# Fluid Flow and Electromagnetic Fields Modeling for Geophysical Subsurface Sensing

## Abstract

Crosswell electromagnetic (EM) fields measurement has been widely applied in oil industry which has a reservoir scale detection range. One of the limitation of this technique is that the EM singals are usually too weak to be detected. In order to overcome the disadvantage, nanoparticles (NP) designed with high contrast EM properties (conductivity, permittivity and magnetic permeability) are introduced to enhance the signals. They are injected into the formation and moving with the fluids. The movement of NP with the flow in a porous medium is modeled by solving the flow transport equations. The 3-D spectral-element time domain method (SETD) based on Gauss-Lobatto-Legendre (GLL) polynomials is employed to solve the flow equation to obtain the NP concentration distribution as a function of time. Since the method shows a spectral accuracy, i.e., the error decreases exponentially with the order of basis functions, less unknown is needed to achieve a given accuracy with a high order basis function.

The injected fluids with high contrast NP increase the electric conductivity and magnetic permeability in the flooded zone. The effective EM properties of the mixed fluids are calculated by the mixing theory, e.g., Bruggeman mixing rule. The increased EM property values produce higher EM signals in the receivers. The EM fields are then modeled by the volume integral equations (VIE), thus realizing the coupling of fluid flow and EM measurements. Based on the coupling, the detection range of the high contrast NP can be analyzed. Different types of NP are investigated under both electric and magnetic dipole sources.

The magnetic contrast NP excited by a magnetic dipole source can generate a detectable signal, while the electric contrast NP is more sensitive to an electric dipole. Using a magnetic dipole source, it is hard to generate detectable signals at receivers with high dielectric particles, however, increasing the frequency will improve the signals.

The coupling technique can also be used to evaluate the heterogeneity of the formation. When the high contrast agents are injected into a heterogeneous medium, e.g., with a low permeable region. The EM responses at the four producers are different. The signals at the producer near the barrier are lower than the other producers, since the fluids containing the high contrast NP is blocked. The proposed multiphysics coupling technique of fluid flow and EM measurements provides guidance for NP field application and help monitor the flow movement in reservoirs.

One of the applications of the high contrast agents is used for hydraulic fractures detection. Hydraulic fracturing is a technique to crack rocks by pumping high pressure fluid into a segment of a well. The created fractures serve as a pathway to release the hydrocarbon resource such as oil or natural gas from the rock. It is an efficient technology to increase the oil/gas production in tight formations. Successful fracture imaging is important to evaluate the created fractures. This is a part of a large project of the Advanced Energy Consortium (AEC) to image large-scale hydraulic fractures in deep underground with high contrast proppants injection. A group of small-scaled fracturing field tests are performed by AEC to investigate the feasibility of injecting high contrast proppants to detect fractures. The injected proppants are designed with high conductivity and permittivity to generate detectable signals at electrode-type sensors. To map the created fractures, an efficient 3-D EM inversion method with physical constraint on the inverted unknowns is developed to simultaneously reconstruct conductivity and permittivity profiles.

The inversion solver is firstly applied to a theoretical model with the noise-polluted synthetic data to reconstruct the fracture, and then applied to two hydraulic fracturing field tests with injected high conductive proppants, Loresco Coke Breeze and steel shot. The fracture conductivity and permittivity are reconstructed based on the scattered voltage signals which are the difference between the post-fracturing and pre-fracturing data. The post-fracturing data are the signals measured after the fracturing and the pre-fracturing data are measured before the fracturing. The difference signals are regarded as from the created fracture. The reconstructed fracture profiles are compared with the coring data to show the reliability of the inversion results. Their good agreement demonstrates the effectiveness of the inverse solver to estimate the fracture size and location.

In recent several decades, EM fields from layered media have attracted considerable attention concerning various applications including geophysical exploration, microwave remote sensing, wave propagation, microstrip circuits, antennas, etc. Especially, the EM waves in anisotropic laminates are of much concern. For geophysical problems, anisotropy happens commonly in many formations, e.g., shale formation. To accurately evaluate the anisotropic medium, a forward solver capable of handling arbitrary anisotropy is needed.

In this work, the formulations for EM fields in multilayered general anisotropic media are derived. Maxwell's equations in the spectral domain are written into a first-order differential (in $z$) equation concerning the transverse electric and magnetic field components in the spectral domain. The equation can be solved to obtain the EM fields in a homogeneous anisotropic medium. For fields in layered anisotropic media, the local transmission and reflection matrices, the global reflection matrices and the recursion relations of the wave amplitudes at interfaces are derived and used to express the EM fields in arbitrary layers. The electric and magnetic dipole sources can locate in arbitrary layers and the medium can possess an arbitrary anisotropy.

To transform the spectral domain solution into the spatial domain involves the inverse Fourier transform which needs integration from $-\infty$ to $+\infty$. The speed of integration calculation depends on the decaying of the integrands. The singular behavior of the fields in the close vicinity of the dipole source needs to be considered since the integrand usually converges very slowly. In this work, it is handled by subtracting the direct fields in the spectral domain, since the direct fields contribute most of the singular problem. The contributions of the subtracted part in the spatial domain are calculated and added afterwards. An example is modeled to show the convergence of the integrands with / without the singularity subtraction. The subtraction makes the integrands decaying rapidly as functions of $k_x$ and $k_y$.

To validate the algorithm, a multilayer full anisotropic medium is modeled and compared with the finite element method (FEM) results. It is also applied to the geophysical EM well logging by modeling the triaxial induction logging tool. The responses in vertical and deviated wells are computed and compared with FEM results. The good agreement between the two results further validates the algorithm and shows the capability of modeling induction logging tools in multilayered general anisotropic media.

The scattering of EM fields from anisotropic objects has been studied intensively in recent years. Most of the work studies the inhomogeneities in homogeneous isotropic background media and a few work has been done on uniaxial anisotropic media. This work extends to inhomogeneities embedded in layered general anisotropic media. The volume integral equation based on the electric dyadic Green's function is derived and solved efficiently with the fast Fourier transform (FFT) based BCGS method. The FFT technique is employed to calculate the convolution and correlation efficiently involved in the integral equation which reduces the computation cost from $O(N^2)$ to $O (N log N)$. A series of numerical examples are modeled and compared with FEM results to validate the algorithm.

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## Citation

Hu, Yunyun (2018). *Fluid Flow and Electromagnetic Fields Modeling for Geophysical Subsurface Sensing*. Dissertation, Duke University. Retrieved from https://hdl.handle.net/10161/17514.

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