Dimensionless Pressure Derivative Research Articles

Precision in Defining Diagnostic Plots Optimizes Well-Test Results

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 The paper has not been peer reviewed.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

Effects of Reservoir Angle on Vertical Well Performance

Petroleum engineers are desperately needed to maintain sustained clean oil production for a long time without running into flow restrictions caused by external boundaries. These boundaries could take the shape of sealing faults, which are the most frequent in real-world situations, constant-pressure boundaries, or other reservoir heterogeneities, such as fractures, layers, and lateral discontinuities, to name a few. This study aims to examine the impact of reservoir angle on the efficiency of vertical oil wells. In this study, empirical mathematical models were used to describe vertical oil wells constrained by two sealing faults in a reservoir supported by a strong aquifer. The mathematical models developed by (1) are applied by computer software. It was determined after evaluation that lower reservoir angles provide greater values of pressure derivatives, which is indicative of good productivity and performance. To evaluate the impact of the reservoir angle on dimensionless pressure and pressure derivatives, a sensitivity analysis was also conducted utilizing a fixed value of dimensionless time to study various reservoir angle values. However, it was shown that the dimensionless pressure, PD, is unaffected by the reservoir angle. The dimensionless pressure derivative (P'D) was shown to have an inverse relationship with the reservoir angles. To verify the accuracy of the results, a comparison analysis between the findings from the constructed computer model, OBLIQ, and those found in the literature was carried out. The analysis' findings demonstrated that the study's finding is consistent with those in the literature.

Performance of a horizontal well in a bounded anisotropic reservoir: Part I: Mathematical analysis

Abstract To enhance the productivity of horizontal wells, it is of necessity to ensure that they perform optimally. This requires an understanding of how the reservoir’s geometry, anisotropy and well design affect the pressure response. Mathematical formulations can be used to simulate pressure response in the wellbore and the data obtained can be analysed to obtain well and reservoir parameters that can aid performance and evaluation. In this study, a mathematical model that can be used to approximate pressure response in a horizontal well is formulated, and a detailed mathematical analysis that can be used to obtain well and reservoir parameters are provided. A horizontal well inside a rectangular drainage volume with sealed boundaries is considered and the effect of each boundary on pressure throughout its productive life is studied. In the analysis, investigations on how the reservoir parameters can be approximated over a given period of production are conducted. This is achieved by identification of the appropriate source and Green’s functions. These source functions allow us to formulate a mathematical model for dimensionless pressure. Considering the diagnostic plots for both dimensionless pressure and dimensionless pressure derivative, mathematical analysis studies the possible behaviour of the plots. Analysis indicates that the reservoir anisotropy can be approximated during the infinite-acting flow at early times when other parameters are known. Further, when the first boundary is felt, in this case the vertical boundary, the horizontal permeability can be approximated during the transition flow periods at middle times. Finally, at late times when all the boundaries have been felt and a pseudosteady state flow is evident, reservoir dimensions can be approximated. These results can significantly improve well test analysis and enhance the performance evaluation of a horizontal well.

Analysis of flow characteristics of the dual media shale gas reservoir with the elastic outer boundary

In order to more accurately describe the seepage characteristics of shale gas reservoirs, in this paper, an elastic outer boundary condition is introduced, and a new dual-media shale gas seepage model is established to describe the seepage characteristics of shale gas reservoirs more accurately, while considering the adsorption and desorption process. Combining Laplace transformation and Similar Structure Method, the solution of the percolation model is obtained in Laplace space. Furthermore, the solutions are inverted into real space with the Stehfest numerical inversion method. Type curves of dimensionless pressure and dimensionless pressure derivative can be used to evaluate the reservoir characteristics. The results show that the conventional three kinds of outer boundary conditions (infinite, constant pressure and closed) are actually three special cases of elastic outer boundary. The introduction of elastic outer boundary conditions not only expands the scope of data interpretation, but also closer to the actual situation of the outer boundary of the reservoir. The theory of similar structure greatly simplifies the complex process of solving the model and will have a far-reaching impact on the development of well test analysis software in the future.

A Semi-Analytical Analysis of Gas Slippage Effect in Pressure Transient Behavior of Non-Condensate Gas Reservoirs with the Different Boundary Condition

The conventional models utilized to study flow behavior in a Non-Condensate gas reservoir did not consider the effects of gas slippage and different outer boundary conditions. For gas wells with a constant production flow rate in a bounded oil or gas reservoir, commonly, the outer boundary conditions are infinite boundary conditions or zero flux. In this study, the dimensionless pseudo pressure and dimensionless derivative of pseudo pressure in the presence of 4 different outer boundary conditions (infinite reservoir, constant pressure, exponential, and power-law) besides effects of gas slippage, wellbore storage, and skin factor are analyzed. To do this, the dimensionless pseudo pressure partial differential equation of radial flow was derived from the combination of continuity equation with Darcy’s law, the equation of state, compressibility equation, and dimensionless parameters. Then the derived partial differential equation is solved analytically in the Laplace domain. The obtained results of this work have important significance to understand the effects of different conditions on the transient pressure behavior of Non-Condensate gas reservoirs.

Pressure distribution of a horizontal well in a bounded reservoir with constant pressure top and bottom

The purpose of this study is to develop a mathematical model using Source and Green’s functions for a Horizontal Wells in a Bounded Reservoir with Constant Pressure at the Top and Bottom for the interpretation of pressure responses in the reservoir based on dimensionless pressure and pressure derivative. Reservoir and well parameters investigated revealed what sets of reservoir/ well parameters combination that will prolong infinite activity of the reservoir before steady state sets in. Results show that dimensionless lateral extent does not directly affect the dimensionless pressure and dimensionless pressure derivative for very short well lengths as used in this paper. Dimensionless pressure increases with reservoir pay thickness and delay the time for steady state conditions. In fact external fluid invasion is strongly affected by the size of the pay thickness, thus the minimum time for steady state period to set in is according to the relation TD ≥ LD/5.
 Keywords: Bounded, reservoir, steady-state conditions, horizontal well, constant pressure.

Pressure behaviour of a horizontal well sandwiched between two parallel sealing faults

Generally, reservoir fluid flow is governed by diffusivity equation and solution to this equation helps to investigate pressure behaviour under certain reservoir and wellbore boundary conditions. In this paper however, the analytical solution method of Green and Source function is deployed to determine the performance of a horizontal well located between two parallel sealing faults, assuming simple rectangular reservoir geometry. Also, the dimensionless pressure and derivative approach is applied for all computations as it prevents the problem of unit conversions, reduces longer expressions and it helps to handle numerical values. The pressure expression derived from this work reveals that a maximum of two flow periods occur for the stated reservoir model. It was found out that an inverse relationship exists between dimensionless pressure and dimensionless length while pressure increased with thickness. Also high vertical permeability shortens the effect of the early radial flow period experienced by the horizontal well, thereby increasing productivity index. Finally, it was discovered that increased perforation length reduces the production potential of the horizontal well.
 Keywords: Dimensionless pressure, pressure derivatives, heterogeneity, pressure performance, reservoir and wellbore characterization.

Computer Aided Pressure Transient Analysis of a Layered Reservoir System with a Constant Pressure Boundary

A Layered reservoir system with a constant pressure bottom boundary has immense potential to produce oil and gas, using horizontal wells may further increase productivity because it offers a larger surface area for fluid withdrawal. Test analysis in this system is tedious, time consuming and interpretation may be confusing or erroneous. Hence an efficient algorithm is needed to generate accurate pressure response of the wells to facilitate quick and easy analysis and prediction. This paper presents an interactive computer program that computes the dimensionless pressure and dimensionless pressure derivatives of horizontal wells in a two layered reservoir system with a constant pressure boundary. Program’s codes and structure of the reservoir model solution were written utilising the exponential integral solution, Gauss Laguerre and Gauss Lagrange numerical methods. The program has features that gave visual interpretation (graphical user interfaces (GUI)) as it computes pressure and derivative values when the interface between the layers is either permeable (crossflow reservoir) or impermeable (no crossflow reservoir). Examples illustrating its use in model identification and well test analysis of layered system with constant pressure boundary are also included in this work. Hence it will serve as a good learning tool because it is user friendly, fast and results have acceptable level of accuracy.

Pressure Analysis for Volume Fracturing Vertical Well considering Low-Velocity Non-Darcy Flow and Stress Sensitivity

In general, there is stress sensitivity damage in tight reservoirs and fractures. Furthermore, the flow in tight reservoirs is the low-velocity non-Darcy flow. Currently, few researches of pressure analysis for volume fracturing vertical well are conducted simultaneously considering the low-velocity non-Darcy flow and stress sensitivity. In the paper, a novel flow model of a volume fractured vertical well is proposed and solved numerically. Firstly, the threshold pressure gradient, permeability modulus, and experimental data are, respectively, utilized to characterize the low-velocity non-Darcy flow, matrix stress sensitivity, and fracture stress sensitivity. Then, a two-region composite reservoir is established to simulate the vertical well with volume fracturing. After that, the logarithm meshing method is used to discrete the composite reservoir, and the flow model is solved by the method of finite difference and IMPES. Finally, the model verification is conducted, and the effects of the low-velocity non-Darcy flow and stress sensitivity on the pressure and pressure derivative are analyzed. The six flow regimes are identified by the dimensionless pressure and pressure derivative curve. They are, respectively, the fracture linear flow regime, early transition flow regime, radial flow regime, crossflow regime, advanced transition flow regime, and boundary controlling flow regime. The stress sensitivity and threshold pressure gradient have a great effect on the dimensionless pressure and pressure derivative. With the increase of reservoir stress sensitivity, the pressure and pressure derivative are upward at the advanced transition flow and boundary controlling regimes. However, the pressure and pressure derivative are downward at the advanced transition flow and boundary controlling regimes when the fracture sensitivity increases. An increase in the threshold pressure gradient results in a high dimensionless pressure and pressure derivative. This work reveals the effects of low-velocity non-Darcy flow and stress sensitivity on pressure and provides a more accurate reference for reservoir engineers in pressure analysis when developing a tight reservoir by using the volume fracturing vertical well.

Seepage model for multiple-fractured horizontal wells in coal bed methane reservoirs

ABSTRACT Due to the features of low permeability and porosity, the horizontal well and fracturing treatment are effective ways to develop coalbed methane reservoirs, which have the large drain area leading to high producing rate and recovery. The seepage researches in conventional gas reservoir are sufficient, but there are few researches about the flow behavior and pressure distribution in coal bed methane (CBM) reservoirs, especially for multiple-fractured horizontal wells. In this work, based on the seepage theory, point-source function, and Laplace transforms, the mathematic models of gas flow in multiple-fractured wells are derived considering threshold pressure gradient and unsteady flowing. Subsequently, the type curves of multiple-fractured horizontal well in CBM reservoirs are obtained; there are main eight regimes. Finally, the mainly parameters affecting the dimensionless pressure and dimensionless pressure derivative are analyzed. The percolation characteristics of CBM reservoir are more complex than that of conventional gas reservoir for the presence of adsorption/desorption mainly affecting the seepage features in Middle-later Period and boundary behavior. This method does not only enrich our understanding of multiple-fractured horizontal well transport behaviors in coalbed methane reservoirs, but also has an important guiding significance for determining reasonable development program with multiple-fractured wells in the development of coalbed methane reservoir.

Analysis of Factors Affecting Bottom Hole Pressure in Tight Gas Reservoir

In order to solve the problem of low accuracy of bottom hole pressure in tight gas reservoir, this paper presents a new way to analyze the bottom hole pressure. By combining literature research and starting from nonlinear motion equations, the model of bottom hole pressure was established through using various mathematical methods such as separation variable method, identity transformation method and differential discrete method. This paper presents a high-precision and high-efficiency method for solving bottom hole pressure in tight gas reservoirs, and a dynamic calculation method for permeability and conductivity is given. Through the sensitivity analysis of the influencing factors of dimensionless pressure and pressure derivative, it is concluded that the larger the power law index is, the larger the warpage of the dimensionless pressure and pressure derivative curve is. The larger the skin coefficient and the wellbore storage coefficient is, the earlier the fluid enters into the quasi-steady-state seepage. When the tight gas reservoir contains closed edges, the resolution points of the dimensionless pressure and pressure derivative curves are obvious with crossing. On the contrary, when the tight gas reservoir contains the constant pressure boundary, the resolution points of the dimensionless pressure and pressure derivative curves are not obvious without crossing.

A multi-linear transient pressure model for multistage fractured horizontal well in tight oil reservoirs with considering threshold pressure gradient and stress sensitivity

In general, tight oil reservoirs has a numerous tiny and complicated throats, and ultra-low permeability, which leads to a low-velocity non-Darcy flow when the fluid flows in tight reservoirs. The permeability of tight oil reservoirs is no longer a constant, but a variable, which is dependent on reservoir pressure. In other word, the tight oil reservoir is stress sensitive media. The low-velocity non-Darcy flow and stress sensitivity will greatly affect the bottom hole pressure response. So, the low-velocity non-Darcy flow and stress sensitivity should be taken into consideration when researching the transient pressure behaviors of multistage fractured horizontal wells. Currently, there are many researches on transient pressure behaviors of the multistage fractured horizontal well, but none of these researches has taken the low-velocity non-Darcy flow and stress sensitivity into consideration simultaneously. In this paper, threshold pressure gradient and permeability modulus are used to characterize the low-velocity non-Darcy flow and stress sensitivity, respectively. And, a multi-linear transient pressure model of multistage fractured horizontal well with considering low-velocity non-Darcy flow and stress sensitivity is established. The Pedrosa's transformation, perturbation theory, Laplace transformation and stehfest numerical inverse are used to solve this transient pressure model. The novel model is verified by comparing the result of proposed model with that of Ekaterina Stalagorova's model. According to the dimensionless pressure derivative curve, the flow process can be identified as six regimes: ①the main fracture linear flow stage, ②the first transition flow stage, ③dual-linear flow stage, ④the second transition flow stage, ⑤the formation linear flow stage, ⑥the boundary flow stage. Finally, the effects of threshold pressure gradient, permeability modulus and fracture density of stimulated reservoir on dimensionless pressure derivative are analyzed. This research provides theoretical support for correctly understanding the flow law of multistage fractured horizontal well.

Study on characteristics of well-test type curves for composite reservoir with sealing faults

Abstract The pressure response for the composite reservoirs with a sealing fault locating in inner and outer region is different, which neglected by previous researchers, would cause significant errors during well-test interpretations. Based on seepage theory, a well-test model of two-region radial composite reservoir with infinite outer boundary has been built in this study considering wellbore storage and skin effects. The solutions for this model and characteristics of the type curves have been analyzed by applying the method of mirror image, Laplace transformation and superposition principle, including a straight fault, a perpendicular fault and parallel faults cases. The study shows that the dimensionless pressure derivative curves would be obviously different in two cases: the well to fault distance is larger, and smaller than the half length of the inner-region radius. Therefore, type curves are presented with reasonable parameters to analyze the distance effect on the dynamic pressure response. The results in this study are of great significance for guiding the oil and gas composite reservoirs' production and optimizing the hydrocarbon recovery.

A Novel Approach To Detect Interacting Behavior Between Hydraulic Fracture and Natural Fracture by Use of Semianalytical Pressure-Transient Model

Summary The detection of interacting behavior between the hydraulic fracture (HF) and the natural fracture (NF) is of significance to accurately and efficiently characterize an underground complex-fracture system induced by hydraulic-fracturing technology. This work develops a semianalytical pressure-transient model in the Laplace domain to detect interacting behavior between HF and NF depending on pressure-transient characteristics. Our results have shown that no matter what the flow state (compressible or incompressible flow) within a hydraulically induced fracture system, we can easily detect interacting behavior between HF and NF depending on whether the “dip” shape occurs at the formation radial-flow regime. Referring to sensitivity analysis, distance between NF and well, horizontal distance between NF and HF, and NF length are the three most sensitive factors to detect fracture-interacting behavior. Depending on interference analysis, although the pressure-transient characteristics of a pseudosteady-state dual-porosity model can interfere with our proposed methodology, the difference between our model and a pseudosteady-state dual-porosity system lies in whether the value of the horizontal line of dimensionless pressure derivative is equal to 0.5 at the formation radial-flow regime. Our work obtains some innovative insights into detecting fracture-interacting behavior, and the valuable results can provide significant guidance for refracturing operations and fracture detection in an underground fracture system.

Pressure transient analysis of non-planar asymmetric fractures connected to vertical wellbores in hydrocarbon reservoirs

The paper developed a new constant-rate solution for non-planar asymmetric fracture connected to a vertical wellbore in hydrocarbon reservoirs. The non-planar asymmetric fracture has been modeled by combination of two unequal length wings with varied intersection angles. The type curves were drawn in the case of varied fracture conductivities, non-planar angles, and asymmetric factors. The results indicated that fracture conductivity, non-planar angle, and asymmetric factor have coefficient effects on the pressure response, which makes the dimensionless wellbore pressure and pressure derivative curves deviate from the conventional type curves of vertical fractured wells. The dimensionless wellbore pressure and pressure derivative curves are divided into five flow periods. The non-planar angle and asymmetry factor have their unique influences on the shape of type curves during different periods under the different fracture conductivities. The termination of bilinear flow and formation linear flow happened increasingly early with the decrease of non-planar angle or the increase of asymmetry factor. As the fracture conductivity increases, the effect of non-planar angle on the curves moves from the early flow period to the late flow period while the trend of effect of asymmetric factor is quite opposite. The new method lays the foundation of the pressure analysis of the complicated fracture system in developing hydrocarbon reservoirs.

Comparative evaluation of pressure distribution between horizontal and vertical wells in a reservoir (edge water drive)

This paper presents comparative analysis between the pressure behavior of a horizontal well and a vertical well both subject to edge water drive. Green and source function were used to evaluate the performance of horizontal well and vertical well, eventually, compared based dimensionless pressures and pressure derivatives computed by varying the reservoir geometry. Results presented as type curves show that the rate of decline of the pressure derivative curve is more sharper and sensitive in vertical well than horizontal well of the same geometry indicating a shorter period of clean oil production in vertical well than the horizontal well. The period of radial flow as shown by the horizontal flattened part of the type curve indicate that for all reservoir geometry, the horizontal well will be able to sustain longer period of the radial flow than vertical well given infinite conductivity condition. http://dx.doi.org/10.4314/njt.v36i2.19

Composite linear flow model for multi-fractured horizontal wells in heterogeneous shale reservoir

Multi-stage fracturing is currently key technique to develop shale reservoirs. Different analytical models have been proposed to fast investigate post-fracturing pressure- and rate-transient behaviors, and hence, estimate key parameters that affect well performance. However, previous analytical models generally neglect reservoir heterogeneity or typical seepage characters of shale, such as adsorption/desorption, gas slippage, and diffusion effects. This paper presents an analytical model for pressure- and rate-transient analysis of multi-stage fractured shale reservoir, considering heterogeneity, typical seepage characters and, specifically, fluids flow from upper/lower reservoir when vertical fractures partially penetrate the formation.This model is similar to five-region-flow model, but subdivides the reservoir into seven parts, namely, two upper/lower-reservoir regions, two outer-reservoir regions, two inner-reservoir regions, and hydraulic fracture region, which are all transient dual porosity media except the hydraulic fracture. As reservoir heterogeneity along the horizontal wellbore is included, the fracture distribution can be various. Fracture interference is simulated by locating a no-flow boundary between two adjacent fractures. The actual locations of these no-flow boundaries of a specific heterogeneous reservoir are determined based on the pressure value which varies with time and space. Thus, the two sides of this boundary has minimum pressure difference, satisfying the no-flow assumption. Adsorption/desorption, gas slippage and diffusion effects are included for rigorous modeling of flow in shale.This model is validated by comparing with commercial well testing software, obtaining a good match in most flow regimes. Log-log dimensionless pressure, pressure derivative and production type curves are generated to conduct sensitivity analysis. Results suggest that larger desorption coefficient causes smaller pressure and its derivative value as a larger proportion of gas is desorbed in formation and contributes to productivity. Solutions from Azari's (1990, 1991) work, where the effect of fracture height is merely treated as a skin factor are investigated as well. Results show that our model is more accurate in partially penetrating cases, and errors of Azari's method become particularly noticeable in early-middle time response. The influence of other parameters, such as matrix permeability, matrix block size, secondary fracture permeability and hydraulic fracture conductivity, are also discussed. Optimal fracture pattern is selected based on cumulative production. Besides, field data are analyzed and compared graphically with modeling solutions, and reliable results are obtained.As numerical and semianalytical methods require extensive computing processing, this model is a practical alternative to predict well-testing results and select optimal well pattern of shale reservoirs. Reservoir heterogeneities in vertical direction can be further added to our model by vertically subdividing the reservoir into more parts.

A new semi-analytical model for simulating the effectively stimulated volume of fractured wells in tight reservoirs

The interaction of hydraulic fractures and natural fractures results in a complex fracture network system around a producing well, which is the primary contributing region for production in unconventional reservoirs. Currently, most models for simulating fluid flow in formations penetrated by fractured wells are based on the assumption of a homogeneous dual–porosity medium.In this paper, a new semi-analytical model is presented for vertically fractured wells with stimulated reservoir volumes (SRV). Specifically, we employ fractal porosity and permeability to describe the heterogeneous distribution of porous media in a SRV. An approach is given for estimating the size and equivalent permeability of the fractal SRV for vertically fractured wells in tight reservoirs. The line source function, Laplace transformation, integral of the modified Bessel function and Stehfest numerical inversion algorithms are used to solve the composite model. Based on the Duhamel principle, the dimensionless pressure responses of tight oil and gas wells in the Laplace domain and time domain are obtained. Flow regime diagnostics of vertically fractured wells in tight reservoirs are characterized. Sensitivity analysis, including the effects of SRV parameters and fractal parameters on pressure responses and linear flow, are performed. The results show that seven primary regimes can be divided into pressure response curves. The size and equivalent permeability of the fractal SRV can be calculated by the dimensionless pressure derivative curves. In a complex fractal reservoir, the larger the connectivity index is and the smaller the fractal dimension is, the more significant the heterogeneity of the fractal SRV is and the larger the slopes of linear flow are, which leads to a smaller equivalent permeability of the fractal SRV radius. The presented model and the results in this paper can enrich the pressure transient analysis models for vertically fractured wells in unconventional reservoirs.

The Effects of Reservoir Area Extent on the Performance of a Vertical Well in a Reservoir Subject to Bottom Water Drive

Reservoir performance can be understood from system type curves. The type curve gives vivid information about maximum pressure drops, magnitude of near wellbore effects, reservoir fluid and wellbore properties needed to ascertain the strength of available drive mechanism, maximum withdrawal rates and remaining fluid in real time. This paper investigates the effects of reservoir area extent on the performance of a reservoir, subject to active bottom water, when it is completed with a vertical well. Type curves of dimensionless pressures and dimensionless pressure derivatives were produced for various dimensionless values of area extent of the reservoir. These type curves were developed from solutions to flow equations using relevant source and Green’s functions. From the results, it can be observed that the larger the reservoir area extent, the larger the dimensionless pressure drop, the longer the time it takes to attain steady state. This is validated from the pressure derivative curve, which shows that reservoirs with large area extent are characterized by longer period of radial flow and subsequently delay in the attainment of steady state, thus prolonging the arrival of bottom water.

Numerical Simulation of Unsteady-State Flow in Dual Porous Coalbed Methane Horizontal Wells with Complex Boundary Conditions

The bottom-hole pressure response which can reflect the gas flow characteristics is important to study. A mathematical model for description of gas from porous coalbed methane (CBM) reservoirs with complex boundary conditions flowing into horizontal wells has been developed. Meanwhile, basic solution of boundary elements has been acquired by combination of Lord Kelvin point source solution, the integral of Bessel function, and Poisson superimpose formula for CBM horizontal wells with complex boundary conditions. Using this model, type curves of dimensionless pressure and pressure derivative are obtained, and flow characteristics of horizontal wells in complex boundary reservoirs and relevant factors are accordingly analyzed.