Boundary-dominated Flow Research Articles

Application of dynamic material balance to evaluate oil wells in Buzurgan oil field

The Material Balance Equation is a crucial tool utilized in reservoir studies to evaluate fluids and rock properties at static pressures. The Flowing and Dynamic Material Balance methods offer a significant advantage by avoiding the requirement to shut down wells, as they use flowing pressure instead of static pressure under constant or variable flow rates. The concept of "Dynamic Material Balance" involves converting the bottom hole flowing pressure at any point at any given time to the average reservoir pressure at that point. This allows for the use of classical material balance calculations and the development of classical material balance plots. In this study, the Dynamic Material Balance and Agrawal Type Curve techniques were used to estimate average reservoir pressures, initial hydrocarbon in place, and ultimate oil recovery for a well in the Mishrif reservoir, the main reservoir in the Buzurgan oil field. Many wells in this field experience problems such as high-pressure decline or continuous water production, necessitating ongoing evaluation. While the Dynamic Material Balance method focuses on boundary-dominated flow data, the Agrawal-type curve technique analyzes data from both transient and boundary flow periods. Agarwal decline curves were constructed using relationships of pseudo pressure normalized production, material balance pseudo time, and dimensionless variables in well-test analysis. The results from both methods showed comparable results with an absolute percentage error of (0.738) %, (3.07) %, and 5.7% for oil-in-place, drainage area, and average reservoir pressure, respectively. This strong correlation between the Dynamic Material Balance and Type Curve results indicates their accuracy and reliability.

Augmented Depletion Development Technology Assessed Across US Shale Plays

_ This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 217792, “Assessment of Augmented Depletion Development Technology Across US Shale Plays,” by Ahmed Merzoug, SPE, Texas A&M University, and Vibhas J. Pandey, SPE, ConocoPhillips. The paper has not been peer reviewed. _ The concept of augmented depletion development (ADD) was introduced to the industry through a pilot project conducted in the Bakken play in the US. These wells are openhole wells drilled within the fractured network of a pad. They are not fracture stimulated deliberately but do produce and contribute to the overall production of the lease. In this study, numerical modeling is used to understand the potential performance of these wells in several plays across the US. The learnings from this study can be used as a benchmark for operators that are planning on implementing such wells in their field development programs. Theoretical Basis of Study The concept of ADD can be modeled using a similar approach to the analytical solutions developed for flow into a multistage fractured horizontal well. In these wells, several regimes can be identified, such as transient linear flow and boundary-dominated flow. In previous work in the literature that proposed a method for analyzing production data of linear flow of fractured gas wells, fractures were assumed to have infinite dimensionless conductivity. Those researchers proposed a solution for a constant bottomhole-pressure liquid flow that is described in the present complete paper by a series of equations. For the case used by the present authors, a well spacing of 300 ft is assumed, implying that the perpendicular distance from the wellbore to a corresponding boundary is 150 ft, which is half the spacing between the fractured well and the ADD well (assuming each well drains half the fracture length). To elaborate further for better understanding of production, a plot of product of cumulative dimensionless rate and formation permeability (as a proxy for cumulative rate) while assuming all other properties behave the same shows that, with higher permeability, the recovery potential not only increases but also appears to accelerate. This supports the concept that, during the time of production of ADD wells, it is preferable to have higher permeabilities that can support faster production in zones that the primary wells may not drain effectively. The results of the analytical equation are for only one fracture but can be readily extended to multifractured horizontal wells using superposition methods. Methodology This study uses physics-based numerical modeling to simulate the performance of ADD wells in different plays. The numerical code simulates wellbore flow, hydraulic fracturing, geomechanics, and reservoir flow, all in the same package. The fracture and reservoir mesh are independent. The flow between the matrix and the reservoir is calculated using the 1D submesh approximation to reduce the computation expenses with higher accuracy. The hydraulic fracturing propagation algorithm uses a tip-tracking algorithm to allow the use of larger grid size without compromising accuracy in layered formations. The fracture conductivity is calculated with stress changes. The conductivity profile changes from a propped fracture into an unpropped fracture. The unpropped fracture conductivity is an underlying assumption used in this study. The collision between fractures initiated from different wells also is an assumption in this study.

Fast Optimization of the Net Present Value of Unconventional Wells Using Rapid Rate-Transient Analysis

Summary Decline-curve analysis (DCA) is the industry standard to predict hydrocarbon production to then estimate the net present value (NPV) of unconventional wells. Conventional DCA assumes the flowing bottomhole pressure (BHP) is constant. This is an unrealistic assumption for many unconventional wells that can lead to incorrect estimates of ultimate recovery and thus erroneous NPV calculations. This work illustrates the application of a novel technique that combines variable BHP conditions with decline-curve models [Rapid rate-transient analysis (RTA)] to estimate the NPV and select the optimal cluster spacing and number of fractures for a given well. We compare the results of DCA and Rapid RTA to estimate and optimize the cluster spacing for a tight oil and shale gas well. The Rapid RTA results show marked differences in terms of the shape and the optimal value of the NPV function from the simpler DCA method. For the two cases illustrated, DCA results are a direct consequence of the rate-time analysis failing to correctly detect the onset of boundary-dominated flow (BDF). In contrast, Rapid RTA correctly detects the onset of BDF and presents a defined maximum value of the NPV vs. cluster spacing (number of fractures) function. This optimal value is a trade-off between the fracture treatment cost, the speed of recovery of hydrocarbons, and the value of money with time (discount rate). The major contribution of this work is the implementation of the Rapid RTA technique allowing fast computation of pressure-rate-time analysis and NPV calculations with computational times comparable with DCA for optimizing fracture cluster spacing and the total number of fractures of unconventional wells. We have developed a web-based application to give readers a hands-on experience of this new technique and to analyze and optimize the NPV of unconventional wells.

Applications of Asymptotic Solutions of the Diffusivity Equation to Infinite Acting Pressure Transient Analysis

Summary Understanding how pressure propagates in a reservoir is fundamental to the interpretation of pressure and rate transient measurements at a well. Unconventional reservoirs provide unique technical challenges as the simple geometries and flow regimes [wellbore storage (WBS) and radial, linear, spherical, and boundary-dominated flow] applied in well test analysis are now replaced by nonideal flow patterns due to complex multistage fracture completions, nonplanar fractures, and the interaction of flow with the reservoir heterogeneity. In this paper, we introduce an asymptotic solution technique for the diffusivity equation applied to pressure transient analysis (PTA), in which the 3D depletion geometry is mapped to an equivalent 1D streamtube. This allows the potentially complex pressure depletion geometry within the reservoir to be treated as the primary unknown in an interpretation, compared with the usual method of interpretation in which the depletion geometry is assumed and parameters of the formation and well are the unknown properties. The construction is based upon the solution to the Eikonal equation, derived from the diffusivity equation in heterogeneous reservoirs. We develop a Green’s function that provides analytic solutions to the pressure transient equations for which the geometry of the flow pattern is abstracted from the transient solution. The analytic formulation provides an explicit solution for many well test pressure transient characteristics such as the well test semi-log pressure derivative (WTD), the depth of investigation (DOI), and the stabilized zone (SZ) (or dynamic drainage area), with new definitions for the limit of detectability (LOD), the transient drainage volume, and the pseudosteady-state (PSS) limit. Generalizations of the Green’s function approach to bounded reservoirs are possible (Wang et al. 2017) but are beyond the scope of the current study. We validate our approach against well-known PTA solutions solved using the Laplace transform, including pressure transients with WBS and skin. Our study concludes with a discussion of applications to unconventional reservoir performance analysis for which reference solutions do not otherwise exist.

Improved saturation-pressure relationship and multiphase pseudo-pressure calculations for retrograde gas reservoir production under boundary-dominated flow

For reservoirs containing fluids with complex flow characteristics, the calculation of multiphase pseudo-pressure plays a significant role in production data analysis (PDA) practices. In the case of retrograde gas reservoirs/gas condensate reservoirs under multiphase flow conditions, the lack of a reliable and complete saturation-pressure (So-p) relationship presents a significant challenge in obtaining the appropriate value of multiphase pseudo-pressure. Gas condensate reservoirs are one of the viable and lower-emission hydrocarbon sources for meeting the ever-increasing global energy demand. The economic value of producing gas condensates has inspired numerous research into employing innovative production techniques in developing gas condensate fields. Hence, this study proposes an improved saturation-pressure relationship alongside a Fractional Phase Mobility (FPM) criterion to mitigate potential errors in the multiphase pseudo-pressure calculation for gas condensate reservoirs under boundary-dominated flow. This work introduces an improved So-p relationship derived using a Rescaled Constant Volume Depletion (CVD) method for estimating multiphase pseudo-pressures. To demonstrate the effectiveness of the proposed approach, we provide direct comparisons with the Classical CVD method and numerical simulation results for multiple gas condensate cases subjected to constant and variable bottomhole pressure (BHP) schedules. In contrast to the classical approach, the Rescaled CVD method consistently delivers superior performance for multiphase pseudo-pressure estimations across all cases, thus supporting its potential as a robust and accurate approach for gas condensate reservoir modeling.

Skin Factor Consideration in Decline Curve Analysis

Summary This paper introduces an approach for the impact of skin factor on the decline curve analysis of hydraulically fractured reservoirs. The objective is to consider this impact in the production forecasting and the ultimate recovery estimation. The approach focuses on reducing the uncertainties that could be raised from this impact on the production history and increasing the accuracy of the predicted flow rates. It proposes an easy and promising tool for the decline curve analysis that could be applied confidently to different oil- and gas-producing wells and different reservoirs. This approach utilizes the rate-normalized flow rate derivative β behavior of the fractured reservoirs. This derivative demonstrates a constant behavior with time for each flow regime when the production history has not undergone the impact of the skin factor. However, the constant behavior no longer exists when this impact has influenced the production history. Instead, a power-law type model governs the relationship between the flow rate derivative and production time. New analytical flow rate decline curve models, exponential-type, are derived from the flow rate derivative power-law type models for the flow regimes. Different models for calculating the skin factor are developed for the three linear flow regimes that could be observed during the transient state flow conditions. The proposed flow rate models are used to simulate the production history and forecast the future performance. Moreover, the hydraulic fracture conductivity can be calculated using these models as well as the flow rate loss caused by skin factor. Several case studies are examined by the proposed approach where the production history is used to characterize the dominant flow regimes. The study has reached several observations and conclusions. The impact of skin factor is seen clearly throughout transient state flow regimes; however, this impact declines sharply before reaching pseudosteady-state flow (boundary-dominated flow regime). The impact of the skin factor alternates the constant behavior of the flow rate derivative with time to a power-law type relationship. A straight line of a slope (0.5) is diagnosed during hydraulic fracture and formation linear flow regime on the log-log plot of the flow rate derivative β and time, while the bilinear flow regime demonstrates a straight line of a slope (0.25). Because of the skin factor, exponential decline curve models replace the power-law type models of the flow rate during the abovementioned flow regimes. These models exhibit an excellent match between the calculated flow rate and the production history. The maximum flow rate loss occurs during very early production time even though the skin factor during this time is less than the intermediate production time. This study presents a novel approach for the decline curve analysis taking into account the impact of skin factor. The novelty is represented by considering the flow regimes in the production forecasting of hydraulically fractured reservoirs. This approach is smoothly applied to predict the future performance with no need to know the wellbore and reservoir parameters. It can be used to predict the declining flow rate for a constant or varied bottomhole flowing pressure.

A transient production prediction method for tight condensate gas wells with multiphase flow

Considering the phase behaviors in condensate gas reservoirs and the oil-gas two-phase linear flow and boundary-dominated flow in the reservoir, a method for predicting the relationship between oil saturation and pressure in the full-path of tight condensate gas well is proposed, and a model for predicting the transient production from tight condensate gas wells with multiphase flow is established. The research indicates that the relationship curve between condensate oil saturation and pressure is crucial for calculating the pseudo-pressure. In the early stage of production or in areas far from the wellbore with high reservoir pressure, the condensate oil saturation can be calculated using early-stage production dynamic data through material balance models. In the late stage of production or in areas close to the wellbore with low reservoir pressure, the condensate oil saturation can be calculated using the data of constant composition expansion test. In the middle stages of production or when reservoir pressure is at an intermediate level, the data obtained from the previous two stages can be interpolated to form a complete full-path relationship curve between oil saturation and pressure. Through simulation and field application, the new method is verified to be reliable and practical. It can be applied for prediction of middle-stage and late-stage production of tight condensate gas wells and assessment of single-well recoverable reserves.

Analytical Method for Production Forecasting During Boundary-Dominated Flow in Unconventional Volatile Oil Reservoirs Using Single-Phase Analogy

Abstract This paper presents an analytical methodology for production forecasting of wells exhibiting boundary-dominated flow in unconventional volatile oil reservoirs. An analytical model using single-phase analogy is established to predict the production rate of a multi-fractured horizontal well under multiphase flow. Pseudo-variables are employed to linearize governing differential flow equations. Therefore, the analytical model for efficiently handling multiphase flow is essentially an advanced adaptation of existing single-phase models, in which we derive new multiphase fluid properties. To calculate the pseudo-variables accurately, we summarize accessible methods for the determination of saturation–pressure (S–P) relation and further provide an appropriate way to calculate multiphase pseudo-variables in unconventional volatile oil reservoirs based on simulation results. The simulation study indicates that the S–P relation given by constant volume depletion data for near-critical volatile oils leads to satisfactory results, and the S–P relation derived from the tank-type model is appropriate for ordinary volatile oils. The analytical solution and associated methods were validated through comparison with results from a compositional simulator; the excellent agreement during boundary-dominated flow demonstrated the accuracy of the analytical methodology. The analytical methodology can greatly reduce computation and is justified to make production forecasting in unconventional volatile oil reservoirs, this tractable methodology should be attractive to the industry.

Rate transient analysis of wells exhibiting two-phase (gas+water) flowing by use of pseudopressure-based approach

The primary focus of rate transient analysis (RTA) is put on the estimation of the reserves of gas/water reservoirs. RTA based on the single-phase assumption is limited to gas/water two-phase flow in the development of the material-balance time function. This work aims to propose a practical approach for analyzing production data from wells exhibiting gas/water two-phase flow during both infinite-acting flow (IAF) and boundary-dominated flow (BDF) periods. In this work, we present a comprehensive pseudopressure-based solution associated with the constitutive saturation/pressure (Sw/p) relationship. The pseudovariables, namely pseudopressure and β-factor, have the ability of handling the two-phase nonlinearities in the governing flow equations of gas and water phases and generating the required benchmark analytical solutions for liquid flow equation. By integrating the liquid-type solutions with the principle of Duhamel convolution, a set of gas/water-phase productivity equations are recast for wells producing at different operating conditions, including constant bottomhole-pressure, constant rate and steplike change in bottomhole-pressure/rate. The presented solutions provide a theoretical framework for developing two-phase RTA based on two-phase material-balance pseudotime (MBPT). The RTA method is validated against synthetic cases generated from numerical simulation, and the dependability of two-phase MBPT is also corroborated in detail. A field case example is analyzed to demonstrate the practical applicability of this method. The RTA method provided herein accounts for all flowing phases in the reservoir, and serve as an alternative to numerical model history matching for estimating original gas in place (OGIP) and original water in place (OWIP).

Decline-curve analysis with decline rate for improving reserves estimation in gas reservoirs

Over decades, the rate decline analysis has been the industry's bedrock for reserve estimation. The Arps method has gained credibility since its introduction in 1945, particularly in oil reservoirs. However, gas reservoirs present challenges due to handling increasingly deeper geologic horizons with higher pressure and low-permeability formations, rock compaction, and overpressured reservoirs, among other challenges. This study shows a novel decline-curve analysis involving the decline rate rather than the flow rate. The proposed methodology applies to production data from a gas reservoir with varying operating conditions during the boundary-dominated flow (BDF) period. The method does not involve any pseudo-functions.The proposed method is independent of the variation of flow rate and pressure. The PVT data (viscosity, compressibility, z-factor) aids the construction of a type-curve involving the viscosity-compressibility ratio with the recovery factor without any reservoir-specific information. The decline rate with adjustment for changing operating conditions leads to a single decline-curve plot involving the viscosity-compressibility ratio (λ) and dimensionless pressure (pD) without any correlating parameter. Two synthetic examples verified the proposed approach, and two field datasets validated it. Overall, the proposed method strengthens the reserve estimation with an independent process for meeting corporate and regulatory requirements in conventional and tight-gas reservoirs.

A Robust Density-Based Approach to Production Data Analysis of Oil/Water Multiphase Flow System

Abstract This paper presents a novel approach to estimate reserves of oil and water reservoirs undergoing boundary-dominated flow conditions in a simplified yet accurate manner. The methodology incorporates rescaled density-based exponential models and is based on the coupling of two-phase oil and water material balances with multiphase well deliverability equations. Current multiphase production data analysis methods employed for reserve calculations, including density-based approach, are subjected to the determination of saturation–pressure relationship, multiphase pseudo-pressure, and pseudo-time, as well as the iterative nature of its own algorithm. The herein proposed approach circumvents the need for pseudo-variables calculations, thus precluding the determination of saturation–pressure relationship and removing the iterative nature often present in state-of-the-art approaches. The proposed model is validated by comparing its predictions to numerical models with both constant and variable bottomhole pressure constrains, and has been found to match closely. For all cases, relative errors are found to be less than 1%.

Novel Approach for Rate Transient Analysis of Fractured Wells from Carbonate Reservoirs with Heterogeneous Natural Fractures.

This paper developed a new methodology for rate transient analysis of fractured wells in carbonate reservoirs. Both the heterogeneity and dual-permeability flow behavior are incorporated into the proposed model, and the fractured carbonate reservoir was simulated with a two-zone composite model. In each zone, a traditional dual-porosity model was applied to describe the characteristics of the natural fractures and matrix. With the Laplace transform, we derived the solution of the mathematical model and plotted new type curves for transient rate decline analysis. Then, the flow regimes were divided and analyzed based on the new type curves. The influences of several critical parameters on transient rate response were also examined. A field case was studied further to demonstrate the precision and application of the proposed method. The results show that the new type curves are mainly composed of eight flow stages. The difference in physical properties (k 2,1, η2,1) between the two zones significantly impacts the transition and boundary-dominated flow regimes. When the values of k 2,1 and η2,1 are smaller, the derivative curve of the transition flow stage will move down, and the duration of this stage on the derivative curve is longer, while the duration of the boundary dominant flow stage will decrease. The dimensionless radial radius of the inner zone (r 1D) can significantly influence the transition flow regime. When r 1D is larger, the production rate and its derivative curve of the transition flow stage will move up, and the duration of this stage will be longer. The results also show that the proposed methodology can effectively fit the field production data. This method can be applied in well productivity evaluation for fractured carbonate reservoirs.

Assessing reservoir connectivity with pressure/rate surveillance data

Estimating reservoir connectivity is critical for assessing infill-drilling prospects and initiating fluid injection in enhanced oil recovery operations. Several methods have appeared in the literature over decades to meet these business needs, given that all tools, including seismic imaging, have limitations. Besides imaging, geochemical fingerprinting constitutes a powerful tool to gauge the compartmentalization question. However, real-time pressure/rate surveillance data allows assessing interwell connectivity vis-à-vis the overall drainage volume. This study presents a simplified approach to using the reciprocal-productivity index (RPI) vs. the total-material-balance time ( t ¯ T M B ) plot. This tool exhibits the same slope for those wells in the same compartment beyond the start of the boundary-dominated flow (BDF) period. The wells showing different slopes imply that they are in separate drainage volumes. The early-time transient period remains muted to minimize confusion on this Cartesian plot. We validated the proposed tool's efficacy with 2D and 3D models with increased degrees of reservoir complexity, followed by the verification phase with four field examples. The use of other analytical tools complemented our findings.

Performance forecasting of hydraulically fractured horizontal wells using dynamic-drainage-area concept with correct distance of investigation

Performance forecasting of multi-fractured horizontal wells (MFHWs) completed in tight/shale oil or gas reservoirs is of great significance in the development of unconventional resources. Dynamic drainage area (DDA) concept has emerged as a production forecasting methodology for unconventional reservoirs. This paper investigated the DDA method and derived correct expressions of distance of investigation (DOI) for both constant production rate and constant bottom-hole pressure cases by integrating material balance and deliverability equation within the range of DOI. The modified DDA method with correct DOI coefficients provided in this paper permits direct calculation of dynamic performance at arbitrary time steps before the end of transient linear flow. At the same time smooth production forecasting of MFHWs from transient linear to boundary-dominated flow is realized without modification by extra coefficient. Hybrid models such as DDA plus dual-exponential and DDA plus hyperbolic are presented, which can be applied quickly and easily to MFHWs in unconventional reservoirs as alternatives to complex numerical simulation. Meanwhile, average pressure in the range of DDA can be readily obtained with correct DOI coefficients, avoiding complex iterative calculations. The reliability and practicability of this solution have been demonstrated by synthetic and field cases in this work.

Gas relative permeability evaluation in tight rocks using rate-transient analysis (RTA) techniques

The evaluation of effective and relative gas permeability is critical for modeling hydrocarbon recovery from low-permeability (unconventional) reservoirs, and for evaluating the potential for clean energy pathways such as carbon capture and sequestration and hydrogen storage in subsurface. Nevertheless, due to the low flow rates and long experimental times associated with low-permeability (nanodarcy/microdarcy range) sample testing, measuring relative permeability through conventional techniques is technically challenging, time-consuming, and expensive. Additionally, and importantly, none of the routine laboratory techniques for measuring relative permeability can mimic the boundary conditions under which wells are produced from, or injected into, in the field.For the first time, effective and relative gas permeability of partially-saturated low-permeability sedimentary rocks were investigated using state-of-the-art core analysis techniques based on rate-transient analysis (RTA). Conventional and new straight-line analysis methods, commonly applied to production data in the field, were used to estimate effective gas permeability of core plugs under variable stress conditions and water saturations. Five low-permeability siltstone and organic/clay-rich samples, differing in mineralogy, porosity, and pore size distribution, were analyzed for proof-of-concept testing.The observed flow regimes consist of transient linear flow followed by boundary-dominated flow regimes, as previously reported for field data. Satisfactory correlation (average coefficient of variation less than 7%) between permeability estimates using different RTA techniques demonstrates the applicability of this method for effective and relative permeability assessment of partially-saturated ultra-tight reservoirs (down to less than 1.0∙10−5 md). Importantly, test times after the initiation of the production phase, were on the order of minutes, which is significantly shorter than those commonly achievable using conventional techniques (e.g., hours using pulse-decay gas permeability testing). Effective gas permeability (from 1.5∙10−3 to 7.5∙10−6 md), and slip factor (from 715 to −286 psia−1), decreased with increasing water saturation (0–53%), reflecting the lower contribution of slip flow to gas transport. Effective gas permeability decreased with increasing effective stress (500–4000 psia), likely due to reduced effective (transport) pore throat size and increased tortuosity. The reduction of slip factor and the observed stress dependency with water saturation is attributed to the channel flow mechanism.

An Explicit Method for Estimating the Average Reservoir Pressure and Gas in Place Based on the Gas Property Polynomial Function

The fast and accurate determination of average reservoir pressure (ARP) and gas reserves is vital for the analysis and forecasting of gas well performance. An explicit and relatively rigorous method for estimating the ARP and gas in place, based on the dynamic material balance equation and the gas property polynomial function (GPPF), is presented to circumvent such drawbacks as high cost, empiricism, poor adaptability to variable production schedules, and the necessity for iteration processes, by which current approaches usually have been limited. The gas flow model is solved by introducing pseudofunctions and considering the compressibility effects of rock and irreducible water, and the pressure-rate correlations during boundary-dominated flow (BDF) which constitute the theoretical proofs of this method are derived from the superposition principle coupled with the constant rate solution. Production data under different production scenarios prove it effective. The error of estimation for formation pressure and gas reserves hardly ever goes beyond 4% given that the BDF condition is generally satisfied by the rate and bottom hole flowing pressure (BHP) data. We employ the GPPF to capture the nonlinear variations of gas viscosity andZ-factor, helpful to implement a quick conversion between pressure and pseudopressure and to solve the integral equation concerning pseudopressure and thus valid in overcoming the limitations of previous methods concerning pressure or pressure squared. The proposed methodology boasts its simpleness and practicability, which not only dispenses with iterations on pseudotime but also applies to various production systems of gas wells under constant BHP, constant rate, or variable BHP/variable rate conditions.

Flow diagnosis in variable bottom hole pressure multi-well horizontal pad with well interference using rate/pressure transient analysis

Horizontal multi-well pads are frequently used in unconventional reservoirs. Along with infill wells and hydraulic fracturing, interference between multiple multi-fractured horizontal wells (MFHWs) has become a major concern. The current rate transient analysis (RTA) makes the assumption that the unconventional formation contains a single MFHW. This study introduces a novel multi-MFHW solution and associated analysis methodology for analyzing the performance of targeted well rates in a multi-MFHW system.The constant bottom-hole pressure (BHP) condition and the Laplace transform can be used to obtain multi-MFHW solutions for transient flow. We investigated interference between various fractures and MFHWs using the superposition of various constant BHP solutions. The variable BHP of the targeted well is calculated using a variable dimensionless BHP function in the Laplace domain without performing any convolution or deconvolution. The proposed method is rigorously validated using a commercial numerical simulator for cases involving offset MFHWs and multi-MFHW with variable BHP. With this multi-MFHW analysis, we can analyze a target well in the pad using the total material balance of the multi-MFHW system. Offset well interference frequently occurs following the onset of infinite-acting radial flow (IARF) in the target well's hydraulic fracture. It results in an increase in the pressure derivative curves for elliptical flow and IARF, as well as the rate-normalized pressure (RNP) derivative. Inverse semi-log derivatives exhibit the inverse trend. The proposed deviation pressure integral and RNP can be used to diagnose the flow region caused by the offset well's flow rate in a unique manner, displaying the horizontal line, V-shaped dip, and unit slope, respectively, during IARF, cross flow, and boundary-dominated flow (BDF). Sensitivity analysis of well spacing demonstrates that as well spacing increases, the “transition flow” between wells transitions from elliptical to formation linear flow and can exhibit transitional flow characteristics in more common cases.

A dynamic fracture model combining with Laplace transform and synchronous iteration method for transient behavior analysis of a four-zone system

A robust and generic semi-analytical model with the capture of dynamic fracture behaviors under long-term pumping rate condition is developed for transient flow analysis in this four-zone system. To characterize the variations of fracture length, the improved pressure-dependent dynamic fracture factor is introduced and incorporated into this model. The Laplace domain with predetermined time discretization and synchronous iteration technique are deployed to obtain the solutions of pressure and pumping rate with the least computational burden. Meanwhile, the stability and convergence of this semi-analytical model can be met within the discrete time. Furthermore, the pressure and pumping rate solutions of this model under the static fracture condition are in good agreement with those of the constant length fracture, which verifies the reliability of this model.The dynamic fracture model with the four -zone system develops five flow stages on the dimensionless pressure and its derivative curves: fracture bilinear flow, dynamic fracture flow, inner area linear flow, transitional flow and boundary-dominated flow. The dynamic fracture behavior is mainly reflected in the obviously positive upwards and pressure derivative curve presents a higher than unit slope. Moreover, the significant decrease of flow rate indicates that fracture closure could result in a rapid decline in pumping rate, which might be the reason why the pumping rate decrease quickly during the long-term pumping period. As the shrinkage coefficient increases, fractures are closed from abruptly to gradually, and typical signatures of fracture closure may be hidden if the shrinkage factor is too small, which may be caused by the instantaneous pressure depletion in the fractures. With the decrease of retained fracture coefficient, there is greater in the positive upward of transient pressure behavior and a lower in pumping rate. As a result, the flow rate will be seriously overestimated if the fracture closure is not considered especially for the tight formation.

Pressure transient analysis of horizontal wells in multibranched fault-karst carbonate reservoirs: Model and application in SHB oilfield

Current flow models for fault-karst reservoirs are mostly described as a single fault formation, which cannot be applied in recent-developed multibranched fault-karst reservoirs. This paper established a novel analytical model to investigate pressure response behavior of a horizontal well in multibranched fault-karst reservoirs. The model is able to describe the influence of the physical properties and spatial structure of fracture-cave system on pressure transient response. The flow model considers different flow behaviors in each region, which includes Darcy flow (gravity included) in fault-fracture, large-scale storage flow in karst-cave, and Poiseuille-law-based horizontal laminar flow in the horizontal wellbore, respectively. These assumptions enable the model to match complex situations in multibranched fault-karst reservoirs. Then, the model was retrograded to compare with a single fault-karst reservoir model to verify its accuracy. Further, the solutions were graphed on log-log plots, and we discussed the effect of fluids mobility, formation storability, and structure characteristics (e.g., length, angle, depth, distance) of fracture-cave branches on transient pressure responses. Results show that (a) the number of fracture-cave branches in a reservoir can be directly observed by counting the number of V-shaped appearances on the pressure derivative curve. (b) The exact shut-in time when V-shape appears is affected by volume and distance between two neighboring fracture-cave branches. (c)The characteristics of the V-shape are affected by fluid mobility, formation storability, and length of fracture region. (d) The slope of the pressure derivative curve in the boundary-dominated flow regime can be used to evaluate the gravity effect. (e) The pressure response behavior exhibits a near-well effect when a horizontal well commingled production in the multibranched fault-karst reservoir. Finally, we applied our model and resulting observations to analyze pressure build-up data tested from SHB Oilfield, which demonstrated a workflow to identify the number of fault-karst branches and also to estimate reservoir properties.

A multi-fractured well performance model for unconventional reservoirs

Due to the ultra-low permeability of unconventional reservoirs, transient state prevails for a considerable period. Despite this, fracture interference can cause an apparent no-flow boundary. Consequently, the Duong's model, which was developed for transient-state period, yields unreliable estimates during the late-time period. In this paper, the Duong's model is modified to account for boundary effects caused by fracture interference and/or unstimulated reservoir regions that serve as no-flow boundaries. Specifically, an empirical correction function, which assumes an exponential decline, has been used as a “modifier” to extend the Duong's model to boundary-dominated flow period. The correction function ensures that during boundary-dominated flow period, an exponential-decline behaviour dominates. The proposed rate-decline model encompasses a gamma function, which converges at large times. Results show that a fractured-well production behaviour is characterised by a decaying power-law during early-time period and tends to exponential decline during late-time period. The results also suggest that although the conventional Duong's model gives good estimates during the transient-state period, it yields optimistic estimates during the boundary-dominated flow period. The proposed model gives a good match and estimates not only in the transient-state period, but also in the boundary-dominated flow period. A major advantage of the proposed model is that it converges to estimated ultimate recovery at large times without imposing any rate and time limits. A good agreement of the estimated ultimate recovery with analytical and semi-analytical models was obtained. Also, results suggest that the proposed model gives conservative estimates. The proposed model will be useful for analysing and predicting both the early- and late-time production performance of a multi-fractured well producing from an unconventional reservoir.