Long-term dynamics for well productivity index for nonlinear flows in porous media

Abstract

Motivated by the reservoir engineering concept of the well Productivity Index (PI) we study a time dependent functional for general non-linear Forchheimer equation. The PI of the well characterizes the well capacity with respect to drainage area of the well. Unlike the linear case for which this concept is well developed, there are only a few recent publications dedicated to the PI for nonlinear case. In this paper the PI is comprehensively studied both theoretically and numerically. The impact of the nonlinearity of the flow filtration on the value of the PI is analyzed. Exact formula for the so called "skin factor" in radial case is derived depending on the rate of the flow, the order of nonlinearity and the geometric parameters. Dynamics of the PI for the class of boundary conditions is studied and its convergence to the specific value of steady state PI was justified. Developed framework is applied to obtain non-linear analog of Peaceman formula for the well-block pressure in unstructured grid. Numerical simulations sustain theoretical results.

Time asymptotics of non-darcy flows controlled by total flux on the boundary

“…Moreover, as was shown in [19], under the above conditions, the diffusive capacity defined on such a fully transient pressure also converges to the time-invariant pseudosteady state one. Note that the function γ(t) in the split boundary condition is not specified, but is controlled by a given constant rate Q s .…”

“…For the generalized Forchheimer equations this result was obtained in [15,19] and is recently used by the reservoir engineers in their research [20]. However, for arbitrary initial data one cannot expect the pseudo-steady state pressure distribution and the constant in time diffusive capacity.…”

“…As was proved in [15,19], the pseudo-steady state pressure p s (x, t) serves as a pseudoattractor for the fully transient pressure p(x, t) with arbitrary initial data provided the same constant total boundary flux Q s in both regimes and the split trace p(x, t)| Γ i = γ(t) + ψ(x) on the boundary. Moreover, as was shown in [19], under the above conditions, the diffusive capacity defined on such a fully transient pressure also converges to the time-invariant pseudosteady state one.…”

“…To prove the main result, in Section 4 we prove different a priori estimates for the pressure and its time and spatial derivatives. The proof of a priori bounds for the solution of the initial boundary problem in Section 4 requires more refined analysis than that used in our previous paper [19].…”

Nonlinear stability in microfluidic porous convection problems

“…407-415. Within the fields of porous media, convection with LTNE effects, and various other areas of partial differential equations, structural stability has recently been focussed on by Aulisa et al [4,5], Capone [7], Celebi et al [24], Chirita et al [25], Ciarletta et al [26], D'Amore [28], De Angelis and Renno [29], Gentile and Straughan [36], Hoang et al [43], Hoang and Ibragimov [41,42], Kalantarov and Zelik [44], Kandem [45], Kang and Park [46], Kelliher et al [48], Knops and Payne [49], Knops and Payne [50], Layton and Rebholz [53], Li et al [54], Lin and Payne [55][56][57], Liu [58,59], Liu et al [60,61], Ouyang and Yang [70], Passarella et al [71], Payne [72][73][74], Payne et al [75], Payne and Straughan [76][77][78][79][80][81], Rionero and Vergori [98], Salvadori and Visentin [99], Ugurlu [112], You et al [114].…”