Index

Note: Page numbers followed by f indicate figures, and t indicate tables.

A

Approximate analytic solutions, diffusivity crossflows 
arbitrary functions 294
area crossflow rate 71–72, 294
asymptotic steady value 294
dependent variables 292
dimensionless crossflow velocity 71
exponential integral function 293
flow conditions 291
homogeneous equations 295
long time period 69–71
middle time period 69
net crossflow velocity 71, 294
numerical calculations 72
pressure differences 70, 293
short time period 69–70
steady peak value 294

B

Buildup pressure 143–144
Buildup well tests 
high diffusivity layer 
area crossflow rate 42f, 43–44
wellbore pressure 40–42, 40f
low diffusivity layer 
area crossflow rate 46f, 47
wellbore pressure 44–46, 44f
wellbore pressure behavior 95–98, 96–97f

C

Commingled production method 
constant pressure 260, 261t
data match curve 260, 261f
layered reserve estimation, two adjacent points 260–262, 262t
performace 259
physical and mathematical model 259–260, 259f
Commingled reservoir 
constant pressure outer boundary condition 59–61
dimensionless wellbore pressure 60
governing equation 59
infinite outer boundary condition 59
max effective hole-diameter mathematical model 61, 61–62t
no-flow outer boundary condition 59, 61
pressure curves and pressure derivative response 62–63, 62–63f
real wellbore radii model 61, 61–62t
wellbore boundary conditions 59–60
Computer program 272
Delphi programming language 272
design 272, 273f
downhole flow rate 272
multilayer interpretation theory 272
Constant surface rate 
buildup case 208–209
f=αD(pwD1 – pwD2) 205, 323–325
k˜=0si1_e 205–206, 323
known function f(tD) 322
known function f(pw1, pw2) 204
limiting behavior, f=αD(pwD1 – pwD2) 206–208
nonlinear parameter estimation 209–210
unknown leakage rate qD2 322
unknown function f(pw1, pw2) 204–205
Crossflow reservoir 62–63, 62–63f

D

Delphi programming language 272
Diffusivity crossflows 
approximate analytic solutions 
arbitrary functions 294
area crossflow rate 71–72, 294
asymptotic steady value 294
dependent variables 292
dimensionless crossflow velocity 71
exponential integral function 293
flow conditions 291
homogeneous equations 295
long time period 69–71
middle time period 69
net crossflow velocity 71, 294
numerical calculations 72
for pressure differences 70, 293
short time period 69–70
steady peak value 294
dimensionless expressions 66–68
numerical solutions 
five-layer cases 83–86
n = 3 and n = 5 cases 74–75
three-layer cases 75–82
semipermeable wall model 66, 67f
Drawdown tests 
dimensionless crossflow velocity 94
dimensionless expressions 93
four-layer reservoir 
approximate condition 94–95
area crossflow rate behavior 98–102, 98–101f
boundary condition 94–95
g – ln(2dei) function behavior 102–106, 103–104f, 105t
g(tD) function behavior 102
wellbore pressure behavior 95–98, 96–97f
high diffusivity layer 
area crossflow rate 41f, 43
pressure difference 41f, 43
wellbore pressure 40–42, 40f
low diffusivity layer 
area crossflow rate 45f, 46–47
pressure difference 45f, 46
new drawdown test 106–107
n-layer reservoir 92, 92f

F

Factor analysis, reserve estimation 
dynamic reserve estimation 
crossflow effect 255–256, 256f
interlayer heterogeneity reservoir 254, 254–255f
permeability, reserve estimation 256, 257f
simulation parameters 253–254
Five-layer reservoir 
area crossflow rates 86
wellbore pressure behavior 83–85, 84–85f
Four-layer reservoir 
approximate condition 94–95
area crossflow rate behavior 98–102, 98–101f
boundary condition 94–95
crossflow, different boundary pressures 297–298
g – ln(2dei) function behavior 102–106, 103–104f, 105t
g(tD) function behavior 102
wellbore pressure behavior 95–98, 96–97f
Fundamental differential equations 
gas reservoir 10
oil reservoir 
characteristic dimension 6
Darcy’s law 6
law of mass conservation 7
local mass crossflow velocity 4
n-layer reservoir 6, 7f
semipermeability 5–6
for steady and incompressible flow 7–8
unsteady flow 8

G

Gas reservoirs 
differential equations, linearization 11–12
fundamental differential equations 10
semipermeable wall model 
assumptions 9
local crossflow velocity 9
two-layer model 5f, 9

H

Horner time scale 40–42, 40f
Horner plot 143–144

I

Incompressible flow, crossflow behavior 
different boundary pressures 12–18, 15–17f
permeability change 18–22, 18–21f
Interlayer crossflow 235, 243, 255
Interpretation method 
buildup curves 154
drawdown curve 151–154
n-layered reservoirs 155–157
Interpretation theory  See also Unsteady crossflow (USC) model
constant surface rate 
buildup case 208–209
f=αD(pwD1 – pwD2) 205, 323–325
k˜=0si1_e 205–206, 323
known function f(tD) 322
known function f(pw1, pw2) 204
limiting behavior, f=αD(pwD1 – pwD2) 206–208
nonlinear parameter estimation 209–210
unknown leakage rate qD2 322
unknown function f(pw1, pw2) 204–205
different parameters, drawdown curves 
leakage rates, wellbore pressure 215f, 217
linear leakage rate, wellbore pressure 210–212, 210–211f
semipermeability, wellbore pressure 213f, 216
skin factor, wellbore pressure 213–214f, 216
storativity, low-permeability layer 212f, 213–215
w1, wellbore pressure 214–215f, 216–217
wellbore storage, wellbore pressure 212f, 215–216
micro-annulus 194
model description 194–196, 197t
q(t) is known 
case for k˜=0si1_e 202–204, 319–320
dimension and dimensionless Laplace transformation 197–198
f=αD(pwD1 – pwD2) 201
governing equations 315
known leakage rate 319
known function f(pw1, pw2) 200
parameter estimation 199
reservoir parameters 199
unknown function f(pw1, pw2) 200–201
unknown function f(pwD1, pwD2) 318–319
steady-state crossflow 193
vertical interference testing 192

L

Leaky aquifer model  See Pseudosteady crossflow (PSC) model

M

Material balance equation 
commingled production method 
constant pressure 260, 261t
data match curve 260, 261f
layered reserve calculation, two adjacent points 260–262, 262t
performance 259
physical and mathematical model 259–260, 259f
supplying region 
deliverability equation and recharge equation 265
mathematical model 264–265
numerical simulation method 263
performance forecaste 267–268, 268f
physical model 263–264, 264f
p/Z curve 266–267, 267f
real model, two-tank region gas reservoir 265–266, 266f
semipermeable wall model 262–263
two-layer gas reservoir 
with crossflow, mathematical model 251
dynamic reserve estimation 248, 258f
factor analysis, reserve estimation 253–256
layer with low permeability 257–258, 257f
physical model 249–250, 249f
production performances, multilayer reservoirs 249
p/Z curve 252–253
single-layer production, low permeability 258
without crossflow, mathematical model 250–251
Multilayer reservoirs, crossflow 
interpretation method 
buildup curves 154
drawdown curve 151–154
n-layered reservoirs 155–157
mathematical model 
Darcy’s law 135
dimensionless parameters 135
formation properties, well test theory 134–135
individual-layer well test 135, 135f
one drawdown test 134
test and closed layers 135
parameters of individual layers, transient tests 
drawdown tests 173–174
n-layer case 180–182
two-layer case, buildup period 176–177
two-layer case, drawdown period 174–176
pressure-test analysis 160
semipermeabilities determination, steady rate 
constant total rate 185–186
dimensionless parameters 186
simulation drawdown data 188, 188t
steady layer rates 185–186
vertical permeability 189
wellbore storage 185–186
simulation results 
area crossflow rate 149
area crossflow rate distribution, different times 149–150, 151f
boundary crossflow 149
crossflow velocity distribution, different times 149, 150f
diffusivity crossflow 149
dimensionless crossflow velocity 149
dimensionless parameters 146–148
finite difference method 145
Horner plots 149
long time period 146–148
short time period 146–148
theoretical drawdown and buildup curves, four-layered reservoir 146–148, 146–147f
transition period 146–148
unsteady flow 
buildup case 143–144
early transition period 137–139
long time period 139–142
n-layered reservoirs 144–145
semipermeability determination 142–143
short time period 136
unsteady-flow, skin factor 
approximate solution, long time period 166–167, 302–305
approximate solution, short time period 163–165, 300–302
area crossflow rate, layer 1 for different factors 172–173, 172f
area crossflow rate, layer 1 for same factors 172–173, 173f
assumptions and three flow periods 161–163
buildup case 167–168
change of rates, different skin factors 168–169, 169f
change of rates, same skin factors 168–169, 170f
crossflow velocity, layer 1 for different factors 170–171, 170f
crossflow velocity, layer 1 for same factors 171–172, 172f
crossflow velocity, layer 2 for different factors 170–171, 171f
crossflow velocity, layer 3 for different factors 170–171, 171f
drawdown and buildup curves, different skin factors 168, 168f
drawdown and buildup curves, same skin factors 168, 169f

N

n-layered crossflow system 3
n-layer reservoirs 
analytical solution 112
area crossflow rate 116
boundary conditions 111
constant wellbore pressure 111, 111f
crossflow 
by different boundary pressures 116–117
by different diffusivities 115–116
diffusivity crossflows  See (Diffusivity crossflows)
drawdown tests 92, 92f
fundamental differential equations 6, 7f
initial conditions 111–112
interpretation method 155–157
kh-weighted pressure 112
kh-weighted wellbore pressure 114
parameters of individual layers, transient tests 180–182
T and tD relationship 113–114, 113f
unsteady flow 144–145

O

Oil reservoir 
fundamental differential equations 
characteristic dimension 6
Darcy’s law 6
law of mass conservation 7
local mass crossflow velocity 4
n-layer reservoir 6, 7f
semipermeability 5–6
for steady and incompressible flow 7–8
unsteady flow 8
semipermeable wall model 
assumptions 4
two-layer model 4, 5f
VE concept 4
vertical resistance 4

P

Partially perforated two-layer reservoir 
analytical solution 56–58
assumptions 38–39
buildup well tests, high diffusivity layer 
area crossflow rate 42f, 43–44
wellbore pressure 40–42, 40f
buildup well tests, low diffusivity layer 
area crossflow rate 46f, 47
wellbore pressure 44–46, 44f
crossflow causes 39
dimensionless crossflow velocity 39
dimensionless form equations 39
drawdown tests, high diffusivity layer 
area crossflow rate 41f, 43
pressure difference 41f, 43
wellbore pressure 40–42, 40f
drawdown tests, low diffusivity layer 
area crossflow rate 45f, 46–47
pressure difference 45f, 46
parameters estimation 
advantage 47, 52
dimensionless wellbore pressure 47
drawdown cases 47–49, 48–49f
skin factors 52
ψ and β correlation 49, 50f, 51–52, 51t
Pseudosteady crossflow (PSC) model 223–224

R

Reservoir parameter estimation 229–230

S

Semipermeability determination 
crosspoint of two straight lines 142–143
steady rate 
constant total rate 185–186
dimensionless parameters 186
simulation drawdown data 188, 188t
steady layer rates 185–186
vertical permeability 189
wellbore storage 185–186
steady wellbore pressure differences 142
Semipermeable wall (SW) model 224–225, 249–250, 262–263
gas reservoirs 
assumptions 9
local crossflow velocity 9
two-layer model 5f, 9
oil reservoir 
assumptions 4
two-layer model 4, 5f
VE concept 4
vertical resistance 4
Single-well chemical tracer test 66
Short time period 
change of rates, different skin factors 164–165, 165f
diffusivity crossflows 69–70
dimensional synthetic diffusivity 163–164
dimensionless rate of layer 163
drawdown curves, different skin factors 164–165, 165f
drawdown curves, two-layer reservoir 164–165, 166f
effective skin factor 163
f=αD(pwD1 – pwD2) 231
two-layer reservoirs 309
unsteady flow 163–165
wellbore pressure 163
SW model  See Semipermeable wall (SW) model
Synthetic oil well 
flow profile, stabilized rate 275
interpretation results 276, 278t
multirate analysis, three-layer reservoir 276, 277f
PLT test, stabilized flow rate 276, 277t
pressure history match, three-layer reservoir 276, 278f
rate and pressure history 276, 276t
surface flow rate 275–276

T

Three-layer reservoir 
analytical solution 
constant pressure outer boundary condition 87
governing equation 86
infinite outer boundary condition 87
initial condition 86
no-flow outer boundary condition 87
pressure and rate derivation 87–89
wellbore boundary conditions 87
area crossflow rates 
distribution of 80–82, 81–82f
peak value 79, 79–80f
similarity of 78–80, 78f
crossflow velocity behavior 82, 83f
flow rate behavior 75–77, 76f
kh-weighted pressure distribution 77–78, 77f
wellbore pressure behavior 75, 76f
well test design 
cross-sectional view 273, 273f
injection water well, rate and pressure history 274, 275f
layer properties 274, 274t
oil well, rate and pressure history 274, 274f
rate schedules of test 274, 274t
Transient well test methods 110
Two-layer gas reservoir 
with crossflow, mathematical model 251
dynamic reserve estimation 248, 258f
factor analysis, reserve estimation 253–256
layer with low permeability 257–258, 257f
physical model 249–250, 249f
production performances, multilayer reservoirs 249
p/Z curve 252–253
single-layer production, low permeability 258
without crossflow, mathematical model 250–251
Two-layer reservoir  See also Two-layer gas reservoir
analytical solution 
commingled reservoir 59–63, 61–62t
real wellbore radii model 53–54
approximate solution 
extension of 313–314
for long time period 309–311
for short time period 309
area crossflow distributions 119–122, 121–122f
buildup case 312
constant pressure condition 
constant pressure outer boundary condition 124
dimensionless governing equation 124
dimensionless pressure 125–127
infinite outer boundary condition 124
initial condition 124
no-flow outer boundary condition 124
production ratio curves 127–129, 127–130f
skin factor inner boundary condition 127
wellbore boundary conditions 125
constant rate and pressure, boundary conditions 289
kh-weighed pressure vs. time 118, 118–120f
maximum effective hole-diameter 53–54
parameters of individual layers, transient tests 
buildup period 176–177
drawdown period 174–176
partially perforated two-layer reservoir  See (Partially perforated two-layer reservoir)
pressure difference distributions vs. time 118, 120–121f
pressure drawdown and buildup behavior 
area crossflow rate 35–37
assumptions 28–29
governing equations 29
initial and boundary conditions 30
isolated homogeneous layer 27–28
mass crossflow rate 29
numerical solution 30–35, 32–34f
qci peak value vs. time 123–124, 123f
terminal rate vs. time 117–118, 117f

U

Unsteady crossflow (USC) model 
comparison of pwD2 pressure 243, 244f
dimensionless parameters 225
f=αD(pwD1 – pwD2) 
early time period 230–231, 335–336
long time period 231–232, 337–338
short time period 231, 336–337
interpretation method of parameters 
interpretation results 235–242, 242t
pwD2 235, 240–241t, 243
steps 232–234
tDdpwD1/dtD 235, 238–239t, 243
wellbore pressures pwD1 234–235, 236–237t, 243
K(x) function 227
Laplace transform 226
leakage function 
f=αD(pwD1, pwD2) 331–332
linear form 227–229, 331
nonlinear unknown function 229–230, 332–333
model description 
boundary conditions 221
conservation of matter 222
flow problem 221
leakage rate, micro-annulus behind casing 223
leaky vertical interference well test 220–221, 221f
PSC model 223–224
SW model 224–225
Unsteady flow 
buildup case 143–144
early transition period 137–139
fundamental differential equations 8
long time period 139–142
n-layered reservoirs 144–145
semipermeability determination 142–143
skin factor 
approximate solution, long time period 166–167
approximate solution, short time period 163–165
area crossflow rate, layer 1 for different factors 172–173, 172f
area crossflow rate, layer 1 for same factors 172–173, 173f
assumptions and three flow periods 161–163
buildup case 167–168
change of rates, different skin factors 168–169, 169f
change of rates, same skin factors 168–169, 170f
crossflow velocity, layer 1 for different factors 170–171, 170f
crossflow velocity, layer 1 for same factors 171–172, 172f
crossflow velocity, layer 2 for different factors 170–171, 171f
crossflow velocity, layer 3 for different factors 170–171, 171f
drawdown and buildup curves, different skin factors 168, 168f
drawdown and buildup curves, same skin factors 168, 169f
short time period 136
USC model  See Unsteady crossflow (USC) model

V

Vertical equilibrium (VE) concept 4
Vertical permeability 175, 189, 255, 256f

W

Wellbore pressure 
different layers 136
leakage rates 215f, 217
linear leakage rate 210–212, 210–211f
semipermeability 213f, 216
skin factor 213–214f, 216
short time period 163
w1 214–215f, 216–217
wellbore storage 212f, 215–216
Well test analysis 
gas field data 
geological information 278–279
log–log plot, Sebei gas field 278–279, 281f
multilayer test 279–281
synthetic injection water well 278, 279–280t
synthetic oil well 
flow profile, stabilized rate 275
interpretation results 276, 278t
multirate analysis, three-layer reservoir 276, 277f
PLT test, stabilized flow rate 276, 277t
pressure history match, three-layer reservoir 276, 278f
rate and pressure history 276, 276t
surface flow rate 275–276
Well test design, three-layer reservoir 
cross-sectional view 273, 273f
injection water well, rate and pressure history 274, 275f
layer properties 274, 274t
oil well, rate and pressure history 274, 274f
rate schedules of test 274, 274t
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