Numerical Modelling of Electroacoustic Logging Including Joule Heating
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rical Modelling of Electroacoustic Logging Including Joule Heating B. D. Plyushchenkova, *† and A. A. Nikitinb, ** a
Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, 125047 Russia b Faculty of Geology, Lomonosov Moscow State University, Moscow, 119991 Russia *e-mail: [email protected] **e-mail: [email protected] Received June 27, 2019; revised June 27, 2019; accepted September 9, 2019
Abstract—Modification of Pride’s equations, describing the interrelated process of acoustic and electromagnetic waves’ propagation in saturated porous medium, arising due to the electrokinetic effect, is proposed. This modification allows taking into account Joule heating during the propagation of acoustic oscillations, generated by electromagnetic source, and is implemented by adding “thermoelastic terms” into the state equations for porous fluid and formation. Temperature change is determined by heat conduction equation. Method of constructing of corresponding finite-difference scheme in axially symmetric case is described. The results of numerical simulations are presented, which show that at low frequencies the Stoneley wave can be used to estimate formation permeability, and the fast acoustic wave can be used for this purpose at high frequencies, when the influence of Joule heating can be neglected. However, if the conductivity of porous or especially borehole fluid is high enough, then the thermoelastic wave arises due to Joule heating, whose contribution is hard to account for due to the nonlinearity of such an effect. Keywords: electrokinetic effect, Joule heating, numerical modelling, electroacoustic logging DOI: 10.1134/S2070048220050166
INTRODUCTION In a porous medium saturated with a fluid electrolyte the electric double layer (EDL) is formed on the contact of solid and liquid phases [1]. The solid phase (matrix) is assumed to be dielectric, and EDL’s surface is usually charged negatively. In the porous medium EDL is represented by adsorption and diffuse layers. The adsorption layer, two or three molecules thick, is located on the matrix’s surface and consists of positive (hydrated) ions, which are immovable due to short range Coulomb forces. The diffuse layer is significantly thicker and over-saturated with positive ions, which can be move due to long range Coulomb forces. When the acoustic field creates the motion of the pore fluid relative to the matrix, the diffuse layer oscillates due to the fluid viscosity. As a result, the streaming electric current is induced, which in turn creates an electromagnetic field. On the contrary, when the electromagnetic field acts on the diffuse layer, the fluid viscosity also causes it to move relative to the matrix, thereby an acoustic field is formed. These interconnected phenomena are called acoustoelectric and electroacoustic conversions, respectively, and, in general, the electrokinetic effect. Since the electrokinetic effect correlates with the formation properties, these phenomena were studied experimentally [2, 3] for the purpose of geophy
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