Infinite-conductivity Fractures Research Articles

Productivity Model and Production Decline Characteristics of Multi-Fracturing Horizontal Well

By using the potential theory and the superposition principle, Productivity model of multi-fracturing horizontal well is established under the condition of infinite conductivity fractures. It solves the problem by using numerical methods and analyzes the influence of fracture numbers, fracture length and fracture spacing on well production by using actual reservoir parameters. Finally, it analyzes the production decline characteristics of multi-fracturing horizontal well based on the change of production curve. In the early period of production, the well production increases with the increase of fracture numbers, fracture length and fracture spacing. In the late period, the fracture parameters have less effect on well production. Early-stage production of multi-fracturing horizontal well obeys to the modified-hyperbolic decline model, while the relationship between ratio of Q, Qi and time is exponential model in the late-stage. It has some significance for design of horizontal well fracturing parameters and production forecasts in unconventional reservoirs.

Productivity calculation and distribution of staged multi-cluster fractured horizontal wells

Applying the potential theory and the superposition principle, the authors derived the linear productivity equations of staged multi-cluster fractured horizontal well considering finite and infinite conductivity fractures, respectively. The cluster yields and total yield were obtained by solving these equations numerically. A concrete example is used to analyze the productivity distribution of the fractured horizontal well and the influence of fracture parameters on it. When the fracture conductivities are infinite, the total productivity decreases significantly with the decrease of all fractures' half-lengths; outer fractures near the endpoints of the horizontal wells contribute most to the total productivity, and fractures on the edges contribute more than the middle ones within the same stage. When the fracture conductivities are finite, the contribution of outer fractures near the endpoints of the horizontal wells to the total productivity decreases, and the contribution of middle fractures within each stage increases significantly, but the overall distribution characteristics are in accord with the former case. Fracture conductivities have a great influence on the productivity, but they are not linearly correlated. Three stages and two clusters per stage can ensure high productivity, and during fracturing design, one should try to increase fracture half-lengths and fracture conductivities, especially for outer fractures near the endpoints of the horizontal well, and there exists an optimal value for the fracture conductivities.

A novel binomial deliverability equation for fractured gas well considering non-Darcy effects

Hydraulic fracturing is one of the most effective methods to improve the productivity of low permeability reservoirs. Now, the effects of non-Darcy flow mainly caused by high velocity gas in hydraulic fracture are paid increasing attention to. But the binomial deliverability equation influenced by non-Darcy effects has never been seen in any paper for the difficulty in coupling linear flow model in the fracture and radial flow model in the reservoir.Based on equivalent radius models for hydraulic fracture, the authors put forward the concept of CTER, which is short for conductive thickness between equivalent radii. The value of equivalent radius for finite conductivity fracture is smaller than that of infinite conductivity fracture because of the existence of fracture conductivity. CTER whose permeability is equal to reservoir permeability is a theoretical radial conductive thickness existing between equivalent radius models for finite conductivity fracture and infinite conductivity fracture. By making use of CTER, we deduce the novel binomial deliverability equation with respect to the inertial non-Darcy effects in the fracture. With the help of the new concept and the new equation, it is found that non-Darcy effect caused by high flow rate of gas in the fracture influences high permeability reservoirs (>5 mD) more dramatically than those with low permeability (<1 mD).

Pressure-Transient Behavior of Continuously and Discretely Fractured Reservoirs

SummaryFractures are common features of many well-known reservoirs. Naturally fractured reservoirs contain fractures in igneous, metamorphic, and sedimentary formations. Faults in many naturally fractured carbonate reservoirs often have high-permeability zones, and are connected to many fractures with varying conductivities. Furthermore, in many naturally fractured reservoirs, faults and fractures can be discrete (i.e., not a connected-network fracture system). New semianalytical solutions are used to understand the pressure behavior of naturally fractured reservoirs containing a network of discrete and/or connected (continuous) finite- and infinite-conductivity fractures. We present an extensive literature review of the pressure-transient behavior of fractured reservoirs. First, we show that the Warren and Root (1963) dual-porosity model is a fictitious homogeneous porous medium because it does not contain any fractures. Second, by use of the new solutions, we show that for most naturally fractured reservoirs, the Warren and Root (1963) dual-porosity model is inappropriate and fundamentally incomplete for the interpretation of pressure-transient well tests because it does not capture the behavior of these reservoirs.We examined many field well tests published in the literature. With few exceptions, none of them shows the behavior of the Warren and Root (1963) dual-porosity model. These examples exhibit very diverse pressure behaviors of discretely and continuously fractured reservoirs. Unlike the single derivative shape of the Warren and Root (1963) model, the derivatives of these examples exhibit many different flow regimes depending on fracture distribution and on their intensity and conductivity. We show these flow regimes with our new model for discretely and continuously fractured reservoirs. Most well tests published in the literature do not exhibit the Warren and Root (1963) dual-porosity reservoir-model behavior. If we interpret them by use of this dual-porosity model, then the estimated permeability, skin factor, interporosity flow coefficient (λ), and storativity ratio (ω) will not represent the actual reservoir parameters.

On the Liquid-Flow Analog To Evaluate Gas Wells Producing in Shales

Summary Drawing on links to the analog considered by Al-Hussainy et al. (1966), we present a corresponding analog to correlate solutions for a fractured well producing at a constant pressure. A solution in terms of the similarity transformation for the pressure distribution in a linear reservoir filled with a real gas provides the basis. This solution is particularly suited to demonstrate that anomalous results will be obtained when long linear-flow trends typical of shales produced through a horizontal well consisting of multiple, infinite-conductivity fractures are evaluated in classical terms. The basis for the liquid-flow analog is re-examined by considering 2D numerical solutions for a fractured well producing a gas reservoir at a constant pressure. A method to correlate the nonlinear solutions with the corresponding liquid-flow solutions for fractured wells producing at a constant pressure during the infinite-acting period is provided. The phrase “analog” used here represents attempts to match values of both the well response and its derivative for a 2D system during transient flow. This correlation enables analysts to obtain estimates that are accurate in the manner of Al-Hussainy et al. (1966). An example illustrates the application of this recommendation for a horizontal well producing a shale reservoir through multiple hydraulic fractures.

Transient Pressure Behavior of Reservoirs with Discrete Conductive Faults and Fractures

Fractures and faults are common features of many well-known reservoirs. They create traps, serve as conduits to oil and gas migration, and can behave as barriers or baffles to fluid flow. Naturally fractured reservoirs consist of fractures in igneous, metamorphic, sedimentary rocks (matrix), and formations. In most sedimentary formations both fractures and matrix contribute to flow and storage, but in igneous and metamorphic rocks only fractures contribute to flow and storage, and the matrix has almost zero permeability and porosity. In this study, we present a mesh-free semianalytical solution for pressure transient behavior in a 2D infinite reservoir containing a network of discrete and/or connected finite- and infinite-conductivity fractures. The proposed solution methodology is based on an analytical-element method and thus can be easily extended to incorporate other reservoir features such as sealing or leaky faults, domains with altered petrophysical properties (for example, fluid permeability or reservoir porosity), and complicated reservoir boundaries. It is shown that the pressure behavior of discretely fractured reservoirs is considerably different from the well-known Warren and Root dual-porosity reservoir model behavior. The pressure behavior of discretely fractured reservoirs shows many different flow regimes depending on fracture distribution, its intensity and conductivity. In some cases, they also exhibit a dual-porosity reservoir model behavior.

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 Completed With a Vertical Penny-Shaped Fracture

Summary In this work, we present a simple polynomial relationship for the effective well radius of a well completed with a vertical, infinite-conductivity, penny-shaped hydraulic fracture as a function of fracture diameter. It is based on an analytical solution for steady-state, single-phase flow to a circular, constant-pressure surface in an infinite porous medium. This solution is extended to a vertical, penny-shaped fracture in the center of a plane circular reservoir. The effective well radius increases with increasing fracture diameter to approximately 0.2 times the fracture diameter for a fracture diameter equal to the reservoir thickness. As a rule of thumb, the productivity of a well with an infinite-conductivity, penny-shaped fracture exceeds the openhole productivity for fracture diameters larger than one-third of the reservoir thickness. The productivity of a well with a fracture diameter equal to the reservoir thickness is approximately twice the openhole productivity. The adverse effect of fracture conductivity can be estimated by a simple formula that relates fracture efficiency to the productivity index of a well with an infinite-conductivity fracture and to the ratio of the flow capacity of the fracture to that of the reservoir.

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.

A simplified analytical method for estimating the productivity of a horizontal well producing at constant rate or constant pressure

In this work, we present an analytical method for estimating the productivity of an infinite-conductivity horizontal well in a closed, box-shaped reservoir producing at either a constant rate or a constant pressure. It is based on approximate analytical solutions for the 2D inflow performance of an infinite-conductivity fracture in a rectangular drainage area and the near-well pressure drop for flow to a circular well in a vertical cross-section of the reservoir. The method can accommodate reservoirs with arbitrary horizontal and vertical aspect ratios, off-center wells, permeability anisotropy and skin factors. Being simple and straightforward, the method can be readily implemented in a standard spreadsheet program. Well productivities for depletion at constant rate are always higher than at constant pressure. The difference can be as much as 22% for a fully penetrating well in a thin, isotropic reservoir.

Effect of Complex Reservoir Geometries and Completion Practices on Production Analysis

This article, written by Assistant Technology Editor Karen Bybee, contains highlights of paper SPE 111285, "Evaluation of the Effect of Complex Reservoir Geometries and Completion Practices on Production Analysis," by S.A. Cox, SPE, R.P. Sutton, SPE, and R.P. Stolz, SPE, Marathon Oil Co.; R.D. Barree, SPE, Barree &amp; Assocs.; and M.W. Conway, SPE, Stim- Lab, prepared for the 2007 SPE Eastern Regional Meeting, Lexington, Kentucky, 17–19 October. The paper has not been peer reviewed. Complex reservoir geometries can influence the results obtained from transient-production-decline analysis. For tight gas reservoirs, it is common to observe limited drainage areas and linear-flow geometries. In some cases, these results are inconsistent with the expected geological, structural, and depositional character of the reservoir. Complex reservoir geometries and flow conditions such as liquid loading can contribute to this phenomenon. Numerical-simulation cases will be used to generate the control data sets to demonstrate these effects. The full-length paper presents seven cases showing the effects of complicated reservoir flow conditions on the results obtained from production analysis. Introduction Production analysis commonly is used to evaluate completion efficiency and effective drainage area of producing wells. The technique is a type-curve-matching technique. Flowing bottomhole pressures and production rates are treated as a long-term production/drawdown test, providing an estimate of effective drain-age area, flow capacity, and effective fracture half-length through type-curve analysis. This technique, applicable for both oil and gas producers, originally was presented for a single-layer reservoir. The technique is applied to layered systems by assuming average properties for the productive zones. Production-analysis techniques are well suited for the analysis of tight gas reservoirs. Tight gas reservoirs typically stay in transient flow for an extended period of time, and therefore daily pressure and rate data can be used to analyze the transient behavior of the reservoir around the producer. Tight gas wells typically drain a limited area and often experience linear flow for an extended period of time after completion. Simulation Numerical-simulation cases were constructed to develop production and pressure profiles used for the analysis. The base case for this analysis was a fracture-stimulated radial-flow model. The reservoir geometry for the single-layer cases was a square, and for the multilayer case, the top layer was a high-permeability channel. Production rate from the models was controlled by a 350-psi constant tubing pressure with a 10,000-Mscf/D maximum rate limit. The base case was run for 2 years. Base Case The base case represents a well producing from the center of a 40-acre square reservoir. The results of this case demonstrate the ability of the production-analysis technique to match simulation results. Fig. 1 is the type-curve match for this case. It should be noted that the match assumes an infinite-conductivity fracture and that an acceptable match also was obtained by use of a finite-conductivity model with a 200-ft fracture half-length, Xf, and a 75-md-ft fracture conductivity. Logarithmic gridding was used in the simulation cases to capture the transient flow in the reservoir and eliminate numerical dispersion.

Stabilized Productivity of a Hydraulically Fractured Well Producing at Constant Pressure

Summary This paper describes a simple and easy-to-construct numerical model for the calculation of the stabilized productivity of a hydraulically fractured well producing at a constant well pressure. The model takes into account both Darcy and non-Darcy pressure losses in the fracture. Dimensionless charts are presented that illustrate productivity improvement as a function of fracture length, fracture conductivity, and non-Darcy flow. For dimensionless fracture lengths in excess of 0.2, constant-pressure productivities are significantly lower than constant-rate productivities as predicted, for example, by the McGuire-Sikora productivity improvement chart. The maximum difference is 20% for an infinite-conductivity fracture with a length of unity. Both fracture conductivity and non-Darcy flow adversely affect well productivity; the reduction in productiv-ity is larger for longer fractures.

A new method for pressure test analysis of a vertically fractured well producing commingled zones in bounded square reservoirs

Although hydraulically or naturally fractured wells located in stratified bounded reservoirs are common, reliable techniques available to analyze the pressure test data for such reservoirs are lacking. This paper presents a mathematical model that describes the pressure behavior of a vertically fractured well located in a stratified, bounded, square reservoir. The fracture can be either a uniform flux or an infinite conductivity fracture. It was found that the dimensionless pressure function and its derivative and the fractional production rate from the different layers are mainly controlled by the fracture penetration into the formation, and that transmissibility and storativity affect the fractional production rate and the pressure derivative but have little effect on the dimensionless pressure function. Type curves of dimensionless pressure and dimensionless pressure derivative can be used to evaluate the reservoir characteristics. The selection of the appropriate type curve is guided by the behavior of the layer fractional production rate obtained from flow rate survey carried out during well testing. Type curves for uniform flux and infinite conductivity fractures exhibit similar features. Two examples are presented to demonstrate the application of the new method of analysis presented in this paper.

Interpretation of stress damage on fracture conductivity

This paper focuses on the investigation of pressure responses in a well producing through a composite system of a stress-sensitive vertical symmetric or asymmetric fracture interacting with a stress-sensitive permeable-porous medium. Results of this investigation demonstrate that when the fracture conductivity is stress dependent, the use of conventional techniques to evaluate fractured wells may lead to incorrect estimates of the fracture-formation properties. For finite conductivity fractures, the stress effects in the formation have a negligible influence on the absolute value of the slope that defines the pseudo-bilinear flow regime. During this flow regime only the pressure sensitivity in the fracture and the fracture conductivity govern this flow period. For cases that involve highly conductive fractures, it was found that the slope of 0.5 which defines the linear flow regime is not influenced by the stress effect of the fracture and formation. However, stress in highly conductive fractures has a measureable influence in the change of conductivity. The occurrence of the stress damage in finite and infinite conductivity fractures is due to the partial closure fracture or partial choke fracture effect and can be anticipated in terms of pressure or time level by using the equations provided in this work. The results derived from this research have a major impact in improving the analysis of fractured wells pressure responses, as well as in forecasting of fracture closure occurrence and in programming stimualtion operations in fractured wells. An example with synthetic data is presented to illustrate these new findings.

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.

Fracturing Aspects of Horizontal Wells

Summary. This paper discusses the main reservoir engineering and fracture mechanics aspects of fracturing horizontal wells. Specifically, the paper discusses fracture orientation with respect to a horizontal wellbore, locating a horizontal well to optimize fracture height, determining the optimum number of fractures intercepting a horizontal well, and the mechanism of fluid flow into a fractured horizontal well. Introduction Interest in horizontal well drilling and completions has increased during the last few years. The significant advances in drilling and monitoring technology have made it possible to drill, guide, and monitor horizontal holes, making horizontal drilling not only possible but also consistently successful. Most wells have been completed as drainholes. These drainholes have been used in primary production and in EOR. Papers on drilling, completion, well testing, and increased production of horizontal vs. vertical wells have been presented in the petroleum literature. Many papers have dealt with steady-state production increase of horizontal wells over vertical wells Graphs and equations have been presented for calculating steady-state ratios for both fractured and unfractured wells. Ref. 2 provides a recent review of this technology. Other authors have studied the transient behavior of pressure response during a drawdown or a buildup of a drainhole. The literature lacks comprehensive studies on fracturing horizontal wells, and none of the studies cited above discussed this subject. Only Karcher et al. studied production increase caused by multiple fractures intercepting a horizontal hole. Using a numerical simulator, Karcher studied steady-state behavior of infinite-conductivity fractures. Stability of horizontal holes during drilling is another important aspect of horizontal well technology. It has been found that the degree of stability of horizontal holes depends on the relative magnitude of the three principal stresses and the orientation of the wellbore with respect to the minimum horizontal stress. Although productivity of horizontal wells could be two to five times higher than productivity of vertical wells, fracturing a horizontal well may further enhance its productivity, especially when formation permeability is low. Presence of shale streaks or low vertical permeability that impedes fluid flow in the vertical direction could make fracturing a horizontal well a necessity. This paper discusses fracturing horizontal wells from both reservoir engineering and fracture mechanics points of view. Our goal is to shed some light on important aspects of fracturing horizontal wells. Stress Magnitude and Orientation The first parameter to be determined is the fracture orientation with respect to the wellbore. Because fractures are always perpendicular to the least principal stress, the questions actually concern wellbore- and stress-orientation measurements. In what direction will induced fractures occur? What is the anticipated fracture geometry? What is the optimum length of the perforation interval? What is the optimum treatment size? What are the expected fracturing pressures? Data necessary for planning a fracturing treatment are the mechanical properties of the formation, the orientation and magnitude of the least principal stress, the variation in stresses above and below the target formaation, and the leakoff characteristics of the formation. It is commonly accepted that, at depths usually encountered in the oil field, the least principal stress is a horizontal stress. It also can be shown that the induced fracture will be oriented perpendicular to the least principal stress. The result is that a fracture created by a treatment will be in a vertical plane. If the horizontal segment is drilled in the direction of the least stress, several vertical fractures may be spaced along its axis wherever perforations are present. This spacing is one of the design parameters to be selected. Usually, this is investigated with numerical simulators. If the horizontal segment is drilled perpendicular to the least stress, one vertical fracture will be created parallel to the well. Figs. 1 and 2 show fracture direction vs. well direction. When the wellbore is not in one of these two major directions, several scenarios may occur, depending on the angle between the wellbore and the stress direction and on the perforation distribution and density. JPT P. 966⁁

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.

Elliptical Flow Equations for Vertically Fractured Gas Wells

Summary A set of elliptical equations is developed for use in understanding and evaluating vertically fractured gas wells in low-permeability gas sands. The equations are designed for use with short-term flow tests at either constant-rate or constant-pressure test conditions. Introduction Much new Rocky Mountain-area gas is being produced from tight gas sands. These wells often prove uneconomical unless they are fractured hydraulically. After the wells are fractured, testing generally is required to ensure that sustained production will justify the cost of a pipeline connection. Current practice generally involves a flow test ranging from several hours to several days. A corresponding pressure buildup often ranges from a day to a week or more. Using a well test involving hours to forecast many years' performance can be very misleading if not done with care.A basic understanding of elliptical equations can yield a great deal of useful information from short flow and buildup tests. It also helps set limits on what information can and cannot be expected from tests. Elliptical Equation The elliptical flow equations result from the basic fluid flow equations in porous media with fracture geometry. The fracture is characterized by the distance it extends from the wellbore. It extends from -Xf to +Xf for a total length of 2Xf. Gas flows from the formation into the fracture and is conducted by the fracture to the wellbore. For an infinite conductivity fracture, the inner boundary is one of uniform pressure along the fracture face. Flow may be either at a constant rate, a constant pressure, or a combination of the two.Three points on a given isobar demonstrate steadystate elliptical geometry. Two points are at a distance of +A and -A from the wellbore. The third is at a distance B from the wellbore (Fig. 1). Conformal mapping with the coordinate transformation w = arcsin (Z) is used to transform the X-Y plane into a U - V plane. The transformed system is linear with known solutions for the given conditions. At the external boundary parallel to the fracture, Ve = A′ = B' = arccosh (A) = arcsinh (B). Rearranging this equation yields Ve = ln (A + B) = -ln (A-B) or A2 - B2 = Xf2. This defines an ellipse. The steady-state isobars are concentric ellipses. A is a major semiaxis. B is a minor semiaxis. In this paper, A and B refer to the external ellipse that defines the drainage area. Axf and Bxf refer to the inner ellipse that is the fracture. Fluid flow occurs along the streamlines that are orthogonal to the ellipses. The driving force is the real gas pseudopressure. Steady-State Elliptical Flow Equations Development of steady-state equations will solve few problems but lends to an understanding of unsteady-state problems. The generalized steady-state equation is (1) Expressed in X - Y coordinates, the equation is (2a) Boundary conditions are (2b) JPT P. 2489^

Vertical Fracture Height: Effect on Transient Flow Behavior

Abstract Although a number of studies have examined the performance of vertically fractured wells, no performance of vertically fractured wells, no analytical study has examined the effect of fracture height on transient flow behavior and production capacity. The purpose of this paper is to close this gap in the knowledge about fractured well behavior. This paper presents an analytical study of the pressure behavior of a well producing at a constant pressure behavior of a well producing at a constant rate through one vertical fracture from an infinitely large reservoir with impermeable upper and lower boundaries. The fracture height is less than the formation thickness. The characteristics of the dimensionless wellbore pressure drop vs dimensionless time curves are examined for both uniform-flux and infinite-conductivity fractures. Two fracture locations (center and top) in the producing interval are considered. Curves also are presented for calculating the additional pressure drop that results because the fracture height is less than the formation thickness. Other parameters of interest examined here are (1) the ratio of vertical to horizontal permeability and (2) the ratio of fracture height to permeability and (2) the ratio of fracture height to formation thickness. The application of theoretical results to well test analysis also is discussed. Introduction Hydraulic fracturing, introduced in 1949, has provided the petroleum industry with an inexpensive provided the petroleum industry with an inexpensive way to increase the fluid production or injection capacity of wells. The success of many marginal wells can be directly attributed to hydraulic fracturing. Because of the many wells that have been hydraulically fractured, the study of the flow behavior of wells intersecting vertical, horizontal, and inclined fractures has received considerable attention. As a result of these studies, it is possible to predict and analyze pressure behavior of fractured wells and to compute production increases caused by fracture treatments. All studies on vertical and inclined fractures cited assume that the fracture extends over the entire vertical extent of the formation. Field observations, however, indicate that in Some instances this assumption is not valid. Also, the fracture height through which fluid is actually produced may not be equal to the created fracture produced may not be equal to the created fracture height. There is no discussion in the literature of the transient pressure behavior of a reservoir producing through a well with a fracture that does producing through a well with a fracture that does not extend throughout the vertical extent of the formation. This paper first examines the effect of fracture height on transient pressure behavior of a vertically fractured well producing at a constant rate. Second, we present information regarding production rate changes as a function of fracture height. Third, we delineate conditions under which it would be possible to recognize (by pressure analysis) that possible to recognize (by pressure analysis) that the thickness and fracture height are different. REMARKS ABOUT THE NOMENCLATURE In the petroleum engineering literature, the term, "a partially penetrating well," has been used to describe the situation where a well does not penetrate the entire thickness of the formation. penetrate the entire thickness of the formation. The term, "penetration ratio," has been defined as the ratio of the length of the open interval to the formation thickness. Unfortunately, this term also has been used to define the ratio of the fracture half-length to the drainage length. Among other definitions, the word "penetrate" implies "to enter or pierce; to make way into another body." In this context, using the term, "penetration ratio," to describe both situations is correct. Confusion regarding this terminology has not come up, since all studies on the pressure behavior of vertically fractured wells have assumed that the fracture extends over the entire extent of the formation. In this study, however, we need to clarify this term, even though we are examining only the pressure behavior of vertically fractured wells in an pressure behavior of vertically fractured wells in an infinite reservoir. SPEJ P. 265

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