture-axis
By comparing the equation (4-12) with radial to flow solution (2-3), we can easily associate the proportionality constant to the flow variables
From well fracture solution:
By adding to the pressure due to the skin and radial convergence to fracture solutions (4-12), we can estimate the well pressure. We have already found the pressure loss due to radial convergence for long wells (4-5). Using this we get the pressure profile approximation
(4-13)
4.2.2 Productivity index for short horizontal wells
For a reservoir area that can be approximated by an equivalent radius (A=pre2), the productivity index may be approximated by re-arranging the expression above
(4-14)
The result above expresses the ratio: inflow rate/ pressure drop outer boundary to wellbore. For pseudo steady state production and areal radial flow towards the horizontal well, eq. (4-14) should ideally be modified by: -3/4, to account for the difference between pressure at outer boundary and averaged reservoir pressure, as is done for radial flow towards vertical well. This is appropriate for very short horizontal wells. However, using (4-14) as approximation for somewhat longer wells, such an adjustment may cause serious errors and is therefore usually omitted.
4.3 Productivity Index at various reservoir geometries
The figure below shows a relatively short well in an elongated drainage area. The influx of this well conflict with the requirements for "well short" and "long well."
Figure 4.8 Well in rectangular drainage area
Babu & Odeh (1989) developed a method to predict the productivity index for wells with drainage areas corresponding to fig. 4.8. The method is based on following formula
(4-15)
Where:
K: constant; for SI-units:
CH: geometry factor, function of width / height and vertical / horizontal permeability
: corrections for anisotropy, reservoir geometry, well length and location; correlations developed from numerical calculations.
Productivity index may alternatively be predicted based on source-sink solution for flow towards fractures Gringarten et al (1974). After some development the productivity index can the be expressed based on the formula for long, horizontal wells (4-9), with a correction factor: fa
(4-16)
The correction factor depends on the ratio reservoir length / width and length well / reservoir length: , illustrated below.
Figure 4.9 Correction factor, the numerical estimation of equation (4-16)
The blue lines in Figure 4.9 shows numerically calculated results, while the dashed lines show the approximation by (4-17) below.
(4-17)
Productivity index estimated by (4-16) assume well trajectory along the middle of the reservoir layer. Other trajectories can be included by skin factors developed in the next chapter
Literature
Muskat, M.: The flow of Homogenous Fluids through Porous Media
McGraw-Hill, Ann Arbor, Michigan 1937
Butler, R.: Horizontal Wells for the Recovery of Oil, Gas and Bitumen.
Calgary, 1994
Karcher, B., Gig, F.M., & Combe, J.:
"Some Practical Formula To Predict Horizontal Well Behaviour"
SPE 15430, 61. Annual Tech. Conf., New Orleans, Oct.5-8, 1986
Joshi, S.D. "Augmentation of Well Productivity With Slant and Horizontal Wells"
J. Pet. Tech., June 1988, 729
Babu, D.K. & Odeh, AS: "Productivity of a Horizontal Well"
SPE Reservoir Engineering, Nov. 1989, 417
Gringarten, A.,C.,Ramey, H.,J.,Raghavan, R.: ”Unsteady-State Pressure Distributions Created by a Well With a Single Infinite-Conductivity Vertical Fracture” SPEJ, August 1974, 347