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This paper presents analytic solutions of the pressure transient behavior of a well intersected by a finite-conductivity fracture in an infinite-acting, or in cylindrically or rectangularly bounded finite reservoirs. These solutions include the practical effects of reservoir permeability anisotropy and dual porosity behavior. Those solutions are analytic, and thus do not require discretization in space.
The analytical solutions of the finite-conductivity fracture transient behavior presented in this paper eliminate the numerical difficulties associated with other mathematically rigorous finite-conductivity fracture solutions that have been reported in the literature. Both the pressure and rats transient responses can be accurately evaluated using the finite conductivity fracture solutions presented in this paper. This is especially important for low-conductivity fractures, for which the pressure and rate transient behavior is often difficult to evaluate accurately using the solutions available in the literature.
Abstract Recovery factor for multi-fractured horizontal wells (MFHWs) at development spacing in tight reservoirs is closely related to the effective horizontal and vertical extents of the hydraulic fractures. Direct measurement of pressure depletion away from the existing producers can be used to estimate the extent of the hydraulic fractures. Monitoring wells equipped with downhole gauges, DFITs from multiple new wells close to an existing (parent) well, and calculation of formation pressure from drilling data are among the methods used for pressure depletion mapping. This study focuses on acquisition of pressure depletion data using multi-well diagnostic fracture injection tests (DFITs), analysis of the results using reservoir simulation, and integration of the results with production data analysis of the parent well using rate-transient analysis (RTA) and reservoir simulation. In this method, DFITs are run on all the new wells close to an existing (parent) well and the data is analyzed to estimate reservoir pressure at each DFIT location. A combination of the DFIT results provides a map of pressure depletion around the existing well, while production data analysis of the parent well provides fracture conductivity and surface area and formation permeability. Furthermore, reservoir simulation is tuned such that it can also match the pressure depletion map by adjusting the system permeability and fracture geometry of the parent well. The workflow of this study was applied to two field case from Montney formation in Western Canadian Sedimentary Basin. In Field Case 1, DFIT results from nine new wells were used to map the pressure depletion away from the toe fracture of a parent well (four wells toeing toward the parent well and five wells in the same direction as the parent). RTA and reservoir simulation are used to analyze the production data of the parent well qualitatively and quantitatively. The reservoir model is then used to match the pressure depletion map and the production data of the parent well and the outputs of the model includes hydraulic fracture half-lengths on both sides of the parent well, formation permeability, fracture surface area and fracture conductivity. In Field Case 2, the production data from an existing well and DFIT result from a new well toeing toward the existing wells were incorporated into a reservoir simulation model. The model outputs include system permeability and fracture surface area. It is recommended to try the method for more cases in a specific reservoir area to get a statistical understanding of the system permeability and fracture geometry for different completion designs. This study provides a practical and cost-effective approach for pressure depletion mapping using multi-well DFITs and the analysis of the resulting data using reservoir simulation and RTA. The study also encourages the practitioners to take every opportunity to run DFITs and gather pressure data from as many well as possible with focus on child wells.
Summary. This paper presents a new general analytic formulation forpressure-transient behavior of commingled systems (layered reservoirspressure-transient behavior of commingled systems (layered reservoirs withoutcrossflow). The formulation includes the effects of surface and downholeflow-rate variations and of wellbore storage resulting from the wellborevolume's location below the flow-rate measuring point (at any location in thewellbore, including the surface and sandface). The method can be applied to avariety of layered reservoirs. Each individual subzone (reservoir) in thesystem can be different, with different initial and outer-boundary conditions,and each zone can start to produce at a different time. The well completion foreach layer can also be different.
New Laplace domain solutions are presented for partially penetrating slantedwells and partially penetrated wells with and without a gas cap. The solutionfor slanted wells is based on that of Cinco-Ley et al. but includes the correcteffective wellbore radius for the case of an anisotropic formation. Solutionsto a few selected commingled systems are also presented to explore theapplication of the formulation.
Hydrocarbon reservoirs that lie on top of each other are usually separatedby shale zones or nonpermeable or semipermeable formations. The Sadlerochit,Shublik, and Sag River formations of the Prudhoe Bay field are good examples ofsuch a system (Fig. 1). These layers do not communicate in terms of fluid flowthrough the formation but may be produced by the same wellbore. These types ofreservoirs are called commingled systems. The wellbore in commingled systemsmay be vertical, horizontal, inclined, fractured, or partially penetrated.Individual layers may be homogeneous, heterogeneous, or fractured and can havedifferent initial and outer-boundary conditions: infinite extent, constantpressure, no flow, or mixed. pressure, no flow, or mixed. During the last 30years, many papers have appeared in the petroleum literature about the behaviorof commingled layered petroleum literature about the behavior of commingledlayered reservoirs. With a few exceptions, most of these papers assume thateach layer is a radial system that is either infinite or bounded by no-flow orconstant-pressure conditions at the drainage radius. Ehlig-Economides andJoseph conducted an extensive survey on layered systems, and Mavor and Walkupused the parallel-resistance concept to present solutions for commingledreservoirs in which the initial pressure present solutions for commingledreservoirs in which the initial pressure of each reservoir or layer is thesame.
This paper presents generalized analytical solutions for commingled systemsin which each reservoir or layer can have a different initial pressure ordifferent initial pressure distribution. The formalism is combined withwellbore storage in a way that allows the initial wellbore pressure and initiallayer pressures at the sandface to be different from each other. Theformulation is in the Laplace transform domain and allows the response of theentire system to be computed if individual layer solutions are known. For thisreason, we also present new Laplace space solutions for single layers for somewell/reservoir systems of interest.
Let us assume that n reservoirs or layers with different initial pressuredistributions are commingled so that they have a common pressure distributionsare commingled so that they have a common wellbore pressure, pw. If we let qibe the flow rate from the ith layer and q be the total flow rate, then wehave
q may or may not be constant, but even when it is constant, the individualflow rates will generally vary as functions of time. In the Laplace domain Eq.1 becomes
In the ith reservoir, the relationship between flow rate (input) andpressure (output) at the wellbore can be described as a convolution pressure(output) at the wellbore can be described as a convolution operation (see theAppendix).
where phi wi(t) is the contribution from the initial pressure distributionof the ith reservoir, and Gwi(t) is the impulse response of the ith reservoir.In the Laplace domain this becomes
Solving Eq. 4 for the average of qi gives
Substituting for avg of qi in Eq. 2 and solving for the avg of pw yields
Note that Eq. 6 may be written as
Thus, in the time domain the solution is of the convolution form
where phi(t) and G(t) are inverse Laplace transforms of the avg of phi(s)and the avg of G(s), respectively.
The convolution integral (Eq. 10) and its Laplace transform (Eq. 7) providea general framework for treating commingled reservoirs of any type witharbitrary initial pressure distributions, time-varying flow rates, andarbitrary boundary conditions. The only requirement is the ability to solve theindividual layer problems in the Laplace domain. Note, however, that in somecases (particularly when initial layer pressures are different), the flow rateqi (t) in one or more of the layers may be negative, even if the total flowrate q(t) is positive. In these cases, the formulation is correct only if eachlayer positive. In these cases, the formulation is correct only if each layerhas the same fluid viscosity.
The above formulation uses Gwi(t) and phi wi(t) and their Laplace transformsas the basic quantities to describe each layer and the effect of its initialconditions. Precise definitions of these quantities are given in the Appendix.The quantity Gwi(t) is the impulse response function of a single layer, and isrelated to the usual constant-rate response by
A model recently presented by Cinco et al. for the transient pressure behavior of wells with finite conductivity vertical fractures was modified to include the effects of wellbore storage and fracture damage. An infinitesimal skin was considered around the fracture, and it was handled as a dimensionless factor defined as (pi/2)(wd/xf)[(k/kd) - 1].
It was found that the well behavior is importantly affected by the fracture damage. When plotted as a function of log pwD vs lot tD for short plotted as a function of log pwD vs lot tD for short times, results show flat, almost horizontal lines that later become concave upward curves asymptotically approaching the curve for undamaged fractures. This behavior is shown even by slightly damaged fractures. It also was found that important information about the fracture characteristics may not be determined when wellbore storage effects are present. present
It has been shown that the increase in the productivity of a well created by hydraulic productivity of a well created by hydraulic fracturing depends on fracture characteristics, such as fracture conductivity, length, penetration, and also on a possible damage to the penetration, and also on a possible damage to the formation immediately surrounding the fracture. During the last few years, there has been a continuously increasing interest in the determination of the characteristics and orientation of fractures by means of transient pressure analysis. Most of these methods consider the fracture to be of infinite conductivity or of uniform flux; others consider finite conductivity fractures. Generally, these methods assume that there is no skin damage around the fracture. Evans proposed a pressure analysis technique considering fracture skin damage. He assumed the flow from the formation to the fracture to be linear, passing through two porous media in series, one being the damaged zone around the fracture and the other the undamaged formation. Ramey and Gringarten discussed the transient well behavior of vertically fractured wells with large wellbore storage, and suggested a matching technique for analyzing pressure data. Recently, Raghavan discussed pressure analysis techniques for vertically fractured wells, including the effects of wellbore storage and skin. He assumed the fracture to be of uniform flux, and presented general characteristics of the pressure transient behavior for these systems.
The purpose of this study is to present solutions for the transient wellbore pressure behavior of a well crossed by a finite conductivity vertical fracture, considering the effect of a damaged zone around the fracture and wellbore storage. It is also intended to show the general flow characteristics of these fractured systems.
MATHEMATICAL MODELS AND METHODS OF SOLUTION
The transient flow toward a well with a finite conductivity vertical fracture surrounded by a damaged zone was studied by using a modified version of the model presented by Cinco et al. The following assumptions were considered.
1. An infinite, homogeneous, isotropic reservoir of permeability k, porosity phi, and thickness h.
2. The formation is produced through a vertically fractured well. The wellbore is intersected by a fully penetrating vertical fracture of permeability kf, porosity cf, width w, and permeability kf, porosity cf, width w, and half-length xf. All production of fluid is via the fracture.
3. There is a zone of reduced permeability caused by fracturing fluid loss around the fracture. This region has a permeability ks and width ws.
4. The porous medium contains a slightly compressible fluid of viscosity mu and compressibility c.
5. All formation, fracture and fluid properties are independent of pressure.
6. Gravity effects are negligible and pressure gradients are small everywhere.