Infinite-conductivity Fractures Research Articles

Generalized Analytical Solutions of Vertically Fractured Wells in Commingled Reservoirs: Field Case Study

Summary Accurate identification of the individual-layer parameters for vertically fractured wells in commingled reservoirs is essential for development plan design, reservoir numerical simulation, and stimulation measure selection. Different semi-analytical and numerical models are generally applied in multilayer transient testing (MLT) analysis to determine the properties of individual layer. However, these approaches require numerous computations and are complicated to program due to the fracture and reservoir discretization. This work thus presents the generalized analytical solutions of vertically fractured wells in infinite, closed, or constant-pressure commingled reservoirs with both computational and functional simplicity. The fully analytical solutions are derived based on the early-time approximate solutions of infinite-conductivity fracture and trilinear flow models, infinite-conductivity fracture solutions, pressure superposition principle, and Duhamel principle. A systematic verification by employing a standardized well testing software and trilinear flow model is conducted to ensure the general application accuracy of the presented solutions. The results show that the developed analytical solutions are valid when the dimensionless fracture conductivity is more than 2 (FcD > 2) with an average absolute percent deviation (AAD) of ~2% for pressure and that is ~4% for pressure derivative. The developed analytical solutions also exhibit improvements in early-time pressure and derivative calculation. Finally, a field case of a four-layer fractured well is interpreted by the developed solutions and well testing software to illustrate the feasibility. The interpretation results of two methods are nearly identical, with only a minor difference. The developed analytical solutions are computationally accurate while maintaining functional simplicity and can be considered as an alternative to the current semi-analytical and numerical approaches in MLT analysis.

Precision in Defining Diagnostic Plots Optimizes Well-Test Results

_ This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 206108, “Diagnostic Plots of Pressure-Transient, Rate-Transient, and Diagnostic Fracture Injection/Falloff Tests,” by David Craig, SPE, Occidental, and Thomas Blasingame, SPE, Texas A&M University. The paper has not been peer reviewed. _ All transient-test interpretation methods use diagnostic plots for the identification of wellbore or fracture storage distortion, flow regimes, and other parameters. The associated diagnostic plots are not interchangeable between such tests. The objective of this work is to clearly define the appropriate diagnostic plots for each type of transient test. Introduction In the complete paper, the authors review constant-rate drawdown solutions for a few specific cases of interest and the plotting function used to match buildup data to the drawdown solutions. They extend their review to the cases of buildup or falloff analysis following short flow periods and illustrate the differences between analysis of buildup or falloffs following short and long flow periods. After covering pressure-transient testing, they discuss rate-transient diagnostic plots and the ambiguity observed during analysis of multifractured horizontal well diagnostic plots and the uncertainty of analyst straight lines. Finally, they discuss and demonstrate the correct plotting functions and flow-regime interpretation for diagnostic fracture injection and falloff tests. Multiple examples from the literature are included as part of the discussion. Pressure-Transient Diagnostic Plots Pressure-transient-testing diagnostic plots evolved from the solutions to constant-rate drawdown problems. The authors’ intent is to review infinite-acting solutions for radial flow with wellbore storage and skin and flow through an infinite-conductivity fracture. On a log-log plot, the characteristic slopes, detailed in the complete paper, are unit slope, ½ slope, and zero slope. Many authors have proposed plotting functions for overlaying the observed pressure response from transient tests other than a constant-rate drawdown on the drawdown solutions. In many cases, the slopes identifying storage distortion and flow regimes from drawdown solutions will match and can be used to identify storage and flow regimes from pressure-buildup tests. A situation in which pressure-buildup or falloff test plotting functions fail to match the drawdown solution is when the flow period before shut-in is short. Short-flow transient tests have been reviewed by multiple authors. In lower-permeability reservoirs, pressure-transient tests with short flow periods are very common, and, in unconventional reservoirs, short-flow transient tests are the most-common transient test. Diagnostic plots constructed based on the characteristic shapes of the dimensionless pressure and the dimensionless pressure derivative with respect to the natural logarithm of dimensionless time for constant-rate drawdown solutions cannot be applied in the same manner for short-flow transient tests. With short-flow transient tests, the observed derivative of pressure difference with respect to the natural logarithm of time matches the negative product of dimensionless time and the second derivative of dimensionless pressure with respect to dimensionless time. Additionally, the commonly known impulse derivative, which is defined as the product of time and the derivative of pressure difference with respect to the natural logarithm of time, will match the negative product of dimensionless time squared and the second derivative of dimensionless pressure with respect to dimensionless time.

A new enriched method for extended finite element modeling of fluid flow in fractured reservoirs

Fractured porous media are always characterized by complex fractures on multiple scales. Conventional reservoirs simulation methods always fail to capture the flow dynamics of multiscale fracture networks without mesh refinement. In this work, the fluid flux is approximated using the extended finite element method (XFEM) to accurately represent discontinuities and singularities that develop as a result of the much larger permeability within the fractures than in the matrix. We consider two individual cases: infinite-conductivity fractures (pressure gradient is neglected along the fracture) and finite-conductivity fractures (pressure gradient is present along the fracture). Benchmark problems describing fluid flow around a single fracture for the two cases are both solved to demonstrate optimal convergence of the method and the improved accuracy compared to a traditional finite element approximation. The new method provides a far more accurate representation of the pressure-gradient singularity near the fracture tip. To demonstrate the robustness of the method, the pressure fields surrounding an arbitrary network of intersecting fractures for the two cases are modeled, illustrating those enrichments representing multiple fractures can be superposed in a straightforward matter to simulate fluid flow in complex fracture networks. Finally, an engineering case for transient oil production from a horizontal well in a fractured reservoir is presented to demonstrate promising prospects of the proposed method.

Analytical well-test model for hydraulicly fractured wells with multiwell interference in double porosity gas reservoirs

Interference resulting from infill wells in multiwell systems becomes a common issue in interpreting well tests data from hydraulicly fractured wells in tight carbonate gas reservoirs. Current single-well pressure transient approaches may falsely interpret well interferences as the composite reservoir or boundary effects, leading to significant errors in reservoir parameter estimation. Additionally, the most semi-analytical and numerical solutions used for fractured wells are computationally demanding and inconvenient to use. This work presents a novel and simple analytical well-test model for hydraulicly fractured wells with multiwell interference in double porosity gas reservoirs with significantly less computational time and effort than the existing methods. The analytical solution for a hydraulically fractured well with finite conductivity fracture in a multiwell system with well interference is presented based on the superposition principle, the early-time approximate solutions of trilinear flow and the infinite-conductivity fracture model. A systematic verification is conducted to ensure the validity of the presented solution. The solution is valid when the dimensionless fracture conductivity is greater than 1 (FcD>1) with ∼1% error in pressure and derivative. Furthermore, we offer a criterion of identifying well interference using the pressure derivative of the late-time approximation of the proposed analytical solution. Finally, a case study is conducted in the Sichuan Basin, indicating the proposed solution's applicability.

A Method to Improve Computational Efficiency of Productivity Evaluation with Rectangular Coalbed Methane Reservoir

Computational efficiency is the key factor to be considered in the productivity evaluation of rectangular coalbed methane reservoir. There are three main factors affecting the calculation speed: the nonlinearity of the material balance equation of coalbed methane reservoir, the poor conductivity of fractures cannot be considered as infinite conductivity fractures, and the Duhamel convolution is needed in history fitting and boundary image inversion. At present, there is no method to quickly evaluate the productivity of finite conductivity fracture model in rectangular coalbed methane reservoir. Diffusion equation of matrix is generated by the Fick diffusion law. The Darcy seepage law is used to build the seepage equation of fractured system in coalbed methane reservoir. In order to transform the calculation result of infinite conductivity fracture into finite conductivity fracture, fracture conductivity factor is employed in this paper. The applicability of fracture conductivity factor in the whole production process is clarified. It is clear that the factor is prone to calculation errors when the time is small, and the calculation fluctuates greatly. According to the characteristics of the Riley method and discrete method, an accurate and efficient analytical solution calculation process is designed. This will make the calculation results accurate. A production evaluation method of rectangular coalbed methane reservoirs with fractured vertical well and finite conductivity fracture is proposed. The purpose of quickly and accurately predict well production capacity is reached. The geological parameters are recombined, and new coalbed methane reservoir flow parameters are defined. Through parameter sensitivity analysis, the influence of different flow characteristic parameters on gas production is clarified. The dimensionless transfer constant and dimensionless storage capacity affect the appearance time of desorption and diffusion and the storage capacity of the fracture system, respectively. The dimensionless desorption constant describes the strength of desorption and diffusion. The influence of fracture conductivity factor on production is studied. It is clarified that its impacts are different in the early stage and the later stage of production. There is a limit to the fracture conductivity factor. When the limit is exceeded, the fracture conductivity factors no longer affect the production of a single well. The findings of this study can understand the percolation stage of finite conductivity fractured wells with rectangular coalbed methane reservoir and can also guide fracturing design and writing in the field. The research results enrich the productivity evaluation model of coalbed methane reservoir. In the end, a set of production evaluation method is put forward suitable for the well in rectangular coalbed methane reservoirs with fractured vertical well and finite conductivity fracture. In this paper, the influence of fracture conductivity on single well productivity in rectangular coalbed methane reservoir is quantitatively evaluated for the first time. By improving the calculation method and optimizing the calculation path, the productivity evaluation calculation speed of finite conductivity fractured wells in rectangular coalbed methane reservoir is optimized without affecting the calculation accuracy. The new method can be applied directly to productivity evaluation software, which has the significance of popularization.

A 2-D analytical solution to infinite conductivity fracture injection pressure transient analysis in non-Newtonian/Newtonian composite infinite reservoirs

Two fluid regions are usually created during either water or polymer injection into an oil reservoir for enhanced oil recovery. The two fluid regions create a composite reservoir system that requires analytical solution that can be used to evaluate the injection performance. In this study, we present a 2-D solution for non-Newtonian fluid in the inner zone coupled with infinite boundary in the outer zone using power law model. The outer zone fluid can either be Newtonian or non-Newtonian. Moreover, the mathematical model is formulated such that either zone can be with non-Newtonian power law fluid or Newtonian fluid. The infinite conductivity fractured solution obtained was compared with the ones obtained using the radial line source solution that had already been developed in the literatures, and a perfect match was obtained. An application was made to three examples that have been presented in a previous work, and satisfactory results were obtained. The dimensionless pressure drops during the early linear flow regime for all values of flow indices are the same, but with deviation during radial flow regime.

Rate Decline Analysis of Horizontal Wells with Multiple Variable Conductivity and Uneven Distributed Fractures

In this paper, a new rate decline analysis model of horizontal wells with variable conductivity and uneven distribution of multiple fractures is proposed. By Laplace transformation, point source integration, and superposition principle, solutions of multiple infinite conductivity fractures in closed reservoirs are obtained. By coupling Fredholm integral equation of variable conductivity, linear equations of variable conductivity fractures in Laplace space are obtained. Gauss-Newton iteration, Duhamel convolution, and Stehfest numerical inversion method are used to obtain the bottom hole production solution. The accuracy of the results is verified by comparing with Eclipse software simulation. Then, the influence of some important reservoir and fracture parameters on the production is analyzed. The calculative results show that the smaller the fracture spacing is, the earlier the fracture begins to decline, the more the production will decrease; the change of different fracture length with the total fracture length unchanged has almost no effect on the production; the angle between fracture and x -axis has an important effect on the production; the smaller the angle between fracture and x -axis is, the stronger the interference between fractures is, the higher the production; the initial fracture conductivity affects the early production behavior, and the higher the initial fracture conductivity, the higher the production; the larger the fracture declines index, the lower the production, but the decreasing range gradually decreases with the increase of the decline index; the larger the reservoir drainage radius, the later the energy depletion stage, the higher the production. At last, a good fitting effect is obtained by fitting an example of oil field. The model proposed in this paper enriches the model base of rate decline analysis of fractured horizontal wells and lays a theoretical foundation for efficient development and practice of tight reservoirs.

Transient pressure behavior for unconventional gas wells with finite-conductivity fractures

Horizontal wells and hydraulic fracturing technologies are the only way that makes commercial production from the low and ultra-low permeability unconventional plays possible. The development of analytical and semi-analytical tools to analyze transient behavior of multi-fractured horizontal gas wells have been largely biased towards solving the simplified problem of one-dimensional (1D) Cartesian gas flow in matrix domain, which are only applicable to the production from infinite-conductivity fractures. This study presents a novel semi-analytical method for transient behavior analysis of horizontal gas wells producing from finite-conductivity fractures. Unlike the available transient analysis tools for finite-conductivity fractures which are developed based on empirical and approximate solution, the proposed solution is rigorously derived by solving the governing gas diffusivity equations written for gas flow in two contiguous regions of matrix and fracture simultaneously. Nonlinear, pressure-dependent gas properties remaining in pseudopressure-based diffusivity equations—namely viscosity and compressibility—are rigorously and straightforwardly captured in both fracture and matrix systems. The validity of proposed solution is verified against commercial numerical simulator (COMSOL Multiphysics®) for a series of synthetic cases considering constant and variable -rate production constraints. The case studies presented in this paper also showcase the applicability of proposed solution to the analysis of transient behavior of multi-fractured horizontal gas wells in shale plays under the following two state-of-the-art interpretations: 1) production from finite-conductivity hydraulic fractures in a single-porosity matrix; and 2) production from infinite-conductivity hydraulic fractures in a naturally-fractured matrix.

Semi-analytical solution for productivity evaluation of a multi-fractured horizontal well in a bounded dual-porosity reservoir

Development of horizontal drilling and multi-stage fracturing technologies makes the production of unconventional reservoirs economically and practically feasible. We study the productivity of the multi-fractured horizontal well (MFHW) in a bounded dual-porosity formation. The infinite conductivity fracture case is modeled by combining the Laplace transform (LT) and finite Fourier cosine transform (FFCT). The model is implemented to finite conductivity cases by applying the distributed volumetric sources (DVS) method. The model outputs are verified by comparing them with the results of the separation of variables technique. Different possible flow regimes are observed. The capability of the finite conductivity solution method for optimizing the fracture geometry is approved. The effects of the interporosity coefficient and storativity ratio as the dual-porosity parameters on the productivity of a bounded dual-porosity reservoir are studied by conducting a sensitivity analysis. Similar to single-porosity reservoirs, it is found that the optimum fracture design based on the pseudo-steady state flow regime leads to a higher productivity in the transient flow regimes in the dual-porosity cases. Finally, the proposed approach is used to perform an economic optimization of the hydraulic fracture design. This research study assists researchers and engineers to make better/more accurate fracture design in various applications such as production from unconventional petroleum reservoirs and remediation in fractured underground porous systems.

Pressure-Transient Responses of Fractures With Variable Conductivity and Asymmetric Well Location

Summary Fractures often influence production in hydrocarbon reservoirs, yet the pressure transients observed in the wells might not show the conventional well-test signatures. In this case, the effect of fractures on production would be misinterpreted or even completely missed. The heterogeneous nature of fractured reservoirs makes them difficult to characterize and develop. In addition, the location of a producer within the fracture network also affects the pressure response; however, conventional well-test analysis assumes that the producer is located in symmetrical fracture networks. In this paper we investigate the effects of variations in fracture conductivity and location of the producer in the fracture network on the pressure-transient responses. To overcome the limitations of the dual-porosity (DP) model, this study uses a discrete fracture/matrix (DFM) modeling technique and an unstructured-grid reservoir simulator to generate pressure transients in all analyzed fracture networks. Our rigorous and systematic geoengineering work flow enables us to correlate the pressure transients to the known geological features of the simulated reservoir model. We observed that the simulated pressure transients vary significantly depending on the location of the producer in the fracture network and the properties of the fractures that the producer intercepts. Our findings enable us to interpret some unconventional features of intersecting fractures with variable conductivity. We observed that the behavior of two intersecting fractures, in which the well asymmetrically intercepts a finite-conductivity fracture, can be similar to that of a well intercepting a fracture in a connected fracture network with uniform fracture conductivity. Furthermore, a well intercepting a finite-conductivity fracture in naturally fractured reservoirs (NFRs) with both finite- and infinite-conductivity fractures would yield a DP response (V-shape) that might otherwise be absent if the fracture network is assumed to have uniform conductivity.

Well-flow behavior of reorientation fracture considering permeability anisotropy

PurposeIn petroleum industry, hydraulic fracturing is essential to enhance oil productivity. The hydraulic fractures are usually generated in the process of hydraulic fracturing. Although some mathematical models were proposed to analyze the well-flow behavior of conventional fracture, there are few models to depict unconventional fracture like reorientation fracture. To figure out the effect of reorientation fracture on production enhancement and guide the further on-site operating, this paper aims to investigate the well-flow behavior of vertical reorientation fracture in horizontal permeability anisotropic reservoir.Design/methodology/approachBased on the governing equation considering horizontal permeability anisotropy, the mathematical models for reorientation fractures in infinite reservoir are developed by using the principle of superposition. Furthermore, a rectangular closed drainage area is also considered to investigate the well-flow behavior of reorientation fracture, and the mathematical models are developed by using Green’s and source functions.FindingsComputational results indicate that the flux distribution of infinite conductivity fracture is uniform at very early times. After a period, it will stabilize eventually. High permeability anisotropy and small inclination angle of reorientation will cause significant end point effect in the infinite conductivity fracture. The reorientation fractures with small inclination angle in high anisotropic reservoir are capable of improving 1-1.5 times more oil productivity in total.Originality/valueThis paper develops the mathematical methods to study the well-flow behavior for unconventional fracture, especially for reorientation fracture. The results validate the production enhancement effect of reorientation fracture and identify the sensitive parameters of productivity.

Section-Based Approach Optimizes Unconventional-Reservoir-Fracture Spacing

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 186107, “Optimization of Spacing and Penetration Ratio for Infinite-Conductivity Fractures in Unconventional Reservoirs: A Section-Based Approach,” by S. Liu, SPE, and P.P. Valko, SPE, Texas A&M University. The paper has been peer reviewed and published in the December 2017 SPE Journal. In this paper, the authors consider the development plan of shale gas or tight oil with multiple multistage fractured laterals in a large square drainage area that they call a “section” (usually 640 acres in the US). A section-based optimization of the fracture array is proposed to provide some general statements regarding spacing of wells and fractures. The approach is dependent on a reliable and efficient productivity-index (PI) calculation for the boundary-dominated state (BDS). The information derived from the section-based optimization method and the efficient and reliable algorithm for PI calculation should help the design of multistage fracturing in shale-gas or ultralow-permeability oil formations. Introduction During the last decade, multiple studies have been conducted on various aspects of fracture design. Most of these use a numerical simulation that requires a large number of inputs, which, at the early stages of development, may be in-accessible or burdened by substantial uncertainty. Many of these works use quite sophisticated nonlinear search algorithms. In this work, however, only that part of the problem that can be addressed in rather general terms is considered. Therefore, simplifying assumptions are made to a point at which numerical results can be delivered. The results will provide some insight and useful bounds on achievable productivity. The methodology is dependent on an accurate numerical solution of a well-defined eigenvalue problem and, as such, can provide reference values for various research fields relying heavily on the diffusivity equation. In the complete paper, a method that combines the finite-element method (FEM) with Richardson extrapolation is introduced to calculate the single-fracture PI both for constant-pressure and constant-rate wellbore conditions. Then, with the reliable PI for a single fracture, a convenient section-based optimization is conducted involving only two integer decision variables: the number of columns and the number of rows of the fracture array. Because the constant-pressure wellbore condition is more realistic, the overall PI of the section in the BDS flow regime is considered as the objective function. Two parameters are suggested as the main decisive factors: the dimensionless total fracture length and the feasible range of fracture half-length. In the present approach, the actual optimization takes a simple form of enumeration of a limited number of discrete cases; therefore, advanced nonlinear search algorithms are not involved. Those algorithms become irreplaceable when petrophysical and economic details are incorporated and the structure of the objective function loses its transparent nature, but such situations are beyond the focus of the present work.

Technology Focus: New-Frontier Reservoirs I (July 2018)

Technology Focus Recently, a lot of discussion has revolved around the benefits of multilaterals in unconventional reservoirs. But is this truly the next breakthrough for the industry, or is it all hype? No doubt enormous need exists for a technology such as this to improve reservoir recovery and reduce well cost. The current modus operandi in many plays is to drill horizontal hydraulically fractured wells. The results, however, continue to show that these well types are not effective at draining the reservoir. This is a result of complex hydraulic-fracture geometries with very short propped heights and lengths. Fracture complexity is a natural phenomenon now proved by coring and is likely caused by reservoir stratigraphy and geomechanics. Completion engineers typically counteract this, to some degree, by pumping more fracture treatments in the horizontals. This ineffective drainage points to the benefits of greater reservoir contact with a multilateral. As well as improved drainage, multilateral technology also can reduce well cost by avoiding the necessity to drill top hole for the second (or third) lateral and requiring only one completion and one set of surface equipment. Multilateral technology has been around for many years; however, what has changed in recent times is the reliability of these systems, with more than 98% reliability quoted in a recent publication. Hydraulically fracturing the laterals is now possible as well. Companies such as Occidental, BP, and ConocoPhillips are evaluating the potential of this technology in onshore unconventional plays. In fact, in the fourth quarter of 2017, Occidental successfully performed its first pilot. Enormous need for this technology exists, and time will tell whether it is the next major step in subsurface technology to improve the unit cost of production. Recommended additional reading at OnePetro: www.onepetro.org. SPE 188231 Modeling Early-Time Rate Decline in Unconventional Reservoirs Using Machine Learning Techniques by Aditya Vyas, Texas A&M University, et al. SPE 186107 Optimization of Spacing and Penetration Ratio for Infinite-Conductivity Fractures in Unconventional Reservoirs: A Section-Based Approach by S. Liu, Texas A&M University, et al. SPE 186092 Rate Dependence of Bilinear Flow in Unconventional Gas Reservoirs by M.S. Kanfar, University of Calgary, et al.

PRIDICTION OF THE PRODUCTIVITY OF HYDRAULIC FRACTURED VERTICAL WELL WITH INFINITE CONDUCTIVITY FRACTURE

Hydraulic fracturing is used to significantly improve well productivity, particularly in reservoirs with low permeability. In carrying out this operation in the reservoir creates a large branched system of cracks. Increase filtration connection of the wells to remote areas. This article will present the results of numerical solution of the problem related to the definition of increasing the productivity of vertical wells after hydraulic fracturing, with infinite conductivity fracture. As a result, productivity index increases with penetration ratio distance for all aspect ratios.

Estimation of characteristics of naturally fractured reservoirs by pressure transient analysis

PurposeThis paper aims to analyze the pressure behavior in dual porosity reservoirs using different techniques in an attempt to correctly characterize reservoir properties. Pressure transient tests in naturally fractured reservoirs often exhibit non-uniform responses.Design/methodology/approachThe pressure transient tests in naturally fractured reservoirs were analyzed using conventional semi-log analysis, type curve matching (using commercial software) and Tiab’s direct synthesis (TDS) technique. In addition, the TDS method was applied in case of a naturally fractured formation with a vertical hydraulic fracture. These techniques were applied to a single-layer, naturally fractured reservoir under pseudosteady state matrix flow. By studying the unique characteristics of the different flow regimes appear on the pressure and pressure derivative curves, various reservoir characteristics can be obtained such as permeability, skin factor and fracture properties.FindingsFor naturally fractured reservoirs, a comparison between the results semi-log analysis, software matching and TDS method is presented. In case of wellbore storage, early time flow regime can be obscured that lead to incomplete semi-log analysis. Furthermore, the type curve matching usually gives a non-uniqueness solution, as it needs all the flow regimes to be observed. However, the direct synthesis method used analytical equation to calculate reservoir and well parameters without type curve matching. For naturally fractured reservoirs with a vertical fracture, the pressure behavior of wells crossed by a uniform flux and infinite conductivity fracture is analyzed using TDS technique. The different flow regimes on the pressure derivative curve were used to calculate the fracture half-length in addition to other reservoir properties.Originality/valueThe results of different field cases showed that TDS technique offers several advantages compared to semi-log analysis and type curve matching. It can be used even if some flow regimes are not observed. Direct synthesis results are accurate compared to the available core data and the software matching results. It can be used to confirm the software matching results and to give reliable reservoir characteristics when there is lack of data.

Optimization of Spacing and Penetration Ratio for Infinite-Conductivity Fractures in Unconventional Reservoirs: A Section-Based Approach

Summary In this paper, we consider the development plan of shale gas or tight oil with multiple multistage fractured laterals in a large square drainage area that we call a “section” (usually 640 acres in the US). We propose a convenient section-based optimization of the fracture array with two integer variables, the number of columns (horizontal laterals) and rows (fractures created in a lateral), to provide some general statements regarding spacing of wells and fractures. The approach is dependent on a reliable and efficient productivity-index (PI) calculation for the boundary-dominated state (BDS). The dimensionless PI is obtained by solving a time-independent eigenvalue problem by use of the finite-element method (FEM) combined with the Richardson extrapolation. The results of the case study demonstrate two decisive factors: the dimensionless total fracture length, related to the total amount of proppant and fracturing fluid available for the section, and the feasible range of actual fracture half-lengths, related to current fracturing-technology limitations. Under the constraint of dimensionless total fracture length, increasing the number of columns (horizontal laterals) increases the total PI but with only diminishing returns, whereas the optimal fracture-penetration ratio decreases somewhat, but is still near unity. When adding the technological constraint of a limited range of fracture half-lengths that can be routinely and reliably created, only a few choices remain admissible, and the optimal decision can be easily made. These general statements for the ideal homogeneous and isotropic formation can serve as a reference in the more-detailed optimization works. In other words, we offer a first-pass method for decision making in early stages when detailed inputs are not yet available. The information derived from the section-based optimization method and the efficient and reliable algorithm for PI calculation should help the design of multistage fracturing in shale-gas or ultralow-permeability oil formations.

Modeling sulfur plugging of fractured wells under non-Darcy and non-equilibrium sulfur deposition

The issue of sulfur plugging has been a concern for engineers for a long time. Some conventional models for vertical wells have been presented by authors; however, analytical models of sulfur plugging for fractured wells have not been reported. Based on non-Darcy and non-equilibrium theory, this article presents a new sulfur deposition prediction model for fractured wells, and a new concept, named sulfur deposition equivalent wellbore radius (SDER), was induced into new equations to describe fractured well characteristics. In a conventional gas reservoir, the fracture conductivity is constant, but here it is changeable due to the presence of sulfur deposition. The SDER value will increase with the increasing fracture conductivity but will reach a stable point at a certain fracture conductivity value. The value of that stable point is equal to 0.5, which is just the value of the equivalent wellbore radius with infinite conductivity fractures. The influence of some important parameters on sulfur deposition was investigated, including gas production, relaxation time, water saturation, gas temperature, fracture conductivity, initial permeability, flow pressure, and radial distance. The results show that the plugging of the formation by deposited sulfur occurs later during the non-equilibrium process, occurs earlier when the flow rate is as high as expected, occurs earlier as the water saturation increases, occurs earlier under high temperature conditions, occurs earlier as the non-Darcy flow becomes stronger, and occurs earlier as the fracture conductivity becomes smaller.

Producing-Gas/Oil-Ratio Behavior of Multifractured Horizontal Wells in Tight Oil Reservoirs

Summary Horizontal wells with hydraulic fractures in tight oil reservoirs show producing-gas/oil-ratio (GOR) behavior that is very different from conventional, higher-permeability reservoirs. This paper explains the reasons for the observed behavior by use of reservoir simulation, with field examples from the STACK and SCOOP plays of the Anadarko Basin in central Oklahoma. A framework for interpreting observed GOR behavior in tight black-oil reservoirs is modeled after the following stages in a well's history. Some stages may not be visible because of the degree of undersaturation, flowing-bottomhole-pressure schedule, finite-conductivity fractures, and duration of the transient-flow period. Stage 1: Early GOR is constant at the initial solution GOR (Rsi) while bottomhole flowing pressure is above the bubblepoint. Stage 2: A rise in GOR as bottomhole flowing pressure declines to less than the bubblepoint. Stage 3: The transient GOR “plateau”, which is characteristic of transient linear flow. Stage 4: A continuous rise in GOR during boundary-dominated flow. Fundamental differences between linear and radial flow, which cause the dependence of GOR on flowing bottomhole pressure, are explained by use of simulation. During transient linear flow, the GOR response to changes in flowing bottomhole pressure is independent of permeability for infinite-conductivity fractures, but not for finite-conductivity fractures. Several practical observations are made. Knowing Rsi and the transient-GOR-plateau level in an area can help one interpret where a well is in its GOR history. Rate-transient-analysis (RTA) diagnostic plots are altered by rising GOR, and sometimes show an early unit slope. During boundary-dominated flow, GOR is more a function of cumulative production than of time; wells with closer fracture spacing have a faster GOR rise with time, but also recover oil more quickly. If compound linear flow develops, GOR can decline late in the well life. The Meramec and Woodford formations in STACK can be history matched without invoking a suppressed bubblepoint caused by pore-proximity effects. The critical gas saturation in the Meramec appears to be in the range of 0–5%. Technical contributions include a framework for interpreting GOR behavior over well life; the effect of changing bottomhole flowing pressure on GOR; the effect of fracture spacing, conductivity, and half-length on GOR; and the effect of GOR on RTA diagnostic plots.

Analysis of Pressure Transient Tests in Naturally Fractured Reservoirs

Pressure transient tests in naturally fractured reservoirs often exhibit non-uniform responses. Different techniques can be used to analyze the pressure behavior in dual porosity reservoirs in an attempt to correctly characterize reservoir properties. In this paper, the pressure transient tests in naturally fractured reservoirs were analyzed using conventional semi-log analysis, type curve matching (using commercial software) and Tiab’s direct synthesis (TDS) technique. In addition, the TDS method was applied in case of a naturally fractured formation with a vertical hydraulic fracture. These techniques were applied to a single layer naturally fractured reservoir under pseudosteady state matrix flow. By studying the unique characteristics of the different flow regimes appear on the pressure and pressure derivative curves, various reservoir characteristics can be obtained such as permeability, skin factor, and fracture properties. For naturally fractured reservoirs, a comparison between the results semi-log analysis, software matching, and TDS method is presented. In case of wellbore storage, early time flow regime can be obscured that lead to incomplete semi-log analysis. Furthermore, the type curve matching usually gives a non-uniqueness solution as it needs all the flow regimes to be observed. However, the direct synthesis method used analytical equation to calculate reservoir and well parameters without type curve matching. For naturally fractured reservoirs with a vertical fracture, the pressure behavior of wells crossed by a uniform flux and infinite conductivity fracture is analyzed using TDS technique. The different flow regimes on the pressure derivative curve were used to calculate the fracture half-length in addition to other reservoir properties. The results of different cases showed that TDS technique offers several advantages compared to semi-log analysis and type curve matching. It can be used even if some flow regimes are not observed. Direct synthesis results are accurate compared to the available core data and the software matching results.

Pressure-Transient Tests and Flow Regimes in Fractured Reservoirs

Summary Fractures are common features in many well-known reservoirs. Naturally fractured reservoirs include fractured igneous, metamorphic, and sedimentary rocks (matrix). Faults in many naturally fractured carbonate reservoirs often have high-permeability zones, and are connected to numerous fractures that have varying conductivities. Furthermore, in many naturally fractured reservoirs, faults and fractures can be discrete (rather than connected-network dual-porosity systems). In this paper, we investigate the pressure-transient behavior of continuously and discretely naturally fractured reservoirs with semianalytical solutions. These fractured reservoirs can contain periodically or arbitrarily distributed finite- and/or infinite-conductivity fractures with different lengths and orientations. Unlike the single-derivative shape of the Warren and Root (1963) model, fractured reservoirs exhibit diverse pressure behaviors as well as more than 10 flow regimes. There are seven important factors that dominate the pressure-transient test as well as flow-regime behaviors of fractured reservoirs: (1) fractures intersect the wellbore parallel to its axis, with a dipping angle of 90° (vertical fractures), including hydraulic fractures; (2) fractures intersect the wellbore with dipping angles from 0° to less than 90°; (3) fractures are in the vicinity of the wellbore; (4) fractures have extremely high or low fracture and fault conductivities; (5) fractures have various sizes and distributions; (6) fractures have high and low matrix block permeabilities; and (7) fractures are damaged (skin zone) as a result of drilling and completion operations and fluids. All flow regimes associated with these factors are shown for a number of continuously and discretely fractured reservoirs with different well and fracture configurations. For a few cases, these flow regimes were compared with those from the field data. We performed history matching of the pressure-transient data generated from our discretely and continuously fractured reservoir models with the Warren and Root (1963) dual-porosity-type models, and it is shown that they yield incorrect reservoir parameters.