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Abstract Characterization of hydraulic fracture system in multi-fractured horizontal wells (MFHW) is one of the key steps in well spacing optimization of tight and shale reservoirs. Different methods have been proposed in the industry including core-through, micro-seismic, off-set pressure data monitoring during hydraulic fracturing, pressure depletion mapping, rate-transient analysis, pressure-transient analysis, and pressure interference test. Pressure interference test for a production and monitoring well pair includes flowing the production well at a stable rate while keeping the monitoring well shut-in and recording its pressure. In this study, the coupled flow of gas in hydraulic fractures and matrix systems during pressure interference test is modeled using an analytical method. The model is based on Laplace transform combined with pseudo-pressure and pseudo-time. The model is validated against numerical simulation to make sure the inter-well communication test is reasonably represented. Two key parameters were introduced and calculated with time using the analytical model including pressure drawdown ratio and pressure decline ratio. The model is applied to two field cases from Montney formation. In this case, two wells in the gas condensate region of Montney were selected for a pressure interference test. The monitoring well was equipped with downhole gauges. As the producing well was opened for production, the bottom-hole pressure of the monitoring well started declining at much lower rate than the production well. The pressure decline rate in the monitoring well eventually approached that of the producing well after days of production. This whole process was modeled using the analytical model of this study by adjusting the conductivity of the communicating fractures between the well pairs. This study provides a practical analytical tool for quantitative analysis of the interference test in MFHWs. This model can be integrated with other tools for improved characterization of hydraulic fracture systems in tight and shale reservoirs.
L. F. Bittencourt Neto, Jessica (Pontifical Catholic University of Rio de Janeiro) | Vieira Bela, Renan (Pontifical Catholic University of Rio de Janeiro) | Pesco, Sinesio (Pontifical Catholic University of Rio de Janeiro) | Barreto, Abelardo (Pontifical Catholic University of Rio de Janeiro)
Injectivity testing is a procedure used by the oil industry whose purpose, when injecting a fluid, is to obtain information about the characteristics of an oil reservoir. Such characteristics can be estimated from the pressure response obtained during the test. This work presents a solution to the well pressure during an injectivity test, considering a piston flow, using a radially composed reservoir model. It is considered that the reservoir may have a two-phase flow with two regions (water and oil) or three regions. In this case it is considered that the region near the well represents a damaged zone (skin zone). From the known analytical solution for pressure in the Laplace space, it is possible to use a matrix representation of the problem the well pressure calculation. With the aid of Stehfest algorithm, it is possible to perform the numerical inversion of the solution to the real field. From this problem, it was possible to develop a code that simulates the injectivity test and the pressure behavior over time.
ABSTRACT: Indentation of a poroelastic solid by a smooth rigid sphere is analyzed within the framework of Biot's theory. The particular cases when the spherical indenter is loaded instantaneously to a fixed depth with the surface of the semi-infinite domain being either fully undrained or in a mixed drainage condition are solved. Constituents of the poroelastic medium are assumed to be slightly compressible. The solution procedure based on the McNamee-Gibson displacement function method is adopted in this work. Problem formulation and the solution procedure are first introduced. Effect of poroelasticity on the contact stress and the transient indentation force response is then discussed. The asymptotic behaviors for the normalized transient force at early and late times are derived. Though derivation of these fully coupled solutions requires the aid of a variety of mathematical techniques, the normalized transient force responses are remarkably simple and show only weak dependence on one derived material constant, which lend itself to convenient use for poroelastic characterization of geomaterials such as rocks in the laboratory.
Over the last decade, spherical indentation has been developed into a laboratory testing technique to characterize poroelasticity of fully saturated soft and biological materials such as polymeric gels and hydrated bones via either displacement- or force-controlled tests. In a displacement-controlled load relaxation test, the indenter is pressed instantaneously to a fixed depth and held until the indentation force approaches a horizontal asymptote (Hu et al., 2010; Kalcioglu et al., 2012), whereas in a step force loading or ramp-hold test, the indentation force is kept constant after reaching a prescribed level (Oyen, 2008; Galli and Oyen, 2008). In theory, for a step loading test, since both the solid and fluid phases can be considered incompressible for these materials, elastic constants can be determined from the early and late time responses following the classical Hertzian solution, while the hydraulic diffusivity can be obtained from the transient response by matching the measured indentation force or displacement as a function of time against a master curve. Such master curves for various indenter shapes have been previously constructed through finite element simulations (Hui et al., 2006; Lin and Hu, 2006; Lin et al., 2007; Galli and Oyen, 2009; Hu et al., 2010; Lai and Hu, 2017) and also semi-analytically for spherical indentation with step force loading (Agbezuge and Deresiewicz, 1974). In general, after the indentation force or displacement is normalized by the early and late time asymptotes, these master curves can be fitted by elementary functions.
Summary In this study, theoretical models have been formulated, validated, and applied to evaluate the transient pressure behavior of a horizontal well with multiple fractures in a tight formation by taking stress‐sensitive fracture conductivity into account. On the basis of the superposition principle in the Laplace domain, we propose a coupled matrix/fracture‐flow model with consideration of the stress‐sensitivity effect in fractures, which strengthens the nonlinearity of the governing equations. More specifically, a new slab‐source function in the Laplace domain was developed to describe the transient pressure responses caused by fluid flow from the matrix to the fracture, and a new solution was derived to describe the fluid flow in the fracture under the stress‐sensitivity effect. Subsequently, a semianalytical method was applied by discretizing each hydraulic fracture into small segments, and a linearization scheme and an iteration method are adopted to deal with the nonlinear problem in the Laplace domain. Meanwhile, a modified superposition principle was proposed and applied to generate the pressure distributions for buildup tests with consideration of stress‐sensitive fracture conductivity. Furthermore, pressure responses and their corresponding derivative type curves were generated to examine the effect of stress‐sensitive conductivity. For pressure‐drawdown tests, it is found that gradual increases in both pressure drop and pressure derivative occur over time because of the partial closure of the fractures. The stress‐sensitivity effect in fractures becomes more evident with a smaller fracture conductivity and a larger fracture‐permeability modulus. From the pressure‐buildup curves, a one‐fourth‐slope line characteristic of the bilinear‐flow period and constant derivatives of 0.5 representing a pseudoradial‐flow regime can be clearly observed. Only fracture conductivity near the wellbore at the shut‐in time can be estimated from the buildup pressures obtained in this work, whereas pressure-buildup analysis derived from the traditional superposition principle will result in an erroneous evaluation of the stress‐sensitive fracture conductivity. It is also found that the effect of permeability hysteresis in the fractures has a negligible impact on the pressure-buildup responses.
Wu, Yonghui (China University of Petroleum, Beijing) | Cheng, Linsong (China University of Petroleum, Beijing) | Huang, Shijun (China University of Petroleum, Beijing) | Fang, Sidong (Sinopec Petroleum Exploration and Production Research Institute) | Killough, John Edwin (Texas A&M University) | Jia, Pin (China University of Petroleum, Beijing) | Wang, Suran (China University of Petroleum, Beijing)
Abstract A major concern with hydraulic fracturing in tight formation is the fracturing fluid-induced formation damage (FFIFD) for the high capillary pressure and the presence of water-sensitive clays. Analytical models are good choices for formation damage evaluation comparing to computationally expensive numerical simulations. However, many analytical models are limited to single-phase flow and the FFIFD is seldom addressed. This paper presents a semi-analytical model for this problem with the consideration of both two-phase flow and FFIFD. A triple-porosity model is modified to capture the formation damage caused by fracturing fluid, and mainly two modifications are made. First, two-phase flow is assumed in the fractures to capture the choking effects. In addition, a low permeability fracturing fluid invasion layer (FFIL) is used to characterize leakoff caused clay swelling and polymer adsorption in the matrix pores. The analytical solution is obtained in the Laplace domain, and a successive iteration is used to update the dynamic parameters by coupling the flowing material balance equations. The precision of the semi-analytical model is validated using the commercial numerical simulator Eclipse. Several synthetic cases and a field case are studied, the results show that gas production is greatly affected by FFIFD. The effect of two-phase flow on gas production is mainly within the early several days. The permeability and width of FFIL have a significant effect on gas production. Gas production rate in the early hundreds of days will be much larger with thinner FFIL and larger permeability, but it will decline sharper later for the faster depletion. The field case shows that different models can match the production data, but the interpreted parameters and the physical meaning are totally different. In addition, a good match will be obtained with FFIFD and two-phase flow effects considered in the model. A simple yet versatile semi-analytical model is proposed in this paper for production prediction and analysis with the consideration of both FFIFD and two-phase flow. It is a quite comprehensive model to allow for a seamless way to analyze production data in the flow-back and production stage. It can also serve as an alternative to computationally expensive numerical simulations to evaluate the formation damage for unconventional reservoirs.
Zhu, Langtao (China University of Petroleum, Beijing) | Liao, Xinwei (China University of Petroleum, Beijing) | Chen, Zhiming (China University of Petroleum, Beijing) | Cheng, Xuyang (China University of Petroleum, Beijing)
Summary The technique of stimulated reservoir volume (SRV) is becoming a key measure to enhance vertical-well productivity and ultimate recovery in tight oil reservoirs. After the SRV treatment, microseismic monitoring results strongly show approximately rectangular stimulated reservoir volumes with a single biwing fracture. However, most published works assumed that the SRV is approximately circular, and few works have considered a rectangular SRV. In this paper, a new analytical pressure-transient solution for vertically fractured wells (VFWs), with rectangular SRV and a single biwing fracture, is derived under constant rate. First, a two-region composite model is established to model the VFW. The inner region is simplified as rectangle dual-porosity with a single biwing fracture. In addition, this model also considers the effect of other multiple factors including stress-sensitivity effect of permeability and unsteady crossflow between matrix and fracture. Then, the trilinear flow model is extended to handle the rectangle boundary and biwing fracture; the Pedrosa's perturbation and Laplace transformation are used to solve the nonlinear equation; the Kazemi's method is adopted to simulate the unsteady crossflow; and the pressure equation for VFW is solved. After that, this analytical solution is applied to a real case from Ordos Basin to conduct equation validation. Finally, sensitivity studies are conducted to evaluate the effect of some critical parameters on the pressure behavior of VFW. The results of model validation show that there is close agreement. Results from this study show that the special flow regimes for a VFW are: (1) linear flow dominated by the biwing fracture with high conductivity, (2) SRV-width-effect flow, and (3) inner boundary-dominated flow. The linear flow is stronger with the increase of biwing-fracture length, biwing-fracture conductivity, and SRV width. As the permeability of SRV increases, the inner boundary-dominated flow becomes more dominant. Moreover, the SRV linear flow will become stronger as the SRV volume increases. This work provides a significant reference for reservoir engineers in pressure-transient analysis as well as fracturing evaluations of vertically fractured wells in tight oil reservoirs.
ABSTRACT: Indentation of a poroelastic solid by a spherical-tip tool is analyzed within the framework of Biot’s theory. We seek the response of the indentation force as well as the field variables as functions of time when the rigid indenter is loaded instantaneously to a fixed depth. We consider the particular case when the surface of the semi-infinite domain is permeable and under a drained condition. Compressibility of both the fluid and solid phases is taken into account. The solution procedure based on the McNamee- Gibson displacement function method is adopted. One of the difficulties in solving this class of problems analytically is in evaluating integrals with oscillatory kernels over an unbounded interval. We show that such issues can be overcome by the use of a series of special functions. Problem formulation and the solution procedure are first introduced. Implications of the poroelastic solution for incipient failure in form of tensile crack initiation and plastic deformation are then discussed. An interesting outcome from this analysis is that if the indentation forces at the undrained and drained limits are known, relaxation of the indentation force with time can be used to determine the diffusion coefficient of a porous medium.
The process of indentation by a rigid tool has been widely studied for its versatility as an experimental technique to probe constitutive properties of materials of various kinds across multiple scales (Marshall et al., 2015) and also for its relevance in understanding the rock fragmentation mechanisms for predicting the efficiency of mechanical excavation (Cook et al., 1998).
In this work, indentation of a poroelastic solid by a spherical-tip tool is analyzed within the framework of Biot’s theory (Biot, 1941; Detournay and Cheng and 1993). We seek the response of the indentation force as well as the field variables as functions of time when the rigid indenter is loaded instantaneously to a fixed depth. We consider the particular case when the surface of the semi-infinite domain is permeable and under a drained condition. Compressibility of both the fluid and solid phases is taken into account.
Previous studies on Laplace domain waveform inversion (WI) lack guidelines to determine efficient Laplace constants. This paper presents presents such guidelines when considering a given source-receiver geometry. For better understanding of this method, we describe the Green's function of the Laplace domain assuming homogeneous media and analyze the wavepath of the Laplace domain using the Green’s function. We demonstrate that the continuity of the imaginary wavenumber coverage of the wavepath in Laplace domain should be maintained to improve the resolution of the Laplace domain inversion result. Using the proposed Laplace constant selection strategy, the Laplace constants can be chosen to maintain the continuity of the vertical imaginary wavenumber of the local wavepath and minimize the redundancy of the vertical imaginary wavenumber. A 1D model test shows that the proposed Laplace constant selection strategy has better performance than the conventional Laplace constant selection strategy using a fixed interval.
Presentation Date: Tuesday, September 26, 2017
Start Time: 2:15 PM
Location: Exhibit Hall C, E-P Station 3
Presentation Type: EPOSTER
Abstract An anomalous flowrate feature (often a " hump" or even a " spike") is characteristically observed at early-times during flowback performance in multi-fractured horizontal wells (MFHW) completed in ultra-low permeability (shale) reservoirs prior to the onset of a characteristic reservoir flow regime (i.e., linear or bilinear flow). The flowrate feature tends to occur in all fluid phases and this feature is thought to be attributed to the " clean-up" behavior following well stimulation and/or the phase behavior of the fluid as it flows along the well path. The guiding principle of this work is that this anomalous flowrate feature can be represented by decaying skin effects, a changing wellbore storage effect, or a combination of both decaying skin effects and changing wellbore storage effects. The goal of this work is to provide a proof-of-concept which considers the simplified case of a vertical well with a single vertical fracture to develop a series of time-dependent skin and wellbore storage models that can effectively be used to characterize the early-time flowrate behavior observed in practice. For this study, we forced a constant wellbore flowing pressure constraint, and while we recognize that this constraint is not truly met in practice, we believe that this approach can serve as a base model for diagnostics/interpretative analyses. Based on the work developed by Fair (1981) and Larsen and Kviljo (1990), our procedure is to couple a series of time-dependent wellbore storage and skin effect models with a set of " power law" reservoir flow models (i.e., linear flow, bilinear flow and a generalized power-law flow model). Specifically, we combine the time-dependent wellbore storage and skin effect models with the constant rate solution reservoir flow models, then apply the convolution integral to produce the constant pressure condition — all in the Laplace domain. In order to generate various scenarios of production performance, we use the Gaver-Wynn-Rho Algorithm implemented in Mathematica to numerically invert the Laplace domain solutions into the real time domain. A generalized workflow is provided to demonstrate the addition of time-dependent wellbore storage and skin effects to any prescribed reservoir model. Using the various wellbore storage and skin time-dependent models proposed in this work, we observe that each of these models, individually and in combination, provide behavior indicative of early-time flowrates observed in the field. In short, we demonstrate that each time-dependent model has unique characteristics, which could, in concept, allow for characterization of flow behavior in the fracture prior to the onset of an undistorted " reservoir" flow geometry (i.e., formation linear or bilinear flow).
Libing, Fu (Research Institute of Petroleum Exploration & Development, CNPC) | Zifei, Fan (Research Institute of Petroleum Exploration & Development, CNPC) | Qingying, Hou (China University of Geosciences) | Jun, Ni (Research Institute of Petroleum Exploration & Development, CNPC) | Lun, Zhao (Research Institute of Petroleum Exploration & Development, CNPC)
Abstract The effect of bottom-hole pressure and formation pressure due to a partially penetrating well (PPW) is different from that for an open hole well. In order to analyze the effect of imperfection on pressure response type curves, this paper presents a 3D symmetry porous flow model for circularly partially penetrating wells. Laplace transform and Fourier transform and Bessel functions are applied to obtain the analytical solution of the model. The pressure response and pressure distribution are obtained and the influence on flow regime surrounding the well and pressure response caused by partial penetration are analyzed. Research results show that when the imperfect area tends to zero, the solution of the model can be reduced to the traditional model of the perfect wells presented by Theis, demonstrating the correctness of the solution. The early-time pressure is significantly lower than the case of complete well. The pressure difference between a partially penetrating well and a completely penetrating well decreases with time increasing. Without considering the variation of spatial distribution of flow field due to imperfect well it may bring about errors of formation parameters calculated by perfect well model. Those conclusions improve the seepage model and provide theoretical guidance for the transient pressure data interpretation, formation parameters calculation and productivity prediction of partially penetrating wells. The presented research content furthers the theory of well test analysis, and builds theoretical foundation for the technologies of well testing interpretation and reservoir numerical simulation.