Transient Stresses and Displacement Around a Wellbore Due to Fluid Flow in Transversely Isotropic, Porous Media: II. Finite Reservoirs

1968 ◽  
Vol 8 (01) ◽  
pp. 79-86 ◽  
Author(s):  
M.S. Seth ◽  
K.E. Gray
Keyword(s):  
Porous Media ◽  
Radial Displacement ◽  
Outer Boundary ◽  
Stress Gradient ◽  
Fluid Pressure ◽  
Radial Distance ◽  
Transversely Isotropic ◽  
Vertical Stress ◽  
State Flow ◽  
Constant Rate

Abstract In Part 1 of this work,1 equations of elasticity were formulated for transversely isotropic, axisymmetric, homogeneous, porous media exhibiting pore fluid pressure. Equations of elasticity and the thermal analogy method were used to determine transient horizontal, tangential, and vertical stresses and radial displacement in a semi-infinite cylindrical region when either a constant rate of pressure or a constant rate of flow is maintained at the wellbore. In this paper, the approach presented earlier is extended to finite reservoirs for the cases ofsteady-state flow,constant pressures at the well bore and outer boundary andconstant pressure at the wellbore and no flow at the outer boundary. Results of this work show that radial and tangential stress gradients are high near the wellbore but diminish rapidly away from the well; the vertical stress gradient behaves in the same way but is less severe. Radial stresses are compressive or neutral, whereas tangential and vertical stresses may be tensile, neutral or compressive, depending upon the boundary conditions, the physical properties of the system and the radial distance involved (vertical stresses are always compressive in an unbounded system1). For constant boundary pressures, both radial and tangential stresses increase with time whereas they both decrease for a closed outer boundary and constant pressure at the wellbore. The vertical stress decreases with time for both systems. For steady-state systems, radial displacement may be positive or negative, depending upon the dimensions of the system, the pressure differential and the porosity. Radial displacement may be positive or negative for a closed outer boundary but is positive for constant pressures at both boundaries. INTRODUCTION The importance, utility and complexity of a realistic appraisal of the stress state at and local to a wellbore were indicated in Part 1. In this paper the analytical approach presented earlier is extended to finite, cylindrical reservoir geometry for the cases ofsteady-state flow,constant pressures at wellbore and outer boundary andconstant pressure at the wellbore and no flow at the outer boundary. Other than the outer boundary of the reservoir being finite, the physical system and assumptions pertinent thereto are the same as before. The reader may wish to review the mathematical development through Eq. 49 of Part 1 before proceeding here.

Transient Stresses and Displacement Around a Wellbore Due to Fluid Flow in Transversely Isotropic, Porous Media: I. Infinite Reservoirs

1968 ◽  
Vol 8 (01) ◽  
pp. 63-78 ◽  
Author(s):  
M.S. Seth ◽  
K.E. Gray
Keyword(s):  
Porous Media ◽  
Fluid Flow ◽  
Porous Body ◽  
Radial Displacement ◽  
Fluid Pressure ◽  
Transversely Isotropic ◽  
The State ◽  
Tangential Stresses ◽  
State Of Stress ◽  
Rock Formations

Abstract Equations of elasticity for transversely isotropic, axisymmetric, homogeneous, porous media exhibiting pore fluid pressure were formulated. Using an analogy between thermal and porous body stresses, it was shown that the solution for a transversely isotropic porous body may be obtained by incorporating body forces and the stresses due to a boundary load into the corresponding solution for the thermal stress problem. Equations of elasticity and the thermal analogy method were used to determine transient horizontal, tangential, and vertical s tresses and radial displacement in a semi-infinite cylindrical region when either a constant pressure or a constant rate of flow is maintained at the wellbore. The vertical and tangential displacements are zero from the conditions of the problem. A numerical analysis was made of the solutions obtained by using a digital computer to determine the relative influence of each system variable. Considering rock as a porous body with internal fluid pressure generally gives results significantly different than considering the rock to be nonporous; the directional character of rocks leads to significant differences as compared to results based upon the common assumption of isotropy. Stress gradients are high near the wellbore but die out away from the well. Radial stresses are compressive or neutral, whereas tangential stresses are tensile, neutral or compressive, depending upon the boundary conditions and physical properties of the system. Vertical stresses are compressive for an unbounded system. For constant wellbore injection rate, the vertical stress is proportional to the rate of fluid injection and decreases with time, whereas the radial and tangential stresses increase with time. At a given location, the radial displacement generally is very dependent upon time. INTRODUCTION A realistic appraisal of the state of stress in subsurface rock formations would be of considerable interest and use to the petroleum industry. For example, knowing the state of stress in proximity to a wellbore would be of fundamental importance in designing a fracturing operation or, more important, of clearly understanding the conditions necessary to produce rock failure of desired dimensions and geometry. Understanding conditions necessary for rock failure at the wellbore would also be of utility in a preventive sense. For example, borehole stability is an important consideration for many rock formations, and knowledge of the stress state at and near the wellbore under conditions of substantial pressure gradient due to fluid flow would be of great value. When fluid flows through a porous body which is initially at some uniform stress level, the following forces generate stresses at any point in the body.Forces due to nonuniform pressure distribution. With increasing pressure the elements of a body are compressed. Such compression cannot proceed freely in a continuum when the pressure is not uniform throughout, and thus, stresses due to flow of fluid are set up.Pore fluid pressure. This gives rise to normal stresses whose value at any point is the product of areal porosity and the fluid pressure. Although the fluid exerts uniform pressure, the stresses it creates in an anisotropic body may not be the same in all directions since the areal porosity in an anisotropic porous body is a direction-dependent quantity. This consideration leads to the concept of directional porosity.

Anisotropic Porochemoelectroelastic Solution for an Inclined Wellbore Drilled in Shale

2013 ◽  
Vol 80 (2) ◽  
Author(s):  
Minh H. Tran ◽  
Younane N. Abousleiman
Keyword(s):  
Porous Media ◽  
Fluid Pressure ◽  
Transversely Isotropic ◽  
Biological Tissues ◽  
Analytical Models ◽  
Combined Effects ◽  
Saturated Porous Media ◽  
Stress Distributions ◽  
Shale Formations ◽  
Electrically Charged

The porochemoelectroelastic analytical models have been used to describe the response of chemically active and electrically charged saturated porous media such as clay soils, shales, and biological tissues. However, existing studies have ignored the anisotropic nature commonly observed on these porous media. In this work, the anisotropic porochemoelectroelastic theory is presented. Then, the solution for an inclined wellbore drilled in transversely isotropic shale formations subjected to anisotropic far-field stresses with time-dependent down-hole fluid pressure and fluid activity is derived. Numerical examples illustrating the combined effects of porochemoelectroelastic behavior and anisotropy on wellbore responses are also included. The analysis shows that ignoring either the porochemoelectroelastic effects or the formation anisotropy leads to inaccurate prediction of the near-wellbore pore pressure and effective stress distributions. Finally, wellbore responses during a leak-off test conducted soon after drilling are analyzed to demonstrate the versatility of the solution in simulating complex down-hole conditions.

Charactervation of Adhesion in Thin-Film Materials by the Blister Test

1992 ◽  
Vol 276 ◽  
Author(s):  
Y Z. Chu ◽  
H. S. Jeong ◽  
R. C. White ◽  
C. J. Durning
Keyword(s):  
Fluid Pressure ◽  
Structural Features ◽  
Adhesion Energy ◽  
Pressure Data ◽  
Blister Test ◽  
Constant Rate ◽  
Polymer Metal ◽  
Metal Interfaces ◽  
Peel Tests ◽  
Microscopic Dynamic

ABSTRACTIn this work a blister test is applied to study the adhesion of thin films to substrates. In the blister test one injects a fluid at constant rate at the interface between the substrate and an overlayer to create a “blister”. The fluid pressure is measured as function of time. An analysis gives a reliable way of calculating the adhesion energy Ga. from the time-dependent pressure data. The method was applied to a variety of systems including polymer/polymer, polymer/silicon and polymer/metal interfaces. The results show that the test is very sensitive and is able to determine small adhesion energies inaccessible in conventional peel tests. This work demonstrates that the blister test provides a means of relating the mechanical strength of an interface to its microscopic dynamic and structural features.

scholarly journals Decayless Kink Oscillations Excited by Random Driving: Motion in Transitional Layer

Solar Physics ◽  
2021 ◽  
Vol 296 (8) ◽  
Author(s):  
M. S. Ruderman ◽  
N. S. Petrukhin ◽  
E. Pelinovsky
Keyword(s):  
Coronal Loop ◽  
Radial Displacement ◽  
Oscillation Amplitude ◽  
Radial Distance ◽  
Magnetic Pressure ◽  
Pressure Perturbation ◽  
Transitional Layer ◽  
Thin Boundary Layer ◽  
Plasma Displacement ◽  
Thin Tube

AbstractIn this article we study the plasma motion in the transitional layer of a coronal loop randomly driven at one of its footpoints in the thin-tube and thin-boundary-layer (TTTB) approximation. We introduce the average of the square of a random function with respect to time. This average can be considered as the square of the oscillation amplitude of this quantity. Then we calculate the oscillation amplitudes of the radial and azimuthal plasma displacement as well as the perturbation of the magnetic pressure. We find that the amplitudes of the plasma radial displacement and the magnetic-pressure perturbation do not change across the transitional layer. The amplitude of the plasma radial displacement is of the same order as the driver amplitude. The amplitude of the magnetic-pressure perturbation is of the order of the driver amplitude times the ratio of the loop radius to the loop length squared. The amplitude of the plasma azimuthal displacement is of the order of the driver amplitude times $\text{Re}^{1/6}$ Re 1 / 6 , where Re is the Reynolds number. It has a peak at the position in the transitional layer where the local Alfvén frequency coincides with the fundamental frequency of the loop kink oscillation. The ratio of the amplitude near this position and far from it is of the order of $\ell$ ℓ , where $\ell$ ℓ is the ratio of thickness of the transitional layer to the loop radius. We calculate the dependence of the plasma azimuthal displacement on the radial distance in the transitional layer in a particular case where the density profile in this layer is linear.

Revisiting Current Techniques for Analyzing Reservoir Performance: A New Approach for Horizontal-Well Pseudosteady-State Productivity Index

SPE Journal ◽  
2018 ◽  
Vol 24 (01) ◽  
pp. 71-91 ◽  
Author(s):  
Salam Al-Rbeawi
Keyword(s):  
Porous Media ◽  
Pressure Drop ◽  
Transient State ◽  
Darcy Flow ◽  
Analytical Models ◽  
Productivity Index ◽  
Pressure Derivative ◽  
Starting Time ◽  
State Flow ◽  
State Pressure

Summary The objective of this paper is to revisit currently used techniques for analyzing reservoir performance and characterizing the horizontal-well productivity index (PI) in finite-acting oil and gas reservoirs. This paper introduces a new practical and integrated approach for determining the starting time of pseudosteady-state flow and constant-behavior PI. The new approach focuses on the fact that the derivative of PI vanishes to zero when pseudosteady-state flow is developed. At this point, the derivative of transient-state pressure drop and that of pseudosteady-state pressure drop become mathematically identical. This point indicates the starting time of pseudosteady-state flow as well as the constant value of pseudosteady-state PI. The reservoirs of interest in this study are homogeneous and heterogamous, single and dual porous media, undergoing Darcy and non-Darcy flow in the drainage area, and finite-acting, depleted by horizontal wells. The flow in these reservoirs is either single-phase oil flow or single-phase gas flow. Several analytical models are used in this study for describing pressure and pressure-derivative behavior considering different reservoir configurations and wellbore types. These models are developed for heterogeneous and homogeneous formations consisting of single and dual porous media (naturally fractured reservoirs) and experiencing Darcy and non-Darcy flow. Two pressure terms are assembled in these models; the first pressure term represents the time-dependent pressure drop caused by transient-state flow, and the second pressure term represents time-invariant pressure drop controlled by the reservoir boundary. Transient-state PI and pseudosteady-state PI are calculated using the difference between these two pressures assuming constant wellbore flow rate. The analytical models for the pressure derivatives of these two pressure terms are generated. Using the concept that the derivative of constant PI converges to zero, these two pressure derivatives become mathematically equal at a certain production time. This point indicates the starting time of pseudosteady-state flow and the constant behavior of PI. The outcomes of this study are summarized as the following: Understanding pressure, pressure derivative, and PI behavior of bounded reservoirs drained by horizontal wells during transient- and pseudosteady-state production Investigating the effects of different reservoir configurations, wellbore lengths, reservoir homogeneity or heterogeneity, reservoirs as single or dual porous media, and flow pattern in porous media whether it has undergone Darcy or non-Darcy flow Applying the concept of the PI derivative to determine the starting time of pseudosteady-state stabilized PI The novel points in this study are the following: The derivative of the PI can be used to precisely indicate the starting time of pseudosteady-state flow and the constant behavior of PI. The starting time of pseudosteady-state flow determined by the convergence of transient- and pseudosteady-state pressure derivative or by the PI curve is always less than that determined from the curves of total pressure drop and its derivative. Non-Darcy flow may significantly affect the transient-state PI, but pseudosteady-state PI is slightly affected by non-Darcy flow. The starting time of pseudosteady-state flow is not influenced by non-Darcy flow. The convergence of transient- and pseudosteady-state pressure derivatives is affected by reservoir configurations, wellbore lengths, and porous-media characteristics.

Finite Element Contact Analysis of a Cartilage Layer Exhibiting Tension-Compression Nonlinearity

1999 ◽  
Author(s):  
Gerard A. Ateshian ◽  
Michael A. Soltz
Keyword(s):  
Finite Element ◽  
Porous Media ◽  
Experimental Studies ◽  
Transversely Isotropic ◽  
Element Solution ◽  
Biphasic Model ◽  
Cartilage Mechanics ◽  
State Of Stress ◽  
Cartilage Layer ◽  
Diarthrodial Joints

Abstract Experimental studies have demonstrated that the moduli of articular cartilage in compression are one to two orders of magnitude smaller than in tension. However, only a few analyses of cartilage mechanics have been performed which account for this tension-compression nonlinearity (Soulhat et al., 1998; Ateshian and Soltz, 1999a,b). In order to understand the state of stress under loading conditions which simulate the physiologic environment of diarthrodial joints, and the possible implications for tissue failure and the pathomechanics of osteoarthritis, it is important to determine whether the tension-compression nonlinearity of cartilage significantly affects our current understanding of its response in contact mechanics. Most analyses of cartilage contact have employed linear isotropic or transversely isotropic models for cartilage, either within the context of elasticity theory or porous media theories. In this study, we present a finite element solution for the contact of a rigid spherical impermeable sphere against a cartilage layer supported on a rigid impermeable subchondral bone foundation, where cartilage is modeled using our recently proposed biphasic conewise linear elasticity model (Ateshian and Soltz, 1999a,b). A comparison is also provided with the more frequently used linear isotropic biphasic model, under similar conditions.

Stabilized Productivity of a Hydraulically Fractured Well Producing at Constant Pressure

SPE Journal ◽  
2006 ◽  
Vol 11 (01) ◽  
pp. 120-131 ◽  
Author(s):  
Jacques Hagoort
Keyword(s):  
Flow Conditions ◽  
Productivity Improvement ◽  
Fracture Conductivity ◽  
Fracture Length ◽  
State Flow ◽  
Production Improvement ◽  
Constant Rate ◽  
Infinite Conductivity ◽  
State Production ◽  
Fractured Well

Summary This paper describes a simple and easy-to-construct numerical model for the calculation of the stabilized productivity of a hydraulically fractured well producing at a constant well pressure. The model takes into account both Darcy and non-Darcy pressure losses in the fracture. Dimensionless charts are presented that illustrate productivity improvement as a function of fracture length, fracture conductivity, and non-Darcy flow. For dimensionless fracture lengths in excess of 0.2, constant-pressure productivities are significantly lower than constant-rate productivities as predicted, for example, by the McGuire-Sikora productivity improvement chart. The maximum difference is 20% for an infinite-conductivity fracture with a length of unity. Both fracture conductivity and non-Darcy flow adversely affect well productivity; the reduction in productivity is larger for longer fractures. Introduction The productivity of a well is commonly expressed by a productivity index defined as the ratio of production rate and difference between average reservoir pressure and well pressure. Stabilized productivity refers to production from a well in the semisteady-state flow regime (i.e., the regime beyond the initial transient regime), during which flow in the reservoir is dominated by the reservoir boundaries. In the past, most studies on the stabilized productivity of hydraulically fractured wells were about steady-state production or semisteady-state production at a constant rate. As we shall demonstrate in this paper, the type of well boundary condition has a significant effect on productivity, especially for long fractures. For production by pressure depletion, characterized by declining production rates, constant well pressure is a more appropriate boundary condition. In the late 1950s, McGuire and Sikora (1960) presented a productivity improvement chart for fully penetrating fractured wells producing at a constant rate under semisteady-state flow conditions based on electrical analog model experiments. The chart shows production improvement vs. fracture conductivity for various fracture lengths. The McGuire-Sikora chart is a classic in the fracturing literature and is being used to this day. In the early 1960s, Prats (1961) presented a theoretical study on the productivity of a fully penetrating fractured well under steady-state flow conditions. He showed that the effect of a fracture can be represented by an apparent or effective wellbore radius, which depends on fracture length and fracture conductivity. For fractures that are relatively small and have an infinite conductivity, the effective wellbore radius is equal to half the fracture half-length. In a follow-up study, Prats et al. (1962) demonstrated that this result also holds for stabilized flow of a slightly compressible liquid. In the mid-1970s, Holditch presented a production improvement chart (included in Lee 1989) based on experiments with a numerical reservoir simulator, which essentially confirmed the earlier results of McGuire and Sikora. Although based on production at constant rate, the McGuire-Sikora and Holditch charts are also being used for production at declining production rates (Lee 1989).

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