Injectivity of Completions for Soft Formations
A. Settari, TAURUS Reservoir Solutions Ltd., Calgary (updated 01/05/01)
Introduction
Part of the work in Task 3 is to investigate two aspects of various completion configurations for soft formations.
a)Choice of the best completion method for a specific reservoir, and
b)the injectivity of the specific completion configuration.
The first task was addressed in the Completions Workshop and resulted in a Completion Selection Tool which is essentially a “knowledge base” of the Sponsors and accounts for factors such as sand production, screen plugging, operational considerations, etc. This report deals with some aspects of the calculation/estimation of the injectivity of various completions, assuming that there is no mechanical or chemical damage or plugging due to PWRI. This will establish a baseline for comparison, on which the effects specific to PW need to be superimposed.
On the whole, it is believed that the completion selection criteria will override injectivity considerations, which will be therefore secondary to completion choice. Accordingly, the various calculations are not treated in detail, except for some aspects that are not generally recognized.
1.Injectivity of Different Well Completions in Matrix mode
1.1Openhole Completion
This is the reference case. The most convenient way of expressing the effect of other completions is to compare their injectivity to an openhole completion. The injectivity can be calculated from radial flow equations. Note that assumptions about boundaries must be made; the usual case is to assume a constant pressure outside boundary. As a consequence, the absolute value of the injectivity depends slightly on the outside radius, re. For liquid, single phase, steady-state flow, this is the familiar equation (e.g., Pucknell and Mason, 1992).
P = 141.2 qLL BL [ln(re/rw) + S + D qL] /(kR hR) (1)
where P is the drawdown (Pe – Pwf), qL is the rate, BL is the fluid formation volume factor (FVF), re and rw are the outside and wellbore radii, S is the mechanical skin, D is the non-Darcy skin, kR is the reservoir permeability and hR is the reservoir thickness. The constant 141.2 applies for metric units (qL in m3/d, L in cp, kR in md, hR in m). For an openhole completion, the only component of mechanical skin is due to permeability damage or enhancement (no perforations), denoted by Sk.
If the area of the injection well has not been completely waterflooded to the external radius, multiphase injectivity estimates can be obtained by simulation or from analytical extension of the above equation for a radially composite mobility system. Note that in this case the injectivity varies with time.
1.2 Openhole Screens and Slotted Liners
The additional pressure drop in wire wrap screens is probably small (without plugging) although at extremely high rates there may be entrance effects and possibly turbulence. Generally, the pressure drop across the screen is small and the screen does not alter the radial flow pattern in the formation. Asadi and Penney (2000) measurements give pressure drop of 0.05-0.1 psia at an injection rate of 6 – 18 BPD/ft of length. Figure 1 shows their data extrapolated to higher rates using the assumption that the pressure drop is proportional to velocity squared.
Figure 1.Pressure drop through clean screens extrapolated from data of Asadi and Penny (2000)
Therefore, openhole screen completions should be similar in terms of Injectivity Index to openhole completions except at extremely high rates. It must be stressed that the results on Figure 1 are extrapolated; operator experience suggests that the screen pressure drop is not nearly quadratic. Also, the screens tested by Asadi and Penny are not typical wire mesh screens which would have lower resistance.
Slotted liners have also small pressure drop across the liner (when clean), but cause convergent flow to the slot in the formation, which is the largest contribution to slotted liner skin (also called the slot factor). The skin as a function of slot open area or density, and width is shown in Figure 2 (Figures 3 and 4 from Kaiser et al., 2000).
Figure 2.Skin factors for slotted screens.
Another factor increasing the skin is present if there are sections of the liner without slots. If the area ratio of the slotted sections to entire liner area is B, this effect can be estimated (according to Kaiser et al., 2000) by a “partial coverage” skin of
SB = ln(re/rw)(1-B)/B
Manufacturers of screens and liners should also provide data on screen skin, Sscr, and then Equation (1) can be used with S = Sk + Sscr.
1.3 Cased and Perforated Completion
For perforated completions, the mechanical skin is due to the combination of the geometry of the perforations and any previous damage (assumed to be radial). In addition, matrix acidizing after completion will contribute a negative component, primarily due to permeability enhancement around perforations.
Extensive literature exists for predicting the perforation skin Sp from different perforating geometries, in both laminar and turbulent flow. This includes finite element simulations by Tariq (1987), semi-analytical methods for laminar and turbulent flow (Karakas and Tariq, 1988; McLeod, 1982) and finite difference near-wellbore reservoir simulation (Behie and Settari, 1993). Typically, the results are expressed in terms of the productivity ratio, PR, defined as the productivity of the perforated completion divided by the productivity of an openhole, single phase, undamaged completion. This concept can be applied directly to injection wells. If the flow rate in the actual completion is q and the corresponding pressure drop is P = Pe – Pw, the injectivity ratio IR is given by:
(3)
where rb is the outside radius of the model which was used to generate the result (not necessarily the same as re). C is a conversion constant. For field units, if q is in bbls/d, k in md, h in ft and in cp, C=1866.9; for metric units, if q is in m3/d, k in md, h in m and in cp, C=149.19. Typically the IR is correlated with perforation phasing, shots per foot (spf) as well as perforation length and diameter. The majority of this data is for single phase flow (without or with turbulence). For multiphase flow it is recommended that the results be obtained by simulation (Behie and Settari, 1993) because the analytical techniques (Perez and Kelkar, 1988) are too simplified.
For laminar flow, IR is independent of rate. The effect of reservoir turbulence is theoretically dependent on rate. Numerical work (Behie and Settari, 1993) showed that the reduction of IR due to reservoir turbulence in liquid flow is small for permeabilities up to 800 md. This is shown in Figure 3 for a perforating pattern with 0° phasing, 4 spf, fluid viscosity 0.7 cP, perforation diameter rp = 0.4 inch and wellbore diameter rw = 0.5 ft. Much larger effects are possible in gas injection.
Figure 3.An example of the effect of reservoir turbulence on cased hole injectivity for clean empty perforations.
As a final note, the perforation program evaluated in Figure 3 would leave a positive skin compared to openhole. Large shot density and tunnel length are needed to achieve or exceed an openhole Injectivity Index. The equivalent skin corresponding to a given IR is
S = ln(re/rw) (1- IR)/IR(3)
The data on Figure 1 translates into skins of approximately 1 to 4.
Filled or Collapsed Perforations
The above calculations apply for clean perforations, i.e., when the perforation tunnel remains empty. In PWRI injection, there is a possibility of solids accumulation in the perf tunnel itself. Since the solids are very fine, the permeability in the tunnel can be potentially very small. In addition, in soft formation, there is a possibility of the perforation collapse if formation failure is reached. The collapsed region may be dilated and therefore have a higher porosity and permeability than the formation. However, there may be a compacted lower permeability zone around the dilated zone. These zones will probably have a different shape from the original tunnel.
The IR of filled or collapsed perforations can be significantly lower compared to clean perforations. Calculations were done in this project using the model described in Behie and Settari (1993) with the perforation filled by a material with different permeability. The results are shown in Figure 4 for laminar flow, for permeability of 438 md, porosity of 25% and perforation diameter of 0.4 inch.
Figure 4.Injectivity of perforated completion with filled perforations.
Note that the (standard) case of the empty perforation is obtained as a limiting case of kp > k.
In addition, turbulence in the perforation can now play a significant role. The turbulence can be also included in the simulations using the Behie and Settari model. Such calculations have been performed for the empty as well as filled perforations for the same data as for Figure 4 (k=438 md). The results are best expresses as a ratio of the IR with turbulence to IR obtained without turbulence. The results are shown in Figure 5.
Figure 5.Effect of turbulence on injectivity of perforated completion with filled or collapsed perforations, k=438 md.
The total IR for a filled perforation is then obtained by taking the value from Figure 4 and multiplying it by the factor from Figure 5. This can lead to significant skin factors. For example, taking kp=k, IR= 0.32 x 0.3 = 0.096 and from Equation (3), taking ln(re/rw) ~8, S= 75.
1.4 Cased Hole Screens
The addition of the screen introduces additional pressure drop as discussed above for openhole screens. For clean screens, the completion PI should be close to the same completion without a screen.
1.5 Gravel Packed Cased Hole
This case has been treated in detail by Pucknell and Mason (1992). A cased gravel packed completion has poorer injectivity than the alternatives. Additional pressure drops result from the gravel pack layer between the screen and casing, and from the gravel packing the perforation tunnel itself. The latter is expected to be more significant.
The resistance between the screen and the casing can be expressed as laminar and non-Darcy skin contributions, Ss and Ds:
Ss = (kR/kgr) ln ( rc/rs)(4)
where kgr is the gravel pack permeability, rc is the inner radius of the casing and rs the outer radius of the screen.
Ds = Dconstgr (1/rs – 1/rc) (5)
where:
gr is the turbulence factor of the gravel and
Dcnst = 1.02 x 10-14 B kR hR /(heff2)
is the constant term in Equation (5). The height heff is not explained in Pucknell and Mason (1992), but it is understood to be an effective height, which corrects the radial flow equation for the converging nature of the flow. A more accurate method accounting for the convergence of the flow towards the perforation tunnels is given by Yildiz and Langlinais (1991). The turbulence factor can be correlated with gravel permeability in the same manner as for fracturing proppants and turbulence measurements for proppants can be used to estimate gr.
The effect of gravel permeability in the perforation tunnel is the largest factor reducing gravel pack injectivity. It can be expressed in terms of an “effective length” of a perforation, Lpe, defined as the length of a perforation without the gravel pack, which would have the same IR as the actual gravel packed perforation with a length Lp. Numerical solutions with the simulator described by Behie and Settari (1993) were used by Pucknell and Mason to correlate the ratio Lpe/Lp with (kgr/kR)(rp/Lp)3/2. This correlation is shown in Figure 6. Once Lpe is known, the standard methods for empty perforations can be used to calculate the perforation skin.
The effect of turbulence in the gravel packed tunnel can be evaluated by the same numerical model, but to our knowledge, there is no correlation currently available.
The above method can be used for “intact” perforations. A different calculation method was developed by Pucknell and Mason (1992) for “collapsed” perforations. They approximate the collapsed geometry by hemispheres filled with gravel, with size determined from the perforation geometry. The total pressure drop is determined using radial flow in the reservoir up to the envelope of the hemispheres and then using hemispherical flow from the outside radius of the hemisphere to the radius of the perforation.
Figure 6.Effect of gravel packing on the effective perforation length.
1.6 Propped Fracture (Injection Below Fracture Pressure)
It may be difficult to operate this type of completion, because the injection pressure must be kept below the current fracture pressure at all times. This means that either sophisticated prediction and monitoring tools should be employed, or a sufficient safety margin in injection pressure must be maintained, which reduces achievable injection rates.
However, a propped fracture completion in a cased and perforated hole can be common in converted producers.
For long-term injectivity estimates, the effect of a fracture can be expressed by a fracture skin Sf in the radial flow equation, or converted to an equivalent wellbore radius rwe according to:
rwe = rw e-Sf (6)
For an infinite conductivity fracture, the well known result is rwe = Lf/ 2, which gives a skin
Sf = ln (2 rw/Lf)(7)
However, since PWRI often occurs in high permeability formations, the conductivity of the fracture may not be infinite. In that case, Equation. (7) overpredicts injectivity. A simple adjustment can be made by using the result of Cinco-Ley and Samaniego for finite conductivity fractures (Cinco-Ley and Samaniego, 1981), which gives rwe/Lf as a function of dimensionless fracture conductivity FcD
FcD = kf bf/(kR Lf), (8)
where kf is the fracture permeability and bf is the fracture width (assumed constant along the length). The function rwe/Lf = f(FcD) is shown in Figure 7. As a rule of thumb, fractures with an FcD greater than 10 can be considered to be infinite-acting (i.e., of infinite conductivity).
Figure 7.Correction to the infinite conductivity fracture skin calculation.
Note that all of these results assume a vertical fracture with the same height as the reservoir pay, in a homogeneous reservoir and in single phase flow. Also, these methods cannot be used to look at short-term (transient) data. Finally, the method of Figure 7 - for correcting the fracture skin for finite conductivity of the fracture - becomes unreliable when the fracture (proppant) and reservoir permeabilities are of the same order of magnitude. This is easily seen by considering the case of kf = kR in which the skin should be zero regardless of the value of FcD ( = bf/Lf in this case). In such cases, numerical modeling should be employed.
Effect of turbulent flow
In gas wells, turbulence can reduce the benefits of fracturing to the extent that a fractured well only achieves the performance of an unfractured well without turbulence (the concept of “neutral skin”; refer to Stark et al., 1998 ). Turbulence is usually thought to have a negligible effect for liquid flow. However, careful analysis shows that it can be significant at high rates, as shown for production wells in Bale et al. (1994).
A general correlation for predicting the reduction in injectivity due to turbulence has been developed recently by TAURUS and Statoil (Bale, 1999) and made available to the PWRI project. The correlation is based on a large database of fine-grid, accurate solutions of steady-state, single phase flow. The final correlation is quite complex and it is expressed as:
IR = (II)turb / (II)noturb = f(QD, FcD, Fc, kR) (9)
where QD is a dimensionless injection rate and Fc = kfbf is fracture conductivity.
An example of the effect of turbulence for data typical of Ewing Bank FracPack completions is shown in Figure 8. The case is based on Marathon's presentation at the Soft Formations Workshop (Angel, 1999) and the data used are as follows:
Proppant...... 20/40 Econoprop or frac sand,
Proppant Diameter...... 0.0252 inches,
Proppant Porosity...... 0.4,
Proppant Density...... 165.3 lbm/ft3,
Proppant Permeability (Stressed)...... 180,000 mD * 0.6 (gel reduction) = 108,000 mD,
Fracture Turbulence Factor, ...... 32789.00 (1/psia),
Fracture Geometry...... penny-shaped (approximated by a square),
Reservoir Height...... 100 ft,
Reservoir Permeability...... 1600 md
Effective Fluid Viscosity.....0.5 cP (corresponding to an injection temperature of 60-80°F),
Injection Rates...... from 5,000 to 35,000 BPD (50 – 350 BPD/ft of formation).
Based on our experience with fracturing design, it is difficult to achieve average proppant coverage of more than 3-4 lb/ft2. Results on Figure 8 indicate that at 2 lb/ft2, turbulence can account for up to 11% reduction in the Injectivity Index, while at 3 lb/ft2 the effect decreases to 5% reduction. However, more serious effects are present in transversely fractured high angle wells (Settari, 2000).
Finally, it should be noted that the correlations used in the above example have been developed for a permeability range of 20 – 200 md. For higher permeability (as in the example data), the actual permeability value was used for calculating the dimensionless groups, but the correlating functions for 200 md were used. Extension for high permeability sands is possible.
Figure 8.Effect of turbulence on typical PWRI Fracpack completions.
2. Injectivity in fracture mode
Induced (dynamic) fractures are commonly propagated in PWRI. Equivalent fracture permeability, from the theory of laminar flow between parallel plates, is kf = b2/12; this estimate is usually too high by orders of magnitude due to roughness, tortuosity, etc. ,Even then, the conductivity of an open fracture is usually high enough to be infinite-acting.
For a given fracture length, estimates for a propped fracture with infinite conductivity could be used to calculate skin. However, this value will be misleading for two reasons:
a)induced fractures are growing and a true steady state Injectivity Index does not exist unless the fracture stabilizes in length
b)The standard definition of Injectivity Index is not applicable for a well with a propagating fracture.
2.1 How To Measure Injectivity For A Well With Dynamic Fracture?
The relation between bottomhole pressure and rate below fracture pressure (including a well with propped or acid fracture) is a function of completion geometry and reservoir properties. One can define the true Injectivity Index as:
II = dQ/dpwf = (Q1 – Q2) / (pwf1 – pwf2) (10)
where Q1 and Q2 are the stabilized rates corresponding to flowing pressures pwf1 and pwf2. This definition is the analog of the definition below fracture pressure.
However, if we use this II to compute the expected rate for a “drawdown” (pwf – pR) the standard equation
Q = II (pwf – pR) (11)
clearly does not give the correct answer. Conversely, one could interpret II from the long-term p and Q data by simply applying Equation (11) and that is how much of the PWRI data is analyzed. However, the II computed from Eqn. (11) is not independent of rate.