Analytic Two-Dimensional Models of Water Cresting Before Breakthrough for Horizontal Wells
- Francois M. Giger (Inst. Francais du Petrole)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- November 1989
- Document Type
- Journal Paper
- 409 - 416
- 1989. Society of Petroleum Engineers
- 1.6.6 Directional Drilling, 4.6 Natural Gas, 5.3.2 Multiphase Flow, 2 Well Completion, 4.1.5 Processing Equipment, 4.1.2 Separation and Treating, 4.2 Pipelines, Flowlines and Risers, 5.7.2 Recovery Factors
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Production with horizontal wells offers a new approach to reducing water-coning effects during oil production. This paper evaluates what effect water cresting under a horizontal well will have and determines the critical flow rate by fully analytic. two-dimensional (2D) methods developed in a vertical plane perpendicular to the axis of the horizontal well. Three production mechanisms are considered: lateral edge drive,, gas-cap drive, and bottomwater drive. The theory gives the critical flow rate value in each case. A major achievement of this theory is the proof of a decreasing critical flow rate value with cumulative production by the bottomwater-drive mechanism. Results are presented in graphs for quick use. Case studies exemplify a simple evaluation method to help solve practical engineering problems.
Whatever method is envisaged today for producing an oil deposit that has a contact surface with water or gas near the well, a socalled "critical" flow rate exists that cannot be exceeded without leading rapidly to the entry of water or gas into the well. In vertical producing wells, the effect is called "water coning," a frequently encountered phenomenon that introduces vital constraints on the production rate of the field and the oil recovery, Physical theory explains that this phenomenon of deformation of the water/oil contact (WOC) surface is caused by the pressure sink created by dynamic flow of oil toward the drainage system. Its effect is intuitively understood because of the higher convergence of flow stream-lines and rapid increase of the pressure gradient. For vertical wells, where flow is of the radial/circular type in first approximation, sharp gradients appear in the neighborhood of the well. Its trace is limited to a single point in a horizontal plane. For the reasons stated above, the gradients induce water to rise in the well's immediate area. Reference to the horizontal plane is justified by the fact that it extends parallel to the initial oil/water plane, following the largest dimensions of the deposit. This observation leads us to expect that a horizontal drain, open over its entire length and for which the trace on a horizontal plane is a segment of a straight line, will give a much more favorable flow in respect to both water occurrences (because the gradients are shallower) and the sweep efficiency (because of the disappearance of the individual points of the production system with vertical wells). The aim of this paper is to present analytical solutions that describe the shape of the water crest created by a horizontal drain used for producing an oil deposit bounded by a WOC surface. Actually, the rise of water below a horizontal well does not create a cone but a crest, and "water cresting" is used here to describe the phenomenon. The horizontal well is situated near the roof of the reservoir for the lateral-edge-drive and bottomwater-drive cases. The calculations provide an evaluation of the critical flow rate.
Physics of the Water/Oil Surface
Physical Schematic Representation. The diagrammatic examples studied deal with the formation of water crests when the oil contained in a reservoir stratum is produced by means of an infinite drain situated near the horizontal roof of the stratum. The flows are approximated by a piston-type flow scheme. Phenomena of capillarity are not taken into account. Displacement efficiency is assumed to be complete. The nature of the problem enables us to deal with it in the plane perpendicular to the axis of the drain. The problem is therefore a permanent and incompressible 2D fluid-flow problem in a porous medium. The velocity field therefore derives from a scalar potential, the Laplace operator of which is zero.
Expression in the Form of Equations. The state of pressure equilibrium on each side of the oil/water separation surface can be written with the assistance of the Darcy equation (Eq. 1) to take into account the dynamic pressure potential created by the flow of oil and the difference between the hydrostatic pressures in the oil and in the water resulting from the difference in their densities (Eq. 2):
By introducing the angle that the separation surface makes with the horizontal plane at one point, U, the local equation of this surface is expressed as
The problems can be solved by the hodograph method, as explained by Muskat or Kidder. Appendix A describes the resolution method.
Lateral Edge Drive
Case Description. Consider the part of the reservoir shown in cross section in Fig. 1. The horizontal well is near the top of the oil column to provide the maximum distance from the initial WOC. The aquifer is almost nonactive and the production mechanism is a symmetric lateral drive in the oil zone. As the well goes on steady production, a water crest forms below the horizontal well. The flow conditions at the borders are Neumann conditions with no flow on Line A-C to represent the impervious top of the reservoir and Eq. 3 for the free oil/water boundary. Dirichlet conditions of defined potential value are applied to the well and to the edges. We want to determine how the shape of the water crest is related to the linear flow rate.
Shape of the WOC. The general solution is obtained without any approximation by improving an approach proposed by Efros (see Appendix A). The approach provides the shape of the oil/water separation line with the following parametric expression, where 0 is the parameter:
where x=abscissa and - N-=ordinate of a moving point describing the boundary of the water crest, h=vertical distance between the top of the water crest and the top of the reservoir, and c = positive parameter discussed later. The volumetric rate of oil production from the well per unit length, q is
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