Pseudo-steady State Flow Research Articles

Development of new analytical model for series and parallel triple porosity models and providing transient shape factor between different regions

Fluid flow architecture in fractured reservoirs is mainly based on the dual porosity model. In this model, the matrix block is considered as a fluid storage source and fracture as a flow path. Nevertheless, in naturally fractured reservoirs, different properties of matrix blocks and fracture systems have caused the triple porosity model to be more applicable than the traditional dual porosity model.The existed triple porosity models presented so far are mainly based on pseudo steady state flow in the matrix block and vug and via a semi-analytic form. In this paper, two analytical models of triple porosity model are presented with consideration of transient flow; the first model being: the series flow of the matrix block to the vug and the vug to the fracture, and the second model: transient parallel flow of the matrix block and vug to the fracture. In this paper, for the first time, an analytical and transient shape factor between different regions is put forward.Using the developed new solution, various flow regimes in triple porosity model have been identified. In the series triple porosity model, changing the parameters of vug and matrix did not affect the early production while it influenced the rate of the middle and lately times. The shape factor between the matrix and vug was independent of the physical properties of vug while the shape factor between vug and fracture was influenced by the physical properties of vug and matrix. In the parallel triple porosity model, the shape factor of matrix-fracture and vug-fracture were not dependent on the properties of the other regions of rock. Moreover, a decrease in inter-porosity coefficient and an increase in storativity ratio increased the shape factor in transient time.Additionally, in order to validate the analytical model, the results of the model are compared with the numerical simulations. This study presents a new analytical and transient shape factor which finds application in simulation of triple porosity formations.

A Mathematical Model for Predicting Long-Term Productivity of Modern Multifractured Shale-Gas/Oil Wells

SummaryModern multifractured shale-gas/oil wells are horizontal wells completed with simultaneous-fracturing, zipper-fracturing, and (in particular) modified-zipper-fracturing techniques. An analytical model was developed in this study for predicting the long-term productivity of these wells under conditions of pseudosteady-state (PSSS) flow, considering the cross-bilinear flow in the rock matrix and hydraulic fractures. Performance of the model was verified with the well-productivity data obtained from a shale-gas well and a shale-oil well. Sensitivity analyses were performed to identify key parameters of hydraulic fracturing affecting well productivity. The conducted field case studies show that the analytical model overpredicts shale-gas-well productivity by 2.3% and underpredicts shale-oil productivity by 7.4%. A sensitivity analysis with the model indicates that well productivity increases with reduced fracture spacing, increased fracture length, and increased fracture width, but not proportionally. Whenever operational restrictions permit, more fractures with high density should be created in the hydraulic-fracturing process to maximize well productivity. The benefit of increasing fracture width should diminish as the fracture width becomes large. Increasing fracture length by pumping more fracturing fluid can increase well-production rate nearly proportionally. Therefore, it is desirable to create long fractures by pumping high volumes of fracturing fluid in the hydraulic-fracturing process.

A new method to determine reasonable well pattern density in low-permeability gas reservoirs

The productivity evaluation of gas reservoirs is used to determine the optimal number of well pattern, which is an important part of the preparation of gas reservoir development plan. Using integral transformation and asymptotic analyses to solve the pseudo steady-state flow model in closed rectangular gas reservoir of constant thickness, analytic solution of pressure distribution of eccentric well can be obtained. This paper presents mathematical model of multiple wells producing from a closed gas reservoir by applying the principle of pressure superposition. The pseudo steady-state dimensionless production index is introduced to solve the multiple well mathematical models under the constant pressure, and the number of production wells and cumulative production in different production interval are obtained. The results of the calculation shows that the curve of the cumulative production and economic benefits have a maximum value when the producing time, gas price, and drilling costs are given, so it can determine the optimal number of wells and provide the basis for making gas reservoir development plan.

Integrated deterministic approaches for productivity index of reservoirs depleted by horizontal wells and undergone multiphase flow conditions

This paper introduces new integrated approaches for estimating pseudo-steady state productivity index (PI) and shape factor of oil and gas reservoirs drained by horizontal wells and dominated by multiphase flow conditions. These approaches couple PVT data, relative permeability curves, and pressure distribution models during pseudo-steady state flow (PSS). The objective is eliminating the uncertainties of applying single phase flow models and substantially comprising the realistic parameters that govern multiphase flow conditions. The importance of this study is represented by introducing the impact of multiphase flow (Gas, Oil, Water) to the productivity index and inflow performance relationship of reservoirs depleted by horizontal wellbores.This study is conducted throughout three tasks. The first includes reviewing currently used techniques for calculating pseudo-steady state PI and shape factor of bounded oil and gas reservoirs assuming single phase flow is the dominant flow pattern in the porous media. The second introduces new deterministic models for reservoir mobility and compressibility and new multiphase flow pressure function [f(P)mp]. While the third focuses on calculating PI and shape factor of reservoirs depleted by horizontal wellbores and predicting inflow performance relationship (IPR) considering multiphase flow conditions.The outcomes of this study can be summarized as: 1) Developing deterministic models by multi-regression analysis for reservoir mobility and compressibility using PVT data and relative permeability curves. 2) Generating multiphase flow reservoir pressure function [f(P)mp] that introduces the impact of multiphase flow conditions to PI model. 3) Applying the new approaches for calculating multiphase flow PI and shape factor and thereafter generating multiphase flow IPR. The study has pointed out: 1) The fact that the PI of constant Sandface flow rate is greater than constant wellbore pressure is confirmed for single and multiphase flow conditions. 2) PI and shape factor are time invariant at late production time and depends only on reservoir geometry and wellbore type. 3) PI of multiphase flow is less than the index of single phase flow while shape factor exhibits higher value for multiphase flow compared to the one for single phase flow. 4) Multiphase flow negatively changes the IPR either by increasing the pressure drop or by declining the flow rate. 5) Most of the deterministic models developed in this study show high accuracy (high R-squared value, R2≥90% ) for the changes in fluid properties with reservoir pressure.

Fishbone type horizontal wellbore completion: A study for pressure behavior, flow regimes, and productivity index

This paper focuses on studying pressure behavior, flow regimes, and productivity index of oil and gas reservoirs depleted by horizontal wells with a fishbone type completion technology. The main objective is introducing an integrated approach for the favorable configuration and distribution of multilateral horizontal wells extending in porous media at arbitrary directions and arbitrary permeability trajectories. The study aims to demonstrate the advantages and disadvantages of fishbone completion in terms of transient and pseudo-steady state productivity index considering single phase flow in the porous media. It is also an attempt to propose new analytical models for the expected flow regimes and new practical solutions for estimating stabilized productivity index for different scenarios of fishbone completion.In this study, pressure distribution in the porous media is described by several analytical models considering different reservoir geometries and different multilateral horizontal wellbore configurations. These models are solved for single and dual porous media where naturally induced fractures exist. Lateral and spatial permeability are considered in these models as well as non-Darcy flow impact close to the vicinity of the wellbores in gas reservoirs. The solutions of these models are used to generate analytical models for the flow regimes that can be observed during the entire production life. Transient and stabilized productivity indices are quantified for the case of constant production rate using the observed pressure responses of fishbone type horizontal wells. The starting time of pseudo-steady state and the stabilized pseudo-steady state productivity index are used to assess the optimal configuration for fishbone completion.The observations are summarized in: 1) Pressure behavior, flow regimes, and productivity index of fishbone type horizontal wells are impacted by main wellbore and laterals distribution, however, laterals distribution in both sides of the main wellbore may not have an impact in isotropic formations. 2) Azimuth and vertical deviation angles from the main horizontal wellbore and lateral and spatial anisotropies have similar impacts because they change hypothetically the lateral length. 3) The impact of azimuth angle might be maximized by the lateral anisotropy while spatial anisotropy could maximize the impact of vertical deviation angle. 4) The conjunction points of the laterals and the main wellbore may not have a significant impact. 5) Productivity indices of fishbone type horizontal wellbores during early and late production times are less than the ones obtained by single horizontal well, however, almost identical indices are seen at intermediate production time when linear flow regime is observed. 6) The starting time of pseudo-steady state flow is controlled by the main wellbore length more than the laterals’ characteristics unless these laterals extend to reservoir boundaries.The conclusions summed up from the results of this study are: 1) Eventhough fishbone type completion could be significant solution for a lot of operation problems of long horizontal wellbore such as excessive pressure drop and wellbore instability, it is not necessary delivering optimum reservoir performance. 2) Flow regimes and flow patterns (transient or pseudo-steady state) are affected by the characteristics and distributions of the main wellbore and laterals in the porous media.

Pressure-rate convolution and deconvolution response for fractured conventional and unconventional reservoirs using new decline rate model

Abstract This paper introduces new approach for pressure-rate convolution and deconvolution analysis of multi-stages hydraulically fractured conventional and unconventional reservoirs. This approach demonstrates the impact of variable Sand face flow rate on reservoir performance. A new model for P/R deconvolution is used to convert pressure pulse from variable flow rate to single and constant rate response. The target of this study is fractal reservoirs with and without stimulated and unstimulated reservoir volume. Multi-linear flow regimes approach is used to describe pressure behavior in the reservoirs while decline flow rate behavior is described by newly proposed model in this study. This model depicts, instead of van Everdingen model, indirectly the declining rate with time by using pressure responses with production time. Decline flow rate behavior simulated by linear and bi-linear flow models are also studied and compared with the one obtained by the new model. Several analytical models are used in this study by applying P/R convolution and deconvolution technique and solved for constant and variable flow rate considering different reservoir configurations and operating conditions. The results are interpreted and analyzed for better understanding pressure behaviors, flow regime types, and productivity index trends for continuously changing flow rate especially at early production time. Estimating stimulated reservoir volume ( V s r v ) is considered one of the applications of convolved pressure since it is calculated from pseudo-steady state flow when late time boundary dominated flow regime is reached. The outcomes of this study can be summarized as: 1) Introducing new approach for pressure-rate convolution and deconvolution technique for multi-stages hydraulically fractured reservoirs by applying new decline flow rate model that indirectly simulates variable flow rate with time. 2) Generating analytical models for dimensionless pressure and flow rate for constant and variable flow rate using the concept of P/R convolution and deconvolution. 3) Comparing the result of newly proposed models with the results obtained by applying van Everdingen model for decline rate behavior. 4) Studying the applicability of linear and bi-linear flow models in converting variable flow rate pressure response to single and constant flow rate pressure response. 5) Applying the deconvolution technique to simulate pressure response at late production time to estimate stimulated reservoir volume. The most interesting points are: 1) The main difference in wellbore pressure behavior between variable and constant flow rate can be seen at early production time, however intermediate production time could also show very limited changes for the case of variable rate wellbore pressure. 2) A unit slope line flow regime could be developed for varied flow rate pressure response at very early production time similar to the wellbore storage dominated flow regime. 3) Productivity index calculated by the proposed models for variable flow rate is greater than the index for constant flow rate. 4) The impact of petrophysical properties of porous media and hydraulic fracture characteristics on pressure response are similar in the two cases of variable and constant flow rate. 5) The decline rate models for linear and bi-linear flow are not applicable in pressure deconvolution technique.

Practical solution for stabilized pseudo-steady state productivity index of multiple vertical wells depleting closed reservoirs

Abstract The objective of this paper is introducing practical solutions for stabilized pseudo-steady state productivity index and pressure behaviors and flow regimes of multiple vertical wells depleting closed rectangular reservoirs. It introduces full understanding of reservoir performance during transient and pseudo-steady state flow using pressure and pressure derivative patterns and productivity index and productivity index derivative schemes. The practical solutions proposed in this study are relied on a new approach for accurate estimation of starting time of pseudo-steady state flow and stabilized productivity index by applying productivity index derivative that converges mathematically to zero at the beginning of this flow. It is also an attempt for pointing out the optimal reservoir configuration that can give maximum stabilized pseudo-steady state productivity index for different wellbore types and distributions in the drainage area. Several analytical models are used in this study for describing pressure drop and productivity index behavior of multiple vertical wells considering different reservoir configurations and different wellbore types and locations. These wells were assumed either fully or partially penetrate the formations. These pressure models are justified to generate two pressure derivatives, one represents the derivative of time dependent pressure drop and the second represents the derivative of time-invariant pressure drop. The two derivatives are used for determining the starting time of pseudo-steady state flow and stabilized productivity index when both converge and mathematically become identical. These models are used also to generate several plots for time-invariant or stabilized pseudo-steady state productivity index for different reservoir geometries considering different wellbore conditions. The maximum stabilized productivity index is illustrated in these plots and several analytical models for the expected flow regimes are developed using pressure and pressure derivative behaviors of different reservoirs and wellbore conditions. The outcomes of this study are summarized in: 1) Developing new analytical solutions for pressure distribution in porous media drained by multiple vertical wells. 2) Developing new practical solution for estimating stabilized pseudo-steady state productivity index. 3) Understanding pressure, pressure derivative, and productivity index behavior of finite acting reservoir depleted by multiple vertical wells during transient and pseudo-steady state production. 4) Investigating the impacts of different reservoir configurations and wellbore sizes and locations as well as partial penetration on stabilized pseudo-steady state productivity index. The novel points in this study are: 1) The optimum reservoir configuration that gives the maximum stabilized productivity index is the rectangular shape reservoir with reservoir length to width ratio of ( 2 − 4 ) . 2) Starting time of pseudo-steady state and stabilized productivity index are impacted by wellbore numbers, sizes, and locations. 3) Starting time of pseudo-steady state flow is not affected by partial penetration ratio, however, stabilized productivity index is affected by this ratio. 4) Five spots pattern gives the maximum productivity index compared with other wellbore distribution patterns. 5) Productivity index of diagonally distributed wellbores is greater than the index of wellbores distributed along reservoir length.

Productivity determination for cyclic production using steady state harmonic theory – Application to artificial lift wells

In this paper we derive new solutions to the problem of harmonic flow in a bounded porous medium with inclusion of wellbore storage and skin. We present new solutions to varying radial boundary conditions and pay special attention to the cases of pseudo steady and steady state flow. These solutions have a new, special application to cyclically produced wells undergoing artificial lift under these conditions.It is normally difficult for engineers to determine the Productivity Index (PI) or permeability of production wells undergoing cyclic operation. The relationship between bottomhole pressure, cycles, run time % and formation properties is quite non-linear, which we will illustrate here. However, the new theory allows simple determination of PI for such wells. The results will have application to artificially lifted wells or those wells intentionally pulsed in bounded reservoirs. A calculation method is shown for this type of problem along with field application.

Semi-analytical solution for pressure transient analysis of a hydraulically fractured vertical well in a bounded dual-porosity reservoir

We study the role of a hydraulic fracture on the pressure transient behavior of a vertical well producing from a bounded (or finite) dual-porosity formation. A combination of Laplace transform (LT) and the finite Fourier cosine transform (FFCT) are used to solve the diffusivity equation. The presented analysis allows identification of five flow regimes, including: 1) early linear flow, 2) volumetric depletion of natural fractures, 3) natural-fracture radial flow, 4) transition from natural-fracture radial flow to total (natural fractures and matrix) radial flow, and 5) pseudo-steady state flow. The results reveal that the interporosity flow coefficient, storativity ratio, natural-fracture permeability anisotropy, and reservoir size play significant roles on the identified flow regimes compared to the hydraulically fractured well location and reservoir shape. The developed solution can be useful for well test analysis by generating a new set of type curves or can be applicable to a forward model for estimating parameters of reservoir. This study presents a new semi-analytical solution which finds application in well testing of hydraulically fractured wells in dual-porosity formations.

Revisiting Current Techniques for Analyzing Reservoir Performance: A New Approach for Horizontal-Well Pseudosteady-State Productivity Index

Summary The objective of this paper is to revisit currently used techniques for analyzing reservoir performance and characterizing the horizontal-well productivity index (PI) in finite-acting oil and gas reservoirs. This paper introduces a new practical and integrated approach for determining the starting time of pseudosteady-state flow and constant-behavior PI. The new approach focuses on the fact that the derivative of PI vanishes to zero when pseudosteady-state flow is developed. At this point, the derivative of transient-state pressure drop and that of pseudosteady-state pressure drop become mathematically identical. This point indicates the starting time of pseudosteady-state flow as well as the constant value of pseudosteady-state PI. The reservoirs of interest in this study are homogeneous and heterogamous, single and dual porous media, undergoing Darcy and non-Darcy flow in the drainage area, and finite-acting, depleted by horizontal wells. The flow in these reservoirs is either single-phase oil flow or single-phase gas flow. Several analytical models are used in this study for describing pressure and pressure-derivative behavior considering different reservoir configurations and wellbore types. These models are developed for heterogeneous and homogeneous formations consisting of single and dual porous media (naturally fractured reservoirs) and experiencing Darcy and non-Darcy flow. Two pressure terms are assembled in these models; the first pressure term represents the time-dependent pressure drop caused by transient-state flow, and the second pressure term represents time-invariant pressure drop controlled by the reservoir boundary. Transient-state PI and pseudosteady-state PI are calculated using the difference between these two pressures assuming constant wellbore flow rate. The analytical models for the pressure derivatives of these two pressure terms are generated. Using the concept that the derivative of constant PI converges to zero, these two pressure derivatives become mathematically equal at a certain production time. This point indicates the starting time of pseudosteady-state flow and the constant behavior of PI. The outcomes of this study are summarized as the following: Understanding pressure, pressure derivative, and PI behavior of bounded reservoirs drained by horizontal wells during transient- and pseudosteady-state production Investigating the effects of different reservoir configurations, wellbore lengths, reservoir homogeneity or heterogeneity, reservoirs as single or dual porous media, and flow pattern in porous media whether it has undergone Darcy or non-Darcy flow Applying the concept of the PI derivative to determine the starting time of pseudosteady-state stabilized PI The novel points in this study are the following: The derivative of the PI can be used to precisely indicate the starting time of pseudosteady-state flow and the constant behavior of PI. The starting time of pseudosteady-state flow determined by the convergence of transient- and pseudosteady-state pressure derivative or by the PI curve is always less than that determined from the curves of total pressure drop and its derivative. Non-Darcy flow may significantly affect the transient-state PI, but pseudosteady-state PI is slightly affected by non-Darcy flow. The starting time of pseudosteady-state flow is not influenced by non-Darcy flow. The convergence of transient- and pseudosteady-state pressure derivatives is affected by reservoir configurations, wellbore lengths, and porous-media characteristics.

A modified predictive model for estimating gas flow rate in horizontal drain hole

Accurate prediction of the flow rate of horizontal gas well is necessary for economic feasibility, planning and development of gas field. Most of the early models assumed that the production from the horizontal well is infinitely conductive except few recent models. Some recent models reported in the literature for estimating flow rate in horizontal well where the pressure losses due to friction along horizontal drain hole was considered. An improved model that checks the impact of all possible well bore pressure losses on gas production rate of horizontal well is reported. The neglected impact of well bore pressure losses due to fluid accumulation and kinetic energy in the past models is thought to be a conceivable reason for the inconsistency between computed rates from the models and rates got from production tests. The new model was validated using the same field contextual investigation utilized by Guo et al. and outcome got from the new model yields more satisfactory results. A more realistic results that evident all flow phenomena in gas well include the initial unsteady, pseudo-steady and steady state flow condition hence flow rate at any given production time has been established for flow of gas along horizontal well. The outcomes of the study demonstrate that the percentage deviation of the new model at steady state flow condition is less than 5.0% compared with 11.05% acquired from Guo et al. model following by 259.7% from Furui's model, and 1118.2% got from Joshi's model. This work gives field operators a precise and helpful device for prediction and assessment of production in a gas horizontal well.

Integrated analysis of pressure response using pressure-rate convolution and deconvolution techniques for varied flow rate production in fractured formations

This paper introduces an integrated analysis for pressure transient behavior of conventional and unconventional multi-porous media reservoirs considering varied flow rate conditions. It focuses on the applications of pressure-rate convolution and deconvolution techniques for analyzing pressure records of homogenous single porous media, double porous media, and triple porous media reservoirs. The tasks covered in this paper are: Deconvloving pressure response, characterizing and developing analytical models for new flow regime that represents the impact of varied flow rate at early production time, estimating productivity index, and calculating stimulated reservoir volume (Vsrv). The reservoirs of interests in this study are depleted by horizontal wells intersecting multiple hydraulic fractures and composed of stimulated and unstimulated reservoir volume with different configurations and geometries.Multi-linear flow regimes approach is used to describe pressure behavior in the reservoirs while declining flow rate scheme is described by van Everdingen model. These two models are assembled to deconvolve pressure responses from varied to single flow rate condition. The deconvolved pressure and pressure derivative are used in characterizing the new flow regime that is observed for the case of varied flow rate during early production time and generating analytical models in dimensionless form and field units for oil and gas reservoirs. They are also used to calculate stimulated reservoir volume using very late production time, pseudo-steady state flow is reached, without the need for running well test to this time. Productivity index for constant and varied flow rate are calculated and compared using pressure responses of the two cases. The methodology used in this paper has experienced reservoirs with hydraulic fractures of different characteristics, stimulated reservoir volume (Vsrv) between hydraulic fractures, and unstimulated reservoir volume (Vusrv) in the outer drainage are beyond fracture tips.The outcomes of this study can be summarized as: 1) Introducing an integrated analysis of pressure behavior using pressure-rate convolution and deconvolution techniques for hydraulically fractured conventional and unconventional reservoirs. 2) Deconvloving pressure response from varied to single flow rate condition by applying multi-linear flow regimes approach collaboratively with decline flow rate model proposed by van Everdingen. 3) Characterizing the impact of variable production rate on pressure response for different reservoir types, configurations, and geometries as well as different hydraulic fracture characteristics. The most interesting points are: 1) The main differences in wellbore pressure behavior between variable and constant flow rate can be seen at early production time. 2) The new flow regime that could be developed for varied flow rate at very early production time is characterized by slope of (1.30−1.35) on pressure derivative curve. 3) Productivity index calculated by deconvolved pressure for variable flow rate is slightly less than the index for constant flow rate at early production time, however, the index has the same value for the two conditions at late production time.

Analytical Solution to Study Depletion/Injection Rate on Induced Wellbore Stresses in an Anisotropic Stress Field

During production or injection, the state of stresses within the reservoir as well as around the wellbore changes. It is therefore important to evaluate the impact of the induced stresses on stability of wellbores. This study proposes an analytical solution to estimate the influence of production or injection rate on stresses around a wellbore in an anisotropic stress field. For considering the effect of production/injection rate, pseudo steady state flow was assumed within the reservoir that occurs more frequently than transient or steady state flow in a reservoir with an expanding drainage radius. Until now stress equations have not been developed in which the effect of production/injection rate being included. The results showed that the impact of the depletion and injection rate on induced stresses around the wellbore is significant and should be considered in the geomechanical analysis especially in low permeability reservoirs. Application of the proposed solution on a typical sandstone reservoir showed that in relatively high production rate, the radial and tangential stresses near wellbore decrease whereas the vertical and pore pressure increase. An opposite trend was observed for the effect of injection rate on near wellbore stresses. This results are consistent with field observation and means that in rapid production, the vertical stress will increase and horizontal stresses will decrease leading to surface subsidence.

New approach for predicting multiple fractured horizontal wells performance in tight reservoirs

Multiple fractured horizontal wells (MFHWs) are considered as the most effective stimulation technique to improve recovery from low permeability reservoirs particularly tight and shale assets. Understanding of the complex flow behaviour and predicting Productivity Index (PI) of these wells are vital for exploitation of unconventional reservoirs.The analytical or semi analytical models previously proposed for their PI calculations cannot accurately describe the flow behaviour around MFHWs mainly due to lack of capturing the complexity of the flow especially the fracture-to-fracture interference effects. Many of them are also too complex and/or not general enough limiting their use. The fine grid three-dimensional (3D) simulation approach is also costly and cumbersome. In this work, we followed a new approach to develop a new equation that can predict MFHWs performance under pseudo-steady state (PSS) flow conditions in tight reservoirs.A programming code, producing the include files, was coupled with a fine grid 3D commercial reservoir simulator to generate a large data bank. For these simulations, the pertinent parameters (matrix permeability, the number of fractures and fracture permeability, spacing, width, length and conductivity) were varied over wide practical ranges based on the Latin Hypercube sampling method.The individual impact of the parameters on PI, as the output variable, was evaluated by a statistical analysis technique under different prevailing conditions. It is shown, for instance, that increasing the fracture width and permeability does not result in a significant monotonic increase in PI while changing fracture length, spacing and numbers influence PI greatly.A new equation is then proposed that relates MFHWs-PI to a limited number of parameters by applying symbolic regression technique. Here, the total productivity index of MFHWs is related to the PI of the horizontal well with a single fracture, the number of fractures and dimensionless fracture spacing. The cross-validation results show that the proposed equation is general, reliable and simple for prediction purposes because it benefits from limited and appropriate dimensionless numbers with good values of fitting indices.This study expands our understanding of flow behaviour in tight reservoirs and provides an invaluable engineering tool that can facilitate simulation of flow around MFHWs, their optimum design and their well performance prediction.

Prediction of productivity of high energy gas-fractured oil wells

The use of fresh water as a fracturing fluid has limitations such as limited water availability in arid areas and negative impacts of water on oil and gas production in formations with high clay content. Alternatives to water for well fracturing include tailored energetic materials such as novel explosives. High energy gas fracturing (HEGF) creates radial fractures with fracture-orientations independent of formation stress anisotropy and heterogeneity. This eliminates the requirement of in-situ stress orientation for designing multi-hydraulic water-fracturing horizontal wells. Assuming uniformly distributed radial fractures around wellbore, an analytical well productivity model was derived in this study to predict productivity of HEGF-completed oil wells under pseudo steady state flow condition. Field case studies and sensitivity analyses were performed with the analytical model. Result of field case studies indicates that the analytical model over-predicts productivity of HEGF-completed wells by about 10%. Sensitivity analysis with the analytical model shows that the productivity of HEGF-completed wells reaches a maximum value at an optimal number of radial fractures around the wellbore. The productivity of the HEGF-completed wells increases non-linearly with fracture conductivity. But the benefit of increasing fracture conductivity levels out beyond fracture conductivity 2000 md-inch for the typical case investigated in this study.

Estimating the properties of naturally fractured reservoirs using rate transient decline curve analysis

Transient rate decline curve analysis for constant pressure production is presented in this paper for a naturally fractured reservoir. This approach is based on exponential and constant bottom-hole pressure solution. Based on this method, when ln (flow rate) is plotted versus time, two straight lines are obtained which can be used for estimating different parameters of a naturally fractured reservoir. Parameters such as storage capacity ratio (ω), reservoir drainage area (A), reservoir shape factor (CA), fracture permeability (kf), interporosity flow parameter (λ) and the other parameters can be determined by this approach. The equations are based on a model originally presented by Warren and Root and extended by Da Prat et al. and Mavor and Cinco-Ley. The proposed method has been developed to be used for naturally fractured reservoirs with different geometries. This method does not involve the use of any chart and by using the pseudo steady state flow regime, the influence of wellbore storage on the value of the parameters obtained from this technique is negligible. In this technique, all the parameters can be obtained directly while in conventional approaches like type curve matching method, parameters such as ω and λ should be obtained by other methods like build-up test analysis and this is one of the most important advantages of this method that could save time during reservoir analyses. Different simulated and field examples were used for testing the proposed technique. Comparison between the obtained results by this approach and the results of type curve matching method shows a high performance of decline curves in well testing.

Hydraulically Fractured Formations: Parameters Controlling Performance and Maximum Number of Fractures

Horizontal wells essentially increase the area of contact between wellbores and reservoir fluids to some extent. Hydraulic fractures increase this area significantly and develop the vertical permeability. Because of these two techniques, well deliverability or productivity index can be increased to the limit required by the worldwide needs. Several models have been derived for the productivity index of fractured formations and the maximum number of fractures for both finite and infinite reservoirs. The models were developed based on the idea that the total pressure drop in the wellbore can be estimated as the sum of different pressure drops caused by different flow regimes. This pressure drop is necessary for the fluid to flow from the reservoir toward the wellbore. It is well known that the developed flow regimes in the area around the horizontal wells or the hydraulic fractures are not the same as the flow regimes at far distance from wellbores, which is close to the outer boundaries. Therefore, four flow regimes were expected to develop in infinite acting reservoir: pseudo radial flow at the outer boundaries, elliptical flow in the area between wellbores and the regions close to the outer boundaries, formation linear flow in the area between fractures toward wellbores and fractures, and finally fractures linear flow, while pseudo-steady state flow was the expected flow regime for the case of limited reservoirs. Each one of these flow regimes contributes to the total pressure drop necessary for producing certain flow rate in addition to the pressure drop caused by the damage zones resulted from horizontal well drilling and completion, hydraulic fracturing process, and fluid flow chocking effect. In this study, the effects of the anisotropy, fracture dimensions, radius of drainage area, number of fractures and fracture conductivity on productivity index had been investigated. A novel approach for the maximum number of fractures necessary for a specific productivity index was introduced in this paper. The model had been examined for two field cases taken from literatures. The calculated flow rates by this model showed good agreement with the measured flow rates.

Analysis of pressure behaviors and flow regimes of naturally and hydraulically fractured unconventional gas reservoirs using multi-linear flow regimes approach

This paper targets the applicability of multi-linear flow regimes approach in gas reservoirs having multiple porous media: matrix, natural fractures, and hydraulic fractures. The approach attempts to understand pressure profiles and flow regimes developed in some of unconventional shale layers that have already been naturally fractured. In this paper, the main focus is given to fluid flow phenomenon from the matrix towards naturally induced fractures and from these fractures to the hydraulic fractures. Studying the impact of matrix, natural fracture and hydraulic fracture parameters on reservoir performance is another objective for this paper.Purposefully, several mathematical models have been developed to investigate transportation phenomenon of reservoir fluid in porous medium. The porous medium is assumed to be finite rectangular reservoir having different configurations. It is divided into two zones; inner porous medium close to the horizontal wellbore where multiple-hydraulic fractures propagate to some distance and outer porous medium where naturally induced fractures exist in the drainage area close to the outer boundary. Reservoir fluids, in the outer zone, are assumed to flow from the matrix to natural fractures and then from natural fractures to the inner zone. In the inner porous medium, fluids flow from matrix to natural fractures in addition to fluids flowing from the outer porous medium. Then all fluids flow by natural fractures to hydraulic fractures and finally from hydraulic fractures to the wellbore.The mathematical models for fluid flow in the two zones of the reservoir have been solved and three main flow regimes have been characterized. For each flow regime, an analytical model is used to describe its pressure profile with time and production performance. Bilinear flow regime, typically observed at early time, refers to simultaneous linear flow inside hydraulic fractures and flow from inner porous medium to the hydraulic fractures while linear flow describes fluids flow from inner zone to hydraulic fractures and late linear or boundary dominated flow regime describes fluids flow from the outer to the inner porous media.The main outcomes of this study are: 1) The ability to characterize reservoirs with triple porous media using multi-linear flow regimes approach. 2) Understanding the impact on pressure distribution of inner porous medium caused by the characteristics of hydraulic fractures such as fracture width and conductivity in addition to hydraulic fracture storativity. 3) Understanding the effect of reservoir configuration, matrix, and natural fractures characteristics on pressure distribution in the outer porous media. 4) Pressure distribution and flow regimes obtained by applying transient flow approach are not similar to those obtained by applying pseudo-steady state flow approach. 5) Characterization of different linear flow regimes, controlled by different parameters, with respect to production time.

A comprehensive model combining Laplace-transform finite-difference and boundary-element method for the flow behavior of a two-zone system with discrete fracture network

This paper provides a comprehensive model for the flow behavior of a two-zone system with discrete fracture network. The discrete fracture network within the inner zone is represented explicitly by fracture segments. The Laplace-transform finite-difference method is used to numerically model discrete fracture network flow, with sufficient flexibility to consider arbitrary fracture geometries and conductivity distributions. Boundary-element method and line-source functions in the Laplace domain are employed to derive a semi-analytical flow solution for the two-zone system. By imposing the continuity of flux and pressure on discrete fracture surfaces, the semi-analytical two-zone system flow model and the numerical fracture flow model are coupled dynamically. The main advantage of the approach occurring in the Laplace domain is that simulation can be done with nodes only for discrete fractures and elements for boundaries and at predetermined, discrete times. Thus, stability and convergence problems caused by time discretization are avoided and the burden of gridding and computation is decreased without loss of important fracture characteristics. The model is validated by comparison with the results from an analytical solution and a fully numerical solution.Flow regime analysis shows that a two-zone system with discrete fracture network may develop six flow regimes: fracture linear flow, bilinear flow, inner zone linear flow, inner zone pseudosteady-state flow, outer zone pseudoradial flow and outer zone boundary-dominated flow. Especially, local solutions for the inner-zone linear flow have the same form with that of a finite conductivity planar fracture and can be correlated with the total length of discrete fractures and an intercept term. In the inner zone pseudosteady-state flow period, the discrete fractures, along with the boundary of the inner zone, will act as virtual closed boundaries, due to the pressure interference caused by fracture network and the mobility contrast of the two zones. The dimensionless fracture conductivity of the fracture network determines the characteristics of the bilinear flow and the inner zone linear flow. The mobility ratio, M, and storability ratio, Fs, primarily influence the flow behavior of the middle to later time. For a larger M, the ending time of the inner zone pseudosteady-state flow will be advanced and the beginning time of the outer-zone pseudoradial flow will be delayed. And the larger M also results in increasing of the values of the transient responses in these two periods. Duration of the outer-zone pseudoradial flow becomes longer, as Fs increases; the development of the outer-zone boundary-dominated flow is therefore postponed. Finally, idealization of dual-porosity model for the two zones will introduce two dips on the pressure derivatives on the log/log plot. Depending on the interporosity flow parameter, the time of the two dips varies. However, the dips are not very appreciable for the transient dual-porosity model, when the permeability of matrix system or fracture density is relatively high.

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.