Finite-conductivity Vertical Fracture Research Articles

Practical pressure-transient analysis solutions for a well intercepted by finite conductivity vertical fracture in naturally fractured reservoirs

The semi-analytical or numerical solutions are the most used for pressure-transient analysis (PTA) for finite-conductivity fractured wells in naturally fractured reservoir now. However, the semi-analytical solutions are time-consuming and easily lead to inaccuracy at low fracture conductivity with less discretized segments. The numerical solutions are more time-consuming and usually face convergence and stability problems. These solutions are not convenient for practical well test analysis in Oilfield. Our motivation for this paper is to develop simply, quickly and stably analytical PTA solutions for finite-conductivity fractured wells in naturally fractured reservoir that can serve the well test analysis in Oilfield. We first derive a transformation method which can extend finite-conductivity fracture solutions in homogeneous reservoir to dual-porosity reservoir. Thereafter, analytical solutions for finite-conductivity fractured wells in homogeneous reservoir are presented. Then we provide a series of analytical finite-conductivity fracture solutions in dual-porosity reservoir under different boundary conditions based on the transformation method and homogeneous reservoir solutions. Six solutions given in the published literatures are used to verify the proposed solutions under different dimensionless fracture conductivity (FcD). Results indicate that (a) the proposed solutions are valid for FcD>1 with maximum error (5%) and this error will quickly decreases to around 1%; (b) the proposed solutions are applicable for infinite, closed and constant-pressure boundary conditions; (c) the proposed solutions can be extended to triple-porosity naturally fractured reservoirs after changing the f(s) function; (d) the proposed solutions are accurate and can greatly shorten the time of well test interpretation comparing to conventional solutions. Finally, the pressure-buildup data from Shunbei Oilfield is interpreted using the proposed solutions to verify its feasibility and practicability. The findings of this work can help for better understanding of its potential in quick well test analysis for field cases.

A quick analysis approach for early-time well test data of vertically fractured wells: Applications in Changqing oilfield, China

The increasing development activities for low-permeability even ultra-low permeability reservoirs worldwide present many challenges to well test interpretation. On the one hand, the increasing well number puts higher requirements on the reduction of interpretation time. On the other hand, much test data locates at the early-time regimes, so incomplete flow regimes pose serious non-uniqueness. To solve the problems above, we first introduce an analytical solution with simple form, fast calculation speed, and accurate results for finite-conductivity vertical fracture model on the basis of our previous work, which is further used to shorten the inversion time and obtain short-time asymptotic solution in the real space. We verify its accuracy by the semi-analytical model proposed by Cinco-Ley et al. and numerical model built in KAPPA. Results indicate that the analytical solution is valid when FcD>1. Then, the wellbore storage (CD) and skin (S) are extracted by the short-time asymptotic solution of proposed analytical solution to construct a novel type curve. The type curve can aid the conventional interpretation to obtain more reliable CD and S, thus reducing the non-uniqueness. We mainly divided the type curve into three typical regimes. The first regime is a straight line with zero-slope at early time which is dominated by CD. Under the larger CD, the duration of straight line lasts longer and the position is lower on the log-log plot. The second is a transition between two straight-line regimes which reflects the S. The transition appears the later under the larger S. The third is also a constant-slope straight line at late time and we think it is no use for well test inversion. Finally, the methodology is applied in Changqing oilfield cases, which illustrate its applicability. This work can help for better understanding of its potential in improve the interpretation efficiency and reduce the non-uniqueness.

A hybrid analytic solution for a well with a finite-conductivity vertical fracture

In the existing well test models of finite-conductivity fracture, the semi-analytical solution has a particularly slow inversion speed, and the analytical model may encounter some computation problems, such as convergence and inaccurate results. To overcome the disadvantages above, we establish an influence function of the fracture conductivity (IFFC) in the Laplace space based on both the solution for infinite-conductivity vertical fracture and the solution for trilinear finite-conductivity vertical fracture, then develop an approximate analytic model with simple form, fast calculation speed, and accurate results for a well with a finite-conductivity vertical fracture. The IFFC increases as the dimensionless time increases and decreases as the fracture conductivity (FcD) increases, eventually tends to a constant. We can obtain the approximate time when bilinear flow and linear flow appear and end from the curve of the IFFC versus time. Additionally, the approximate model is verified with different solutions given in the literature. Results indicate our model is valid for FcD>1. In particular, the IFFC is also applicable for closed boundary and constant pressure boundary. Finally, the recorded pressure data is interpreted by the model proposed in this paper. Comparison with well test software demonstrates the accuracy of the developed model. The findings of this work can help for better understanding of its potential in the quick interpretation of well test data from the oilfield.

Well test analysis for a well with stress-sensitive natural fracture permeability and stress-dependent hydraulic fracture conductivities

The commonly used methods in dual-porosity systems assume constant reservoir permeability over an entire range of formation pressure and constant hydraulic fracture conductivity inside the fractures. In this paper, the hydraulic fracture conductivity and natural fracture permeability were considered as a function of pressure. Based on finite-conductivity vertical fracture bilinear flow model in stress sensitive dual-porosity reservoirs, the analytical solutions were solved by perturbation and Laplace transform method. The features of typical curves were studied. The pressure and pressure derivative curves with stress-sensitivity were compared with the curves without stress-sensitivity. The influences of inter-porosity coefficient, storativity ratio, wellbore storage coefficient, permeability modulus and the hydraulic fracture conductivity on the typical curves were analysed with different seepage regimes. The results show that both the pressure and pressure derivative curves move upward in pressure-sensitive formations, especially for the end of the curves. The upward trend behaves more obviously due to stress sensitivity of the fractures. Field examples were applied to verify the validity of the proposed model. The novel well test model improves the accuracy of well test interpretation results, which is essential to the design and evaluation of stimulation treatments in these reservoirs. [Received: August 21, 2017; Accepted: September 10, 2018]

A generalization to transient bilinear flows

A set of analytical solutions to transient pressure and double-porosity (bilinear) flows in the Cartesian domain for constant and variable flow rated wells having hydraulic fractures are presented. Both finite and infinite-fracture reservoir systems with transient linear-flows for the fracture and the formation in wells intercepted by a finite-conductivity vertical fracture are considered. A new method to obtain Laplace-transformed solutions for bilinear flows leading to predictions of new regions of flow behavior is presented. The accurate match of measured production data of a hydraullicaly-fractured well in northeast Mexico using the proposed solutions shows the adequacy of the new approach.

INTERFERENCE TEST TO FRACTURED INJECTION WELLS: MATHEMATICAL MODEL AND FIELD CASE

This article is devoted to examine the pressure behavior during well tests at the system with two finite-conductivity vertical hydraulic fractures. An approximate numerical solution of the problem is obtained, simu-lation of studies by interference test is conducted. This was done by correlating the amplitude of the pressure in the observation well at the distance between two fractures with the fracture flow conductivity. The established mathematical model is used to interpret field case which have been get by method of interference test on a deposit with the low permeable reservoir.

Transient responses of multifractured systems with discrete secondary fractures in unconventional reservoirs

This paper provides a semi-analytical model to analyze the transient responses of multifractured systems with discrete secondary fractures in unconventional reservoirs. The model idealizes the stimulated reservoir volume as two regions: a primary hydraulic-fracture and discrete secondary fractures. Transient pressure and derivative responses are obtained by coupling an analytical model for matrix flow and a numerical model for fracture flow. This approach can consider the fracture characteristics in detail. For example, the discrete secondary fractures may interconnect the primary hydraulic-fracture with different inclination, nonuniform fracture spacing and varying conductivity. The semi-analytical model was verified in comparison with the results of a fully numerical simulation model.Transient flow behavior of the system was analyzed in detail. The study has shown that a multifractured system with discrete secondary fractures may exhibit four flow regimes: bilinear flow, fluid feed flow, matrix linear flow and pseudosteady-state flow. In the fluid feed flow period, the secondary fractures act as supply sources to feed the primary hydraulic-fracture to generate a dip on the type curves. Local solutions for the matrix linear flow are similar to that for a finite conductivity vertical fracture and can be correlated with the ratio of the total length of the secondary fractures to that of the primary hydraulic-fracture and an intercept term. With the increasing of number of secondary fracture, the ending time of the bilinear flow will be advanced and the fluid feed flow period will come earlier. The dimensionless secondary fracture length determines the slope of the straight line of the matrix linear flow on square-root-dimensionless-time plot. As the secondary fracture inclination decreases, the duration of the matrix linear flow period is shortened and the beginning time of the pseudosteady-state flow period is delayed. In addition, the secondary fracture conductivity affects the depth of the dip and the beginning time of the matrix linear flow period. However, the influence is not appreciable on the log/log graph.

A new method in predicting productivity of multi-stage fractured horizontal well in tight gas reservoirs

The generally accomplished technique for horizontal wells in tight gas reservoirs is by multi-stage hydraulic fracturing, not to mention, the flow characteristics of a horizontal well with multiple transverse fractures are very intricate. Conventional methods, well as an evaluation unit, are difficult to accurately predict production capacity of each fracture and productivity differences between wells with a different number of fractures. Thus, a single fracture sets the minimum evaluation unit, matrix, fractures, and lateral wellbore model that are then combined integrally to approximate horizontal well with multiple transverse hydraulic fractures in tight gas reservoirs. This paper presents a new semi-analytical methodology for predicting the production capacity of a horizontal well with multiple transverse hydraulic fractures in tight gas reservoirs. Firstly, a mathematical flow model used as a medium, which is disturbed by finite conductivity vertical fractures and rectangular shaped boundaries, is established and explained by the Fourier integral transform. Then the idea of a single stage fracture analysis is incorporated to establish linear flow model within a single fracture with a variable rate. The Fredholm integral numerical solution is applicable for the fracture conductivity function. Finally, the pipe flow model along the lateral wellbore is adapted to couple multi-stages fracture mathematical models, and the equation group of predicting productivity of a multi-stage fractured horizontal well. The whole flow process from the matrix to bottom-hole and production interference between adjacent fractures is also established. Meanwhile, the corresponding iterative algorithm of the equations is given. In this case analysis, the productions of each well and fracture are calculated under the different bottom-hole flowing pressure, and this method also contributes to obtaining the distribution of pressure drop and production for every horizontal segment and its changes with effective fracture half-length and conductivity. Application of this technology will provide gas reservoir engineers a better tool to predict well and fracture productivity, besides optimizing transverse hydraulic fractures configuration and conductivity along the lateral wellbore.

Inflow performance of wells intersecting long fractures with transient pressure and variable flow-rate behavior

Abstract A mathematical model which predicts the performance of vertical and horizontal wells intersecting fractures is presented. Based on the formulation originally proposed by Guo and Schechter (1997) , this model predicts the transient pressure behavior with variable flow-rate both during the increasing pressure stage and in the declining pressure and flow-rate situation. The flow in the matrix and that in the fracture are coupled to allow the calculation of the pressure distribution in the matrix and in the fracture, as functions of space and time. The model is consistent with transient pressure analyses in finite-conductivity vertical fractures ( Cinco-Ley and Samaniego, 1981 ), and presents a useful tool for productivity analyses of hydraulically fractured wells during the rapid decline stage. Comparisons with real well data are analyzed and discussed.

Analytical modeling of acceleration-induced conductivity damage in a propped hydraulic fracture of a high-pressure gas well

The compressible pseudoradial and pseudosteady state flow in a well intersected by a finite-conductivity vertical fracture is analyzed. A modified dimensionless fracture conductivity reflecting the acceleration effect is introduced from a comparison of the mass rate of an accelerating flow and that of a Darcy flow at the same pressure drawdown. It is shown that a gas flow with the acceleration effect can be choked at a moderate drawdown pressure. The damage to the dimensionless fracture conductivity caused by the acceleration effect is significant when the fracture permeability is greater than 20D, the reservoir permeability is greater than 5mD, the fracture length is less than 200 m or the reservoir pressure is higher than 100 MPa. In a low permeability reservoir (<1 mD), the Forchheimer drag is significant, while the acceleration effect is negligible.

Numerical well test for well with finite conductivity vertical fracture in coalbed

A new well test model is developed for the hydraulic fractured well in coalbed by considering the following aspects: methane desorption phenomena, finite conductivity vertical fractures, and asymmetry of the fracture about the well. A new parameter is introduced to describe the storage of the fracture, which is named as a combined fracture storage. Another new concept called the fracture asymmetry coefficient is used to define the asymmetry of the fracture about the well. Finite element method (FEM) is used to solve the new mathematical model. The well test type curves and pressure fields are obtained and analyzed. The effects of the combined fracture storage, desorption factor, fracture conductivity, and fracture asymmetry coefficient on the well test type curves are discussed in detail. In order to verify the new model, a set of field well test data is analyzed.

The Inversion of Formation Parameters after Volume Fracturing

Volume fracturing is the key technology that has made the development of tight oil formations economical. Well testing is often used to obtain formation parameters in oil field after volume fracturing in tight oil formations, which is time-consuming and accuracy is low. Based on a new analytical solution methodology, this paper presents a simple and efficient approach by matching the production data to obtain formation parameters, just like formation permeability, fracture conductivity, dimensionless conductivity, propped length and so on. The numerical solution is applicable for finite-conductivity vertical fractures in rectangular shaped reservoirs. The mathematical formulation is based on the method of images with no flow boundaries for symmetrical patterns.

Pressure transient analysis of multi-stage fractured horizontal wells in boxed reservoirs

Abstract A new analytical solution of fluid unsteady flow was presented for wells with finite-conductivity vertical fractures by determining impact functions of fracture conductivity. Based on the new solution, the transient pressure distribution of a fractured horizontal well with multiple finite-conductivity vertical fractures was obtained using the principle of superposition. Calculation results show the pressure transient behavior of fractured horizontal wells under the condition of constant bottom rate. In the ideal fracture arrangement, flow patterns occur in the order of early bilinear flow → early linear flow → middle radial flow → middle linear flow → late pseudo steady state. Uneven length of fractures with equal conductivity and uniform arrangement may affect the transform of early linear flows from earlier bilinear flows and the premature birth of middle radial flows. Nonuniform arrangement of fractures with even length and equal conductivity may influence middle radial flows. Different conductivity of fractures with equal length and uniform arrangement shortens the period of bilinear flows and lengthens the linear flows. Increase in the number of fractures with even length, equal conductivity and uniform arrangement accelerates pressure drops in the reservoir, shortens the period of middle radial flows and lengthens the middle linear flows. Increase in length of all fractures shortens the period of middle radial flows and results in absence of radial flows. It is demonstrated that the horizontal well with multiple vertical fractures may improve fluid flow patterns.

A Semianalytic Solution for Flow in Finite-Conductivity Vertical Fractures by Use of Fractal Theory

SummaryThe exploitation of unconventional reservoirs complements the practice of hydraulic fracturing, and with an ever-increasing demand in energy, this practice is set to experience significant growth in the coming years. Sophisticated analytic models are needed to accurately describe fluid flow in a hydraulic fracture, and the problem has been approached from different directions in the past 3 decades—starting with the work of Gringarten et al. (1974) for an infinite-conductivity case, followed by contributions from Cinco-Ley et al. (1978), Lee and Brockenbrough (1986), Ozkan and Raghavan (1991), and Blasingame and Poe (1993) for a finite-conductivity case. This topic remains an active area of research and, for the more-complicated physical scenarios such as multiple transverse fractures in ultratight reservoirs, answers are currently being sought.Starting with the seminal work of Chang and Yortsos (1990), fractal theory has been successfully applied to pressure-transient testing, although with an emphasis on the effects of natural fractures in pressure/rate behavior. In this paper, we begin by performing a rigorous analytical and numerical study of the fractal diffusivity equation (FDE), and we show that it is more fundamental than the classic linear and radial diffusivity equations. Thus, we combine the FDE with the trilinear flow model (Lee and Brockenbrough 1986), culminating in a new semianalytic solution for flow in a finite-conductivity vertical fracture that we name the “fractal-fracture solution (FFS).” This new solution is instantaneous and comparable in accuracy with the Blasingame and Poe solution (1993). In addition, this is the first time that fractal theory is used in fluid flow in a porous medium to address a problem not related to reservoir heterogeneity. Ultimately, this project is a demonstration of the untapped potential of fractal theory; our approach is flexible, and we believe that the same methodology could be extended to different applications.One objective of this work is to develop a fast and accurate semianalytical solution for flow in a single vertical fracture that fully penetrates a homogeneous infinite-acting reservoir. This would be the first time that fractal theory is used to study a problem that is not related to naturally fractured reservoirs or reservoir heterogeneity. In addition, as part of the development process, we revisit the fundamentals of fractals in reservoir engineering and show that the underlying FDE possesses some interesting qualities that have not yet been comprehensively addressed in the literature.

Rate Decline Curves Analysis of a Vertical Fractured Well With Fracture Face Damage

Rate decline analysis is a significant method for predicting well performance. Previous studies on rate decline analysis of fractured wells are all based on homogeneous reservoirs rather than homogeneous ones considering fracture face damage. In this article, a well model intercepted by a finite conductivity vertical fracture with fracture face damage is established to investigate how face damage factor affects the productivity of fractured well. Calculative results show that in transient flow, dimensionless rate decreases with the increase of fracture face damage and in pseudo steady-state flow, all curves under different face damage factors coincide with each other. Then, a new pseudo steady-state analytic formula and its validation are presented. Finally, new Blasingame type curves are established. It is shown that the existence of fracture damage would decrease the rate when time is relatively small, so fracture damage is an essential factor that we should consider for type curves analysis. Compared with traditional type curves, new type curves could solve the problem of both variable rate and variable pressure drop for fractured wells with fracture face damage factor. A gas reservoir example is performed to demonstrate the methodology of new type curves analysis and its validation for calculating important formation parameters.

Application de la courbe diagnostique aux essais de pompage en milieu karstique

Diagnostic plots applied to pumping tests in karst systems. Pumping tests conducted on wells intersecting karst heterogeneities such as the conduit network are difficult to interpret. Nevertheless, this case can be solved by assimilating the horizontal karst conduit to a finite-conductivity vertical fracture. In this case, several flow patterns corresponding to the respective contributions of karst subsystems (fractured matrix, small conduits, and main karst drainage network) can be identified on the diagnostic plot of the drawdown derivative. This is illustrated on two examples from Mediterranean karst systems in southern France. A pumping test on a well intersecting the main karst drainage network of the Cent-Fonts karst system shows (i) a preliminary contribution of the karst conduit storage capacity followed by (ii) linear flows into the fractured matrix. A pumping test on a well intersecting a small karst conduit of the Corbières karst system shows the existence of(i) bi-linear flow within both the karst conduit and the fractured matrix at early times, followed by (ii) radial flows within the fractured matrix and (Hi) finally the contribution of a major karst cavity. The use of diagnostic plots allows identifying the various flow regimes during pumping tests, corresponding to the response of the individual karst aquifer subsystems. This is helpful for improving the understanding of the structure of the karst aquifer and flow exchanges between subsystems.

Productivity analysis of horizontal wells intercepted by multiple finite-conductivity fractures

Horizontal wells in the anisotropic reservoirs can be stimulated by hydraulic fracturing in order to create multiple finite-conductivity vertical fractures. Several methods for evaluating the productivity of the horizontal wells have been presented in the literature. With such methods, however, it is still difficult to obtain an accurate result. This paper firstly presents the dimensionless conductivity theory of vertical fractures. Then models for calculating the equivalent wellbore radius and the skin factor due to flow convergence to the well bore are proposed after analyzing the steady-state flow in porous reservoirs. By applying the superposition principle to the pressure drop, a new method for evaluating the productivity of horizontal wells intercepted by multiple finite-conductivity fractures is developed. The influence of fracture conductivity and fracture half length on the horizontal well productivity is quantitatively analyzed with a synthetic case. Optimum fracture number and fracture space are further discussed in this study. The results prove that the method outlined here should be useful to design optimum fracturing of horizontal wells.

Practical Well Test Analysis of a Hydraulically Fractured Low Permeability Gas Reservoir: A Case History

The primary objective of hydraulic fracturing is to create a propped fracture with sufficient conductivity and length to amplify or at least optimize well performance of low permeability tight gas reservoir. The oil industry has suggested that hydraulically fractured tight gas wells performance is hindered significantly by non-Darcy flow effect.This work will present an investigation of non-Darcy flow effect to hydraulically fractured gas wells performance and provide the development, validation, and application of actual well test analysis for wells with a finite conductivity vertical fracture.Also, this work presents the results obtained in the study of actual post frac modified isochronal test data of gas wells intersected by a finite conductivity vertical fracture in a tight low permeability gas reservoir. In addition, the estimation of reservoir properties and fracture properties were carried out to construct a simple analytical model, which used for rate prediction.The effect of non-Darcy flow in fractures is clearly seen in the tests data and will lead to limiting production especially on higher chokes (after one inch). Therefore, non-Darcy effects should be considered in design of hydraulic fracture treatments, otherwise the design might be far from optimal

A Hybrid Numerical/Analytical Model of a Finite-Conductivity Vertical Fracture Intercepted by a Horizontal Well

Summary This paper presents a hybrid numerical/analytical model for the pressure-transient response of a finite-conductivity fracture intercepted by a horizontal well. The model dynamically couples a numerical fracture model with an analytical reservoir model. This approach allows us to include finer details of the fracture characteristics while keeping the computational work manageable. For example, the fracture may have irregular shape, nonuniform width, and variable conductivity, and the well may not intersect the fracture at its geometric center. In this paper, we use the hybrid model to investigate the effects of fracture properties on the pressure-transient characteristics of a single, finite-conductivity horizontal-well fracture. The single horizontal-well-fracture model can be extended easily to multiply fractured horizontal wells by superposition. The model also can be used to compute the pseudoskin caused by the effects of nonideal fracture geometry, variable conductivity, and flow choking around the wellbore and to investigate the influence of fracture properties on the performance of horizontal wells.

Effect of boundary conditions on pressure behavior of finite-conductivity fractures in bounded stratified reservoirs

In this study, a mathematical model was developed to model the pressure behavior of a well located in a bounded multilayer reservoir and crossed by a finite-conductivity vertical fracture. It was found that the dimensionless pressure function and its derivative strongly depend on fracture conductivity and fracture extension during early times. The effect of reservoir heterogeneity on the pressure function is negligible compared to that on the pressure derivative. Both functions exhibit four flow periods: bilinear, formation linear, pseudoradial and pseudosteady-state which are separated by transition periods. One or more of these flow periods may be missing. Data obtained from a long test and which are characterized by a unit slope line indicate that the well is intercepted by deeply extended fractures. It has been found that the fractional production rates of different layers are a good measure of reservoir and fracture characteristics. Flowmeter survey data can be used to eliminate the non-uniqueness problem when using the type curves presented in this study.