scholarly journals A mathematical model and semi-analytical solution for transient pressure of vertical fracture with varying conductivity in three crossflow rectangular layers

2018 ◽  
Vol 37 (1) ◽  
pp. 230-250 ◽  
Author(s):  
Jie Liu ◽  
Pengcheng Liu ◽  
Shunming Li ◽  
Xiaodong Wang
Keyword(s):  
Mathematical Model ◽  
Analytical Solution ◽  
Flow Characteristics ◽  
Well Test ◽  
Transient Pressure ◽  
Well Test Analysis ◽  
Test Analysis ◽  
Fracture Conductivity ◽  
Vertical Fracture ◽  
Flow Phase

This paper first describes a mathematical model of a vertical fracture with constant conductivity in three crossflow rectangular layers. Then, three forms of vertical fracture (linear, logarithmic, and exponential variations) with varying conductivity are introduced to this mathematical model. A novel mathematical model and its semi-analytical solution of a vertical fracture with varying conductivity intercepting a three-separate-layered crossflow reservoir is developed and executed. Results show that the transient pressures are divided into three stages: the linear-flow phase, the medium unsteady-flow stage, and the later pseudo-steady-flow phase. The parameters of the fracture, reservoir, and the multi-permeability medium directly influence the direction, transition, and shape of the transient pressure. Meanwhile, the fracture conductivity is higher near the well bottom and is smaller at the tip of the fracture for the varying conductivity. Therefore, there are many more differences between varying conductivity and constant conductivity. Varying conductivity can correctly reflect the flow characteristics of a vertical fractured well during well-test analysis.

Condensate Bank Characterization from Well Test Data and Fluid PVT Properties

2006 ◽  
Vol 9 (05) ◽  
pp. 596-611 ◽  
Author(s):  
Manijeh Bozorgzadeh ◽  
Alain C. Gringarten
Keyword(s):  
Capillary Number ◽  
Gas Condensate ◽  
Test Interpretation ◽  
Well Test ◽  
Transient Pressure ◽  
Compositional Simulation ◽  
Well Test Analysis ◽  
Test Analysis ◽  
Inertia Effects ◽  
The North

Summary Published well-test analyses in gas/condensate reservoirs in which the pressure has dropped below the dewpoint are usually based on a two- or three-region radial composite well-test interpretation model to represent condensate dropout around the wellbore and initial gas in place away from the well. Gas/condensate-specific results from well-test analysis are the mobility and storativity ratios between the regions and the condensate-bank radius. For a given region, however, well-test analysis cannot uncouple the storativity ratio from the region radius, and the storativity ratio must be estimated independently to obtain the correct bank radius. In most cases, the storativity ratio is calculated incorrectly, which explains why condensate bank radii from well-test analysis often differ greatly from those obtained by numerical compositional simulation. In this study, a new method is introduced to estimate the storativity ratios between the different zones from buildup data when the saturation profile does not change during the buildup. Application of the method is illustrated with the analysis of a transient-pressure test in a gas/condensate field in the North Sea. The analysis uses single-phase pseudo pressures and two- and three-zone radial composite well-test interpretation models to yield the condensate-bank radius. The calculated condensate-bank radius is validated by verifying analytical well-test analyses with compositional simulations that include capillary number and inertia effects. Introduction and Background When the bottomhole flowing pressure falls below the dewpoint in a gas/condensate reservoir, retrograde condensation occurs, and a bank of condensate builds up around the producing well. This process creates concentric zones with different liquid saturations around the well (Fevang and Whitson 1996; Kniazeff and Nvaille 1965; Economides et al. 1987). The zone away from the well, where the reservoir pressure is still above the dewpoint, contains the original gas. The condensate bank around the wellbore contains two phases, reservoir gas and liquid condensate, and has a reduced gas mobility, except in the immediate vicinity of the well at high production rates, where the relative permeability to gas is greater than in the bank because of capillary number effects (Danesh et al. 1994; Boom et al. 1995; Henderson et al. 1998; Mott et al. 1999).

Analysis of Horizontal-Well Responses: Contemporary vs. Conventional

2001 ◽  
Vol 4 (04) ◽  
pp. 260-269 ◽  
Author(s):  
Erdal Ozkan
Keyword(s):  
Horizontal Well ◽  
Horizontal Wells ◽  
Well Test ◽  
Transient Pressure ◽  
Vertical Wells ◽  
Well Test Analysis ◽  
Test Analysis ◽  
Response Models ◽  
In The Beginning ◽  
Well Models

Summary Most of the conventional horizontal-well transient-response models were developed during the 1980's. These models visualized horizontal wells as vertical wells rotated 90°. In the beginning of the 1990's, it was realized that horizontal wells deserve genuine models and concepts. Wellbore conductivity, nonuniform skin effect, selective completion, and multiple laterals are a few of the new concepts. Although well-established analysis procedures are yet to be developed, some contemporary horizontal-well models are now available. The contemporary models, however, are generally sophisticated. The basic objective of this paper is to answer two important questions:When should we use the contemporary models? andHow much error do we make by using the conventional models? This objective is accomplished by considering examples and comparing the results of the contemporary and conventional approaches. Introduction Since the early 1980's, horizontal wells have been extremely popular in the oil industry and have gained an impeccable standing among the conventional well completions. The rapid increase in the applications of horizontal-well technology brought an impetuous development of the procedures to evaluate the performances of horizontal wells. These procedures, however, used the vertical-well concepts almost indiscriminately to analyze the horizontal-well transient-pressure responses.1–14 Among these concepts were 1) the assumptions of a line-source well and an infinite-conductivity wellbore, 2) a single lateral withdrawing fluids along its entire length, and 3) a skin region that is uniformly distributed along the well. It should be realized that for the lengths, production rates, and configurations of horizontal wells drilled in the 1980's, these concepts were usually justifiable. The increased lengths of horizontal wells, high production rates, sectional and multilateral completions, and the vast variety of other new applications toward the end of the 1980's made us question the validity of the horizontal-well models and the well-test concepts adopted from vertical wells. The interest in improved horizontal-well models also flourished on the grounds of high productivities of horizontal wells. It was realized that, in many cases, a few percent of the production rate of a reasonably long horizontal well could amount to the cumulative production rate of a few vertical wells. In addition, the productivity-reducing effects were additive; that is, a slight reduction in the productivity here and there could add up to a sizeable loss of the well's production capacity. Furthermore, the low oil prices also created an economic environment where the marginal gains and losses in the productivity may decisively affect the economics of many projects. In the beginning of the 1990's, a new wave of developing horizontal-well solutions under more realistic conditions gained impetus.15–25 As a result, some contemporary models are available today for those who want to challenge the limitations of the conventional horizontal-well models. Unfortunately, the rigor is accomplished at the expense of complexity. Furthermore, even when a rigorous model is available, well-established analysis procedures are usually yet to be developed. This paper presents a critique of the conventional and contemporary horizontal well-test-analysis procedures. The main objective of this assessment is to answer the two fundamental questions horizontal-well-test analysts are currently facing:When is the use of contemporary analysis methods essential? andIf the conventional analysis methods are used, what are the margins of error? Background: The Conventional Methods The standard models of horizontal-well-test analysis have been developed mostly during the 1980's.1-4,8,9 Despite the differences in the development of these models, the basic assumptions and the final solutions are similar. Fig. 1 is a sketch of the horizontal well-reservoir system considered in the pressure-transient-response models. A horizontal well of length Lh is assumed to be located in an infinite slab reservoir of thickness h. The elevation of the horizontal well from the bottom boundary of the formation (well eccentricity) is denoted by zw. The top and bottom reservoir boundaries are usually assumed to be impermeable, although some models consider constant-pressure boundaries.14,15 Before discussing the characteristic features of the conventional horizontal-well transient-pressure-response models, we must first define the dimensionless variables to be used in our discussion. We define the dimensionless pressure, time, and distance in the conventional manner except that we use the horizontal-well half-length, Lh/2, as the reference length in the system. These variables are defined, respectively, by the following expressions.Equation 1Equation 2Equation 3Equation 4 In Eqs. 1 through 3, k=the harmonic average of the principal permeabilities that are assumed to be in the directions of the coordinate axes (). We also define the dimensionless horizontal-well length, wellbore radius, and well eccentricity (distance from the bottom boundary of the formation) as follows.Equation 5Equation 6Equation 7 In Eq. 6, rw, eq=the equivalent radius of the horizontal well in an anisotropic reservoir.26

Reservoir Characterization Constrained to Well-Test Data: A Field Example

2000 ◽  
Vol 3 (04) ◽  
pp. 325-334 ◽  
Author(s):  
J.L. Landa ◽  
R.N. Horne ◽  
M.M. Kamal ◽  
C.D. Jenkins
Keyword(s):  
Mathematical Model ◽  
Test Data ◽  
Pressure Response ◽  
Well Testing ◽  
Pressure Measurements ◽  
Well Test ◽  
Well Test Analysis ◽  
Test Analysis ◽  
Reservoir Description ◽  
Different Sources

Summary In this paper we present a method to integrate well test, production, shut-in pressure, log, core, and geological data to obtain a reservoir description for the Pagerungan field, offshore Indonesia. The method computes spatial distributions of permeability and porosity and generates a pressure response for comparison to field data. This technique produced a good match with well-test data from three wells and seven shut-in pressures. The permeability and porosity distributions also provide a reasonable explanation of the observed effects of a nearby aquifer on individual wells. As a final step, the method is compared to an alternate technique (object modeling) that models the reservoir as a two-dimensional channel. Introduction The Pagerungan field has been under commercial production since 1994. This field was chosen to test a method of integrating dynamic well data and reservoir description data because the reservoir has only produced single phase gas, one zone in the reservoir is responsible for most of the production, and good quality well-test, core, and log data are available for most wells. The method that was used to perform the inversion of the spatial distribution of permeability and porosity uses a parameter estimation technique that calculates the gradients of the calculated reservoir pressure response with respect to the permeability and porosity in each of the cells of a reservoir simulation grid. The method is a derivative of the gradient simulator1 approach and is described in Appendices A and B. The objective is to find sets of distributions of permeability and porosity such that the calculated response of the reservoir closely matches the pressure measurements. In addition, the distributions of permeability and porosity must satisfy certain constraints given by the geological model and by other information known about the reservoir. Statement of Theory and Definitions The process of obtaining a reservoir description involves using a great amount of data from different sources. It is generally agreed that a reservoir description will be more complete and reliable when it is the outcome of a process that can use the maximum possible number of data from different sources. This is usually referred to in the literature as "data Integration." Reservoir data can be classified as "static" or "dynamic" depending on their connection to the movement or flow of fluids in the reservoir. Data that have originated from geology, logs, core analysis, seismic and geostatistics can be generally classified as static; whereas the information originating from well testing and the production performance of the reservoir can be classified as dynamic. So far, most of the success in data integration has been obtained with static information. Remarkably, it has not yet become common to completely or systematically integrate dynamic data with static data. A number of researchers,2–5 are studying this problem at present. This work represents one step in that direction. Well Testing as a Tool for Reservoir Description. Traditional well-test analysis provides good insight into the average properties of the reservoir in the vicinity of a well. Well testing can also identify the major features of relatively simple reservoirs, such as faults, fractures, double porosity, channels, pinchouts, etc. in the near well area. The difficulties with this approach begin when it is necessary to use the well-test data on a larger scale, such as in the context of obtaining a reservoir description. One of the main reasons for these difficulties is that traditional well-test analysis handles transient pressure data collected at a single well at a time, and is restricted to a small time range. As a result, traditional well-test analysis does not make use of "pressure" events separated in historical time. The use of several single and multiple well tests to describe reservoir heterogeneity has been reported in the literature,6 however, this approach is not applied commonly because of the extensive efforts needed to obtain a reservoir description. The method presented in this paper uses a numerical model of the reservoir to overcome these shortcomings. It will be shown that pressure transients can be used effectively to infer reservoir properties at the scale of reservoir description. Well-test data, both complete tests and occasional spot pressure measurements, will be used to this effect. The well-test information allows us to infer properties close to the wells and, when combined with the shut-in pressures (spot pressure), boundary information and permeability-porosity correlations, provides the larger scale description. General Description of the Method The proposed method is similar to other parameter estimation methods and thus consists of the following major items: the mathematical model, the objective function and the minimization algorithm. Mathematical Model. Because of the complexity of the reservoir description, the reservoir response must be computed numerically. Therefore, the pressure response is found using a numerical simulator. The reservoir is discretized into blocks. The objective is to find a suitable permeability-porosity distribution so that values of these parameters can be assigned to each of the blocks.

Semianalytical Modeling of Complex-Geometry Reservoirs

2002 ◽  
Vol 5 (06) ◽  
pp. 437-446 ◽  
Author(s):  
Gang Zhao ◽  
Leslie G. Thompson
Keyword(s):  
Analytical Solutions ◽  
Complex Geometry ◽  
Well Test ◽  
Transient Pressure ◽  
Well Test Analysis ◽  
Test Analysis ◽  
Simple Geometry ◽  
Reservoir Systems ◽  
Fluid Transfer ◽  
Source Sink

Summary Complex geometry reservoirs can be encountered in the field for a variety of depositional and tectonic processes. For example, fluvial depositional environments may produce interbranching channel reservoirs or reservoirs consisting of relatively high-permeability channels in communication with low-permeability splays. This paper presents a general methodology for computing pressure responses and flow characteristics in complex geometry reservoirs. The proposed method consists of decomposing the original complex-geometry reservoir into a set of simple-geometry reservoirs, which interact with each other by transfer of fluid and equality of pressure over the regions where they are in hydraulic contact. Analytical solutions are written for each of the simple reservoir components in terms of the unknown pressures and fluxes at their boundaries, and the coupled systems are solved for the desired wellbore pressure responses. The method of sources and sinks is used to compute the pressure response in the Laplace domain, and the results are inverted numerically with the Stehfest Inversion algorithm.1 We present fast, accurate methods of taking numerical Laplace transforms of the source/sink solutions that make the computations reasonably fast and efficient. The proposed methodology can be extended to any system (infinite or bounded) in which the Laplace-space solution can be written easily in terms of integrals of real-space source/sink functions, including production at constant bottomhole pressure, wellbore storage effects, or naturally fractured systems. We demonstrate the applicability of the method by modeling branching channels and channel/splay systems. Introduction Classical well-test analysis has long been used as a valuable tool in characterizing reservoirs using transient pressure vs. time behavior. In most classical well-test models, the reservoir is idealized as a homogeneous single- or dual-porosity system with a simple reservoir and well geometry. This is done to facilitate generation of analytical solutions to the reservoir problem. In fluvial-deltaic reservoir systems, however, the complex geometry precludes idealizing the reservoir as a simple-shaped homogeneous system and, in general, the transient pressure responses do not resemble those of classical simple-geometry systems. Fluvial-deltaic reservoir systems are one example of general complex-geometry reservoirs for which classical well-test analysis models may not be applicable. Except for work presented by Larsen2,3 on a network of interconnected linear reservoirs, there appears to have been very little presented in the literature on modeling or describing the pressure transient pressure behavior of these types of reservoirs. Larsen's work was concerned primarily with the long-time productivity of a network of intersecting linear reservoirs, with no special consideration given to the geometry at the regions of intersection. Larsen's work indicated that proper understanding of reservoir type is critical in situations in which extrapolation of short-time test data to possible late-time production characteristics is attempted. Reservoirs with sealing or partially sealing faults are another type of complex-geometry reservoirs that has received attention in the literature.4–9 Both numerical and analytical solutions have been presented to model the pressure behavior in faulted reservoirs; however, the focus of these studies has been on the effect of the sealing and/or nonsealing faults, with little attention given to the physics of the fluid transfer between the communicating reservoirs. In this work, we attempt to rigorously account for the effects of the hydraulic contact between the connected reservoir components. Modeling Philosophy and Methodology In this section, we illustrate our general modeling approach by applying it to a simple reservoir system. The physical model considered in Fig. 1 consists of two semi-infinite reservoirs separated over most of their extent by a hydraulically sealing barrier. One has flow properties of a "channel," which we will refer to as Region 1, while the other has properties of a "splay" (Region 2). The reservoirs are in hydraulic contact only over a small area. A well is producing from one of the reservoirs. The first step in the modeling procedure is to identify simplegeometry homogeneous reservoir components that make up the complex system. In the preceding example, the complex reservoir is composed of two semi-infinite reservoirs: Region 1 and Region 2. Considering Region 1 only, the reservoir behaves as if it contains two wells: the original producer and a planar injection well at the area of communication between the two reservoirs. Similarly, considering Region 2 only, it behaves as if there were a single producing planar fractured well at the area of communication between the two reservoirs. Because all the fluid leaving Region 2 is entering Region 1, the production and injection rates of the apparent planar "fractured" wells in each of the regions must be the same. The second step in the modeling process involves separating the various reservoir components from one another at their junction(s) by applying no-flow boundaries. For simple reservoir systems that contain wells, this is achieved by creating a mirror image to the no-flow boundary; that is, the inserted no-flow boundary becomes a plane of symmetry for the extended system. For the problem under consideration, Region 1 and Region 2 are each reflected using the fault as a mirror (see Figs. 2 and 3). Planar injection or production wells are added along the plane of symmetry to account for fluid transfer from one system to the other. The separate systems are then coupled by equating rates and pressures at the junctions and solving (in Laplace space) for the unknown flux distributions along the introduced planar wells. Finally, the flux profiles from the preceding step are substituted into the pressure equation for the well of interest to obtain the desired pressure response. We illustrate the procedure in more detail for the simple problem under consideration.

Model and method of well test analysis for wells with vertical fracture

2005 ◽  
Vol 26 (5) ◽  
pp. 571-578 ◽  
Author(s):  
Guo Da-li ◽  
Zeng Xiao-hui ◽  
Zhao Jin-zhou ◽  
Liu Ci-qun
Keyword(s):  
Well Test ◽  
Well Test Analysis ◽  
Test Analysis ◽  
Vertical Fracture

scholarly journals Derivation and application of mathematical model for well test analysis with variable skin factor in hydrocarbon reservoirs

AIP Advances ◽  
2016 ◽  
Vol 6 (6) ◽  
pp. 065324 ◽  
Author(s):  
Pengcheng Liu ◽  
Wenhui Li ◽  
Jing Xia ◽  
Yuwei Jiao ◽  
Aifang Bie
Keyword(s):  
Mathematical Model ◽  
Hydrocarbon Reservoirs ◽  
Well Test ◽  
Well Test Analysis ◽  
Test Analysis ◽  
Skin Factor

scholarly journals Analysis of Pressure and Production Transient Characteristics of Composite Reservoir with Moving Boundary

Energies ◽  
2019 ◽  
Vol 13 (1) ◽  
pp. 34
Author(s):  
Zhixue Sun ◽  
Xugang Yang ◽  
Yanxin Jin ◽  
Shubin Shi ◽  
Minglu Wu
Keyword(s):  
Mathematical Model ◽  
Oil Recovery ◽  
Moving Boundary ◽  
Initial Boundary ◽  
Well Test ◽  
Well Test Analysis ◽  
Test Analysis ◽  
Wellbore Storage ◽  
Boundary Velocity ◽  
The Mathematical Model

The mathematical model of composite reservoir has been widely used in well test analysis. In the process of oil recovery, due to the injection or replacement of the displacement agent, the model boundary can be moved. At present, the mathematical model of a composite reservoir with a moving boundary is less frequently studied and cannot meet industrial demand. In this paper, a mathematical model of a composite reservoir with a moving boundary is developed, with consideration of wellbore storage and skin effects. The characteristics of pressure transient in moving boundary composite reservoir are studied, and the influences of parameters, such as initial boundary radius, moving boundary velocity, skin factor, wellbore storage coefficient, diffusion coefficient ratio, and mobility ratio on pressure and production, are analyzed. The moving boundary effects are noticeable mainly in the middle and late production stages. The proposed model provides a novel theoretical basis for well test analysis in these types of reservoirs.

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