Uniform Flux Fracture Research Articles

A new semi-analytical method for calculating well productivity near discrete fractures

Highly permeable discrete fractures are often observed in sandstone and carbonate formations. However, the impacts of these fractures on the productivity index of a vertical well are poorly understood. In this paper, we present a new uniform-flux fracture solution derived by integrating an eccentric point source in a pseudo-steady state within a circular reservoir. Moreover, we establish a new model of fluid flow in a uniform-flux fracture. We then integrate the fracture solution and flow model into a semi-analytical model capable of calculating the productivity index of a vertical well near discrete fractures in a circular reservoir. Finally, we discuss the effect of discrete fractures on the productivity index in detail.Our results show that, for a single-fracture-well system, the well-fracture distance and fracture conductivity exert significant influences on the productivity index of the well. The maximum productivity index of a vertical well can be achieved if the well is drilled on a discrete fracture, and the productivity index decreases with increasing well-fracture distance and falling fracture conductivity. The effect of fracture conductivity on the productivity index of a vertical well can be ignored if the distance between the well and the fracture is greater than a certain value. At large conductivity values, the dimensionless fracture-well distance d1D has a much greater impact on the productivity index than the fracture conductivity.For a double-fracture-well system, strong interference between fractures will reduce the increment of the productivity index associated with the second fracture if two parallel fractures are very close. If the distance is larger, the effect of fractures on the productivity index can be ignored. The closest fracture to the vertical well contributes the most to the productivity. The second, further fracture contributes relatively little (less than 9% for parallel fractures and less than 25% for orthogonal fractures). This paper provides insight into factors affecting the productivity index of a discrete well-fracture system.

A Generic analytical solution for modelling pumping tests in wells intersecting fractures

The behaviour of transient flow due to pumping in fractured rocks has been studied for at least the past 80 years. Analytical solutions were proposed for solving the issue of a well intersecting and pumping from one vertical, horizontal or inclined fracture in homogeneous aquifers, but their domain of application—even if covering various fracture geometries—was restricted to isotropic or anisotropic aquifers, whose potential boundaries had to be parallel or orthogonal to the fracture direction. The issue thus remains unsolved for many field cases. For example, a well intersecting and pumping a fracture in a multilayer or a dual-porosity aquifer, where intersected fractures are not necessarily parallel or orthogonal to aquifer boundaries, where several fractures with various orientations intersect the well, or the effect of pumping not only in fractures, but also in the aquifer through the screened interval of the well.Using a mathematical demonstration, we show that integrating the well-known Theis analytical solution (Theis, 1935) along the fracture axis is identical to the equally well-known analytical solution of Gringarten et al. (1974) for a uniform-flux fracture fully penetrating a homogeneous aquifer. This result implies that any existing line- or point-source solution can be used for implementing one or more discrete fractures that are intersected by the well. Several theoretical examples are presented and discussed: a single vertical fracture in a dual-porosity aquifer or in a multi-layer system (with a partially intersecting fracture); one and two inclined fractures in a leaky-aquifer system with pumping either only from the fracture(s), or also from the aquifer between fracture(s) in the screened interval of the well. For the cases with several pumping sources, analytical solutions of flowrate contribution from each individual source (fractures and well) are presented, and the drawdown behaviour according to the length of the pumped screened interval of the well is discussed. Other advantages of this proposed generic analytical solution are also given.The application of this solution to field data should provide additional field information on fracture geometry, as well as identifying the connectivity between the pumped fractures and other aquifers.

A simple and accurate calculation method for finite-conductivity fracture

A simple, rapid and accurate method was developed to calculate the performance of a well intercepted by finite-conductivity fractures. A conductivity influence function (CIF) in Laplace domain has been proposed to build bridges between the finite-conductivity fracture and the uniform-flux fracture. Results show that CIF increases as the dimensionless time before a critical point while it keeps constant after the critical point. A long time approximate solution of CIF in the real domain has been obtained to calculate the shape factor of a finite-conductivity fracture in a bounded rectangular reservoir. Results show that our method can be used for rapid computation of finite-conductivity fractures in different boundary conditions. The method can easily accurately analyze and evaluate the performances of a well intercepted by finite-conductivity fractures.

A semianalytical approach to estimate fracture closure and formation damage of vertically fractured wells in tight gas reservoir

This paper presents a new semianalytical method to estimate the fracture closure and formation damage by formation evaluation in tight gas reservoirs. A new well interpretation model is established by assuming that the fracture half-length can be divided into N segments with different fracture conductivity and flux density. New semi-analytical solutions and type curves are developed based on this model, while sensitivity analysis is also conducted. The pressure and pressure derivative behavior show that the multi-segment fracture model exists new flow regimes which are termed first interference formation linear flow and second interference formation linear flow and can be characterized by straight line with slope m1 and m2 on the pseudo bottom-hole pressure derivative curve. The case study in field tests highlights the accuracy of this new model in the interpretation of formation and fracture characteristics compared with conventional uniform-flux fracture model, especially the effective fracture half-length. The extent of fracture closure and formation damage evaluated by this new method can be further applied to enhance tight gas recovery.

The Establishment and Evaluation of an Effective Driving Pressure System in a Rectangular Five-Spot Pattern

For rapid production decline in low-permeability reservoirs, and for pressure transmission discipline between injection and production wells, effective driving pressure system is proposed. An effective driving pressure system can supplement enough energy to layer on time to curb production decline. The calculation of pressure drop at a point in infinite rectangular five-spot pattern is simplified into pressure drop overlay of two bounded rectangular reservoirs by mirror image and inverse mirror image theory. Layer pressure caused by a well with uniform flux fracture in the center of a bounded rectangular reservoir is gained by integrating instantaneous source function. Bottomhole pressure (BHP) of a well with infinite-conductivity fracture in infinite five-spot pattern is gained by pressure overlay while injection and production wells work at the same time. The method of establishing and the model of evaluating effective driving pressure system are gained by demanding for time limits on BHP of the fractured well. Results show that the efficiency of an injection-production pressure system is in directly proportionate to reservoir thickness and permeability, producing pressure drop; is inversely proportionate to production rate and oil viscosity; and is also related to length of fractured crack, well spacing, and well draining. The bigger the production rate, the smaller the terminal well draining while other factors are given; The bigger the permeability, the bigger the production rate while other factors are given; optimizing relationship between permeability, production rate, well spacing, and well draining makes pressure system between injection and production wells stable, and an effective driving pressure system be established.

The Productivity of a Well with a Vertical Infinite-Conductivity Fracture in a Rectangular Closed Reservoir

Summary In this work, we have evaluated the productivity index (PI) of a well with a vertical, infinite-conductivity fracture in a closed rectangular reservoir for a wide range of fracture lengths, reservoir aspect ratios, and fracture eccentricities. The reported PIs are based on a rigorous analytical solution for the semi-steady-state pressure distribution created by a uniform-flux fracture in combination with Muskat’s method to convert the uniform-flux to a uniform-pressure solution. The PIs based on Earlougher’s shape factors are too optimistic. The equivalent-pressure (EP) method works very well at small fracture lengths but overestimates the PI at medium and large fracture lengths. The average-pressure (AP) method is too pessimistic at all fracture lengths.

Analysis of pressure and pressure derivative without type-curve matching: Vertically fractured wells in closed systems

This paper discusses a new technique for interpreting log-log plots of pressure and pressure derivatives of vertically fractured wells in closed systems without using type curve matching. For the uniform flux fracture case, pressure derivative plots for various x e x f ratios reveal three dominant flow regimes. During early times the flow of fluids is linear and can be identified by a straight line of slope 0.5. The linear flow line is used to calculate the half-fracture length. The infinite acting radial flow regime, which can be identified by a horizontal straight line, is dominant for x e x f > 8 . This flow regime is used to calculate permeability and skin. The third straight line, which corresponds to the pseudosteady flow regime, has a unit slope. This line is used to calculate the drainage area and shape factor. For the infinite conductivity fracture case, pressure derivative plots reveal a fourth dominant flow regime, called here the bi-radial flow. This flow regime, which can be identified by a straight line of slope 0.36, also can be used to calculate the half-fracture length and permeability.

Pressure-Derivative Type Curves, for Vertically Fractured Wells

Summary This paper presents a technique for interpreting pressure-derivative type curves for a vertically fractured well located in an infinite reservoir. These type curves can be used to match pressure test data to determine the formation permeability, the half-length of the fracture, and the nature of the fracture—i.e., uniform-flux fracture or infinite-conductivity fracture. At long producing times, the buildup type curves are identical to the drawdown type curves for both types of fractures.

Analysis of Pressure Data for Fractured Wells: The Constant-Pressure Outer Boundary

Abstract Analysis of flowing and shut-in pressure behavior of a fractured well in a developed live-spot fluid injection-production pattern is presented. An idealization of this situation, a fractured well located at the center of a constant pressure square, is discussed. Both infinite-conductivity and uniform-flux fracture cases are considered. Application of log-log and semilog methods to determine formation permeability, fracture length, and average reservoir pressure A discussed. Introduction The analysis of pressure data in fractured wells has recovered considerable attention because of the large number of wells bat have been hydraulically fractured or that intersect natural fractures. All these studies, however were restricted to wells producing from infinite reservoirs or to cases producing from infinite reservoirs or to cases where the fractured well is located in a closed reservoir. In some cases, these results were not compatible with production performance and reservoir characteristics when applied to fractured injection wells. The literature did not consider a fractured well located in a drainage area with a constant-pressure outer boundary. The most common example of such a system would be a fractured well in a developed injection-production pattern. We studied pressure behavior (drawdown, buildup, injectivity, and falloff) for a fractured well located in a region where the outer boundaries are maintained at a constant pressure. The results apply to a fractured well in a five-slot injectionproduction pattern and also should be applicable to a fractured well in a water drive reservoir. We found important differences from other systems previously reported. previously reported. We first examined drawdown behavior for a fractured well located at the center of a constant-pressure square. Both infinite-conductivity and uniform-flux solutions were considered. The drawdown solutions then were used to examine buildup behavior by applying the superposition concept. Average reservoir pressure as a function of fracture penetration ratio (ratio of drainage length to fracture length) and dimensionless time also was tabulated. This represented important new information because, as shown by Kumar and Ramey, determination of average reservoir pressure for the constant-pressure outer boundary system was not as simple as that for the closed case since fluid crossed the outer boundary in an unknown quantity during both drawdown (injection) and buildup (falloff). MATHEMATICAL MODEL This study employed the usual assumptions of a homogeneous, isotropic reservoir in the form of a rectangular drainage region completely filled with a slightly compressible fluid of constant viscosity. Pressure gradients were small everywhere and Pressure gradients were small everywhere and gravity effects were neglected. The outer boundary of the system was at constant pressure and was equal to the initial pressure of the system. The plane of the fracture was located symmetrically plane of the fracture was located symmetrically within the reservoir, parallel to one of the sides of the boundary (Fig. 1). The fracture extended throughout the vertical extent of the formation and fluid was produced only through the fracture at a constant rate. Both the uniform-flux and the infinite-conductivity fracture solutions were considered. P. 139

Determination of the Orientation of a Vertical Fracture by Interference Tests

An analytic solution for the pressure distribution around a fractured well in a homogeneous, infinite reservoir is presented in terms of tabulated functions. This solution can be used to determine fracture orientation from interference tests. Introduction The advent of costly tertiary recovery programs has resulted in the need to determine flow patterns created by fractures intersecting wellbores. This requires a knowledge of the compass orientation and length of fractures. Several papers have discussed determination of fracture length by pressure analysis. However, methods for determining the compass orientation using pressure data have not received much attention. This paper pressure data have not received much attention. This paper discusses a method for determining the compass orientation using pressure data obtained at shut-in observation wells due to pressure data obtained at shut-in observation wells due to production or injection at the fractured well. production or injection at the fractured well. In 1960 Elkins and Skov reported a method for determining the orientation of natural fractures. They assumed the system of natural fractures to behave like an anisotropic system and used the classical line-source solution to analyze, pressure behavior. More recently, Pierce et al. demonstrated the use of pulse testing for calculating fracture length and orientation. Other studies on the determination of fracture orientation include (1) use of acoustic signals; (2) measurement of pressure response at observation wells during fracture initiation and propagation; and (3) use of inflation impression packers and television cameras.Even though the problem has been recognized for quite some time, no general method exists for determining pressure distribution in the area drained by a vertically fractured well. The objective of this paper is to present an analytic expression, in terms of simple tabulated functions, for the pressure distribution in a uniform, homogeneous, infinite reservoir drained by a vertically fractured well. This solution may be used to analyze pressure data in adjacent observation wells to determine fracture orientation in a manner analogous to standard interference tests. It also may be used to analyze test data obtained by pulsing a vertically fractured well. pulsing a vertically fractured well. Theory The mathematical model considered in this study is identical to the uniform-flux fracture examined by Gringarten et al. We assume that a single vertical fracture intersects a wellbore located in an infinite, homogeneous, porous medium (Fig. 1). The surface production rate is assumed to be constant and all the production is obtained by means of the fracture. Fluid enters production is obtained by means of the fracture. Fluid enters the fracture at the same rate per unit area of the fracture (uniform-flux fracture). JPT P. 73