It has long been known that any activity that results in changes in subsurface pressure, such as hydrocarbon production or waste or water reinjection, also causes underground deformations and movement, which can be described in terms of volumetric changes. Such deformations induce surface movement, which has a significant environmental impact. Induced surface deformations are measurable as vertical displacements; horizontal displacements; and tilts, which are the gradient of the surface deformation. The initial component of this study is a numerical model developed in C++ to predict and calculate surface deformations based on assumed subsurface volumetric changes occurring in a reservoir. The model is based on the unidirectional expansion technique using equations from Okada’s theory of dislocations (Okada, 1985). A second numerical model calculates subsurface volumetric changes based on surface deformation measurements, commonly referred to as solving for the inverse case. The inverse case is an ill-posed problem because the input is comprised of measured values that contain error. A regularization technique was therefore developed to help solve the ill-posed problem. A variety of surface deformation data sets were analyzed in order to determine the surface deformation input data that would produce the best solution and the optimum reconstruction of the initial subsurface volumetric changes. Tilt measurements, although very small, were found to be much better input than vertical displacement data for finding the inverse solution. Even in an ideal case with 0 % error, tilts result in a smaller RMSE (about 12 % smaller in the case studied) and thus a better resolution. In realistic cases with error, adding only 0.55 % of the maximum random error in the surface displacement data affects the back-calculated results to a significant extent: the RMSE increased by more than 13 times in the case studied. However, in an identical case using tilt measurements as input, adding 20 % of the maximum surface tilt value as random error increased the RMSE by 7 times, and remodelling the initial distribution of the volumetric changes in the subsurface was still possible. The required area of observation can also be reduced if tilt measurements are used. The optimal input includes tilt measurements in both directions: dz/dx and dz/dy. iv With respect to the number of observation points chosen, when tilts are used with an error of 0 %, very good resolution is obtainable using only 0.4 % of the unknowns as the number of benchmarks. For example, using only 10 observation points for a reservoir with 2500 elements, or unknowns resulted in an acceptable reconstruction. With respect to the sensitivity of the inverse solution to the depth of the reservoir and to the geometry of the observation grid, the deeper the reservoir, the more ill-posed the problem. The geometry of the benchmarks also has a significant effect on the solution of the inverse problem.

It has long been known that any activity that results in changes in subsurface pressure, such as hydrocarbon production or waste or water reinjection, also causes underground deformations and movement, which can be described in terms of volumetric changes. Such deformations induce surface movement, which has a significant environmental impact. Induced surface deformations are measurable as vertical displacements; horizontal displacements; and tilts, which are the gradient of the surface deformation. The initial component of this study is a numerical model developed in C++ to predict and calculate surface deformations based on assumed subsurface volumetric changes occurring in a reservoir. The model is based on the unidirectional expansion technique using equations from Okada’s theory of dislocations (Okada, 1985). A second numerical model calculates subsurface volumetric changes based on surface deformation measurements, commonly referred to as solving for the inverse case. The inverse case is an ill-posed problem because the input is comprised of measured values that contain error. A regularization technique was therefore developed to help solve the ill-posed problem. A variety of surface deformation data sets were analyzed in order to determine the surface deformation input data that would produce the best solution and the optimum reconstruction of the initial subsurface volumetric changes. Tilt measurements, although very small, were found to be much better input than vertical displacement data for finding the inverse solution. Even in an ideal case with 0 % error, tilts result in a smaller RMSE (about 12 % smaller in the case studied) and thus a better resolution. In realistic cases with error, adding only 0.55 % of the maximum random error in the surface displacement data affects the back-calculated results to a significant extent: the RMSE increased by more than 13 times in the case studied. However, in an identical case using tilt measurements as input, adding 20 % of the maximum surface tilt value as random error increased the RMSE by 7 times, and remodelling the initial distribution of the volumetric changes in the subsurface was still possible. The required area of observation can also be reduced if tilt measurements are used. The optimal input includes tilt measurements in both directions: dz/dx and dz/dy. iv With respect to the number of observation points chosen, when tilts are used with an error of 0 %, very good resolution is obtainable using only 0.4 % of the unknowns as the number of benchmarks. For example, using only 10 observation points for a reservoir with 2500 elements, or unknowns resulted in an acceptable reconstruction. With respect to the sensitivity of the inverse solution to the depth of the reservoir and to the geometry of the observation grid, the deeper the reservoir, the more ill-posed the problem. The geometry of the benchmarks also has a significant effect on the solution of the inverse problem.

Magneto-sensitive elastomers (MSEs) establish a special class of smart materials, which are able to change their shape and mechanical behavior under external magnetic field. Nowadays, MSEs are one of the most perspective smart materials, since they can be used for design of functionally integrated lightweight structures in sensors, robotics, actuators and damper applications. MSEs typically consist of micron-sized magnetizable particles (e.g. carbonyl iron) dispersed within a non-magnetic elastomeric matrix. The spatial distribution of magnetic particles in MSEs can be either isotropic or anisotropic, depending on whether they have been aligned by an applied magnetic field before the cross-linking of the polymer. Depending on the magnetic properties of the particles, their shape, size and spatial distribution, the MSEs can exhibit different mechanical behavior. Most experimental studies show that MSEs with isotropic distribution of magnetic particles demonstrate a uniaxial expansion along the magnetic field. On the other side, it was shown experimentally that MSEs with anisotropic particle distributions demonstrate a uniaxial contraction along the magnetic field. Also, the experimental works show that the shear moduli of MSEs increase with increasing strength of the magnetic field and depend on the magnetic properties, volume fraction and spatial distribution of particles. Different analytical approaches were used in theoretical studies of the mechanical behavior of MSEs. They can be roughly classified as phenomenological, continuum-mechanics and microscopic approaches. In the phenomenological approaches, the expansion into a series of the shear modulus as a function of the strength of the magnetic field has been proposed, the coefficients of the expansion being considered as phenomenological fitting parameters. In the continuum-mechanics approach, an MSE is considered as continuous magnetic media. It allows us to determine the shape and the change in volume of a spherical MSE sample, placed in a uniform magnetic field. However, this approach is restricted to homogeneous particle distributions. The microscopic approach has a clear advantage, while a discrete particle distribution and pair-wise interactions between induced magnetic dipoles can be considered explicitly. The aim of the present work is to develop a microscopic theory, which properly describes the mechanical behavior of MSEs in the external magnetic field. The theory takes a microscopic structure, finite shape of the samples and magneto-mechanical coupling between particle positions and sample deformation explicitly into account.

Magneto-sensitive elastomers (MSEs) establish a special class of smart materials, which are able to change their shape and mechanical behavior under external magnetic field. Nowadays, MSEs are one of the most perspective smart materials, since they can be used for design of functionally integrated lightweight structures in sensors, robotics, actuators and damper applications. MSEs typically consist of micron-sized magnetizable particles (e.g. carbonyl iron) dispersed within a non-magnetic elastomeric matrix. The spatial distribution of magnetic particles in MSEs can be either isotropic or anisotropic, depending on whether they have been aligned by an applied magnetic field before the cross-linking of the polymer. Depending on the magnetic properties of the particles, their shape, size and spatial distribution, the MSEs can exhibit different mechanical behavior. Most experimental studies show that MSEs with isotropic distribution of magnetic particles demonstrate a uniaxial expansion along the magnetic field. On the other side, it was shown experimentally that MSEs with anisotropic particle distributions demonstrate a uniaxial contraction along the magnetic field. Also, the experimental works show that the shear moduli of MSEs increase with increasing strength of the magnetic field and depend on the magnetic properties, volume fraction and spatial distribution of particles. Different analytical approaches were used in theoretical studies of the mechanical behavior of MSEs. They can be roughly classified as phenomenological, continuum-mechanics and microscopic approaches. In the phenomenological approaches, the expansion into a series of the shear modulus as a function of the strength of the magnetic field has been proposed, the coefficients of the expansion being considered as phenomenological fitting parameters. In the continuum-mechanics approach, an MSE is considered as continuous magnetic media. It allows us to determine the shape and the change in volume of a spherical MSE sample, placed in a uniform magnetic field. However, this approach is restricted to homogeneous particle distributions. The microscopic approach has a clear advantage, while a discrete particle distribution and pair-wise interactions between induced magnetic dipoles can be considered explicitly. The aim of the present work is to develop a microscopic theory, which properly describes the mechanical behavior of MSEs in the external magnetic field. The theory takes a microscopic structure, finite shape of the samples and magneto-mechanical coupling between particle positions and sample deformation explicitly into account.

A new theoretical formalism is developed for the study of the mechanical behaviour of magneto-sensitive elastomers (MSEs) under a uniform external magnetic field. This formalism allows us to combine macroscopic continuum-mechanics and microscopic approaches for complex analysis of MSEs with different shapes and with different particle distributions. It is shown that starting from a model based on an explicit discrete particle distribution one can separate the magnetic field inside the MSE into two contributions: one which depends on the shape of the sample with finite size and the other, which depends on the local spatial particle distribution. The magneto-induced deformation and the change of elastic modulus are found to be either positive or negative, their dependences on the magnetic field being determined by a non-trivial interplay between these two contributions. Mechanical properties are studied for two opposite types of coupling between the particle distribution and the magneto-induced deformation: absence of elastic coupling and presence of strong affine coupling. Predictions of a new formalism are in a qualitative agreement with existing experimental data.

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