Wellbore Storage Effects Research Articles

Nonlinear Flow Characteristics and Horizontal Well Pressure Transient Analysis for Low-Permeability Offshore Reservoirs

Threshold pressure gradient (TPG) and stress sensitivity which cause the nonlinear flow in low permeability reservoirs were carried out by experiments. Firstly, the investigation of existing conditions of TPG for oil flow in irreducible water saturation low-permeability reservoirs was conducted and discussed, using the cores from a real offshore oilfield in China. The existence of TPG was proven. The relationship between TPG and absolute permeability was obtained by laboratory tests. TPG increases with decreasing absolute permeability. Then, stress sensitivity experiment was carried out through depressurizing experiment and step-up pressure experiment. Permeability modulus which characterizes stress sensitivity increases with decreasing absolute permeability. Consequently, a horizontal well pressure transient analysis mathematical model considering threshold pressure gradient and stress sensitivity was established on the basis of mass and momentum conservation equations. The finite element method (FEM) was presented to solve the model. Influencing factors, such as TPG, permeability modulus, skin factor, wellbore storage, horizontal length, horizontal position, and boundary effect on pressure and pressure derivative curves, were also discussed. Results analysis demonstrates that the pressure transient curves are different from Darcy’s model when considering the nonlinear flow characteristics. Both TPG and permeability modulus lead to more energy consumption and the reservoir pressure decreases more than Darcy’s model.

Erratum to: Pressure Transient Analysis for Multi-stage Fractured Horizontal Wells in Shale Gas Reservoirs

This article presents the PTA on the multi-stage fractured horizontal well in shale gas reservoirs incorporating desorption and diffusive flow in the matrix. Currently, most PTA models are simply based on Darcy flow both in natural fractures and matrix without considering the mechanisms of desorption and diffusion in shale matrix. Source function and Laplace transform with the numerical discrete method are employed to solve the mathematical model. The solution is presented in the Laplace domain so that the wellbore storage effect and skin factor can be easily incorporated by convolution. Type curves are plotted with Stehfest algorithm and different flow regimes are identified. The presented model could be used to interpret pressure signals more accurately for shale gas reservoirs.

On the aquitard–aquifer interface flow and the drawdown sensitivity with a partially penetrating pumping well in an anisotropic leaky confined aquifer

Summary A mathematical model for describing groundwater flow to a partially penetrating pumping well of a finite diameter in an anisotropic leaky confined aquifer is developed. The model accounts for the jointed effects of aquitard storage, aquifer anisotropy, and wellbore storage by treating the aquitard leakage as a boundary condition at the aquitard–aquifer interface rather than a volumetric source/sink term in the governing equation, which has never developed before. A new semi-analytical solution for the model is obtained by the Laplace transform in conjunction with separation of variables. Specific attention was paid on the flow across the aquitard–aquifer interface, which is of concern if aquitard and aquifer have different pore water chemistry. Moreover, Laplace-domain and steady-state solutions are obtained to calculate the rate and volume of (total) leakage through the aquitard–aquifer interface due to pump in a partially penetrating well, which is also useful for engineers to manager water resources. The sensitivity analyses for the drawdown illustrate that the drawdown is most sensitive to the well partial penetration. It is apparently sensitive to the aquifer anisotropic ratio over the entire time of pumping. It is moderately sensitive to the aquitard/aquifer specific storage ratio at the intermediate times only. It is moderately sensitive to the aquitard/aquifer vertical hydraulic conductivity ratio and the aquitard/aquifer thickness ratio with the identical influence at late times.

The transient behaviour of CO 2 flow with phase transition in injection wells during geological storage – Application to a case study

Abstract This article presents a study on the transient behaviour of CO 2 in wellbores during injection of CO 2 for geological storage or enhanced recovery of oil or gas. To deal with the transient flow equations, we developed a novel modelling scheme that fully includes the storage/inertial terms, and tested the scheme by simulating observations from a field trial of CO 2 enhanced coal bed methane recovery. Because the field trial involved short periods of CO 2 injection, transient thermal and wellbore storage effects played a significant role in the wellbore׳s hydraulics. It was shown that our model׳s simulation presented a satisfactory agreement with the field trial data. A further sensitivity analysis demonstrated the importance of representing wellbore transients. The results and observations from this work include the following. (1) A set of field measurements for low temperature (below −20 °C) carbon dioxide injection is presented, which can serve as an experimental benchmark for verification and validation of theoretical and numerical models. (2) A modelling scheme for transient flows is developed, which shows that for the transient flow concerned the storage /inertia terms in the flow equations must be included and that large errors can result if they are ignored. (3) Unlike a long-term steady injection problem, for short-term injection or at an early stage of injection the thermal exchange between the injecting carbon dioxide and its surroundings is mainly dependent on the thermal scenario in the immediate vicinity of the wellbore, and the geothermal gradient of far-field is less important in this situation. (4) When the injected carbon dioxide is in a two-phase state, the Joule–Thomson coefficient may no longer be the best approach to characterising the dependence of temperature on pressure. Instead, the phase transition and the heat transfer processes may play an equal or a more important role in determining the pressure and the temperature profiles of the fluid within the wellbore.

Effect of Knudsen diffusion and Langmuir adsorption on pressure transient response in tight- and shale-gas reservoirs

Abstract Klinkenberg number is not constant in tight- and shale-gas reservoir. We first derived an expression of b / p in Klinkenberg apparent permeability correlation based on Knudsen number, which is a function of µ/p in an isothermal shale-reservoir system. For pressure in [15 MPa, 30 MPa], µ/p changes slowly and is very small, and the permeability correction factor is 1.15 for 17 MPa, 1.14 for 20 MPa, 1.12 for 28 MPa for the case in this paper. To characterize the physics of multi-transport mechanisms, and gas adsorption and desorption, a mathematical model for flow in tight- and shale-gas reservoirs with wellbore storage effect is used to understand transient pressure response. A full implicit numerical simulation based on PEBI gridding is developed to quantify the transient pressure behaviors for flow in tight- and shale-gas reservoirs. Based on numerical results, we firstly find that the adsorption makes the curves of the bottom-hole pressure (BHP) have a seeming singular point. The intrinsic permeability determines the position of the seeming singular point, and the ultimate adsorption capacity (UAC) determines the size of the angle around the seeming singular point. Because the turning points appear at early time flow, the position and angle of the seeming singular point may give us an economic and quick methodology to approximately estimate the UAC and intrinsic permeability for the tight- and shale-gas reservoir. This phenomenon can help us to understand the behavior of the flow in tight- and shale-gas reservoir better.

A novel semi-analytical model for horizontal fractures with non-Darcy flow

Abstract Based on Darcy law in matrix and non-Darcy law in horizontal fractures, a novel semi-analytical model is presented to investigate effects of non-Darcy on flow behavior. The proposed semi-analytical model convergences rapidly and it is convenient to add wellbore storage and skin effect in this model. The effects of the parameters on dimensionless pressure curves and their derivative curves are discussed in detail, including Forchheimer number, dimensionless formation thickness, fracture conductivity, and fracture diffusivity factor. Calculative results show that a big Forchheimer number, a big dimensionless formation thickness and a low conductivity will lead to big pressure depletion at early time. However, flow characteristics in entire flow regime cannot be determined by diffusivity factor. Then effects of the Forchheimer on the flow rate profiles are also discussed, indicating that as the time increases the effect of non-Darcy on flow behavior becomes weak. The proposed model could be well applied to make well test data analysis for a finite conductivity horizontal fracture with non-Darcy flow.

A new changing wellbore storage model for pressure oscillation in pressure buildup test

The early characteristic of the pressure buildup test is usually dominated by wellbore storage effect. These wellbore phase redistribution effects, which are transient in nature, including phase re-injection, bubble flow and hammering effect, sometimes lead to the pressure oscillation in early pressure buildup. And, these phenomena are characterized with the changing wellbore storage effect. But the present changing wellbore storage models vary with time monotonically, which could not describe these periodic processes.This paper presents a new changing wellbore storage model for analysis of the pressure oscillation process at wellbore or near wellbore region during the pressure buildup test. First, the physical processes of the phase re-injection, bubble flow and hammering effect are analyzed for reasons of the pressure oscillation. Then, a periodic correlation is introduced to describe these processes based on experiments. Finally, a new model is proposed by taking account of the combination of the Fair model and a cosine function.Then, the Stehfest algorithm and Crump algorithm are used to simulate the process of pressure oscillation. The results show that Stehfest algorithm has a significant error on high frequency pressure oscillation, but Crump algorithm could simulate the whole process of pressure oscillation more accurately than that of Stehfest algorithm.When the pressure oscillation occurs in the well test, the pressure and pressure derivative of the proposed model could match the experimental data and oil field data very well. With this model, the pressure oscillation in the pressure buildup test can be understood and predicted better. And the proposed model is a supplement to the phase redistribution model and yields an insight into the corresponding impact on well test transient pressure behavior.

Non-Darcian flow toward a larger-diameter partially penetrating well in a confined aquifer

In this study, non-Darcian flow to a larger-diameter partially penetrating well in a confined aquifer was investigated. The flow in the horizontal direction was assumed to be non-Darcian and described by the Izbash equation, and the flow in the vertical direction was assumed to be Darcian. A linearization procedure was used to approximate the nonlinear governing equation. The Laplace transform associated with the finite cosine Fourier transform was used to solve such non-Darcian flow model. Both the drawdowns inside the well and in the aquifer were analyzed under different conditions. The results indicated that the drawdowns inside the well were generally the same at early times under different conditions, and the features of the drawdowns inside the well at late times were similar to those of the drawdowns in the aquifer. The drawdown in the aquifer for the non-Darcian flow case was larger at early times and smaller at late times than their counterparts of Darcian flow case. The drawdowns for a partially penetrating well were the same as those of a fully penetrating well at early times, and were larger than those for a fully penetrating well at late times. A longer well screen resulted in a smaller drawdown in the aquifer at late times. A larger power index n in the Izbash equation resulted in a larger drawdown in the aquifer at early times and led to a smaller drawdown in the aquifer at late times. A larger well radius led to a smaller drawdown at early times, but it had little impact on the drawdown at late times. The wellbore storage effect disappears earlier when n is larger.

Analysis of radial composite systems based on fractal theory and fractional calculus

This paper analyzes the mathematical modeling of a two-region composite reservoir using the concepts of fractal geometry and fractional calculus. Heterogeneity of the reservoir is considered based on fractal geometry. Fractional calculus is used to consider production history in fractal reservoirs. An analytical solution is derived for the pressure-transient behavior of a well in a radial composite system when wellbore storage and skin effects are significant. Some new type-curves are developed under three outer boundary conditions: infinite, closed and constant-pressure. These new type curves can be used to analyze well-tests from a variety of enhanced oil recovery projects, geothermal reservoirs and acidization projects, more accurately.

Determination of minimum permeability plateau and characteristic length in porous media with non-Darcy flow behavior

Abstract A mathematical model has been formulated to determine the minimum permeability plateau and characteristic length in porous media with non-Darcy flow behaviour. More specifically, two new parameters are first defined to respectively include effect of minimum permeability plateau and characteristic length on non-Darcy flow behaviour across the completion with consideration of wellbore storage and skin effects. The newly formulated mathematical model is solved by discretizing the convolution integral and material balance equation, and then validated by simplifying it to the traditional Forchheimer model. The pressure together with its corresponding derivative type curves for both buildup and drawdown tests can be reproduced to examine effects of both wellbore storage and non-Darcy flow across the completion. The pressure derivative response with non-Darcy flow across the completion has been found to have a steep slope since the non-Darcy effect is increased after the wellbore-storage-dominated period. Such steep derivatives resulting from the increased non-Darcy effect cannot be observed from the existing models based on Darcy׳s law. In particular, the non-Darcy parameter depending on flow rate is found to dominate the shape of the derivative curves. Compared to the drawdown derivative curves, there exists a larger gap in the buildup derivative curves, indicating that buildup tests are more appropriate than drawdown tests for type curve matching non-Darcy flow behaviour.

Numerical solution of the nonlinear diffusivity equation in heterogeneous reservoirs with wellbore phase redistribution

Abstract We consider the application of the Finite Element Method (FEM) for numerical pressure transient analysis under conditions where no reliable analytical solution is available. Pressure transient analysis is normally based on various analytical solutions of the linear one-dimensional diffusion equation under restrictive assumptions about the formation and its boundaries. For example, the formation is either assumed isotropic or a restrictive a priori assumption is made about its heterogeneity. The wellbore storage effect is also often considered without regard to the possibility of phase redistribution. In many practical situations such restrictions are not justified and analytical solutions do not exist. Here we present a numerical solution of the nonlinear diffusion equation based on the FEM that can be used without any restrictive a priori assumptions. Through the use of the weak formulation of the FEM, solution can be obtained for a heterogeneous medium with discontinuous or nonlinear properties. The weak formulation also enables the handling of time dependent boundary conditions and hence problems involving wellbore storage with significant phase redistribution. The speed and accuracy of the numerical technique is first confirmed by comparison with simple test cases that admit an analytical solution. The practical utility of the proposed method is then demonstrated for a number of test cases that involve discontinuous and nonlinear formation properties and/or wellbore storage with phase redistribution.

Explicit Deconvolution of Well Test Data Dominated by Wellbore Storage

This paper addresses some methods for interpretation of oil and gas well test data distorted by wellbore storage effects. Using these techniques, we can deconvolve pressure and rate data from drawdown and buildup tests dominated by wellbore storage. Some of these methods have the advantage of deconvolving the pressure data without rate measurement. The two important methods that are applied in this study are an explicit deconvolution method and a modification of material balance deconvolution method. In cases with no rate measurements, we use a blind deconvolution method to restore the pressure response free of wellbore storage effects. Our techniques detect the afterflow/unloading rate function with explicit deconvolution of the observed pressure data. The presented techniques can unveil the early time behavior of a reservoir system masked by wellbore storage effects and thus provide powerful tools to improve pressure transient test interpretation. Each method has been validated using both synthetic data and field cases and each method should be considered valid for practical applications.

Characteristic Value Method of Well Test Analysis for Horizontal Gas Well

This paper presents a study of characteristic value method of well test analysis for horizontal gas well. Owing to the complicated seepage flow mechanism in horizontal gas well and the difficulty in the analysis of transient pressure test data, this paper establishes the mathematical models of well test analysis for horizontal gas well with different inner and outer boundary conditions. On the basis of obtaining the solutions of the mathematical models, several type curves are plotted with Stehfest inversion algorithm. For gas reservoir with closed outer boundary in vertical direction and infinite outer boundary in horizontal direction, while considering the effect of wellbore storage and skin effect, the pseudopressure behavior of the horizontal gas well can manifest four characteristic periods: pure wellbore storage period, early vertical radial flow period, early linear flow period, and late horizontal pseudoradial flow period. For gas reservoir with closed outer boundary both in vertical and horizontal directions, the pseudopressure behavior of the horizontal gas well adds the pseudosteady state flow period which appears after the boundary response. For gas reservoir with closed outer boundary in vertical direction and constant pressure outer boundary in horizontal direction, the pseudopressure behavior of the horizontal gas well adds the steady state flow period which appears after the boundary response. According to the characteristic lines which are manifested by pseudopressure derivative curve of each flow period, formulas are developed to obtain horizontal permeability, vertical permeability, skin factor, reservoir pressure, and pore volume of the gas reservoir, and thus the characteristic value method of well test analysis for horizontal gas well is established. Finally, the example study verifies that the new method is reliable. Characteristic value method of well test analysis for horizontal gas well makes the well test analysis process more simple and the results more accurate.

Pressure Data Analysis and Multilayer Modeling of a Gas-Condansate Reservoir

Kailastila gas field located at Golapgonj, Sylhet is one of the largest gas fields in Bangladesh. It produces a high amount of condensate along with natural gas. For the high values of GOR, it may be treated as a wet gas at reservoir condition. Three main sand reservoirs are confirmed in this field (upper, middle & lower).In this study, it has been shown that reservoir parameters of this gas field can be obtained for multilayered rectangular reservoir with formation cross-flow using pressure and their semi log derivative on a set of dimensionless type curve.The effects of the reservoir parameters such as permeability, skin, storage coefficient, and others such as reservoir areal extent and layering on the wellbore response, pressure are investigated.Shut in pressures are used in calculating permeability, skin factor, average reservoir pressure, wellbore storage effect and other reservoir properties. The direction of the formation cross flow is determined, first by the layer permeability and later by the skin factor.Finally, it is recommended to perform diagnostic analysis along with multilayer modeling to extract better results.Reservoir can also be considered as a multilayer cylindrical and can also use these models for other fields.

Well Test Analysis of Naturally Fractured Vuggy Reservoirs with an Analytical Triple Porosity – Double Permeability Model and a Global Optimization Method

The aim of this work is to study the automatic characterization of Naturally Fractured Vuggy Reservoirs <i>via<i/> well test analysis, using a triple porosity-dual permeability model. The inter-porosity flow parameters, the storativity ratios, as well as the permeability ratio, the wellbore storage effect, the skin and the total permeability will be identified as parameters of the model. In this work, we will perform the well test interpretation in Laplace space, using numerical algorithms to transfer the discrete real data given in fully dimensional time to Laplace space. The well test interpretation problem in Laplace space has been posed as a nonlinear least squares optimization problem with box constraints and a linear inequality constraint, which is usually solved using local Newton type methods with a trust region. However, local methods as the one used in our work called TRON or the well-known Levenberg-Marquardt method, are often not able to find an optimal solution with a good fit of the data. Also well test analysis with the triple porosity-double permeability model, like most inverse problems, can yield multiple solutions with good match to the data. To deal with these specific characteristics, we will use a global optimization algorithm called the Tunneling Method (TM). In the design of the algorithm, we take into account issues of the problem like the fact that the parameter estimation has to be done with high precision, the presence of noise in the measurements and the need to solve the problem computationally fast. We demonstrate that the use of the TM in this study, showed to be an efficient and robust alternative to solve the well test characterization, as several optimal solutions, with very good match to the data were obtained.

Numerical well testing interpretation model and applications in crossflow double-layer reservoirs by polymer flooding.

This work presents numerical well testing interpretation model and analysis techniques to evaluate formation by using pressure transient data acquired with logging tools in crossflow double-layer reservoirs by polymer flooding. A well testing model is established based on rheology experiments and by considering shear, diffusion, convection, inaccessible pore volume (IPV), permeability reduction, wellbore storage effect, and skin factors. The type curves were then developed based on this model, and parameter sensitivity is analyzed. Our research shows that the type curves have five segments with different flow status: (I) wellbore storage section, (II) intermediate flow section (transient section), (III) mid-radial flow section, (IV) crossflow section (from low permeability layer to high permeability layer), and (V) systematic radial flow section. The polymer flooding field tests prove that our model can accurately determine formation parameters in crossflow double-layer reservoirs by polymer flooding. Moreover, formation damage caused by polymer flooding can also be evaluated by comparison of the interpreted permeability with initial layered permeability before polymer flooding. Comparison of the analysis of numerical solution based on flow mechanism with observed polymer flooding field test data highlights the potential for the application of this interpretation method in formation evaluation and enhanced oil recovery (EOR).

Laplace-Transform Finite-Difference and Quasistationary Solution Method for Water-Injection/Falloff Tests

Summary In this work, we present a method for efficiently and accurately simulating the pressure-transient behavior of oil/water flow associated with water-injection/falloff tests. The method uses the Laplace-transform finite-difference (LTFD) method coupled with the well-known Buckley-Leverett frontal-advance formula to solve the radial diffusivity equation describing slightly compressible oil/water two-phase flow. The method is semianalytical in time, and as a result, the issue of time discretization in the finite-difference approximation method is eliminated. Thus, stability and convergence problems caused by time discretization are avoided. Two approaches are presented and compared in terms of accuracy for simulating the tests with multiple-rate injection and falloff periods: One is based on solving the initial-boundary-value (IVB) problem with the initial condition attained from the end of the previous flow period, and the other is based on the conventional superposition on the basis of the single-phase flow of a slightly compressible fluid. The former is shown to always provide a more accurate and efficient solution. The method is quite general in that it allows one to incorporate the effect of wellbore storage and thick-skin and finite outer-boundary conditions. The accuracy of the method was evaluated by considering various synthetic test cases with favorable and unfavorable mobility ratios and by comparing the pressure and pressure-derivative signatures with a commercial black-oil simulator, and an excellent agreement was seen.

Type Curves Analysis for Asymmetrically Fractured Wells

In this paper, a new constant rate solution for asymmetrically fractured wells was proposed to analyze the effect of fracture asymmetry on type curves. Calculative results showed that for a small wellbore storage coefficient or for the low fracture conductivity, the effect of fracture asymmetry on early flow was very strong. The existence of the fracture asymmetry would cause bigger pressure depletion and make the starting time of linear flow occur earlier. Then, new type curves were established for different fracture asymmetry factor and different fracture conductivity. It was shown that a bigger fracture asymmetry factor and low fracture conductivity would prolong the time of wellbore storage effects. Therefore, to reduce wellbore storage effects, it was essential to keep higher fracture conductivity and fracture symmetry during the hydraulic fracturing design. Finally, a case example is performed to demonstrate the methodology of new type curves analysis and its validation for calculating important formation parameters.

Similar Constructing Method for Solving the Model of the Plane Radial Flow of Dual Permeability Reservoir

A seepage model of producing at a constant rate, based on the three outer boundary (infinite, closed, constant value) conditions and regardless of the well-bore storage and skin effects, is established for the problem of the plane radial flow of dual permeability reservoirs. Firstly, we get dimensionless model by introducing the dimensionless variables. Second, we obtain a boundary value problem of ordinary differential equation in the Laplace space by using the Laplace transformation. Finally, we prove that the solution to the boundary value problem has similar structural formula. Therefore a new method for solving such seepage model is obtainedSimilar Constructive Method (shortened as SCM). And then, according to the modified numerical inversion formula of Stehfest, we draw the curves of bottom-hole under the three kinds of outer boundary conditions by using MATLAB.

Minidrillstem Tests To Characterize Formation Deliverability in the Bakken

Summary Characterization of reservoir deliverability is fundamental for the economic development of any field. In the Bakken, there is a need for reliable pressure-transient tests to provide the effective formation permeability of the fracture/matrix system in this formation for deliverability calculations. This effective permeability can then be compared with laboratory-measured core permeability of the matrix rock samples. This comparison is the basis for planning early-production options and subsequent enhanced-oil-recovery (EOR) decisions. This comparison is particularly important because of the influence of massive hydraulic-fracture stimulation on reservoir performance. Determining well-deliverability potential by conventional drill-stem tests (DSTs) or traditional wireline formation tests (WFTs) in the past has resulted in mixed success in the Bakken. On the other hand, the mini-DST has increased the reliability of pressure-transient tests because the formation interval of interest is produced at a precise constant rate into a chamber between two inflated packers with negligible wellbore-storage effects. Also, the operation of a mini-DST tool requires much less time than the classical DST, and multiple zones can be tested sequentially to assess the individual-zone deliverability. The mini-DST tool uses the conventional WFT configured with a dual-packer module and downhole pump. Tests are conducted by inflating the dual-packer module to isolate a 3-ft interval of the wellbore. The formation fluid is pumped out from the packer-isolated wellbore interval to conduct pressure-drawdown and -buildup tests in the interval. An overlay of all the pressure-drawdown and -buildup results from various intervals is compared on a single plot to identify the most productive interval. Finally, conventional pressure-transient analyses are performed to interpret all pressure-drawdown and -buildup tests. In this paper, we present several field tests that were analyzed both by the preceding procedure and by numerical simulation. The analyses of several mini-DST results have provided insight into a better understanding of the flow mechanism in the Bakken both during primary production and in forecasting various improved-recovery and EOR proposals. The results can also serve as a basis for test design in similar low-permeability reservoirs elsewhere.