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# Estimating horizontal well productivity

There are two fundamental problems that make accurately estimating the productivity of a horizontal well more difficult than estimating the productivity of a vertical well. The theoretical models available have a number of simplifying assumptions and the data required for even these simplified models are not likely to be available. Still, we must make estimates and decisions based on those estimates. In this page, two productivity models that have proved useful in practice are discussed. The first, published by Babu and Odeh in 1989, is limited to single-horizontal wells. The second, published by Economides, Brand, and Frick in 1996, is more general and is useful for multilateral wells.

## Babu-Odeh productivity model

Babu and Odeh obtained a rigorous solution to the diffusivity equation for a well in a box-shaped reservoir, subject to certain limiting assumptions. The assumptions include the following:

• Fluid flows to the well uniformly at all points along the wellbore (uniform flux) and the well is completed uniformly.
• The sides of the drainage volume are aligned with the principal permeability direction.
• The wellbore is parallel to the sides of the drainage area and is oriented parallel to one direction of principal permeability and perpendicular to the other two.
• The boundaries of the reservoir are all no-flow boundaries and the well reaches stabilized, pseudosteady-state flow.
• The formation damage around the wellbore is uniform at all points along the wellbore.

Fig. 1 introduces the nomenclature in the Babu and Odeh solution. The solution is quite complex but is approximated accurately with an equation written in the same form as the pseudosteady-state flow equation for a vertical oil well producing a single-phase, slightly incompressible liquid. ....................(1) ....................(2)

Table 1 gives equations to estimate CH and sp. Two examples adapted from Babu and Odeh illustrate the application of these equations.

### Example 1

A horizontal well 1,000 ft long (Lw) is drilled in a box-shaped drainage volume 4,000 ft long (aH), 2,000 ft wide (bH), and 100 ft thick (h). The well is off-center in the y -direction (parallel to the well), so dy = 250 ft and Dy = 750 ft. The well is also off-center in the x-direction so that dx = 1,000 ft and Dx = 3,000 ft. Finally, the well is centered in the z-direction so that dz = Dz = 50 ft. The wellbore radius is 0.25 ft; kx = ky = 200 md and kz = 50 md. Fluid properties are Bo = 1.25 RB/STB and μ = 1 cp. Calculate the productivity index.

Solution. From Eq. A-38 in Table 1,

Thus, use Case 1 equations (Eqs. A-26 through A-31 in Table 1) to calculate sp.

To calculate pxy, determine ym, Lw/2bH, (4ymLw)/2bH, and (4ym + Lw)/2bH.

Thus,

Then,

Then, sp = pxyz + pxy =4.50+6.54=11.0, and

### Example 2

A horizontal well is drilled in a box-shaped reservoir with the following characteristics: Lw = 1,000 ft, aH = 2,000 ft long, bH = 4,000 ft wide, and hw = 2,000 ft thick. The well is off-center in the y-direction (dy = 1,000 ft; Dy = 2,000 ft), centered in the x-direction (dx = Dx = 1,000 ft), and off-center in the z-direction (dz = 50 ft; Dz = 150 ft). Permeabilities are kx = ky =100 md and kz = 20 md. Wellbore radius is 0.25 ft, Bo = 1.25 RB/STB, μ = 1 cp, and sd = 0. Find the productivity index, J.

Solution. From Eq. A-38 (Table 1),

Note that

Thus, use Case 2 equations to calculate sp.

To calculate py, determine ym. From Eq. A-29 (Table 1),

From Eq. A-35 (Table 1),

Thus, sp = 16.79 + 7.90 + 7.02 = 31.7. Then, from Eq. A-37 (Table 1),

## Economides et al. method

Economides et al. presented a more general method to estimate productivity index for a horizontal well. The method has the advantage that it is applicable to multilateral wells in the same plane and is not limited to wells aligned with principal permeabilities. It includes solutions for wells with no pressure drop in the wellbore (infinite conductivity, as opposed to wells with uniform flux). It has the disadvantage that it requires interpolation in a table in which only certain drainage area shapes are given.

The basic working equation for the productivity index in this method is ....................(3)

where Σs refers to damage skin, turbulence, and other pseudoskin factors. In Eq. 3, ....................(4)

where ....................(5)

and se, describing eccentricity effects in the vertical direction, is ....................(6)

se = 0 when a well is centered in the vertical plane. This convergence skin differs only slightly from that used by Babu and Odeh. The difference is 0.25 ln (kx / kz) + h / Lw[2dz / h - 1/2(2dz / h)2 - 2/3], which is usually small (< 0.5). Table 2 gives values of CH for several drainage areas and multilateral configurations. The equations as written are for isotropic reservoirs. Certain variable transformations are required before substituting into the working equation: ....................(7) ....................(8)

where ....................(9)

and ....................(10)

ϕ is the azimuth of the well trajectory (relative to the y-axis). Reservoir dimensions: ....................(11) ....................(12) ....................(13)

and ....................(14)

Two examples, one from an isotropic reservoir and one from an anisotropic reservoir, illustrate this method.

### Example 3

Economides et al. provide this example. Consider a horizontal well 1,500 ft long in a reservoir with bH = 2,000 ft, aH = 4,000 ft, h = 20 ft, rw = 0.4, kx = ky = kz = 10 md, Bo = 1.25 RB/STB, and μ = 1 cp. Assume that the well is centered vertically so that se = 0. Also, assume Σs = 0.

Solution. From Eq. 6,

(As a matter of interest, the Babu and Odeh sc for this case is also 2.07.) From Table 2, for 2bH = aH and Lw/bH = 1,500/2,000 = 0.75, CH = 2.53. From Eq. 4,

Then, from Eq. 3,

### Example 4

Rework the Babu-Odeh Example 2 using the Economides et al. method.

Solution. First transform the variables. From Eqs. 9 and 10,

Because the well is parallel to the x-axis, ϕ = 0, and

From Eq. 14,

From Eq. 7,

From Eq. 8,

From Eqs. 11 through 13,

Thus, the equivalent system is a rectangular-shaped drainage area twice as long parallel to the wellbore (3,050 ft) as perpendicular, with L′/bH ′ = 765/3,058 = 0.25. In the original example, the well was off-center in the horizontal plane; here, assume that a centered well is an adequate approximation. From Table 2, CH = 3.19.

From Eq. 6,

Then,

(The Babu-Odeh sc is 5.60 for this case.) Then, from Eq. 4,

Finally, from Eq. 3,

The result is slightly larger than the result using the Babu-Odeh method (J = 25.6 STB/D/psi). At least part of the reason for the difference is that, in this example, it was necessary to assume that the well was centered in its drainage volume, which was not true in the original example. The optimal location of a horizontal well to maximize productivity is to center it within its drainage volume.

## Comparison of recent and older horizontal well models

Ozkan compared "contemporary" (generally 1990s) and "conventional" horizontal well models in a paper published in 2001. He pointed out that the older models are used for both pressure-transient test analysis and for estimating well productivity. Ozkan stressed three limitations of the conventional models, which include the Babu-Odeh model and other pioneering work.

Conventional models usually assume that the horizontal well is parallel to one of the principal permeability directions (preferably the minimum permeability direction in the horizontal plane). In many cases, this is not true. In fact, in many cases the principal permeability directions are unknown. When the principal permeability directions are known, corrections to length are possible (as in the Economides et al. model); if they are not known, there is no way to correct the analysis. Contemporary models show that the error in permeability estimates approaches 50% when the deviation angle exceeds 50°. Unfortunately, the models also indicate that there is nothing in a well’s response that provides any indication that the assumption that the well is parallel to a principal permeability direction is incorrect.

Ozkan pointed out that the damaged region around a horizontal well probably is nonuniform with distance (perhaps with the greatest damage near the heel of the well and the least near the toe, because filtrate invasion is of much longer duration near the heel). If there is variable permeability along the path of the well, the situation is even more complicated. Some contemporary models can take this variation into account; however, most conventional models cannot. Conventional models usually assume (implicitly) uniform skin effect along the wellbore. However, the contemporary models will not be helpful if the skin distribution along the length is unknown.

Ozkan notes that it is a common practice to complete horizontal wells selectively. Also, in other cases, some segment of the well may not be open to flow of reservoir fluids because of relatively low permeabilities or relatively large local skin effects. The absolute amount of the well that is open to flow and the location of the open intervals affect the pressure response in the well. Some contemporary well models can take these effects into account, but, again, the capabilities of the newer models may be limited if the location and length of the open intervals is unknown.

Many models assume negligible pressure drop in the wellbore (infinite conductivity). Others assume the same flow rate per unit length at all points along the well bore (uniform flux). In fact, there is likely to be finite pressure drop in the wellbore, resulting in neither uniform flow nor infinite conductivity. Contemporary models in which a reservoir model is coupled to a wellbore model can take these effects into account.

Unfortunately, contemporary horizontal well models have not led to simple, easily applied methods of well-test analysis or of predicting well productivity. Further, their full utility depends on availability of detailed well and reservoir description data. At present, the major use of such models may be to quantify the possible errors that arise from uncertainty and to be used to history-match observed information when sufficient data are available.

## Nomenclature

 aH = total width of reservoir perpendicular to the wellbore, ft aH′ = modified total width of reservoir perpendicular to the wellbore, ft bH = length in direction parallel to wellbore, ft bH′ = modified length in direction parallel to wellbore, ft B = formation volume factor, res vol/surface vol dx = shortest distance between horizontal well and x boundary, ft dy = shortest distance between tip of horizontal well and y boundary, ft dz = shortest distance between horizontal well and z boundary, ft h = net formation thickness, ft J = productivity index, STB/D, psi = average permeability, md kx = permeability in x-direction, md ky = permeability in y-direction, md kz = permeability in z-direction, md Lw = completed length of horizontal well, ft = volumetric average or static drainage-area pressure, psi pw = BHP in wellbore, psi pwf = flowing BHP, psi pxy = parameter in horizontal well analysis equations pxyz = parameter in horizontal well analysis equations py = parameter in horizontal well analysis equations pD = 0.00708 kh(pi – p)/qBμ, dimensionless pressure as defined for constant-rate production rsp = radius of source or inner boundary of spherical flow pattern, ft rw = wellbore radius, ft sc = convergence skin, dimensionless sd = skin caused by formation damage, dimensionless se = skin caused by eccentric effects, dimensionless sp = skin resulting from an incompletely perforated interval, dimensionless α = exponent in deliverability equation α = parameter characteristic of system geometry in dual-porosity system β = turbulence factor β′ = transition parameter γm = matrix density μ = viscosity, cp μw = water viscosity, cp = viscosity evaluated at , cp ϕ = porosity, dimensionless