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Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Latin America and Caribbean Petroleum Engineering Conference, May 17–19, 2017

Paper Number: SPE-185455-MS

... effects of turbulence,

**partial****penetration**, wellbore storage and reservoir damage in wells, especially during the transient flow period, as an attempt to reduce erroneous estimation of reservoir parameters during well test analysis. Many models to address the reservoir response of**partial****penetration**...
Abstract

The increasing complexity of reservoir responses, especially with unconventional reservoirs, calls for more accurate methods of reservoir characterization and well test interpretation. Though the unconventional gas reservoirs are characterized with low flow rates, these can be high enough to exceed the laminar flow range as described by Forchheimer number given in [ 1 ]. Moreover, prolonged transient flow in such unconventional reservoirs makes the use Odehs steady state or pseudo-steady state models ([ 2 ] and [ 3 ] respectively) inappropriate. This paper investigates the combined effects of turbulence, partial penetration, wellbore storage and reservoir damage in wells, especially during the transient flow period, as an attempt to reduce erroneous estimation of reservoir parameters during well test analysis. Many models to address the reservoir response of partial penetration wells have evolved with time as seen in the works of [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ], [ 7 ] and [ 8 ]. Most of the models however, do not address the commingled effects of partial penetration with other wellbore or reservoir boundary conditions. For this reason, the combined effects of partial penetration with different wellbore and reservoir conditions are investigated. The model was developed based on the composite reservoir concept to incoporate the skin effects analogue Odeh [ 3 ] pseudo-steady state model. Flow in the reservoir is addressed in both the radial and vertical directions and by transforming the pressure diffusivity equations using the Laplace and finite Fourier transforms, a diffusivity equation analogue [ 6 ] can be derived. Turbulence flow in the reservoir was addressed using the Forchheimer model with transient turbulent flow incorporated analgue Kome [ 9 ] and modified to addressed partial penetration effect of the wellbore flowing conditions by transforming the wellbore conditions using the finite Fourier cosine and sine transforms. With the model, the influence of each of the reservoir parameters/ wellbore flowing conditions are depicted. By considering turbulent flow during the transient phase of production, the partial penetration skin also becomes influenced by turbulence effects. By incorporation the composite reservoir model for the damage skin, the partial penetration effects does not only limit to the vertical and radial permeabilities of the undamaged zone, but also addresses the effects of alterations in vertical permeability in the skin zone. As seen in the work of Kome, 2017, the permeability conventionally derived during IARF for the transient turbulent regime is a function of the Forchheimer number hence approaches will be made as to how to elimate these effects for the partially penetrated well. The novel model clearly shows the effects of different wellbore flowing conditions and reservoir parameters for a partially penetrated well, especially during the transient flow period and the possible errors made in the past with respect to permeability and damage skin

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Heavy and Extra Heavy Oil Conference: Latin America, September 24–26, 2014

Paper Number: SPE-171060-MS

... and environmental concerns in a high biodiversity region. This study provides the necessary test design considerations to calculate, from a conventional pressure transient analysis, the timing for the start and the end of all the flow regimes happening in a

**partial****penetration**test; thus, some recommendations were...
Abstract

Recently, a big heavy oil province has been discovered in a remote area of the Amazonian rainforest in Peru. These fields tested heavy oil production (< 12° API); commercial decision on developing this area will take into account results of drilling appraisal wells, evacuation of the heavy oil from the Peruvian jungle to the cost and environmental concerns in a high biodiversity region. This study provides the necessary test design considerations to calculate, from a conventional pressure transient analysis, the timing for the start and the end of all the flow regimes happening in a partial penetration test; thus, some recommendations were drawn to reduce wellbore storage and ensure the hemispherical flow regime appears, as longer as possible during the test, to measure vertical permeability. A partial penetration test was carried out in an exploratory well supported by this methodology; anisotropy and productivity index was calculated. In addition, a comparison analysis was done to calculate dimensional productivity index and predict production performance for different well types by simulation, and precise decision-making process towards optimizing well trajectories for exploiting reservoirs to produce.

Journal Articles

*J Ship Prod Des*6 (01): 27–34.

Paper Number: SNAME-JSP-1990-6-1-27

Published: 01 February 1990

...Edward T. Gaines This paper reviews development of weld design equations which can be used to analyze bevel

**partial**-**penetration**tee joints. The method developed herein follows closely the development of equations for the design of square-edge**partial**-**penetration**tee joints which was presented...
Abstract

This paper reviews development of weld design equations which can be used to analyze bevel partial-penetration tee joints. The method developed herein follows closely the development of equations for the design of square-edge partial-penetration tee joints which was presented at the 1986 Ship Production Symposium. For U.S. Navy ship design, technical authority is vested in the Naval Sea Systems Command (NAVSEA). The published NAVSEA design criteria for partial-penetration tee joints are so conservative that it is mathematically impossible to design a conventional, 100 percent efficient partial-penetration bevel tee joint. With a more rigorous engineering analysis, the alternative method for bevel joints outlined in this paper might be an acceptable replacement for the simple, though unduly conservative existing design criteria.

Journal Articles

Journal:
SPE Reservoir Engineering

Publisher: Society of Petroleum Engineers (SPE)

*SPE Res Eng*2 (02): 227–234.

Paper Number: SPE-13956-PA

Published: 01 May 1987

...Paul Papatzacos Approximate

**Partial**-**Penetration**Pseudoskin for Infinite- Pseudoskin for Infinite- Conductivity Wells Summary. This paper presents a simple formula for the pseudoskin factor of a well with restricted flow entry where infinite conductivity is taken into account analytically...
Abstract

Approximate Partial-Penetration Pseudoskin for Infinite- Pseudoskin for Infinite- Conductivity Wells Summary. This paper presents a simple formula for the pseudoskin factor of a well with restricted flow entry where infinite conductivity is taken into account analytically. Comparisons with previously published results are shown graphically and in tabulated form. Introduction The concept of pseudoskin appears naturally in calculations of the pressure drop in an infinite slab reservoir with a well that is either partially penetrating or has limited flow entry. It is, in the semilog flow regime, the additional pressure drop that arises at the wellbore when the interval open to flow is smaller than the reservoir thickness. Ref. 1 provides a comprehensive list and analysis of the extensive literature on the subject before 1975. For papers published since 1975, see Ref. 2 and its references. papers published since 1975, see Ref. 2 and its references. The analytical calculation of the pressure drop is relatively straightforward for uniform-flux wells. For infinite-conductivity wells, however, there arise mathematical difficulties first identified by Muskat. The earliest results on partial-penetration pseudoskin were presented graphically by Brons and Marting. The calculations were based on a uniform-flux well and Musket's method to approximate the infinite-conductivity situation (see Ref. 1). Streltsova-Adams presented explicit equations for pseudoskin caused by restricted flow entry in the form pseudoskin caused by restricted flow entry in the form of infinite series. Streltsova-Adams uses a uniform-flux well and calculates an average pressure drop at the wellbore by integrating along the interval open to flow. Finally, Odeh presented a simple equation. Refs. 5 and 6 assume anisotropic reservoirs. In Ref. 5, pseudoskin is expressed in terms of three dimensionless parameters (see Fig. 1): (1) (2) and (3) An isotropic reservoir was assumed in Ref. 4, but Refs. 7 and 8 show that the Brons-Marting charts can be used for anisotropic reservoirs, provided one uses r instead of r /h . Numerical values of pseudoskin for chosen values of h, r, and h where infinite conductivity has been accounted for numerically can be found in Refs. 2 and 7. Both references will be considered later in this paper as a basis for comparison. Result of the Theoretical Analysis The theoretical analysis presented in this paper is based on a result, derived in Ref. 9, concerning an infinite-conductivity well in an infinite reservoir. This result is used, together with the method of images, to obtain the steady-state pressure drop at the wellbore of a well with restricted flow entry in a slab reservoir. The following equation is thus found, expressing pseudoskin in terms (4) where (5) and (6) The derivation of Eq. 4 is given in Appendix A. As mentioned previously, it relies on the method of images, which uses, in addition to the physical well, an infinite number of image wells to generate the no-flow condition at the top and bottom of the reservoir. The steady-state pressure drop caused by any well is known. The real pressure drop caused by any well is known. The real well contributes a pressure drop that is constant along its own wellbore. Each image well, however, contributes a pressure drop that necessarily varies along the wellbore pressure drop that necessarily varies along the wellbore of the real well so that the method of images will not yield the exact infinite-conductivity solution. It will be shown, however, that Eq. 4 gives a good approximation in most cases of practical interest. SPERE P. 227

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE California Regional Meeting, March 27–29, 1985

Paper Number: SPE-13627-MS

...SPE 13627 SPE Society of Petroleum Engineers Pressure Transient Analysis for Single Well/Reservoir Configurations (Considering

**Partial****Penetration**, Mixed Boundary, Wellbore Storage, and Skin Effects) by B.C. Theuveny, U. of Alaska; W.A. Jack, Sohio Alaska Petroleum Co.; and M.J. Economides, * U...
Abstract

Pressure Transient Analysis for Single Well/Reservoir Pressure Transient Analysis for Single Well/Reservoir Configurations (Considering Partial Penetration, Mixed Boundary, Wellbore Storage, and Skin Effects) Abstract This work presents a set of solutions for pressure transients analysis demonstrating the flexibility of Green's functions in solving a wide range of problems. problems. The impact of partial completions and off centered wells is presented along with a first-time incorporation of wellbore storage and skin effects. Custom type-curves tailored to fit any well, reservoir and boundary configuration may be generated using the methods outlined here. An associated computer model is fully interactive and can be run on a small computer, readily available to most reservoir engineers. Introduction Green's functions have been used by a number of investigators to describe reservoir pressure transients in petroleum and geothermal reservoirs. Wegner (1983) developed an automated procedure to generate analytical solutions for a variety of boundary configurations. He compiled, in a concise form, various existing Green's functions. His computer code provides "custom type curves", suited to each well as completed in each reservoir. A small desk-top computer could run it, creating a fully interactive scheme. In order to rationalize the present work, a general presentation of the conventional approach used in pressure transients analysis is offered. The more direct and more easily understood solutions, are those in Laplace space, first presented by van Everdingen and Hurst (1949) and which are used by most investigators of pressure transients analysis. The main drawback from these methods is the difficulty to extend the results obtained for a particular geometry, to a generally similar, but different type of geometry or boundary conditions. This means that for each individual problem, some analytical work is required and more importantly, from an operational point of view, it is necessary that different cases should be treated differently. This is essential in identifying the trends of the pressure response and a major step towards pressure response and a major step towards the uniqueness of the solution. Another drawback, that has not been clearly pinpointed in the literature is the problem related to the inversion of the problem related to the inversion of the Laplace solutions. Once the solution to a particular problem is obtained in Laplace particular problem is obtained in Laplace space, it needs to be transformed back in the real time domain. Two main approaches have been used so far: an analytical inversion using Mellin's formula and a numerical inversion of the Laplace transform. Stablest (1970) developed a numerical algorithm that has been used widely. The use of Mellin's formula requires substantial and complex analytical work. It has the advantage of providing closed form solutions, usually involving infinite intergrals. However, there is no systematic treatment of the inversion in the petroleum literature. Each inversion petroleum literature. Each inversion attempt must be studied carefully and independently, thus forbidding the use of a simple computer code. In the past, the Stablest algorithm has been considered, as the panacea for too many times. P. 387

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*36 (12): 2197–2210.

Paper Number: SPE-12178-PA

Published: 01 December 1984

...A.C. Reynolds; J.C. Chen; R. Raghavan Summary In this study we examine the pseudoskin factor caused by

**partial****penetration**in a two-layer reservoir when only**partial****penetration**in a two-layer reservoir when only one layer is open to flow. We show that the pseudoskin factor can be correlated...
Abstract

Summary In this study we examine the pseudoskin factor caused by partial penetration in a two-layer reservoir when only partial penetration in a two-layer reservoir when only one layer is open to flow. We show that the pseudoskin factor can be correlated as a unique function of three reservoir parameters and in many cases can be correlated accurately as a function of two reservoir parameters. We present graphs and procedures for estimating the pseudoskin factor. pseudoskin factor. Introduction Frequently, wells are perforated over only a portion of the productive zone to delay or to prevent water and/or gas coning. This situation is called "partial penetration" or "restricted entry."This type of well completion has received considerable attention in both the petroleum and groundwater hydrology literature. The problem was studied as early as 1949 by Muskat and has subsequently been studied in a plethora of papers including Refs. 2 through 10. Desirable information papers including Refs. 2 through 10. Desirable information on this problem includes procedures for the analysis of well-test data and the evaluation of the productivity loss from partial penetration or restricted entry. This work provides information on the second topic; specifically, it provides information on the second topic; specifically, it provides methods for evaluating the pseudo skin factor caused provides methods for evaluating the pseudo skin factor caused by partial penetration. As shown in Refs. 1 through 6, the pseudo skin factor determines the productivity decrease pseudo skin factor determines the productivity decrease resulting from partial penetration. The results of Refs. 1 through 10 consider only single-phase flow, whereas in reality partial penetration or restricted entry exists to prevent the partial penetration or restricted entry exists to prevent the production of an undesirable fluid under multiphase flow production of an undesirable fluid under multiphase flow conditions i.e., to prevent or delay water and/or gas coming. Ref. 11, clearly indicates that the single-phase correlations for the pseudo skin factor do not accurately predict the pseudo skin factor caused by partial penetration under multiphase flow conditions. The results of Ref. 11, which are for an oil/ water system, indicate that if horizontal saturation gradients are negligible, then the two-phase flow situation should be analogous to single-phase flow in a layered reservoir with the oil, transition, and water zones each representing a separate layer. If the transition zone is small, the two-phase flow partial penetration situation should resemble single-phase flow partial penetration situation should resemble single-phase flow in a two-layer reservoir. This provides the motivation for the single-phase flow two-layer reservoir problem considered here. Our main objective is to present methods for estimating the pseudo skin factor in a two-layer reservoir with only one layer pseudo skin factor in a two-layer reservoir with only one layer open to flow. If this can be done, the true skin factor can be estimated by subtracting the pseudo skin factor from the total skin factor computed by semilog analysis techniques. The results of Refs. 1 through 6 are restricted to single-layer, single-phase flow problems. It is relevant to point out that a partially penetrating well in a layered reservoir has been studied previously; see Refs. 7 and 9. However, to our knowledge, no one has identified the key parametric groups that uniquely determine the pseudo skin factor caused by partial penetration in a two-layer reservoir or provided methods for computing this pseudo skin factor. The intended contribution of this paper is pseudo skin factor. The intended contribution of this paper is to provide this knowledge. Pseudo skin Computation Pseudo skin Computation Hereafter we refer to the pseudo skin factor caused by partial penetration as simply the pseudo skin factor. In this major penetration as simply the pseudo skin factor. In this major section, the mathematical model and the procedure used to compute this factor are discussed. Mathematical Model. To compute the pseudo skin factor, we consider a single well in the center of a two-layer cylindrical reservoir with impermeable top, bottom, and outer boundaries. The layers are assumed homogeneous and contain a slightly compressible fluid of constant viscosity. The initial pressure is assumed uniform throughout the reservoir. Gravitational and wellbore storage effects are neglected. Each layer can be either isotropic or anisotropic, but the vertical permeabilities of the two layers are nonzero so that cross flow can occur between the two layers. The well is produced at a constant rate. Since the reservoir pressure exhibits radial symmetry, the mathematical model is two-dimensional i.e., an r-z model (Fig. 1). The pseudo skin factor results presented in this work were obtained by generating the pressure response for the model shown in Fig. 1 under the assumption that only Layer 1 is perforated; that is, all production is from Layer 1 and all perforated; that is, all production is from Layer 1 and all of Layer 1 is open to flow. JPT p. 2197

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 23–26, 1979

Paper Number: SPE-8204-MS

... for determining the skin factor due to

**partial****penetration**. The importance of the ratio of vertical to horizontal permeability is considered in the analysis. An example illustrates the application of the method. Introduction An altered zone of reduced permeability surrounding a wellbore is quantified in terms...
Abstract

Abstract There is considerable disagreement in the literature concerning the interpretation of wellbore damage (the skin effect) in partially completed wells. The skin factor is composed of two components, one of which is indicative of actual formation damage and the other of which results from an additional pressure drop due to the partial completion. The skin factor measured in a buildup test does not reflect the simple sum of these two components. Rather, the effect of the actual damage is accentuated by the partial completion. A method is presented to aid in the interpretation of well tests which yields estimates for both contributions to the skin factor. Analysis techniques are derived for steady-state flow and these are corroborated for transient flow by using simulated well test results. The simulations also provide a basis for comparing the various correlations for determining the skin factor due to partial penetration. The importance of the ratio of vertical to horizontal permeability is considered in the analysis. An example illustrates the application of the method. Introduction An altered zone of reduced permeability surrounding a wellbore is quantified in terms of the skin factor, s. This factor is related to the excess steady-state pressure drop 1,2 in a flowing well as: Equation 1 The flow efficiency, E, is expressed in terms of Î"p s and the total drawdownÎ"p w , as: Equation 2 The skin factor has been related 3 to an altered zone near the wellbore (Figure 1). If a zone of reduced permeability, k a , extends a distance, r a , into the formation, then the skin factor, s, is: Equation 3 where r w is the wellbore radius and k is the bulk formation permeability. It is the skin factor described by Equation 3 that is a measaure of the true formation damage caused by drilling and completion practices and other factors. This type of flow impediment is the target of acid treatment and other workover procedures. Thus, in order to evaluate completion practices and to recommend workover procedures, it is necessary to accurately determine the 'true' skin factor due to formation damage. In subsequent discussion this skin due to true damage will be designated as s d . The value of s d can be somewhat difficult to measure directly. A well is often completed over only a portion of the producing formation (Figure 2). This leads to a flow construction which is detected as an additional pressure drop or skin effect. Several authors 4,5,6,7,8 have developed analytical descriptions of the skin factor, s p , due to partial penetration in an otherwise undamaged reservoir. In particular, Nisle 9 has developed a formulation for simulating well tests within an infinite acting reservoir with a partial completion. The concept of a skin factor is not introduced.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Fall Technical Conference and Exhibition, October 9–12, 1977

Paper Number: SPE-6753-MS

... Abstract Although there have been many studies on unsteady behavior of wells with a

**partially**-**penetrating**wellbore, there has been no study of the combined effects of wellbore damage and storage on the behavior of**partially**-**penetrating**wells. The purpose of this study is to fill this existing...
Abstract

Abstract Although there have been many studies on unsteady behavior of wells with a partially-penetrating wellbore, there has been no study of the combined effects of wellbore damage and storage on the behavior of partially-penetrating wells. The purpose of this study is to fill this existing gap in knowledge. Specifically, this study concerned a bounded, anisotropic, cylindrical reservoir with a partially-penetrating, infinite-conductivity partially-penetrating, infinite-conductivity cylindrical inner boundary. This inner boundary also contained wellbore storage and a flux-dependent, infinitesimal (zero storage capacity) skin effect. An analysis of pressure versus time indicated the possible existence of two semi-log straight line periods assuming a very small wellbore storage effect. Either straight line might be interpreted to yield horizontal permeability and the skin effect. Vertical permeability might be evaluated from correlations relating either the ending time of the first straight line, the intersection time of the two straight lines, or the beginning time to the second straight line to the effective vertical permeability. All three methods will fail if permeability. All three methods will fail if wellbore storage is significant. Introduction In many oil and gas reservoirs, producing wells are completed as partially-penetrating wells; that is, only a portion of the zone is perforated. This may be done for many reasons, but the most common one is to prevent or delay the intrusion of unwanted fluids (gas or water) into the wellbore. Partial-penetration will cause performance which, Partial-penetration will cause performance which, if not properly evaluated, can be mistaken for formation damage and can lead to errors in the interpretation of well-test data. The purpose of this study is to reach general conclusions concerning the effects of wellbore damage and storage on the behavior of partially-penetrating wells, and to determine how these partially-penetrating wells, and to determine how these effects combine to influence the interpretation of short-time well tests. The combined effects of partial-penetration, wellbore storage, and partial-penetration, wellbore storage, and wellbore damage have never been studied. The partially-penetrating well problem was first studied by Muskat for steady-state conditions. He calculated pressure distributions and productive capacities for an anisotropic system, productive capacities for an anisotropic system, and concluded that the productivity depended slightly on the directional permeability ratio (kz/kr greater than 0.1). In 1958, Nisle used the instantaneous point source solution to the diffusivity equation to solve the constant flux, isotropic, partial-penetration problem. He constructed synthetic partial-penetration problem. He constructed synthetic pressure buildup curves for various penetration pressure buildup curves for various penetration ratios, and found that theoretical buildup curves consisted of two semi-log straight line portions: an early time straight line having a slope inversely proportional to the flow capacity of the open proportional to the flow capacity of the open interval khw, and a later semi-log straight line which had a slope inversely proportional to the flow capacity of the entire formation kh. Nisle showed that it was theoretically possible to calculate the penetration ratio from the ratio of the slope of the late part to that of the early part of the buildup curve. From the calculated penetration ratio and the thickness of the known penetration ratio and the thickness of the known producing interval, the effective formation thickness producing interval, the effective formation thickness might be obtained. Later Brons and Marting computed pseudoskin effects caused by either a partially-penetrating or limited-entry line source partially-penetrating or limited-entry line source well. Their results compared closely with the steady-state solutions of Muskat. Odeh used a finite cosine transform to arrive at a solution for the steady-state flow problem where the open interval was located anywhere within the producing formation. Hantush solved the transient, anisotropic, partial-penetration problem by the successive use of LaPlace and Fourier transforms for the infinite reservoir case. He assumed the wellbore radius vanishing and the flux to be uniform for each point along the vertical section open to flow.

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*14 (04): 413–426.

Paper Number: SPE-3819-PA

Published: 01 August 1974

... fracture of finite thickness at any position within a producing interval in an infinitely large reservoir with impermeable upper and lower boundaries. This general solution also contained solutions for the cases of a single, plane (zero thickness) horizontal fracture,

**partial****penetration**of the producing...
Abstract

Although there have been many studies on unsteady behavior of wells with vertical fractures, and although there was at one time a controversy concerning the occurrence of horizontal or vertical fractures as a result of hydraulic fracturing, to date there bas been published no study of the unsteady behavior of a well containing a horizontal fracture. This is particularly surprising because such a study might have indicated significant differences between the performance of wells with horizontal fractures and those with vertical fractures. The purpose of this study was to fill that existing gap in knowledge of fractured-well behavior.An analytical solution was developed by means of the concept of instantaneous sources and Green's functions. The analytical solution modeled the behavior of constant-rate production from a well containing a single, horizontal fracture of finite thickness at any position within a producing interval in an infinitely large reservoir with impermeable upper and lower boundaries. This general solution also contained solutions for the cases of a single, plane (zero thickness) horizontal fracture, partial penetration of the producing formation, and limited flow entry throughout a producing interval. Although those are interesting solutions, the main purpose of this study was to investigate the horizontal fracture case. The analytical solution for this case was evaluated by computer to produce tables of dimensionless pressures vs dimensionless times sufficient for well-test analysis purposes. A careful analysis of the general solution for a horizontal fracture indicated the existence of four different flow periods. It appears that during the first period all production originates within the fracture, causing a typical storage-controlled period. This period is followed by a period of vertical, linear flow. There Then follows a transitional period, after which flow appears essentially radial. During the last period, the pressure is The same as that created by a line-source well with a skin effect. The skin effect is independent of time, but does depend upon the position of the pressure point It was found that there is a radius of influence beyond which flow is essentially radial for all times. Approximating solutions and appropriate time limits for approximate solutions were derived. Introduction Hydraulic fracturing has been used for improving well productivity for the last 20 years and is generally recognized as a major development in well-completion technology. There was considerable discussion in the early 1950's about the orientation and the number of fractures created by this type of well stimulation. It is now generally agreed that a vertical fracture will result if the least principal stress in the formation is horizontal, whereas a horizontal fracture will be created if the least principal stress is vertical. Further, data collected and reported by Zemanek et al. shows that hydraulic fracturing usually results in one vertical fracture, the plane of which includes the axis of the wellbore. This conclusion appears widely held today. Thus, most studies of the flow behavior for fractured wells consider vertical fractures only.However, the existence of horizontal fractures has been paved in some cases, and various authors have considered them. The steady-state behavior of horizontally fractured wells has been studied numerically by Hartsock and Warren. Their model assumed the reservoir to be homogeneous, of constant thickness, of anisotropic permeabilities, and completely penetrated by a well of small radius. A single, horizontal, symmetrical fracture of negligible thickness and finite conductivity was located at the center of the formation. Radial flow was assumed beyond a critical radius four times as large as be fracture radius, and there was no flow across the drainage radius. The only flow into the well itself was through the fracture. SPEJ P. 413^

Journal Articles

Journal:
Transactions of the AIME

Publisher: Society of Petroleum Engineers (SPE)

*Trans.*213 (01): 85–90.

Paper Number: SPE-971-G

Published: 01 December 1958

... not exist in the case of a well which

**partially****penetrates**the producing formation. As a result of this lack of symmetry the use of the classic theory in such cases become questionable. In this paper the mathematical theory has been extended to include the case of**partially****penetrating**wells. Numerical...
Abstract

Published in Petroleum Transactions, AIME, Volume 213, 1958, pages 85–90. Abstract The classic theory of pressure build-up in shut-in oil wells as developed by Horner and van Everdingen is based on two-dimensional radial symmetry in the well-reservoir system. Such symmetry does not exist in the case of a well which partially penetrates the producing formation. As a result of this lack of symmetry the use of the classic theory in such cases become questionable. In this paper the mathematical theory has been extended to include the case of partially penetrating wells. Numerical solutions illustrating the application of the equations are presented. The effect of partial penetration on pressure build-up is shown by a comparison of synthetic pressure build-up curves derived from the numerical solution of the equation for partially penetrating wells for various degrees of penetration. It is shown that partial penetration is detectable from the characteristic shape of the pressure build-up curve and that formation productivity may be calculated from the pressure build-up data in a manner identical to that described by the classic theory. Introduction The use of pressure build-up data on shut-in oil wells is a well-established technique for the measurement of reservoir productivity. The work of Horner and van Everdingen is well known. Since the publication of their pioneering work, others have elaborated and extended the technique. These authors have dealt exclusively with the case of wells that completely penetrate the producing formation. Two-dimensional radial symmetry is achieved, thereby, and the problem is greatly simplified. This ideal situation is seldom encountered in practice. But, in spite of this, the technique is often employed anyway. The question, therefore, arises: how much confidence can be placed in the results of such calculations when the data are taken on wells that only partially penetrate the producing formation? It is the purpose of this work to answer this question. Only the single-phase fluid case will be considered.

Journal Articles

Journal:
SPE Journal

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*24 (02): 811–833.

Paper Number: SPE-194189-PA

Published: 31 December 2018

... in a

**partially****penetrating**inclined fracture (PPIF) in these formations. It is necessary for the petroleum industry to conduct a pressure-transient analysis on such fractures to properly understand the major mechanisms governing the oil production from them. In this work, we develop a semianalytical model...
Abstract

Summary Field studies have shown that, if an inclined fracture has a significant inclination angle from the vertical direction or the fracture has a poor growth along the inclined direction, this fracture probably cannot fully penetrate the formation, resulting in a partially penetrating inclined fracture (PPIF) in these formations. It is necessary for the petroleum industry to conduct a pressure-transient analysis on such fractures to properly understand the major mechanisms governing the oil production from them. In this work, we develop a semianalytical model to characterize the pressure-transient behavior of a finite-conductivity PPIF. We discretize the fracture into small panels, and each of these panels is treated as a plane source. The fluid flow in the fracture system is numerically characterized with a finite-difference method, whereas the fluid flow in the matrix system is analytically characterized on the basis of the Green's-function method. As such, a semianalytical model for characterizing the transient-flow behavior of a PPIF can be readily constructed by coupling the transient flow in the fracture and that in the matrix. With the aid of the proposed model, we conduct a detailed study on the transient-flow behavior of the PPIFs. Our calculation results show that a PPIF with a finite conductivity in a bounded reservoir can exhibit the following flow regimes: wellbore afterflow, fracture radial flow, bilinear flow, inclined-formation linear flow, vertical elliptical flow, vertical pseudoradial flow, inclined pseudoradial flow, horizontal-formation linear flow, horizontal elliptical flow, horizontal pseudoradial flow, and boundary-dominated flow. A negative-slope period can appear on the pressure-derivative curve, which is attributed to a converging flow near the wellbore. Even with a small dimensionless fracture conductivity, a PPIF can exhibit a horizontal-formation linear flow. In addition to PPIFs, the proposed model also can be used to simulate the pressure-transient behavior of fully penetrating vertical fractures (FPVFs), partially penetrating vertical fractures (PPVFs), fully penetrating inclined fractures (FPIFs), and horizontal fractures (HFs).

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Oil and Gas India Conference and Exhibition, April 4–6, 2017

Paper Number: SPE-185347-MS

... Abstract The effect of bottom-hole pressure and formation pressure due to a

**partially****penetrating**well (PPW) is different from that for an open hole well. In order to analyze the effect of imperfection on pressure response type curves, this paper presents a 3D symmetry porous flow model...
Abstract

The effect of bottom-hole pressure and formation pressure due to a partially penetrating well (PPW) is different from that for an open hole well. In order to analyze the effect of imperfection on pressure response type curves, this paper presents a 3D symmetry porous flow model for circularly partially penetrating wells. Laplace transform and Fourier transform and Bessel functions are applied to obtain the analytical solution of the model. The pressure response and pressure distribution are obtained and the influence on flow regime surrounding the well and pressure response caused by partial penetration are analyzed. Research results show that when the imperfect area tends to zero, the solution of the model can be reduced to the traditional model of the perfect wells presented by Theis, demonstrating the correctness of the solution. The early-time pressure is significantly lower than the case of complete well. The pressure difference between a partially penetrating well and a completely penetrating well decreases with time increasing. Without considering the variation of spatial distribution of flow field due to imperfect well it may bring about errors of formation parameters calculated by perfect well model. Those conclusions improve the seepage model and provide theoretical guidance for the transient pressure data interpretation, formation parameters calculation and productivity prediction of partially penetrating wells. The presented research content furthers the theory of well test analysis, and builds theoretical foundation for the technologies of well testing interpretation and reservoir numerical simulation.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Latin American and Caribbean Petroleum Engineering Conference, November 18–20, 2015

Paper Number: SPE-177041-MS

... Abstract In this work, Babu and Odeh's model for estimating flow rate from

**partially****penetrating**systems is studied. A VBA code is written in order to simulate the model and its results are compared against simulations using Eclipse Software. Sensitivity to different parameters is analyzed...
Abstract

In this work, Babu and Odeh's model for estimating flow rate from partially penetrating systems is studied. A VBA code is written in order to simulate the model and its results are compared against simulations using Eclipse Software. Sensitivity to different parameters is analyzed. The results obtained are compared against Eclipse runs, it is observed that the model is reasonably accurate. It is concluded that the model is accurate within its simplistic assumptions, but it is in these same assumptions that it might yield very different results than reality itself.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Production and Operations Symposium, March 23–26, 2013

Paper Number: SPE-164500-MS

... are not always fully

**penetrating**the formations. This paper introduces a new technique for analyzing the pressure behavior of a horizontal well with multiple vertical and inclined**partially****penetrating**hydraulic fractures. The hydraulic fractures in this model could be longitudinal or transverse, vertical...
Abstract

Horizontal wells with multiple hydraulic fractures have become a common occurrence in the oil and gas industry, especially in tight formations. Published models assume that hydraulic fractures are fully penetrating the formations. However, studies have shown that fractures are not always fully penetrating the formations. This paper introduces a new technique for analyzing the pressure behavior of a horizontal well with multiple vertical and inclined partially penetrating hydraulic fractures. The hydraulic fractures in this model could be longitudinal or transverse, vertical or inclined, symmetrical or asymmetrical. The fractures are propagated in isotropic or anisotropic formations and considered having different dimensions and different spacing. This technique, based on pressure and pressure derivative concept, can be used to calculate various reservoir parameters, including directional permeability, fracture length and percentage of penetration. The study has shown that the pressure behavior of small penetration rate is similar to the horizontal wells without hydraulic fractures. A type curve matching technique has been applied using the plots of the pressure and pressure derivative curves. A set of type curves, which will be included in the paper, have been generated for the partially penetrating hydraulic fractures associated to the horizontal wells with different penetration rates. A step-by-step procedure for analyzing pressure tests using these type curves is also included in the paper for several numerical examples.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 21–24, 2008

Paper Number: SPE-116733-MS

... permeability reservoir complex reservoir pseudo skin pressure drop homogeneous reservoir hydraulic fracture linear flow regime fractured reservoir Abstract Although the effect of

**partial****penetration**of an infinite conductivity hydraulic fracture has been considered in a homogeneous reservoir...
Abstract

Abstract Although the effect of partial penetration of an infinite conductivity hydraulic fracture has been considered in a homogeneous reservoir, there is no study in similar problem in naturally fractured reservoirs. This paper presents the analysis of the solution to such problem in naturally fractured reservoirs. The method of analysis with or without type curve that enables us to evaluate the permeability in the three principal axes directions is also presented. The solution to the mathematical model was obtained in Laplace domain with elliptical flow model. Several type curves were generated to study the pressure behavior. Both the early linear and pseudo-radial flow regimes are observed. The duration of the early linear flow regime is a function of the natural fractures storativity ratio, interporosity flow coefficient and the dimensionless hydraulic fracture's height. The effect of the dimensionless hydraulic fracture's height on the duration of the linear flow becomes negligible as its dimensionless height approaches unity. Therefore there is no single unique value of a dimensionless time for the end of the linear flow regime as in the case of homogeneous reservoir. Raghavan et al (1978) determined the end of the linear flow regime in fully penetrating hydraulic fractures in homogeneous reservoir to be 0.016. This value is based on the dimensionless pressure drop only. In this study, this value was found to be 0.01 and it was evaluated with pressure derivative curve which is more accurate. Two simulated examples were used to validate the method of the analysis developed. The results obtained are in agreement with the input data. Introduction Hydraulic fracturing in the oil industry has contributed sizable reserves to the overall hydrocarbon reserves in the world. All the tight hydrocarbon reservoirs have to be fractured before they can be producible. These reservoirs are often produced with fully penetrating hydraulic fracture. A fully penetrating hydraulic fracture in a reservoir with water and hydrocarbon in contact will lead to an early or immediate water production. The only method of preventing unwanted fluid at the wellbore in a hydraulic fracture is to carry out partially penetrating hydraulic fracturing. Anderson and Stahl (1967) have shown by actual measurement that hydraulic fracture may not penetrate the entire formation thickness even when it is intended to do so. According to Tinsley et al (1969), the entire height of the hydraulic fracture may not be producing in addition to partial penetration. Moreover, not all the fractured height is propped open by propants. The unpropped height may be healed and close completely. Therefore micro seismic and production logging tools are necessary to determine the effective height of the fracture. Raghavan et al (1978) first presented the solution to a partially penetrating hydraulic fracture in a homogeneous reservoir. Their solution is based on the Green's function product solution technique presented by Gringarten and Ramey (1973). Several type curves for evaluation were presented without any example. Rodriguez et al (1984) presented type curve method of analysis for finite and infinite conductivity based on a numerical method for homogeneous isotropic system. They did not investigate the effect of vertical position on the wellbore pressure. The effect of the transition flow regime on the duration of the linear flow regime makes it necessary to study the behavior of transient flow in naturally fractured reservoirs. Moreover, the method employed by Raghavan and Rodriguez cannot be applied directly to naturally fractured reservoirs because of the transfer function. The problem has to be solved in Laplace domain before inversion to the real time domain. In this study, the elliptical flow model was applied to compute the dimensionless pressure of a partially penetrating hydraulic fracture at the wellbore. The effect of the vertical position of the fracture on the computed wellbore pressure was fully investigated.

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*J Can Pet Technol*47 (05).

Paper Number: PETSOC-08-05-63

Published: 01 May 2008

...K. Slimani; D. Tiab Often and for many reasons the wellbore does not completely

**penetrate**the entire formation, yielding a unique early-time pressure behaviour. Some of the main reasons for**partial****penetration**, in both fractured and unfractured formations, are to prevent or delay the intrusion...
Abstract

Often and for many reasons the wellbore does not completely penetrate the entire formation, yielding a unique early-time pressure behaviour. Some of the main reasons for partial penetration, in both fractured and unfractured formations, are to prevent or delay the intrusion of unwanted fluids into the wellbore, i.e., water coning. The transient flow behaviour in these types of completions is different and more complex compared to that of a fully penetrating well. This paper proposes a method for identifying, on the pressure and pressure derivative curves, the unique characteristics of the different flow regimes resulting from these types of completions and to obtain various reservoir parameters, such as vertical and horizontal permeability, fracture properties and various skin factors. Both naturally fractured and unfractured (homogeneous) reservoirs have been investigated. For a naturally fractured formation, the type curves of the pressure and pressure derivative reveal that the combination of partial penetration and dual-porosity effects yields unique finger prints at early and transition periods. These unique characteristics are used to calculate several reservoir parameters, including the storage capacity ratio, interporosity flow coefficient, permeability and pseudoskin. Equations have been developed for calculating the skin for three partial completion cases: top, centre and bottom. The analytical solution was obtained by combining the partially penetrating well model in a homogeneous reservoir with the pseudo-steady model for a naturally fractured reservoir (NFR). The interpretation of pressure tests is performed using the TDS (Tiab's Direct Synthesis) technique for analyzing log-log pressure and pressure derivative plots. The TDS technique uses analytical equations to determine reservoir and well characteristics without using type-curve matching. These characteristics are obtained from unique fingerprints, such as flow regime lines and points of intersection of these lines, which are found on the log-log plot of pressure and pressure derivative. Two numerical examples are included to illustrate the application of the. proposed technique. Introduction Consider a vertical well partially penetrating a naturally fractured reservoir, i.e., only a portion of hydrocarbon-bearing formations is perforated. The naturally fractured reservoir has an infinite radial extent. The Warren and Root (1) model is used in which the matrix blocks are replaced by a system of uniform rectangular parallelepipeds with identical properties. The fractures are assumed to be parallel with the principal axes. FIGURE 1: Different types of partially penetrating vertical wells based on the position in the perforated interval h w . Available in Full Paper The pressure solution is derived using the Laplace transformation and the separation of variables technique as proposed by Bui et al. (2) . This solution is expressed as an infinite Fourier-Bessel series in Laplace domain. The theory for a partially penetrating well in a homogenous reservoir developed by Yildiz and Bassiouni (3) is used for comparison purposes. The analytical solution for constant flow rate in Laplace space was inverted into real dimensionless pressure using the Stehfest algorithm (4) . Pressure Derivative Behaviour Four types of partial penetration or completion schemes are considered (as shown in Figure 1). A plot of the dimensionless pressure derivative t D * P' D versus t D is shown in Figures 2 and 3.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the International Oil Conference and Exhibition in Mexico, August 31–September 2, 2006

Paper Number: SPE-104059-MS

... Abstract Often and for many reasons the wellbore does not completely

**penetrate**the entire formation, yielding a unique early-time pressure behavior. Some of the main reasons for**partial****penetration**, in both fractured and unfractured formations, are to prevent or delay the intrusion of unwanted...
Abstract

Abstract Often and for many reasons the wellbore does not completely penetrate the entire formation, yielding a unique early-time pressure behavior. Some of the main reasons for partial penetration, in both fractured and unfractured formations, are to prevent or delay the intrusion of unwanted fluids into the wellbore, i.e. water coning. A similar early-time pressure behavior may be due to the presence of plugged perforations. Drilling problems associated with high mud losses when the well encounters fractures, often prevent well penetration of the total formation thickness. Penetration in naturally fractured reservoirs is usually minimal (10 to 20%), but with the right mud, it can reach 50% and in some cases 100%. Such well completions are referred to as limited-entry, restricted-entry or partially penetrating wells. The transient flow behavior in these types of completions is different and more complex compared to that of a fully penetrating well. This paper proposes a method for identifying, on the pressure and pressure derivative curves, the unique characteristics of the different flow regimes resulting from these types of completions and to obtain various reservoir parameters, such as vertical and horizontal permeability, fracture properties, and various skin factors. Both naturally fractured and unfractured (homogeneous) reservoirs have been investigated. For unfractured and homogeneous formations, a spherical or hemispherical flow regime occurs prior to the radial flow regime whenever the penetration ratio is twenty percent or less. A half-slope line on the pressure derivative is the unique characteristic identifying the presence of the spherical flow. This straight line can be used to calculate spherical permeability and spherical skin values. These parameters are then used to estimate vertical permeability, anisotropy index and skin. For a naturally fractured formation, the type curves of the pressure and pressure derivative reveal that the combination of partial penetration and dual-porosity effects yields unique finger prints at early and transition periods. These unique characteristics are used to calculate several reservoir parameters, including the storage capacity ratio, interporosity flow coefficient, permeability and pseudo-skin. Equations have been developed for calculating the skin for three partial completion cases: top, center and bottom. The analytical solution was obtained by combining the partially penetrating well model in a homogeneous reservoir with the pseudo-steady model for a naturally fractured reservoir. The interpretation of pressure tests in both systems, i.e. fractured and unfractured reservoirs, is performed using Tiab's Direct Synthesis (TDS) technique for analyzing log-log pressure and pressure derivative plots. TDS uses analytical equations to determine reservoir and well characteristics without using type-curve matching. These characteristics are obtained from unique fingerprints, such as flow regime lines and points of intersection of these lines, that are found on the log-log plot of pressure and pressure derivative. It isapplied to both drawdown and buildup tests. Several numerical examples are included to illustrate the step-by-step application of the proposed technique. Introduction Over the last four decades, naturally fractured reservoirs have been a topic of continuous research due to the fact that many producing fields of the world are found in such type of formations. These reservoirs differ in geological and petrophysical properties from homogeneous reservoirs. Additionally, in many oil and gas reservoirs the producing wells are completed as partially penetrating wells; that is, only a portion of the pay zone is perforated. This may be done for a variety of reasons, but the most common one is to prevent or delay the intrusion of unwanted fluids into the wellbore. Generally, pressure behavior of a partially penetrating vertical well in naturally fractured reservoirs has been considered and interpreted as fully penetrating, isotropic fracture permeability with the existing of only the mechanical skin. But in the reality, the partial penetration effect causes a characteristic shape on the pressure derivative curves (which allowsestimation of some reservoir parameters) at early and transition time and differ from that of fully penetrating. Furthermore, it causes an additional pressure drop near the wellbore that is known as the pseudo-skin.

Proceedings Papers

Publisher: Petroleum Society of Canada

Paper presented at the Canadian International Petroleum Conference, June 7–9, 2005

Paper Number: PETSOC-2005-263

...1 PAPER 2005-263 Pressure Transient Analysis of

**Partially****Penetrating**Wells in a Naturally Fractured Reservoir K. SLIMANI Sonatrach/AMT/PED D. TIAB Oklahoma University This paper is to be presented at the Petroleum Society s 6th Canadian International Petroleum Conference (56th Annual Technical...
Abstract

Abstract Often and for many reasons the wellbore does not completely penetrate the entire formation, yielding a unique early-time pressure behavior. Some of the main reasons for partial penetration, in both fractured and unfractured formations, are to prevent or delay the intrusion of unwanted fluids into the wellbore, i.e. water coning.. The transient flow behavior in these types of completions is different and more complex compared to that of a fully penetrating well. This paper proposes a method for identifying, on the pressure and pressure derivative curves, the unique characteristics of the different flow regimes resulting from these types of completions and to obtain various reservoir parameters, such as vertical and horizontal permeability, fracture properties, and various skin factors. Both naturally fractured and unfractured (homogeneous) reservoirs have been investigated. For unfractured and homogeneous formations, a spherical or hemispherical flow regime occurs prior to the radial flow regime whenever the penetration ratio is twenty percent or less. A half-slope line on the pressure derivative is the unique characteristic identifying the presence of the spherical flow. This straight line can be used to calculate spherical permeability and spherical skin values. These parameters are then used to estimate vertical permeability, anisotropy index and skin. For a naturally fractured formation, the type curves of the pressure and pressure derivative reveal that the combination of partial penetration and dual-porosity effects yields unique finger prints at early and transition periods. These unique characteristics are used to calculate several reservoir parameters, including the storage capacity ratio, interporosity flow coefficient, permeability and pseudo-skin. Equations have been developed for calculating the skin for three partial completion cases: top, center and bottom. The analytical solution was obtained by combining the partially penetrating well model in a homogeneous reservoir with the pseudo-steady model for a naturally fractured reservoir. The interpretation of pressure tests in both systems, i.e. fractured and unfractured reservoirs, is performed using Tiab's Direct Synthesis (TDS) technique for analyzing log-log pressure and pressure derivative plots. TDS uses analytical equations to determine reservoir and well characteristics without using typecurve matching. These characteristics are obtained from unique fingerprints, such as flow regime lines and points of intersection of these lines, that are found on the log-log plot of pressure and pressure derivative. It is applied to both drawdown and buildup tests. Several numerical examples are included to illustrate the step-by-step application of the proposed technique. Introduction Over the last four decades, naturally fractured reservoirs have been a topic of continuous research due to the fact that many producing fields of the world are found in such type of formations. These reservoirs differ in geological and petrophysical properties from homogeneous reservoirs. Additionally, in many oil and gas reservoirs the producing wells are completed as partially penetrating wells; that is, only a portion of the pay zone is perforated. This may be done for a variety of raisons, but the most common one is to prevent or delay the intrusion of unwanted fluids into the wellbore.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE International Petroleum Conference in Mexico, November 7–9, 2004

Paper Number: SPE-92116-MS

..., an extension of the Warren and Root model, existing an interaction between matrix, vugs, and fracture systems. The new model assumes a single

**partially****penetrating**well producing from an undersaturated system considering several boundary conditions. The inner boundary condition at the wellbore can...
Abstract

Abstract This work presents results on both pressure and production responses during transient and boundary-dominated flow periods, in naturally fractured reservoirs with vuggy porosity. It is considered that triple-porosity systems may not be treated as dual-porosity reservoirs, mainly because fractures and vugs have different geological origin and therefore their interaction with matrix blocks do not have to be similar. New solutions are presented for the case where there is no primary flow through vugs, an extension of the Warren and Root model, existing an interaction between matrix, vugs, and fracture systems. The new model assumes a single partially penetrating well producing from an undersaturated system considering several boundary conditions. The inner boundary condition at the wellbore can be considered either a fixed flux or constant pressure boundary. The upper, lower, and outer boundaries can be modeled at constant pressure or prescribed flux. These boundaries can be closed, or the influx can be modeled via a step rate or a ramp rate function. The step rate function could represent a waterflood effect and the ramp rate function a natural waterdrive or gas drive cap effect. The transient pressure response is also analyzed by considering an infinite outer boundary. In transient well tests and decline curve analysis, the effects of triple-porosity for different upper and lower boundaries conditions are analyzed, besides considering closed upper and lower boundaries as it has been done in the literature. It is shown that for triple porosity systems the derivative function may exhibit different behavior to that of double-porosity reservoirs. It is demonstrated that the presence of vugs and caves may have a definitive influence on transient well test and decline curve behaviors. Synthetic and field examples are presented to illustrate the methodology proposed in this work. Introduction In most oil wells, it is common to perforate only a fraction of the oil-bearing formation thickness to delay or reduce gas and/or water coning. The transient pressure response of partially penetrated wells, producing from a homogeneous reservoir with top and bottom impermeable boundaries, has been the subject of study of many researchers, most of these reviewing the pseudoskin effect due to flow convergence near the wellbore. However, when a gas cap and/or an active aquifer are present it is expected to have a vertical pressure support as oil is produced. Abbaszadeh and Hegeman 1 , presented solutions for partially penetrated slanted wells in infinite homogeneous reservoirs with various upper and lower boundary conditions, including no-flow and constant pressure at the gas/oil and/or oil/water interface. Ozkan and Raghavan 2 derived point-source solutions for vertical wells in infinite and bounded systems, in the Laplace domain. Streltsova-Adams 3 extended the analytical solution given by Hantush 4 including the effect of an overlying gas cap layer. In a more recent work, Ozkan and Raghavan 5 presented a solution for a limited-entry slanted well in an infinite reservoir with closed top and bottom boundaries. These authors used the Laplace transformation, which could be easily extended to naturally fractured systems. Bui et al . 6 studied the transient behavior of limited-entry wells in naturally fractured reservoir with the double-porosity formulation of Warren and Root 7 . These authors considered impermeable upper and lower boundaries.

Proceedings Papers

Publisher: Petroleum Society of Canada

Paper presented at the Canadian International Petroleum Conference, June 10–12, 2003

Paper Number: PETSOC-2003-009

... Abstract With the similar methods in Ref.[1, 2, 3, 4, 5], taking a

**partially****penetrating**well as a uniform line sink in three dimensional space, by developing necessary mathematical analysis, this paper presents**partially****penetrating**wells pressure drawdown formulae in a circular cylinder...
Abstract

Abstract With the similar methods in Ref.[1, 2, 3, 4, 5], taking a partially penetrating well as a uniform line sink in three dimensional space, by developing necessary mathematical analysis, this paper presents partially penetrating wells pressure drawdown formulae in a circular cylinder drainage volume with constant pressure at edge boundary. We also provide the formulae to calculate pseudo-skin factor due to partial penetration. And if the producing well length is equal to the pay zone thickness, the equations of fully penetrating wells are obtained. Introduction For both fully and partially penetrating vertical wells, steady-state and unsteady-state pressure-transient testings are useful tools for evaluating in-situ reservoir and wellbore parameters that describe the production characteristics of a well. The use of transient well testing for determining reservoir parameters and well productivity has become common, in the past years, analytic solutions have been presented for the pressure behavior of partially penetrating vertical wells. The problem of fluid flow into wells with partial penetration has received much attention in the past years, both in ground-water hydrology ([ 7, 8, 9, 10]) and in petroleum engineering ([ 12, 13, 14, 15, 16, 17, 18]). In many oil and gas reservoirs the producing wells are completed as partially penetrating wells; that is, only a portion of the pay zone is perforated. This may be done for a variety of reasons, but the most common one is to prevent or delay the unwanted fluids into the wellbore. The exact solution of the partial penetration problem presents great analytical problems because the boundary conditions that the solutions of the partial differential equations must satisfy are mixed; i.e., on one of the boundaries the pressure is specified on one portion and the flux on the other. This difficult occurs at the wellbore, for the flux over the nonproductive section of the well is zero, the potential over the perforated interval must be constant. This problem may be overcome in the case of constant rate production by making the assumption that the flux into the well is uniform over the entire perforated interval, so that on the wellbore the flux is specified over the total formation thickness. This approximation naturally leads to an error in the solution since the potential (pressure) will not be uniform over the perforated interval, but it has been shown that this occurrence is not too significant. Many different techniques have been used for solving the partial penetration problem, namely, the pointsource solution([6]), a thermal model([10]), finite difference method([11]), Fourier, Hankel and Laplace transforms ([14, 15, 16]), Green's functions([17]). The analytical expressions and the numerical results obtained for reservoir pressures by different methods were essentially identical, however, there are some differences between the values of wellbore pressures computed from numerical models and those obtained from analytical solutions ([18]) - The primary goal of this study is to present new pressure drawdown formulae of partially penetrating wells. Analytical solutions are derived by making the assumption of uniform fluid withdrawal along the portion of the wellbore open to flow.

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