Development of Fractional Derivative-Based Mass and Heat Transport Model

PROCEEDINGS, Thirty-Eighth Workshop on Geothermal Reservoir Engineering 
Stanford University, Stanford, California, February 11-13, 2013 
SGP-TR-198 
DEVELOPMENT OF FRACTIONAL DERIVATIVE-BASED
MASS AND HEAT TRANSPORT MODEL

1
Anna SUZUKI, 

2
Roland N. HORNE, 
1
Hiroshi MAKITA, 
1
Yuichi NIIBORI, 
3
Sergei A. FOMIN,
4
Vladimir A. CHUGUNOV, 
1
Toshiyuki HASHIDA 
1
Tohoku University,
6-6-11-707, Aramaki-Aza-Aoba,Aoba 
Sendai 980-8579, Japan 
e-mail: anna.suzuki@rift.mech.tohoku.ac.jp 
2 
Stanford Geothermal Program 
Department of Energy Resources Engineering, 367 
Panama St., 
Stanford, CA 94305-2220, USA
e-mail: horne@stanford.edu 
3
California State University
Chico, CA 95929, USA 
e-mail: sfomin@csuchico.edu 
4
Kazan Federal University
Kazan 420008, Russia 
e-mail: Vladimir.Chugunov@ksu.ru 
ABSTRACT 
A numerical scheme to evaluate the effect of cold 
water injection in a geothermal reservoir has been 
proposed. The governing equations are derived based 
on non-Fickian mass transport model. We assume 
conduction into surrounding rocks using fractional 
derivative in time, as well as non-Fickian diffusion 
into the surrounding rocks. The constitutive 
parameters in the heat transfer model are able to 
determine mechanisms of heat-fluid–rock interaction 
within the reservoir of a complex geological 
structure. 
Numerical investigations have shown that topography 
of a reservoir wall affects tracer responses and 
temperature distributions. The long tails can be found 
in tracer response curves in a reservoir with rough 
surfaces. The permeability difference between a 
reservoir and surrounding rocks leads to retardation 
of tracer and thermal breakthrough.