Jing Lu

Evaluation of Fracture Asymmetry of Finite-Conductivity Fractured Wells

Nearly all commercial hydraulic fracture design models are based on the assumption that a single fracture is initiated and propagated identically and symmetrically about the wellbore, i.e., the fracture growth and proppant transport occurs symmetrically with respect to the well. However, asymmetrical fractures have been observed in hundreds of hydraulic fracturing treatments and reported to be a more realistic outcome of hydraulic fracturing. The asymmetry ratio (length of short fracture wing divided by length of long wing) inftuenced the production rate adversely. In the worst case, the production rate could be reduced to that of an unfractured well. Several authors observed asymmetrically propagated hydraulic fractures in which one wing could be ten times longer than the other. Most pressure transient analysis techniques of hydraulically fractured wells assume the fracture is symmetric about the well axis for the sake of simplicity in developing mathematical solution. This study extends the work by Rodriguez to evaluate fracture asymmetry of finite-conductivity fracture wells producing at a constant-rate. The analysis presented by Rodriguez only involves the slopes of the straight lines that characterize the bilinear, linear and radial flow from the conventional Cartesian and semilog plots of pressure drop versus time. This study also uses the Tiabs direct synthesis (TDS) technique to analyze the linear and bilinear flow regimes in order to find the asymmetry factor of the fractured well. With the fracture conductivity estimated from the bilinear flow region, dimensionless fracture conductivity and the asymmetry ratio are calculated. A technique for estimating the fracture asymmetry ratio from a graph is presented. An equation relating the asymmetry ratio and dimensionless fracture conductivity is also presented. This equation assumes that the linear and/or bilinear flow regime is observed. However, using the TDS technique, the asymmetry ratio can be estimated even in the absence of bilinear or linear flow period. It is concluded that the relative position of the well in the fracture, i.e., the asymmetry condition, is an important consideration for the fracture characterization. A log-log plot of pressure derivative can be used to estimate the fracture asymmetry in a well intersected with a finite-conductivity asymmetric fracture. The analysis using pressure derivative plot does not necessarily require the radial flow period data to calculate the asymmetric factor.

PRESSURE BEHAVIOR OF HORIZONTAL WELLS IN DUAL-POROSITY, DUAL-PERMEABILITY NATURALLY-FRACTURED RESERVOIRS

The pressure behavior of horizontal wells in dual-porosity, dual-permeability naturally-fractured reservoirs is presented. The proposed equation is obtained by double Fourier transformation and Laplace transformation. The results calculated for combinations of various dimensionless characterizing parameters, including the permeability ratio between matrix and fracture systems, have revealed the unique behavior of naturally fractured reservoirs when the flow state within the matrix blocks is taken into account. It is concluded that, for the flow within matrix blocks will weaken the essential nature of fluid flow through a dual-porosity, single permeability medium revealed by Warren-Root model. This paper also presents the application of “Tiab’s Direct Synthesis” technique to horizontal wells in an infinite-acting dual-porosity, dualpermeability naturally-fractured reservoirs with pseudosteady state interporosity flow.

Steady State Productivity Equations for a Vertical Well in Anisotropic Sector Fault, Channel, and Rectangular Reservoirs

This paper presents steady state productivity equations for a fully penetrating vertical well in the following three anisotropic systems: (a) sector fault, (b) channel, and (c) rectangular reservoir using a uniform line sink model. The new equations, which are based on conformal mapping method, are simple, accurate, and easy to use in field practice. If the well is in a sector fault reservoir, the productivity is a function of the angle of the sector, wellbore location angle, off-vertex distance, and drainage radius. If the well is in a channel reservoir with two parallel impermeable lateral boundaries, well flow rate reaches a maximum value when the well is located in the middle of the channel width. If the well is in a rectangular reservoir with constant pressure lateral boundaries, a new equation is provided to calculate the productivity of the well arbitrarily located in the anisotropic reservoir for the case where the flow rate of an off-center well is bigger than that of a centered well. It is concluded that, for a vertical well, different steady state productivity equations should be used in different reservoir geometries. DOI: 10.1115/1.3066429

Productivity Formulae of an Infinite-conductivity Hydraulically Fractured Well Producing at Constant Wellbore Pressure Based on Numerical Solutions of a Weakly Singular Integral Equation of the First Kind

In order to increase productivity, it is important to study the performance of a hydraulically fractured well producing at constant wellbore pressure. This paper constructs a new productivity formula, which is obtained by solving a weakly singular integral equation of the first kind, for an infinite-conductivity hydraulically fractured well producing at constant pressure. And the two key components of this paper are a weakly singular integral equation of the first kind and a steady-state productivity formula. A new midrectangle algorithm and a Galerkin method are presented in order to solve the weakly singular integral equation. The numerical results of these two methods are in accordance with each other. And then the solutions of the weakly singular integral equation are utilized for the productivity formula of hydraulic fractured wells producing at constant pressure, which provide fast analytical tools to evaluate production performance of infinite-conductivity fractured wells. The paper also shows equipotential threads, which are generated from the numerical results, with different fluid potential values. These threads can be approximately taken as a family of ellipses whose focuses are the two endpoints of the fracture, which is in accordance with the regular assumption in Kuchuk and Brigham, 1979.

Theories of Arps' Decline Curve Exponent and Loss Ratio, for Saturated Reservoirs

Rate decline curve analysis is an essential tool in predicting reservoir performance and in estimating reservoir properties. In its most basic form, decline curve analysis is to a large extent based on Arps’ empirical models that have little theoretical basis. The use of historical production data to predict future performance is the focus of the empirical approach of decline analysis while the theoretical approach focuses on the derivation of relationships between the empirical model parameters and reservoir rock/fluid properties; thereby establishing a theoretical basis for the empirical models. Such relationships are useful in formulating techniques for reservoir properties estimation using production data. nMany previous attempts at establishing relationships between the empirical parameters and the rock/fluid properties have been concerned primarily with the exponential decline of single phase oil reservoirs. A previous attempt to establish the theories of hyperbolic decline of saturated reservoirs (multiphase) have yielded an expression relating the Arps’ decline exponent, to nrock/fluid properties. However, the values of exponent computed from the expression are not constant through time, whereas, the empirically-determined exponent b is a constant value. nThis work utilizes basic concepts of compressibility and mobility to justify the dynamic behaviour of the values obtained from the existing theoretical expression of the previous theory; to prove that the expression, though rigorously derived, is not the theoretical equivalence of the empirical Arps’ b-exponent; and finally, to properly to offer a new logical perspective to the previous theory relating b-exponent to rock and fluid property. Ultimately, this work presents, for the first time, a new consistent theoretical expression for the Arps’ exponent, b. The derivation of the new expression is still founded on the concept of Loss Ratio, as in previous attempts; however, this latest attempt utilizes the cumulative derivative of the Loss Ratio, instead of the instantaneous derivative implied in the previous attempt. nThe new expression derived in this work have been applied to a number of saturated reservoir models and found to yield values of b-exponent that are constant through time and are equivalent to the empirically-determined b-exponent.

Pressure Drop Equations for a Partially Penetrating Vertical Well in a Circular Cylinder Drainage Volume

Taking a partially penetrating vertical well as a uniform line sinknin three-dimensional space, by developing necessary mathematicalnanalysis, this paper presents unsteady-state pressure drop equationsnfor an off-center partially penetrating vertical well in a circularncylinder drainage volume with constant pressure at outer boundary.nFirst, the point sink solution to the diffusivity equation isnderived, then using superposition principle, pressure drop equationsnfor a uniform line sink model are obtained. This paper also gives annequation to calculate pseudoskin factor due to partial penetration.nThe proposed equations provide fast analytical tools to evaluate thenperformance of a vertical well which is located arbitrarily in ancircular cylinder drainage volume. It is concluded that the well off-center distance has significantneffect on well pressure drop behavior, but it does not have any effect onnpseudoskin factor due to partial penetration. Because the outernboundary is at constant pressure, when producing time isnsufficiently long, steady-state is definitely reached. When wellnproducing length is equal to payzone thickness, the pressure dropnequations for a fully penetrating well are obtained.

Steady-State Productivity Equations for a Multiple-Wells System in Sector Fault Reservoirs

Summary Currently in the oil industry, pseudosteady-state productivity equations for a multiple-wells system are used in all reservoir systems, regardless of the outer boundary conditions. However, if the reservoir is under edgewaterdrive, pseudosteady state is no longer applicable. When the producing time is sufficiently long, productivity equations based on the steady state are required. This paper presents steady-state productivity equations for a multiple-wells system in homogeneous, anisotropic sector fault reservoirs. Taking fully penetrating vertical wells as uniform line sinks, and solving a square matrix of dimension n, where n is the number of wells, simple, reasonably accurate multiple-wells-system productivity equations are obtained. The proposed equations relate the production-rate vector to the pressure-drawdown vector and are applicable to a multiple-wells system, which is located arbitrarily in a sector fault reservoir. The analytical solutions are verified with reservoir numerical simulation in several examples. This paper also gives an equation for calculating skin factors of each well in steady state. The analytical solutions perform well and match well with numerical solutions, and the benefit of the analytical model can be emphasized when the reservoir data deviate from their idealistic representation. It is concluded that the proposed equations provide a fast analytical tool to evaluate the performance of a multiplewells system in a sector fault reservoir.

Steady-State Productivity Equations for a Multiple-Wells System in Sector Fault Reservoirs and Channel Reservoirs

Currently in the oil industry, pseudo-steady state productivity equations for a multiple wells system are used in all reservoir systems, regardless of the outer boundary conditions. However, if the reservoir is under edge water drive or with infinite lateral extension, pseudo-steady state is no longer applicable. When producing time is sufficiently long, productivity equations based on the steady state are required. This paper presents steady-state productivity equations for a multiple-wells system in homogeneous, anisotropic sector fault reservoirs and channel reservoirs. Taking fully penetrating vertical wells as uniform line sinks, and solving a square matrix of dimension n, where n is the number of wells, simple, reasonably accurate multiple-wells system productivity equations are obtained. The proposed equations which relate the production rate vector to the pressure drawdown vector provide a fast analytical tool to evaluate the performance of multiple wells, which are located arbitrarily in a sector fault reservoir or a channel reservoir. This paper also gives an equation for calculating skin factors of each well. It is concluded that, for a given number of wells, well pattern, anisotropic permeabilities, skin factor, pressure difference between reservoir outer boundary and flowing bottomhole pressure, have significant effects on single well productivity and total productivity of the multiple-wells system. In a sector fault reservoir, the production rates are increasing functions of the sector angle; in a channel reservoir, the production rates are increasing functions of the reservoir width. Introduction Well productivity is one of primary concerns in field development and provides the basis for field development strategy. To determine the economical feasibility of drilling a well, petroleum engineers need reliable methods to estimate its expected productivity. We often relate the productivity evaluation to the long time performance behavior of a well, that is, the behavior during pseudo-steady state or steady-state flow. Substituting Darcys equation into the equation of continuity, the productivity equation of a fully penetrating vertical well in a homogeneous, isotropic permeability circular reservoir is: ) / ln( ) /( ) ( 2 w e w e R R B P P KH D w F Q μ π − = (1) where e P is the outer boundary pressure, w P is the flowing bottomhole pressure, and D F is the unit conversion factor (Butler, 1994). In oil field units, 001127 . 0 = D F ; in practical SI units, 4 . 86 = D F The introduction of field and practical SI units is in Appendix A, units conversion factors can be found in Appendix B. Ge (1982) introduced the following equation, which accounts for asymmetrical positioning of a well within its circular drainage area, to calculate the productivity for an off-center well,