Evan Schankee Um

3D time-domain simulation of electromagnetic diffusion phenomena: A finite-element electric-field approach

We present a finite-element time-domain FETD approach for the simulation of 3D electromagnetic EM diffusion phenomena. The finite-element algorithm efficiently simulates transient electric fields and the time derivatives of magneticfields in general anisotropic earth media excited by multiple arbitrarily configured electric dipoles with various signal waveforms. To compute transient electromagnetic fields,theelectricfielddiffusionequationistransformedinto asystemofdifferentialequationsviaGalerkin’smethodwith homogeneous Dirichlet boundary conditions. To ensure numerical stability and an efficient time step, the system of the differential equations is discretized in time using an implicit backward Euler scheme. The resultant FETD matrix-vector equation is solved using a sparse direct solver along with a fill-inreducedorderingtechnique.Whenadvancingthesolution in time, the FETD algorithm adjusts the time step by examining whether or not the current step size can be doubled without unacceptably affecting the accuracy of the solution. To simulate a step-off source waveform, the 3D FETD algorithm also incorporates a 3D finite-element direct current FEDCalgorithmthatsolvesPoisson’sequationusingasecondarypotentialmethodforageneralanisotropicearthmodel. Examples of controlled-source FETD simulations are compared with analytic and/or 3D finite-difference time-domain solutions and are used to confirm the accuracy and efficiencyofthe3DFETDalgorithm.

Fracture Propagation, Fluid Flow, and Geomechanics of Water-Based Hydraulic Fracturing in Shale Gas Systems and Electromagnetic Geophysical Monitoring of Fluid Migration

Author(s): Kim, Jihoon; Um, Evan; Moridis, George | Abstract: We investigate fracture propagation induced by hydraulic fracturing with water injection, using numerical simulation. For rigorous, full 3D modeling, we employ a numerical method that can model failure resulting from tensile and shear stresses, dynamic nonlinear permeability, leak-off in all directions, and thermo-poro-mechanical effects with the double porosity approach. Our numerical results indicate that fracture propagation is not the same as propagation of the water front, because fracturing is governed by geomechanics, whereas water saturation is determined by fluid flow. At early times, the water saturation front is almost identical to the fracture tip, suggesting that the fracture is mostly filled with injected water. However, at late times, advance of the water front is retarded compared to fracture propagation, yielding a significant gap between the water front and the fracture top, which is filled with reservoir gas. We also find considerable leak-off of water to the reservoir. The inconsistency between the fracture volume and the volume of injected water cannot properly calculate the fracture length, when it is estimated based on the simple assumption that the fracture is fully saturated with injected water. As an example of flow-geomechanical responses, we identify pressure fluctuation under constant water injection, because hydraulic fracturing is itself a set of many failure processes, in which pressure consistently drops when failure occurs, but fluctuation decreases as the fracture length grows. We also study application of electromagnetic (EM) geophysical methods, because these methods are highly sensitive to changes in porosity and pore-fluid properties due to water injection into gas reservoirs. Employing a 3D finite-element EM geophysical simulator, we evaluate the sensitivity of the crosswell EM method for monitoring fluid movements in shaly reservoirs. For this sensitivity evaluation, reservoir models are generated through the coupled flow-geomechanical simulator and are transformed via a rock-physics model into electrical conductivity models. It is shown that anomalous conductivity distribution in the resulting models is closely related to injected water saturation, but not closely related to newly created unsaturated fractures. Our numerical modeling experiments demonstrate that the crosswell EM method can be highly sensitive to conductivity changes that directly indicate the migration pathways of the injected fluid. Accordingly, the EM method can serve as an effective monitoring tool for distribution of injected fluids (i.e., migration pathways) during hydraulic fracturing operations

A tetrahedral mesh generation approach for 3D marine controlled-source electromagnetic modeling

Abstract 3D finite-element (FE) mesh generation is a major hurdle for marine controlled-source electromagnetic (CSEM) modeling. In this paper, we present a FE discretization operator (FEDO) that automatically converts a 3D finite-difference (FD) model into reliable and efficient tetrahedral FE meshes for CSEM modeling. FEDO sets up wireframes of a background seabed model that precisely honors the seafloor topography. The wireframes are then partitioned into multiple regions. Outer regions of the wireframes are discretized with coarse tetrahedral elements whose maximum size is as large as a skin depth of the regions. We demonstrate that such coarse meshes can produce accurate FE solutions because numerical dispersion errors of tetrahedral meshes do not accumulate but oscillates. In contrast, central regions of the wireframes are discretized with fine tetrahedral elements to describe complex geology in detail. The conductivity distribution is mapped from FD to FE meshes in a volume-averaged sense. To avoid excessive mesh refinement around receivers, we introduce an effective receiver size. Major advantages of FEDO are summarized as follow. First, FEDO automatically generates reliable and economic tetrahedral FE meshes without adaptive meshing or interactive CAD workflows. Second, FEDO produces FE meshes that precisely honor the boundaries of the seafloor topography. Third, FEDO derives multiple sets of FE meshes from a given FD model. Each FE mesh is optimized for a different set of sources and receivers and is fed to a subgroup of processors on a parallel computer. This divide and conquer approach improves the parallel scalability of the FE solution. Both accuracy and effectiveness of FEDO are demonstrated with various CSEM examples.

On the Physics of the Marine-Time-Domain Controlled Source Electromagnetic Method for Detecting Hydrocarbon Reservoirs

Summary This paper addresses the physics of the marine time-domain controlled source electromagnetic (TDCSEM) method. Like the marine frequency-domain controlled source electromagnetic (FDCSEM) method, the TDCSEM method is sensitive to resistive hydrocarbon reservoirs. Employing forward modeling techniques is helpful to better understand the physics of the TDCSEM method.

A finite element algorithm for 3-D transient electromagnetic modeling

Summary We present a 3-D finite-element time-domain (FETD) algorithm for the simulation of electromagnetic (EM) diffusion phenomena. The algorithm simulates transient electric fields and time derivatives of the magnetic fields for a general anisotropic earth. In order to compute transient fields, the electric field wave equation is transformed into a system of ordinary differential equations (ODE) via a Galerkin method with Dirichlet boundary conditions. To ensure both numerical stability and an efficient time step size, the system of ODE is discretized in time using the implicit backward Euler scheme. The resultant FETD matrix-vector equation is solved using a sparse direct solver with a fill-in reducing algorithm. When advancing the solution in time, the algorithm adjusts the tine step by examining if or not a current step size can be doubled without affecting the accuracy of the solution. Instead of directly solving another FETD matrix-vector equation for transient magnetic fields, Faraday’s law is employed to compute time-derivatives of magnetic fields only at receiver positions. The accuracy and efficiency of the FETD algorithm are demonstrated using time-domain controlled source EM (TD-CSEM) simulations.

A Lorenz-gauged finite-element solution for transient CSEM modeling

Summary We present a Lorenz-gauged finite-element (FE) solution for transient controlled-source electromagnetic (CSEM) modeling. Using the time-derivative of the Lorenz gauge, we split the single vector-and-scalar potential equation into a diffusion equation for vector potentials and Poisson’s equation for scalar potentials. The diffusion equation is considered as a primary equation and is solved for the timederivatives of vector potentials at every time step. In contrast, the Poisson’s equation is considered as an auxiliary equation and is evaluated only at the time steps where the electric fields are sampled. The Lorenz-gauged diffusion equation does not include the scalar potential term and therefore, has the minimum number of unknowns. The diffusion equation is converted into a system of FE equations using Galerkin method and edge elements. The resulting system can be solved with an iterative solver in the static limit and is advanced efficiently with the adaptive time-step doubling method.

On the Physics of Seabed Logging over 3D Hydrocarbon Reservoirs

Summary We present numerical modeling of seabed logging (SBL) for 3-D hydrocarbon detection. We show that the SBL results for the 3-D reservoir are very sensitive to survey configuration. Three dimensional forward modeling and plotting of the fields and currents in the sub-seafloor sediments is helpful to optimize SBL survey configuration and understand its response.