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Most of the drilled wells are conducted on acidizing in order to eliminate the permeability reduction due to mud filtration. This type of formation damage is more sever in naturally fractured wells due to the high conductivity of fracture. If the radius of the damage due to lost circulation of drilling fluid is determined, the volume of acid needed to conduct any acid job could be calculated. Also, prediction of the near-wellbore conditions, such as mud filtrate radius, is important for accurate interpretation of the well-logs used for measuring and monitoring the properties of the near-wellbore formations. In this paper a mathematical model is developed to investigate the amount of mud filtration during drilling operation in a well with a finite conductivity vertical fracture in an infinite slab reservoir. Convection-dispersion filtrate transport model is developed in which the drilling mud can be mixed with the formation fluid. A systematic step-by-step procedure illustrating the methodology of the proposed model for the analysis of mud filtration in a finite conductivity fractured well is presented. This model is formulated for linear flow in both fracture and matrix. The effect of the filter cake is simulated by means of an empirically decaying filter rate equation. The developed model is solved by numerical analysis. The consistency of the numerical solution is checked and the best situation is considered. The sensitive analysis has been done on all the parameters of the model and the effect of each parameter such as wellbore pressure, fracture permeability, fracture diffusion coefficient, matrix permeability, matrix diffusion coefficient, matrix porosity, viscosity, constants related to the filter cake formation and fracture width on the amount of filtration are investigated. By means of the proposed model, the mud filtration can be plotted against position in both fracture and matrix for different wellbore and reservoir properties. The position in the fracture and matrix at which the curve of concentration reaches zero could be considered to represent skin radius. This radius could be used for the determination of the acid volume which is needed for the acidizing operation. Also, this model is a useful tool for accurate interpretation of the resistivity profiles which are essential for the development of efficient well-log interpretation.

A mathematical model was developed to study the behavior of a well with variable finite-conductivity and skin fracture, for a vertical fracture in an infinite slab reservoir. The solution is valid for slightly compressible fluid and gas fluid applying the pseudo-pressure concept. The constant rate with wellbore storage and constant pressure are the production modes considered. Curves type are used to showsthe effect of the changes of conductivity and skin near the well and in the extreme of the fracture. A field test was analyzed to detect the changes of the properties. This model is a tool to analyze real cases when the fracture properties are affected during the productive life of the well.

This operation was designed out of a need for hydraulic stimulation - whithno height increase - in a highly homogeneous medium with a water table (SierrasBlancas Formation, YPF.Nq.Bco x-3 ST well). From this, an artificial lowerbarrier was placed to control growth in the direction of the water and acrosslinked gel pad was pumped (followed by sand concentrations carried by alinear gel for sand settling) to give rise -upon fracture closure - to adehydrated gel zone of poor permeability. Furthermore, a water inhibitor wasused as an additive for improved water control. Furthermore, a water inhibitorwas used as an additive for improved water control. As an additional advantage, Bottom Screen Out (BSO) pumping might result inan increased fracture length which in turn would lead to a better productionresponse from stimulation, a "sine qua non" condition in lowpermeability reservoirs.

Abstract The pressure transient behavior of hydraulically fractured wells has been the subject of considerable study over the recent years. Several investigators have presented solutions of the fundamental equations, identified qualitative diagnostic trends and suggested integral techniques. This paper presents a new method to solve the problem of analysis of well test data of the hydraulically fractured well. The new mathematical model for the vertically fractured well in the dual-porosity and homogeneous reservoirs was founded by using the elliptic flow model and the integral method of conservation mass. The solution for the new model was obtained in real space. The type curves for the vertically fractured well in both homogeneous and dual-porosity reservoirs with and without wellbore storage and skin effect were calculated. In order to consider the wellbore storage and skin effect, the techniques of Laplace numerical transformation and Laplace numeral inversion were used. The special results for homogeneous reservoir have been identified by compared with the results of other authors, such as the infinite conductivity solution of A.C. Gringarten, the finite conductivity solution of H. Cinco-ley and the finite conductivity solution of M.J. Economides which considered the wellbore storage. They all have the same identities. Two field examples are used to illustrate the use of the new method. The characteristics of the type curves for both homogeneous and dual-porosity reservoir are described. For the dual-porosity reservoir, the type curve presents seven flow periods: the wellbore storage controlled period, the skin effect controlled period, the bilinear flow period, the first transient period, the dual-porosity characteristic flow period, the second transient flow period, and the radial or pseudo-radial flow period. The solutions have the following advantages: The solution may be used for both homogenous and dual-porosity reservoirs, and for both infinite and finite conductivity vertical fracture. The range of the dimensionless conductivity is from 0.001 to 500. Even more, the type curve is quickly calculated by using this method than any others. The new sets of type curves can be directly used in the oil field. Introduction With the development of low-permeability reservoirs, hydraulic stimulation is more widely used in the oil field than before. However, it is more difficult to analyze and evaluate the test data from the vertically fractured well. P. 533

Abstract In this paper, we develop a simple, closed form approximation for the Laplace transform solution for the case of a well with a finite conductivity vertical fracture in an infinite-acting reservoir. Our hybrid solution is based on a coupling of the solution for a trilinear finite conductivity vertical fracture model (which does not model radial flow) and the solution for a uniform flux/infinite conductivity vertical fracture (which does model pseudoradial flow). These solutions are readily obtained from the literature. Overall, we consider our solution to be valid for and we show that our solution gives less than 1 percent error in both PD and PD' for. We suggest that our hybrid solution is not valid for and do not recommend its use for under any circumstances. We have verified this solution against four different solutions given in the literature. Each comparison was excellent which suggests that our simplified solution is more than adequate for practical applications. In particular, we provide verification for constant rate and constant pressure production for values of between 0.25 and 10,000. We also show that our solution is capable of producing very accurate derivative functions. In addition, by reproducing the literature solutions so well, we also verified that individual flow regimes (formation/fracture bilinear flow, formation linear flow, and pseudoradial flow) are all modeled accurately by our new solution. Introduction Our motivation for this paper is to provide the technical audience with an accurate, but computationally simple, approximation for the Laplace transform solution for the case of a well with a finite conductivity vertical fracture in an infinite-acting reservoir. We have the following objectives for this work–To develop a rigorous closed form solution that can be used to accurately model a single well with a finite conductivity vertical fracture in an infinite-acting homogeneous reservoir.–To provide evidence of systematic verification to warrant application of this method for general practice.–To illustrate the application of this approximate solution to generate solutions for wells produced at both constant rates and constant pressures.–To identify and discuss any limitations of this solution and to give recommendations for computational considerations. SOLUTIONS FOR A WELL WITH A VERTICAL FRACTURE IN AN INFINITE-ACTING RESERVOIR Uniform Flux/Infinite Conductivity Solutions The original development of the uniform flux/infinite conductivity solutions for a vertically fractured well was performed by Gringarten, et al. The most important aspect of that work is that it provides working relations and theoretical considerations for fractured wells as well as the foundation and motivation for later efforts regarding finite conductivity vertical fracture solutions. Prior to the development of solutions for a well with a finite conductivity vertical fracture, the application of the uniform flux and infinite conductivity solutions often drew heated debate as to which was the most appropriate physical model. It is generally agreed that the uniform flux and infinite conductivity are "best case" scenarios whereas the finite conductivity model is seen as the most appropriate for general practice. However, the uniform flux and infinite conductivity vertical fracture solutions do have a theoretical tie to the finite conductivity fracture case. As such, it is theoretically consistent to use the uniform flux and infinite conductivity vertical fracture solutions to develop relations for finite conductivity vertical fracture solutions. Such is the case in our work. As background for the development of our solutions, we need to familiarize ourselves with the solutions for the uniform flux and infinite conductivity cases and the computational aspects of these solutions.. Gringarten, Ramey, and Raghavan Real Space Solution: From Gringarten, et al. we have the following result for the constant rate behavior of an infinite conductivity or uniform flux vertical fracture P. 89^

Abstract A numerical method, based on an Integral Finite Difference approach, is presented to investigate wells intercepting fractures in general and vertical fractures in particular. Such features as finite conductivity, wellbore storage, damage, and fracture deformability and its influence as permeability are easily handled. The advantage of the numerical approach is that it is based on fewer assumptions than analytic solutions and hence has greater generality. Illustrative examples are given to validate the method against known solutions. New results are presented to demonstrate the applicability of the method to problems not apparently considered in the literature so far. Introduction The problem of fluid flow to a well intercepting a single vertical fracture H of considerable interest in the field of petroleum engineering. The transient pressure response of such a well to fluid production has been investigated by previous workers production has been investigated by previous workers using analytical solutions, some of which are evaluated numerically. Despite their power, analytical methods have certain limitations. To overcome these limitations, numerical methods in conjunction with fast computing devices can be utilized with great advantage. The purpose of this paper is to describe and demonstrate the power of a fairly general numerical model in studying fluid flow to a well intercepting a vertical fracture. The Problem The problem of interest is schematically represented in Fig. 1. The reservoir is assumed to be homogeneous, horizontal, a really infinite and of thickness h. A vertical fracture of width w, length 2Xf and extending, throughout the thickness of the reservoir and is fully penetrated by a well of radius rw. It is assumed that the well produces fluid at a prescribed (usually constant) rate. The problem is to predict the pressure transient problem is to predict the pressure transient behavior of the system. in general and the well in particular. particular. Previous Work Previous Work Although interest in wells intercepting vertical fractures dates back to 1960, the first comprehensive study of the pressure transient response of such a well was performed by Gringarten et all who studied the case of flow to an infinitely conducting vertical fracture, with a hypothetical zero-radius well at its center. This and a few subsequent studies considered not only an infinite reservoir but also various types of bounded reservoirs. Later, Cinco-Ley et al extended the Green's functions approach of Gringarten et all to consider the mare realistic cases of finite conductivity vertical fractures. In their studies, Cinco-Ley et al coupled the Green's fraction solution of the earlier work with a one dimensional, linear solution for fluid flow within the fracture, the coupling procedure being subject to continuity of fluxes at the procedure being subject to continuity of fluxes at the fracture surface. To evaluate their solutions, Cinco-Ley et al used a numerical method. In subsequent studies, Cinco-Ley and Samaniego extended the solutions to include effects of wellbore storage and formation damage around the fracture. Although they do discretize the fracture into several segments in their numerical evaluation, Cinco-Ley et al's method is basically analytic or at best semi-analytic in nature. Limitations of Analytic Approach Use analytic approach has formed the backbone of well test interpretation and reservoir analysis since the 30's and has provided innumerable valuable insights in understanding reservoir response. Yet, tractable analytic solutions can be obtained only if the system under consideration is simple.

American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. Abstract Numerical simulation of a well in a vertically fractured reservoir is described. Real gas potential is the dependent variable. A fractional step numerical method which simultaneously integrates the difference expressions of the reservoir and fracture in an economical manner is used. Simulations of well performance with constant well potential and at several fracture extents are shown and discussed. Introduction The increasing demand for an adequate supply of natural gas has caused the development of extremely low permeability gas reservoirs. This is now possible because hydraulic fracturing can produce fractures with radii of 500 feet or more and thus stimulate production to levels that are economic. Estimation of the benefits to result from the hydraulic generation of a fracture in a particular formation is desirable. It is the authors' belief that the numerical model given here is sufficiently novel and advantageous to merit attention. Real gas potential is the dependent variable. The fracture is taken to occupy a negligible volume when compared to the volume of the reservoir. Then there is not the difficult problem of giving numerical representation of the small fracture width and the large reservoir extent in the one model. Yet one-dimensional Darcy flow is modeled in the fracture and two-dimensional Darcy flow is modeled in the reservoir. The numerical replacements of those differential expressions are solved simultaneously with relative ease. The numerical method used is from a class of methods called fractional step methods. Some recent results show how these methods lend themselves to the treatment of third type boundary conditions such as occur for this problem. Also, it is recognized that numerical methods of this type may yield solutions that exhibit peculiar behavior near boundaries. peculiar behavior near boundaries.

Abstract This study simulates the pressure transient behavior of flow of Newtonian/Newtonian and nonNewtonian/Newtonian fluid composite systems in porous media with a finite-conductivity vertical fracture. The main objective of this study is to determine the location of the flood front, reservoir rock and fluid properties and fracture parameters via history match ing of the test data. It is assumed that the inject fluid front propagates in the form of an ellipse who focal length is equivalent to the fracture length, and single-phase flow conditions exist behind and ah of the flood front. The viscosity of the non-Newton fluid in the flooded region was formulated as a function of the average flow rate. Both the injected and the reservoir fluids are treated as being slightly compressible. It is assumed that flow inside the matrix is two-dimensional, and one-dimensional inside the fracture. Two different fluid flow equations, one for the matrix and the other for the fracture, are used to describe the single-phase flow conditions. The equations are coupled through a fluid loss term from the fracture into the matrix and/or from the matrix into the fracture. Four different fluid loss models in three different flow geometries (elliptical, linear and radial) are considered. These two non-linear equations are linearized numerically, and the resulting two systems of linear equations are solved simultaneously using the alternating direction implicit procedure. Extensive testing and validation of this generalized formulation was conducted by using it to study nine special cases previously reported in the literature. A parametric investigation was conducted using dimensionless groups such as non-Newtonian power-law fluid exponent, fracture conductivity, elliptical interface location and mobility and diffusivity ratio. The results indicate that as the power-law index increases, the dimensionless bottom-hole pressure increases for a given injection rate. In cases of mobility ratios greater than unity, smaller elliptical interface locations, which represent initial amounts of injected fluid in the reservoir, give greater dimensionless bottomhole pressures, and in cases of mobility ratios less than unity, smaller elliptical interface locations give smaller dimensionless bottomhole pressures. Greater mobility ratios yield greater dimensionless bottomhole pressures; increasing dimensionless fracture conductivities result in decreased bottomhole dimensionless pressures. It is observed that this decrease is not significant for dimensionless conductivities larger than five hundred. Introduction Same enhanced oil recovery processes such as polymer flooding involve the injection of non-Newtonian fluids into a reservoir to increase oil recovery. The primary function of such a fluid injection scheme is to improve the mobility ratio. Uniform flood front, less pronounced viscous fingering and decrease in by-passing of the least permeable sections of the reservoir are some of the attributes of the non-Newtonian fluid injection. Thus, both the areal sweep efficiency and the displacement efficiency can be significantly improved. The injection of polymer solutions, and plain water injection processes are examples of non-Newtonian/Newtonian and Newtonian/ Newtonian fluid composite systems, respectively, considered in this study. Problems involving a non-Newtonian fluid have been studied in the literature/ Other EOR techniques such as steam injection and in-situ combustion invariably result in Newtonian/ Newtonian fluid composite system in the reservoir. P. 249^

Abstract A major problem in stimulating some formations has been the high production of formation water following fracturing treatments. This is generally considered to be due to the lack of a sufficient barrier to help prevent downward growth of the fracture from the hydrocarbon producing zone into the water producing interval. Thus, with conventional treatments, the undesirable water producing zone may be fractured and propped producing zone may be fractured and propped during the fracturing treatment of the hydrocarbon producing zone. A density controlled fracturing technique has been developed that is capable of placing the proppant in the upper portion of a producing zone. The technique utilizes a high producing zone. The technique utilizes a high density pad followed by a lower density treating fluid containing the proppant. This sequence allows the proppant to be preferentially placed in the upper portion preferentially placed in the upper portion of a vertical fracture. Laboratory tests have been conducted with a fracture model that demonstrated layering of fluids of different densities. Field results of several density controlled fracturing treatments are presented. Results demonstrate that this method is capable of increasing hydrocarbon production with little or no increase in the production of formation water. Introduction Water producing zones are often closely associated with hydrocarbon bearing formations. In some cases, the water is separated from hydrocarbons by a barrier, while in others there is no separation and the two phases are in direct contact. When hydraulic phases are in direct contact. When hydraulic fracturing is used to stimulate hydrocarbon bearing formations that are in direct contact with water, the fracture very likely will penetrate into the water bearing zone. The penetrate into the water bearing zone. The Edwards Limestone Trend of South Texas is an example of such a formation. The Edwards is approximately 1,000 feet (305 m) in thickness with hydrocarbon accumulations occurring near the top. Even when the hydrocarbon producing formation and water are separated by a third zone, a hydraulic fracture may grow out of the producing zone, through the separating layer, and into the water producing interval. Whether or not this will occur may depend on the type and thickness of the formations and the type of fracturing treatment employed. Conventional stimulation techniques on these types of formations often result in high volumes of produced water. Water production may be high immediately after production may be high immediately after the treatment, or the well may gradually water out over a period of a few weeks or months. CONVENTIONAL FRACTURING TREATMENTS When relatively low viscosity fracturing fluids such as uncrosslinked guar or cellulose are employed to stimulate formations, an equilibrium bed height of proppant is deposited in the bottom of the proppant is deposited in the bottom of the fracture. In instances where the fracture penetrates a water producing zone, the lower water zone may receive a large amount of proppant (Fig. 1). Gamma ray logs run before and after a treatment of this type indicated proppant some 35 feet (10.7 m) below the bottom perforation (Fig. 5). When high viscosity systems such as emulsions or crosslinked gels are utilized as fracturing fluids, proppant settling may be greatly reduced or eliminated. In this instance, the proppant is essentially distributed from top to bottom of a vertical fracture as shown in Fig. 2.

Members SPE-AIME Abstract This paper discusses the development of a novel fracture treatment technique which uses a nonreactive, buoyant diverter to control upward propagation of vertical fracture during the treatment. It is necessary to control such growth to prevent fracture penetration into undesirable zones, such as gas cap, penetration into undesirable zones, such as gas cap, water-bearing or nonproductive zones, and to promote maximum fracture extension within the producing interval Results of field evaluations indicate that the new treatment technique is successful. It offers a unique solution to preventing upward fracture growth that is independent of the formation properties and geologic stress environment. Introduction The ability to control vertical propagation of fractures has always been of considerable interest in hydraulic fracture treatment design. Uncontrolled vertical growth can lead to problematic production of water or gas. In addition, optimum lateral extension of the fracture cannot be achieved because the fluid, proppant and pumping energy are wasted in propagating proppant and pumping energy are wasted in propagating the fracture out-of-zone. Studies to identify parameters which are considered to be influential to the containment of hydraulic fractures are well documented. Most of the authors have concluded that in-situ minimum horizontal stress contrast between the pay zone and the barrier is the dominant factor in arresting or at least restricting vertical growth of fractures. Warpinski et al. found that a stress difference of 200 to 500 psi (1.4 to 3.5 MPa) between a basal ash-flow layer and an ash-fall tuff layer was necessary to contain the fracture within the ash-fall tuff. Teufel and Clark concluded that an increase of 700 psi (5 MPa) in horizontal stress was required for complete fracture containment in a number of limestones and sandstones. These experimentally derived stress differences serve only to indicate the magnitude of the stress contrast and thus should not be generalized. Inherent properties of the interface also contribute to fracture containment. Teufel and Clark suggested that if shear strength of the interface is small relative to the tensile strength and minimum horizontal compressive stress of the bounding layer, vertical fractures may turn into interfacial fractures at the interface and containment is, in effect, achieved. However, this type of containment may be very temporary. Hanson et al. showed experimentally that after the fracture is laterally displaced at the interface, it can still grow into the bounding layer if the frictional properties of the interface change. Daneshy found that at shallow depth, a weakly bonded interface may stop fracture growth. Others indicated that a stress discontinuity (i.e., fault) may terminate the fracture. Contrast in material properties (elastic modulus, porosity, density, etc.) and pore pressure generally porosity, density, etc.) and pore pressure generally has only secondary effects on containment. Elastic modulus is mare influential than the other material properties; however, it may only be able to reduce the properties; however, it may only be able to reduce the width of the out-of-zone fracture, not stop it. Treatment parameters which may influence fracture geometry are fluid viscosity and density, proppant concentration, pump rate and placement of perforations. The number and placement of perforations can be manipulated to ensure that the fracture is initiated in-zone, but once the fracture has propagated away from the wellbore, the material properties and stress environment will again be the governing factors. By varying fluid properties, proppant concentration and/or injection rate, frictional pressure drop along the fracture faces might be kept below the minimum horizontal stress difference between the confining zone and the pay zone. Nolte and Smith provided a method to detect the possible onset of provided a method to detect the possible onset of undesirable growth of fracture height from treating pressures which may allow corrective actions to be pressures which may allow corrective actions to be taken during the treatment or, at least, provide useful information to design subsequent treatments for similar wells.

Abstract In this investigation, a new method is presented for predicting both the non-Darcy flow presented for predicting both the non-Darcy flow coefficient and vertical fracture length based upon single-point, variable flow drawdown tests of shallow, low permeability gas reservoirs. The mathematical model simultaneously incorporates the effects of non-Darcy flow and variable rate expressed as a continuous function of time, thus providing a corrected pressure-drawdown versus time providing a corrected pressure-drawdown versus time relationship. Pascal and Quillian described a method of analyzing variable flow tests where rate variation, usually declining during the test, is considered a continuous function of time. Solutions proposed in the foregoing article offered credible values for permeability and skin when matched with well production histories. Pascal, Quillian and production histories. Pascal, Quillian and Kingston have now expanded upon the basic concepts presented under reference 1 to provide solutions presented under reference 1 to provide solutions for vertical fracture length and the non-Darcy flow coefficient, using single-point variable flow data, resulting in a comprehensive model for analyzing transient pressure behavior. A correlation between the non-Darcy flow coefficient and permeability is also proposed. In low permeability gas reservoirs, the effects of variable (declining) rate become less apparent as flow time is extended. Conversely, the influence of non-Darcy flow becomes more dominant with increasing flow time. The model is illustrated using several field cases from hydraulically fractured gas wells in a low permeability reservoir. Introduction The geometry of the mathematical model developed herein is similar to that commonly expressed in literature. Hydraulically fractured, low permeability reservoirs exhibit linear flow geometry during early time. In this case, non-Darcy flow in the reservoir is described by the following basic equations: (1) and (2) Where m(p) is the real gas pseudopressure and q is mass velocity, q = pv.

Abstract A method is presented for the determination of the orientation of fully penetrating vertical fractures by means of analysis of transient pressure data recorded at the active well and at two observation wells due to production or injection at the active fractured well. The fracture is considered to be of finite conductivity. The method discussed in thus work is an extension of the method of Uraiet et al. for the determination of compass orientation of vertical fractures. They considered uniform flux fractures. Results of this study show that when the pressure interference data, on the presence of finite pressure interference data, on the presence of finite conductivity fractures, are matched to the uniform flux-type curves, a match may not be obtained correctly. Introduction It is important to know the flow patterns created after a number of wells in a reservoir have been hydraulically fractured. This is of prime interest for enhanced recovery projects. Also, the development of low-permeability reservoirs using fracturing techniques requires an accurate knowledge of the dimensions and orientation of the fractures. This is essential for well spacing purposes if interference effects are to be minimized. Several methods have been suggested for determining the orientation of fractures. Elkins and Skov discussed a method for the determination of the orientation of natural fractures in a reservoir. They considered the fractured system (the reservoir) to be of anisotropic permeability and used the line-source solution to find the orientation of the major fracture trend. Reynolds et al. and Fraser and Pettit presented a method to obtain data about the fractures presented a method to obtain data about the fractures by means of an inflatable formation packer. They could get information about the type of fracture, vertical, horizontal or inclined, and the penetration and orientation. Pierce et al. described a way to us use pulse testing to determine the fracture length and orientation. Power et al. showed how acoustic, seismic, and surface electric-potential measurements can be applied to detect fracture orientation. Recently, Uraiet et al. discussed a method for determining the fracture orientation using transient pressure data recorded at the active fractured well, pressure data recorded at the active fractured well, production or injection, and at the observation wells. production or injection, and at the observation wells. They considered the fracture to be of uniform flux. The purpose of this study is to extend the method recently presented by Uraiet et al. to the case of finite conductivity fractures, and provide type curves for the analysis of pressure interference data. It is also intended to show what kind of problems may arise when pressure interference data, created by an active finite conductivity fractured well, are matched to the uniform flux-type curves. MATHEMATICAL FORMULATION The mathematical model considered in thus study is the finite conductivity vertical fracture model of Cinco et al. It is assumed that the porous medium is isotropic, homogeneous, horizontal, of uniform thickness h, permeability k, and porosity phi. All formation properties are assumed to be independent of pressure. The reservoir contains a slightly pressure. The reservoir contains a slightly compressible fluid of compressibility c and viscosity mu, both properties being constant. Fluid is produced through properties being constant. Fluid is produced through a vertically fractured well intersected by a fully penetrating finite conductivity fracture of penetrating finite conductivity fracture of half length xf, width w, and permeability kf. These fracture characteristics are constants and fluid entering the wellbore comes only via the fracture. The system defined under the above assumptions is shown in Fig. 1. It is also assumed that gravity effects are negligible and that Darcy flow occurs in the porous medium and in the fracture. It can be shown that the dimensionless pressure drop at a point (xD, yD) in the reservoir, created by the fractured active well can be expressed approximately by the following equation. PD(XD,YD,TDM) = PD(XD,YD,TDM) =

This paper was prepared for the SPE-European Spring Meeting 1974 of the Society of Petroleum Engineers of AIME, held in Amsterdam, the Netherlands, May 29–30, 1974. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made. Discussion of this paper is invited. Three copies of any discussion should be sent to the Netherland Section of the Society of Petroleum Engineers, P. O. Box 228, The Hague, the Netherlands. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Abstract An experimental and theoretical study has been made of the build-up of a proppant bank in a non-leaking fracture of constant width during a conventional fracturing treatment with a low-viscous fracturing fluid. The results indicate that the build-up of a proppant bank can be considered to take place in proppant bank can be considered to take place in three consecutive phases. During the first phase the bank builds up gradually as a function of time until an equilibrium height H(e) is reached near the well bore and the bank stops-growing at this point as a result of the erosion caused by the increased fluid drag forces on the proppant particles. During the second phase the bank grows only in height until it reaches He over its full length. Finally, in the third phase the bank grows only in length and the injected proppant saltates over the full length of, the bank towards the bank's front where it settles, increasing the length of the bank in the direction of flow. The analytical relations derived for each of these phases in bank build-up have been confirmed by experiments in a transparent model. The observed disturbance of the proppant bank close to the well bore, induced by the jetting effect of the injected proppant slurry, might be a major 'bottle-neck' in obtaining effective fractures. This phenomenon should be investigated extensively. phenomenon should be investigated extensively. Introduction Productivity improvement in oil and/or gas wells Productivity improvement in oil and/or gas wells obtained with a conventional hydraulic fracturing treatment depends mainly on fracture length and fracture conductivity. Both length and conductivity are directly related to the efficiency with which a fracture can be filled with proppant material. During a conventional treatment with a low-viscous fracturing fluid the injected proppant material will settle and form a bank at the bottom of the fracture. Only that part of the fracture where a bank is formed contributes to actual fracture length and conductivity, because this part remains open after the fracture is allowed to close. Therefore, knowledge of the dimensions of the bank formed and its rate of build-up is required for a proper hydraulic fracturing design. We have studied proper hydraulic fracturing design. We have studied this both theoretically and experimentally. 2. THEORETICAL DESCRIPTION OF PROPPANTBANK BUILD-UP IN A RECTILINEAR VERTICAL FRACTURE In the description of a proppant-bank build-up we considered a rectilinear vertical fracture of uniform width and height extending from line well bore into a non-dipping formation. It was assumed that no fluid losses occur through the fracture faces. The proppant material is considered to be of uniforms size and homogeneously suspended in a Newtonian incompressible fracturing fluid with a constant temperature.