Stability, Accuracy, and Efficiency of Sequential Methods for Coupled Flow
and Geomechanics
J. Kim, SPE, and H.A. Tchelepi, SPE, Stanford University, and R. Juanes, SPE, Massachusetts Institute of Technology Copyright ? 2011 Society of Petroleum Engineers
This paper (SPE 119084) was accepted for presentation at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, USA, 2–4 February 2009, and revised for publication. Original manuscript received for review 14 November 2008. Revised manuscript received for review 24 June 2010. Paper peer approved 19 July 2010.
Summary
We perform detailed stability and convergence analyses of sequen-tial-implicit solution methods for coupled fluid flow and reservoir geomechanics. We analyze four different sequential-implicit solu-tion strategies, where each subproblem (flow and mechanics) is solved implicitly: two schemes in which the mechanical problem is solved first—namely, the drained and undrained splits—and two schemes in which the flow problem is solved first—namely, the fixed-strain and fixed-stress splits. The von Neumann method is used to obtain the linear-stability criteria of the four sequential schemes, and numerical simulations are used to test the validity and sharpness of these criteria for representative problems. The analysis indicates that the drained and fixed-strain splits, which are commonly used, are conditionally stable and that the stability limits depend only on the strength of coupling between flow and mechanics and are independent of the timestep size. Therefore, the drained and fixed-strain schemes cannot be used when the cou-pling between flow and mechanics is strong. Moreover, numerical solutions obtained using the drained and fixed-strain sequential schemes suffer from oscillations, even when the stability limit is honored. For problems where the deformation may be plastic (non-linear) in nature, the drained and fixed-strain sequential schemes become unstable when the system enters the plastic regime. On the other hand, the undrained and fixed-stress sequential schemes are unconditionally stable regardless of the coupling strength, and they do not suffer from oscillations. While both the undrained and fixed-stress schemes are unconditionally stable, for the cases investigated we found that the fixed-stress split converges more rapidly than the undrained split. On the basis of these findings, we strongly recommend the fixed-stress sequential-implicit method for modeling coupled flow and geomechanics in reservoirs.Introduction
Reservoir geomechanics is concerned with the study of fluid flow and the mechanical response of the reservoir. Reservoir geomechani-cal behavior plays a critical role in compaction drive, subsidence, well failure, and stress-dependent permeability, as well as tar-sand and heavy-oil production [see, for example, Lewis and Sukirman (1993), Settari and Mourits (1998), Settari and Walters (2001), Thomas et al. (2003), Li and Chalaturnyk (2005), Dean et al. (2006), and Jha and Juanes (2007)]. The reservoir-simulation community has traditionally emphasized flow modeling and oversimplified the mechanical response of the formation through the use of the rock compressibility, taken as a constant coefficient or a simple function of porosity. In order to quantify the deformation and stress fields because of changes in the fluid pressure field and to account for stress-dependent permeability effects, rigorous and efficient model-ing of the coupling between flow and geomechanics is required.In recent years, the interactions between flow and geomechanics have been modeled using various coupling schemes (Settari and Mourits 1998; Settari and Walters 2001; Mainguy and Longuemare
2002; Minkoff et al. 2003; Thomas et al. 2003; Tran et al. 2004, 2005; Dean et al. 2006; Jha and Juanes 2007). Coupling methods are classified into four types: fully coupled, iteratively coupled, explicitly coupled, and loosely coupled (Settari and Walters 2001; Dean et al. 2006). The characteristics of the coupling methods are 1. Fully coupled. The coupled governing equations of flow and geomechanics are solved simultaneously at every timestep (Lewis and Sukirman 1993; Wan et al. 2003; G ai 2004; Phillips and Wheeler 2007a, 2007b; Jeannin et al. 2006). For nonlinear problems, an iterative (e.g., Newton-Raphson) scheme is usually employed to compute the numerical solution. The fully coupled method is unconditionally stable, but it is computationally very expensive. Development of a fully coupled flow/mechanics reser-voir simulator, which is needed for this approach, is quite costly. 2. Iteratively coupled. These are sequential (staggered) solution schemes. Either the flow or the mechanical problem is solved first, then the other problem is solved using the intermediate solution information (Prevost 1997; Settari and Mourits 1998; Settari and Walters 2001; Mainguy and Longuemare 2002; Thomas et al. 2003; Tran et al. 2004; G ai 2004; Tran et al. 2005; Wheeler and G ai 2007; Jha and Juanes 2007; Jeannin et al. 2006). For each timestep, several iterations are performed, each involving sequential updating of the flow and mechanics problems until the solution converges to within an acceptable tolerance. For a given timestep, at convergence, the fully coupled and sequential solutions are expected to be the same if they both employ the same spatial discretization schemes of the flow and mechanics problems. In principle, a sequential scheme offers several advantages (Mainguy and Longuemare 2002), including working with separate modules for flow and mechanics, each with its own advanced numerics and engineering functionality. Sequential treatment also facilitates the use of different computational domains for the flow and mechani-cal problems. In practice, the domain of the mechanical problem is usually larger than that for flow because the details of the stress and strain fields at reservoir boundaries can be an important part of the problem (Settari and Walters 2001; Thomas et al. 2003). 3. Explicitly coupled. This is also called the noniterative (sin-gle-pass) sequential method. This is a special case of the iteratively coupled method, where only one iteration is taken (Park 1983; Zienkiewicz et al. 1988; Armero and Simo 1992; Armero 1999). 4. Loosely coupled. The coupling between the two problems is resolved only after a certain number of flow timesteps (Bevillon and Masson 2000; Gai et al. 2005; Dean et al. 2006; Samier and Gennaro 2007). This method can be less costly compared with the other strategies. However, reliable estimates of when to update the mechanical response are required.
Given the enormous software development and computational cost of a fully coupled flow/mechanics approach, it is desirable to develop sequential-implicit coupling methods that can be competi-tive with the fully implicit method (i.e., simultaneous solution of the flow and mechanics problems), in terms of numerical stability and computational efficiency. Sequential or staggered coupling schemes offer wide flexibility from a software-engineering per-spective, and they facilitate the use of specialized numerical methods for dealing with mechanics and flow problems (Felippa and Park 1980; Settari and Mourits 1998). As opposed to the fully coupled approach, with sequential-implicit schemes, one can use separate software modules for the mechanics and flow problems, whereby the two modules communicate through a well-
defined interface (Felippa and Park 1980; Samier and G ennaro 2007). Then, the robustness and efficiency of each module—flow and geomechanics—are available for the coupled flow/mechanics problem. In order for a sequential-implicit simulation framework to succeed, it should possess stability and convergence properties that are competitive with the corresponding fully coupled strategy.
We analyze the stability and convergence behaviors of sequen-tial (staggered) -implicit schemes for coupled mechanics and flow in oil reservoirs. In addition to building on the developments in the reservoir-simulation community, we take advantage of the sig-nificant efforts in the geotechnical and computational-mechanics communities in pursuit of stable and efficient sequential-implicit schemes for coupled poromechanics (or the analogous thermome-chanics) problems (Park 1983; Zienkiewicz et al. 1988; Huang and Zienkiewicz 1998; Farhat et al. 1991; Armero and Simo 1992; Armero 1999).
Most of the sequential methods developed in the geotechnical community assume that the mechanical problem is solved first. In this context, two different schemes have been used. One method is called the drained split [the isothermal split in the thermoelastic problem (Armero and Simo 1992)], and the other is the undrained split (Zienkiewicz et al. 1988; Armero 1999; Jha and Juanes 2007) [the adiabatic split in the thermoelastic problem (Armero and Simo 1992)]. The drained-split scheme freezes the pressure when solving the mechanical-deformation problem. This scheme is only conditionally stable, even though each of the subproblems is solved implicitly (Armero 1999). The undrained-split strategy, on the other hand, freezes the fluid mass content when solving the mechanics problem. Armero (1999) showed that the undrained split honors the dissipative character of the continuum problem of coupled mechanics and flow in porous media and that it is uncon-ditionally stable with respect to time evolution, independently of the schemes used for spatial discretization. Of course, numerical solution strategies must employ appropriate spatial discretization schemes for the flow (pressure) and mechanics (displacement vec-tor). The undrained-split scheme has been applied successfully to solve coupled linear (Zienkiewicz et al. 1988; Huang and Zienkie-wicz 1998) and nonlinear (Armero 1999) problems.
In the reservoir-engineering community, the focus has largely been on extending general-purpose (robust behavior with wide engineering functionality) reservoir-flow simulators to account for geomechanical deformation effects. As a result, sequential coupling schemes are popular. However, not all sequential cou-pling schemes are created equal. An obvious split corresponds to freezing the displacements during the solution of the flow problem and then solving the mechanics problem with the updated pressure field. Unfortunately, this “fixed-strain” scheme is conditionally stable. Here, we show that the fixed-strain split has stability char-acteristics that are quite similar to those of the “drained” split.
Settari and Mourits (1998) described a sequential-implicit coupling strategy, in which the flow and mechanics problems com-municate through a porosity correction. Since then, published sim-ulation results and engineering experience have shown that their method has good stability and convergence properties, especially for linear poroelasticity problems. However, no formal stability analysis of the Settari and Mourits scheme was available. Here, we perform such a stability analysis. Specifically, we show that their scheme is a special case of the “fixed-stress” split, which we prove to be unconditionally stable. Moreover, we demonstrate that the fixed-stress split is also quite effective for nonlinear deformation problems. Wheeler and Gai (2007) employed advanced numerical methods for the spatial discretization of coupled poroelasticity and single-phase flow. Specifically, they used a finite-element method for flow and a continuous G alerkin scheme for the displace-ment-vector field. They used a sequential scheme to couple the flow and mechanics problems, in which they first solve the flow problem by lagging the stress field and then solve the mechanics problem. Their coupling scheme is quite similar to that used by Settari and Mourits (1998) and enjoys good stability properties. Wheeler and G ai (2007) provide a priori convergence-rate esti-mates of this fixed-stress coupling scheme for single-phase flow and poroelasticity.
Mainguy and Longuemare (2002) described three different strat-
egies to sequentially couple compressible fluid flow in the presence of poroelastic deformation in oil reservoirs. For each sequential strategy, they showed the corresponding porosity-correction term that must be used to account for geomechanical effects. Specifi-cally, they derived the porosity correction in terms of (1) volumetric strain, (2) pore volume, and (3) total mean stress. While no stabil-ity analysis of the various sequential schemes was provided, they demonstrated that the pore-compressibility factor can be used as a relaxation parameter to speed up the convergence rate of sequential-implicit schemes (Mainguy and Longuemare 2002).
There is a growing recognition that different sequential-implicit solution strategies for coupling fluid flow and geomechanical deformation in reservoirs often lead to vastly different behaviors in terms of stability, accuracy, and efficiency. Here, we perform detailed analysis of both the stability and convergence properties of sequential schemes for coupling flow and geomechanical deforma-tion. Moreover, detailed analysis of the applicability of sequential schemes to coupled nonlinear deformation (e.g., plasticity) and flow is also performed. Specifically, we analyze the stability and convergence behaviors of four sequential coupling schemes: drained, undrained, fixed strain, and fixed stress. Both poroelastic and poroelastoplastic mechanics are analyzed in the presence of a slightly compressible, single-phase fluid.
Our stability analysis shows that the drained and fixed-strain split methods are conditionally stable; moreover, their stability limit is independent of timestep size and depends only on the cou-pling strength. Therefore, problems with strong coupling cannot be solved by the drained or fixed-strain schemes, regardless of the timestep size. On the other hand, we show that the undrained and fixed-stress split methods are unconditionally stable and that they are free of oscillations with respect to time. We also show that the undrained and fixed-stress sequential-implicit methods are conver-gent and monotonic for nonlinear elastoplastic problems. Mathematical Model
We adopt a classical continuum representation where the fluid and solid are viewed as overlapping continua. The physical model is based on poroelasticity and poroelastoplasticity theories (Coussy 1995). We assume isothermal single-phase flow of a slightly compressible fluid, small deformation (i.e., infinitesimal trans-formation), isotropic geomaterial, and no stress dependence of flow properties, such as porosity or permeability. The governing equations for coupled flow and reservoir geomechanics come from the conservation of mass and linear momentum. Under the quasistatic assumption for earth displacements, the governing equation for mechanical deformation of the solid/fluid system can be expressed as
Div?+?
b
g=0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(1) where Div(·) is the divergence operator, ? is the Cauchy total
stress tensor, g is the gravity vector, ?
b
= ??
f
+(1??)?
s
is the bulk density, ?
f
is fluid density, ?
s
is the density of the solid phase, and ? is the true porosity. The true porosity is defined as the ratio of the pore volume to the bulk volume in the deformed configuration.
A stress/strain relation must be specified for the mechanical behav-ior of the porous medium. Changes in total stress and fluid pressure are related to changes in strain and fluid content by Biot’s theory (Biot 1941; G eertsma 1957; Coussy 1995; Lewis and Schrefler 1998; Borja 2006). Following Coussy (1995), the poroelasticity equations can be written as
??
???
()
00
=:
C1
dr
b p p
ε . . . . . . . . . . . . . . . . . . . . . . .(2)
1
=
1
,0
00
?
f
v
m m b
M
p p
?
()+?
()
ε, . . . . . . . . . . . . . . . . . . . .(3)
where subscript 0 refers to the reference state, C
dr
is the rank-4 drained elasticity tensor, 1 is the rank-2 identity tensor, p is fluid
pressure, m is fluid mass per unit bulk volume, M is the Biot modulus, and b is the Biot coefficient. Note that we use the conven-tion that tensile stress is positive. Assuming that the infinitesimal transformation assumption is applicable, the linearized strain ten-sor ε can be expressed as
ε==12
Grad u Grad u Grad u s t +(). . . . . . . . . . . . . . . . . . .(4)
Note that (Coussy 1995)
1=00M c b K f s
??+? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(5)
b K K dr
s
=1?, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(6)where c f is the fluid compressibility (1/K f ), K f is the bulk modulus
of the fluid, K s is the bulk modulus of the solid grain, and K dr is the
drained bulk modulus. The strain and stress tensors, respectively,
can be expressed in terms of their volumetric and deviatoric parts as follows:
ε=1
3εv 1e + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(7)σ=?v 1s +, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(8)where εv = tr(ε) is the volumetric strain (trace of the strain ten-sor), e is the deviatoric part of the strain tensor, ?v =1
3
tr σ() is the volumetric (mean) total stress, and s is the deviatoric total stress tensor.
Under the assumption of small deformation, the fluid mass-conservation equation is
??+m
t f f Div q =,0?, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(9)where q is the fluid mass flux and f is a volumetric source term.
The fluid volume flux (velocity), v = q /?f,0, relative to the deform-ing skeleton, is given by Darcy’s law (Coussy 1995):
v k
Grad g =1??()B p f f ??, . . . . . . . . . . . . . . . . . . . . . . . . .(10)where k is the absolute-permeability tensor, ? is fluid viscosity,
and B f = ?f,0/?f is the formation volume factor of the fluid (Aziz
and Settari 1979).
Using Eq. 3, we write Eq. 9 in terms of pressure and volumetric strain:
1=,0M p t b t f v
f ?+?+εDiv
q . . . . . . . . . . . . . . . . . . . . . . . .(11)By noting the relation between volumetric stress and strain, ??v v dr v b p p K ?()+?(),00=ε, . . . . . . . . . . . . . . . . . . . . .(12)we write Eq. 11 in terms of pressure and the total (mean) stress:1=2,0M b K p t b K t
f dr dr v f +???????+?+?Div q
. . . . . . . . . . . . . .(13)Eqs. 11 and 13 are two equivalent expressions of mass con-servation (i.e., flow problem) in a deforming porous medium. Substitution of Darcy’s law in Eqs. 11 and 13 leads, respectively,
to pressure/strain and pressure/stress forms of the flow equation. These two forms of mass conservation are equivalent. However, using one form vs. the other in a sequential-implicit scheme to couple flow with mechanics leads to very different stability and convergence behaviors. Later, we use these expressions (Eqs. 11 and 13) to show the exact forms of the operator-splitting strategies (solving two subproblems—flow and mechanics—sequentially) studied in this paper.
To complete the description of the coupled flow and geome-chanics mathematical problem, we need to specify initial and boundary conditions. For the flow problem, we consider the bound-ary conditions p p = (prescribed pressure) on ?p , and v n ?=v (prescribed volumetric flux) on ?v , where n is the outward unit
normal to the boundary ??. For well-posedness, we assume that ??p v ∩?= and ???p v ∪?=.The boundary conditions for the mechanical problem are u u = (prescribed displacement) on ?u and ??n t = (prescribed traction) on ??. Again, we assume ??u ∩??=, and ???u ∪??=.
Initialization of the geomechanical model is a difficult task in itself (Fredrich et al. 2000). The initial stress field should satisfy
mechanical equilibrium and be consistent with the history of
the stress/strain paths. Here, we take the initial condition of the
coupled problem as p p t |==00 and ??|==00t .
Discretization
The finite-volume method is widely used in the reservoir-flow-simulation community (Aziz and Settari 1979). On the other
hand, nodal-based finite-element methods are quite popular in
the geotechnical engineering and thermomechanics communities
(Zienkiewicz et al. 1988; Lewis and Sukirman 1993; Lewis and
Schrefler 1998; Armero and Simo 1992; Armero 1999; Wan et al.
2003; White and Borja 2008). In the finite-volume method for the flow problem, the pressure is located at the cell center. In the
nodal-based finite-element method for the mechanical problem,
the displacement vector is located at vertices (Hughes 1987). This
space-discretization strategy has the following characteristics:
local mass conservation at the element level; a continuous displace-ment field, which allows for tracking the deformation; and conver-gent approximations with the lowest-order discretization (Jha and Juanes 2007). Because we assume a slightly compressible fluid, the given space discretization provides a stable pressure field (Phillips
and Wheeler 2007a, 2007b). It is well known that, for incompress-ible (solid/fluid) systems, nodal-based finite-element methods
incur spurious pressure oscillations if equal-order approximations
of pressure and displacement (e.g., piecewise continuous interpola-tion) are used (Vermeer and Verruijt 1981; Murad and Loula 1994; White and Borja 2008). Stabilization techniques for such spurious pressure oscillations have been studied by several authors (Murad
and Loula 1992, 1994; Wan 2002; Truty and Zimmermann 2006;
White and Borja 2008).
We partition the domain into nonoverlapping elements (grid-blocks), ??==1∪j n j elem , where n elem is the number of elements. Let
Q ?()L 2? and U ?()[]H d
1? (where d = 2, 3 is the number of space dimensions) be the functional spaces of the solution for pressure p and displacements u . Let Q 0 and U 0 be the correspond-ing function spaces for the test functions ? and ?, for flow and mechanics, respectively (Jha and Juanes 2007); and let Q h , Q h ,0, U h ,
and U h ,0 be the corresponding finite-dimensional subspaces. Then,
the discrete approximation of the weak form of the governing equa-tions (Eqs. 1 and 9) becomes: Find u h h h h p ,()∈×U Q such that ?
?
????∫∫∫?+??∈Grad g t s
h
h h b h h h ?
????:=,d d d ??
,U 0
. . . . . . . . . . . . . . . . . . . . . . .(14)
1
=,,0
??
???f h
h
h h h h h m t
f ?
????
?∫∫∫??+?∈d Div d d v Q ,0. . . . . . . . . . . . . . . . . . . .(15)
The pressure and displacement fields are approximated as fol-lows:
p P h j n j j =
=1
elem
∑? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(16)
u U h b n b
b
=
=1
node
∑?, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(17)
where n node is the number of nodes, P j are the element pressures, and U b are the displacement vectors at the element nodes (vertices).We restrict our analysis to pressure shape functions that are
piecewise constant, so that ?j takes a constant value of unity over Element j and 0 at all other elements. Therefore, Eq. 15 can be
interpreted as a mass-conservation statement element by element. The second term can be integrated by parts to arrive at the sum of integral fluxes V h,ij between Element i and its adjacent Elements j : ?????
∫∫∑∫??????i h i h i j n ij
h Div d d face v v n v n ===1
i j n h ij V d face
?==1
,?∑. . . . . . . . . . . . . . . . . . . . . . .(18)
The interelement flux can be evaluated using a two-point or a multipoint flux approximation (Aavatsmark 2002).
The displacement interpolation functions are the usual C 0-con-tinuous isoparametric functions, such that ?b takes a value of unity at Node b and zero at all other nodes. Inserting the interpolation
from Eqs. 16 and 17 and testing Eqs. 14 and 15 against each individual shape function, the semidiscrete finite-element/finite-volume equations can be written as
??????∫∫∫+?B g t a T h a b a
a n ?d d d node ==1,,????,… . . . . . . . . . . . . . . . . . . . . . . . .19)?????i i
i v j n
h ij
i
M P t b t V ∫∫∑∫??+???1==1
,d d face
εf i n d elem ?,=1,,?…, . . . . . . . . . . . . . . . . .(20)where Eq. 20 is obtained from Eqs. 3, 15, and 18.
The matrix B a is the linearized strain operator, which, in two dimensions, takes the form
B a x a y a y a x a =00??????????????
????. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(21)The stress and strain tensors are expressed in compact engineering notation (Hughes 1987). For example, in two dimensions, ?h h xx h yy h xy h h xx h yy =,
=2,,,,,??????????
???εεεεh xy ,??????
?
?
??. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(22)The stress/strain relation for linear poroelasticity takes the form ???h h h h h bp =,
=′?′1D ??ε, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(23)
where ?? is the effective stress tensor and D is the elasticity matrix, which, for 2D plane-strain conditions, reads
D =
111211111
111E ?()+()?()??????????
????
????
???
???
??????
?
??
?
???
1, . . . . . . . . . . . . .(24)where E is Young’s modulus, and ? is Poisson’s ratio.
The coupled equations of quasistatic poromechanics form an elliptic/parabolic system of equations. A fully discrete system of
equations can be obtained by discretizing the mass accumulation term in Eqs. 19 and 20 with respect to time. Throughout this paper,
the backward Euler method is used for time discretization.Solution Strategies
A primary objective of this work is to analyze the stability of
sequential-implicit solution schemes for coupled flow and mechan-ics in porous media, where the two problems of flow and mechan-ics are solved in sequence such that each problem is solved using implicit time discretization. We analyze four sequential-implicit solution strategies—namely, drained, undrained, fixed-strain, and
fixed-stress. Because we use the fully coupled scheme as reference, we also summarize its stability property.
The drained and undrained splits solve the mechanical problem first, and then they solve the fluid-flow problem (Fig. 1a). In contrast,
the fixed-strain and fixed-stress splits solve for the fluid-flow problem first and then they solve the mechanical problem (Fig. 1b).
(a) (b)
Fig. 1—Iteratively coupled methods for flow and geomechanics. (a) Drained and undrained split methods. (b) Fixed-strain and fixed-stress split methods. The symbol (˙) denotes time derivative, ?( )/ ?t .
Fully Coupled Method. Let us denote by A the operator of the original problem (Eqs. 1 and 9). The discrete approximation of this operator corresponding to the fully coupled method can be represented as
u p u p n n fc n n fc ????????→??????
??++A A 11,where :==,0Div Div ?++??
????b f m g 0
q f . . . . . . . . .(25)where (˙) denotes time derivative. Using a backward Euler time
discretization in Eqs. 19 and 20, the residual form of the fully discrete coupled equations is
R B g t a u a T h n a b n a n =111
?????
?∫∫∫+++????d d d ????
,a n =1,,…node . . . . . . . . . . . . . . . . . . .(26)R M
P P b t i p i i n i n i v n v n =111?????∫∫++?()+?()?d d εεj n h ij n i n V t f i n =1
,11=1,,face elem d ∑∫++?????,…, . . . . . . . . .(27)where R a u and R i p
are the residuals for mechanics (Node a ) and flow
(Element i ), respectively. The superscript n indicates the time level. The set of Eqs. 26 and 27 is to be solved for the nodal displace-ments u U n b n
++{}
11
= and element pressures p n j n P ++{}
11= (a total of d×n node +n elem unknowns). Given an approximation of the solution [u n+1(k ), p n+1(k )], where (k ) denotes the iteration level, Newton’s method leads to the following system of equations for the corrections: K L L F J
u p R R ????????????????+()
T
n k u
p ??1,=????
?
?+()
n k 1,, . . . . . . . . . . . . . . . . .(28)
where J is the Jacobian matrix, K is the stiffness matrix, L is the
coupling poromechanics matrix, and F = Q +?t T is the flow matrix (Q is the compressibility matrix, and T is the transmissibility
matrix). The entries of the different matrices are
K B DB ab a T b
=?
?∫d , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(29)L Grad ib i b T
b =?
?∫()??d , . . . . . . . . . . . . . . . . . . . . . . . .(30)
and
Q ij i j M =1?
?∫???d , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(31)
and T ij is the transmissibility between Gridblocks i and j . The fully
coupled method computes the Jacobian matrix J and determines ?u and ?p simultaneously, iterating until convergence.
Drained Split. In this scheme, the solution is obtained sequentially by ? rst solving the mechanics problem and then the ? ow prob-lem. The pressure ? eld is frozen when the mechanical problem is solved. The drained-split approximation of the operator A can be written as
u p u p u p n n dr u n n dr p n n ????????→????????→+++A A 111,
:=,=0?????
?
??+where Div A A dr u b dr p p ???g 0:=,:,0 m f f +?????Div determined q ?ε. . . . . . . . . . .(32)One solves the mechanical problem with no pressure change, then the fluid-flow problem is solved with a frozen displacement field.
We write the drained split as
K L L F u p K 0L F u p ?????????????????????????T ????=???????????
??0L 00u p T ??, . . . . . .(33)where we have dropped the explicit reference to the timestep n +1 and iteration (k ). In the drained split (?p = 0), we solve the mechanics problem first: K ?u = ?R u . Then, the flow problem is solved: F ?p = ?R p ?L ?u . In this scheme, the fluid is allowed to flow when the mechanical problem is solved.
Undrained Split. In contrast to the drained method, the undrained
split uses a different pressure predictor for the mechanical prob-lem, which is computed by imposing that the ? uid mass in each
gridblock remains constant during the mechanical step (?m = 0). The original operator A is split as follows: u p u p u
n n ud u n n ud p n ????????→????????→+++A A 1121
p g 0n ud u
b u m +????????+1,:=,=0
where Div A A ???d p f m :=,:,0 +?????Div determined q ?f ε. . . . . . . . . . .(34)The undrained strategy allows the pressure to change locally when the mechanical problem is solved. From Eq. 3, the undrained con-dition (?m = 0) yields 0=1
b M
p v ??ε+
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(35)and the pressure is updated locally in each element using p
p bM n n v n v
n +
+??(
)
1
2
1
2=εε. . . . . . . . . . . . . . . . . . . . . . . . . .(36)
For the mechanical problem, Eq. 2 is discretized as follows:
??n dr n n b p p +++???()
1
201
2120=:C 1ε. . . . . . . . . . . . . . . .(37)
After substituting Eq. 36 into Eq. 37, the mechanical problem can be expressed in terms of displacements using the undrained bulk modulus, C C 11ud dr b M =2+?. We write the undrained split as follows:
K L L
F u p K +L Q L 0L F u ?????????????
??
?????T T ????=1p L Q L L 00u p ??????????
??????????T T 1??. . . . . . . .(38)Note that the undrained condition is written as Q ?p +L ?u = 0.
This implies (K +L T Q ?1L )?u = ?R u during the mechanical step. The matrix multiplication for calculating L T Q ?1L is not required
because this calculation simply entails using the undrained moduli. Then, the fluid-flow problem is solved with F ?p = ?R p ?L ?u . The additional computational cost is negligible because the calculation of p n +1/2 is explicit.Fixed-Strain Split. In this scheme, the ? ow problem is solved
? rst. The original operator A is approximated and split in the following manner:u p u p u n n sn p n n sn u n ????????→????????→+++A A 1211p q n sn p
f m
f +??????
??+1,0,:=,where Div A ??εv sn u b p =0:=,:A Div determined ?+??????g 0. . . . . . . . . . . .(39)
For the fixed-strain split, ? εv =0, which implies that εεv n v n
+1/2=, so
the volumetric strain term b v ε in the accumulation term of Eq. 11
for the flow problem is computed explicitly. We write the fixed-strain split as follows:
K L L F u p K L 0F u p ????????????????????????T T ????=??????
????????
??00L 0u p ??. . . . .(40)We first solve the flow problem F ?p = ?R p while freezing the strain field (i.e., ? ε
=0, or equivalently ?L ?u = 0). Then, we solve the mechanical problem as K ?u = ?R u +L T ?p . It is worth noting that
the mechanical problem uses the drained rock properties, and that
the pressure corrections act as loads (Settari and Mourits 1998).
Fixed-Stress Split. In this case, the ? ow problem is solved ? rst but now freezing the total mean stress ? eld (???? v v n v n
=0=1/2?+), so the volumetric stress term b K dr v /() ? in the accumulation term of Eq. 13 for the ? ow problem is computed explicitly.
The original operator A is decomposed as follows:
u p u p u n n ss p n n ss u
n ????????→????????→+++A A 1211p q n ss p f m
f +????????+1,0,:=,where Div A ???v ss u b p =0:=,:A Div determined ?+??????
g 0. . . . . . . . . . .(41)If we fix the rate of the entire stress tensor field during the solution of the flow problem, the condition to be satisfied is ?L ?u +LK ?1L T ?p = 0. We write the fixed-stress split as K L L
F u p K L 0F LK L ?????????????
=?+??????
??T T
T
??1??????????????????????????u p 00L LK L u p 1T . . . . . . .(42)Thus, in the fixed-stress split, we first solve the flow problem with (F +LK ?1L T )?p = ?R p . However, the full matrix inversion and multiplication LK ?1L T is not required because rate of mean stress is kept constant by introducing the term b 2/K dr locally in each ele-ment (see Eq. 13). Once the flow problem is solved, we solve the mechanical problem exactly as for the fixed-strain split.
Fixed-Strain and Fixed-Stress Methods vs. the Pore-Compress-ibility Approach. In traditional reservoir simulation, the pressure equation typically employs a pore compressibility c p , expressed as
??00=c
c p
t
f f
p +()
??+Div v . . . . . . . . . . . . . . . . . . . . . . . .(43)The pore compressibility is not an intrinsic parameter of the rock because it depends on the deformation scenario and the boundary conditions of the coupled problem. It is used in traditional reservoir simulation as a simplified way to account for changes in the state of stress and strain in the reservoir (Settari and Mourits 1998; Set-tari and Walters 2001). From Eqs. 5, 11, and 43, the fixed-strain
split takes ?0|c p sn as (b ??0)/K s , and b v ε
as a correction source term from mechanical effects. Similarly, from Equations 5, 13, and 43, the fixed-stress split takes ??002|=//c b K b K p ss s dr ?()+, and b K dr v / ? as a correction source term from the mechanical solution. The expression for the pore compressibility associated with the fixed-stress split coincides with the one proposed by Set-tari and Mourits (1998) (although for linear poroelasticity only). Other values of c p can be used in order to enhance the stability and convergence of a sequential-implicit scheme. This possibility was studied, in the context of linear poroelasticity, by Bevillon and Masson (2000) and Mainguy and Longuemare (2002).
In order to account for geomechanical effects properly, a cor-rection needs to be included as a source term. This source term is known as porosity correction ?? (Mainguy and Longuemare
2002) and takes the following two equivalent expressions: ??=00??c b K p
b p s v +?????
?
??() ε,from Eq.11 . . . . . . . . . .(44) =1100???c K b K p K K p s dr dr s v +?????????????
?? ,f rom Eq.13(). . . . . . . . . . . . . . . . . . . . . . . .(45)Even though the pore compressibility has been recognized as a stabilization term, a complete stability analysis and comparison
study of sequential methods including plasticity is lacking. In the
next section, we analyze the stability and accuracy of the four sequential methods presented here.Stability Analysis for Linear Poroelasticity Von Neumann Stability Analysis. We use the von Neumann method to analyze the stability of sequential-implicit coupling strategies. This method is frequently used to analyze the stability of linear, or linearized, problems with respect to time [see, for example, Strikwerda (2004), Miga et al. (1998), and Wan et al.
(2005)]. The details of the stability analysis are given elsewhere (Kim 2010). Here, we simply report the expressions (either
explicit or quadratic equations) of the ampli? cation factors ? for
1D problems.Fully coupled:
??fc dr dr dr K M b K M b K k t =0,2
2+++?1
2112h ?()≤???????
cos ?. . . . . . . . . . .(46)Drained split:
K M K k t h dr dr +?()???????1
212cos ?
?×???????????dr dr dr K M b b 2
22=0
. . . . . . . . . . . . . . . . . . . . . .(47)Undrained split:
??ud dr K b M M b Mk t h =0,1222++?11
21222?()+++()?()
cos cos ???K b M M K b M k t h dr dr ?≤???
??
?
???1. . . . . . . . . . . . . . . . . . . . . . . .(48)
Fixed-strain split:
K M K k t h dr dr +?()???1
212cos ????????????
?????sn
dr sn K M b b 222=0.
. . . . . . . . . . . . . . . . . . .(49)Fixed-stress split:
?ss dr dr dr K M b K M b K =0,2
2+++k t h ???1
2112
?()≤?????
??cos . . . . . . . . . . . .(50)
In these expressions, ?t and h are the timestep size and spatial grid spacing, respectively, and ? is the phase parameter in the von Neumann method.
The amplification factors are less than unity and non-negative for the fully coupled (?fc ), undrained (?ud ), and fixed-stress (?ss ) schemes; as a result, unconditional stability and nonoscillatory solutions as a function of time are expected for these methods. Note that the amplification factors of the fixed-stress split (Eq. 50) and the fully coupled scheme (Eq. 46) are identical. Numerical simula-tions, which are discussed later, confirm that the fixed-stress split enjoys excellent stability and convergence properties.
Eqs. 47 and 49 indicate that the amplification factors of the drained and fixed-strain splits are the same. Therefore, the drained and fixed-strain methods are conditionally stable and they share the same stability limit—namely,
?=
12b M
K dr
≤, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(51)where ? is referred to as the coupling strength and is given by the ratio of the bulk stiffness of the fluid to that of the solid skeleton. The stability results can be extended from one to multiple dimen-sions because the coupling between flow and mechanics is taken as a volumetric response, which is a scalar quantity. In one dimension, K dr is the constrained modulus. In the 2D plane-strain case, K dr is 14221D 1T
ps ; in three dimensions, K dr is the drained bulk modulus, which is given by 19
331D 1T dr , where 12=1,1,0T [], 13=1,1,1,0,0,0T [],
D ps is a 3×3 matrix given by Eq. 24, and D dr is a 6×6 matrix involv-ing the drained moduli (Hughes 1987). The coupling strength can be extended to multidimensional elastoplasticity, as follows:
?=
19233
b M T
ep 1D 1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(52)
where D ep is the elastoplastic tangent modulus (in compact engi-neering notation) in three dimensions.
It is important to note that the drained and fixed-strain methods have negative amplification factors, which implies the possibility
of oscillatory numerical behavior as a function of time, even when the stability limit is honored. This is confirmed by the numerical solutions (Kim 2010).
In reservoir-engineering applications, the fluid system (usually oil and water) is assumed to be (at minimum) slightly compressible. Nevertheless, it is important to understand the stability characteris-tics of coupled flow-mechanics problems in the limit of incompress-ible fluids. When both the fluid and solid (the grains that makes up the rock) are incompressible, the Biot modulus M is infinite. As a result, the coupling strength ?, as defined here, is infinite. In this case, the fixed-stress split has max ?()<1 indicating stability. In contrast, the undrained split in this case has distinct amplification factors with max ?()=1, which indicates bounded error propaga-tion (stability) but with expected convergence difficulties (Kim 2010). Note that, in this (incompressible) limit case, both the drained and fixed-strain methods are unconditionally unstable.Numerical Simulations. Numerical solutions of simple 1D prob-lems con? rm that the undrained and ? xed-stress methods are unconditionally stable and convergent (Kim 2010). These 1D solutions also con? rm that the stability criteria for the drained and ? xed-strain splits are valid and sharp. That is, small violations of the stability limit lead to unstable solutions. Moreover, consistent with Eq. 51, the numerical solutions con? rm that the drained and ? xed-strain stability limit depends on the coupling strength only and is independent of the timestep size. The details of these simple 1D test cases are given elsewhere (Kim 2010).
Next, we describe the results from numerical simulations of three test cases.
? Case 1—Injection and production in a 1D poroelastic medium. The driving force is from injection and production (Fig. 2). The focus is on the behavior as a function of coupling strength. Dila-tion and compaction take place around the injection and produc-tion wells.
? Case 2—Mandel’s problem in a 2D elastic medium. The driv-ing force is provided by the sideburden (Fig. 3a).
? Case 3—Fluid-production scenario in two dimensions with elastoplastic behavior described by the modified Cam-clay model. Compaction of the reservoir occurs because of production (Fig. 3b). Plasticity leads to significant compaction.
Next, the results for the first two cases, which assume linear poroelastic behavior, are presented. The results are based on one iteration per timestep (i.e., solve one problem implicitly, then solve the other problem, also implicitly), unless noted otherwise.
Case 1. The numerical values of the parameters for Case 1 are given in Table 1. Here b = 1.0 (i.e., incompressible solid grains, K s = ∞). Given the coupling strength ?, M = ?K dr /b 2. Fig. 4 shows a comparison between the undrained and fixed-stress schemes for Case 1 with high coupling strength. As shown in the figure, the fixed-stress scheme converges after one iteration per timestep. This behavior is consistent with the fact that the amplification factors of the fully coupled and fixed-stress split methods are identical. On the other hand, several iterations are required in order for the undrained method to match (within a very small tolerance) the fully coupled solution. For this linear coupled problem, the fixed-stress split takes two iterations to match the fully coupled solution when the exact K dr is used (Kim 2010). In problems with more complex boundary conditions, estimating K dr can be quite difficult and is, in fact, part of the problem. In such problems, the fixed-stress scheme may require several iterations to converge.
Case 2—Mandel’s Problem. Mandel’s problem is commonly used to validate simulators of coupled flow and geomechanics. A detailed description and analytical solution of Mandel’s prob-lem is presented in Abousleiman et al. (1996). The input param-eters are listed in Table 2. Fig. 5 shows that the drained method (Fig. 5a) is stable when ? is less than unity, while it is unstable when ? is greater than unity. Furthermore, severe oscillations are observed even though the drained split is stable. Because of the oscillations, the early-time solution is not computed properly in the drained split even though the late-time solution converges to the analytical results. On the other hand, the undrained method shows unconditional stability and yields a numerical solution that
Overburden 2.125 MPa
No flow Production Observation (5th from the top)
No gravity 15 Gridblocks
Injection No flow
Δz
L z
P i =2.125 MPa Fig. 2—1D problem with injection and production wells (Case 1).
matches the analytical solution for all time. Fig. 6 indicates that the fixed-strain split (Fig. 6a) is stable for ? < 1, while it is unstable when ? > 1. Similar to the drained split method, the fixed-strain split method can suffer from severe oscillations at early time. The fixed-stress split, on the other hand, is stable and nonoscillatory under all conditions.
)b (
)a (Overburden 2.125 MPa
Overburden 2.125 MPa
2.125 MPa Sideburden
No flow
No flow
No flow
No flow
No flow
No flow
No flow
No gravity
No gravity
4×25 Gridblocks
4×2.125 MPa Sideburden
5×5 Gridblocks Production and
observation well (3,3)
Drainage boundary
P i =2.125 MPa u x =const P i =2.125 MPa
P bc =2.125 MPa
Fig. 3—(a ) Ma ndel’s problem in 2D with ela stic deforma tion (Ca se 2). (b) 2D problem driven by single-well production in a n elastoplastic medium (Case 3).
0.050.10.150.20.25
0.650.7
0.750.80.850.90.951τ=12.12
Dimensionless time (pore volume produced)
D i m e n s i o n l e s s p r e s s u r e (P /P i
)
Fixed stress 1 iteration Fully coupled
Undrained 10 iterations Undrained 1 iteration
Fig. 4—Beha vior of the undra ined a nd fixed-stress splits for
cases with very high coupling strength: Case 1 with ? = 12.12.
The fixed-stress method requires a single iteration per timestep to match the fully coupled solution, while many more iterations
are required for the undrained method.
The Mandel-Cryer effect, where a rise in the pressure during early time is observed, can be captured by properly constructed sequential methods. All solutions from the fully coupled, und-rained, and fixed-stress methods are in good agreement with the analytical solution.
Stability Analysis: Nonlinear Poroelastoplasticity
Coupling With Elastoplasticity. The coupling between the mechanical and ? ow problems in the elastoplastic regime under isothermal conditions is modeled by the following constitutive equations (Coussy 1995):
????()??()00=:C 1dr p b p p εε . . . . . . . . . . . . . . . . .(53)
1=1
,0
0,0??f p v p v m m b M p p ?()??()+?()εε, . . . . . . . . .(54)
where εp is the linearized plastic strain tensor from inelasticity, εp,v
= tr(εp ), and ?p is the plastic porosity. The plastic porosity and plas-tic strain can be related to each other by assuming that ??p
p v =,ε. Here, we assume that ? = b (Armero 1999). Note that ? ≈ b ≈ 1 if the solid grains are incompressible. The relation between the total stress ? and total strain ε is written as ????()??()??()000=:=:C 1
C 1dr p ep b p p b p p εεε, . . . . . . . . . . . . . . . .(55)
where C ep is the rank-4 elastoplastic tangent tensor. The compact engineering notations for C ep and 1 are D ep and 13T , respectively (this notation is also employed in Eq. 52). Elastoplasticity renders the mechanical problem highly nonlinear.
Stability analysis of slightly compressible flow and nonlinear mechanical deformation can be performed using the von Neumann method through linearization of the problem. The difference with respect to the linear problem is that the coefficients of the stability conditions, such as D ep and K dr , are linearized around values from the previous iteration (using Newton’s method, for example).
Our analysis of the linearized systems shows that the fully coupled, undrained, and fixed-stress methods yield unconditional stability regardless of D ep and K dr . However, the drained and fixed-strain methods show strong dependence on the coupling strength defined in Eq. 52. This implies that the drained and fixed-strain methods can be unstable as the medium begins to yield (plastic regime), even if they satisfy the stability conditions at the begin-ning of a simulation (elastic regime).
For nonlinear problems, one can apply a sequential solution method before or after linearization. In this paper, we are inter-ested in sequential implicit/implicit schemes, where each iteration involves solving two problems in sequence, such that each problem, which may be nonlinear, is solved implicitly. For a given timestep, a single-pass strategy would entail solving the first subproblem
)b (
)a (0
246
8
00.10.20.30.40.50.60.70.80.9Dimensionless time (t d =4c v t /(Lz )2
)
D i m e n s i o n l e s s p r e s s u r e (P ?P i )/P i
Mandel’s problem, τ=0.90
Analytic solution Fully coupled Undrained Drained
12345
6
00.1
0.20.30.40.50.60.70.80.9
Analytic solution Fully coupled Undrained Drained
Dimensionless time (t d =4c v t /(Lz )2
)
D i m e n s i o n l e s s p r e s s u r e (P ?P i )/P i
Mandel’s problem, τ=1.10
Fig. 5—Case 2 (Mandel’s problem). Evolution of the pressure at the observation point as a function of dimensionless time. The results of the fully coupled, drained, and undrained methods are shown for two values of the coupling strength: (a) ? = 0.90; (b)
? = 1.10.??t c t
L d v z =42(), where c ? is the consolidation coefficient defined as c k K c v dr f =+()1/??
and L z is the vertical length of the
reservoir domain.
implicitly (subject to a given tolerance), updating the appropriate
terms to set up the second problem, and solving the second problem implicitly. We then move to the next timestep. One can also iterate by repeating the implicit/implicit solution sequence.
Numerical Simulations. To test the validity of the results obtained from our linearized stability analysis, we perform numerical experi-ments for Case 3. We adopt an associated plasticity formulation (Simo 1991; Simo and Hughes 1998; Coussy 1995). The yield func-tion f Y of the modi? ed Cam-clay model is (Borja and Lee 1990)f q M p Y mcc
v v
co ==02
2′+′′?()??, . . . . . . . . . . . . . . . . . . . . . .(56)where q ? is the deviatoric effective stress, ′?v is the volumetric effective stress, M mcc is the slope of the critical state line, and p co is the preconsolidation pressure. The input parameters for Case 3 are listed in Table 3. The parameter ? is the virgin compression index, and ? is the swell index. The schematic of the return mapping for
the modified Cam-clay model is illustrated in Fig. 7 [refer to Borja and Lee (1990) for more details]. We employ full iteration with a consistent return-mapping algorithm (Simo and Hughes 1998).Fig. 8 shows that the drained and fixed-strain split methods
are stable during the early-time elastic regime, which has a weak
coupling strength (? < 1). However, when plasticity is reached,
the solution by the drained method is no longer stable because the coupling strength increases dramatically. Fig. 9 depicts the variation of the coupling strength during the simulation. It is clear that, when the coupling strength increases beyond unity, the solu-tion by the drained and fixed-strain methods becomes unstable. No
solution by the drained and fixed-strain split methods is possible
after plasticity. On the other hand, the fully coupled, undrained,
and fixed-stress methods provide stable results well into the plastic regime, as shown in Figure 8. In particular, the solution of the fully
)b (
)a (0
246
8
00.10.20.30.40.50.60.70.80.9
Analytic solution Fully coupled Fixed stress Fixed strain
123456
00.1
0.20.30.40.50.60.70.80.9
Analytic solution Fully coupled Fixed stress Fixed strain
Dimensionless time (t d =4c v t /(Lz )2
)
D i m e n s i o n l e s s p r e s s u r e (P ?P i )/P i
Mandel’s problem, τ=0.90
Dimensionless time (t d =4c v t /(Lz )2)
D i m e n s i o n l e s s p r e s s u r e (P ?P i )/P i Mandel’s problem, τ=1.10
Fig. 6—Case 2 (Mandel’s problem). Evolution of the pressure at the observation point, as a function of dimensionless time. Shown are the results for the fully coupled, fixed-strain, and fixed-stress splits. (a) ? = 0.90; (b) ? = 1.10.
trial
Fig. 7—Schema tic of the return ma pping a lgorithm for the
modified Cam-clay model.
coupled method is matched by the undrained and fixed-stress splits after only two iterations.
We also show the results using full Newton iterations (i.e., a sequential solution strategy is employed to solve the linear system of equations associated with each Newton iteration of the fully coupled system). Fig. 10 shows that, using such a strategy, the drained and fixed-strain splits are unstable as we enter the plastic regime, even though they are stable in the elastic regime. On the other hand, the undrained and fixed-stress splits are stable, even for plastic deformation.
(a)
(b)
0.010.020.030.040.40.50.60.70.80.91Dimensionless time (pore volume injected)
D i m e n s i o n l e s s p r e s s u r e (P /P i )
Plasticity with hardening
Drained
Undrained (1 iteration)Undrained (2 iterations)Fully coupled
0.010.020.030.040.40.50.60.70.80.91Dimensionless time (pore volume injected)
D i m e n s i o n l e s s p r e s s u r e (P /P i )
Plasticity with hardening
Fixed strain
Fixed stress (1 iteration)Fixed stress (2 iterations)Fully coupled
Fig. 8—Case 3 (2D production test in the elastoplastic regime). Evolution of the dimensionless pressure at the observation point is shown a s a function of dimensionless time (pore volumes produced). (a ) Fully coupled, dra ined, a nd undra ined numerica l solutions. (b) Fully coupled, fixed-strain, and fixed-stress solutions. The undrained and fixed-stress splits yield stable solution, while the drained and fixed-strain splits become unstable as the problem enters the plastic regime.
00.010.020.030.040.05
01
23
4
56
78Dimensionless time (pore volume produced)C o u p l i n g s t r e n g t h (τ)
Variation of τ during simulation
Fig. 9—Variation of the coupling strength ? during the course
of the simula tion for Ca se 3. The coupling strength, initia lly less than unity, jumps to a very high value when the medium enters the plastic regime, rendering the drained and fixed-strain methods unstable.
Summary and Conclusions
We employed the von Neumann method to analyze the stabil-ity properties of several sequential-implicit solution strategies for coupled flow and mechanical deformation in oil reservoirs. Detailed numerical simulations of several representative problems were used to test the validity of the stability analysis. The study is limited to single-phase flow of a slightly compressible fluid; however, both elastic and elastoplastic material behaviors are investigated. The four sequential methods investigated fall into two categories: those that solve the mechanical problem first (drained and undrained splits) and those that solve the flow problem first (fixed-strain and fixed-stress splits).
The drained split, in which the pressure is frozen while solving the mechanics problem, and the fixed-strain split, in which the
displacements are frozen while solving the flow problem, are the
obvious sequential solution strategies. As we show quite clearly
in this work, however, these two schemes suffer from severe
stability and convergence problems. Specifically, the drained and fixed-strain split methods are conditionally stable; moreover, their stability limit (? < 1) depends on the coupling strength only and is independent of timestep size. Thus, physical problems with high
coupling strength (i.e., ? > 1) cannot be solved by the drained or
fixed-strain split methods, regardless of the timestep size. Even when they are stable, the drained and fixed-strain splits display oscillatory behavior, which is implied by their negative amplifica-tion factors.
In contrast, the undrained and fixed-stress methods show unconditional stability for both elasticity and elastoplasticity.
These sequential methods can be applied safely to model porome-chanical problems of practical interest, such as compressible solid
grains and plasticity with hardening. Moreover, in addition to the amplification factors being less than unity, they are positive; as a result, the numerical solutions do not exhibit oscillations in time.
While the undrained split and fixed-stress split have similar stability properties, we have found that, for the problems we
studied, the fixed-stress scheme converges much faster than the
undrained method. This behavior was observed across a wide range of coupling strengths. On the basis of these findings, we strongly recommend the fixed-stress split over the other sequential-implicit methods for modeling coupled geomechanics and flow in
oil reservoirs.
Nomenclature b = Biot’s coef? cient
B a = linearized strain operator B f = formation volume factor of the ? uid c f = ? uid compressibility, 1/K f c p = pore compressibility c v = consolidation coef? cient
C dr = rank-4 drained elasticity tensor C ep = rank-4 elastoplastic tangent tensor C ud = rank-4 undrained elasticity tensor
D ps = e lasticity matrix for the plane-strain case in two dimen-sions D dr = d rained modulus matrix (compact engineering notation
for C dr ) in three dimensions
D ep = e lastoplastic tangent modulus matrix (compact engineer-ing notation for C ep ) in three dimensions Div(·) = divergence operator e = deviatoric part of the strain tensor
e P d = L 2 norm o
f the error for the dimensionless pressure E = Young’s modulus f = volumetric source term for ? ow f Y = yield function for elastoplasticity F = ? ow matrix
g = gravity vector
h = grid spacing used in the von Neumann method J = Jacobian matrix k = absolute permeability tensor K = stiffness matrix K dr = d rained bulk modulus used for the de? nition of the cou-pling strength
K f = bulk modulus of the ? uid
K s = bulk modulus of the solid grain L = coupling poromechanics matrix m = ? uid mass per unit bulk volume M = Biot’s modulus M mcc = s lope of the critical state line of the modi? ed Cam-clay
model
n elem = number of elements n node = number of nodes p = ? uid pressure p = boundary pressure p co = p reconsolidation pressure of the modi? ed Cam-clay
model
p i = initial pressure P j = pressure at Element j q = ? uid mass ? ux (? uid mass ? ow rate per unit area and
time)
q ? = deviatoric effective stress Q = compressibility matrix Q inj = injection rate Q prod = production rate
R a u ,R j p
= residuals for mechanics (Node a ) and ? ow (Element j ) s = deviatoric total stress tensor T = transmissibility matrix
T ij = transmissibility between Gridblocks i and j u = displacement
U b = displacement vector at the Node b v = ? uid velocity V h,ij = ? ux between Gridblocks i and j ?t = timestep size
?x = grid spacing on the x axis ?z = grid spacing on the z axis ?dr = ampli? cation factor of the drained split ?fc = ampli? cation factor of the fully coupled method ?sn = ampli? cation factor of the ? xed-strain split ?ss = ampli? cation factor of the ? xed-stress split ?ud = ampli? cation factor of the undrained split ε = linearized strain tensor εp = linearized plastic-strain tensor εv = volumetric strain (the trace of the strain tensor) εp,v = v olumetric plastic strain (the trace of the plastic-strain
tensor)
? = test function for mechanics ? = phase parameter in the von Neumann method
Fig. 10—Case 4 (2D production test in elastoplastic regime). Shown is the evolution of the dimensionless pressure at the observa-tion point against dimensionless time (pore volumes produced). (a) Fully coupled, drained, and undrained numerical solutions. (b) Fully coupled, fixed-strain, and fixed-stress solutions. The model enters the plastic regime at t d ≈ 0.018. Beyond this point, the drained and fixed-strain methods become unstable and fail to produce a solution at all.
)b (
)a (0
0.010.020.030.040.05
Dimensionless time (pore volume produced)Plasticity with hardening
0.010.020.030.040.05
0.30.40.50.60.70.80.91Dimensionless time (pore volume produced)
Plasticity with hardening
D i m e n s i o n l e s s p r e s s u r e (P /P i )
D i m e n s i o n l e s s p r e s s u r e (P /P i )
? = swell index of the modi? ed Cam-clay model ? = v irgin compression index of the modi? ed Cam-clay model ? = ? uid viscosity ? = Poisson’s ratio ?b = bulk density ?f = ? uid density ?s = density of the solid phase ? = Cauchy total stress tensor ?? = effective stress tensor ? = overburden
?v = volumetric (mean) total stress ′?v = volumetric effective stress ? = coupling strength ? = test function for ? ow ? = true porosity, Euler’s porosity ?p = plastic porosity ? = Lagrange’s porosity 1 = rank-2 identity tensor (·)0 = reference state
(·)d = number of space dimensions (·)d = dimensionless quantity (·)n = time level (·)k = iteration level (˙) = time derivative
Acknowledgments This work was supported by Foundation CMG (Computer Modeling
Group), the Stanford University Petroleum Research Institute for Res-ervoir Simulation (SUPRI-B), and the Massachusetts Institute of Tech-nology Reed Research Fund and ARCO Chair in Energy Studies.
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Jihoon Kim is a geologic al postdoc toral fellow in the Earth Sc ienc es Division at Lawrenc e Berkeley National Laboratory. He studies geomec hanic al responses on geothermal and hydrate reservoirs. He holds a PhD degree in petroleum engi-neering from Stanford University. Hamdi A. Tchelepi is associate professor in the Department of Energy Resources Engineering and is codirector of the Center for Computational Earth and Environmental Sciences at Stanford University. He is interested in building c omputational frameworks for reservoir simula-tion and subsurfac e c arbon sequestration. Spec ific researc h interests include modeling unstable flow processes, stochastic numeric al methods for unc ertainty quantific ation, and multi-scale formulations for multiphase flow and transport. He holds a PhD degree in petroleum engineering from Stanford University. Ruben Juanes is the ARCO Associate Professor in Energy Studies, in the Department of Civil and Environmental Engineering at the Massachusetts Institute of Technology. Before joining MIT in 2006, he was ac ting assistant professor at Stanford University (2003–05), and assistant professor at The University of Texas at Austin (2006). He leads a research group in the area of multi-phase flow in porous media, with application to energy-driven problems, inc luding oil and gas rec overy, methane hydrates, and geologic al c arbon sequestration. He holds MS and PhD degrees from the University of California at Berkeley.
广州大学学生实验报告 实验室：电子信息楼 317EDA 2017 年 10 月 2 日 学院机电学院年级、专 业、班 电信 151 姓名苏伟强学号1507400051 实验课 程名称 可编程逻辑器件及硬件描述语言实验成绩 实验项 目名称 实验4 2位十进制高精度数字频率计设计指导老师 秦剑 一实验目的 1 熟悉原理图输入法中74系列等宏功能元件的使用方法，掌握更复杂的原理图层次化设计技术和数字系统设计方法。 2 完成2位十进制频率计的设计，学会利用实验系统上的FPGA/CPLD验证较复杂设计项目的方法。 二实验原理 1 若某一信号在T秒时间里重复变化了N 次，则根据频率的定义可知该信号的频率fs 为：fs=N/T 通常测量时间T取1秒或它的十进制时间。 三实验设备 1 FPGA 实验箱，quarteus软件 四实验内容和结果 1 2位十进制计数器设计 1.1 设计原理图：新建quarteus工程，新建block diagram/schematic File文件，绘制原理图，命名为conter8,如图1，保存，编译，注意：ql[3..0]输出的低4位（十进制的个位）， qh[3..0]输出的高4位（十进制的十位） 图片11.2 系统仿真：如图2建立波形图进行波形仿真，如图可以看到完全符合设计要求，当clk输入时钟信号时，clr有清零功能，当enb高电平时允许计数，低电平禁止计数，当低4位计数到9时向高4位进1 图2 1.3 生成元件符号：File->create/updata->create symbol file for current file,保存，命名为conter8,如图3为元件符号（block symbol file 文件）： 图3 2 频率计主结构电路设计 2.1 绘制原理图：关闭原理的工程，新建工程，命名为ft_top,新建原理图文件，在project navigator的file 选项卡，右键file->add file to the project->libraries->project library name添加之前conters8工程的目录在该目录下，这样做的目的是因为我们会用到里面的conters8进行原理图绘制，绘制原理图，如图4，为了显示更多的过程信息，我们将74374的输出也作为output,重新绘制了原理图，图5 图4
测量中的重要概念——精确度，准确度，敏感度和分辨率 问题简述：在测量中经常会遇到测量精确度(accuracy)、准确度(precision)、敏感度(sensitivity)以及分辨率(resolution)的概念，它们的含义是什么，以及在何种程度上会影响到测量结果，是不是分辨率越高精确度就越好，本文就这些内容作一个探讨。 问题解答：对于精确度(accuracy)和准确度(precision)，简单来说，精确度表征的是测量结果与真实值偏差的多少，准确度则是指多次测量结果的一致性如何。以下图为例，我们将测量比作打靶。精确度越高，多次测量结果取平均值就越接近真实值；准确度越高，多次测量结果越一致。 工程应用中，准确度(precision)也是一个十分重要的指标。由于实际现场存在许多不可预期因素，测量结果的精确度总是会随着时间、温度、湿度、光线强度等因素的变化而发生变化。但如果测量的准确度足够高，即测量结果的一致性较好，就可以通过一定的方式对测量结果进行校正，减小系统误差，提高精确度。 在测量系统中，分辨率(resolution)和敏感度(sensitivity)也是常见指标。以NI 的M 系列数据采集卡为例。下图是NI 6259 的部分技术参数： 可以看到，6259 模拟输入的分辨率是16 位，即采用的是16 位的ADC。那么在满量程下(-10,10V)，ADC 的码宽为20/2^16=305μV ，通常我们也将该值称为1LSB（1LSB = V FSR/2N，其中V FSR为满量程电压，N 是ADC 的分辨率）。在满量程下，6259 的精确度为
1920μV。敏感度是采集卡所能感知到的最小电压变化值。它是噪声的函数。 数据采集卡可能在基准电压，可编程仪器放大器(PGIA)，ADC 等处引入测量误差，如下图所示。 NI 的数据采集卡精确度遵循以下计算公式： 精确度= 读数×增益误差+ 量程×偏移误差+ 噪声不确定度 增益误差= 残余增益误差+ 增益温度系数×上次内部校准至今的温度改变+ 参考温度系数×上次外部校准至今的温度改变 偏移误差= 残余偏移误差+ 偏置温度系数×上次内部校准的温度改变+ INL_误差 可以在625X 的技术手册中查找公式中的各项参数，如下表所示： 其中增益误差主要由于放大器的非线性引起，而ADC 的分辨率主要影响INL(Integral nonlinearity)误差（积分非线性误差）。 DNL(Differential nonlinearity)误差定义（微分非线性误差）为实际量化台阶与对应于1LSB 的理想值之间的差异(见下图)。对于一个理想ADC，跳变值之间的间隔为精确的1LSB。若DNL误差指标≤1LSB，就意味着传输函数具有保证的单调性，没有丢码。当一个ADC 的数字量输出随着模拟输入信号的增加而增加时(或保持不变)，就称其具有单调性，相应传输函数曲线的斜率没有变号。
第二章：精度指标与误差传播 内容及学习要求 本章详细讨论偶然误差分布的规律性，衡量精度的绝对指标－中误差，相对指标－权及其确定权的实用方法；方差、协因数定义及其传播律等问题。本章内容是是测量平差的理论基础，也是本课程的重点之一。学习本章要求深刻理解精度指标的含义，掌握权、协方差、协因数概念，确定权及根据已知协方差、协因数的观测值求其函数的方差、协因数的方法（协因数、协方差传播律）。 §2-1概述 概括本章内容，其主线是偶然误差的统计规律→衡量单个随机变量的精度指标－方差→衡量随机向量的精度指标－协方差阵→求观测值向量函数的精度指标－协方差传播律→精度的相对指标-权。 §2-2偶然误差的规律性 本小节阐述偶然误差的统计规律性，提出偶然误差服从正态分布的结论 任何一个观测值，客观上总是存在一个真正代表其值的量，这一数值就称观测值的真`值。从概率统计的观点看，当观测量仅含偶然误差时，真值就是其数学期望。 某一随机变量的数学期望为：i n i i p x X E ∑== 1 )( 或 ?+∞ ∞ -=dx x xf X E )()( 期望的实质是一种理论平均值,可用无穷观测,以概率为权,取加权平均值的概念理解.dx x f )(表示x 出现在小区间dx 的概率。 设对n 个量进行了观测，观测值为。 、、、n L L L ???21其相应的真值分别为。 、、、n L L L ???21令i i i i L L ?-=?， 即真误差。由于假定测量平差所处理的观测值只含偶然误差，所以真误差i ?就是偶然误差。用向量形式表述为： ? ????????????=?n b L L L L 211、?????? ????????=?n n L L L L ..211、?? ?????????????=??n n .211 则有：111???-=?n n n L L 注意：本教程中凡是不加说明，即没有下标说明的向量都是列向量，若表示行向量则加以转置符号表示，如：T T T B A L 、、等。 对单个的偶然误差而言，大小和符号都没有规律，及事先完全不可预知。但从大量测量实践中知道，在相同的观测条件下，偶然误差就总体而言，有一定的统计规律，表现为如下几点： 1、 误差绝对值有一定限值 2、 绝对值小的比大的多 3、 绝对值相等的正负误差出现的个数相等或接近。 教材中分别列举两个实例，以358和421个三角形闭合差的分析结果验证了上述结论（闭合差是理论值与观测值之差，故是真误差）。注意：统计规律只有当有较多的观测量时，才能得出正确结论。 为了形象地刻画误差分布情况，以横坐标表示误差的大小，纵坐标采用单位区间频率（出现在某区间内的频率，等于该区间内出现的误差个数i v 除误差总个数n ，而采用单位频率 i i nd V ?为纵坐标值，使曲线（直方图）趋势不因区间间隔不同而变化）。根据统计规律可知，在相同条件下所得一组独立观测值，n 足够大时，误差出现在各个区间的频率总是稳定在某一常数（理论频率）附近，n 越大；稳定程度越高。n 趋于∞，则频率等于概率（理论频率）。令区间长度0→?d ，则长方条顶形成的折线变成光滑曲线，称概率曲线。
《综合课程设计》 一．数字频率计的设计 姓名：万咬春学号2005142135 一、课程设计的目的 通过本课程设计使学生进一步巩固光纤通信、单片机原理与技术的基本概念、基本理论、分析问题的基本方法；增强学生的软件编程实现能力和解决实际问题的能力，使学生能有效地将理论和实际紧密结合，拓展学生在工程实践方面的专业知识和相关技能。 二、课程设计的内容和要求 1．课程设计内容 （硬件类）频率测量仪的设计 2．课程设计要求 频率测量仪的设计 要求学生能够熟练地用单片机中定时/计数、中断等技术，针对周期性信号的特点，采用不同的算法，编程实现对信号频率的测量，将测量的结果显示在LCD 1602 上，并运用Proteus软件绘制电路原理图，进行仿真验证。 三．实验原理 可用两种方法测待测信号的频率 方法一：（定时1s测信号脉冲次数） 用一个定时计数器做定时中断，定时1s，另一定时计数器仅做计数器使用，初始化完毕后同时开启两个定时计数器，直到产生1s中断，产生1s中断后立即关闭T0和T1（起保护程序和数据的作用）取出计数器寄存器内的值就是1s内待测信号的下跳沿次数即待测信号的频率。用相关函数显示完毕后再开启T0和T1这样即可进入下一轮测量。 原理示意图如下：
实验原理分析： 1．根据该实验原理待测信号的频率不应该大于计数器的最大值65535，也就是说待测信号应小于65535Hz。 2．实验的误差应当是均与的与待测信号的频率无关。 方法二（测信号正半周期） 对于1：1占空比的方波，仅用一个定时计数器做计数器，外部中断引脚作待测信号输入口，置计数器为外部中断引脚控制（外部中断引脚为“1”切TRx=1计数器开始计数）。单片机初始化完毕后程序等待半个正半周期（以便准确打开TRx）打开TRx，这时只要INTx （外部中断引脚）为高电平计数器即不断计数，低电平则不计数，待信号从高电平后计数器终止计数，关闭TRx保护计数器寄存器的值，该值即为待测信号一个正半周期的单片机机器周期数，即可求出待测信号的周期：待测信号周期T=2*cnt/(12/fsoc) cnt为测得待测信号的一个正半周期机器周期数；fsoc为单片机的晶振。所以待测信号的频率f=1/T。 原理示意图如下： 实验原理分析： 1．根据该实验原理该方法只适用于1：1占空比的方波信号，要测非1：1占空比的方波信号 2．由于有执行f=1/（2*cnt/(12/fsoc)）的浮点运算，而数据类型转换时未用LCD 浮点显示，故测得的频率将会被取整，如1234.893Hz理论显示为1234Hz，测 得结果会有一定程度的偏小。也就是说测量结果与信号频率的奇偶有一定关 系。 3．由于计数器的寄存器取值在1~65535之间，用该原理时，待测信号的频率小于单片机周期的1/12时，单片机方可较标准的测得待测信号的正半周期。故用 该原理测得信号的最高频率理论应为fsoc/12 如12MHZ的单片机为1MHz。 而最小频率为f=1/（2*65535/(12/fsoc)）如12MHZ的单片机为8Hz。 四．实验内容及步骤 1. 仿真模型的构建 数字方波频率计的设计总体可分为两个模块。一是信号频率测量，二是将测得的频率数据显示在1602液晶显示模块上。因此可搭建单片机最小系统构建构建频率计的仿真模型。原理图，仿真模型的总原理图如下：
目录 前言．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．1 第一章 FPGA及Verilog HDL．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．2 1.1 FPGA简介．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．2 1.2 Verilog HDL 概述．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．2 第二章数字频率计的设计原理．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．3 2.1 设计要求．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．3 2.2 频率测量．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．3 2.3．系统的硬件框架设计．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．4 2.4系统设计与方案论证．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．5 第三章数字频率计的设计．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．8 3.1系统设计顶层电路原理图．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．8 3.2频率计的VHDL设计．...．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．9 第四章软件的测试．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．15 4.1测试的环境——MAX+plusII．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．15 4.2调试和器件编程．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．15 4.3频率测试．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．16
如何理解电子测量仪器的精度指标 精确度是衡量电子测量仪器性能最重要的指标，通常由读数精度、量程精度两部分组成。本文结合几个具体案例，讲述误差的产生、计算以及标定方法，正确理解精度指标能够帮助您选择合适的仪器仪表。 一、测量误差的定义 误差常见的表示方法有：绝对误差、相对误差、引用误差。 1）绝对误差：测量值x*与其被测真值x之差称为近似值x*的绝对误差，简称ε。 计算公式：绝对误差 = 测量值 - 真实值； 2）相对误差：测量所造成的绝对误差与被测量（约定）真值之比乘以100%所得的数值，以百分数表示。 计算公式：相对误差 =（测量值 - 真实值）/真实值×100%（即绝对误差占真实值的百分比）； 3）测量的绝对误差与仪表的满量程值之比，称为仪表的引用误差，它常以百分数表示。引用误差=（绝对误差的最大值/仪表量程）×100% 引用误差越小，仪表的准确度越高，而引用误差与仪表的量程范围有关，所以在使用同一准确度的仪表时，往往采取压缩量程范围，以减小测量误差 举个例子，使用万用表测得电压1.005V，假定电压真实值为1V，万用表量程10V，精度（引用误差）0.1%F.S，此时万用表测试误差是否在允许范围内？ 分析过程如下： 绝对误差：E = 1.005V - 1V = +0.005V； 相对误差：δ=0.005V/1V×100%=0.5%； 万用表引用误差：10V×0.1%F.S=0.1V； 因为绝对误差0.005V<0.1V，所以10V量程引用误差0.1%F.S的万用表，测量1V相对误差为0.5%，仍在误差允许范围内。 二、测量误差的产生 绝对误差客观存在但人们无法确定得到，且绝对误差不可避免，相对误差可以尽量减少。误差组成成分可分为随机误差与系统误差，即：误差=测量结果-真值=随机误差+系统误差因此任意一个误差均可分解为系统误差和随机误差的代数和系统误差： 1）系统误差（Systematic error） 定义：在重复性条件下，对同一被测量进行无限多次测量所得结果的平均值与被测量的真值之差。 产生原因：由于测量工具（或测量仪器）本身固有误差、测量原理或测量方法本身理论的缺陷、实验操作及实验人员本身心理生理条件的制约而带来的测量误差。 特性：是在相同测量条件下、重复测量所得测量结果总是偏大或偏小，且误差数值一定或按一定规律变化。 优化方法：方法通常可以改变测量工具或测量方法，还可以对测量结果考虑修正值。 2）随机误差。 定义：随机误差又叫偶然误差，是指测量结果与同一待测量的大量重复测量的平均结果之差。产生原因：即使在完全消除系统误差这种理想情况下，多次重复测量同一测量对象，仍会由于各种偶然的、无法预测的不确定因素干扰而产生测量误差。 特点：是对同一测量对象多次重复测量，测量结果的误差呈现无规则涨落，可能是正偏差，也可能是负偏差，且误差绝对值起伏无规则。但误差的分布服从统计规律，表现出以下三个
等精度数字频率计的设计 李艳秋 摘要 基于传统测频原理的频率计的测量精度将随着被测信号频率的下降而降低，在实用中有很大的局限性，而等精度频率计不但有较高的测量精度，而且在整个测频区域内保持恒定的测试精度。运用等精度测量原理，结合单片机技术设计了一种数字频率计，由于采用了屏蔽驱动电路及数字均值滤波等技术措施，因而能在较宽定的频率范围和幅度范围内对频率，周期，脉宽，占空比等参数进行测量，并可通过调整闸门时间预置测量精度。选取的这种综合测量法作为数字频率计的测量算法，提出了基于FPGA 的数字频率计的设计方案。给出了该设计方案的实际测量效果，证明该设计方案切实可行，能达到较高的频率测量精度。 关键词等精度测量，单片机，频率计，闸门时间，FPGA Ⅱ
ABSTRACT Along with is measured based on the traditional frequency measurement principle frequency meter measuring accuracy the signalling frequency the drop but to reduce, in is practical has the very big limitation, but and so on the precision frequency meter not only has teaches the high measuring accuracy, moreover maintains the constant test precision in the entire frequency measurement region. Using and so on the precision survey principle, unified the monolithic integrated circuit technical design one kind of numeral frequency meter, because has used the shield actuation electric circuit and technical measure and so on digital average value filter, thus could in compared in the frequency range and the scope scope which the width decided to the frequency, the cycle, the pulse width, occupied parameter and so on spatial ratio carries on the survey, and might through the adjustment strobe time initialization measuring accuracy. Selection this kind of synthesis measured the mensuration took the digital frequency meter the survey algorithm, proposed based on the FPGA digital frequency meter design proposal. Has produced this design proposal actual survey effect, proved this design proposal is practical and feasible, can achieve the high frequency measurement precision Keywords Precision survey, microcontroller, frequency meter, strobe time，field programmable gate array Ⅱ
公差与配合（摘自GB1800～1804－79）1．基本偏差系列及配合种类 .2．标准公差值及孔和轴的极限偏差值 基本尺寸 mm 公差等级 IT5 IT6 IT7 IT8 IT9 IT10 IT11 IT12 >6～10 >10～18 >18～30 >30～50 >50～80 >80～120 >120～180 >180～250 >250～315 >315～400 >400～500 6 8 9 11 13 15 18 20 23 25 27 9 11 13 16 19 22 25 29 32 36 40 15 18 21 25 30 35 40 46 52 57 63 22 27 33 39 46 54 63 72 81 89 97 36 43 52 62 74 87 100 115 130 140 155 58 70 84 100 120 140 160 185 210 230 250 90 110 130 160 190 220 250 290 320 360 400 150 180 210 250 300 350 400 460 520 570 630
孔的极限差值（基本尺寸由大于10至315mm）μm
轴的极限偏差（基本尺寸由于大于10至315mm）
公差带级 >10～18>18～30 >30～50 >50～80 >80～120>120～180 >180～250>250～315 K 5 +9 +1 +11 +2 +13 +2 +15 +2 +18 +3 +21 +3 +24 +4 +27 +4 ▼6 +12 +1 +15 +2 +18 +2 +21 +2 +25 +3 +28 +3 +33 +3 +36 +4 7 +19 +1 +23 +2 +27 +2 +32 +2 +38 +3 +43 +3 +50 +4 +56 +4 M 5 +15 +7 +17 +8 +20 +9 +24 +11 +28 +13 +33 +15 +37 +17 +43 +20 6 +18 +7 +21 +8 +25 +9 +30 +11 +35 +13 +40 +15 +46 +17 +52 +20 7 +25 +7 +29 +8 +34 +9 +41 +11 +48 +13 +55 +15 +63 +17 +72 +20 N 5 +20 +12 +24 +15 +28 +17 +33 +22 +38 +23 +45 +27 +51 +31 +57 +34 ▼6 +23 +12 +28 +15 +33 +17 +39 +20 +45 +23 +52 +27 +60 +31 +66 +34 7 +30 +12 +36 +15 +42 +17 +50 +20 +58 +23 +67 +27 +77 +31 +86 +34 p 5 +26 +18 +31 +22 +37 +26 +45 +32 +52 +37 +61 +43 +70 +50 +79 +56 ▼6 +29 +18 +35 +22 +42 +26 +51 +32 +59 +37 +68 +43 +79 +50 +88 +56 7 +36 +18 +43 +22 +51 +26 +62 +32 +72 +37 +83 +43 +96 +50 +108 +56 注：标注▼者为优先公差等级，应优先选用。 形状和位置公差（摘自GB1182～1184－80） 形位公差符号 分类形状公差位置公差 项目直线 度 平面 度 圆度 圆柱 度 平行 度 垂直 度 倾斜 度 同轴 度 对称 度 位置 度 圆跳 动 全跳动 符号
分辨率与精度的区别 2010-10-07 10:28:37 很多人对于精度和分辨率的概念不清楚，这里我做一下总结，希望大家不要混淆。 我们搞编码器制做和销售的，经常跟“精度”与“分辨率”打交道，这个问题不是三言两语能搞得清楚的，在这里只作抛砖引玉了。 简单点说，“精度”是用来描述物理量的准确程度的，而“分辨率”是用来描述刻度划分的。从定义上看，这两个量应该是风马牛不相及的。（是不是有朋友感到愕然^_^）。很多卖传感器的JS就是利用这一点来糊弄人的了。简单做个比喻：有这么一把常见的塑料尺（中学生用的那种），它的量程是10厘米，上面有100个刻度，最小能读出1毫米的有效值。那么我们就说这把尺子的分辨率是1毫米，或者量程的1%；而它的实际精度就不得而知了（算是0.1毫米吧）。当我们用火来烤一下它，并且把它拉长一段，然后再考察一下它。我们不难发现，它还有有100个刻度，它的“分辨率”还是1毫米，跟原来一样！然而，您还会认为它的精度还是原来的0.1毫米么？（这个例子是引用网上的，个人觉得比喻的很形象！） 所以在这里利用这个例子帮大家把这两个概念理一下，以后大家就可以理直气壮的说精度和分辨率了，而不是将精度理解为分辨率。呵呵，希望对大家有用！^_^ 加工精度是加工后零件表面的实际尺寸、形状、位置三种几何参数与图纸要求的理想几何参数的符合程度。理想的几何参数，对尺寸而言，就是平均尺寸；对表面几何形状而言，就是绝对的圆、圆柱、平面、锥面和直线等；对表面之间的相互位置而言，就是绝对的平行、垂直、同轴、对称等。零件实际几何参数与理想几何参数的偏离数值称为加工误差。加工精度与加工误差都是评价加工表面几何参数的术语。加工精度用公差等级衡量，等级值越小，其精度越高；加工误差用数值表示，数值越大，其误差越大。加工精度高，就是加工误差小，反之亦然。 任何加工方法所得到的实际参数都不会绝对准确，从零件的功能看，只要加工误差在零件图要求的公差范围内，就认为保证了加工精度。 机器的质量取决于零件的加工质量和机器的装配质量，零件加工质量包含零件加工精度和表面质量两大部分。 机械加工精度是指零件加工后的实际几何参数（尺寸、形状和位置）与理想几何参数相符合的程度。它们之间的差异称为加工误差。加工误差的大小反映了加工精度的高低。误差越大加工精度越低，误差越小加工精度越高。 加工精度包括三个方面内容： 尺寸精度指加工后零件的实际尺寸与零件尺寸的公差带中心的相符合程度。 形状精度指加工后的零件表面的实际几何形状与理想的几何形状的相符合程度。 位置精度指加工后零件有关表面之间的实际位置与理想 精度就是结果值与结果真值的差值。 精度Accuracy 观测结果、计算值或估计值与真值（或被认为是真值）之间的接近程度。每一种物理量要用数值表示时，必须先要制定一种标准，并选定一种单位(unit)。标准及单位的制定，是为了沟通人与人之间对于物理现象的认识。这种标准的制定，通常是根据人们对于所要测量的物理量的认识与了解，并且要考虑这标准是否容易复制，或测量的过程是否容易操作等实际问题。由
建筑物沉降观测精度指标及评定方法 摘要：本文结合相关标准，探讨了建筑物沉降观测精度指标的含义及其估算方法，并对沉降观测结果的精度评定进行了研究。 关键词：建筑物；沉降观测；精度评定；精度指标 0 引言 沉降观测的精度要求取决于观测的目的、该建筑物的允许变形值以及建筑物的结构与基础类型[1]。由于沉降观测的精度直接影响到观测成果的可靠性和精确性，因此精度指标的确定及评定是沉降观测中的一个重要环节。然而，在现实工作中，建筑物沉降观测的精度评定经常被忽视，不少测量工作者甚至不清楚精度指标的含义及精度评定的方法。本文结合标准《建筑物沉降观测方法》DGJ32/J18-2006及《建筑变形测量规范》JGJ8-2007的要求，对建筑物的精度指标及评定进行深入探讨，弄清精度指标的概念及精度评定的方法。 1 基本概念 在测量中，由于受到测量仪器、观测者、外界条件等种种因素的影响，产生误差是不可避免的。测量误差分为偶尔误差和系统误差两大类，所谓精度，就是描述偶然误差分布的参数，精度越高，表示偶然误差的离散度越小，观测成果越可靠，反之亦然。 为了衡量观测精度的高低，利用一些数字反映误差分布的离散程度，这些数字称为衡量精度的指标，较常用的精度指标为方差和中误差，计算公式如下： （1） （2） 方差和中误差是表征精度的绝对数字指标，权、协因数（权倒数）则是表征精度的相对数字指标。设有观测值，对应的方差为，如选定任一常数，协因数的计算公式为： （3） 则称为的协因数或权倒数，为单位权中误差。对于水准测量，常用每公里观测高差中误差或者每测站高差中误差作为单位权中误差。 2 建筑物沉降观测精度指标及评定方法 2.1 精度指标
等精度数字频率计的设计 （Design of equal precision digital frequency meter）作者：李欢（电子工程学院光信息科学与技术 1103班） 指导教师：惠战强 摘要：伴随着集成电路(IC)技术的发展，电子设计自动化(EDA)逐渐成为重要的设计手段，已经广泛应用于模拟与数字电路系统等许多领域。电子设计自动化是一种实现电系统或电子产品自动化设计的技术，它与电子技术、微电子技术的发展密切相关，它吸收了计算机科学领域的大多数最新研究成果，以高性能的计算机作为工作平台，促进了工程发展。 数字频率计是一种基本的测量仪器。它被广泛应用于航天、电子、测控等领域。采用等精度频率测量方法具有测量精度保持恒定，不随所测信号的变化而变化的特点。本文首先综述了EDA技术的发展概况，FPGA/CPLD开发的涵义、优缺点，VHDL语言的历史及其优点，然后介绍了频率测量的一般原理。 关键字：电子设计自动化；VHDL语言；频率测量；数字频率计 Abstract The Electronic Design Automation (EDA) technology has become an important design method of analog and digital circuit system as the integrated circuit's growing. The EDA technology, which is closely connected with the electronic technology, microelectronics technology and computer science, can be used in designing electronic product automatically. Digital frequency meter is a basic measuring instruments. It is widely used in aerospace, electronics, monitoring and other fields. With equal precision frequency measurement accuracy to maintain a constant, and not with the measured signal varies.We firstly present some background information of EDA, FPGA/CPLD and VHDL;then introduced the general principle of frequency measurement. Keywords: Electronic Design Automation,VHDL, Frequency measurement,digital frequency meter.
最近做了一块板子，当然考虑到元器件的选型了，由于指标中要求精度比较高，所以对于AD的选型很慎重。很多人对于精度和分辨率的概念不清楚，这里我做一下总结，希望大家不要混淆。我们搞电子开发的，经常跟“精度”与“分辨率”打交道，这个问题不是三言两语能搞得清楚的，在这里只作抛砖引玉了。 简单点说，“精度”是用来描述物理量的准确程度的，而“分辨率”是用来描述刻度划分的。从定义上看，这两个量应该是风马牛不相及的。（是不是有朋友感到愕然^_^）。很多卖传感器的JS就是利用这一点来糊弄人的了。简单做个比喻：有这么一把常见的塑料尺（中学生用的那种），它的量程是10厘米，上面有100个刻度，最小能读出1毫米的有效值。那么我们就说这把尺子的分辨率是1毫米，或者量程的1%；而它的实际精度就不得而知了（算是0.1毫米吧）。当我们用火来烤一下它，并且把它拉长一段，然后再考察一下它。我们不难发现，它还有有100个刻度，它的“分辨率”还是1毫米，跟原来一样！然而，您还会认为它的精度还是原来的0.1毫米么？（这个例子是引用网上的，个人觉得比喻的很形象！） 回到电子技术上，我们考察一个常用的数字温度传感器：AD7416。供应商只是大肆宣扬它有10位的AD，分辨率是1/1024。那么，很多人就会这么欣喜：哇塞，如果测量温度0-100摄氏度，100/1024……约等于0.098摄氏度！这么高的精度，足够用了。但是我们去浏览一下AD7416的数据手册，居然发现里面赫然写着：测量精度0.25摄氏度！所以说分辨率跟精度完全是两回事，在这个温度传感器里，只要你愿意，你甚至可以用一个14位的AD， 获得1/16384的分辨率，但是测量值的精度还是0.25摄氏度^_^ AD的参考电压为VREF，则AD理论上能测到的最小电压值为分辨率*VREF。实际上还跟精度有关系。 所以很多朋友一谈到精度，马上就和分辨率联系起来了，包括有些项目负责人，只会在那里说：这个系统精度要求很高啊，你们AD的位数至少要多少多少啊…… 其实，仔细浏览一下AD的数据手册，会发现跟精度有关的有两个很重要的指标：DNL和INL。似乎知道这两个指标的朋友并不多，所以在这里很有必要解释一下。 DNL：DifferencialNonLiner——微分非线性度 INL：IntergerNonLiner——积分非线性度（精度主要用这个值来表示） 他表示了ADC器件在所有的数值点上对应的模拟值，和真实值之间误差最大的那一点的误差值。也就是，输出数值偏离线性最大的距离。单位是LSB（即最低位所表示的量）。 当然，像有的AD如△—∑系列的AD，也用Linearity error 来表示精度。 为什么有的AD很贵，就是因为INL很低。分辨率同为12bit的两个ADC，一个INL＝±3LSB，而一个做到了±1.5LSB，那么他们的价格可能相差一倍。 所以在这里帮大家把这两个概念理一下，以后大家就可以理直气壮的说精度和分辨率了，而不是将精度理解为分辨率。呵呵，希望对大家有用！^_^
光电与通信工程学院课程设计报告书 课设名称：等精度频率计 年级专业及班级： 姓名： 学号：
一、课程设计目的 1、进一步熟悉 Quartus Ⅱ的软件使用方法，熟悉 keil 软件使用； 2、熟悉单片机与可编程逻辑器件的开发流程及硬件测试方法； 3、掌握等精度频率计设计的基本原理。 4、掌握独立系统设计及调试方法，提高系统设计能力。 实验设备 EDA最小系统板一块（康芯）、PC机一台、示波器一台、信号发生器一台、万用表一个。 二、设计任务 利用单片机与FPGA设计一款等精度频率计，待测脉冲的检测及计数部分由FPGA实现，FPGA的计数结果送由单片机进行计算，并将最终频率结果显示在数码管上。要求该频率计具有较高的测量精度，且在整个频率区域能保持恒定的测试精度，具体指标如下： a）具有频率测试功能：测频范围 100Hz~5MHz。测频精度：相对误差恒为基准频率的万分之一。 b）具有脉宽测试功能：测试范围 10μs~1s，测试精度：0.1μs。 c）具有占空比测试功能：测试精度1%~99%。 d）具有相位测试功能。 （注：任务a 为基本要求，任务 b、c、d 为提高要求） 三、基本原理 基于传统测频原理的频率计的测量精度将随被测信号频率的下降而降低，在实用中有较大的局限性，而等精度频率计不但具有较高的测量精度，而且在整个频率区域能保持恒定的测试精度。 3.1 等精度测频原理 等精度频率计主控结构如图 1 所示
预置门控信号 CL 选择为 0.1~1s 之间（通过测试实验得出结论：CL 在这个 范围内选择时间宽度对测频精度几乎没有影响）。BZH 和 TF 分别是 2 个高速计数器，BZH 对标准频率信号（频率为 Fs）进行计数，设计数结果为 Ns；TF 对被测信号（频率为Fx）进行计数，计数结果为 Nx，则有 MUX64-8 模块并不是必须的，可根据实际设计进行取舍。分析测频计测控时序，着重分析 START的作用，完成等精度频率计设计。 3.2 FPGA 模块 FPGA模块所要完成的功能如图 1 所示，由于单片机的速度慢，不能直接测量高频信号，所以使用高速 FPGA 为测频核心。100MHZ 的标准频率信号由FPGA 内部的 PLL 倍频实现，待测信号 TCLK 为方波，由信号发生器给出待测方波信号（注意：该方波信号带有直流偏置，没有负电压，幅值3.3V）。预制
广州致远电子股份有限公司 示波器的垂直精度与垂直分辨率 示波器的垂直世界 类别 内容 关键词 垂直精度、垂直分辨率 摘 要 示波器的垂直精度与垂直分辨率解析
修订历史
目录 1. 概述 (1) 1.1垂直精度 (1) 1.2垂直分辨率解析 (1) 1.3算法提高分辨率 (1) 1.3.1几个基本概念 (1) 1.3.2平均算法 (2) 1.3.3高分辨率算法 (3) 2. 小结 (4) 3. 免责声明 (5)
1. 概述 数字存储示波器与万用表相比，测量电压到底是谁更准确呢？当然是万用表，但是为什么大家还会经常使用示波器来进行测量呢？ 1.1 垂直精度 提到测量问题，就会涉及到测量精度。用数字存储示波器测量模拟波形第一步就是用ADC将连续的模拟波形信号转换成量化的数字信号，最常用的是8位ADC，也就说对于任何一个波形值都是用256个0和1来重组。 当我们用同一个示波器在不同垂直档位下测量同一信号时，一般情况下得到的测量结果是不一样的，事实上，它涉及到垂直精度的问题，假设当垂直档位为500mV/div时，示波器垂直方向有8格，则其垂直精度分别为（500mV*8）/256=15.625 mV，也就是小于15.625 mV 的电压不会准确测量出来，测量同一个信号，在垂直档位为50mV/div的情况下，即（50mV*8）/256=1.5625 mV，垂直精度就达到了1.5625 mV，小于该垂直精度的电压值是不能测量出来的，即数字测量仪器都是存在采集的量化误差的，只能说ADC的位数越高，量化误差就会越小，但它只能无限减小，并不能消除。 所以当我们在对波形进行测量时，尽量使波形占满示波器屏幕，目的就是为了提高垂直精度，使测量结果更准确。 图1.1 垂直精度示意图 1.2 垂直分辨率解析 我们通常用示波器的垂直分辨率来描述数字示波器中ADC的位数，即位数越高，垂直分辨率越高，该分辨率由硬件决定，一旦确定无法改变。而示波器整个系统的有效位数（ENOB）形成的分辨率与前者不同，它可以由8位变为12位，甚至16位！ 示波器整个系统的有效位数（ENOB），它限制着测量系统区分和表示小信号的能力，该能力用噪声失真比（SINAD）表示，其值越大代表信号的噪声干扰越小，有效位数（ENOB）与噪声失真比（SINAD）之间的关系为： SINAD（噪声失真比，单位：dB）=6.02* ENOB（有效位数）+1.76 根据该数学关系式可知，SINAD（噪声失真比）大约每增加6 dB，ENOB（有效位数）就能增加1bit。所以提高信噪比，就能提高所谓的系统等效分辨率。 但是只要ADC位数不变，无论怎样提高所谓的分辨率归根结底都是对ADC采样后的数据进行数字信号处理，最终只能是在“软件”上提高了分辨率，并不能达到硬件上实现的性能，因为软件算法提高分辨率会产生副作用，影响采样率等关键指标，波形显示可能会发生失真现象等等。 1.3 改善等效分辨率 示波器都是如何通过改变算法来实现提高分辨率的呢？ 1.3.1 几个基本概念 我们将ADC转换成的离散数字信号称为采样点，相邻采样点之间的时间称为采样时间
FPGA/SOPC课程设计报告书 课题名称：等精度数字频率计的设计与实现 姓名： 学号： 院系：电子与信息工程系 专业：电子信息工程 指导教师： 时间：2012年6月
课程设计项目成绩评定表设计项目成绩评定表
课程设计报告书目录 设计报告书目录 一、设计目的 (1) 二、设计思路 (1) 三、设计过程 (1) 3.1、系统方案论证 (1) 3.2、模块电路设计 (3) 四、系统调试与结果 (4) 五、主要元器件与设备 (6) 六、课程设计体会与建议 (6) 6.1、设计体会 (6) 6.2、设计建议 (7) 七、参考文献 (7)
一、设计目的 1、熟悉DE2_70电路板的引脚安排。 2、掌握芯片的逻辑功能及使用方法。 3、了解面包板结构及其接线方法。 4、了解等精度数字频率计的组成及工作原理。 5、熟悉等精度数字频率计的设计与制作。 二、设计思路 1、设计测量计算频率电路。 2、设计译码显示电路。 三、设计过程 3.1、系统方案论证 等精度测频的实现方法可简化为如图1 所示的框图。图中CNT l和CNT2 是两个可控计数器, 标准频率( fs ) 信号从CN Tl的时钟输入端CLK 输入, 经整 形后的被测信号( fx ) 从CNT 2 的时钟输入端CLK输入。每个计数器中的CEN 输入端为使能端, 用来控制计数器计数。当预置闸门信号为高电平( 预置时间开始) 时, 被测信号的上升沿通过D触发器的输入端, 同时启动两个计数器计数; 同样, 当预置闸门信号为低电平( 预置时间结束) 时, 被测信号的上升沿通过 D触发器的输出端, 使计数器停止计数。 图1 等精度数字频率计设计原理
ADC制造商在数据手册中定义ADC性能的方式令人困惑，并且可能会在应用开发中导致错误的推断。最大的困惑也许就是“分辨率”和“精确度”了——即Resolution和Accuracy，这是两个不同的参数，却经常被混用，但事实上，分辨率并不能代表精确度，反之亦然。本文提出并解释了ADC“分辨率”和“精确度”，它们与动态范围、噪声层的关系，以及在诸如计量等应用中的含义。 ADC动态范围，精确度和分辨率 动态范围被定义为系统可测量到的最小和最大信号的比例。 最大信号可为峰间值，零到峰(Zero-to-Peak)值或均方根(RMS)满量程。其中任何一个都会给出不同值。例如，对于一个1V正弦波来说： 峰间(满量程)值＝2V 零到峰值＝1V RMS满量程＝0.707×峰值振幅＝0.707×1V＝0.707V 最小信号通常为RMS噪声，这是在未应用信号时测量的信号的均方根值。测量得到的RMS 噪声级别将取决于测量时使用的带宽。每当带宽翻倍，记录的噪声将增长1.41或3dB。 因此，一定要注意动态范围数字始终与某个带宽相关，而后者通常未被指定，这使记录的值变得没有意义。 器件的信噪比(SNR)和动态范围多数时候被定义为同一个值，即： 动态范围＝SNR ＝RMS满量程/RMS噪声 并且经常使用dB作为单位，即 动态范围(dB) ＝SNR(dB) ＝20*Log10 (RMS满量程/RMS噪声) 与使用RMS满量程相反，一些制造商为了使图表看上去更漂亮，引用零到峰或峰间值，这使得最终的动态范围或SNR增加了3dB或9dB，因此我们需要仔细研究规范以避免误解。 在讨论ADC性能时，分辨率和精确度是经常被混用的两个术语。一定要注意，分辨率并不能代表精确度，反之亦然。 ADC分辨率由数字化输入信号时所使用的比特数决定。对于16位器件，总电压范围被表示为216 (65536)个独立的数字值或输出代码。因此，系统可以测量的绝对最小电平表示为1比特，或ADC电压范围的1/65536。 A/D转换器的精确度是指对于给定模拟输入，实际数字输出与理论预期数字输出之间的接近度。换而言之，转换器的精确度决定了数字输出代码中有多少个比特表示有关输入信号的有用信息。