Introduction to Reservoir Simulation as Practiced in Industry Knut{Andreas Lie SINTEF Digital, Mathematics and Cybernetics / Department of Mathematical Sciences, NTNU, Norway Computational Issues in Oil Field Applications Tutorials March 21{24, 2017, IPAM, UCLA 1 / 52 Petroleum reservoirs Naturally occurring flammable liquid/gases found in geological formations I Originating from organic sediments that have been compressed and 'cooked' to form hydrocarbons that migrated upward in sedimentary rocks until limited by a trapping structure I Found in shallow reservoirs on land and deep under the seabed I Only 30% of the reserves are 'conventional'; remaining 70% include shale oil and gas, heavy oil, extra heavy oil, and oil sands. Uses of (refined) petroleum: I Fuel (gas, liquid, solid) I Alkenes manufactured into plastics and compounds I Lubricants, wax, paraffin wax I Pesticides and fertilizers for Johan Sverdrup, new Norwegian 'elephant' discovery, 2011. agriculture Expected to be producing for the next 30+ years 2 / 52 Production processes Gas Oil Caprock Aquifer w/brine Primary production { puncturing the 'balloon' When the first well is drilled and opened for production, trapped hydrocarbon starts flowing toward the well because of over-pressure 3 / 52 Production processes Gas injection Gas Oil Water injection Secondary production { maintaining reservoir flow As pressure drops, less hydrocarbon is flowing. To maintain pressure and push more profitable hydrocarbons out, one starts injecting water or gas into the reservoir, possibly in an alternating fashion from the same well. 3 / 52 Production processes Gas injection Gas Oil Water injection Enhanced oil recovery Even more crude oil can be extracted by gas injection (CO2, natural gas, or nitrogen), chemical injection (foam, polymer, surfactants), microbial injection, or thermal recovery (cyclic steam, steam flooding, in-situ combustion), etc. 3 / 52 Why reservoir simulation? To estimate reserves and support economic and operational decisions To this end, reservoir engineers need to: I understand reservoir and fluid behavior I quantify uncertainty I test hypotheses and compare scenarios I assimilate data I optimize recovery processes 4 / 52 1 a geological model { volumetric grid with cell/face properties describing the porous rock formation 1 0.8 0.6 @t(φbwSw) + r · (bw~uw) = bwqw 2 0.4 a flow model { describes how fluids flow @t[φ(bwSo + bgrvSg)] + r · (bo~uo + bgrv~ug) = boqo + bgrvqg in a porous medium (conservation laws + 0.2 @t[φ(bgSg + borsSo)] r · appropriate closure relations) + (bg~ug + bors~uo) = bgqg0 + borsqo 0 0.2 0.4 0.6 0.8 1 3 a well model { describes flow in and out of the reservoir, in the wellbore, flow control devices, surface facilities Reservoir models Somewhat simplified, consist of three parts: 5 / 52 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 @t(φbwSw) + r · (bw~uw) = bwqw 2 a flow model { describes how fluids flow @t[φ(bwSo + bgrvSg)] in a porous medium (conservation laws + + r · (bo~uo + bgrv~ug) = boqo + bgrvqg @t[φ(bgSg + borsSo)] appropriate closure relations) + r · (bg~ug + bors~uo) = bgqg + borsqo 3 a well model { describes flow in and out of the reservoir, in the wellbore, flow control devices, surface facilities Reservoir models Somewhat simplified, consist of three parts: 1 a geological model { volumetric grid with cell/face properties describing the porous rock formation 5 / 52 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 3 a well model { describes flow in and out of the reservoir, in the wellbore, flow control devices, surface facilities Reservoir models Somewhat simplified, consist of three parts: 1 a geological model { volumetric grid with cell/face properties describing the porous rock formation @t(φbwSw) + r · (bw~uw) = bwqw 2 a flow model { describes how fluids flow @t[φ(bwSo + bgrvSg)] in a porous medium (conservation laws + + r · (bo~uo + bgrv~ug) = boqo + bgrvqg @t[φ(bgSg + borsSo)] appropriate closure relations) + r · (bg~ug + bors~uo) = bgqg + borsqo 5 / 52 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Reservoir models Somewhat simplified, consist of three parts: 1 a geological model { volumetric grid with cell/face properties describing the porous rock formation @t(φbwSw) + r · (bw~uw) = bwqw 2 a flow model { describes how fluids flow @t[φ(bwSo + bgrvSg)] in a porous medium (conservation laws + + r · (bo~uo + bgrv~ug) = boqo + bgrvqg @t[φ(bgSg + borsSo)] appropriate closure relations) + r · (bg~ug + bors~uo) = bgqg + borsqo 3 a well model { describes flow in and out of the reservoir, in the wellbore, flow control devices, surface facilities 5 / 52 Geologic model: sedimentary rocks Mineral particles broken off by weathering and erosion Transported by wind or water to a place where they settle and accumulate into a sediment, building up in lakes, rivers, sand deltas, lagoons, choral reefs, etc 6 / 52 Geologic model: sedimentary rocks Erosion Deposition Flood plain Mud Sand Gravel Layered structure with different mixtures of rock types with varying grain size, mineral type, and clay content Thin beds that stretch hundreds or thousands of meters, typically horizontally or at a small angle. Gradually buried deeper and consolidated 6 / 52 Geologic model: sedimentary rocks Normal dip-slip fault Reverse dip-slip fault Strike-slip fault Geological activity will later fold, stretch, and fracture the consolidated rock 6 / 52 Geologic model: sedimentary rocks Structural trap: anticline Stratigraphic traps Sandstone encased Gas in mudstone Unconformity Oil Pinch out Impermeable rock Permeable rock with brine Fault trap Salt dome Fault Impermeable salt 6 / 52 Geologic model: sedimentary rocks Outcrops of sedimentary rocks from Svalbard, Norway. Length scale: ∼100 m 6 / 52 Geologic model: sedimentary rocks Layered geological structures typically occur on both large and small scales 6 / 52 Porous media flow { a multiscale problem The scales that impact fluid flow in subsurface rocks range from I the micrometer scale of pores and pore channels I via dm-m scale of well bores and laminae sediments I to sedimentary structures that stretch across entire reservoirs Porous rocks are heterogeneous at all length scales (no scale separation) −! 7 / 52 Porous media flow { a multiscale problem −! 7 / 52 Flow model: representative elementary volume Porosity: V φ = v Vv + Vr The assumption of a repre- sentative elementary volume (REV) is essential in macro- scale modeling of porous media. Here illustrated for porosity. 8 / 52 I Darcy's law: ~u = −K(rp − ρgrz) empirical law for describing processes on an unresolved scale. Similar to Fourier's law (heat) [1822], Ohm's law (electric current) [1827], Fick's law (concentration) [1855], except that we now have two driving forces Governing equations for fluid flow In its simplest form { two main principles I Conservation of mass @ Z I Z m dx + F~ · ~nds = r dx @t V @V V m=mass, F~ =flow rate, r=fluid sources 9 / 52 Governing equations for fluid flow In its simplest form { two main principles I Conservation of mass @ Z I Z m dx + F~ · ~nds = r dx @t V @V V m=mass, F~ =flow rate, r=fluid sources I Darcy's law: ~u = −K(rp − ρgrz) empirical law for describing processes on an unresolved scale. Similar to Fourier's law (heat) [1822], Ohm's law (electric current) [1827], Fick's law (concentration) [1855], except that we now have two driving forces 9 / 52 Darcy's law and permeability In reservoir engineering: K ~u = − rp − ρgrz µ Intrinsic permeability K measures ability to transmit fluids Anisotropic and diagonal by nature, full tensor due to averaging. Reported in units Darcy: 1 d = 9:869233 · 10−13 m2 Fluid velocity: Darcy's law is formulated for volumetric flux, i.e., volume of fluid per total area per time. The fluid velocity is volume per area occupied by fluid per time, i.e., ~u . ~v = φ Theoretical basis (M. K. Hubbert, 1956): Darcy's law derived from the Navier{Stokes equations by averaging, neglecting intertial and viscous effects 10 / 52 Assume constant density ρ, unit fluid viscosity µ, and neglect gravity g −! flow equation on mixed form r · ~u = q; ~u = −Krp or as a Poisson equation with variable coefficients −∇Krp = q Single-phase, incompressible flow Model equations for single-phase flow: @(φρ) K + r · ρ~u = q; ~u = − rp − ρgrz @t µ 11 / 52 Single-phase, incompressible flow Model equations for single-phase flow: @(φρ) K + r · ρ~u = q; ~u = − rp − ρgrz @t µ Assume constant density ρ, unit fluid viscosity µ, and neglect gravity g −! flow equation on mixed form r · ~u = q; ~u = −Krp or as a Poisson equation with variable coefficients −∇Krp = q 11 / 52 Insert into conservation equation @(φρ) K = r · ρ rp @t µ @p c ρ ρ (c + c )φρ = f rp · Krp + r · (Krp) r f @t µ µ If cf is sufficiently small, so that cf rp · Krp r · (Krp), we get @p 1 = r · Krp;c = c + c @t µφc r f Single-phase, slightly compressible flow Introduce compressibilities for rock and fluid dφ dρ = c φ, = c ρ dp r dp f 12 / 52 If cf is sufficiently small, so that cf rp · Krp r · (Krp), we get @p 1 = r · Krp;c = c + c @t µφc r f Single-phase, slightly compressible flow Introduce compressibilities for rock and fluid dφ dρ = c φ, = c ρ dp r dp f Insert into conservation equation @(φρ) K = r · ρ rp @t µ @p c ρ ρ (c + c )φρ = f rp · Krp + r · (Krp) r f @t µ µ 12 / 52 Single-phase, slightly compressible flow Introduce compressibilities for rock and fluid dφ dρ = c φ, = c ρ dp r dp f Insert into conservation equation @(φρ) K = r · ρ rp @t µ @p c ρ ρ (c + c )φρ = f rp · Krp + r · (Krp) r f @t µ µ If cf is sufficiently small, so that cf rp · Krp r · (Krp), we get @p 1 = r · Krp;c = c + c @t µφc r f 12 / 52 Mass conservation per grid cell: