scholarly journals A Single Equation to Depict Bottomhole Pressure Behavior for a Uniform Flux Hydraulic Fractured Well

2022 ◽  
Vol 12 (2) ◽  
pp. 817
Author(s):  
Jang Hyun Lee ◽  
Juhairi Aris Bin Muhamad Shuhili
Keyword(s):  
Radial Flow ◽  
Early Stage ◽  
Error Function ◽  
Well Test ◽  
Transition Flow ◽  
Linear Flow ◽  
Pressure Transient ◽  
Flow Equations ◽  
Transition Flow Regime ◽  
Fractured Well

Pressure transient analysis for a vertically hydraulically fractured well is evaluated using two different equations, which cater for linear flow at the early stage and radial flow in the later stage. However, there are three different stages that take place for an analysis of pressure transient, namely linear, transition and pseudo-radial flow. The transition flow regime is usually studied by numerical, inclusive methods or approximated analytically, for which no specific equation has been built, using the linear and radial equations. Neither of the approaches are fully analytical. The numerical, inclusive approach results in separate calculations for the different flow regimes because the equation cannot cater for all of the regimes, while the analytical approach results in a difficult inversion process to compute well test-derived properties such as permeability. There are two types of flow patterns in the fracture, which are uniform and non-uniform, called infinite conductivity in a high conductivity fracture. The study was conducted by utilizing an analogous study of linear flow equations. Instead of using the conventional error function, the exponential integral with an infinite number of wells was used. The results obtained from the developed analytical solution matched the numerical results, which proved that the equation was representative of the case. In conclusion, the generated analytical equation can be directly used as a substitute for current methods of analyzing uniform flow in a hydraulically fractured well.

A Novel Infinite-Acting-Radial-Flow Analysis Procedure for Estimating Permeability Anisotropy From an Observation-Probe Pressure Response at a Vertical, Horizontal, or Inclined Wellbore

2014 ◽  
Vol 17 (02) ◽  
pp. 152-164 ◽  
Author(s):  
M.. Onur ◽  
P.S.. S. Hegeman ◽  
I.M.. M. Gök
Keyword(s):  
Radial Flow ◽  
Flow Analysis ◽  
Single Layer ◽  
Flow Equation ◽  
Observation Point ◽  
Analysis Procedure ◽  
Pressure Transient ◽  
Flow Equations ◽  
Vertical Permeability ◽  
Inclination Angles

Summary This paper presents a new infinite-acting-radial-flow (IARF) analysis procedure for estimating horizontal and vertical permeability solely from pressure-transient data acquired at an observation probe during an interval pressure-transient test (IPTT) conducted with a single-probe, dual-probe, or dual-packer module. The procedure is based on new infinite-acting-radial-flow equations that apply for all inclination angles of the wellbore in a single-layer, 3D anisotropic, homogeneous porous medium. The equations for 2D anisotropic cases are also presented and are derived from the general equations given for the 3D anisotropic case. It is shown that the radial-flow equation presented reduces to Prats' (1970) equation assuming infinite-acting radial flow at an observation point along a vertical wellbore in isotropic or 2D anisotropic formations of finite bed thickness. The applicability of the analysis procedure is demonstrated by considering synthetic and field packer/probe IPTT data. The synthetic IPTT examples include horizontal- and slanted-well cases, but the field IPTT is for a vertical well. The results indicate that the procedure provides reliable estimates of horizontal and vertical permeability solely from observation-probe pressure data during radial flow for vertical, horizontal, and inclined wellbores. Most importantly, the analysis does not require that both spherical and radial flow prevail at the observation probe during the test.

Unsteady-State Spherical Flow With Storage and Skin

1985 ◽  
Vol 25 (06) ◽  
pp. 804-822 ◽  
Author(s):  
Jeffrey A. Joseph ◽  
Leonard F. Koederitz
Keyword(s):  
Boundary Condition ◽  
Radial Flow ◽  
Linear Partial Differential Equation ◽  
Flow Profile ◽  
Unsteady State ◽  
Well Test ◽  
Linear Flow ◽  
Dimensionless Groups ◽  
Short Time ◽  
Suitable Expression

Abstract This paper presents short-time interpretation methods for radial-spherical (or radial-hemispherical) flow in homogeneous and isotropic reservoirs inclusive of wellbore storage, wellbore phase redistribution, and damage skin effects. New dimensionless groups are introduced to facilitate the classic transformation from radial flow in the sphere to linear flow in the rod. Analytical expressions, type curves (in log-log and semilog format), and tabulated solutions are presented, both in terms of pressure and rate, for all flow problems considered. A new empirical equation to estimate the duration of wellbore and near-wellbore effects under spherical flow is also proposed. Introduction The majority of the reported research on unsteady-state flow theory applicable to well testing usually assumes a cylindrical (typically a radial-cylindrical) flow profile because this condition is valid for many test situations. Certain well tests, however, are better modeled by assuming a spherical flow symmetry (e.g., wireline formation testing, vertical interference testing, and perhaps even some tests conducted in wellbores that do not fully penetrate the productive horizon or are selectively penetrate the productive horizon or are selectively completed). Plugged perforations or blockage of a large part of an openhole interval may also promote spherical flow. Numerous solutions are available in the literature for almost every conceivable cylindrical flow problem; unfortunately, the companion spherical problem has not received as much attention, and comparatively few papers have been published on this topic. papers have been published on this topic. The most common inner boundary condition in well test analysis is that of a constant production rate. But with the advent of downhole tools capable of the simultaneous measurement of pressures and flow rates, this idealized inner boundary condition has been refined and more sophisticated models have been proposed. Therefore, similar methods must be developed for spherical flow analysis, especially for short-time interpretations. This general problem has recently been addressed elsewhere. Theory The fundamental linear partial differential equation (PDE) describing fluid flow in an infinite medium characterized by a radial-spherical symmetry is (1) The assumptions incorporated into this diffusion equation are similar to those imposed on the radial-cylindrical diffusivity equation and are discussed at length in Ref. 9. In solving Eq. 1, the classic approach is illustrated by Carslaw and Jaeger (later used by Chatas, and Brigham et al.). According to Carslaw and Jaeger, mapping b=pr will always reduce the problem of radial flow in the sphere (Eq. 1) to an equivalent problem of linear flow in the rod for which general solutions are usually known. (For example, see Ref. 17 for particular solutions in petroleum applications.) Note that in this study, we assumed that the medium is spherically isotropic; hence k in Eq. 1 is the constant spherical permeability. This assumption, however, does not preclude analysis in systems possessing simple anisotropy (i.e., uniform but unequal horizontal and vertical permeability components). In this case, k as used in this paper should be replaced by k, an equivalent or average (but constant) spherical permeability. Chatas presented a suitable expression (his Eq. 10) obtained presented a suitable expression (his Eq. 10) obtained from a volume integral. It is desirable to transform Eq. 1 to a nondimensional form, thereby rendering its applicability universal. The following new, dimensionless groups accomplish this and have the added feature that solutions are obtained directly in terms of the dimensionless pressure drop, PD, not the usual b (or bD) groups. ......................(2) .......................(3) .........................(4) The quantity rsw is an equivalent or pseudospherical wellbore radius used to represent the actual cylindrical sink (or source) of radius rw. SPEJ p. 804

Pressure Transient Analysis for Fracture-Cavity Carbonate Reservoirs with Large Scale Fractures-Caves in Series Connection

2021 ◽  
pp. 1-20
Author(s):  
Cuiqiao Xing ◽  
Hongjun Yin ◽  
Hongfei Yuan ◽  
Jing Fu ◽  
Guohan Xu
Keyword(s):  
Transient Analysis ◽  
Carbonate Reservoir ◽  
Carbonate Reservoirs ◽  
Test Model ◽  
Well Test ◽  
Analysis Model ◽  
Pressure Transient Analysis ◽  
Linear Flow ◽  
Pressure Transient ◽  
Well Test Model

Abstract Fractured vuggy carbonate reservoirs are highly heterogeneous and non-continuous, and contains not only erosion pores and fractures but also the vugs. Unfortunately, the current well test model cannot be used to analyze fractured-vuggy carbonate reservoirs, due to the limitations of actual geological characteristics. To solve the above-mentioned problem, a pressure transient analysis model for fracture-cavity carbonate reservoir with radial composite reservoir that the series multi-sacle fractures and caves exist and dual-porosity medium (fracture and erosion pore) is established in this paper, which is suitable for fractured vuggy reservoirs. Laplace transformation is used to alter and solve the mathematical model. The main fractures' linear flow and the radial flow of caves drainage area are solved by coupling. The pressure-transient curves of the bottomhole have been obtained with the numerical inversion algorithms. The typical curves for well test model which has been established are drawn, and flow periods are analyzed. The sensitivity analysis for different parameters is analyzed. The variation characteristic of typical curves is by the theoretical analysis. With the increasing of fracture length, the time of linear flow is increased. While the cave radius is the bigger, the convex and concave of the curve is the larger. As a field example, actual test data is analyzed by the established model. An efficient well test analysis model is developed, and it can be used to interpret the actual pressure data for fracture-cavity carbonate reservoirs.

Pressure-Transient Analysis of a Finite-Conductivity Inclined Fracture Connected to a Slanted Wellbore

SPE Journal ◽  
2016 ◽  
Vol 21 (02) ◽  
pp. 522-537 ◽  
Author(s):  
Pin Jia ◽  
Linsong Cheng ◽  
Shijun Huang ◽  
Hongjun Liu
Keyword(s):  
Inclination Angle ◽  
Transient Analysis ◽  
Radial Flow ◽  
Flow Regimes ◽  
Finite Conductivity ◽  
Transient Responses ◽  
Pressure Transient Analysis ◽  
Linear Flow ◽  
Pressure Transient ◽  
Permeability Anisotropy

Summary The principal focus of this work is on pressure-transient analysis of a finite-conductivity inclined fracture connected to a slanted wellbore, on the basis of a semianalytical model. Detailed analysis of unsteady-state pressure behavior of a fully penetrating inclined fracture in an infinite-slab reservoir was provided. The study has shown that a finite-conductivity inclined fracture may exhibit five flow regimes: bilinear flow, formation linear flow, early radial flow, compound linear flow, and pseudoradial flow. The characteristics of bilinear flow and formation linear flow are predominantly determined by fracture conductivity. In the case of a low formation-thickness/fracture-half-length ratio and small inclination angle, both early radial flow and compound linear flow may be absent. Analytical solutions for transient responses during different flow regimes are similar to that for a fully penetrating vertical fracture and can be correlated with the cosine of fracture-inclination angle with consideration of permeability anisotropy. Effect of inclination angle and reservoir-permeability anisotropy on transient responses is strong, which extends to pseudoradial-flow period. Formation thickness mainly influences the middle to late flow periods. In addition, the pseudoskin factor is also investigated in detail.

Analysis of Short Transient Tests Affected by Changing Wellbore Storage

1998 ◽  
Vol 1 (03) ◽  
pp. 261-267 ◽  
Author(s):  
M. Vasquez-Cruz ◽  
R. Camacho-Velazquez
Keyword(s):  
Exponential Function ◽  
Radial Flow ◽  
Error Function ◽  
Early Time ◽  
Curve Matching ◽  
Well Test ◽  
Flow Conditions ◽  
Type Curve ◽  
Wellbore Storage ◽  
Matching Procedure

Abstract It is well known that wellbore storage and phase redistribution have a direct influence over well testing data, mainly on those recorded at early times of the test. After this early time period, such influence disappears, and the pressure response is dominated by the reservoir and the skin zone properties. However, sometimes the effects of wellbore dynamics last long enough as to completely disguise the reservoir response. The above situation frequently constrains the use of some analysis procedures such as type curve matching, especially if the test did not last long enough as to reach radial flow conditions. In this way, some tests are uninterpretable because of the duration of these wellbore effects. Using as a basis two classical models related to well test affected by changing wellbore storage, this work introduces a new method of analysis for these tests with insufficient duration to reach radial flow conditions. The use of the methodology proposed in this work is illustrated with synthetic examples and a field case. For some synthetic cases the type curve matching procedure may yield completely erroneous values of the parameters, while with the suggested method reasonable estimates are obtained. Introduction Wellbore storage is recognized as a phenomenom that affects the recorded pressure behavior at early times during a well test. Sometimes, the phenomena related to mass balance and fluid momentum that occur inside the wellbore, constrain the application of some analysis procedures, especially if the test does not show transient radial flow conditions, and can even disguise completely the pressure response. Phase redistribution occurs when a producing well (that contains more than one phase) is closed at the surface, and due to gravitational forces these phases segregate from each other, this causes a distorsion on early time data of the test. In 1981, Fair introduced a model which takes into account the phase redistribution phenomenom in the diffusivity equation solution as applied to buildup analysis. He assumed an exponential increase in wellbore storage. More recently, Hegeman et. al., proposed an extension to Fair's model, obtaining a general solution in Laplace space, which allows to add the changing wellbore storage effect for a variety of analytical well-reservoir models. These authors considered two models of changing wellbore storage, the exponential function and the error function. For the error function, the transition period is more abrupt than with the exponential function. In 1992, Fair presented another study on the influence of a wide range of wellbore phenomena including mass-balance and momentum phenomena. Also, in that work wellbore phenomena such as fluid temperature changes (represented by an exponential function as in the case of wellbore storage), phase changes inside the production tubing and inertial effects were considered. Currently, the only available method to estimate the reservoir permeability and skin factor from insufficient transient data (tests affected by changing wellbore storage and that have not reached radial flow period) is the type-curve matching technique. To apply this procedure it is necessary to identify the function (exponential or error function) that fits best the storage effect. Nevertheless, the absence of data in the infinite acting period difficults the matching procedure, resulting in a match with high uncertainty. With the purpose of modeling in a more rigorous way the reservoir-wellbore system, it is necessary an algorithm that couples the mechanistic behavior of the well with that of the reservoir, such as that of Hasan and Kabir, Winterfeld propose the use of a completely numerical scheme that simulates the transient conditions of the system.

Well-Test Characterization of Wedge-Shaped, Faulted Reservoirs

2001 ◽  
Vol 4 (03) ◽  
pp. 221-230 ◽  
Author(s):  
D.D. Charles ◽  
H.H. Rieke ◽  
R. Purushothaman
Keyword(s):  
Gulf Of Mexico ◽  
Test Data ◽  
Radial Flow ◽  
Constant Pressure ◽  
Depositional Environments ◽  
Well Test ◽  
Linear Flow ◽  
Drainage Systems ◽  
Reservoir Models ◽  
Flow Boundaries

Summary Two offshore, wedge-shaped reservoirs in south Louisiana were interpreted with pressure-buildup responses by comparing the results from simulated finite-element model studies. The importance of knowing the correct reservoir shape, and how it is used to interpret the generated boundary-pressure responses, is briefly discussed. Two different 3D computer models incorporating different wedge-shaped geometries simulated the test pressure-buildup response patterns. Variations in the two configurations are topologically expressed as a constant thickness and a nonconstant thickness, with smooth-surface, wedged-shaped reservoir models. The variable-thickness models are pinched-out updip at one end and faulted at the other end. Numerical well-test results demonstrated changes in the relationships between the pressure-derivative profile, the wellbore location, and the extent of partial penetration in the reservoir models. The wells were placed along the perpendicular bisector (top view) at distances starting from the apex at 5, 10, 20, 40, 50, 60, 80, and 90% of the reservoir length. Results demonstrate that boundary distance identification (such as distance, number, and type) based solely on the log-log derivative profile in rectangular and triangular wedge-shaped reservoirs should be strongly discouraged. Partial-penetration effects (PPE's) in wedge-shaped reservoirs are highly dependent on the wellbore location relative to the wedge, and the well-test-data analysis becomes more complex. Introduction The interpretation of the effect of reservoir shape on pressure-transient well-test data needs improvement. It is economically imperative to be able to generate an accurate estimate of reserves and producing potential. This is especially critical for independent operators who wish to participate in deepwater opportunities in the Gulf of Mexico. Proper interpretation of data extracted from cost-effective well tests is an integral part of describing, evaluating, and managing such reservoirs. Well-test information such as average reservoir pressure, transmissivity, pore volume, storativity, formation damage, deliverability, distance to the boundary, and completion efficiency are some of the technical inputs into economic and operational decisions. Several key economic decisions that operators have to make are:Should the reservoir be exploited?How many wells are needed to develop the reservoir?Is artificial lift necessary (and if so, when)? The identification of morphological demarcation components such as impermeable barriers (faults, intersecting faults, facies changes, erosional unconformities, and structural generated depositional pinchouts) and constant-pressure boundaries (aquifer or gas-cap) from well testing help to establish the reservoir boundaries, shape, and volume. One must remember that the geological entrapment structure or sedimentological body does not always define the reservoir's limits. Our present study provides insight into wedge-shaped reservoirs in the Gulf of Mexico. Seismic exploration can define geological shapes in either two or three dimensions in the subsurface. These shapes are expressions of the preserved structural history and depositional environments and are verified by observations of such structures in outcrops and present-day depositional environments. From a sedimentological viewpoint, the following sedimentary deposits can exhibit wedge-shaped geometries. Preserved barchan sand dunes, reworked transgressive sands, barrier-island sands, offshore bars, alluvial fan deposits, delta-front sheet sands, and lenticular channel sands form the more plausible pinchout, wedge-shaped geological models recognized in the Gulf of Mexico sedimentary sequence. Wedge-Shaped Reservoirs Reviewing the petroleum engineering literature, we found very few technical papers addressing wedge-shaped reservoir geometries and their effects on reservoir performance. Their detailed analytical results are discussed and applied to the interpretations of our model results. An overview of the conceptual models is presented as a quick orientation to emphasize some model issues. Horne and Temeng1 were the first to address the problem of recognizing, discriminating, and locating reservoir pinchouts with the Green's functions method proposed by Gringarten and Ramey2 in pressure-transient analysis. The analytical solution considered a dimensionless penetration depth of the well. Their results showed that pinchout boundaries appear similar to constant-pressure boundaries with respect to pressure-drawdown behavior and not as a perpendicular sealing boundary. Yaxley3 presented a set of simple equations for calculating the stabilized inflow performance of a well in infinite rectangular and wedge-shaped drainage systems. The basis for Yaxley's mathematical model is the application of transient linear flow (as opposed to radial flow conditions assumed for the reservoir) and the mathematical difference between a plane source and a line source in linear-flow drainage systems for various rectangular drainage shapes. The equations were derived from transient linear-flow relationships for a well located between parallel no-flow boundaries. This concept was applied to intersecting no-flow boundaries and an outer circular, no-flow, constant-pressure boundary. His approach involved a constant ßr that is interpreted as an extra pressure drop relative to a well of radius ro (radial distance to the well location), which is a result of the distortion of the radial streamline pattern. Chen and Raghavan4 developed a solution to compute pressure distributions in wedge-shaped drainage systems using Laplace transforms. Their mathematical approach overcame existing limitations in some of the previous solutions, which were mentioned earlier. By applying the inversion theorem to the Laplace transformation, they verified that the slope of the pressure profile is inversely proportional to the wedge angle of the drainage system. An examination of their results is important to the interpretation of our own simulated pressure-response issues. Generally, their model solutions showed three radial-flow periods in the absence of wellbore-storage effects. The radial-flow periods showed that:During an initial radial-flow period, neither of the impermeable boundaries registered either singly or jointly.In the second phase, one or two boundaries became evident on the pressure signature.A third radial-flow period exhibited a semi logarithmic slope proportional to p/?o, where ?o=the angle of the wedge.

Injection Rates - The Effect of Mobility Ratio, Area Swept, and Pattern

1961 ◽  
Vol 1 (02) ◽  
pp. 81-91 ◽  
Author(s):  
John C. Deppe
Keyword(s):  
Radial Flow ◽  
Initial Rate ◽  
Approximate Equation ◽  
Mobility Ratio ◽  
Rate Equations ◽  
Sweep Efficiency ◽  
Linear Flow ◽  
Flow Equations ◽  
Well Pattern ◽  
Injection Rates

Abstract A method is presented for calculating approximate injection rates in secondary recovery operations. The method can he applied to cases of unequal fluid mobilities, irregular well patterns and boundary patterns. The steady-state pressure distributions for the four flood patterns reported by Muskat and for five additional patterns reveal that most of the difference in pressure between the injection and producing wells occurs in regions around the wells which can adequately he described as regions of radial flow. This leads to a method of calculating injectivity by approximating the flood pattern with radial flow elements (or a combination of radial- and linear-flow elements for some patterns such as the direct line drive). Irregular and boundary patterns can also be approximated by radial and linear elements. Each of these elements can be described by radial- and linear-flow equations and the results combined as series flow resistances to give an approximate equation for the initial injection rate. The mobility ratio does not affect the initial rate; therefore, if the well pattern is one of those regular patterns for which theoretical rate equations have been derived for unity mobility ratio, the approximate initial rate equation can be improved by adjusting it to match the theoretical equation. The available theoretical rate equations are listed, including five new cases. As the flood progresses, the injectivity changes because in general the flood front will divide the pattern into areas of different fluid mobilities. Simple shapes can be assumed for the flood front so that both areas can be divided into radial- and linear-flow elements. Radial- and linear-flow equations are applied to these elements to account for the change in flow resistance behind the front. To calculate injection rates after breakthrough, it is necessary to know the sweep efficiency at breakthrough. Areal sweep data are available in the literature for a number of patterns, and sufficiently accurate breakthrough sweep efficiency can be estimated from these data if it is not otherwise available.

Pressure transient behavior of a fractured well in multi-region composite reservoirs

2017 ◽  
Vol 158 ◽  
pp. 535-553 ◽  
Author(s):  
Qi Deng ◽  
Ren-Shi Nie ◽  
Yong-Lu Jia ◽  
Quan Guo ◽  
Kai-Jun Jiang ◽  
...  
Keyword(s):  
Transient Behavior ◽  
Pressure Transient ◽  
Fractured Well

scholarly journals Numerical Well Test Analysis for Polymer Flooding considering the Non-Newtonian Behavior

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Jia Zhichun ◽  
Li Daolun ◽  
Yang Jinghai ◽  
Xue Zhenggang ◽  
Lu Detang
Keyword(s):  
Transient Response ◽  
Polymer Concentration ◽  
Shear Thinning ◽  
Polymer Flooding ◽  
Oil Field ◽  
Well Test ◽  
Well Test Analysis ◽  
Test Analysis ◽  
Pressure Transient ◽  
Thinning Effect

Well test analysis for polymer flooding is different from traditional well test analysis because of the non-Newtonian properties of underground flow and other mechanisms involved in polymer flooding. Few of the present works have proposed a numerical approach of pressure transient analysis which fully considers the non-Newtonian effect of real polymer solution and interprets the polymer rheology from details of pressure transient response. In this study, a two-phase four-component fully implicit numerical model incorporating shear thinning effect for polymer flooding based on PEBI (Perpendicular Bisection) grid is developed to study transient pressure responses in polymer flooding reservoirs. Parametric studies are conducted to quantify the effect of shear thinning and polymer concentration on the pressure transient response. Results show that shear thinning effect leads to obvious and characteristic nonsmoothness on pressure derivative curves, and the oscillation amplitude of the shear-thinning-induced nonsmoothness is related to the viscosity change decided by shear thinning effect and polymer concentration. Practical applications are carried out with shut-in data obtained in Daqing oil field, which validates our findings. The proposed method and the findings in this paper show significant importance for well test analysis for polymer flooding and the determination of the polymer in situ rheology.

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