This work proposes a novel boundary-element based approach to model fluid transport in unconventional shale gas reservoirs with complex hydraulic fracture networks. The fluid flow model employed in this work considers multiple fluid transport mechanisms identified in in gas transporting process in shale nanopores including diffusion, sorption Kinetics, Knudsen diffusion, and sorbed-phase surface diffusion. Accordingly, two governing partial differential equations (PDEs) are written for free and sorbed gases. In the proposed method, boundary integral formulations are analytically derived using the fundamental solution of the Laplace Equation for two governing nonlinear PDEs and Green's second identity. The domain integrals considering the nonlinear terms due to multi-mechanism effects, are transformed into boundary integrals employing the dual reciprocity method (DRM). The resulting boundary integral equations for free and sorbed gas later are solved in terms of a series of discrete nodes after coupling with fracture flow model. The validity of proposed solution is verified using several case studies through comparison with a commercial finite-element numerical simulator COMSOL.
Zhang, Miao (The Pennsylvania State University) | Chakraborty, Nirjhor (The Pennsylvania State University) | Karpyn, Zuleima (The Pennsylvania State University) | Emami-Meybodi, Hamid (The Pennsylvania State University) | Ayala, Luis (The Pennsylvania State University)
Nano-scale pores and a dual storage mechanism shared between free and adsorbed gas make the transport behavior in shale gas reservoirs very different from conventional macropore reservoirs. This work explores a straightforward model for the gas transport behavior in shale nanopores, which couples sorption, diffusion, and sorbed-phase surface diffusion phenomena. The model combines two governing equations for free and sorbed gas phase transport processes in nanopores, respectively: a diffusion-based equation for free gas phase transport, and a surface-diffusion equation for the sorbed phase. Mass transfer between the two phases is quantified by kinetic models of sorption. The two governing equations are solved simultaneously using finite element methods (FEM). Model performance is successfully validated by closely matching density propagation profiles of a gas transport experiment obtained by quantitative X-ray computerized tomography (CT) imaging for a Marcellus shale sample. Transport-related parameters estimated from history matching are shown to be consistent with literature data.
Application of superposition principle to nonlinear gas governing equations has been an elusive goal in early-transient production data analysis and has been so far limited to the use of empirical and approximate methods best applicable to boundary-dominated flow conditions. This paper presents a novel and rigorous semi-analytical model that is applicable for the analysis of production data from multifractured horizontal gas wells (MFHWs) producing under early-transient variable rate/pressure production conditions. Nonlinear, pressure-dependent hydraulic diffusivity retained in pseudo-pressure-based gas diffusivity equation is straightforwardly and rigorously captured without approximation. The resulting formulation of superposition applied in nonlinear gas system is written in terms of the classical solution for linear PDE plus an analytical adjustment factor that quantifies the nonlinearity of the system. Numerical examples are presented to test the validity and showcase the capabilities of proposed approach. Comparisons against available empirical and approximate models are also provided for these cases.
In multifractured horizontal wells producing from unconventional reservoirs, linear flow is commonly observed to be the dominant flow regime during early production. This linear flow regime may remain infinite-acting for a long stretch of time because of the ultralow permeability character of these formations. In this study, we first analytically corroborate that constant gas/oil ratio (GOR) should be expected in unconventional multiphase reservoirs (gas condensates and volatile oils below saturation conditions) exhibiting linear flow under constant-bottomhole-pressure (BHP) production constraint during infinite-acting periods (i.e., before pressure transient reaching the reservoir boundary). We propose a semianalytical solution to the multiphase governing partial-differential equations (PDEs) by applying the similarity method—also referred to as the Boltzmann transformation—to transform the system of PDEs to ordinary-differential-equation (ODE) form. The transformed system of ODEs and boundary conditions are solved by means of Runge-Kutta integration. By solving the equations for pressure and saturation, the GOR trend and value can be fully predicted before availability of production data. The results show that a constant GOR effect could be maintained as long as the flow regime remains infinite-acting and its value varies with BHP specifications for a given reservoir and fluid system.
Liquid-rich gases in unconventional reservoir environments can exhibit complex phase and flow behavior due to gas condensation and re-vaporization and differences in phase mobilities that results in compositional variations inside the system. To date, the analysis of in situ and flowing composition variation in unconventional liquid-rich wells has been largely limited to numerical modeling. This work uses an analytical approach to study the in situ and flowing fluid composition of gas condensate wells producing under infinite-acting linear flow--a commonly observed flow regime in hydraulically-fractured horizontal wells in unconventional formations. We propose a semi-analytical solution to the governing partial differential equations (PDEs) written in terms a compositional fluid formulation. The proposed solution is developed using Boltzmann's transformation and is validated by both analytical development and numerical simulation data. Results corroborate that when hydraulically-fractured horizontal wells are producing against a constant bottomhole pressure (BHP) constraint, the producing wellbore fluid composition remains constant as long as the system remains infinite acting, leading to a constant producing gas-oil ratio (GOR). This constant wellstream composition is shown to be very different from in situ composition, which varies according to pressure and production condition inside the reservoir.
Linear flow is a fundamental reservoir-flow geometry typically associated with production from unconventional resources stimulated by means of hydraulic fracturing. Recently, linear flow has been intensively studied following the fast pace of development of unconventional resources. Previous studies have mainly focused on early transient behavior and behavior of composite linear-flow systems. In this work, a density-based analysis method is extended to study decline behavior of the linear-flow system in boundary-dominated flow (BDF). In this study, we first discuss traditional approaches used to model linear flow in gas reservoirs. Second, we show the applicability of the density-based method for gas linear flow both analytically and numerically. Next, late-time solutions are discussed, and the analytical forecasting solution that best describes the BDF behavior is selected for long-term decline-behavior studies. Previously reported results on radial flow as well as early transient-flow effect are also incorporated to provide a more complete understanding of decline behavior and the impact of flow geometry. We show that boundary-dominated responses in linear-flow scenarios fully develop at much later stages of reservoir depletion compared with radial-flow scenarios. As a result, and in marked contrast with radial flow, purely hyperbolic decline behavior may be completely lost in linear-flow scenarios during boundary-dominated conditions. It is demonstrated that most of the recoverable hydrocarbons are produced during the early transient period for linear-flow conditions, whereas most of them are recoverable during the BDF period for radial flow. These results suggest that the availability of accurate early transient models is much more critical for the formulation of linear-flow-decline models than had been traditionally necessary for radial-flow-decline models.
In recent years, constant producing gas-oil-ratio (GOR) trends have been repeatedly observed in many unconventional oil and gas reservoirs. This effect can last many years, even after the depleting system is believed to have crossed its saturation pressure. In this study, we corroborate that constant GOR could indeed be expected in depleting reservoir systems exhibiting early-transient infinite-acting flow under constant bottomhole pressure (BHP) production constraint for both linear and radial flow geometries. This effect had been shown to be present in infinite-acting linear systems by previous authors; but this study shows how this GOR trend and value can be fully predicted prior to availability of production data. Focusing on liquid-rich gas systems, we start with the applicable multiphase governing flow equations and provide semi analytical and rigorous solutions. Similarity theory is applied to transform the resulting system of equations and achieve the multiphase solution. The results show that a constant GOR effect could be maintained as long as the flow regime remains infinite-acting, which is a typical condition to be found in unconventional systems. Also, it is found that, in both linear and radial multiphase systems, this constant or stabilized GOR is shown to be present regardless of condensate phase mobility at sandface conditions.
In this study, we analytically cross examine the consistency among available zero-dimensional material balance equations (MBEs) for liquid-rich gas equations and derive a new simple yet rigorous MBE starting from governing equations applicable to these systems. We propose a new zero-dimensional (i.e. tank) material balance equation that is directly applicable to the analysis of liquid-rich (wet and retrograde) gas reservoirs by expression of the equations in term of an equivalent gas molar density. Following model development, proposed model predictions of gas reservoir behavior with varying condensate content (lean, intermediate and rich) are investigated and critically compared to previous zero-dimensional models. All models are employed to predict reservoir performance given reservoir original-fluids-in-place and compared against benchmark examples created by numerical simulation. Actual field examples are also analyzed using existing and proposed models to test the ability of the proposed models to provide reliable reserve estimations using straight-line methods. The proposed density-based equation is proven to be straightforward to implement since is written in terms of density. This, in turn, allows it be directly expressed as an extension of the dry gas MBE, while not requiring the implementation of two-phase Z-factors.
The state-of-the-art analysis of the production performance of gas wells relies on material-balance concepts combined with pseudopressure and pseudotime for rate-time decline analysis and reserves estimations. In many cases, rock compressibility and reservoir pore-volume (PV) change are either neglected or accounted for by replacing gas compressibility with total compressibility values. In this work, we extend the applicability of a rescaled exponential and density-based decline-analysis approach (Ayala and Ye 2013a, b; Zhang and Ayala 2014a, b) for the decline analysis of gas systems experiencing significant rock-compressibility effects. We formally derive the density-based analytical techniques that rigorously capture formation-compressibility effects during the analysis of gas-well-production data during boundarydominated flow, which proves crucially important for high-pressure and/or large-formation-compressibility gas-reservoir systems. The proposed formulation enables the calculation and correct prediction of well performance and original gas in place (OGIP) by incorporating formation compressibility and the change of reservoir PV effects, which may prove crucially important in high-pressure and/or relatively large-formation-compressibility gas reservoirs. We also present the associated straight-line analysis technique used for OGIP determination on the basis of the density approach applicable to constant-bottomhole-pressure production and variable-flow-rate/pressure-drop systems.
This study demonstrates that production-data analysis of variable bottomhole-flowing-pressure/variable-rate gas wells under boundary- dominated flow (BDF) is possible by use of a density-based approach. In this approach, governing equations are expressed in terms of density variables and dimensionless viscosity/compressibility ratios. Previously, the methodology was successfully used to derive rescaled exponential models for gas-rate-decline analysis of wells primarily producing at constant bottomhole pressure (Ayala and Ye 2013a, b; Ayala and Zhang 2013; Ye and Ayala 2013; Zhang and Ayala 2014). For the case of natural-gas systems experiencing BDF, gas-well-performance analysis has been made largely possible by invoking the concepts of pseudotime, normalized pseudotime, or material-balance pseudotime. The density-based methodology rigorously derived in this study, however, does not use any type of pseudotime calculations, even for variable-rate/variablepressure-drawdown cases. The methodology enables straightforward original-gas-in-place calculations and gas-well-performance forecasting by means of type curves or straight-line analysis. A number of field and numerical case studies are presented to showcase the capabilities of the proposed approach.