Both the Rawlins and Schellhardt and Houpeurt analysis techniques are presented in terms of pseudopressures. Flow-after-flow tests, sometimes called gas backpressure or four-point tests, are conducted by producing the well at a series of different stabilized flow rates and measuring the stabilized BHFP at the sandface. Each different flow rate is established in succession either with or without a very short intermediate shut-in period. Conventional flow-after-flow tests often are conducted with a sequence of increasing flow rates; however, if stabilized flow rates are attained, the rate sequence does not affect the test. Fig 1 illustrates a flow-after-flow test.

This article summarizes the fundamental gas-flow equations, both theoretical and empirical, used to analyze deliverability tests in terms of pseudopressure. The four most common types of gas-well deliverability tests are discussed in separate articles: flow-after-flow, single-point, isochronal, and modified isochronal tests. Deliverability testing refers to the testing of a gas well to measure its production capabilities under specific conditions of reservoir and bottomhole flowing pressures (BHFPs). A common productivity indicator obtained from these tests is the absolute open flow (AOF) potential. The AOF is the maximum rate at which a well could flow against a theoretical atmospheric backpressure at the sandface.

Steady-state-, pseudosteady-state-, and transient-flow concepts are developed, resulting in a variety of specific techniques and empirical relationships for both testing wells and predicting their future performance under different operating conditions. The basis for all well-performance relationships is Darcy's law, which in its fundamental differential form applies to any fluid--gas or liquid. However, different forms of Darcy's law arise for different fluids when flow rates are measured at standard conditions. The different forms of the equations are based on appropriate equations of state (i.e., density as a function of pressure) for a particular fluid. In the resulting equations, presented next, flow rate is taken as being positive in the direction opposite to the pressure gradient, thus dropping the minus sign from Darcy's law. When multiple-line equations are presented, the first will be in fundamental units, the second in oilfield units, and the third in SI units.

Summary The network of induced fractures and their properties control pressure propagation and fluid flow in hydraulically fractured shale reservoirs. We present a novel fully fractal model in which both the spacing and the porosity/permeability of induced fractures are distributed according to fractal dimensions (i.e., fractal decay of fracture density and the associated porosity/permeability away from the main induced fracture). The fractal fracture distribution is general, and handles exponential, linear, power, and uniform distributions. We also developed a new fully fractal diffusivity equation (FDE) using the fractal distribution of fractures and their properties. We then used, for the first time, the semianalytic Bessel spline scheme to solve the developed diffusivity equation. Our proposed model is general and can capture any form of induced-fracture distribution for better analysis of pressure response and production rates at transient-and pseudosteady-state conditions. We compared the unsteady-state and pseudosteady-state pressure responses calculated by our fully fractal model with former models of limited cases: uniform fracture spacing and uniform porosity/permeability [conventional diffusivity equation (CDE)]; variable fracture spacing and uniform porosity/permeability [modified CDE (MCDE)]; and uniform fracture spacing and fractal porosity/permeability distribution (FPPD). We used these models to match and predict the production data of a multifractured horizontal gas well in the Barnett Shale. Our results showed that the fractal distribution of fracture networks and their associated properties better matches the field data. Introduction In recent years, horizontal wells with multistage hydraulic fracturing have been widely used in unconventional reservoirs (Ozkan et al. 2011; Kissinger et al. 2013; Yao et al. 2013; Yuan and Wood 2015; Yuan et al. 2015b). Reservoirs with multifractured horizontal wells have a stimulated reservoir volume around the hydraulic fractures, which is filled with complex networks of induced fractures (Cipolla et al. 2010; Xu et al. 2013; Yao et al. 2013; Moridis and Blasingame 2014; Yuan et al. 2015b).

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.

Summary This paper introduces a new approach for studying productivity-index (PI) behavior of fractured oil and gas reservoirs during transient-and pseudosteady-state conditions. This approach focuses on the fact that PI derivative could vanish at a certain production time, indicating the beginning of pseudosteady state, wherein the PI demonstrates constant value. The reservoirs in this study are considered depleted by horizontal wells intersecting multiple hydraulic fractures where Darcy flow and non-Darcy flow may control flow patterns in the porous media. The PI is calculated assuming constant production rate and considering pressure profile for early- and intermediate-production time when transient condition dominates fluid flow and late-production time when pseudosteady state is reached. The outcomes of this study can be summarized as understanding PI behavior at early- and intermediate-production time when transient flow is dominant in the porous media and late-production time when pseudosteady-state condition is reached; indicating the effect of reservoir configuration on PI and the time when this index approaches constant value; and introducing a study for the influence of non-Darcy flow in the PI. The most-interesting points in this study are the following. First, that PI reaches constant value when the rates of change with time for the two pressure drops—transient and pseudosteady state—are equal. Second, the time for approaching constant PI in a small drainage area is faster than for a large area. Third, that PI is affected by non-Darcy flow at early- and intermediate-production time; however, the effect is not seen at late-production time. Last, that PI could exhibit constant behavior for severe non-Darcy flow at early- and intermediate-production times even though transient-state condition dominates fluid flow in the porous media.

Abstract This paper examines the behavior of heavy oil reservoirs developed with horizontal and multilateral wells. Advanced decline curve analyses were used to characterize flow regimes and estimate the time to pseudosteady-state. Reservoir and well parameters such as the OOIP, Arps "b" exponent, decline rate, reserves, permeability and well productivity indices were also determined. Example analyses are presented for single, dual and triple lateral wells from heavy oil fields located in Venezuela and Canada. All wells exhibit a characteristic extended transient linear flow regime followed by an exponential decline. Similar results were obtained whether the analyses were performed on single, dual or triple lateral wells. Interference between laterals was not observed. Introduction The application of horizontal and multilateral wells is gaining momentum worldwide due to their ability to drain reservoirs more effectively. This advantage is even more pronounced in tight gas or heavy oil reservoirs where low mobility is responsible for long transient flow periods. The relatively new application of these exotic well geometries to such reservoirs provides a challenge in the area of production forecasting because traditional methods and equations were developed based on flow to a vertical well. This paper demonstrates the use of rate-time performance analyses on heavy oil reservoirs developed with horizontal and multilateral wells. Well productivity indices (PI) were calculated from the transient production period by matching the rate-time data to type curves. Permeability-thickness or the equivalent skin factor was calculated based on this PI. Hydrocarbon volume connected to the well, the Arps "b" exponent and the decline rate were calculated from the pseudosteady-state producing period. The decline curve results were also verified using a reservoir simulation flow model. Decline curve analysis was performed on the rate versus time values generated by the flow model to confirm that the model had similar transient and depletion behavior as the actual performance data. Decline Curve Analysis Concepts When a well is first opened to flow, it produces under transient flow conditions. It will remain under this condition until the production from the well affects the entire drainage area. This flow condition is referred to as pseudosteady-state or boundary dominated flow. Transient rate and pressure data are used to calculate permeability-thickness and skin, whereas pseudosteady-state data are used to determine connected OOIP. Constant well pressure solutions used to predict declining production rates as a function of time were first published in 1933 by Moore, Schilthius, and Hurst.[1] Results were presented for infinite, slightly compressible, single phase plane radial systems. The results were presented in graphical form in terms of dimensionless flow rate and dimensionless time as shown in Figure 1.

Summary Early, accurate determination of original gas in place (OGIP) is highly desirable for planning the future of a gas field. This paper presents a case history of an offshore gas field, using a recently developed pressure-rate-time analysis technique to illustrate the effectiveness of the method. In addition, this paper demonstrates the great benefit of permanently installed pressure gauges in obtaining consistent pressure and flow-rate data for the effective use of the technique for subsea completed wells. Introduction Early, accurate determination of OGIP is always desirable in a new gas field development, as this information is critical in decision making for reservoir management. For example, the optimum number of wells or the need for compression at the surface depends on OGIP. In addition, the operator is often required to project production rates in order to secure a gas sales contract. These decisions are typically made early in the life of a field, and erroneous estimates of OGIP can lead to poor decisions that may be costly or impossible to correct later in the field life. Various reserve estimation methods have been published in the literature, ranging from basic material-balance calculation to decline-curve analysis. Recently, Agarwal et al. presented new production decline curves for analyzing well production data by combining type-curve and decline-curve analysis concepts. In this report, we applied Agarwal et al.'s analysis techniques to pressure and rate measurements over a period of approximately 3,000 days at Ballycotton. Subsequent analysis of short-term performance data showed that an accurate estimate of OGIP was established from the first 6 months of production. The results were confirmed independently using reservoir simulation while examining the effects of water influx and production interference from Kinsale Head. Background Information Geologic Setting and Field Description. The Ballycotton gas field lies approximately 25 miles off the coast of Cork, Ireland in the North Celtic Sea basin. It was discovered in March 1989 and has been produced from Well 48/20-2 since July 1991 as a satellite field by means of a single subsea well completion and tieback to the Kinsale Head's Bravo platform. A location map is shown in Fig. 1. The ‘A’ Sand producing interval in Ballycotton and Kinsale Head is the focus of this paper. This sand extends from Kinsale Head gas field in the south to Ballycotton gas field in the north, thus connecting the two fields by a common aquifer. Ref. 3 gives a comprehensive geologic description of the region. The Kinsale Head gas field was discovered in 1971 and was placed on production in October 1978. The existence of gas-processing facilities at Kinsale Head made it commercially feasible to produce Ballycotton, which at the time of discovery was thought to contain 80 Bcf from a volumetric study. About 760 Bcf of gas were produced from the ‘A’ Sand of Kinsale Head by the time Ballycotton was placed on-line. Immediately after Ballycotton was put on production in July 1991, an attempt was made to estimate OGIP from the initial production data. Assuming hydrostatic equilibrium between the two fields, the initial pressure at Ballycotton was estimated from 1978 pressures at Kinsale Head gas field. This pressure, compared to the Ballycotton pressure measured immediately before commercial production, shows a pressure loss of 50 psi, thus reflecting interference resulting from production at Kinsale Head. Pressure interference is transmitted through the common aquifer between Kinsale Head and Ballycotton. Because the complexities of water influx and interference cannot be easily accounted for in a gas/water system, it was difficult to provide an OGIP estimate from the simple material balance. However, a numerical simulation study of a single well in an edge-waterdrive gas reservoir indicated that it was possible to estimate an OGIP from the drawdown data in the early life of a reservoir before the influence of water influx becomes dominant. Later, a full-field model was set up to examine the strength of water influx. This will be discussed in the simulation study section. The Ballycotton completion was the first subsea development in the Celtic Sea. A permanent gauge was installed on the subsea tree in 285 ft of water to monitor wellhead pressure (WHP). From the corresponding gas rate and measured WHP, we calculated bottomhole pressure (BHP) with the Cullender and Smith method.Fig. 2 shows a graph of calculated flowing BHP and gas-flow rate between July 1991 and September 1999. Both pressure and rate are monotonically decreasing. Our goal was to determine the OGIP from the production-performance data with both flowing BHP and gas-flow rate. Historical Development of Analysis Technique. Analysis of the gas-production performance data involves nonlinear effects of gas physical properties and variable-rate production mode during boundary-dominated flow. Recognizing that the partial-differential equation governing the flow of gases is nonlinear, researchers have focused on correlating the gas-flow solutions with the appropriate liquid-flow solutions. Gradually, a new time function was developed to convert the production data of either constant rate or constant BHP into a form that could be analyzed with constant-rate liquid solutions. This section presents the historical development of these techniques. To analyze monotonically declining rate and pressure data to estimate reserves, pseudosteady-state flow must be established in the long-term performance data. Using the principle of superposition, Blasingame and Lee showed that equivalent time, te, defined as cumulative production divided by the corresponding flow rate, could be applied to a liquid system in pseudosteady-state flow. This suggests that long-term performance data can be used to estimate pore volume by plotting the pressure difference (?=pi-pwf) normalized by the corresponding flow rate vs. te. The slope of the graph is used to calculate reserves for a slightly compressible liquid case. Al-Hussainy et al. showed that during the boundary-dominated flow period there are considerable differences between the liquid and gas solutions. It is clear that nonlinear effects become dominant during the pseudosteady-state flow period, when the wellbore pressure depends on the physical properties of the produced gas and the flow rate.

Abstract A dimensionless inflow performance relationship (IPR) curve that is a function of permeability has been developed for unfractured gas wells. Both numerical and analytical methods were used to solve the non-Darcy, gas flow equations. This analysis showed that the current dimensionless IPR curve for unfractured gas wells in pseudosteady-state flow is basically independent of all relevant variables, except permeability. The proposed current dimensionless IPR curve has the same general form as the familiar Vogel dimensionless IPR curve for solution-gas-drive oil reservoirs, with the coefficients of the proposed dimensionless IPR curve being a function of permeability. The proposed current dimensionless IPR curve can be used to predict the deliverability of a gas well using a single-point test, as opposed to a standard four-point deliverability test, if the permeability of the well is known. A future dimensionless IPR curve to predict future gas well deliverability that is a function of permeability is also presented. The time to reach pseudo-steady state flow for an unfractured gas well is two to three multiples of the time predicted by the commonly used, stabilization time equation. Introduction Deliverability testing refers to the testing of a gas well to measure its production capabilities at a given stage of reservoir depletion. Deliverability testing commonly yields a reservoir inflow performance relationship (IPR) curve. An IPR curve relates production rate versus flowing bottomhole pressure for a given reservoir pressure, the reservoir pressure at which the deliverability testing was performed.It is mainly used to predict current gas-well deliverability given a fixed backpressure. Traditionally, deliverability testing of a gas well is accomplished using a four-point backpressure test, an isochronal test, or a modified isochronal test. All of these methods require testing a well at a minimum of four flow rates. These multi-point tests all yield very reliable results but are very expensive in terms of manpower and testing equipment. A method which can predict both current and future gas well deliverability using only a single-point flowrate test is thus, desirable.

This paper (SPE 51023) was revised for publication from paper SPE 37030, first presented at the 1996 SPE Asia Pacific Oil & Gas Conference held in Adelaide, Australia, 28-31 October. Original manuscript received for review 16 September 1996. Revised manuscript received 20 May 1998. Revised manuscript approved 9 June 1998.

Summary

This study presents a new finite-element approach for directly calculating pseudosteady-state flow behavior for wells in depletion systems. The approach allows for spatially dependent reservoir properties, complex reservoir geometries, and multiple wells. Results are verified against long-time transient solutions reported in the literature for several regularly shaped systems. The paper also demonstrates application of the approach to field-scale problems. Results show that this approach provides a fast and accurate method for modeling the long-time behavior of depletion reservoirs. The approach is particularly applicable to single-phase volumetric gas reservoirs.

A bound reservoir with wells producing at constant rate will exhibit pseudosteady-state behavior after the end of typically short-lived infinite-acting and transition flow periods. This study develops a new approach for directly calculating pseudosteady-state flow behavior without solving the full time-dependent form of the diffusivity equation. This approach can be applied to the linearized forms of the diffusivity equation for either single-phase liquid or gas flow. A finite-element method is used that allows for spatially dependent reservoir properties, complex reservoir geometries, and multiple wells. The first part of this paper presents a verification of the approach by comparing results for some regularly shaped systems against full-transient solutions reported in the literature.

For the simulation of field-scale problems with multiple wells of differing production rates, a well model based on a near-wellbore approximation of the pseudopressure distribution during pseudosteady-state is introduced to reduce the concentration of elements near wells. The second part of the paper demonstrates application of the direct pseudosteady-state concept to actual reservoir problems. To account for rate changes during extended production periods, the pseudosteady-state equation was solved successively for each flow period and combined with an overall reservoir material balance analysis.

Results from this study show that this approach provides a fast and accurate method for modeling the long-time behavior of various types of reservoirs under depletion conditions. The approach is particularly applicable to single-phase volumetric gas reservoirs.

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