Pressure Transient Behavior of a Finite-Conductivity Fractured Well with Spatially Varying Fracture Properties

Abstract

The pressure transient behavior of a finite conductivity vertically fractured well is important in reservoir engineering because it provides valuable information concerning the well completion, and provides estimates of the in-situ reservoir properties and the propped fracture dimensions and conductivity. The pressure transient behavior of finite-conductivity wells has typically been analyzed using mathematical models that assume relatively simple fracture properties. Most of these models assume uniform fracture height, width, and proppant permeability distributions with respect to space, and many of these models also assume that fracture storage effects are negligible. This paper presents the pressure transient solution of a well intersected by a finite-conductivity vertical fracture with a lengthwise variation in the fracture geometric and material properties. The general solution of the fracture fluid-flow problem, and an approximate solution of the problem in which the effect of fracture storage has been assumed to be negligible, are presented. Comparisons of the pressure transient behavior predicted with the two finite-conductivity fracture solutions are given. The effects of spatially varying fracture height, width, and material properties are investigated with these pressure transient solutions. The new solutions provide more realistic pressure transient analysis models which may be used to better design, implement, and evaluate hydraulic fracture treatments. They will also provide better agreement between the prediction of the fracture geometry and its post-treatment evaluation. Introduction The pressure transient behavior of finite-conductivity vertical fractures has been studied extensively in recent years in an effort to obtain more reliable estimates of the geometry and the conductivity of the vertical fractures. Many of these studies have used semianalytic solutions to describe the pressure distributions in the reservoir and in the due to the production from the well. The semianalytic models that have been used in these studies have been commonly developed by the application of the Boundary Element Method (BEM). Gringarten et al. introduced the use of the Boundary Element Method for the development of pressure transient solutions of vertically fractured wells by an adaptation of the analogous solutions of heat conduction in solids. Gringarten and Ramey demonstrated that Source and Green's functions could be used to construct useful solutions for pressure transient problems in oil and gas reservoirs. These techniques have subsequently been used extensively in the analyses of the pressure transient behavior of vertically fractured wells. The application of the Boundary Element Method to the study of the pressure transient behavior of finite-conductivity vertical fractures was first made by Cinco-Ley et al. They developed a general semianalytic model in which the fracture geometry and material properties were assumed to be uniform. Their model also considered the fracture height to be equal to the reservoir thickness. This model has been commonly used for the analysis of the pressure transient behavior of wells with finite-conductivity fractures. It has also been a reference with which most of the later approximate models have been compared. P. 445^

Design Criteria for Improved Performance of Fractured Wells

“…For vertical fractures with dimensionless fracture conductivities less than the applicability of this solution procedure, a late-time formulation of the general finiteconductivity fracture solution 8 can be employed to readily compute the dimensionless productivity index solution for dimensionless fracture conductivities as low as 0.1. The low conductivity correlation values could also be derived using finite-difference reservoir simulation models.…”

Asymptotic Description of Vertically Fractured Wells Within the Boundary Element Method

“…........... (6) The term in the second bracket represents the solution of the stationary problem with the distributed influx q(x, t) and the flow rate in the well Q(t). This is the case as the summation here is Green's function of the stationary problem (17,18,21) :…”

“…Great attention has been paid in the past, in particular, to modelling flow in vertically fractured wells (1)(2)(3)(4)(5)(6)(7)(8)(9) .…”

Research for Unsteady Seepage Flow of Asymmetrical Fractured Vertical Wells in Coalbed

“…Xuejun Shan [7] explained the fracture geometry of the 21 CBM fracturing wells, the results show that there are three kinds of cracks in the form of expansion: single-wing cracks, asymmetric double-wing cracks, vertical and horizontal cracks symbiotic type, of which asymmetric two-wing cracks accounted for more than 90%. Many scholars at home and abroad have done a lot of research on this aspect [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24], the domestic aspects, Wei Chen [8] established the evaluation model of finite conductivity and symmetrical fractured vertical wells in coalbed methane reservoirs; Weiping Ouyang [9] uses the finite element method to establish the numerical well test model of vertical fractured well of coalbed methane infinite conductivity, and the double logarithmic well test theory chart is drawn; Haitao Cao [13] used the theory of point source function to deduce modern decreasing plate of CBM symmetrically fractured well production; Baojun Cao [14] established a model for asymmetric fracture productivity of volcanic rocks based on the principle of conformal transformation and equivalent filtrational resistance; Wenjuan Wu [15] took the Chang 6 oil reservoir in Ordos Basin as an example, using the logging data to carry out the geological modeling and studying the three-dimensional stress field established the numerical simulation model of the asymmetric fracturing in the ultra-low permeability oil and gas reservoirs; Jian Xiong [16] derived productivity prediction model of finite conductivity asymmetric vertical fractured wells in low permeability gas reservoirs based on the steady flow theory and conformal transformation. The foreign aspects, Benjamin, J. and Barker [17] studied the pressure dynamics of finite conductivity symmetrical fractured vertical well in coalbed methane with confined boundaries based on the assumption of two dimensional single-phase Darcy flow; K.H.Guppy [18] established a numerical and semi-analytical model to analyze the high velocity non-Darcy's flow behavior of the finite conductivity fractured wells in coalbed methane reservoirs; Fernando Rodriguez …”

“…The foreign aspects, Benjamin, J. and Barker [17] studied the pressure dynamics of finite conductivity symmetrical fractured vertical well in coalbed methane with confined boundaries based on the assumption of two dimensional single-phase Darcy flow; K.H.Guppy [18] established a numerical and semi-analytical model to analyze the high velocity non-Darcy's flow behavior of the finite conductivity fractured wells in coalbed methane reservoirs; Fernando Rodriguez [19] established a semi-analytical model of finite conductivity asymmetrical fractured well in oil reservoirs based on a new solution to the dynamic analysis of quasi-linear flow and bilinear flow pressure; B.D. Poe [20] gave a dynamic analysis model of pressure propagation in finite conductivity fractured wells with the change of spatial properties; Sergio Berumen [21] gave a pressure dynamic analysis of finite conductivity asymmetrically fractured well based on the numerical solution. Lei Wang [22][23][24] gave progressive decrease analysis of finite conductivity symmetrically fractured well of coalbed methane and a well test model of finite conductivity asymmetrically fractured well in oil reservoirs based on Laplace transform and Stehfest numerical inversion.…”