Constant Pressure Boundary Research Articles

A numerical study of sequential and simultaneous hydraulic fracturing in single and multi-lateral horizontal wells

Abstract Horizontal well hydraulic fracturing (HF) technology can help to develop low permeability oil and gas resources. Today, industry uses simultaneous and sequential fracturing as a means to fracture single or multiple (Zipper Frac) horizontal wells as an efficient way to produce oil/gas. In zipper fracturing two or more parallel wells are fractured simultaneously or sequentially to achieve the maximum stimulated reservoir volume. In order to achieve optimum stimulated rock volumes and fracture networks, one must understand the effect of various rock and fluid properties on stimulation to minimize the risk of unwanted fracture geometries. This paper describes the development and application of a 2D coupled displacement discontinuity numerical model for simulating fracture propagation in simultaneous and sequential hydraulic fracture operations for single and multiple parallel wells. The sequential fracturing model considers two different boundary conditions for the previously created fractures. A constant pressure boundary condition along the fracture surface is considered when the flow back is restricted between the stages and a joint model is used when fractures are propped. A series of examples are presented to study the effect of fracture spacing on expected stimulated zone. It is found that fracture path is not only affected by fracture spacing but also by the boundary conditions on the previously created fractures.

Modeling transient pressure behavior of a fractured well for shale gas reservoirs based on the properties of nanopores

Shale gas sources are abundant in China, but gas flow in reservoir cannot be simply represented by Darcy equation anymore and the natural productivity is ultra-low owing to numerous nanoscale pores in shale, so how to model gas flow in nanopores and couple it with hydraulic fracturing become important. This paper developed a dual-porosity model to simulate naturally fractured reservoir based on the research of hydraulic fracturing in which desorption, diffusion and slippage flow in nanopores were considered. The analytical solution of a fractured well in 3D reservoir simulator with laterally closed, infinite and constant pressure boundaries is obtained in Laplace space by employing point source function, and type curves of transient pressure and rate decline behaviors are plotted with Stehfest numerical inversion algorithm. Influence factors have been analyzed combined with the data of shale gas reservoirs. Results show that six flow regimes are recognized for transient pressure behavior. Gas desorbed from pore walls, Knudsen diffusion and slip flow in nanopores have great effect on type curves: the bigger the desorption coefficient is, the smaller the wellbore pressure drop is and the well produces longer; the bigger the diffusion coefficient and slippage factor are, the higher the production rate is and the transition regime finished earlier. Besides, capacitance coefficient, transfer coefficient and boundary also have influence on transient pressure and rate decline behaviors of fractured well.

A Study about Solving the Nonlinear Spherical Seepage Model of Three-Area Composite Reservoir

This paper studies the nonlinear spherical seepage model of three-area composite reservoir under three kinds of outer boundary conditions (infinite boundary, constant pressure boundary and closed boundary) which not considering well-bore storage and skin factor. On the basis of similar structure of solution for differential equation’s boundary value problem, this paper obtains the solution of the nonlinear spherical seepage model of three-area composite reservoir. The study not only benefits to understand the inherent laws of the solution, but also provides the convenience to compile corresponding test software.

Galaxy evolution: modelling the role of non-thermal pressure in the interstellar medium

Galaxy evolution depends strongly on the physics of the interstellar medium (ISM). Motivated by the need to incorporate the properties of the ISM in cosmological simulations we construct a simple method to include the contribution of non-thermal components in the calculation of pressure of interstellar gas. In our method we treat three non-thermal components - turbulence, magnetic fields and cosmic rays - and effectively parametrize their amplitude. We assume that the three components settle into a quasi-steady-state that is governed by the star formation rate, and calibrate their magnitude and density dependence by the observed Radio-FIR correlation, relating synchrotron radiation to star formation rates of galaxies. We implement our model in single cell numerical simulation of a parcel of gas with constant pressure boundary conditions and demonstrate its effect and potential. Then, the non-thermal pressure model is incorporated into RAMSES and hydrodynamic simulations of isolated galaxies with and without the non-thermal pressure model are presented and studied. Specifically, we demonstrate that the inclusion of realistic non-thermal pressure reduces the star formation rate by an order of magnitude and increases the gas depletion time by as much. We conclude that the non-thermal pressure can prolong the star formation epoch and achieve consistency with observations without invoking artificially strong stellar feedback.

Direct pore-to-core up-scaling of displacement processes: Dynamic pore network modeling and experimentation

Summary We present a new dynamic pore network model that is capable of up-scaling two-phase flow processes from pore to core. This dynamic model provides a platform to study various flow processes in porous media at the core scale using the pore-scale physics. The most critical features of this platform include (1) the incorporation of viscous, capillary, and gravity pressure drops in pore-scale displacement thresholds, (2) wetting-phase corner flow in capillary elements with angular cross-sections, (3) adjustments of corner interfaces between wetting and non-wetting phases based on changes in local capillary pressure, (4) simultaneous injection of wetting and non-wetting phases from the inlet of the medium at constant flow rates that makes the study of steady-state processes possible, (5) heavy parallelization using a three-dimensional domain decomposition scheme that enables the study of two-phase flow at the core scale, and (6) constant pressure boundary condition at the outlet. For the validation of the dynamic model, three two-phase miniature core-flooding experiments were performed in a state-of-the-art micro core-flooding system integrated with a high-resolution X-ray micro-CT scanner. The dynamic model was rigorously validated by comparing the predicted local saturation profiles, fractional flow curves, relative permeabilities, and residual oil saturations against their experimental counterparts. The validated dynamic model was then used to study low-IFT and high-viscosity two-phase flow processes and investigate the effect of high capillary number on relative permeabilities and residual oil saturation.

The effect of compressibility on flow start-up of waxy crude oils

The current work presents a mathematical model to simulate the flow start-up of gelled oils in pipelines. The model comprises the conservation equations of mass and momentum and an elasto-viscoplastic constitutive equation to account for thixotropy of the material. The flow is considered one-dimensional, laminar and compressible and the cross-section shear stress is admitted to vary linearly so that the constitutive equation can be integrated along the pipe radius. The balance equations are solved by the method of characteristics, the constitutive equation by the finite difference method and finally they are altogether integrated iteratively. Two kinds of boundary conditions are investigated: a constant inlet pressure and a constant inlet flow rate. Two time-scales are identified in the current study: the pressure wave propagation time and the material response time. It can be anticipated that the ratio of this two times are quite important on the restart time in case of a constant pressure boundary condition and on the magnitude of the maximum pressure for a constant flow rate boundary condition.

Similar Constructive Method of Solution for the Nonlinear Percolation Model in Composite Reservoir

This paper presents a percolation model for the composite reservoir, in which quadratic-gradient effect, well-bore storage, effective radius and three types of outer boundary conditions: constant pressure boundary, closed boundary and infinity boundary are considered. With Laplace transformation, the percolation model was linearized by the substitution of variables and obtained a boundary value problem of the composite modified zero-order Bessel equation. Using the Similar Constructive Method this method, we can gain the distributions of dimensionless reservoir pressure for the composite reservoirs in Laplace space. The similar structures of the solutions are convenient for analyzing the influence of reservoir parameters on pressure and providing significant convenience to the programming of well-test analysis software.

Research on Optimization of Perforated Horizontal Wells with Separated Production Scheme Based on Genetic Algorithm

Based on the separated production scheme of perforated horizontal well, the production is closely related to the number, length and location of horizontal intervals. New reservoir / wellbore coupling models were modeled which included two boundaries: closed boundary and constant pressure boundary. For these coupling models, two new optimization models were modeled, which took the production as the objective function and took the length and location of horizontal intervals as the optimization variables. Then the numerical calculation models were proposed by using the Gauss-Laguerre quadrature formula. Applying the genetic algorithm and writing the numerical stimulation software, the optimized schemes with 2 segments, 3 segments and 4 segments for planning intervals of the perforated horizontal wells were presented. By comparing the optimized schemes, we can see that the horizontal well production was not infinite increasing with the length of horizontal intervals, but only in the suitable length of horizontal intervals to maintain high production. And it is not the more segments the higher production. For the perforated horizontal well in closed reservoir, the production was the highest based on the optimized schemes with 3 segments.

A Study on Solving the Seepage Flow Model of Three-Area Composite Reservoir

This paper studies the seepage flow mathematical model of three-area composite reservoir under three kinds of outer boundary conditions (infinite boundary, constant pressure boundary and closed boundary), in which influences of well-bore storage and skin factor are not taken into consideration. On the basic of theory of similar structure of solution of boundary value problem of differential equation, this paper obtain the solution of the seepage flow model of three-area composite reservoir. The study is not only conducive to further analyze the inherent law of the solution and solve corresponding application problems, but also easy to compile corresponding analysis software.

New Modification on Production Data of Gas Condensate Reservoirs for Rate Transient Analysis

Techniques of production data analysis for single-phase oil and gas reservoirs have advanced significantly over the past few years. These techniques range from traditional (i.e., Arps, 1945; Fetkovich, 1980) to modern methods that account for the variation of operating conditions at the wellbore. However, the application of these later methods to gas condensate reservoirs is a challenge. The authors aimed to extend the applicability of modern production data analysis (single-phase flow) to analyze the production data of a gas condensate reservoir (two-phase flow). A single-phase production model consists of (a) a material balance equation, (b) the solution of diffusivity equation for gas flow subjected to constant rate and constant pressure boundary conditions, and (c) the presentation of a material-balance-time function. Using a compositional reservoir simulator, long-term production data have been generated over a wide range of the gas condensate reservoir parameters. The simulation result is compared with the prediction from the single-phase gas reservoir model, which uses a modern production analysis method. The error level is evaluated and a correlation for treatment of the single-phase analysis method is developed. Finally the results show that the methodology developed can be successfully applied for analysis of production data of a gas condensate reservoir.

Experimental characterization of cement–bentonite interaction using core infiltration techniques and 4D computed tomography

Deep geological storage of radioactive waste foresees cementitious materials as reinforcement of tunnels and as backfill. Bentonite is proposed to enclose spent fuel drums, and as drift seals. The emplacement of cementitious material next to clay material generates an enormous chemical gradient in pore water composition that drives diffusive solute transport. Laboratory studies and reactive transport modeling predict significant mineral alteration at and near interfaces, mainly resulting in a decrease of porosity in bentonite. The goal of this project is to characterize and quantify the cement/bentonite skin effects spatially and temporally in laboratory experiments. A newly developed mobile X-ray transparent core infiltration device was used, which allows performing X-ray computed tomography (CT) periodically without interrupting a running experiment. A pre-saturated cylindrical MX-80 bentonite sample (1920kg/m3 average wet density) is subjected to a confining pressure as a constant total pressure boundary condition. The infiltration of a hyperalkaline (pH 13.4), artificial OPC (ordinary Portland cement) pore water into the bentonite plug alters the mineral assemblage over time as an advancing reaction front. The related changes in X-ray attenuation values are related to changes in phase densities, porosity and local bulk density and are tracked over time periodically by non-destructive CT scans.Mineral precipitation is observed in the inflow filter. Mineral alteration in the first millimeters of the bentonite sample is clearly detected and the reaction front is presently progressing with an average linear velocity that is 8 times slower than that for anions. The reaction zone is characterized by a higher X-ray attenuation compared to the signal of the pre-existing mineralogy. Chemical analysis of the outflow fluid showed initially elevated anion and cation concentrations compared to the infiltration fluid due to anion exclusion effects related to compaction of the bentonite core that was adjusting to the experimental conditions. Subsequently, the OPC fluid is fully buffered, and a gradually decreasing ionic strength is observed as a result of progressive consumption of hydroxide at the mineral reaction front.

Realization of uniform deformation of soil specimen under undrained plane strain condition based on soil–water coupled finite deformation analysis considering inertia forces

Based on a soil–water coupled finite deformation analysis, theoretical considerations and numerical calculations were carried out under the undrained plane strain condition in order to reproduce a uniform deformation field. Rather than the “quasi-static” equation of motion, which does not include inertia forces, a dynamic equation of motion which includes inertia forces was used. At first, a theoretical consideration was carried out to realize uniform deformation for a saturated soil that satisfied the element-wise undrained/constant-volume condition. This presents an “infinitely slow loading” case without ignoring the inertia term based on the u–p formulation. In other words, it can be seen that under general slow loading that is not infinitely slow, a gradient in the pore water pressure will always be produced, resulting in the migration of pore water and loss/collapse of uniformity. This first conclusion is useful for verifying numerical analysis code made in the finite deformation regime. Next, the uniform deformation of a plane strain rectangular soil specimen was measured under constant cell pressure and undrained boundary conditions using a dynamic soil–water coupled analysis in which the SYS Cam-clay model was employed as the elasto-plastic constitutive model for the soil skeleton. In addition, the effects of the loading rates as well as loading applications, with/without inertia forces, on the loss of uniformity in deformation were shown to have a significant influence on the inertia term even though the loss itself was extremely small.

Superposition of steady-state coning solutions for multiple wells with reservoir boundaries

The critical oil (or gas) rate to avoid coning of unwanted fluids into production wells is an important design parameter. Simulation methods are typically used to predict critical rates in reservoirs with complex heterogeneities and boundaries, but they are manpower intensive and prone to errors when large grid blocks are used. Current analytical methods are useful to provide benchmark solutions for simulation, but their assumptions are too restrictive for more general use. Thus, there is a need for improved analytical methods to account for well patterns and more complex boundaries that can serve as additional benchmarks for simulation, and in some cases to predict critical rates. This paper makes analytical solutions more realistic by extending existing single-well analytical solutions to account for multiple wells and common no-flow and constant pressure boundary conditions. A potential function is derived to superimpose existing single well coning solutions for simultaneous two-phase flow. Capillary pressure and relative permeability effects on coning are included in the new potential function. Vertical equilibrium (VE) and steady-state flow are assumed. Comparisons with finely-gridded simulation show good agreement in predicted critical oil rates when steady state and VE are approached. VE and steady-state are approximately achieved when aspect ratios are greater than about 10. The predicted critical rates are conservative for aspect ratios less than 10. The new solutions demonstrate for several examples how the critical rates from one well are affected by reservoir boundaries, and the two-phase production rates of other wells. • Developed new analytical solutions for coning when two phases are flowing simultaneously. • Developed a potential function that allows for superposition so that multiple wells and no-flow and constant potential reservoir boundary conditions can be analyzed. • Demonstrates that boundary conditions and other wells can significantly alter the critical oil rates to avoid coning.

Approximate Analytical Solutions for CO2 Injectivity Into Saline Formations

Summary This paper presents new analytical models to estimate the bottomhole pressure (BHP) of a vertical carbon dioxide (CO2) injection well in a radial, homogeneous, horizontal saline formation. The new models include the effects of multiphase flow, CO2 dissolution in formation brine, and near-well drying out on the BHP. CO2 is injected into the formation at a constant rate. The analytical solutions are presented for three types of formation outer boundary conditions: closed boundary, constant-pressure boundary, and infinite-acting formation. The sensitivity of BHP computations to gas relative permeability, retardation factors, and CO2 compressibility is examined. The predictive capability of the analytical models is tested by use of numerical reservoir simulations. The results show a good agreement between the analytical and numerical computations for all three boundary conditions. Variations in gas compressibility, retardation factors, and gas relative permeability in the drying-out zone are found to have moderate effects on BHP computations. It is demonstrated for several hypothetical but realistic cases that the new models can estimate CO2 injectivity reliably.

Similar Constructive Method for Solving the Nonlinear Spherical Percolation Model in Dual-Porosity Media

This paper presents a spherical percolation model for dual-porosity media reservoir, where the quadratic-gradient term, wellbore storage and three types of outer boundary conditions: constant pressure boundary, closed boundary and infinity boundary were considered. Then a new method: Similar Constructive Method was put forward for solving this type of percolation model. And solutions of the dimensionless reservoir pressure and the dimensionless bottomhole pressure in Laplace space were obtained. It was proved that these solutions had a similar structure. The Similar Constructive Method is an elementary and algebraic method, simple and practical. And the similar structure of solutions can simplify the well test analysis software programming and analyze the reservoir parameter’s affection on pressure conveniently. The present research has a great academic significance and application value in oil-gas field development.

New Type Curves for Modeling Productivity of Horizontal Well With Negative Skin Factors

Summary This paper first establishes a comprehensive semianalytical model for a horizontal well in a homogeneous, dual- or triple-porosity reservoir. By taking a negative-skin approach, including Laplace transform, separation of variables, and inverse Laplace transform, we obtain the solutions in a unified formula for both constant-rate production and constant-wellbore-pressure production. In the model, we consider a set of rich boundary conditions, including infinite reservoir and closed or constant-pressure boundary conditions for the top, bottom, and side of the bounded reservoir. Consequently, we plot a series of new standard log-log type curves and make a clear observation of different flow periods (e.g., early radial flow, hemiradial flow, linear flow, and late pseudoradial flow) from negative-skin type curves. We make a comparison between the solutions using the negative-skin approach and conventional solutions, which shows that there are some small differences between the two methods. We also analyze the type-curve characteristics for horizontal-well production in dual- and triple-porosity reservoirs, which shows that the derivative curves are V-shaped for dual media and W-shaped for triple media. Finally, we perform well-test interpretations for a horizontal well with positive skin and a horizontal well with negative skin and bottomwater, which shows that our new model performs well against the real field data.

The similar structure of solutions in fractal multilayer reservoir including a quadratic gradient term

When the classical nonlinear partial differential equations are used to model the fractal reservoir, based on the assumption of low compressibility fluids, the effects of the quadratic gradient term are ignored, which would be questionable for mixed gas reservoirs and low permeability reservoirs. To consider the influence of the wellbore storage, the nonlinear mathematical flow model of the fractal multilayer reservoir is built in this paper, with three kinds of outer boundaries (infinite boundaries, constant pressure boundaries and closed boundaries). Using the Laplace transform method, the solutions for the dimensionless reservoir pressure and the bottom hole pressure in the Laplace space are obtained. An analysis shows that the solutions involve similar structures even for three different kinds of outer boundaries, and can be unified as a continuous fraction. The unified expression would make it more convenient to analyze the formation parameters, which greatly facilitates the development of the well test analysis software.

Effect of fracture pressure depletion regimes on the dual-porosity shape factor for flow of compressible fluids in fractured porous media

A precise value of the matrix-fracture transfer shape factor is essential for modeling fluid flow in fractured porous media by a dual-porosity approach. The slightly compressible fluid shape factor has been widely investigated in the literature. In a recent study, we have developed a transfer function for flow of a compressible fluid using a constant fracture pressure boundary condition [Ranjbar E, Hassanzadeh H, Matrix-fracture transfer shape factor for modeling flow of a compressible fluid in dual-porosity media. Adv Water Res 2011;34(5):627–39. doi:10.1016/j.advwatres.2011.02.012]. However, for a compressible fluid, the consequence of a pressure depletion boundary condition on the shape factor has not been investigated in the previous studies. The main purpose of this paper is, therefore, to investigate the effect of the fracture pressure depletion regime on the shape factor for single-phase flow of a compressible fluid. In the current study, a model for evaluation of the shape factor is derived using solutions of a nonlinear diffusivity equation subject to different pressure depletion regimes. A combination of the heat integral method, the method of moments and Duhamel’s theorem is used to solve this nonlinear equation. The developed solution is validated by fine-grid numerical simulations. The presented model can recover the shape factor of slightly compressible fluids reported in the literature. This study demonstrates that in the case of a single-phase flow of compressible fluid, the shape factor is a function of the imposed boundary condition in the fracture and its variability with time. It is shown that such dependence can be described by an exponentially declining fracture pressure with different decline exponents. These findings improve our understanding of fluid flow in fractured porous media.

COUPLING MULTIPHASE PORE-SCALE MODELS TO ACCOUNT FOR BOUNDARY CONDITIONS: APPLICATION TO 2D QUASI-STATIC PORE NETWORKS

Accurate predictions of macroscopic multiphase flow properties (relative permeability and capillary pressure) are necessary for modeling flow and transport in subsurface applications, such as hydrocarbon recovery, carbon sequestration and nuclear waste storage. These properties are usually measured experimentally, but pore-scale network modeling has become an efficient alternative for understanding fundamental flow behavior and predicting macroscopic properties. In many cases, network modeling gives excellent agreement with experiment by using models physically representative of real media. Void space within a rock sample can be extracted from high resolution images and converted to a topologically equivalent network of pores and throats. Multiphase fluid transport is then modeled in the network and macroscopic properties extracted from the model. Advancements continue to be made in making multiphase network models (both quasistatic and dynamic) predictive, but one limitation is that arbitrary (e.g., constant pressure) boundary conditions are usually assumed; they do not reflect the local saturations and pressure distributions that are affected by flow and transport in the surrounding media. In this work we demonstrate that transport behavior at the pore scale, and therefore, upscaled macroscopic properties are directly affected by the boundary conditions. Pore-scale drainage in 2D quasi-static networks is modeled by direct coupling to other pore-network models so that the boundary conditions reflect local variations of transport behavior in the surrounding media. Phase saturations are coupled at model boundaries to ensure continuity between adjacent models. Macroscopic petrophysical properties are shown to be largely dependent upon the surrounding media, which are manifested in the form of boundary conditions. The predictive ability of network simulations is thus improved using the novel network coupling scheme.

Development and experimental validation of a new one-dimensional valve boundary condition based on the method of characteristics

One-dimensional (1D) engine gas dynamic simulations are necessary in order to develop control laws for specific valve systems such as variable valve timing or the pneumatic hybrid engine. The keystone of 1D engine models is the valve boundary condition, where the pressure wave phenomenon into a pipe is created and the in-cylinder mass flowrate is determined. Most commercial or research simulation codes embed the method of characteristics (MOC) constant-pressure valve boundary condition from Benson. This constant-pressure model in its original form provides good results in most cases, but this paper shows that it does not converge for low valve opening and pressure ratio values, for which the characteristic lines become vertical. A solution to this problem cannot be found in the literature. The current paper presents a new MOC-based valve boundary condition that obviates convergence problems. The computational time of the proposed model is evaluated and compared with that of Benson’s model. The new valve boundary condition is validated experimentally on a valve gas dynamics test bench and compared with the original constant-pressure model. The study shows that results obtained with the proposed model are similar to data from both experiments and Benson’s model.