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Keywords: vertical fracture

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Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*22 (06): 933–944.

Paper Number: SPE-9346-PA

Published: 01 December 1982

...Naelah A. Mousli; Rajagopal Raghavan; Heber Cinco-Ley; Fernando Samaniego-V. This paper reviews pressure behavior at an observation well intercepted by a

**vertical****fracture**. The active well was assumed either unfractured or intercepted by a fracture parallel to the fracture at the observation well...
Abstract

This paper reviews pressure behavior at an observation well intercepted by a vertical fracture. The active well was assumed either unfractured or intercepted by a fracture parallel to the fracture at the observation well. We show that a vertical fracture at the observation well has a significant influence on the pressure response at that well, and therefore wellbore conditions at the observation well must be considered. New type curves presented can be used to determine the compass orientation of the fracture plane at the observation well. Conditions are delineated under which the fracture at the observation well may influence an interference test. This information should be useful in designing and analyzing tests. The pressure response curve at the observation well has no characteristic features that will reveal the existence of a fracture. The existence of the fracture would have to be known a priori or from independent measurements such as single-well tests. Introduction In this work, we examine interference test data for the influence of a vertical fracture located at the observation well. All studies on the subject of interference testing have been directed toward understanding the effects of reservoir heterogeneity or wellbore conditions at the active (flowing) well. Several correspondents suggested our study because many field tests are conducted when the observation well is fractured. They also indicated that it is not uncommon for both wells (active and observation) to be fractured. To the best of our knowledge, this is the first study to examine the influence of a vertical fracture at the observation well on interference test data. Two conditions at the active well are examined: an active well that is unfractured (plane radial flow) and an active well that intercepts a vertical fracture parallel to the fracture at the observation well. The parameters of interest include effects of the distance between the two wells, compass orientation of the fracture plane with respect to the line joining the two wellbores, and the ratio of the fracture lengths at the active and observation wells if both wells are fractured. The results given here should enable the analyst to interpret the pressure response at the fractured observation well. to interpret the pressure response when both the active and the observation wells are fractured to design tests to account for the existence of a fracture at one or both wells, and to determine quantitatively the orientation and/or length of the fracture at an observation well. We also show that one should not assume a priori that the effect of a fracture on the observation well response will be similar to that of a concentric skin region around the wellbore-i.e., idealizations to incorporate the existence of the fracture, such as the effective wellbore radius concept, may not be applicable. Mathematical Model and Assumptions In this study, we consider the flow of a slightly compressible fluid of constant viscosity in a uniform and homogeneous porous medium of infinite extent. Fluid is produced at a constant surface rate at the active well. Wellbore storage effects are assumed negligible because the main objective of our work is to demonstrate the influence of the fractures. However, note that wellbore storage effects may mask the early-time response at the observation well. Refs. 1 and 2 discuss the influence of wellbore storage on interference test data. We obtained the solutions to the problems considered here by the method of sources and sinks. The fracture at the observation well was assumed to be a plane source of infinite conductivity. SPEJ P. 933^

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*22 (05): 681–698.

Paper Number: SPE-8281-PA

Published: 01 October 1982

...K.H. Guppy; H. Cinco-Ley; H.J. Ramey, Jr.; F. Samaniego-V. Several methods have been proposed in the literature for analyzing drawdown data for the determination of fracture conductivity of

**vertically****fractured**wells. These techniques have paved accurate, but in some cases the fracture conductivity...
Abstract

Several methods have been proposed in the literature for analyzing drawdown data for the determination of fracture conductivity of vertically fractured wells. These techniques have paved accurate, but in some cases the fracture conductivity calculated is much smaller than anticipated. This study shows that producing fractured wells at high flow rates will cause nondarcy effects in the fracture, resulting in a pessimistic fracture conductivity.Numerical and semianalytical models were developed to analyze the unsteady flow behavior of finite conductivity fractures producing at high flow rates. Two methods are presented for determining the true fracture conductivity when drawdown data are available at two different flow rates. The amount of turbulent effects also is quantified by the techniques. Examples are presented to illustrate the solution methods. Introduction The increasing use of hydraulic fracturing as a means of improving the productivity of oil and gas wells in low-permeability formations has resulted in many research efforts aimed at increasing fracturing capabilities as well as evaluating the characteristics of the fracture in the postfracturing period. With the advent of the massive postfracturing period. With the advent of the massive hydraulic fracturing (MHF) treatment in recent years, the need for new solutions for evaluating these systems has increased. The problem with the older solutions was the need for many assumptions to arrive at a simple solution. One of the more common assumptions made in these systems was the use of linear flow to describe the flow within the fracture. In gas wells with finite-conductivity fractures producing at high flow rates, the non-Darcy effect is created within the fracture. Hence, new solutions must be developed for these systems. The objective of this paper is to present a new semianalytical solution to this problem that can be applied both to the linear and to the nondarcy flow regimes within the fracture.Over the years. several methods have been developed to analyze postfracture data. Gringarien et al. first solved the fracture system analytically for three special cases: infinite-conductivity vertical fracture, uniform flux vertical fracture, and horizontal fracture. At that time, its application became quite useful. But since not all systems behaved in this manner, the need for further solutions was warranted. Cinco-L. et al. investigated the general case of finite-conductivity vertical fractures, which included the above solution. as well as fracture conductivities as low as 0.1. This research also led to the need to analyze short-time data to obtain unique solutions. Similar results were obtained by Agarwat et al., who presented a finite-difference solution to this problem, considering both the constant rate as well as the problem, considering both the constant rate as well as the constant pressure cases.One of the first papers written on the effects of non-Darcy flow in fractured systems was by Wattenbarger and Ramey. They investigated the effects of non-Darcy flow in the formation and concluded that these effects cannot be felt if the fracture is long or intermediate in size. They further concluded that the effects of turbulent flow within the fracture were more significant.Holditch and Morse investigated the effect of turbulent flow in a fracture and analyzed the transient behavior of specific conductivities (low, medium, and high), giving a qualitative approach to the solution. They stressed the need for greater detail on these solutions and showed that there was indeed a large reduction in the fracture conductivity when non-Darcy flow was included. Although Holditch and Morse gave a detailed descriptive insight into the flow regime problem, they did not develop any general methods for determining the actual conductivity of the fracture. SPEJ P. 681

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*21 (03): 390–400.

Paper Number: SPE-9344-PA

Published: 01 June 1981

... into a constant-pressure separator or during the reservoir depletion phase, when the rate-decline period occurs. Geothermal reservoirs, which produce fluids that drive backpressure turbines, and open-well production both incorporate the constant-pressure behavior. For finite-conductivity

**vertically****fractured**...
Abstract

In many low-permeability gas reservoirs, producing a well at constant rate is very difficult or, in many cases, impossible. Constant-pressure production is much easier to attain and more realistic in practice. This is seen when production occurs into a constant-pressure separator or during the reservoir depletion phase, when the rate-decline period occurs. Geothermal reservoirs, which produce fluids that drive backpressure turbines, and open-well production both incorporate the constant-pressure behavior. For finite-conductivity vertically fractured systems, solutions for the constant-pressure case have been presented in the literature. In many high-flow-rate wells, however, these solutions may not be useful since high velocities are attained in the fracture, which results in non-Darcy effects within the fracture. In this study, the effects of non-Darcy flow within the fracture are investigated. Unlike the constant-rate case, it was found that the fracture conductivity does not have a constant apparent conductivity but rather an apparent conductivity that varies with time. Semianalytical solutions as well as graphical solutions in the form of type curves are presented to illustrate this effect. An example is presented for analyzing rate data by using both solutions for Darcy and non-Darcy flow within the fracture. This example relies on good reservoir permeability from prefracture data to predict the non-Darcy effect accurately. Introduction To fully analyze the effects of constant-bottomhole-pressure production of hydraulically fractured wells, it is necessary that we understand the pressure behavior of finite-conductivity fracture systems producing at constant rate as well as the effects of non-Darcy flow on gas flow in porous media. Probably one of the most significant contributions in the transient pressure analysis theory for fractured wells was made by Gringarten et al . 1,2 In the 1974 paper, 2 general solutions were made for infinite-conductivity fractures. Cinco et al . 3 found a more general solution for the case of finite-conductivity fractures and further extended this analysis in 1978 to present a graphical technique to estimate fracture conductivity. 4 For the case of constant pressure at the wellbore, solutions were presented in graphical form by Agarwal et al . 5 In his paper, a graph of log (1/ q D ) vs. log ( t Dxf ) can be used to determine the conductivity of the fracture by using type-curve matching. Although such a contribution is of great interest, unique solutions are difficult to obtain. More recently, Guppy et al . 6 showed that the Agarwal et al . solutions may be in error and presented new type curves for the solution to the constant-pressure case assuming Darcy flow in the fracture. That paper developed analytical solutions which can be applied directly to field data so as to calculate the fracture permeability-width ( k f b f ) product.

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*21 (03): 379–389.

Paper Number: SPE-9402-PA

Published: 01 June 1981

...D.J. Jaggernauth; Z.S. Lin; J.A. Lescarboura; K.A. Bishop; C.R. Clark; G.W. Shift Summary A physical model of a gas reservoir having a

**vertical****fracture**with fracture half-length x f =0.0635 m was developed. The model is a right circular cylinder of latex concrete with the radius x e =0.305 m so...
Abstract

Summary A physical model of a gas reservoir having a vertical fracture with fracture half-length x f =0.0635 m was developed. The model is a right circular cylinder of latex concrete with the radius x e =0.305 m so that the x e /x f ratio is 4.8. The producing well is located at the center of the fracture. Experimental drawdown and buildup data taken from this reservoir were analyzed using available theoretical developments from the literature. The effect of pressure on permeability (the Klinkenberg effect) was included in the analysis. Simplex optimization was used in conjunction with unified (drawdown plus buildup) super-positioning to give fracture half-lengths of 0.0631 and 0.0635 m from two sets of experimental data. Corresponding values for permeability for these two sets of data were 0.0605×10 −18 and 0.0624×10 − m 2 , respectively, at a Klinkenberg coefficient of 5900 kPa. The fracture half-length and permeability are shown to be highly correlated. Thus, the results have more uncertainty than would be found in determining parameters by similar methods of analysis for an unfractured system. Bearing this in mind, the agreement between the known fracture half-length and values determined from the analysis of experimental data is excellent. Thus, we have demonstrated the utility of unified analysis as well as the ability to create an artificial fracture. Since the location of the model fracture relative to producing and observation wells is at the discretion of the designer, our model presents a unique opportunity to study various configuration which might be difficult to handle by mathematical modeling alone. Introduction Kurata Thermodynamics Laboratory personnel at the U. of Kansas have developed an apparatus which physically models the behavior of a gas well during drawdown and buildup conditions. This model allows acquisition of data from a porous medium of known characteristics under carefully controlled laboratory conditions. The comparison of these data with results from mathematical models permits checking and, if necessary, modifying the equations that describe flow in porous media. Breit et al . 1 reported data obtained from such a model and described a unified method for analyzing drawdown and/or buildup data. Their unified method is an extension of that given by Odeh and Jones 2 where the sandface rate becomes the afterflow rate after the well is shut in. We have found that it is possible to make a vertical fracture of known length and location relative to the producing well in the physical model reservoir. This paper presents data obtained from a physical model reservoir that contains a vertical fracture with the producing well located at the center of the fracture and an analysis of these data in terms of available theory.

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*21 (01): 5–20.

Paper Number: SPE-8282-PA

Published: 01 February 1981

... the determination of thedrainage volume by type-curve matching appear tobe unrealistic. Introduction No quantitative data are available on the effect of thecompass orientation of a

**vertical****fracture**on pressuretransient data (injection or falloff). This is surprisingsince pressure falloff tests are the principal...
Abstract

This study investigates the flowing and shut-inpressure behavior of a fractured well located in asquare drainage region with the outer boundary at aconstant pressure. The fracture plane lies on one ofthe diagonals of the square. The report shows how toanalyze pressure data for a five-spot pattern when thefracture orientation is most favorable (from theviewpoint of sweep efficiency). Comparisons aremade with studies in the literature that assume anunfavorable fracture orientation. Fractureorientation must be considered in the analysis ofpressure data for the following conditions: smallfracture-penetration ratios, large flowing timesprior to shut-in, and large values of fractureflow capacity. Insights into the application of type-curve analysisto estimate drainage volumes are presented. Claimsin the literature regarding the determination of thedrainage volume by type-curve matching appear tobe unrealistic. Introduction No quantitative data are available on the effect of thecompass orientation of a vertical fracture on pressuretransient data (injection or falloff). This is surprisingsince pressure falloff tests are the principal means ofdetermining the efficacy of a fluid-injectionprogram - e.g., the effective formation flowcapacity, injectivity, skin factor, average reservoirpressure, and position of the flood front. Perhaps the dearth of information on this topic isdue to the fracture lengths being small comparedwith interwell distances in most waterflood orgas-injection projects. If the fracture length is smallcompared with the interwell distance, the orientationof the fracture should have a negligible effect on theshape of the pressure vs. time curve. However, withnew enhanced recovery projects that require closerwell spacing, interwell distances are of the same orderof magnitude as the created fracture length. In such instances, compass orientation of a vertical fracturecan have a significant effect on pressure data. All studies of the transient pressure behavior offractured wells in a bounded drainage region haveassumed that the fracture plane is parallel to theboundaries, which were considered to be eitherclosed or at constant pressure. Raghavan andHadinoto showed that the constant pressureouter-boundary solutions can be applied to a fractured wellin a five-spot injection-production pattern. However, the assumption that the fracture plane is parallel tothe boundaries of the square drainage region impliesthat the fracture is aligned directly with two of theadjacent producers. Clearly, this is only one of themany compass orientations that may exist in thefield; if consideration is given to the sweep efficiency of the flood, then this orientation would be the leastdesirable since the sweep efficiency at breakthroughwill be minimal. The most favorable fractureorientation would be the one in which the fractureplane lies along the diagonal of the square drainageregion (Fig. 1). As already mentioned, the fractureorientation may not have a significant effect ontransient data if the fracture lengths are small - butfor long fracture lengths, the effect of the orientationon the pressure behavior of injection wells can besignificant. Thus, it appears necessary to determinethe effect of fracture orientation on the pressurebehavior of fractured injection wells. SPEJ P. 5^

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*19 (06): 401–410.

Paper Number: SPE-7488-PA

Published: 01 December 1979

...Fikri Kucuk; William E. Brigham This study presents analytical solutions to elliptical flow problems that are applicable to infinite-conductivity

**vertically****fractured**wells, elliptically shaped reservoirs, and anisotropic reservoirs producing at a constant rate or pressure. Type curves and tables...
Abstract

This study presents analytical solutions to elliptical flow problems that are applicable to infinite-conductivity vertically fractured wells, elliptically shaped reservoirs, and anisotropic reservoirs producing at a constant rate or pressure. Type curves and tables are presented for the dimensionless flow rate and the dimensionless wellbore pressure for various inner boundary conditions ranging from K = 1 1, which corresponds to a circle, to K =, which corresponds to a vertical fracture. For elliptical reservoirs, K is the ratio of the major to minor axes of the inner boundary ellipse; for anisotropic reservoirs, it is the square root of the ratio of maximum to minimum permeabilities. Introduction Flow in a homogeneous and isotropic porous medium usually will be radial or linear, depending on the shape of the boundary. But in the area surrounding a vertical fracture, an anisotropic formation, or an aquifer with an elliptical inner boundary, flow will be elliptical.The study of elliptical flow in porous media is more recent than the usual radial and linear flow studies, but even elliptical flow studies date back at least several decades. The earliest discussion of steady-state elliptical flow usually is attributed to Muskat. He presented a steady-state analytical solution for the now from a finite-length line source into an infinitely large reservoir.One of the classic papers on elliptic flow by Prats et al. considered flow of compressible fluids from a vertically fractured well in a closed elliptical reservoir producing at a constant pressure. Prats et al. also producing at a constant pressure. Prats et al. also presented a solution for long times for the presented a solution for long times for the constant-rate case.Gringarten et al. found that older studies by Russell and Truitt (where flow is to a vertically fractured well) are unsuitable for short-time analysis. Gringarten et al. presented analytical solutions for fractures with infinite conductivity and with uniform flux. These solutions were for both closed squares and infinite reservoirs produced at a constant rate.In the last few years considerable work has been done on fracture systems, including numerical solutions and a semianalytical solutions for both finite and infinite fracture conductivities. Most of these studies, however, have not used the concept that the fracture is an elliptical flow system. Nevertheless, the results they obtain are important for well testing.Another problem related to elliptical flow is flow through an anisotropic porous medium. For this problem, a line source solution and a long-time problem, a line source solution and a long-time approximation presented by Earlougher are available for the constant-rate case.The purpose of this paper is to study elliptical flow in a broad sense with regard to reservoir engineering problems and to see whether these problems can be problems and to see whether these problems can be solved and whether elliptical problems can be handled in a unified, consistent manner. Development of Elliptical Flow Models The flow from an isotropic and homogeneous medium to a map usually will be radial, but lack of homogeneity will distort the radial flow geometry. In particular, flow will be elliptical through a porous particular, flow will be elliptical through a porous medium with directional permeability distribution (simple anisotropy). The inner geometry of a well also can distort radial flow geometry. For example, the flow will be elliptical if the well has an infinite-conductivity vertical fracture. Elliptical flow also will be encountered in flow from an aquifer to a reservoir that has an elliptical boundary at the oil/water contact. SPEJ P. 401

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*18 (04): 253–264.

Paper Number: SPE-6014-PA

Published: 01 August 1978

...Heber Cinco L.; F. Samaniego V.; N. Dominguez A. A mathematical model was developed to study the transient behavior of a well with a finite-conductivity

**vertical****fracture**in an infinite slab reservoir. For values of dimensionless time of interest, to >10, the dimensionless wellbore pressure, p...
Abstract

A mathematical model was developed to study the transient behavior of a well with a finite-conductivity vertical fracture in an infinite slab reservoir. For values of dimensionless time of interest, to >10, the dimensionless wellbore pressure, p, can be correlated by the dimensionless group; wk / x k, where w, k, and x are the width, permeability, and half length of the fracture, respectively, and k represents the formation permeability. Results when plotted as a function of P vs log to give, for large t, a 1.151-slope straight line; hence, semilogarithmic pressure analysis methods can be applied. When plotted in terms o/ log P vs log t, a family of curves of characteristic shape result. A type-curve matching procedure can be used to analyze early time transient procedure can be used to analyze early time transient pressure data to obtain the formation and fracture pressure data to obtain the formation and fracture characteristics. Introduction Hydraulic fracturing is an effective technique for increasing the productivity of damaged wells or wells producing from low permeability formations. Much research has been conducted to determine the effect of hydraulic fractures on well performance and transient pressure behavior. The results have been used to improve the design of hydraulic fractures. Many methods have been proposed to determine formation properties and fracture characteristics from transient pressure and flow rate data. These methods have been based on either analytical or numerical solutions of the transient flow of fluids toward fractured wells. Recently, Gringarten et al. made an important contribution to the analysis of transient pressure data of fractured wells. They presented a type-curve analysis and three basic presented a type-curve analysis and three basic solutions: the infinite-fracture conductivity solution (zero pressure drop along a vertical fracture the uniform flux solution for vertical fractures, and the uniform flux solution for horizontal fractures. Although the assumption of an infinite fracture conductivity is adequate for some cases, we must consider a finite conductivity for large or very low flow capacity fractures. Sawyer and Locke studied the transient pressure behavior of finite-conductivity vertical fractures in gas wells. Their solutions cannot be used to analyze transient pressure data because only specific cases were presented. In this study, we wanted to prepare general solutions for the transient pressure behavior of a well intersected by a finite-conductivity vertical fracture. The solutions sought should be useful for short-time or type-curve analysis. We also wanted to show whether conventional methods could be applied to analyze transient pressure data for these conditions. A combination of both methods, as pointed out by Gringarten to al., should permit an pointed out by Gringarten to al., should permit an extraordinary confidence level concerning the analysis of field data. STATEMENT OF THE PROBLEM AND DEVELOPMENT OF FLOW MODELS The transient pressure behavior for a fractured well can be studied by analyzing the solution of the differential equations that describe this phenomenon with proper initial and boundary conditions. To simplify the derivation of flow models, the following assumptions are made. An isotropic, homogeneous, horizontal, infinite, slab reservoir is bounded by an upper and a lower impermeable strata. The reservoir has uniform thickness, h, permeability, k, and porosity, which are independent of pressure. The reservoir contains a slightly compressible fluid of compressibility, c, and viscosity, mu, and both properties are constant. Fluid is produced through a vertically fractured well intersected by a fully penetrating, finite-conductivity fracture of half length, x, width, w, permeability, k, and porosity, phi . These fracture permeability, k, and porosity, phi . These fracture characteristics are constant. Fluid entering the wellbore comes only through the fracture. A system with these assumptions is shown in Fig. 1. In addition, we assume that gravity effects are negligible and also that laminar flow occurs in the system.

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*18 (02): 139–150.

Paper Number: SPE-6015-PA

Published: 01 April 1978

... pressure drop hydraulic fracturing penetration ratio

**vertical****fracture**fracture penetration ratio outer boundary procedure fractured well Trans shut-in time average reservoir pressure graph boundary straight line Analysis of Pressure Data for Fractured Wells: The Constant-Pressure Outer...
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

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*18 (01): 59–74.

Paper Number: SPE-6183-PA

Published: 01 February 1978

...P.J. Closmann; D.M. Phocas This study presents an analysis and related numerical calculations to ascertain the practicality of horizontally fracturing an oil shale formation by thermally biasing underground stresses. Results indicate that stresses induced by beating from a

**vertical****fracture**reduce...
Abstract

This study presents an analysis and related numerical calculations to ascertain the practicality of horizontally fracturing an oil shale formation by thermally biasing underground stresses. Results indicate that stresses induced by beating from a vertical fracture reduce the chances of forming horizontal fractures at the vertical fracture face, as long as the entire system is far from a free surface or pressurized cavity surface, With such surfaces, more favorable conditions for horizontal fracturing may be obtained. Other stress conditions more conducive to horizontal fracturing are suggested, such as those some distance away from the heated fracture and the more favorable stress conditions that result by beating from parallel vertical fractures. These results should be useful to engineers designing thermal recovery processes that require hydraulic fracturing, as well as engineers studying recovery techniques for oil shale. Introduction The use of fracturing to improve productivity of petroleum reservoirs is well established. In petroleum reservoirs is well established. In particular, fractures in thermal recovery operations particular, fractures in thermal recovery operations permit heat to be injected over a wide area into permit heat to be injected over a wide area into an oil-bearing formation. In many cases horizontal fractures are more desirable than vertical ones, since they allow communication between wells to be established more easily. If sufficiently controlled, horizontal fractures permit contact with specially chosen layers of a reservoir. The studies of fracturing of underground formations by Hubbert and Willis indicated that fractures form in a direction normal to the least compressive principal stress. Since the least compressive principal stress. Since the least compressive principal stress is horizontal in most cases, principal stress is horizontal in most cases, fractures are usually vertical. This tendency applies particularly to deep reservoirs. The orientation of particularly to deep reservoirs. The orientation of fractures and the pressures required for fracturing also will be affected by tectonic stresses. Often a vertical stress is determined by the weight of the overburden, although there are exceptions. The use of thermal stresses to modify fracturing pressure and to enhance formation of horizontal pressure and to enhance formation of horizontal fractures was suggested by Matthews et al. They concluded that heating from vertical fractures eventually will allow formation of horizontal fractures by generation of sufficient horizontal stress. In view of the applicability of fracturing to thermal projects in general and the interest in oil shale development in particular, investigation of the stresses developed during heating by injection of hot fluid into a fracture is desirable. Such information will provide an indication of the type of behavior anticipated when heating and applying increasing pressure to a fracture system. This paper considers the stress conditions arising as a paper considers the stress conditions arising as a solution to a thermoelastic boundary value problem. Failure criteria were not considered and could be a subject for future research. McLamore presented some considerations of this nature. Specific effects of the wellbore have not been included in this study. Certain considerations involving wellbore geometry could have a significant effect on the expected values for fracturing pressures. This aspect has been investigated by pressures. This aspect has been investigated by Haimson. In our mathematical solution, a negative stress will be compressive and a positive stress will be tensile. THEORY GENERAL DESCRIPTION Assume that an infinitely long fracture of limited vertical extent is present in the formation initially and that fluid loss through the walls of the fracture is negligible (Fig. 1). SPEJ P. 59

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*14 (04): 347–360.

Paper Number: SPE-4051-PA

Published: 01 August 1974

... or

**vertical****fractures**could produce the same skin effect, but with possibly different short-time pressure data. This could then provide a way to determine the orientation of fractures created by this type of well stimulation. In fact, it is generally agreed that hydraulic fracturing usually results in one...
Abstract

Introduction During the last few years, there has been an explostion of information in the field of well-test analysis. Because of increased physical understanding of transient fluid flow, it is possible to analyze the entire pressure history of a well test, not just long-time data as in conventional analysis. 1 It is now often possible to specify the time of beginning of the correct semilog straight line and determine whether the correct straight lie has been properly identified. It is also possible to identify wellbore storage effects, and the nature of wellbore stimulation as to permeability improvement, or fracturing, and to quantitatively analyze those effects. Such accomplishments have been augmented by attempts to understand the short-time pressure data from well testing - data that were often classified as too complex for analysis. One recent study of short-time pressure behavior 2 showed that it was important to specify the physical nature of the stimulation in considering the behavior of a stimulated well. That is, stating that the van Everdingen-Hurst infinitesimal skin effect was negative was not sufficient to define short-time well behavior. For instance, acidized (but not acid-fractured) and hydraulically fractured wells might not necessarily exhibit the same behavior at early times, even though they could possess the same value of negative skin effect. In the same manner, hydraulic fracturing leading to horizontal or vertical fractures could produce the same skin effect, but with possibly different short-time pressure data. This could then provide a way to determine the orientation of fractures created by this type of well stimulation. In fact, it is generally agreed that hydraulic fracturing usually results in one vertical fracture, the plane of which includes the wellbore. Most studies of the flow behavior for a fractured well consider vertical fractures only. 3–11 Yet it is also agreed that horizontal fractures could occur in shallow formations. Furthermore, it would appear that notch-fracturing would lead to horizontal fractures. Surprisingly, no detailed study of the horizontal fracture case had been performed until recently. 12 A solution to this problem was presented by Gringarten and Ramey. 13 In the course of their study, it was found that a large variety of new transient pressure behavior solutions useful in well and reservoir analysis could be constructed from instantaneous Green's functions. 14 Possibilities included a well with a single vertical fracture in an infinite reservoir, or at any location in a rectangle.

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*12 (04): 306–314.

Paper Number: SPE-3009-PA

Published: 01 August 1972

... and Fast. Therefore, we confine our discussion of previous investigations to those pertinent to the present study of the propagation of

**vertical****fractures**. propagation of**vertical****fractures**. An important theoretical result is Carter's formula for the area of a fracture of constant width formed by injection...
Abstract

This paper treats the propagation of hydraulic fractures of limited vertical extent and elliptic cross-section with the effect of fluid loss included. Numerical and asymptotic approximate solutions in dimensionless form give the fracture length and width at any value of time or any set of physical parameters. The insight provided by The dimensionless parameters. The insight provided by The dimensionless results and approximate solutions should be useful in the design of fracture treatments. Introduction The theory and practice of hydraulic fracturing has been reviewed by Howard and Fast. Therefore, we confine our discussion of previous investigations to those pertinent to the present study of the propagation of vertical fractures. propagation of vertical fractures. An important theoretical result is Carter's formula for the area of a fracture of constant width formed by injection at constant rate with fluid lost to the formation. For a vertical fracture of constant height, Carter's formula gives fracture length as a function of time. In general, Carter's assumption of constant width is not realistic. However, at large values of time the effect of this assumption becomes insignificant since the effect of fluid loss dominates. The width of a vertical fracture was first investigated by Khristianovic and Zheltov under the assumption that the width does not vary in the vertical direction. Thus, a state of plane strain prevails in horizontal planes and the width can be prevails in horizontal planes and the width can be determined as the solution of a plane elasticity problem. An approximate solution is found in Ref. 3 problem. An approximate solution is found in Ref. 3 upon neglect of fluid loss, fracture volume change, and pressure variation along the fracture. The fracture length is determined by the condition of finite stress at the fracture tip. Baron et al. and Geertsma and de Klerk have included the effect of fluid loss in the approach of Ref. 3. Geertsma and de Klerk give simple approximate formulas for fracture length and width. A different approach to the determination of fracture width was taken by Perkins and Kern. They considered a vertically limited fracture under the assumption of plane strain in vertical planes perpendicular to the fracture plane. The perpendicular to the fracture plane. The cross-section of the fracture is found to be elliptical, and the maximum width decreases along the fracture according to a simple formula that contains the fracture length. In the derivation of this formula, fluid loss and fracture volume change are neglected in the continuity equation and the fracture length is not determined. In a subsequent application, a "reasonable" fracture length was assumed. Carter's formula for length and the width formula of Perkins and Kern are both cited by Howard and Fast, and combined use of the two formulas is believed to be common practice. The present theoretical investigation is concerned with vertically limited fractures of the type studied by Perkins and Kern. However, we include the effects of fluid loss and fracture volume change in the continuity equation. Consequently, fracture length is determined as part of the solution. General results for the variation of fracture width and length with time are obtained in dimensionless form by a numerical method. In addition, asymptotic solutions are derived for large and small values of time. The small-time solution is also the exact solution for the case of no fluid loss to the formation. For large values of time our asymptotic formula for fracture length is identical with Carter's formula at large time. Our large-time formula for fracture width differs from the formula of Perkins and Kern by a numerical factor that varies along the fracture. In comparison with our formula, this formulas overestimates the width by 12 percent at the well and 24 percent at the midlength of the fracture. At early times Perkins and Kern's formulas for width in terms of length is again a fair approximation to our result. However, our formula for length differs from Carter's formula, which is not applicable since the neglected width variation is important at early times. The results for the width of a vertically limited fracture as obtained here and in Ref. 6 differ from the results for vertically constant fractures. SPEJ P. 306

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*9 (02): 247–254.

Paper Number: SPE-2232-PA

Published: 01 June 1969

... Displacement tests were performed mostly in cylindrical Berea sandstone cores. In a few cases in which a much lower matrix permeability was desired, Blue jacket sandstone cores were used. The

**vertical****fractures**were formed by cutting the cores with a thin circular blade or by parting them along a bedding plane...
Abstract

Laboratory displacement tests were performed to study oil recovery efficiency by gravity drainage in fractured systems under miscible conditions. The porous media used were cylindrical Berea and Blue porous media used were cylindrical Berea and Blue jacket sandstone cores containing a number of well-defined, artificially formed, vertical and subvertical fractures. Butane and Soltrol 130 were the two miscible fluids used. The purpose of the study was to examine the influences of the displacement rate, fracture density, fracture orientation, fracture permeability, matrix permeability, crossflow, core length and connate permeability, crossflow, core length and connate water on the oil recovery. It was found that displacement rate, matrix permeability and the subvertical fractures affected permeability and the subvertical fractures affected oil recovery most. The critical flow rate, based on the matrix permeability, was found to be a significant factor in the process. For displacement rates below the critical flow rate, the oil recovery efficiency appeared to be unaffected by the density of the subvertical fractures. At the high displacement rates, the fracture density becomes important, with the recovery being the most efficient in the core having the greatest number of the subvertical fractures. The magnitude of the fracture permeability, the fracture orientation, the core permeability, the fracture orientation, the core length and the connate water have little effect on the oil recovery efficiency. Introduction Gravity drainage under miscible conditions from relatively thick reservoirs can be a very efficient recovery process, especially at low flow rates where the gravity forces are dominant and, consequently, the adverse viscous fingers associated with the unfavorable viscosity ratio are minimized or eliminated. Such a process might involve injection of an LPG or some solvent bank at the crest and then driving the bank with dry gas. For some favorable combination of reservoir temperature, pressure and oil composition, there may be no need to inject any solvent bank, since enrichment of the dry gas by the light ends of the crude creates an in-situ solvent bank. Slobod and Howlett have studied the effects of gravity segregation in vertical unconsolidated porous media under miscible conditions in the porous media under miscible conditions in the laboratory. The main variables in their study were the viscosity ratio, the density differences and the rate of flow. In this study, the objective was to find the influence of fractures on gravity drainage under miscible conditions. A greater number of related variables were also studied. LABORATORY STUDY CORE DESCRIPTION Displacement tests were performed mostly in cylindrical Berea sandstone cores. In a few cases in which a much lower matrix permeability was desired, Blue jacket sandstone cores were used. The vertical fractures were formed by cutting the cores with a thin circular blade or by parting them along a bedding plane. The vertical fracture plane always contained the axis of the cylindrical core. The subvertical fractures, cut with a saw, were inclined 45 degrees from the horizontal. The line of intersection between the planes of the vertical and subvertical fractures was parallel to the circular faces of the cores in all cases except one, as depicted by VF3HF-2 in Fig. 1. Fracture geometry of the other cores is also given in Fig. 1. Tables 1 and 2 give the physical properties of the solid and fractured cores, respectively. The suffixes and prefixes shown for each core have been used prefixes shown for each core have been used throughout the paper so that the reader may discern the fracture geometry from the core numbers. SPEJ P. 247

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*8 (03): 260–268.

Paper Number: SPE-1948-PA

Published: 01 September 1968

...D.A.T. Donohue; J.T. Hansford Substantial evidence indicates that many petroleum producing horizons contain naturally occurring, ordered fracture systems and that within a particular geologic zone,

**vertical****fractures**induced in wellbores often will be directed along a particular compass direction...
Abstract

Substantial evidence indicates that many petroleum producing horizons contain naturally occurring, ordered fracture systems and that within a particular geologic zone, vertical fractures induced in wellbores often will be directed along a particular compass direction. Both conditions will seriously alter the fluid displacement behavior within reservoirs. In this study the effect of induced fracture orientation and length on sweep efficiency is determined for a five-spot pattern. In general, it is assumed that all wells are fractured and directed along the same compass direction. Using the electrical analog to steady state, two-dimensional fluid flow in porous media, boundary conditions are obtained from which flood fronts are tracked numerically. The numerical computations require a particle tracking routine for approximating flood front histories. It is shown that recovery is sensitive to the length and orientation of fractures for the pattern studied. With the proper design of fracture-pattern systems, recovery can be enhanced considerably. Introduction Hydraulic fracturing introduced in 1949, gave the industry a rather inexpensive means of increasing the fluid injection or production capacity of wells. It has been used with particular success to increase the production rate of wells completed in tight formations, such as in western Pennsylvania where producers have fractured in depleted or near-depleted fields and observed economic responses. Once the natural energy declines in such a reservoir where all wells have been fractured, waterflooding is generally suggested as means of further increasing recovery. Of the dual objective sought in waterflooding -- high injectivity and high break-through sweep efficiency - the former condition can be obtained if all wells in the flood pattern are fractured; the latter condition should depend on the nature of the fracture system. Considerable theoretical work has been published on the nature of fractures induced in boreholes. Although discussion persists concerning the possibility of forming a horizontal at a given point within the wellbore, it is generally conceded that only vertical fractures will develop below a given depth, i.e., where the fracturing pressure is less than the overburden load. Given the fact that fractures will be vertical in most cases of interest, it is also important to know whether there is order to fracture orientations within a given geological region. Kehle has suggested that in tectonically relaxed areas of uncomplicated geology, the stresses are fairly uniform and all fractures in the region should be parallel. Dunlap arrived at a similar conclusion in a theoretical investigation of localized stress conditions surrounding the borehole. He concluded that most vertical fractures are propagated in a preferred azimuthal direction. Fraser and Pettitt, in extending these theoretical suggestions to a specific field case, used an impression packer to record both a vertical fracture and the orientation of this fracture in the wellbore of a well in the Howard Glasscock field, Tex. Use of this information enhanced the waterflood recovery of the field. Anderson and Stahl also used impression packers on three fractured wells in the Allegheny field, N. Y., and found that the fractures were oriented more or less along the same compass direction. Orientation of the fractures in this manner depends on the stress condition within the formation during fracturing. Elkins and Skov have demonstrated that a natural, oriented, vertical fracture system exists within the Spraberry field. SPEJ P. 260ˆ

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*7 (03): 310–318.

Paper Number: SPE-1710-PA

Published: 01 September 1967

... is a function of the porous elastic constants of the rock, the component of the regional stress normal to the plane of the fracture, the formation fluid pressure and the dimensions of the crack. The same expression may also be used to estimate the

**vertical****fracture**width, provided all other variables are known...
Abstract

A criterion is proposed for the initiation of vertical hydraulic fracturing taking into consideration the three stress fields around the wellbore. These fields arise from nonhydrostatic regional stresses in earth the difference between the fluid pressure in the wellbore and the formation fluid pressure and the radial fluid flow through porous rock from the wellbore into the formation due to this pressure difference. The wellbore fluid pressure required to initiate a fracture (assuming elastic rock and a smooth wellbore wall) is a function o/ the porous elastic constants of the rock, the two unequal horizontal principal regional caresses, the tensile strength of the rock and the formation fluid pressure. A constant injection rate will extend the fracture to a point where equilibrium is reached and then, to keep the fracture open, the pressure required is a function of the porous elastic constants of the rock, the component of the regional stress normal to the plane of the fracture, the formation fluid pressure and the dimensions of the crack. The same expression may also be used to estimate the vertical fracture width, provided all other variables are known. The derived equations for the initiation and extension pressures in vertical fracturing may be employed to solve for the two horizontal, regional, principal stresses in the rock. Introduction Well stimulation by hydraulic fracturing is a common practice today in the petroleum industry. However, this stimulation process is not a guaranteed success; hence, the deep interest shown by the petroleum companies in better 'understanding the mechanism that brings about rock fracturing, fracture extension and productivity increase. Geologists and mining people became interested in hydraulic fracturing from a different point of view: the method may possibly be employed to determine the magnitude and direction of the principal stresses of great depth. Numerous articles in past years have dealt with the theory of hydraulic fracturing, but they all seem to underestimate the effect of stresses around the wellbore due to penetration of some of the injected fluid into the porous formation. Excellent papers on stresses in porous materials due to fluid flow have been published but no real attempt has been made to show the effect of these stresses in the form of a more complete criterion for vertical hydraulic fracturing initiation and extension. This paper is such an attempt. ASSUMPTIONS It is assumed that rock in the oil-bearing formation is elastic, porous, isotropic and homogeneous. The formation is under a nonhydrostatic state of regional stress with one of the principal regional stresses acting parallel to the vertical axis of the wellbore. This assumption is justified in areas where rock formations do not dip at steep angles and where the surface of the earth is relatively flat. This vertical principal regional stress equals the pressure of the overlying rock, i.e. S33= -pD where S33 is the total vertical principal stress (positive for tension), p is average density of the overlying material and D is the depth of the point where S 33 is calculated. The wellbore wall in the formation is considered to be smooth and circular in cross-section. The fluid flow through the porous elastic rock obeys Darcy's law. The whole medium is looked upon as an infinitely long cylinder with its axis along the axis of the wellbore. The radius of the cylinder is also very large. Over the range of depth at which the oil-bearing formation occurs, it will be assumed that any horizontal cross-section of the cylinder is subjected to the same stress distribution, and likewise that it will deform in the same manner. SPEJ P. 310ˆ

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*7 (02): 221–228.

Paper Number: SPE-1718-PA

Published: 01 June 1967

...William L. Huskey; Paul B. Crawford A study has been made to determine the effects of random

**vertical****fractures**existing in a reservoir matrix on the effective matrix properties and producing characteristics of a well. The study indicated that short**vertical****fractures**may distort the isopotentials...
Abstract

A study has been made to determine the effects of random vertical fractures existing in a reservoir matrix on the effective matrix properties and producing characteristics of a well. The study indicated that short vertical fractures may distort the isopotentials by 1 percent in the vicinity of the fracture. A correlation was developed which showed the effect of fracture density on the producing capacity of a well. The correlation indicated that fracture shape had little effect on producing capacity, but that total fracture length was closely correlated with the producing capacity. Effective values of reservoir permeability were found to correlate closely with fracture density when measured parallel to the direction of flow. Introduction In reviewing the petroleum literature, it is found that in most analytical solutions of hydrocarbon reservoir behavior the assumption is made that the rock is uniform throughout although core analysis and well logging surveys indicate that substantially all reservoirs are heterogeneous and some are fractured. Only in recent years have investigators studied the effects of reservoir heterogeneity. Russell discussed the effect of fractures in reservoirs. The literature indicates that fractures greatly affect the exploitation of many reservoirs. This is described by Elkins and Skov and Littlefield et al. Several types of reservoir heterogeneity have been studied by unsteady-state methods. In 1959, Landrum et al. showed that transient phenomena in hydrocarbon reservoirs could be studied by thermal models. Pickering studied the flow of heat in stratified linear reservoirs consisting of plates of different metals separated by sheets of rubber which were suddenly exposed to a low-temperature source at one end. Cotman studied both laterally and vertically heterogeneous thermal model reservoirs with special reference to cross Low. Another type of reservoir heterogeneity involves the presence of highly permeable, circular lenses around or near the wellbore which was studied by Miesch using a thermal and a steady- state model. Many studies have been made of the effects of induced hydraulic fractures on well performance. Work regarding vertical fractures that exist initially in the reservoir was reported by Givens, et al. Since there appears to be little published work on the importance and influence on reservoir performance of fractures that exist out in the rock matrix, this study was conducted to determine the effects of vertical fractures on pressure distribution, producing capacity and the effective permeability of the reservoir rock. The analogy between the flow of electrical current through a conducting body and the flow of fluid in a porous medium provides a convenient method of studying various aspects of petroleum reservoirs. For electrical radial steady-state flow, ........................................(1) For steady-state fluid flow, ........................................(2) With the above analogy, it is only necessary to establish the ratio of the values of the electrical units in terms of the corresponding flow units to make the analogy quantitative. The ratios are shown below. SPEJ P. 221ˆ

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*6 (01): 81–86.

Paper Number: SPE-1356-PA

Published: 01 March 1966

...J.W. Givens; Paul B. Crawford A potentiometric model study has been made of the effect of

**vertical****fractures**existing in the matrix of the reservoir on the flooding or cycling performance. Fractures can have unusual flow characteristics. Fluid entering one side of a fracture can emerge on either...
Abstract

A potentiometric model study has been made of the effect of vertical fractures existing in the matrix of the reservoir on the flooding or cycling performance. Fractures can have unusual flow characteristics. Fluid entering one side of a fracture can emerge on either the same or opposite side of the fracture, depending on the particular streamline. Fracture orientation has a great influence on the sweep efficiency. However, sustained fluid injection may still permit large areas to be swept. Introduction One can find considerable information available in the literature on the effect of fractures originating at the wellbore on sweep efficiencies of waterflooding or gas cycling programs. However, there are few, if any, quantitative data reported in the literature on the effect of vertical or horizontal fractures existing out in the reservoir matrix on secondary recovery performance. Several papers have been presented on the effect of vertical and horizontal fractures on the productivity or conductivity of waterflood and gas injection patterns, but in all cases the fractures initiated at the well and simulated commercial fractures. It is believed that natural fractures may exist out in the reservoir matrix and would effect the displacement performance. The purpose of this report was to study the effect of a few isolated vertical fractures existing out in the reservoir matrix on the performance of waterflood or gas injection patterns. DESCRIPTION AND EQUIPMENT Since the potentiometric model was described by Lee it has been widely used to study numerous types of fluid displacement problems. The potentiometric model can be used for fluid displacement studies when the following assumptions are valid: (a) steady-state conditions exist; (b) the mobility ratio is one; and (c) the capillary and gravitational effects can be neglected. Five-spot and direct line drive square patterns were studied using a copper strip of the desired length and orientation to simulate a vertical fracture in the reservoir matrix. The 20 x 20-in. model was considered to represent only one element of an infinite array of similar patterns. DISCUSSION AND RESULTS FIVE-SPOT PATTERN ONE FRACTURE BETWEEN INJECTION AND PRODUCING WELLS Fig. 1 shows a quadrant of a five-spot pattern with the vertical fracture existing out in the matrix along a line connecting the injection and producing wells. Length of the fracture was equal to 35.4 per cent of the distance between injection and producing wells. In this particular pattern it is noted that fluid breaks through from the injection well to the producing well when the dimensionless time is equivalent to 26.9. Dimensionless time is defined as volume of fluid injected divided by the volume of displaceable fluid in the pattern expressed as a per cent. SPEJ P. 81ˆ

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*2 (02): 87–94.

Paper Number: SPE-98-PA

Published: 01 June 1962

...M. Prats; P. Hazebroek; W.R. Strickler The pressure and production behavior of a homogeneous cylindrical reservoir producing a single fluid through a centrally located

**vertical****fracture**of limited lateral extent was determined by using mathematical methods to solve the appropriate differential...
Abstract

The pressure and production behavior of a homogeneous cylindrical reservoir producing a single fluid through a centrally located vertical fracture of limited lateral extent was determined by using mathematical methods to solve the appropriate differential equation. It is assumed that there is no pressure drop within the fracture - that is, that the fracture capacity is infinite. It was found that the production-rate decline of such a reservoir is constant (except for very early times) when the flowing bottom-hole pressure remains constant. The production-rate decline increases as the fracture length increases. Thus, the lateral extent of fractures can be determined from the production-rate declines before and after fracturing or from the decline rate after fracturing when the properties of the formation and fluids are known. The production behavior over most of the productive life of such a fractured reservoir can be represented by an equivalent radial-flow reservoir of equal volume. The effective well radius of this equivalent reservoir is equal to one-fourth the total fracture length (within 7 per cent); the outer radius of this equivalent reservoir is very nearly equal (within 3.5 per cent) to that of the drainage radius of the fractured well. The effective well radius of a reservoir producing at semisteady state is also very nearly equal to one-fourth the total fracture length. It thus appears that the behavior of vertically fractured reservoirs can be interpreted in terms of simple radial-flow reservoirs of large wellbore. Introduction An earlier report has considered the effect of a vertical fracture on a reservoir producing an incompressible fluid. That investigation of the fractured reservoir producing an incompressible fluid was started because of its simplicity. Thus, pertinent behavior of fractured reservoirs was obtained at an early date, while experience was being gained of value in the solution of more complicated fracture problems. One of these more complicated problems, and the one discussed in this report, considers the effect of a compressible fluid (instead of incompressible fluids) on the production behavior of a fractured reservoir. In the incompressible-fluid work mentioned, it was shown that the production rate after fracturing could be described exactly by an effective well radius equal to one-fourth the fracture length whenever the pressure drop in the fracture was negligible. Because of the simplification in interpretation, it is a matter of much interest to determine whether the production behavior of reservoirs producing a compressible liquid could be described in terms of an effective well radius which remains essentially constant over the producing life of the field. The details of the mathematical investigation are given in the Appendixes. IDEALIZATION AND DESCRIPTION OF THE FRACTURED SYSTEM It is assumed that a horizontal oil-producing layer of constant thickness and of uniform porosity and permeability is bounded above and below by impermeable strata. The reservoir has an impermeable circular cylindrical outer boundary of radius r e. The fracture system is represented by a single, plane, vertical fracture of limited radial extent, bounded by the impermeable matrix above and below the producing layer (reservoir). It is assumed that there is no pressure drop in the fracture due to fluid flow. Fig. 1 indicates the general three-dimensional geometry of the fractured reservoir just described. When gravity effects are neglected, the flow behavior in the reservoir is independent of the vertical position in the oil sand. Thus, the flow behavior in the fractured reservoir is described by the two-dimensional flow behavior in a horizontal cross-section of the reservoir, such as the one shown in Fig. 2. SPEJ P. 87^

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*1 (02): 105–118.

Paper Number: SPE-1575-G

Published: 01 June 1961

...M. Prats The effect of a sand-filled

**vertical****fracture**of limited radial extent and finite capacity (fracture capacity is the product of the permeability and width of the fracture) on the flow behavior of a cylindrical reservoir producing an incompressible fluid through a centrally located well has...
Abstract

The effect of a sand-filled vertical fracture of limited radial extent and finite capacity (fracture capacity is the product of the permeability and width of the fracture) on the flow behavior of a cylindrical reservoir producing an incompressible fluid through a centrally located well has been investigated mathematically. The shape of the lines of equal pressure near the fracture is essentially independent of the size of the reservoir, provided that the field radius is of the order of the fracture length or larger. For reasonable values of production rates and of fluid, reservoir and fracture properties, the total pressure drop between the end of the fracture and the well is generally negligible compared with the pressure drop in the reservoir. With regard to production response, the effect of vertical fractures can be represented by the production response of an equivalent or effective well radius. For a high-capacity fracture, the effective well radius is a quarter of the total fracture length, decreasing with the fracture capacity. When invasion effects are simulated by decreasing the width of the damaged zone with distance from the well, the effect of formation damage around a fracture on the production response is not so serious as indicated by the literature. This suggests that frac fluids with a conventional filter-loss response are better than high-spurt-loss frac fluids, provided the effective permeability of the damaged zone is the same. Introduction This paper considers the effect of the fracture capacity, as well as the formation damage which can result from fracture treatments, on the productive capacity of vertically fractured wells. Other publications, notably those of van Poollen, consider these same effects. In addition to providing more general results for vertical fractures than are available from the literature, the present paper gives the equivalent well radius of fractures having different lengths and Capacities and, also, includes pressure distributions in and around the fractures. The effect of a damaged zone around a fracture on the production response was not found to be so great as that reported by van Poollen. This difference probably stems from the fact that we consider a damaged zone which is widest (but is still small) near the well and thins out toward the extremities of the fracture, whereas van Poollen considers a damaged zone having a uniform width for the entire fracture length. Simple, but adequate, equations which describe the effect of these variables on production response are presented (in Appendixes A and B). Thus, results can easily be extended to values of the variables not specifically considered here.