A Discrete, Arbitrarily Oriented 3D Plane-Source Analytical Solution to the Diffusivity Equation for Modeling Reservoir Fluid Flow

SPE Journal ◽  
2017 ◽  
Vol 22 (05) ◽  
pp. 1609-1623 ◽  
Author(s):  
Anqi Bao ◽  
Randy D. Hazlett ◽  
D.. Krishna Babu
Keyword(s):  
Fluid Flow ◽  
Analytical Solution ◽  
Fluid Transport ◽  
Field Experiments ◽  
Neumann Boundary ◽  
Plane Source ◽  
Multiple Fractures ◽  
Fracture Intersection ◽  
Neumann Function ◽  
Diffusivity Equation

Summary A highly accurate and efficiently computable analytical solution to the diffusivity equation is presented for modeling fluid flow into a 3D, arbitrarily oriented plane sink within a box-shaped, anisotropic medium with Neumann boundary conditions. The plane sink represents a gathering system for a well stimulated by means of hydraulic fracturing. Our plane-source Neumann function arises from analytic double integration of the point-source solution to the diffusivity equation along two vectors, forming a parallelogram. A Neumann boundary condition is achieved by means of the method of images, resulting in triple infinite summations that are reduced with mathematical identities to a combination of closed-form expressions and infinite sums with exponential damping. Our solution forecasts time-dependent behavior of fractured wells, useful in interpreting field experiments for the characterization of fracturing efficacy, reservoir size, and matrix fluid-transport properties. We demonstrate our model with two applications. One is pressure-transient analysis with identified flow regimes from a pressure vs. time plot. The other is pseudosteady-state (PSS) pressure mapping, simulating inflow from multiple fractures along the trajectory of a single horizontal well, which is achieved with superposition theory and adjustment of flux strength of each plane source to achieve a common pressure at each well/fracture intersection.

Study of fluid flow through a channel deformed by piezoelement

2018 ◽  
Vol 13 (3) ◽  
pp. 1-10 ◽  
Author(s):  
I.Sh. Nasibullayev ◽  
E.Sh Nasibullaeva ◽  
O.V. Darintsev
Keyword(s):  
Fluid Flow ◽  
Pressure Drop ◽  
Boundary Conditions ◽  
Flow Rate ◽  
Contact Surface ◽  
Piezoelectric Element ◽  
Neumann Boundary ◽  
Differential Pressure ◽  
Physical Parameters ◽  
Flow Through

The flow of a liquid through a tube deformed by a piezoelectric cell under a harmonic law is studied in this paper. Linear deformations are compared for the Dirichlet and Neumann boundary conditions on the contact surface of the tube and piezoelectric element. The flow of fluid through a deformed channel for two flow regimes is investigated: in a tube with one closed end due to deformation of the tube; for a tube with two open ends due to deformation of the tube and the differential pressure applied to the channel. The flow rate of the liquid is calculated as a function of the frequency of the deformations, the pressure drop and the physical parameters of the liquid.

scholarly journals Semi-Analytical Solution to Assess CO2 Leakage in the Subsurface through Abandoned Wells

Energies ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2452
Author(s):  
Tian Qiao ◽  
Hussein Hoteit ◽  
Marwan Fahs
Keyword(s):  
Analytical Solution ◽  
Storage Site ◽  
Co2 Leakage ◽  
Transient Pressure ◽  
Flow Paths ◽  
Abandoned Wells ◽  
Diffusivity Equation ◽  
Carbon Dioxide Co2 ◽  
Good Agreement

Geological carbon storage is an effective method capable of reducing carbon dioxide (CO2) emissions at significant scales. Subsurface reservoirs with sealing caprocks can provide long-term containment for the injected fluid. Nevertheless, CO2 leakage is a major concern. The presence of abandoned wells penetrating the reservoir caprock may cause leakage flow-paths for CO2 to the overburden. Assessment of time-varying leaky wells is a need. In this paper, we propose a new semi-analytical approach based on pressure-transient analysis to model the behavior of CO2 leakage and corresponding pressure distribution within the storage site and the overburden. Current methods assume instantaneous leakage of CO2 occurring with injection, which is not realistic. In this work, we employ the superposition in time and space to solve the diffusivity equation in 2D radial flow to approximate the transient pressure in the reservoirs. Fluid and rock compressibilities are taken into consideration, which allow calculating the breakthrough time and the leakage rate of CO2 to the overburden accurately. We use numerical simulations to verify the proposed time-dependent semi-analytical solution. The results show good agreement in both pressure and leakage rates. Sensitivity analysis is then conducted to assess different CO2 leakage scenarios to the overburden. The developed semi-analytical solution provides a new simple and practical approach to assess the potential of CO2 leakage outside the storage site. This approach is an alternative to numerical methods when detailed simulations are not feasible. Furthermore, the proposed solution can also be used to verify numerical codes, which often exhibit numerical artifacts.

The transversely isotropic poroelastic wave equation including the Biot and the squirt mechanisms: Theory and application

Geophysics ◽  
1997 ◽  
Vol 62 (1) ◽  
pp. 309-318 ◽  
Author(s):  
Jorge O. Parra
Keyword(s):  
Wave Equation ◽  
Fluid Flow ◽  
Analytical Solution ◽  
Seismic Waves ◽  
Transversely Isotropic ◽  
P Wave ◽  
Fluid Properties ◽  
Permeability Anisotropy ◽  
Attenuation And Dispersion ◽  
Maximum Permeability

The transversely isotropic poroelastic wave equation can be formulated to include the Biot and the squirt‐flow mechanisms to yield a new analytical solution in terms of the elements of the squirt‐flow tensor. The new model gives estimates of the vertical and the horizontal permeabilities, as well as other measurable rock and fluid properties. In particular, the model estimates phase velocity and attenuation of waves traveling at different angles of incidence with respect to the principal axis of anisotropy. The attenuation and dispersion of the fast quasi P‐wave and the quasi SV‐wave are related to the vertical and the horizontal permeabilities. Modeling suggests that the attenuation of both the quasi P‐wave and quasi SV‐wave depend on the direction of permeability. For frequencies from 500 to 4500 Hz, the quasi P‐wave attenuation will be of maximum permeability. To test the theory, interwell seismic waveforms, well logs, and hydraulic conductivity measurements (recorded in the fluvial Gypsy sandstone reservoir, Oklahoma) provide the material and fluid property parameters. For example, the analysis of petrophysical data suggests that the vertical permeability (1 md) is affected by the presence of mudstone and siltstone bodies, which are barriers to vertical fluid movement, and the horizontal permeability (1640 md) is controlled by cross‐bedded and planar‐laminated sandstones. The theoretical dispersion curves based on measurable rock and fluid properties, and the phase velocity curve obtained from seismic signatures, give the ingredients to evaluate the model. Theoretical predictions show the influence of the permeability anisotropy on the dispersion of seismic waves. These dispersion values derived from interwell seismic signatures are consistent with the theoretical model and with the direction of propagation of the seismic waves that travel parallel to the maximum permeability. This analysis with the new analytical solution is the first step toward a quantitative evaluation of the preferential directions of fluid flow in reservoir formation containing hydrocarbons. The results of the present work may lead to the development of algorithms to extract the permeability anisotropy from attenuation and dispersion data (derived from sonic logs and crosswell seismics) to map the fluid flow distribution in a reservoir.

A Unique Approach in Modelling Flow in Tight Oil Unconventional Reservoirs With Viscous, Inertial and Diffusion Forces Contributions

2021 ◽  
Author(s):  
Mohammed Aldhuhoori ◽  
Hadi Belhaj ◽  
Bisweswar Ghosh ◽  
Ryan Fernandes ◽  
Hamda Alkuwaiti ◽  
...  
Keyword(s):  
Fluid Flow ◽  
Fluid Transport ◽  
Synthetic Data ◽  
Diffusion Term ◽  
Flowing Fluid ◽  
New Model ◽  
Forchheimer’S Equation ◽  
Low Viscosity ◽  
Flow Parameters ◽  
Unique Approach

Abstract A model for single-phase fluid flow in tight UCRs was previously produced by modifying the flow Forchheimer’s equation. The new modification addresses the fluid transport phenomena into three scales incorporating a diffusion term. In this study, a new liner model, numerically solved, has been developed and deployed for a gas huff and puff project. The new model has been numerically validated and verified using synthetic data and huff and puff case study. Ideally, the new model suits fluid flow in tight UCRs. The modified Forchheimer’s model presented is solved using the MATLAB numerical method for linear multiphase flow. For the huff & puff case, very simple profiles and flow dynamics of the main flow parameters have been established and a thorough parametric analysis and verifications were performed. It has been observed that the diffusion system becomes more prominent in regulating flow velocity with low permeability of the formation rock and low viscosity of the flowing fluid. The findings indicate a behavioral alignment with a previous hypothesis that matches actual reservoir behavior.

scholarly journals Semi-analytical solution for the problem of extended Pom-Pom fluid flow in a round pipe

2021 ◽  
Vol 2057 (1) ◽  
pp. 012007
Author(s):  
A I Kadyirov ◽  
E K Vachagina
Keyword(s):  
Experimental Data ◽  
Fluid Flow ◽  
Analytical Solution ◽  
Solution Method ◽  
Parametric Method ◽  
Single Equation ◽  
Weissenberg Number ◽  
Normal Stresses ◽  
Rheological Equation ◽  
Rheological Equation Of State

Abstract A semi-analytical solution to the problem of the steady flow of viscoelastic single equation eXended Pom-Pom (XPP) fluid in a round pipe using the four-mode rheological equation of state of XPP is presented. An original parametric method for solving the set problem is used. The resulting method is applicable for solving a similar problem in a flat slit. The developed solution method is tested by comparing it with numerical results and experimental data. Using a polyacrylamide solution as an example, the influence of the Weissenberg number on the axial velocity profiles and the components of normal stresses is studied.

Can We Engineer Better Multistage Horizontal Completions? Evidence of the Importance of Near-wellbore Fracture Geometry from Theory, Lab and Field Experiments

2015 ◽  
Author(s):  
B.. Lecampion ◽  
J.. Desroches ◽  
X.. Weng ◽  
J.. Burghardt ◽  
J.E.. E. Brown
Keyword(s):  
Pressure Drop ◽  
Field Experiments ◽  
Horizontal Wells ◽  
Fracture Plane ◽  
Hydraulic Fractures ◽  
Multiple Fractures ◽  
Fracture Geometry ◽  
Complex Fracture ◽  
Limited Entry ◽  
The Impact

Abstract There is accepted evidence that multistage fracturing of horizontal wells in shale reservoirs results in significant production variation from perforation cluster to perforation cluster. Typically, between 30 and 40% of the clusters do not significantly contribute to production while the majority of the production comes from only 20 to 30% of the clusters. Based on numerical modeling, laboratory and field experiments, we investigate the process of simultaneously initiating and propagating several hydraulic fractures. In particular, we clarify the interplay between the impact of perforation friction and stress shadow on the stability of the propagation of multiple fractures. We show that a sufficiently large perforation pressure drop (limited entry) can counteract the stress interference between different growing fractures. We also discuss the robustness of the current design practices (cluster location, limited entry) in the presence of characterized stress heterogeneities. Laboratory experiments highlight the complexity of the fracture geometry in the near-wellbore region. Such complex fracture path results from local stress perturbations around the well and the perforations, as well as the rock fabric. The fracture complexity (i.e., the merging of multiple fractures and the reorientation towards the preferred far-field fracture plane) induces a strong nonlinear pressure drop on a scale of a few meters. Single entry field experiments in horizontal wells show that this near-wellbore effect is larger in magnitude than perforation friction and is highly variable between clusters, without being predictable. Through a combination of field measurements and modeling, we show that such variability results in a very heterogeneous slurry rate distribution; and therefore, proppant intake between clusters during a stage, even in the presence of limited entry techniques. We also note that the estimated distribution of proppant intake between clusters appears similar to published production log data. We conclude that understanding and accounting for the complex fracture geometry in the near-wellbore is an important missing link to better engineer horizontal well multistage completions.

An Analytical Solution of Micropolar-Newtonian Fluid Flow Through Annular Porous Regions

2020 ◽  
Vol 43 (5) ◽  
pp. 457-462
Author(s):  
Pramod Kumar Yadav ◽  
Jaikanth Yadav Puchakatla ◽  
Sneha Jaiswal
Keyword(s):  
Fluid Flow ◽  
Analytical Solution ◽  
Newtonian Fluid ◽  
Flow Through

Possible Flow Regimes of Adiabatic One-Dimensional Compressible Fluid Flow with Friction in Convergent and Divergent Ducts

1970 ◽  
Vol 185 (1) ◽  
pp. 925-930 ◽  
Author(s):  
A. R. Shouman ◽  
C. E. Garcia
Keyword(s):  
Fluid Flow ◽  
Analytical Solution ◽  
Friction Factor ◽  
Compressible Fluid ◽  
Flow Regimes ◽  
Dimensional Parameter ◽  
One Dimensional ◽  
Specific Heats ◽  
Compressible Fluid Flow ◽  
Half Angle

An analytical solution for the compressible one-dimensional flow in convergent and divergent ducts with friction is obtained. It is found that a non-dimensional parameter, N, can be formed using the friction factor, duct half-angle and the ratio of specific heats of the gas. Seven flow regimes are describable with the solution, based on certain bounds on the magnitude of N. The regimes are discussed and corollary data are presented graphically.

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