Dimensionless Wellbore Pressure Research Articles

INFLUENCE OF DIMENSIONLESS WELLBORE RADIUS ON DIMENSIONLESS PRESSURE AND PRESSURE DERIVATIVES

Effects of wellbore radius on pressure derivatives of a vertical oil well is a major concern to oil and gas industry operators who intend to execute fluid production from a vertical oil well. Reservoir deliverability depends on several factors: the dimensionless wellbore radius. This study examines the influence of the dimensionless wellbore radius on the pressure and pressure derivative. A set of polynomials was implemented to calculate the dimensionless pressure variable and the pressure derivative. The polynomials were put into a computer program developed in the study to create a fast means through which the effect of wellbore radii can be studied easily. However, when several dimensionless radii were used as input parameters for the developed computer program, it was observed that the lower the dimensionless wellbore pressure, the higher the pressure derivative and vice versa. Also, lower dimensionless time yielded higher values of the dimensionless pressure variable. Results from the developed computer model were validated by comparing results from the model to those obtained from a published article while using the same input parameters (tD and rWD). Moreover, the percentage error was estimated to be less than 0.02%.

Pressure transient analysis for multi-vug composite fractured vuggy carbonate reservoirs

Vugs usually exist dispersedly in the formation of fractured vuggy carbonate reservoirs. The vug volume is an important parameter, owing to its significant contribution to oil reserves. However, at present, no reliable method is available for estimating the vug volume. This paper presents an analytical multi-zonal composite reservoir well test model that can determine the vug volume.First, the mathematical model is established according to the simplified physical model, and the analytical solution to the mathematical model is derived using Laplace transform. Second, based on the analytical solution, the type curves of dimensionless wellbore pressure and derivative versus dimensionless time, are plotted through numerical inversion. We use the two-vug model as an example to recognize the flow characteristics, and observe that 10 distinct flow regimes exist with two concaves. Furthermore, by comparing the single-vug and multi-vug models, an interesting finding is that one more concave will be added into the derivative curve when one more vug is present in the formation. The vug volume can be obtained by means of the pressure transient analysis. Also, the characteristics of our model are further exhibited by comparing it with the dual porosity model and the conventional composite model. And, a comparison with the conventional composite model and a field case are conducted to verify the validity and reliability of the proposed model in this study. Finally, the effects of each parameter on the pressure and derivative responses are analyzed. The pressure behavior is influenced by the mobility ratio, storage ratio, vug storage constant, and region radius. The mobility and storage ratios mainly affect the vug storage regimes, while the vug storage constant has an important impact on the concave depth and location, and the radial flow regime is dominated by the region radius.

A New Pseudo Steady-State Constant for a Vertical Well with Finite-Conductivity Fracture

The Pseudo Steady-State (PSS) constant bDpss is defined as the difference between the dimensionless wellbore pressure and dimensionless average pressure of a reservoir with a PSS flow regime. As an important parameter, bDpss has been widely used for decline curve analysis with Type Curves. For a well with a finite-conductivity fracture, bDpss is independent of time and is a function of the penetration ratio of facture and fracture conductivity. In this study, we develop a new semi-analytical solution for bDpss calculations using the PSS function of a circular reservoir. Based on the semi-analytical solution, a new conductivity-influence function (CIF) representing the additional pressure drop caused by the effect of fracture conductivity is presented. A normalized conductivity-influence function (NCIF) is also developed to calculate the CIF. Finally, a new approximate solution is proposed to obtain the bDpss value. This approximate solution is a fast, accurate, and time-saving calculation.

A Finite-Conductivity Horizontal-Well Model for Pressure-Transient Analysis in Multiple-Fractured Horizontal Wells

Summary In this paper, we propose a new model for pressure-transient analysis in multiple-fractured horizontal wells (MFHWs) with consideration of pressure drop along the wellbore. To make the physical model better understood, the whole formation is divided into three parts: (1) reservoir, (2) fracture, and (3) wellbore. With incorporating frictional and acceleration pressure drops, a mathematical model with a finite-conductivity horizontal well (FCHW) is developed. Newton-Raphson iterations are used to solve the mathematical model and obtain the transient-pressure solutions of the MFHW. Model verification is performed by comparing with the solutions from a numerical software. On the basis of the field cases from the Ordos Basin, performance prediction, sensitivity analysis, type-curve matching, and evaluations of uncertainty parameters are conducted. Results show that the contribution of wellbore hydraulics to the total pressure drop increases first and then decreases after reaching the peak value. Ignoring wellbore hydraulics would cause erroneous results during the well-performance forecast. In addition, the dimensionless wellbore pressure of the MFHW increases with an increase in Reynolds number (Re); it decreases as the reservoir/wellbore constant (ChD) increases. Furthermore, the impact of pressure drop on the pressure performance of the MFHW becomes more serious with the increasing Re or the decreasing ChD.

Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs

The paper developed a new constant-rate solution for non-planar asymmetric fracture connected to a vertical wellbore in hydrocarbon reservoirs. The non-planar asymmetric fracture has been modeled by combination of two unequal length wings with varied intersection angles. The type curves were drawn in the case of varied fracture conductivities, non-planar angles, and asymmetric factors. The results indicated that fracture conductivity, non-planar angle, and asymmetric factor have coefficient effects on the pressure response, which makes the dimensionless wellbore pressure and pressure derivative curves deviate from the conventional type curves of vertical fractured wells. The dimensionless wellbore pressure and pressure derivative curves are divided into five flow periods. The non-planar angle and asymmetry factor have their unique influences on the shape of type curves during different periods under the different fracture conductivities. The termination of bilinear flow and formation linear flow happened increasingly early with the decrease of non-planar angle or the increase of asymmetry factor. As the fracture conductivity increases, the effect of non-planar angle on the curves moves from the early flow period to the late flow period while the trend of effect of asymmetric factor is quite opposite. The new method lays the foundation of the pressure analysis of the complicated fracture system in developing hydrocarbon reservoirs.

Derivation and application of mathematical model for well test analysis with variable skin factor in hydrocarbon reservoirs

Skin factor is often regarded as a constant in most of the mathematical model for well test analysis in oilfields, but this is only a kind of simplified treatment with the actual skin factor changeable. This paper defined the average permeability of a damaged area as a function of time by using the definition of skin factor. Therefore a relationship between a variable skin factor and time was established. The variable skin factor derived was introduced into existing traditional models rather than using a constant skin factor, then, this newly derived mathematical model for well test analysis considering variable skin factor was solved by Laplace transform. The dimensionless wellbore pressure and its derivative changed with dimensionless time were plotted with double logarithm and these plots can be used for type curve fitting. The effects of all the parameters in the expression of variable skin factor were analyzed based on the dimensionless wellbore pressure and its derivative. Finally, actual well testing data were used to fit the type curves developed which validates the applicability of the mathematical model from Sheng-2 Block, Shengli Oilfield, China.

Transient pressure responses of vertical fracture with finite conductivity in shale gas reservoir

Shale gas reservoirs usually have extremely low permeability and relatively low porosity. Horizontal wells with massive multi stage hydraulic fracturing treatment has been performed on these reservoirs to achieve economic breakthrough. This paper considered shale gas dispersion and desorption in the matrix, presents a trilinear flow model for finite conductivity on vertical fracture in shale gas reservoir. We applied the Laplace transformation on this model and solved it with the Stehfest numerical inversion method. Typical pressure test curves were plotted, and the effects of sensitive parameters on the dimensionless wellbore pressure were also studied. Adsorption coefficient reflects the adsorption capacity of the matrix on shale gas, which implies the larger adsorption coefficient rate, the higher adsorption capacity, with low desorption capabilities. Fracture conductivity factor denotes the conductivity of fractures; the larger the fracture conductivity factor, the stronger the fracture conductivity capacity, which means limited flow rate within a short time. Inter-porosity transfer coefficient determines the appearance, time and height of the transition stage. If the inter-porosity transfer coefficient became smaller, the transition stage will occur early. Storativity ratio determines the extent of crossflow time; the smaller the storativity ratio is, the earlier the appearance time will be, and crossflow time will be longer.

An Early-Time Model for Drawdown Testing of a Hydrate-Capped Gas Reservoir

Summary Development of natural gas hydrates as an energy resource has gained significant interest during the past decade. Hydrate reservoirs may be found in different geologic settings including deep ocean sediments and arctic areas. Some reservoirs include a free-gas zone beneath the hydrate and such a situation is referred to as a hydrate-capped gas reservoir. Gas production from such a reservoir could result in pressure reduction in the hydrate cap and endothermic decomposition of hydrates. Well testing in conventional reservoirs is used for estimation of reservoir and near-wellbore properties. Drawdown testing in a hydrate-capped gas reservoir needs to account for the effect of gas from decomposing hydrates. This paper presents a 2D (r,z) mathematical model for a constant-rate drawdown test performed in a well completed in the free-gas zone of a hydrate-capped gas reservoir during the earlytime production. Using energy and material balance equations, the effect of endothermic hydrate decomposition appears as an increased compressibility in the resulting governing equation. The solution for the dimensionless wellbore pressure is derived using Laplace and finite Fourier cosine transforms. The solution to the analytical model was compared with a numerical hydrate reservoir simulator across some range of hydrate reservoir parameters. The use of this solution for determination of reservoir properties is demonstrated using a synthetic example. Furthermore, the solution may be used to quantify the contribution of hydrate decomposition on production performance.

Constant rate solutions for a fractured well with an asymmetric fracture

This paper presents solutions for the pressure response on hydraulically fractured wells flowing at constant flow rate through an asymmetric vertical fracture. The pressure behavior of wells intercepting asymmetric fractures of both infinite and finite conductivity was investigated by solving numerically and analytically the mathematical model. The new solutions developed for the dimensionless wellbore pressure under production at constant flow rate are presented in terms of an asymmetry factor ξ. New curves for these systems were generated and the deviation from the classical solution was readily detected. Some qualitative criteria to interpret the intensity of this effect are provided. Results of our investigation indicated that at early times for fractures of moderate conductivity ( C D<5) the characteristic slope of one fourth is present, except for cases of strong asymmetry (0.85< ξ≤1) where no evidence of straight line having one fourth slope was observed. However, it was also detected that at intermediate fracture conductivities (5< C D<50), even the case of complete asymmetry shows the characteristic slope of one fourth. It was also observed that as the asymmetry factor increases, the end of the bilinear flow occurs earlier. Our results are relevant in improving the fracture characterization of fractured wells.

Analytical And Statistical Analyses of Pseudo Skin Factors For Horizontal Wells

Abstract The pressure behaviour of horizontal wells in bounded anisotropic reservoirs has been investigated in this paper. New equations for calculating the effective length of the horizontal well, mechanical skin factor, shape factor, and productivity index of horizontal wells are presented. It is determined that the performance of a horizontal well of half-length Xf1 with mechanical skin can be replaced by the performance of an equivalent horizontal well of half-length Xf2 with no mechanical skin. The statistical analysis showed that pseudo skin factor for a horizontal well in a given drainage area can be directly correlated with formation thickness, and natural logarithms of horizontal well half-length and permeability anisotropy. These statistical correlations can easily be used to predict skin damage around horizontal wellbores, to evaluate stimulation treatment of horizontal wells, and to compare their performance with unstimulated vertical wells without the need to use complex numerical and/or analytical solutions which require extensive computer use. Introduction The growing interest in field development using horizontal well drilling necessitates development of new concepts for improved performance prediction of these wells. To predict the productivity index of horizontal wells requires the determination of the pressure drop due to formation damage around such wells. Long duration of formation exposure to drilling fluids may be the main reason for the unexpected failure of some horizontal wells(1, 2) drilled in favourable reservoirs. The combined effect of the formation damage and the presence of impermeable barriers and low permeability regions in the vicinity of some sections of the horizontal wellbore will manifest itself as mechanical skin damage in well test analysis if the total drilled (or perforated) length of the horizontal section is used in the calculations. The total skin factor obtained from a well test is the summation of the mechanical skin factor and pseudo skin factor. The purpose of this paper is to present new analytical equations for calculating pseudo skin factor and mechanical skin factor for horizontal wells during the unsteady state and pseudo steady state flow periods. The ne\v equations can be used to determine:the fraction of horizontal well length contributing to unrestricted production, andhorizontal well productivity index based on the equivalent non-damaged length of the horizontal wellbore. Furthermore, statistical correlations for pseudo skin factor, mechanical skin factor, and the fraction of horizontal well length contributing to unrestricted production are presented that can be used to evaluate the performance of horizontal wells without the need to use complex numerical and/or analytical solutions. Determination of Horizontal Well Productivity Index Malekzadeh and Abdelgawad(3) presented the dimensionless wellbore pressure, pseudo skin factor, and effective horizontal well length for horizontal wells in bounded anisotropic reservoirs, shown in Figure 1, as: Equation 1 (available in full paper) where the dimensionless parameters are defined as: Equation 2–12 (available in full paper) The late time approximation to the full solution, Equation (1) was derived as(3): Equation 13 (available in full paper)

Well-test Analysis For a Wen In a Finite, Circular Reservoir

Abstract This study presents drawdown and buildup pressure derivative type-curves for a well producing at a constant rate from the centre of a finite, circular reservoir. Early time response (wellbore storage and skin effects) is correlated by C De2s, and late time response (outer boundary effects) by reD2/CD. The outer boundary may be closed or at a constant pressure. Design relations are developed for the time to the beginning and the end of infinite-acting radial flow. Producing time effects on buildup response are also discussed. Introduction Transient pressure response for a well producing from a finite Reservoir of a circular, square, and rectangular drainage shapes has been studied by many investigators(1–8).Mishra and Ramey(9) presented a buildup pressure derivative type-curve for a well with storage and skin, and producing from the centre of a closed, circular reservoir. Their type-curve applies for large producing times such that lpD &amp;gt; lDpss. This work presents drawdown and buildup pressure derivative type-curves for a well producing at a constant rate from the centre of a finite, circular reservoir. The outerboundary may be closed or at a constant pressure. The differences between the responses for a well in a closed, circular reservoir (fully a developed field) and a well in a circular reservoir with a constant-pressure outer boundary (active edge-water drive system or developed five-spot, fluid injection pattern) are discussed. Design relations are developed to estimate the time period which correspond to infinite-acting radial flow regime. Producing time effects on buildup responses are studied using the slope of a dimensionless Agarwal(10) buildup graph. Theory The dimensionless wellbore pressure drop for a constant-rate well with storage and skin may be expressed as(1): Equation (1) Available In Full Paper. where L −1 is the inverse Laplace transform operator. In Equation (1), PD refers to the dimensionless wellbore pressure drop in Laplace transform without storage of skin. For the case of a constant-rate well producing from the centre of a closed circle, the expression for PD is(1). TABLE 1: Dimensionless wellbore drawdown pressure and pressure derivative expressions for a well in a finite, circular reservoir. Illustrations available in full paper. Equation (2) Available In Full Paper. For the case of a constant-rate well producing from the centre of a circular reservoir with a constant-pressure outer boundary, The expression for P D is (1): Equation (3) Available In Full Paper. The dimensionless wellbore pressure drop from Equation (1) was obtained by inverting the Laplace space solution numerically with the Stehfest(11) algorithm. Drawdown Responses Table 1 shows the dimensionless wellbore pressure drop and the semi-log pressure derivative expressions for a well in a finite, circular reservoir during specific flow periods. All expressions in Table 1 may be written as combinations of lD/C D, CDe 2s, and le D2/CD. For example, Equation (4) Available In Full Paper. FIGURE 1: Verification or CDe 2s and re D2/CD as the correlating parameters for the drawdown responses. Illustrations at available in full paper. FIGURE 2: Drawdown pressure derivative type-curve. Illustrations at available in full paper. and

Well Test Analysis for a Well in Rectangular Drainage Area with Constant Pressure or Mixed Boundaries.

The behavior of pressure during drawdown and buildup testing for a well located anywhere in a rectangular drainage area with constant pressure or mixed boundaries was investigated in this study, concentrated on drainage areas of 2:1 and 4:1 rectangles, and dimensionless wellbore pressure and drainage area average pressures are presented in analytical and graphical forms. The time required to reach steady state conditions, the time of the end of semilog straight line and the shape factors for different well locations and boundary conditions are determined. Horner, MDH and extended Muskat plots were obtained for different locations of well and boundary conditions. Also, MBH average and boundary pressure functions were presented in graphical forms. The producing time, before shut-in, affects the behavior of pressure buildup, as long as the system does not reach steady state conditions during production time.

Wellbore Pressures in Reservoirs With Constant-Pressure or Mixed No-Flow/Constant-Pressure Outer Boundary

Abstract This paper reports On a general relationship existing between wellbore pressures in reservoirs with constant pressure or mixed no-flow/constant-pressure outer pressure or mixed no-flow/constant-pressure outer boundary and in related closed reservoirs, and uses this to derive expressions for shape factors, wellbore pressures, and Matthews-Brons-Hazebroek (MBH) functions for static reservoir pressure. Introduction Effects of reservoir shape and well location on wellbore pressures have been studied extensively for closed pressures have been studied extensively for closed reservoirs, but only a few cases have been presented for reservoirs with constant pressure or mixed no-flow/ constant pressure outer boundary. However, the latter group can be increased considerably by the methods presented in this paper, since the wellbore pressures for a wide variety of such reservoirs can be expressed by differences of wellbore pressures from related closed reservoirs. As a consequence, the shape factor, CA, and the dimensionless wellbore pressure in the steady-state period, can be determined readily from existing tables for a large number of cases. Moreover, the end of the infinite-acting period, tDAeia, can be estimated, and a rough upper period, tDAeia, can be estimated, and a rough upper bound can be given for the start of the steady-state period, tDASS. Modifications of standard methods to period, tDASS. Modifications of standard methods to determine the static reservoir pressure from buildup data also are obtained readily. The results of this paper are related closely to those of Earlougher et al. and Ramey et al. Tables from those papers also are used as a basis for this paper. All cases papers also are used as a basis for this paper. All cases that can be handled by direct use of these tables are included in Fig. 1. Tables presented by Earlougher also are used. By using the approach of Ramey and Cobb, it is possible to work with either dimensionless wellbore pressures or with MBH functions. These functions also pressures or with MBH functions. These functions also are used to determine the static pressure from Horner analysis. Theory Only single-well reservoirs that can be generated by regular patterns of producing and injecting (image) wells, with each well having the same rate, q, in absolute value, are considered. Storage and skin effects are not included. The dimensionless pressure drop for the actual well, .....................................(1) can therefore be expressed as a sum of exponential integrals. Here, A denotes drainage area, pi initial fully static pressure, pi flowing wellbore pressure, and .....................................(2) dimensionless time based on the wellbore radius. Dimensionless time based on drainage area, tDA= r2wtD/A, also is used. Consider PwD for such a reservoir with constant pressure or mixed no-flow/constant-pressure outer pressure or mixed no-flow/constant-pressure outer boundary. Since the reservoir can be generated by a regular pattern of producing and injecting wells, a related closed pattern of producing and injecting wells, a related closed reservoir is obtained by letting all wells be producers. By superimposing the two patterns, each injecting well cancels a producing well, and a new pattern of only producing wells is obtained, with each well having rate 2q producing wells is obtained, with each well having rate 2q and draining an area 2A. The superscript "new" refers to the new closed reservoir and "old" refers to the first closed reservoir. It follows that PwD(tD, A) 2P new WD (tD, 2A) -P old wD (tD, A)......(3) Moreover, since .....................................(4) where PDMBH denotes the MBH function, and tD(2A)= (1/2)tDA, it follows that .....................................(5) JPT P. 1613

Transient Flow in Elliptical Systems

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

Transient Pressure Behavior for a Well With a Finite-Conductivity Vertical Fracture

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

Vertical Fracture Height: Effect on Transient Flow Behavior

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

A Simple Method for Determining Well Pressure in Closed Rectangular Reservoirs

In 1968, Earlougher et al., introduced a "buildingblock" concept, using a constart-rate well producing from the center of a closed square, to generate solutions for pressure behavior in several rectangular systems. Tables of the dimensionless pressure drop function for the square system are presented in Ref. 1, along with illustrations of the technique. Their procedure, while a step toward reducing laborious procedure, while a step toward reducing laborious computations, can sometimes be quite tedious. This note describes a further simplification for obtaining dimensionless pressures for these other rectangular systems at the producing well; other points within these systems are not considered. Conveniently, Earlougher et al. have tabulated the Matthews-Brons-Hazebroek pressure correction function, PDMBH, for 16 closed rectangular systems. These correction functions provide the basis from which the dimensionless wellbore pressure drop functions can easily be obtained. In a recent comprehensive review of pressure buildup analysis, it was shown thatwhereSolving Eq. 1 for pD yieldsIt is clear from Eq. 5 that pD is a function of pDMBH, tDA, and A/rw2. Thus, given pDMBH and A/rw2 for a particular system, pD as a function of tDA can be particular system, pD as a function of tDA can be calculated directly. Columns 2 and 3 of Table 1 present the pD and pDMBH data of Ref. 1 for the case of a well in the pDMBH data of Ref. 1 for the case of a well in the center of a square with it value for A/rw of 2,000. The pD data tabulated in Column 4 are obtained by direct substitution of tDA and pDMBH of Columns 1 and 3, respectively, into Eq. 5. As can be seen, the solutions in Columns 2 and 4 are identical. For dimensionless times less than those given in Table 1, pD can be obtained from pD can be obtained fromFor times greater than those given in Table 1, pD can be obtained bywhere CA is the reservoir shape factor, and is be Euler constant - approximately 1.7811. P. 1305