Inflow Performance of Horizontal Wells

Abstract

Summary This paper presents formulas for evaluating the inflow performance of a horizontal well in a rectangular drainage region bounded above and below. The upper boundary may be, either sealed to flow (no flow) or at constant pressure (e.g., gas cap). The well can be placed anywhere within the drainage volume and be of any length. The inflow-performance formulas for horizontal wells presented in the literature make certain limiting assumptions about the well relative to the size of the drainage region, the formation thickness, and the well location. In addition, simplified formulas are derived that are applicable over a wide range of practical interest for a variety of reservoir geometries. Introduction As horizontal-well technology has progressed, several formulas for inflow performance have been presented in the literature.1–3 Severe restrictions, however, were placed on the geometry. Particularly, the well length has been assumed to be long compared with the formation thickness and to be short compared with the dimensions of the drainage area. Also, these formulas require the well to be in the center of the drainage volume. These restrictions are necessary because the solutions are constructed by combining two 1D flow problems. The well is treated as a fully penetrating vertical fracture in the center of a reservoir that is large enough (compared with the length of the well) for radial flow to develop around the fracture before the influence of the lateral boundaries develops. To determine the additional pressure drop that results because the well is not a fracture, the well is treated as though it were in a formation that is only as wide as the well is long (Le., the well crosses the entire width of the reservoir). Moreover, the well is assumed to be centrally located with respect to the upper, lower, and lateral boundaries. Thus, the available formulas are applicable only to very specific geometries. The inflow performance of a horizontal well can be incorrectly estimated when these formulas are used for general cases. The solution presented here is for a well producing from a rectangular region of uniform thickness and requires none of the above assumptions. The well can be placed anywhere within the drainage volume, as shown in Fig. 1, which gives all the pertinent parameters. In addition to a new formula, which gives the inflow performance for a well bounded above and below by no-flow barriers, we also present another formula applicable when the upper boundary may be treated as a constant-pressure boundary, as is the case when a gas cap exists. The constant-pressure-boundary model can also be used when the reservoir is bounded below by an aquifer if the mobility of the water in the aquifer is high compared with the mobility of the fluid in the reservoir and if the aquifer is of large extent. When the well is short compared with the dimensions of the drainage area, a simplified expression may be used. Even when the assumption of the short well is not strictly adhered to, the simple expression appears to work well. With the simplified expression, it is possible to obtain the inflow performance of a horizontal well completed in a reservoir having a geometry other than rectangular. Inflow Performance The inflow performance of a well is related to the steady-state or pseudosteady-state behavior. For a reservoir with no-flow boundaries, the difference between the average pressure of the reservoir and the wellbore pressure approaches a constant value, which is called the pseudosteady-state pressure. If the reservoir is bounded above (or below) by a constant-pressure boundary, then at long times, the difference between the pressure in the well and the pressure at the boundary will become a constant, called the steady-state pressure. When the steady- or pseudosteady-state pressure is normalized with respect to the stabilized well flow rate, it provides a measure of the pressure drawdown required to flow a unit of volume per unit time. The dimensionless pseudosteady-state pressure, pwD, for the no-flow case is defined (in any consistent units) as Equation 1 where p(t)=average reservoir pressure at time t and pw(t)=pressure in the wellbore, excluding the effects of skin. For the constant-pressure-boundary case, Equation 2 where pe=pressure at the constant -pressure boundary. The inflow performance is frequently expressed (in oilfield units) as Equation 3 where kH=kxky and J, or the PI of the well (STB/D-psi) is a direct measure of the well performance. Sm* in Eq. 3 is defined as Equation 4 where Sm, the van Everdingen4 mechanical skin, is described by Equation 5 because pwD has been made dimensionless with respect to the formation thickness, not the length of the well over which the pressure drop owing to skin occurs. p. 319–323