Pseudo-steady State Productivity Index Research Articles

Simple pseudo-steady state productivity index correlations for infinite-conductivity horizontal wells based on a rigorous, semi-analytical model

Simple pseudo-steady state (PSS) productivity index (PI) correlations are developed for horizontal wells producing at a constant rate from a closed, rectangular reservoir based on a rigorous infinite-conductivity model (ICM). The main three correlations are functions of the half-length wellbore segments number (mt) only and mt can be determined by the general correlation, which is a function of dimensionless well length (LD). The correlations are developed by using a nonlinear regression technique (with zero error) based on over thousand semi-analytical runs for a rigorous ICM covering 90 cases of different well and reservoir configurations. The 90 cases are averaged with less than 5% error in the total pressure drop calculation for the most of them to reduce the number of correlations and to remove the slight effect of well penetration ratio (PR). The PSS PI correlations are simply used to calculate mt, average pressure drop at transient-state, average starting time of PSS, and average total pressure drop at the beginning of PSS. At PSS, the reservoir outer boundaries are dominant, and the pressure drop will depend on the temporal and the spatial time-independent functions. The simple PSS PI correlations have been verified, and can be easily implemented in a standard spreadsheet code.The results show that LD is directly proportional to PSS PI and more significant than reservoir thickness (h) on horizontal-well productivity evaluation, which confirms the principle of the economic feasibility for drilling horizontal well in the oil industry. The results also show that the horizontal-well productivity by this study is several times higher than its counterparts by the methods reported in the literature because the method used by this study is based on a rigorous infinite-conductivity solution where no significant pressure drop occurs, while the previous methods are based on approximate infinite-conductivity solutions (AICSs). Therefore, there is a big underestimation in productivity calculation by using the previous methods which crucially reflect on reservoir evaluation. The new correlations give accurate PSS PI for over 3000 cases of diverse well and reservoir configurations, besides giving a rough PSS PI for the other cases.

Oil and Gas Fields Development: Well Spacing Impact on Reservoir Performance Depleted by Multiple Horizontal Wellbores

This paper focuses an insight on the impact of well spacing on the performance of oil and gas reservoirs depleted by multiple horizontal wellbores distributed parallelly or radially in rectangular and circular drainage areas. The objective is to develop an integrated approach for predicting the pressure, flow regimes, flow rate, cumulative production behavior, and estimating the transient and stabilized pseudo-steady state productivity index. The motivation is eliminating the uncertainties in predicting these key parameters that are importantly used in the development plans of the reservoirs. The outcomes of this study can be summarized in the following points: 1) Well spacing does have a significant influence on the performance of all reservoir configurations, however, it is not seen at early production time while it is very clearly observed at late production time. 2) The impact of well spacing on radially distributed wellbores at circular drainage areas is more than parallelly and radially distributed wellbores in rectangular reservoirs. 3) The well spacing causes a well interference that could be characterized by developing a new flow regime called early pseudo-steady state flow that is typically followed by a transient state flow regime. 4) The well interference could be seen early when the well spacing is narrow. It could be observed during intermediate production time. 5) The pressure drop and the productivity index may change significantly after the well interference more than before.

Analysing and characterising horizontal well performance in rectangular closed gas reservoirs considering non-Darcy flow conditions

This paper investigates the impact of non-Darcy flow, represented by rate-dependent skin factor (DQSC), on pressure profiles, flow regimes, and productivity indices of horizontal wells extending in conventional and unconventional gas reservoirs. It introduces a new simplified technique for characterising reservoir performance under different non-Darcy flow impacts. The outcomes of this study are summarised as: 1) understanding the conditions at which non-Darcy flow could have considerable effects on reservoir performance; 2) estimating the deviation in productivity index caused by non-Darcy flow. The most interesting points in this study are: 1) the ability to estimate rate-dependent skin factor from well test analysis by observing early time radial flow regime; 2) all flow regimes, unlike pressure behaviors and productivity index, are not affected by non-Darcy flow; 3) productivity index declines sharply for high rate-dependent skin factor at early production time while pseudo-steady state productivity index is not affected by non-Darcy flow. [Received: April 12, 2018; Accepted: May 9, 2018]

Analysing and characterising horizontal well performance in rectangular closed gas reservoirs considering non-Darcy flow conditions

This paper investigates the impact of non-Darcy flow, represented by rate-dependent skin factor (DQSC), on pressure profiles, flow regimes, and productivity indices of horizontal wells extending in conventional and unconventional gas reservoirs. It introduces a new simplified technique for characterising reservoir performance under different non-Darcy flow impacts. The outcomes of this study are summarised as: 1) understanding the conditions at which non-Darcy flow could have considerable effects on reservoir performance; 2) estimating the deviation in productivity index caused by non-Darcy flow. The most interesting points in this study are: 1) the ability to estimate rate-dependent skin factor from well test analysis by observing early time radial flow regime; 2) all flow regimes, unlike pressure behaviors and productivity index, are not affected by non-Darcy flow; 3) productivity index declines sharply for high rate-dependent skin factor at early production time while pseudo-steady state productivity index is not affected by non-Darcy flow. [Received: April 12, 2018; Accepted: May 9, 2018]

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.

An Analytical Solution of the Pseudosteady State Productivity Index for the Fracture Geometry Optimization of Fractured Wells

The pseudosteady state productivity index is very important for evaluating the production from oil and gas wells. It is usually used as an objective function for the optimization of fractured wells. However, there is no analytical solution for it, especially when the proppant number of the fractured well is greater than 0.1. This paper extends the established fitting solution for proppant numbers less than 0.1 by introducing an explicit expression of the shape factor. It also proposes a new asymptotic solution based on the trilinear-flow model for proppant numbers greater than 0.1. The two solutions are combined to evaluate the pseudosteady state productivity index. The evaluation results are verified by the numerical method. The new solution can be directly used for fracture geometry optimization. The optimization results are consistent with those given by the unified fracture design (UFD) method. Using the analytical solution for the pseudosteady state productivity index, optimization results can be obtained for rectangular drainage areas with arbitrary aspect ratios without requiring any interpolation or extrapolation. Moreover, the new solution provides more rigorous optimization results than the UFD method, especially for fractured horizontal wells.

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.

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.

The optimal reservoir configuration for maximum productivity index of gas reservoirs depleted by horizontal wells under Darcy and non-Darcy flow conditions

A new approach for estimating productivity index of horizontal wells depleting finite gas reservoirs is introduced in this paper. This approach focuses on using reservoir configurations and anisotropy as well as wellbore length for calculating productivity index at pseudo-steady state conditions when production time is no longer affects reservoir performance. The paper also aims to determine the optimal reservoir configurations that could give the maximum productivity index.For this purpose, new analytical models were developed for productivity index of gas reservoirs depleted by horizontal wells based on the solutions of the flow equation (Diffusivity equation) at pseudo-steady state condition. Gas flow in the porous media was considered obeying both Darcy and non-Darcy flow. These models were modified for bounded gas reservoirs when the pressure drawdown becomes time invariant and gas flow rate is constant. The study investigated the effects of reservoir boundaries, anisotropy, wellbore length, wellbore horizontal penetration ratio with respect to reservoir length, and the asymmetricity of wellbore in the formation on the productivity index.The outcomes of this study can be summarized in: 1) Productivity index of gas reservoirs in pseudo-steady state is affected by the dominant flow type in the reservoirs: Darcy or non-Darcy flow. 2) Wellbore length, reservoir configuration, and anisotropy affect the productivity index as well as wellbore asymmetricity. 3) Productivity index of Darcy flow is greater than non-Darcy flow. 4) Wellbore horizontal penetration ratio with respect to reservoir length may have significant impact on productivity index as well as reservoir width. The most interesting points in this study are: 1) The maximum productivity index could be reached for reservoir length of (xeD=0.55−0.65) regardless reservoir width. 2) The anisotropy may affect productivity index for short and moderate wellbores, however, it does not have significant effects in long wellbores. 3) Productivity index might be affected by wellbore location with respect to reservoir width and length but not wellbore location in the vertical direction. 4) Pseudo-steady state productivity index can be estimated by the proposed models using reservoir configuration and wellbore length with a full consideration to the formation anisotropy and wellbore asymmetricity.

Simulation of the mulltizones clastic reservoir: A case study of Upper Qishn Clastic Member, Masila Basin–Yemen

The Upper Qishn Clastic Member is one of the main oil-bearing reservoirs that are located at Masila Basin–Yemen. It produces oil from many zones with different reservoir properties. The aim of this study is to simulate and model the Qishn sandstone reservoir to provide more understanding of its properties. The available, core plugs, petrophysical, PVT, pressure and production datasets, as well as the seismic structural and geologic information, are all integrated and used in the simulation process. Eclipse simulator was used as a powerful tool for reservoir modeling. A simplified approach based on a pseudo steady-state productivity index and a material balance relationship between the aquifer pressure and the cumulative influx, is applied.The petrophysical properties of the Qishn sandstone reservoir are mainly investigated based on the well logging and core plug analyses. Three reservoir zones of good hydrocarbon potentiality are indicated and named from above to below as S1A, S1C and S2. Among of these zones, the S1A zone attains the best petrophysical and reservoir quality properties. It has an average hydrocarbon saturation of more than 65%, high effective porosity up to 20% and good permeability record (66 mD). The reservoir structure is represented by faulted anticline at the middle of the study with a down going decrease in geometry from S1A zone to S2 zone. It is limited by NE-SW and E-W bounding faults, with a weak aquifer connection from the east.The analysis of pressure and PVT data has revealed that the reservoir fluid type is dead oil with very low gas liquid ratio (GLR). The simulation results indicate heterogeneous reservoir associated with weak aquifer, supported by high initial water saturation and high water cut. Initial oil in place is estimated to be around 628 MM BBL, however, the oil recovery during the period of production is very low (<10%) because of the high water cut due to the fractures associated with many faults. Hence, secondary and tertiary methods are needed to enhance the oil recovery. Water flooding is recommended as the first step of oil recovery enhancement by changing some of high water cut wells to injectors.

Predicting productivity index of hydraulically fractured formations

Abstract Productivity index and inflow performance of horizontal wells intersecting multiple hydraulic fractures are of great importance. This importance comes from the fact that the fracturing process has become a common stimulation technique in the petroleum industry. However, few models for the productivity index and inflow performance have been presented in the literatures due to the complexity governing this topic. This paper introduces a new technique for estimating the pseudo-steady state productivity index of horizontal wells intersecting multiple hydraulic fractures. Based on the instantaneous source solutions for the diffusivity equation, seven analytical models have been derived for different source solutions. Four of them represent the effect of the formation height and fracture height (the vertical direction), while the other three represent the solution for the horizontal plane. For vertical hydraulic fractures, the four solutions of the vertical direction, representing the pseudo-skin factor, are almost neglected. The three horizontal plane solutions are the main parameters that control the productivity index and inflow performance of the fractured formations. In this technique, the horizontal wells are acting in finite reservoirs where the pseudo-steady state flow is expected to develop. Reservoir geometry, reservoir properties, and fracture dimensions were considered in this technique. The number of fractures and the spacing between them were also investigated in this study. A new analytical model for estimating the required number of hydraulic fractures has been introduced in this study based on the reservoir drainage area and the surface area of fractures. The models have been used to establish several plots to estimate the shape factor group based on the number of fractures and the half fracture length. This group is one of the main terms in the productivity index model. Several plots for the shape factor of fractured formations have been introduced in this study. The results obtained from the new technique have been compared with the results from previous models. Several numerical examples will be included in the paper.