Borehole Problem Research Articles

Thermo-poroelastodynamic response of a borehole in a saturated porous medium subjected to a non-hydrostatic stress field

A thermo-poroelastodynamic (TPED) model incorporating the thermo-osmosis and thermal-filtration phenomena is used to analyze the transient response of a borehole subject to a non-hydrostatic stress field. The coupled TPED model is developed on the basis of equations of motion, fluid flow, heat transfer and constitutive equations so as to couple the stress wave, pressure wave and thermal wave in early times. Poroelastodynamics (PED) and thermo-elastodynamics (TED) models correspond two particular cases of the current model. The borehole problem is decomposed into the superposition of an axisymmetric mode and an asymmetric mode, corresponding to axisymmetric and deviatoric loadings, respectively. The semi-analytical solutions in the Laplace transform space are inverted numerically to obtain time-dependent displacement, pore pressure, temperature and stresses. Comparisons are performed among different sub-models to examine the influence of thermal effect on the response of pore pressure and stresses as well as the influence of pore water on the response of stress components. A direct comparison among the dynamic, quasi-static and static models shows that inertial effect is very important in the early times and it contributes to the early-time wave behavior.

Projection pursuit adaptation on polynomial chaos expansions

The present work addresses the issue of accurate stochastic approximations in high-dimensional parametric space using tools from uncertainty quantification (UQ). The basis adaptation method and its accelerated algorithm in polynomial chaos expansions (PCE) were recently proposed to construct low-dimensional approximations adapted to specific quantities of interest (QoI). The present paper addresses one difficulty with these adaptations, namely their reliance on quadrature point sampling, which limits the reusability of potentially expensive samples. Projection pursuit (PP) is a statistical tool to find the “interesting” projections in high-dimensional data and thus bypass the curse-of-dimensionality. In the present work, we combine the fundamental ideas of basis adaptation and projection pursuit regression (PPR) to propose a novel method to simultaneously learn the optimal low-dimensional spaces and PCE representation from given data. While this projection pursuit adaptation (PPA) can be entirely data-driven, the constructed approximation exhibits mean-square convergence to the solution of an underlying governing equation and thus captures the supports and probability distributions associated with the physics constraints. The proposed approach is demonstrated on a borehole problem and a structural dynamics problem, demonstrating the versatility of the method and its ability to discover low-dimensional manifolds with high accuracy with limited data. In addition, the method can learn surrogate models for different quantities of interest while reusing the same data set.

Non-local criteria for the borehole problem: Gradient Elasticity versus Finite Fracture Mechanics

Two nonlocal approaches are applied to the borehole geometry, herein simply modelled as a circular hole in an infinite elastic medium, subjected to remote biaxial loading and/or internal pressure. The former approach lies within the framework of Gradient Elasticity (GE). Its characteristic is nonlocal in the elastic material behaviour and local in the failure criterion, hence simply related to the stress concentration factor. The latter approach is the Finite Fracture Mechanics (FFM), a well-consolidated model within the framework of brittle fracture. Its characteristic is local in the elastic material behaviour and non-local in the fracture criterion, since crack onset occurs when two (stress and energy) conditions in front of the stress concentration point are simultaneously met. Although the two approaches have a completely different origin, they present some similarities, both involving a characteristic length. Notably, they lead to almost identical critical load predictions as far as the two internal lengths are properly related. A comparison with experimental data available in the literature is also provided.

A generalized multi-fidelity simulation method using sparse polynomial chaos expansion

The polynomial chaos expansion (PCE) method has received considerable attention in uncertainty quantification (UQ). Nevertheless, it is well known that the computational cost of PCE becomes expensive, or even unaffordable for high-dimensional problems (i.e., curse of dimensionality). To alleviate the computational burden, many multi-fidelity PCE methods have been proposed, which work by approximating the high-fidelity (HF) model with the sum of a low-fidelity (LF) model and a correction function. Although the current multi-fidelity PCE methods have proven to be effective in the examples they tested, their accuracy has a strong dependence on the LF model selection. Aimed at this issue, we develop a generalized multi-fidelity PCE (GMF-PCE) using the control variate method. Specifically, an adjustable coefficient is assigned to the LF model and its optimal value is theoretically derived. Furthermore, a flexible yet still simple enough way is provided to implement the developed method, i.e., the sparse PCE based on the least angle regression (LAR) is employed to approximate the LF model, and then the subset of LF-PCE expansion is automatically detected using LAR and corrected via HF computations. Two classic algebraic examples for UQ, namely the borehole problem and the Ishigami function, as well as an unsaturated flow and heat transport problem are used to examine the performance of GMF-PCE. The results show that with the same computational cost, GMF-PCE can achieve much higher accuracy compared to the sparse LAR-based PCE and MF-PCE (non-generalized version).

An elastoplastic axisymmetric borehole problem using a deformation theory of gradient plasticity

ABSTRACT A deformation theory of gradient plasticity is employed to study elastoplastic axisymmetric boreholes subjected to far-field biaxial tension. The gradient dependence is introduced in the constitutive framework by assuming that the flow stress (i.e., an effective stress measure) depends not only on an effective strain measure but also on the Laplacian thereof. The classical theory of linear elasticity is adopted for the elastic deformations while strain softening is assumed for the plastic part, where an equivalent work hypothesis is used to associate the stress state to the final total strain. Representative stress distributions are illustrated and compared in the context of size effects, which originate from the presence of the gradient term in the governing equations. The influence of the borehole radius on the initiation of plastic deformation, the nominal stress-strain response and the evolution of the elastoplastic interface are examined. Moreover, a maximum principal strain failure criterion is employed to discuss size effects on the onset of macroscopic fracture. The obtained results show that narrower boreholes have higher plastic and failure strength.

Gradient elasticity and size effect for the borehole problem

It has been shown recently that robust gradient models of elasticity offer an effective tool to resolve existing difficulties (e.g., elimination of singularities) or predict new experimentally observed phenomena (e.g., size effects) not captured by classical theory. The price that one pays for it, however, is the need to determine extra phenomenological coefficients and invent new boundary conditions associated with the gradient terms. The modest contribution of this note is to show that even the simplest possible gradient elasticity model may yield entirely different results for different higher-order boundary conditions used. This is demonstrated by considering the borehole problem, i.e., the determination of the state of stress and strain in an externally loaded or internally pressurized body containing a cylindrical hole. The standard practice of using variationally consistent boundary conditions leads to difficulties in physically interpreting them as well as in complex solution formulae without much physical insight. Alternative, mathematically less elegant arguments, as those employed here, lead to much neater formulas, usually consisting of a sum of the classical elasticity solution and an extra gradient term. These may easily be utilized to address engineering applications, ranging from processing/fabrication of metallic specimens at the micron and nanoscales to mining/drilling operations and tunneling excavation at geo-scales. The results are by no means conclusive, only suggestive of the various possibilities and choices that one can make at present. This, in itself, points to the pressing need of further work on the issue of extra boundary conditions with the aid of new novel experiments and accompanied multiscale simulations.

Modeling and analysis of coupled thermal-hydrologic-mechanical processes during lost circulation

Abstract: Wellbore stability remains as one of the major challenges to drilling engineers. Results of recent survey showed about one third of non-productive time of drilling operation is classified as borehole problem, where a significant part is attributed to borehole instability. The thermal regime of a formation in the vicinity of a wellbore affects the stability of the well significantly during drilling operations, especially in High pressure High Temperature (HPHT) wells. This paper describes how the thermal behavior of a tubular-wellbore-reservoir system is altered during mud loss, as well as its consequent impact on near-wellbore state of stresses and critical mud weights. A fully coupled three dimensional thermal-poro-elastic model, integrated with a transient tubular-wellbore-reservoir heat flow model, is developed to evaluate near-wellbore stresses and pore pressure redistribution during mud loss. The forced convective heat transfer coefficients in both conduits-drillpipe and annulus-are rigorously determined as a function of local flow and fluid properties. The results reveal that continuous mud loss destabilizes the wellbore as the pressure window shifts considerably over time. Critical mud weights for both tensile and compressive failures decrease during mud loss over the entire length of the wellbore, in addition to the point of loss. During severe losses, the fracture gradient at the bottom of the well can decrease by over 1 ppg within the first hour. This can intensify the existing fracturing condition and allow the development of new fractures at other locations, which leads to further losses as time progresses. This model enables prediction of a more realistic operating window during lost circulation by taking additional thermally-induced effects into account. In addition, failure indexes during mud loss considering P-ρ-T coupling effect on a compressible mud column in the annulus are investigated with the proposed model. During drilling operations, especially Managed Pressure Drilling (MPD) and offshore applications with moderate to severe mud loss, the operation window should be modified by taking mud loss-induced thermal stresses into account. The mud type, wellbore geometry, circulation rate, and other parameters affect the magnitude of shift in critical mud weight during mud loss, hence evaluations should be based on specific cases.

Cylindrical Borehole Failure in a Transversely Isotropic Poroelastic Medium

Rocks underground often have pores and bedding planes, which are appropriate to be described by the transversely isotropic poroelastic constitutive model. Drilling boreholes in these rocks must be careful, since stresses and pore pressure would change with time, because of the inherent time dependent property of poroelasticity as well as pore fluid diffusion. In order to correlate the behavior of transversely isotropic poroelastic model of borehole in plane strain with the behavior of isotropic poroelastic model, an equivalent isotropic material is built with carefully chosen material constants, and correlation rules are successfully developed. With the solutions for the borehole problem in an isotropic model obtained previously, the solutions to transversely isotropic model can be obtained. Two cases of tensile failure and six cases of shear failure for the borehole are considered. As a result, the allowable borehole working pressure range is formulated by explicit expressions. The failure case, time, and location could also be obtained for any given drilling pressure. Results obtained from the Hooke’s traditional elastic model are compared, and it is found that poroelastic model is necessary in borehole safety check, while Hooke’s model is not on the safe side.

Analyzing wellbore stability in chemically-active anisotropic formations under thermal, hydraulic, mechanical and chemical loadings

Several factors influence the stress state in subsurface rocks. In addition to pore pressure and far-field in-situ stresses, thermal and chemical gradients have substantial bearings on the in-situ stress state during and after drilling operations. Drilling inclined boreholes through laminated formations such as shaley sands presents several challenges. Due to their laminated nature, shales demonstrate high degree of elastic anisotropy, and are often chemically-reactive to drilling fluid. The majority of models utilized in the industry assume a homogeneous isotropic elastic static model that fails to give an accurate depiction of the observed borehole failure. In this work, governing equations for anisotropic porochemothermoelasticity are developed to simulate the drilling of an inclined borehole problem in a transversely isotropic rock. Using the developed constitutive and transport equations, a finite element method based numerical model is constructed to estimate the pore pressure, temperature, solute concentration and stress distribution. Nonlinear conductive-convective heat transfer and diffusive-advective solute transfer models are considered in this work to address both high and low permeability formations. Finally, the model is used to assess time-dependent wellbore stability during and after drilling. The novel pseudo-3D analysis developed in this work is advantageous for real-time operations due to its computational speed and stability.

Anisotropic Diffusive-Advective Porochemoelasticity Modeling for Inclined Boreholes

AbstractIn chemically active rocks, pore pressure and stress state are not the only parameters that are governed by hydromechanical processes. Chemical osmosis is also responsible for changes in the effective stress caused by the presence of chemical potential between the reservoir and the borehole filled with drilling fluid. Subsurface rocks such as shales are highly anisotropic, and assuming an isotropic model will fail to provide an accurate depiction of the in situ stress state. In this work, governing equations for a fully coupled anisotropic nonlinear porochemoelasticity are developed. A numerical model using the FEM is constructed to simulate the drilling of an inclined borehole problem in an anisotropic and chemically active medium. The model accounts for solute transport contributed by diffusion and advection. Several sensitivity analyses are conducted to highlight different features of the chemohydromechanical model. Solute concentration, membrane efficiency, solute-diffusion coefficient, and ad...

HIGH DIMENSIONAL SENSITIVITY ANALYSIS USING SURROGATE MODELING AND HIGH DIMENSIONAL MODEL REPRESENTATION

In this paper, a new non-intrusive method for the propagation of uncertainty and sensitivity analysis is presented. The method is based on the cut-HDMR approach, which is here derived in a different way and new conclusions are presented. The cut-HDMR approach decomposes the stochastic space into sub-domains, which are separately interpolated via a selected interpolation technique. This leads to a dramatic reduction of necessary samples for high dimensional spaces and decreases the influence of the Curse of Dimensionality. The proposed non-intrusive method is based on the coupling of an interpolation technique with the cut-HDMR (high dimension model representation) approach. The new conclusions obtained from the new derivation of the cut-HDMR approach allow one to interpolate each stochastic domain separately, including all stochastic variables and interactions between variables. Moreover, the same conclusions allow one to neglect non-important stochastic domains and therefore, drastically reduce the number of samples to detect and interpolate the higher order interactions. A new sampling strategy is introduced, which is based on a tensor product, but it uses the idea of Smoylak sparse grid for higher domains. For this work, the multi-dimensional Lagrange interpolation technique is selected and is applied for all parts of the cut-HDMR approach. However, the nature of the method allows one to use a combination of various interpolation techniques. The sensitivity analysis is performed on the surrogate model using the Monte Carlo sampling. In this work, the Sobol's approach is followed and sensitivity indices are established for each variable and interaction. Moreover, due to the obtained conclusions, the separate surrogate models allow one to visualize the uncertainty in the high dimensional space via histograms. The usage of a histogram for each stochastic domain allows one to establish full statistical properties of a given stochastic domain. This helps the user to better understand the stochastic propagation for the model of interest. The proposed interpolation technique and sensitivity analysis approach are tested on a simple example and applied on the well-known Borehole problem. Results of the proposed method are compared to the Monte Carlo sampling using the mean value and the standard deviation. Results of the sensitivity analysis of the Borehole case are compared to the literature results and the statistical visualization of each variable is provided.

Unified solution of a borehole problem with size effect

Based on the strain gradient theory and the unified yield criterion, the borehole problem of an elasto-plastic plane strain body containing a traction-free hole subjected to uniform far-field stress is studied. The unified expressions for the plastic region radius and the stress concentration factor considering the size effect are derived on the basis of the unified yield criterion, respectively, which can be reduced to those based on the Tresca, von Mises and twin-shear criteria. The dependences of the plastic region radius and the stress concentration on the yield criteria, the material strain hardening level and the size effect are discussed. As a result, it is concluded that the influences of yield criteria and strain hardening level on the plastic region radius and the stress concentration are significant, and the size effect cannot be ignored when the hole size is in the order of tens of microns.

Loss of uniqueness of numerical solutions of the borehole problem modelled with enhanced media

It is well known that, initial boundary value problems involving constitutive equations modeling the degradation of the strength of materials are not well posed, which renders computations questionable. To overcome this issue it is necessary to enhance the models by incorporating some internal length. It has been shown that such an enhancement restores the objectivity of the computation as spurious mesh dependency is avoided. However, at least for simple problems (e.g. one dimensional ones), it has been proven that uniqueness of the underlying mathematical problem itself is not restored. Moreover numerical modeling of element tests yields several solutions. This paper demonstrates that several numerical solutions can be obtained also for less simple problems, namely the borehole problems. Even when a defect is introduced in the computed problems, different numerical solutions are found. Contrary to the one dimensional problem there is no proof that this loss of uniqueness comes from the underlying mathematical problem. It is our opinion that this is an inherent property of initial boundary value problems where, broadly speaking, strong degradation of the mechanical properties is modeled. In any case, it is necessary to be aware of this issue. For problems involving constitutive equation modeling strength degradation, it is important to try to find other solutions than the one obtained by using routinely a numerical code. The failure patterns of the different solutions found are however similar to experimental observations. This possible loss of uniqueness can then be seen as a counterpart of the difficulties encountered when attempting to reproduce experiments. This is crucial when dealing with geomaterials.

Cylindrical FDTD Analysis of LWD Tools Through Anisotropic Dipping-Layered Earth Media

Electrical logging-while-drilling (LWD) tools are commonly used in oil and gas exploration to estimate the conductivity (resistivity) of adjacent Earth media. In general, Earth media exhibit anisotropic conductivities. This implies that when LWD tools are used for deviated and horizontal drilling, the resulting borehole problem may include dipping-layered media with dipping beds having full 3 times 3 conductivity tensors. To model this problem, we describe a 3-D cylindrical finite-difference time-domain (FDTD) algorithm extended to fully anisotropic conductive media and implemented with cylindrical perfectly matched layers to mimic open-domain problems. The 3-D FDTD algorithm is validated against analytical results in simple formations, showing good agreement, and used to simulate the response of LWD tools through anisotropic dipping beds for various values of anisotropic conductivities and dipping angles

Analyses of Inclined Boreholes in Poroelastic Media

Solutions for the borehole problem based on Biot’s poroelasticity theory have shown that drilling through fluid saturated formations, gives rise to time-dependent stress and pore pressure fields in the vicinity of the borehole. As a natural consequence, borehole stability in such formations is of a time-dependent nature. However, existing analyses are based on two limiting cases viz., constant pore pressure and/or no flux boundary conditions at the borehole wall. Adding time dependency to the pore fluid boundary conditions can simulate realistic field conditions such as those observed during hydraulic fracturing, fluid injection or development of filter cake. Analytical solutions for inclined boreholes with time-dependent pore pressure and flux boundary conditions at the borehole wall are presented in this paper. Analysis is carried out for two special cases of the ramp-type pore pressure and linearly reducing flux boundary conditions. Analytical solutions are supplemented with asymptotic solutions for small and large time analysis. The effects of these conditions on stress concentrations near the borehole wall and their implications on borehole stability are examined in detail.

Analytical Solution of a Borehole Problem Using Strain Gradient Plasticity

An analytical solution is presented for the borehole problem of an elasto-plastic plane strain body containing a traction-free circular hole and subjected to uniform far field stress. A strain gradient plasticity theory is used to describe the constitutive behavior of the material undergoing plastic deformations, whereas the generalized Hooke’s law is invoked to represent the material response in the elastic region. This gradient plasticity theory introduces a higher-order spatial gradient of the effective plastic strain into the yield condition to account for the nonlocal interactions among material points, while leaving other relations in classical plasticity unaltered. The solution gives explicit expressions for the stress, strain, and displacement components. The hole radius enters these expressions not only in nondimensional forms but also with its own dimensional identity, unlike classical plasticity-based solutions. As a result, the current solution can capture the size effect in a quantitative manner. The classical plasticity-based solution of the borehole problem is obtained as a special case of the present solution. Numerical results for the plastic region radius and the stress concentration factor are provided to illustrate the application and significance of the newly derived solution.

Dynamic poroelastic soil column and borehole problem analysis : Vgenopoulou, I; Beskos, D E Soil Dynam Earthq EngngV11, N6, 1992, P335–345

Dynamic poroelastic soil column and borehole problem analysis : Vgenopoulou, I; Beskos, D E Soil Dynam Earthq EngngV11, N6, 1992, P335–345

Dynamics of Saturated Rocks. IV: Column and Borehole Problems

The one-dimensional dynamic column and borehole problems of rock mechanics are solved analytically and numerically on the assumption that the water fully saturated, elastic rock medium is characterized by two degrees of porosity—one due to the pores and the other due to the fissures. Use of Laplace transform with respect to time reduces the governing equations of the column and borehole problems to ordinary differential equations with constant and variable coefficients, respectively. The transformed solution of these problems is obtained analytically for the column and by finite differences for the borehole problem and after a numerical Laplace transform inversion produces the time domain response to suddenly applied or harmonically varying with time loads. The quasi-static counterparts of these problems are analyzed as special cases of the dynamic ones. Numerical studies reveal that inertial effects depend on the kind of the applied load, the influence of porosities and permeabilities is rather small, and that the response is rather close to the one for the single-porosity model.

Dynamic poroelastic soil column and borehole problem analysis

The one-dimensional dynamic column and borehole problems of soil mechanics formulated on the basis of the poroelastic theory of Vardoulakis and Beskos are solved analytically-numerically. The quasi-static counterparts of these problems are analysed as special cases of the dynamic ones. Use of Laplace transform with respect to time reduces the column and borehole problems to ordinary differential equations with constant and variable coefficients, respectively. The transformed solution of these problems is obtained analytically for the column and by finite differences for the borehole problem, and after, a numerical Laplace transform inversion produces the time domain response. Both a suddenly applied and a harmonically varying with time load are considered. It is concluded that the significance of inertial effects depends on the kind of loading and that the degree of saturation for the nearly saturated case greatly affects the response.

Regression models for hydraulic conductivity and field test of the borehole permeameter

The saturated hydraulic conductivity Ks of sediments in the vadose zone is an important parameter in predicting the seepage rates of water and contaminants. The borehole permeameter is an in situ technique to test a relatively large sample size at any depth. Solutions are presented which account for the effects of unsaturated flow. These solutions are derived from a regression analysis of results of numerical simulations in which unsaturated hydraulic conductivity is represented by one or two parameters. The results of a borehole permeameter test in a uniform sand are compared with field ponding tests and air entry permeameter tests. The regression‐based solutions for the borehole permeameter which account for capillarity provide very good agreement with other permeameter results. Depending upon the approach used to solve the borehole problem Ks values determined by methods which neglected and included capillary effects varied by a factor of only about two for the soil tested.